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4,000 | 4,620 | Learning Label Trees for Probabilistic Modelling of
Implicit Feedback
Yee Whye Teh
[email protected]
Gatsby Computational Neuroscience Unit
University College London
Andriy Mnih
[email protected]
Gatsby Computational Neuroscience Unit
University College London
Abstract
User preferences for items can be inferred from either explicit feedback, such as
item ratings, or implicit feedback, such as rental histories. Research in collaborative filtering has concentrated on explicit feedback, resulting in the development
of accurate and scalable models. However, since explicit feedback is often difficult to collect it is important to develop effective models that take advantage of the
more widely available implicit feedback. We introduce a probabilistic approach to
collaborative filtering with implicit feedback based on modelling the user?s item
selection process. In the interests of scalability, we restrict our attention to treestructured distributions over items and develop a principled and efficient algorithm
for learning item trees from data. We also identify a problem with a widely used
protocol for evaluating implicit feedback models and propose a way of addressing
it using a small quantity of explicit feedback data.
1
Introduction
The rapidly growing number of products available online makes it increasingly difficult for users to
choose the ones worth their attention. Recommender systems assist users in making these choices by
ranking the products based on inferred user preferences. Collaborative filtering [6] has become the
approach of choice for building recommender systems due to its ability to infer complex preference
patterns from large collections of user preference data. Most collaborative filtering research deals
with inferring preferences from explicit feedback, for example ratings given to items. As a result,
several effective methods have been developed for this version of the problem. Matrix factorization
based models [13, 5, 12] have emerged as the most popular of these due to their simplicity and superior predictive performance. Such models are also highly scalable because their training algorithms
take advantage of the sparsity of the rating matrix, resulting in training times that are linear in the
number of observed ratings.
However, since explicit feedback is often difficult to collect it is essential to develop effective models
that take advantage of the more abundant implicit feedback, such as logs of user purchases, rentals,
or clicks. The difficulty of modelling implicit feedback comes from the fact that it contains only
positive examples, since users explicitly express their interest, by selecting items, but not their disinterest. Note that not selecting a particular item is not necessarily an expression of disinterest, as it
might also be due to the obscurity of the item, lack of time, or other reasons.
Just like their explicit feedback counterparts, the most successful implicit feedback collaborative
filtering (IFCF) methods are based on matrix factorization [4, 10, 9]. However, instead of a highly
sparse rating matrix, they approximate a dense binary matrix, where each entry indicates whether or
not a particular user selected a particular item. We will collectively refer to such methods as Binary
Matrix Factorization (BMF). Since such approaches treat unobserved user/item pairs as fake negative
examples which can dominate the much less numerous positive examples, the contribution to the
1
objective function from the zero entries is typically downweighted. The matrix being approximated
is no longer sparse, so models of this type are typically trained using batch alternating least squares.
As a result, the training time is cubic in the number of latent factors, which makes these models less
scalable than their explicit feedback counterparts.
Recently [11] introduced a new method, called Bayesian Personalized Ranking (BPR), for modelling
implicit feedback that is based on more realistic assumptions than BMF. Instead of assuming that
users like the selected items and dislike the unselected ones, it assumes that users merely prefer the
former to the latter. The model is presented with selected/unselected item pairs and is trained to rank
the selected items above the unselected ones. Since the number of such pairs is typically very large,
the unselected items are sampled at random.
In this paper we develop a new method that explicitly models the user item selection process using
a probabilistic model that, unlike the existing approaches, can generate new item lists. Like BPR
it assumes that selected items are more interesting than the unselected ones. Unlike BPR, however,
it represents the appeal of items to a user using a probability distribution, producing a complete
ordering of items by probability value. In order to scale to large numbers of items efficiently, we
restrict our attention to tree-structured distributions. Since the accuracy of the resulting models
depends heavily on the choice of the tree structure, we develop an algorithm for learning trees from
data that takes into account the structure of the model the tree will be used with.
We then turn our attention to the task of evaluating implicit feedback models and point out a problem
with a widely used evaluation protocol, which stems from the assumption that all items not selected
by a user are irrelevant. Our proposed solution involves using a small quantity of explicit feedback
to reliably identify the irrelevant items.
2
Modelling item selection
We propose a new approach to collaborative filtering with implicit feedback based on modelling
the item selection process performed by each user. The identities of the items selected by a user
are modelled as independent samples from a user-specific distribution over all available items. The
probability of an item under this distribution reflects the user?s interest in it. Training our model
amounts to performing multinomial density estimation for each user from the observed user/item
pairs without explicitly considering the unobserved pairs.
To make the modelling task more manageable we make two simplifying assumptions. First, we
assume that user preferences do not change with time and model all items chosen by a user as
independent samples from a fixed user-specific distribution. Second, to keep the model as simple
as possible we assume that items are sampled with replacement. We believe that sampling with
replacement is a reasonable approximation to sampling without replacement in this case because
the space of items is large while the number of items selected by a user is relatively small. These
simplifications allow us to model the identities of the items selected by a user as IID samples.
We now outline a simple implementation of the proposed idea which, though impractical for large
datasets, will serve as a basis for developing a more scalable model. As is typical for matrix factorization methods in collaborative filtering, we represent users and items with real-valued vectors of
latent factors. The factor vectors for user u and item i will be denoted by Uu and Vi respectively.
Intuitively, Uu captures the preferences of user u, while Vi encodes the properties of item i. Both
user and item factor vectors are unobserved and so have to be learned from the observed user/item
pairs. The dot product between Uu and Vi quantifies the preference of user u for item i. We define
the probability of user u choosing item i as
exp(Uu> Vi + ci )
P (i|u) = P
,
>
k exp(Uu Vk + ck )
(1)
where ci is the bias parameter that captures the overall popularity of item i and index k ranges over
all items in the inventory. The model can be trained using stochastic gradient ascent [2] on the
log-likelihood by iterating through the user/item pairs in the training set, updating Uu , Vi , and ci
based on the gradient of log P (i|u). The main weakness of the model is that its training time is
linear in the inventory size because computing the gradient of the log-probability of a single item
requires explicitly considering all available items. Though linear time complexity might not seem
2
prohibitive, it severely limits the applicability of the model since collaborative filtering tasks with
tens or even hundreds of thousands of items are now common.
3
Hierarchical item selection model
The linear time complexity of the gradient computation is a consequence of normalization over the
entire inventory in Eq. 1, which is required because the space of items is unstructured. We can speed
up normalization, and thus learning, exponentially by assuming that the space of items has a known
tree structure. We start by supposing that we are given a K-ary tree with items at the leaves and
exactly one item per leaf. For simplicity, we will assume that each item is located at exactly one
leaf. Such a tree is uniquely determined by specifying for each item the path from the root to the leaf
containing the item. Any such path can be represented by the sequence of nodes n = n0 , n1 , ..., nL
it visits, where n0 is always the root node.
By making the choice of the next node stochastic, we can induce a distribution over the leaf nodes in
the tree and thus over items. To allow each user to have a different distribution over items we make
the probability of choosing each child a function of the user?s factor vector. The probability will
also depend on the child node?s factor vector and bias the same way the probability of choosing an
item in Eq. 1 depends on the item?s factor vector and bias. Let C(n) be the set of children of node
n. Then for user u, the probability of moving from node nj to node n on a root-to-leaf tree traversal
is given by
exp Uu> Qn + bn
P (n|nj , u) = P
,
(2)
>
m?C(nj ) exp (Uu Qm + bm )
if n is a child of nj and 0 otherwise. Here Qn and bn are the factor vector and the bias of node n.
The probability of selecting item i is then given by the product of the probabilities of the decisions
that lead from the root to the leaf containing i:
QLi
P (i|u) = j=1
P (nij |nij?1 , u).
(3)
We will call the model defined by Eq. 3 the Collaborative Item Selection (CIS) model. Given a tree
over items, the CIS model can be trained using stochastic gradient ascent in log-likelihood, updating
parameters after each user/item pair.
While the model can use any tree over items, the choice of the tree affects the model?s efficiency and
ability to generalize. Since computing the probability of a single item takes time linear in the item?s
depth in the tree, we want to avoid trees that are too unbalanced. To produce a model that generalizes
well we also want to avoid trees with difficult classification problems at the internal nodes [1], which
correspond to hard-to-predict item paths.
One way to produce a tree that results in relatively easy classification problems is to assign similar
items to the same class, which is the approach of [7] and [14]. However, the similarity metrics used
by these methods are not model-based in the sense that they are not derived from the classifiers that
will be used at the tree nodes. In Section 5 we will develop a scalable model-based algorithm for
learning trees with item paths that are easy to predict using Eq. 2.
4
Related work
The use of tree-structured label spaces to reduce the normalization cost has originated in statistical
language modelling, where it was used to accelerate neural and maximum-entropy language models
[3, 8]. The task of learning trees for efficient probabilistic multiclass classification has received surprisingly little attention. The two algorithms most closely related to the one proposed in this paper
are [1] and [7]. [1] proposed a fully online algorithm for multinomial density estimation that constructs a binary label tree by inserting the previously unseen labels whenever they are encountered.
The location for a new label is found proceeding from the root to a leaf making the left child/right
child decisions based on their probability under the model and a tree balancing penalty. This is the
only tree-learning algorithm we are aware of that takes into account the probabilistic model the tree
is used with. Unfortunately, this approach is very optimistic because it decides on the location for a
new label in the tree based on a single training case and never revisits that decision.
3
The algorithm in [7] was developed for learning trees over words for use in probabilistic language
models. It constructs such trees by performing top-down hierarchical clustering of words, which are
represented by real-valued vectors. The word representations are learned through bootstrapping by
training a language model based on a random tree. This algorithm, unlike the one we propose in
Section 5, does not take into consideration the model the tree is constructed for.
Most work on tree-based multiclass classification deals with non-probabilistic models and does not
apply to the problem we are concerned with in this paper. Of these approaches our algorithm is
most similar to the one in [14], which looks for a tree structure that avoids requiring to discriminate
between easily confused items as much as possible. The main weakness of that approach is the need
for training a flat classifier to produce the confusion matrix needed by the algorithm. As a result, it
is unlikely to scale to large datasets containing tens of thousands of classes.
5
Model-based learning of item trees
5.1 Overview
In this section we develop a scalable algorithm for learning trees that takes into account the parametric form of the model the tree will be used with. At the highest level our approach can be seen as
top-down model-based hierarchical clustering of items. We chose top-down clustering over bottomup clustering because it is the more scalable option. Since finding the best tree is intractable, we take
a greedy approach that constructs the tree one level at a time, learning the lth node of each item path
st
before fixing it and advancing to the (l + 1) node. Because our approach is model-based, it learns
model parameters, i.e. node biases and factor vectors, jointly with the item paths. As a result, at every point during its execution it specifies a complete probabilistic model of the data, which becomes
more expressive with each additional tree level. This makes it possible to monitor the progress of
the algorithm by evaluating the predictions made after learning each level.
For simplicity, our tree-learning algorithm assumes that user factor vectors are known and fixed.
Since these vectors are actually unknown, we learn them by first training a CIS model based on
a random balanced tree. We then extract the user vectors learned by the model and use them to
learn a better tree from the data. Finally, we train a CIS model based on the learned tree, updating
all the parameters, including the user vectors. This three-stage approach is similar to the one used
in [7] to learn trees over words. However, because our tree-learning algorithm is model-based,
we already have a complete probabilistic model at its termination, so we only need to finetune its
parameters instead of learning them from scratch. Finetuning is necessary because the parameters
learned while building the tree are based on the fixed user factor vectors from the random-tree-based
model. Though it is possible to continue alternating between optimizing over the tree structure and
over user vectors, we found the resulting gains too small to be worth the computational cost.
5.2
Learning a level of a tree
We now describe how to learn a level of the tree. Suppose we have learned the first l ? 1 nodes of
each item path and would like to learn the lth node. Let Ui be the set of users who rated item i in
the training set. The contribution made by item i to the log-likelihood is then given by
Q
P
Q
P
P
Li = log u?Ui P (i|u) = u?Ui log j P (nij |nij?1 , u) = u?Ui j log P (nij |nij?1 , u). (4)
The log-likelihood contribution due to a single observation can be expressed as
P
Pl?1
i i
i i
i i
j log P (nj |nj?1 , u) =
j=1 log P (nj |nj?1 , u) + log P (nl |nl?1 , u)+
PLi
i i
j=l+1 log P (nj |nj?1 , u).
(5)
The first term on the RHS depends only on the parameters and path nodes that have already been
learned, so it can be left out of the objective function. The third term is the log-probability of item i
under the subtree rooted at node nil , which depends on the structure and parameters of that subtree,
which we have not learned yet. To emphasize the fact that this term is based on a user-dependent
distribution over items under node nil we will denote it by log P (i|nil , u).
The overall objective function for learning level l is obtained by adding up the contributions of all
items, leaving out the terms that do not depend on the quantities to be learned:
P P
P P
Ll = i u?Ui log P (nil |nil?1 , u) + i u?Ui log P (i|nil , u).
(6)
4
The most direct approach to learning the paths would be to alternate between updating the lth node
in the paths and the corresponding factor vectors and biases. Since jointly optimizing over the lth
node in all item paths is infeasible, we have to resort to incremental updates, maximizing Ll over
the lth node in one item path at a time. Unfortunately, even this operation is intractable because
evaluating each value of nil requires knowing the optimal contribution from the still-to-be-learned
levels of the tree, which is the second term in Eq. 6. In other words, to find the optimal nil we need
to compute
P
i
i
nil = arg maxn?C(nil?1 )
(7)
u?Ui log P (n|nl?1 , u) + F (n, nl?1 ) ,
where we left out the terms that do not depend on nil . The optimal contribution F (nil , nil?1 ) from
the future levels is defined as
P
P
F (nil , nil?1 ) = max k?I(ni ) u?Uk log P (k|nkl , u),
(8)
?
l?1
where I(nil?1 ) is the set of items that are assigned to node nil?1 , and ? is the set of node factor vectors, biases, and tree structures that parameterize the set of distributions {P (k|nkl , u)|k ? I(nil?1 )}.
5.3
Approximating the future
The value of F (nil , nil?1 ) quantifies the difficulty of discriminating between items assigned to node
nil?1 using the best tree structure and parameter setting possible given that item i is assigned to the
child nil of that node. Since F (nil , nil?1 ) in Eq. 8 rules out degenerate solutions where all items
below a node are assigned to the same child of it, leaving F (nil , nil?1 ) out to make the optimization
problem easier is not an option.
We address the intractability of Eq. 7 while avoiding the degenerate solutions by approximating the
user-dependent distributions P (k|nkl , u) by simpler distributions that make it much easier to evaluate
F (nil , nil?1 ) for each candidate value for nil . Since computing F (nil , nil?1 ) requires maximizing
over the free parameters of P (k|nkl , u), choosing a parameterization of P (k|nkl , u) that makes this
maximization easy can greatly speed up this computation. We approximate the tree-structured userdependent P (k|nkl , u) with a flat user-independent distribution P (k|nkl ). The main advantage of
this parameterization is that the optimal P (k|nkl ) can be computed by counting the number of times
each item assigned to node nkl occurs in the training data and normalizing. In other words, when
P (k|nkl ) is used in Eq. 8, the maximum is achieved at
?
? P Ni
if i ? I(nkl )
k
k ) Nm
m?I(n
P (i|nl ) =
(9)
l
?0
otherwise
where Ni is the number of times item i occurs in the training set. The corresponding value for
F (nil , nil?1 ) is given by
P
.
(10)
F (nil , nil?1 ) = k?I(ni ) Nk log P Nk
l?1
m?I(nk ) Nm
l
To show that F (nil , nil?1 ) can be computed in constant time
items under node nil?1 can be written in terms of sums over
F (nil , nil?1 ) =
P
=
P
c?C(nil?1 )
c?C(nil?1 )
P
k?I(c)
P
k?I(c)
Nk log
P
we start by observing that the sum over
items under each of its children:
Nk
m?I(c)
Nk log Nk ?
Nm
P
c?C(nil?1 )
Zc log Zc .
(11)
with Zc = k?I(c) Nk . Since adding a constant to F (nil , nil?1 ) has no effect on the solution of Eq.
7 and the first term in the equation does not depend on nil , we can drop it to get
P
F? (nil , nil?1 ) = ? c?C(ni ) Zc log Zc .
(12)
P
l?1
F? (nil , nil?1 )
To compute
efficiently, we store Zc ?s and the old F? (nil , nil?1 ) value, updating them
whenever an item is moved to a different node. Such updates can be performed in constant time.
5
We now show that the first term in Eq. 7, corresponding to the contribution of the lth path node for
item i, can be computed efficiently. Plugging in the definition of P (n|nj , u) from Eq. 2 we get
P
P
i
>
u?Ui log P (n|nl?1 , u) =
u?Ui Uu Qn + bn + C
>
P
Qn + |Ui |bn + C
(13)
=
u?Ui Uu
where C is a term that does not depend on n and so does not have to be considered when maximizing
over
P n. Since we assume that the user factor vectors are known and fixed, we precompute Ri =
u?Ui Uu for each user, which can be seen as creating a surrogate representation for item i.
Finally, plugging Eq. 13 into Eq. 7 gives us the following update for item nodes:
nil = arg maxn?C(nil?1 ) Ri> Qn + |Ui |bn + F? (n, nil?1 ) .
6
(14)
Evaluating models of implicit feedback
Establishing sensible evaluation protocols for machine learning problems is important because they
effectively define what ?better? performance means and implicitly guide the development of future
methods. Given that the problem of implicit feedback collaborative filtering is relatively new, it is
not surprising that the typical evaluation protocol was adopted from information retrieval. However,
we believe that this protocol is much less appropriate in collaborative filtering than it is in its field
of origin.
Implicit feedback models are typically evaluated using information retrieval metrics, such as Mean
Average Precision (MAP), that require knowing which items are relevant and which are irrelevant
to each user. It is typical to assume that the items the user selected are relevant and all others are not
[10]. However, this approach is problematic because it fails to distinguish between the items the user
really has no interest in (i.e. the truly irrelevant ones) and the relevant items the user simply did not
rate. And while the irrelevant items do tend to be far more numerous than the unobserved relevant
ones, the effect of the latter can still be strong enough to affect model comparison, as we demonstrate
in the next section. To address this issue, we propose using some explicit feedback information to
identify a small number of truly irrelevant items for each user and using them in place of items of
unknown relevance in the evaluation. Thus the models will be evaluated on their ability to rank
the truly relevant items above the truly irrelevant ones, which we believe is the ultimate task of
collaborative filtering. Though this approach does require access to explicit feedback, only a small
quantity of it is necessary, and it is used only for evaluation.
For probabilistic models P (i|u), the most natural performance metrics are log-probability of the
held-out data D and the closely-related perplexity (PPL), the standard metric for language models:
1 P
PPL = exp ?
log P (i|u) .
(15)
|D| (u,i)?D
The model that assigns the correct item probability 1 has the perplexity of 1, while the model that
assigns all N items the same probability (1/N ) has the perplexity of N . Unlike the ranking metrics above, perplexity is computed based on the selected/relevant items alone and does not require
assuming that the unselected items are irrelevant.1
7
Experimental results
First we investigated the impact of using tree-structured distributions over items by comparing the
performance of tree-based CIS models to that of a flat model defined by Eq. 1. We used MovieLens
1M, which is a fairly small dataset, for the comparison in order to be able to train the flat model
within reasonable time. The dataset contains 1M ratings on a scale from 1 to 5 given by 6040 users
to 3952 movies. To simulate the implicit feedback setting, where the presence of a user/item pair
indicates an expression of interest, we kept only the user/item pairs associated with ratings 4 and
above (and discarded the rating values) and split the resulting 575K pairs into a 475K-pair training
set, and a validation and test sets of 50K pairs each. We trained three models with 5-dimensional
1
The implicit assumption here is that the selected items are more relevant than the unselected ones.
6
Table 1: Test set scores in percent on the MovieLens 10M dataset obtained by treating items with low
ratings as irrelevant. Higher scores indicate better performance for all metrics except for perplexity.
Model
CIS (Random)
CIS (LearnedRI)
CIS (LearnedCI)
BPR
BMF
PPL
921
822
820
865
?
MAP
70.68
72.50
72.61
72.75
70.80
P@1
74.65
76.64
76.68
75.75
75.66
P@5
58.02
59.29
59.37
59.15
58.03
P@10
49.91
50.64
50.69
50.63
49.77
R@1
20.66
21.51
21.54
21.50
20.94
R@5
60.02
61.24
61.31
61.43
60.04
R@10
77.31
78.22
78.27
78.39
77.21
Table 2: Test set scores in percent on the MovieLens 10M dataset obtained by treating all unobserved
items as irrelevant.
Model MAP P@1 P@5 P@10 R@1 R@5 R@10
BPR
12.73 14.27 11.56
9.89
3.06 11.55 18.86
BMF
16.13 22.10 16.25 12.94 4.66 15.64 23.55
factor vectors: a flat model, a CIS model with a random balanced binary tree, and a CIS model
with a learned binary tree (as in Section 5). The flat model took 12 hours to train and had the test
set perplexity of 920. Training the random tree model took half an hour, resulting in the perplexity
of 975. The training process for the learned-tree model, which included training a random-tree
model, learning a tree from the resulting user factor vectors, and finetuning all the parameters, took
1 hour. The resulting model performed very well, achieving the test set perplexity of 912. These
results suggest that even when the number of items is relatively small our tree-based approach to
item selection modelling can yield an order-of-magnitude reduction in training times relative to the
flat model without hurting the predictive accuracy.
We then used the larger MovieLens 10M dataset (0-5 rating scale, 69878 users, 10677 movies)
to compare the proposed approach to the existing IFCF methods. As on MovieLens 1M, we kept
only the user/item pairs with ratings 4 and above, producing a 4M-pair training set, a 500K-pair
validation set, and a 500K-pair test set. We compared the models based on their perplexity and
ranking performance as measured by the standard information retrieval metrics: Mean Average
Precision (MAP), Precision@k, and Recall@k. We used the evaluation approach described in the
previous section, which involved having the models rank only the items with known relevance status.
We used the rating values to determine relevance, considering items rated below 3 as irrelevant and
items rated 4 and above as relevant.
We compared our hierarchical item selection model to two state-of-the-art models for implicit feedback: the Bayesian Personalized Ranking model (BPR) and the Binary Matrix Factorization model
(BMF). All models used 25-dimensional factor vectors, as we found that higher-dimensional factor
vectors resulted in only marginal improvements. We included three CIS models based on different binary trees (K = 2) to highlight the effect of tree construction methods. The methods are as
follows: ?Random? generates random balanced trees; ?LearnedRI? is the method from Section 5
with randomly initialized item-node assignments; ?LearnedCI? is the same method with item-node
assignments initialized by clustering surrogate item representations Ri from Section 5.3. Training
a flat item selection model on this dataset was infeasible, as a single pass through the data took six
hours, compared to a mere two minutes for CIS (LearnedCI).
Better performance corresponds to lower values of perplexity and higher values of the other metrics.
Table 1 shows the test scores for the resulting models. In terms of perplexity, CIS (Learned) is the
top performer, with BPR coming in second and CIS (Random) a distant third. Since BMF does not
produce a distribution over items, its performance cannot be naturally measured in terms of PPL. On
the ranking metrics, CIS (Learned) and BPR emerge as the best-performing methods, achieving very
similar scores. BPR has a slight edge over CIS on MAP, while CIS performs better on Precision@1.
BMF and CIS (Random) are the weakest performers, with considerably worse scores than BPR and
CIS (Learned) on all metrics. Comparing the results of CIS (Learned) and CIS (Random) shows that
the of the tree used has a strong effect on the performance of CIS models and that using trees learned
with the proposed algorithm makes CIS competitive with the best collaborative filtering models. The
similar results achieved by CIS (LearnedRI) and CIS (LearnedCI) suggest that that the performance
7
of the resulting model is not particularly sensitive to the initialization scheme of the tree-learning
algorithm.
To understand the behaviour of our tree-learning algorithm better we examined the trees produced
by it. The learned trees looked sensible, with neighbouring leaves typically containing movies from
the same sub-genre and appealing to the same audience. We then determined how discriminative
the decisions were at each level of the tree by replacing the user-dependent distributions under all
nodes at a particular depth by the best user-independent approximations (frequencies of items under
the node). Comparing the perplexity of a model using the tree truncated at level l and at level
l + 1 allowed us to determine how much level l + 1 contributed to the model. In the CIS (Random)
model, the first few and the last few levels had little effect on perplexity and the medium-depth levels
accounted for most of perplexity reduction. In contrast, in the CIS (LearnedRI) model, the effect
of a level on perplexity decreased with level depth, with the first few levels reducing perplexity the
most, which is a consequence of the greedy nature of the tree-learning algorithm.
To highlight the importance of excluding items of unknown relevance when evaluating implicit
feedback models we recomputed the performance metrics treating all items not rated by a user
as irrelevant. As the scores in Table 2 show this seemingly minor modification of the evaluation
protocol makes BMF appear to outperform BPR by a large margin, which, as Table 1 indicates is not
actually the case. In retrospect, these changes in relative performance are not particularly surprising
since the training algorithm for BMF treats unobserved items as negative examples, which perfectly
matches the assumption the evaluation is based on, namely that unobserved items are irrelevant. This
is a clear example of a flawed evaluation protocol favouring an unrealistic modelling assumption.
8
Discussion
We proposed a model that in addition to being competitive with the best implicit feedback models
in terms of predictive accuracy also provides calibrated item selection probabilities for each user,
which quantify the user?s interest in the items. These probabilities allow comparing the degree of
interest in an item across users, making it possible to maximize the total user satisfaction when
item availability is limited. More generally, the probabilities provided by the model can be used in
combination with utility functions for making sophisticated decisions.
Although we introduced our tree-learning algorithm in the context of collaborative filtering, it is
applicable to several other problems. One such problem is statistical language modelling, where
the task is to predict the distribution of the next word in a sentence given its context consisting of
several preceding words. While there already exists an algorithm for learning the structure of treebased language models [7], it constructs trees by clustering word representations, not taking into
account the form of the model that will use these trees. In contrast, our algorithm optimizes the tree
structure and model parameters jointly, which can lead to superior model performance.
The proposed algorithm can also be used to learn trees over labels for multinomial regression models. When the number of labels is large, using a label space with a sensible tree structure can lead
to much faster training and improved generalization. Our algorithm can be applied in this setting
by noticing the correspondence between items and labels, and between user factor vectors and input
vectors. However, unlike in collaborative filtering where user factor vectors have to be learned, in
this case input vectors are observed, which eliminates the need to train a model based on a random
tree before applying the tree-learning algorithm.
We believe that evaluation protocols for implicit feedback models deserve more attention than they
have received. In this paper we observed that one widely used protocol can produce misleading
results due to an unrealistic assumption it makes about item relevance. We proposed using a small
quantity of explicit feedback data to directly estimate item relevance in order to avoid having to
make that assumption.
Acknowledgments
We thank Biljana Petreska and Lloyd Elliot for their helpful comments and the Gatsby Charitable
Foundation for generous funding. 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.
8
References
[1] Alina Beygelzimer, John Langford, Yuri Lifshits, Gregory B. Sorkin, and Alexander L. Strehl.
Conditional probability tree estimation analysis and algorithms. In Proceedings of the 25th
Conference on Uncertainty in Artificial Intelligence, 2009.
[2] L?eon Bottou. Large-scale machine learning with stochastic gradient descent. In Proceedings
of the 19th International Conference on Computational Statistics (COMPSTAT?2010), pages
177?187, 2010.
[3] J. Goodman. Classes for fast maximum entropy training. In Proceedings of ICASSP ?01,
volume 1, pages 561?564, 2001.
[4] Yifan Hu, Yehuda Koren, and Chris Volinsky. Collaborative filtering for implicit feedback
datasets. In Proceedings of the 2008 Eighth IEEE International Conference on Data Mining,
pages 263?272, 2008.
[5] Yehuda Koren. Factorization meets the neighborhood: a multifaceted collaborative filtering
model. In Proceeding of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 426?434, 2008.
[6] Benjamin Marlin. Collaborative filtering: A machine learning perspective. Master?s thesis,
University of Toronto, 2004.
[7] Andriy Mnih and Geoffrey Hinton. A scalable hierarchical distributed language model. In
Advances in Neural Information Processing Systems, volume 21, 2009.
[8] Frederic Morin and Yoshua Bengio. Hierarchical probabilistic neural network language model.
In AISTATS?05, pages 246?252, 2005.
[9] Rong Pan and Martin Scholz. Mind the gaps: weighting the unknown in large-scale one-class
collaborative filtering. In KDD, pages 667?676, 2009.
[10] Rong Pan, Yunhong Zhou, Bin Cao, Nathan Nan Liu, Rajan M. Lukose, Martin Scholz, and
Qiang Yang. One-class collaborative filtering. In ICDM, pages 502?511, 2008.
[11] Steffen Rendle, Christoph Freudenthaler, Zeno Gantner, and Schmidt-Thieme Lars. BPR:
Bayesian personalized ranking from implicit feedback. In UAI ?09, pages 452?461, 2009.
[12] Ruslan Salakhutdinov and Andriy Mnih. Probabilistic matrix factorization. In Advances in
Neural Information Processing Systems, volume 20, 2008.
[13] Nathan Srebro, Jason D. M. Rennie, and Tommi Jaakkola. Maximum-margin matrix factorization. In Advances in Neural Information Processing Systems, 2004.
[14] Jason Weston, Samy Bengio, and David Grangier. Label embedding trees for large multi-class
tasks. In Advances in Neural Information Processing Systems (NIPS), 2010.
9
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4,001 | 4,621 | Effective Split-Merge Monte Carlo Methods for
Nonparametric Models of Sequential Data
Michael C. Hughes1 , Emily B. Fox2 , and Erik B. Sudderth1
1
Department of Computer Science, Brown University, {mhughes,sudderth}@cs.brown.edu
2
Department of Statistics, University of Washington, [email protected]
Abstract
Applications of Bayesian nonparametric methods require learning and inference
algorithms which efficiently explore models of unbounded complexity. We develop new Markov chain Monte Carlo methods for the beta process hidden
Markov model (BP-HMM), enabling discovery of shared activity patterns in large
video and motion capture databases. By introducing split-merge moves based on
sequential allocation, we allow large global changes in the shared feature structure. We also develop data-driven reversible jump moves which more reliably
discover rare or unique behaviors. Our proposals apply to any choice of conjugate
likelihood for observed data, and we show success with multinomial, Gaussian,
and autoregressive emission models. Together, these innovations allow tractable
analysis of hundreds of time series, where previous inference required clever initialization and lengthy burn-in periods for just six sequences.
1
Introduction
Bayesian nonparametric time series models, including various ?infinite? Markov switching processes [1, 2, 3], provide a promising modeling framework for complex sequential data. We focus
on the problem of discovering coherent, short-term activity patterns, or ?behaviors?, shared among
related time series. For example, given collections of videos or human motion capture sequences,
our goal is to (i) identify a concise global library of behaviors that explain the observed motions,
(ii) annotate each sequence with the subset of behaviors used, and (iii) label each timestep with one
active behavior. The beta process hidden Markov model (BP-HMM) [4] offers a promising solution
to such problems, allowing an unbounded set of relevant behaviors to be learned from data.
Learning BP-HMMs from large datasets poses significant computational challenges. Fox et al. [4]
considered a dataset containing only six motion capture sequences and proposed a Markov chain
Monte Carlo (MCMC) method that required careful initialization and tens of thousands of burn-in iterations. Their sampler included innovations like block state sequence resampling [5] and marginalization of some variables. However, like most MCMC samplers, their proposals only modified
small subsets of variables at each step. Additionally, the sampler relied on parameter proposals from
priors, leading to low acceptance rates for high-dimensional data. Alternative single-site MCMC
moves, such as those based on slice sampling [6, 7], can exhibit similarly slow mixing. Our goal
is to expose this pervasive issue with conventional MCMC, and develop new samplers that rapidly
explore the structural uncertainty inherent in Bayesian nonparametric models. While our focus is on
the BP-HMM, the technical innovations underlying our samplers are much more broadly applicable.
We make two complementary improvements to previous BP-HMM samplers [4]. First, we develop
split-merge moves which change many variables simultaneously, allowing rapid improvements in
the discovered behavior library. Our approach builds on previous work on restricted Gibbs proposals [8] and sequential allocation strategies [9], both of which were formulated for static Dirichlet
1
?
?
bk
fi
?i
z i ,1
zi ,2
f1
f2
f3
f4
?
?i
?
?
?k
features
k=1, 2, ...
xi ,1
xi ,2
time series
...
z i ,t
xi ,t
...
z i ,T
xi ,T
f1
f2
f3
f4
i = 1, 2, ? N
z1
z2
z3
z4
merge
split
km
z1
z2
z3
z4
ka kb
Figure 1: Left: The BP-HMM as a directed graphical model. Right: Illustration of our split and merge
proposals, which modify both binary feature assignment matrices F (white indicates present feature) and state
sequences z. We show F, z before (top) and after (bottom) feature km (yellow) is split into ka , kb (red,orange).
An item i with fikm = 1 can have either ka , kb , or both after the split, and its new zi sequence can use any
features available in fi . An item without km cannot possess ka , kb , and its state sequence zi does not change.
process (DP) mixture models [10]. Second, we design data-driven [11] reversible jump moves [12]
which efficiently discover behaviors unique to a single sequence. These data-driven proposals are
especially important for high-dimensional observation sequences. Both innovations apply to any
likelihood model with a conjugate prior; we show success with multinomial models of discrete
video descriptors, and Gaussian autoregressive models of continuous motion capture trajectories.
We begin in Sec. 2 by reviewing the BP-HMM model. We describe previous BP-HMM samplers [4]
in Sec. 3.1, and then develop our novel split-merge and data-driven proposals in Sec. 3.3-3.4. We
evaluate our contributions in Sec. 4, showing improvements over prior work modeling human motions captured both via a marker-based mocap system [4] and video cameras [13].
2
Beta Process Hidden Markov Models
Latent feature models intuitively capture the sparse sharing patterns occurring in collections of
human action sequences. For example, one sequence may contain jogging and boxing, while another
has jogging and dancing. We assign the ith sequence or ?item? with a sparse binary vector fi =
[fi1 , fi2 , . . .] indicating the presence or absence of each feature in the unbounded global collection.
Given N items, corpus-wide assignments are denoted by a binary matrix F whose ith row is fi .1
The feature matrix F is generated by an underlying stochastic process, the beta process (BP) [14]:
?
X
bk ??k .
B | B0 , ?, ? ? BP(?, ?B0 ), B =
(1)
k=1
A realization B of the BP contains infinitely many features k. For each feature, ?k ? B0 marks its
data-generation parameters, while bk ? (0, 1) denotes its inclusion probability. The binary feature
vector for item i is determined by independent Bernoulli draws fik ? Ber(bk ). Marginalizing over
B, the number of active features in item i has distribution Poisson(?), determined by mass parameter ?. The concentration parameter ? controls how often features are shared between items [15].
To apply feature models to time series data, Fox et al. [4] combine the BP with a hidden Markov
model to form the BP-HMM, shown in Fig. 1. The binary vector fi determines a finite set of states
available for item i. Each timestep t is assigned a single state zit = k from the set {k : fik = 1},
determining which parameters ?k generate data xit . Many different data-generation models are
possible. As in [4], for motion capture data we use a first-order Gaussian autoregressive process
with parameters ?k = (Ak , ?k ) drawn from a matrix-normal inverse-Wishart conjugate prior
xit | zit = k, xit?1 ? N (Ak xit?1 , ?k )
Ak , ?k | B0 ? MN W ?1 (?, S0 , R0 )
(2)
To study video, we use a Dirichlet-multinomial model for quantized interest point descriptors [13]
xit | zit = k ? Multinomial(?k )
?k | B0 ? Dirichlet(?0 , ?0 , . . . ?0 )
(3)
1
Throughout this paper, for variables wijk we use w to denote the vector or collection of wijk ?s over the
entire set of subscripts, and wi for the collection over only the omitted subscripts j and k
2
The BP-HMM allows each item independent transition dynamics. The transition distribution ?ij
from each state j for the HMM of item i is built by drawing a set of transition weights ?i , and then
normalizing these over the set of active features fi :
?ijk ? Gamma(? + ??jk , 1),
?ijk fik
?ijk = P
.
` fi` ?ij`
(4)
Here, ?jk = 1 if j = k, and 0 otherwise. This construction assigns positive transition mass ?ijk only
to features k active in fi . The sticky parameter ? places extra expected mass on self-transitions [3],
biasing the model to learn state sequences z with temporally persistent states.
3
MCMC Inference with Split-Merge Proposals
We first summarize the MCMC methods previously proposed for the BP-HMM [4]. We then present
our novel contributions: split-merge moves and data-driven reversible jump proposals. Full algorithmic details for all samplers are found in the supplement, and our code is available online.
3.1
Local Monte Carlo Proposals
Fox et al. [4]?s sampler alternates updates to HMM parameters ? and ?, discrete state sequences z,
and feature assignments F. Fixing F defines a collection of finite HMMs, so each zi can be block
sampled by dynamic programming [5], and then ?, ? drawn from conjugate posteriors.2 Sampling
each item?s features requires separate updates to features shared with some other time series and
features unique to item i. Both updates marginalize state sequences z and inclusion weights b.
For each shared feature k, Fox et al. propose flipping fik to the complementary binary value and
accept or reject according to the Metropolis-Hastings (MH) rule. This local move alters only one
entry in F while holding all others fixed; the split-merge moves of Sec. 3.3 improve upon it.
For unique features, Fox et al. [4] define a reversible pair of birth and death moves which add or
delete features to single sequences. While this approach elegantly avoids approximating the infinite
BP-HMM, their birth proposals use the (typically vague) prior to propose emission parameters ?k?
for new features k ? . We remedy this slow exploration with data-driven proposals in Sec. 3.4.
3.2
Split-Merge Proposals for Dirichlet Process Models
Split-merge MCMC methods were first applied to nonparametric models by Jain and Neal [8] in
work focusing on DP mixture models with conjugate likelihoods. Conjugacy allows samplers to
operate directly on discrete partitions of observations into clusters, marginalizing emission parameters. Jain and Neal present valid proposals that reversibly split a single cluster km into two (ka , kb ),
or merge two clusters into one. Since merges are deterministic, the primary contribution of [8] is a
generic technique ? restricted Gibbs (RG) sampling ? for proposing splits consistent with the data.
To construct an initial split of km , the RG sampler first assigns items in cluster km at random to either
ka or kb . Starting from this partition, the proposal is constructed by performing one-at-a-time Gibbs
updates, forgetting an item?s current cluster and reassigning to either ka or kb conditioned on the
remaining partitioned data. After several sweeps, these Gibbs updates encourage proposed clusters
ka and kb which agree with the data and thus are more likely to be accepted. For non-conjugate
models, more complex RG proposals can be constructed which instantiate emission parameters [16].
Even in small datasets, there can be significant benefits from performing five or more sweeps for
each RG proposal [8]. For large datasets, however, requiring many sweeps for a single proposal
is computationally expensive. An alternative sequential allocation [9] method replaces the random
initialization of RG by using two randomly chosen items to ?anchor? the two new clusters ka , kb .
Remaining items are then sequentially assigned to either ka or kb one-at-a-time, using RG moves
conditioning only on previously assigned data. This creates a proposed partition in agreement with
the data after only one sampling sweep. Recent work has shown some success with sequentiallyallocated split-merge moves for a hierarchical DP topic model [17].
2
Fox et al. [4] contains a small error in the resampling of ?, as detailed and corrected in the supplement.
3
For nonparametric models not based on the DP, split-merge moves are not well studied. Several
authors have considered RG split-merge proposals for beta process models [18, 19, 20]. However,
these papers lack technical details, and do not contain experiments showing improved mixing.
3.3
Split-Merge Proposals for the BP-HMM
We now adapt RG and sequential allocation to define BP-HMM split-merge moves. In the mixture
models considered by prior work [8, 9], each data item i is associated with a single cluster ki ,
so selecting two anchors i, j also identifies two cluster indices ki , kj . However, in feature-based
models such as the BP-HMM, each data item i is associated with a collection of features indexed by
fi . Therefore, our proposals require mechanisms for selecting anchors and for choosing candidate
states to split or merge from fi , fj . Additionally, our proposals must allow changes to state sequences
z to reflect changes in F. Our proposals thus jointly create a new configuration (F? , z? ), collapsing
away HMM parameters ?, ?. Fig. 1 illustrates (F, z) before and after a split move.
Selecting Anchors Following [9], we first randomly select distinct anchor data items i and j. The
fixed choice of i, j defines a split-merge transition kernel satisfying detailed balance. Next, we select
from each anchor one feature it possesses, denoted ki , kj . This choice determines the proposed
move: we merge ki , kj if they are distinct, and split ki = kj into two new features otherwise.
Selecting ki , kj uniformly at random is problematic. First, in datasets with many features choosing
ki = kj is unlikely, making split moves rare. We need to bias the selection process to propose splits
more often. Second, in a well-trained model most feature pairs will not make a sensible merge.
Selecting a pair that explains similar data is crucial for efficiency. We thus develop a proposal
distribution which first draws ki uniformly from fi , and then selects kj given fixed ki as follows:
(
2Cj fjk
if k = ki
qk (ki , kj ) = Unif(ki | fi )q(kj | ki , fj ), q(kj = k | ki , fj ) ?
(5)
m(xki ,xk )
fjk m(xk )m(xk ) o.w.
i
Here, xk is the data assigned to k,P
m(?) denotes the marginal likelihood of data
collapsing away
emission parameters ?, and Cj = kj 6=ki fjkj m(xki , xkj )/ m(xki )m(xkj ) . This construction
gives large mass (2/3) to a split move when possible, and also encourages choices ki 6= kj for a
merge that explain similar data via the marginal likelihood ratio. A large ratio means the model
prefers to explain all data assigned to ki , kj together rather than separately, biasing selection towards promising merge candidates. We find higher acceptance rates for merges under this qk , which
justifies the small cost of computing m(?) from cached sufficient statistics.
Once ki , kj are fixed, we construct the candidate state F? , z? . As shown in Fig. 1, we only alter
f` , z` for items ` which possess either ki or kj . We call this set of items the active set S.
Split Our split proposal is defined in Alg. 1. Sweeping through a random permutation of items
` in the active set S, we draw each item?s assignment to new features ka , kb and resample its state
?
sequence. We sample [f`k
f ? ] from its conditional posterior given previously visited items in
a `kb
S, requiring that ` must possess at least one of the new features. We then block sample its state
sequence z?` given f`? . The dynamic programming recursions underlying these proposals use nonrandom auxiliary parameters: ??` is set to its prior mean, and ??k to its posterior mean given the current
z. For new states k ? ? {ka , kb }, we initialize ??k? from anchor sequences and then update to account
for new data assigned to k ? at each item `. This enables better matching of proposed features to data
?
?
statistics. Finally, we sample f ? , z? for anchor items, enforcing fik
= 1 and fjk
= 1 so the move
a
b
?
?
remains reversible under a merge. This does not force zi to use ka nor zj to use kb .
?
Merge For a merge move, constructing F? is deterministic: we set f`k
= 1 for ` ? S, and 0
m
?
otherwise. We thus need only to sample z` for items in S, using a block sampler as in Alg. 1. Again
? ??, which we emphasize are deterministic tools enabling
this requires auxiliary HMM parameters ?,
collapsed proposals of discrete indicators F? , z? . We never sample ?, ?.
Accept-Reject After drawing a candidate value F? , z? , the final step is to compute a MetropolisHastings acceptance probability min(?, 1). We give the ratio for a split move which creates features
4
Alg. 1 Propose split of feature km into ka , kb given F, z, x, anchor items i, j, set S={`:f`,km =1}
1: fi,[ka ,kb ] ? [1 0]
zi,t:zi,t =km ? ka
use anchor i to create ka
2: fj,[ka ,kb ] ? [0 1]
zj,t:zj,t =km ? kb
use anchor j to create kb
set HMM params deterministically
3: ?? ? E [? | x, z, ?]
??` ? E [?` | ?, ?], ` ? S
initialize set of previously visited items
4: Sprev = {i, j}
5: for non-anchor items ` in random permutation of active set S:
?
?[0 1]
? ??` )
6:
f`,[ka kb ] ? [1 0] ? p(f`,[ka kb ] | fSprev ,[ka kb ] )p(x` | f` , ?,
?
[1 1]
? ??` )
z` ? p(z` | x` , f` , ?,
draw f, z and update ?? for each item
condition on previously visited items
add ` to Sprev
?
9:
for k = ka , kb : ?k ? E [?k | ?, {xnt : znt = k, n ? Sprev }]
[0 1]
[1 0]
finish by sampling f, z for anchors
fj,[ka kb ] ?
10: fi,[ka kb ] ?
7:
8:
[1 1]
? ??i )
11: zi ? p(zi | xi , fi , ?,
[1 1]
? ??j )
zj ? p(zj | xj , fj , ?,
ka , kb from km below. The acceptance ratio for a merge move is the reciprocal of Eq. (6).
p(x, z? , F? ) qmerge (F, z | x, F? , z? , ka , kb ) qk (ka , kb | x, F? , z? , i, j)
(6)
?split =
p(x, z, F)
qsplit (F? , z? | x, F, z, km ) qk (km , km | x, F, z, i, j)
The joint probability p(x, z, F) is only tractable with conjugate likelihoods. Proposals which instantiate emission parameters ?, as in [16], would be required in the non-conjugate case.
3.4
Data-Driven Reversible Jump Birth and Death Proposals
Efficiently adding or deleting unique features is crucial for good mixing. To accept the birth of new
feature k ? = K + 1 for item i, this feature must explain some of the observed data xi at least as well
as existing features 1, 2, . . . K. High-dimensional emission parameters ?k? drawn from a vague prior
are unlikely to match the data at hand. Instead, we suggest a data-driven proposal [11, 13] for ?k? .
First, select at random a subwindow W of the current sequence i. Next, use data in this subwindow
xW = {xit : t ? W } to create a proposal distribution: q? (?) = 12 p(?) + 12 p(? | xW ), which is
a mixture of ??s prior and posterior given xW . This mixture strikes a balance between proposing
promising new features (via the posterior) while also making death moves possible, since the diffuse
prior will place some mass on the reverse birth move.
?
Let Ui denote the number of unique features in fi and ? = ? N ?1+?
. The acceptance probability for
? ? ?
a birth move to candidate state fi , ?i , ? is then min(?birth , 1), where
p(xi | fi? , ?i? , ?? ) Poi(Ui + 1 | ?) p? (?k?? ) qf (fi | fi? )
?birth =
(7)
p(xi | fi , ?i , ?)
Poi(Ui | ?) q? (?k?? ) qf (fi? | fi )
Eq. (7) is similar to the ratio for the birth proposal from the prior, adding only one term to account
for the proposed ?k?? . Note that each choice of W defines a valid pair of birth-death moves satisfying
detailed balance, so we need not account for this choice in the acceptance ratio [21].
4
Experimental Results
Our experiments compare competing inference algorithms for the BP-HMM; for comparisons to
alternative models, see [4]. To evaluate how well our novel moves explore the posterior distribution, we compare three possible methods for adding or removing features: split-merge moves (SM,
Sec. 3.3), data-driven moves (DD, Sec. 3.4), and reversible jump moves using the prior (Prior [4],
Sec. 3.1). All experiments interleave the chosen update with the standard MH updates to shared
features of F and Gibbs updates to HMM parameters ?, ? described in Sec. 3.1.
For each comparison, we run multiple initializations and rank the chains from ?best? to ?worst?
according to joint probability. Each chain is allowed the same amount of computer time.
5
Gaussian
AR
0
SM
DD
Prior
?2
500
1000
1500
2000
2500
cpu time (sec)
3000
?1
SM
DD
Prior
?1.2
?1.4
0
3500
5
?0.8 x 10
log joint prob.
2
?4
0
Multinomial
6
x 10
log joint prob.
log joint prob.
4
4
500
1000
1500
2000
2500
cpu time (sec)
3000
?4.5 x 10
?5
?5.5
SM
DD
Prior
?6
?6.5
0
3500
500
1000
1500
2000
2500
cpu time (sec)
3000
3500
1.5
0.2
1
2
0.5
Worst SM
4
0
0
?0.5
6
?1
?0.2
?1.5
?1
0
8
?1
1
?0.5
0
0.5
1
100
200
300
400
500
100
200
300
400
500
100
200
300
400
500
1.5
0.2
1
2
0.5
Worst DD
4
0
0
?0.5
6
?1
?0.2
?1.5
?1
0
8
?1
1
?0.5
0
0.5
1
1.5
0.2
1
2
0.5
Best Prior
0
0
4
?0.5
6
?1
?0.2
?1.5
?1
0
1
?1
?0.5
0
0.5
1
Figure 2: Feature creation for synthetic data with Gaussian (left), AR (middle), or multinomial (right) likelihoods. Each run begins with one feature used by all items, and must add new features via split-merge proposals
(SM), or reversible-jump moves using data-driven (DD) or prior (Prior) proposals. Top: Log joint probability
versus computation time, for 10 random initializations of each sampling algorithm. Bottom: Emission parameters associated with the last sample after one hour of computation time. Gaussian ? = (?, ?) and AR
? = (A, ?) shown as contour lines in first two dimensions, with location determined by ?, A. Multinomial ?
shown as image where row k gives the emission distribution over vocabulary symbols for state k.
4.1
Synthetic Data
We examined toy datasets generated by a known set of 8 features (behaviors) ?true . To validate
that our contributions apply for many choices of likelihood, we create three datasets: multinomial
?bag-of-words? emissions using 500 vocabulary words, 8?dimensional Gaussian emissions, and a
first-order autoregressive (AR) processes with 5 dimensions. Each dataset has N = 100 sequences.
First, we study how well each method creates necessary features from scratch. We initialize the
sampler with just one feature used by all items, and examine how many true states are recovered
after one hour of computation time across 10 runs. We show trace plots as well as illustrations of
recovered emission parameters ? in Fig. 2. All runs of both SM and DD moves find all true states
within several minutes, while no Prior run recovers all true states, remaining stuck with merged
versions of true features. DD moves add new features most rapidly due to low computational cost.
We next examine whether each inference method can remove unnecessary features. We consider a
different toy dataset of several hundred sequences and a redundant initialization in which 2 copies
of each true state exist. Half of the sequences are initialized with f , z set to corresponding true
values in copy 1, and the second half using copy 2. Using Gaussian and AR likelihoods, all SM runs
merge down to the 8 true states, at best within five minutes, but no DD or Prior run ever reaches
this optimal configuration in the allotted hour. Merge moves enable critical global changes, while
the one-at-a-time updates of [4] (and our DD variant) must take long random walks to completely
delete a popular feature. Further details are provided in the supplementary material.
These results demonstrate the importance of DD birth and split moves for exploring new features,
and merge moves for removing features via proposals of large assignment changes. As such, we
consider a sampler that interleaves SM and DD moves in our subsequent analyses.
4.2
Motion Capture Data
We next apply our improved MCMC methods to motion capture (mocap) sequences from the CMU
database [22]. First, we consider the small dataset examined by [4]: 6 sequences of physical exercises with motion measurements at 12 joint angles, modeled with an AR(1) likelihood. Human
annotation identifies 12 actions, which we take as ground truth. Previous results [4] show that the
BP-HMM outperforms competitors in segmenting these actions, although they report that some true
actions like jogging are split across multiple recovered features (see their Fig. 5). We set likelihood
hyperparameters similarly to [4], with further details provided in the supplement.
6
4
x 10
SM+DD from one
SM+DD from unique5
Prior from unique5
?5.8
1
1.5
2
cpu time (sec)
0.7
2.5
3
3.5
4
x 10
SM+DD from one
SM+DD from unique5
Prior from unique5
0.6
0.5
25
20
20
20
15
15
5
10
5
2
4
2
5
?2
?5
6
?10
8
?15
x
5
5
5
5
5
5
5
5
5
5
5
5
205
200
5
5
5
5
5
5
5
210
205
5
5
5
5
5
5
5
215
210
5
5
5
5
5
5
5
220
215
5
5
5
5
5
5
5
225
220
5
5
5
5
5
5
5
230
225
5
5
5
5
5
5
5
235
230
235
5
5
240
105
5
5
240
105
5
6
5
6
5
5
5
5
110
110
5
5
5
5
5
115
5
115
5
5
5
5
120
120
5
5
5
5
5
5
5
125
5
0
?2
?5
2
?10
4
5
5
5
5
5
5
130
125
5
5
5
5
5
5
5
135
130
5
5
140
135
140
185
140
185
2
4
6
?6
?4
?2
z
5
5
5
5
5
5
185
5
0
x
z
z
200
5
4
x
5
5
0
0
?10
?15
6
10
5
0
5
15
5
10
5
z
10
10
10
10
5
15
15
10
10
?5
0.5
15
y
y
15
30
25
20
y
20
30
25
25
y
20
?6
0
Hamming dist.
25
y
25
?5.6
y
joint log prob.
?5.4
5
5
5
5
5
5
190
5
5
5
5
5
5
5
195
190
5
5
1
5
5
28
28
200
195
5
5
5
5
5
5
5
205
200
5
5
5
5
5
5
5
210
205
210
5
2
215
42
5
2
2
2
2
2
2
2
2
2
2
2
2
215
42
28
14 14 14 14 14 14 14 14 2
2
47
2
2
2
2
52
2
52
2
0
x
2
47
2
?4
?2
57
2
2
2
2
2
2
2
62
57
2
2
2
2
2
2
2
67
62
2
2
2
2
2
2
2
72
67
2
0
2
4
?4
?2
2
0
z
2
2
2
2
2
2
77
72
77
2
2
82
165
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
170
2
?4
?2
4
2
2
2
2
2
2
2
2
2
175
170
2
2
2
2
2
2
2
2
2
2
180
175
2
180
0
2
4
?6
?4
?2
z
x
2
2
185
145
2
2
185
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
150
82
165
2
21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21
25 2
82
145
165
145
2
0
2
x
2
2
2
2
2
2
2
2
2
2
155
150
2
2
1
1
1
1
2
1
1
1
1
1
1
1
160
155
2
1
165
160
1
165
0.4
5
0.3
0.2
0
200
0.5
1
1.5
2
cpu time (sec)
2.5
3
205
5
210
215
5
220
225
5
230
235
5
240
20 20 20 20 20 20 20 20 20 20 20
105
110
115
120
125
130
5
135
5
190
5
195
200
5
205
210
215
42
47
52
57
2
62
67
2
72
77
170
175
180
185
2 26 25 25 25 25 25 2
150
2
2 25 25 25 1
155
160
1
165
3.5
4
x 10
Figure 3: Analysis of 6 motion capture sequences previously considered by [4]. Left: Joint log probability
and Hamming distance (from manually annotated behaviors) for 20 runs of each method over 10 hours. Right:
Examples of arm circles and jogging from 3 sequences, along with estimated zi of last sample from the best
run of each method. SM+DD moves (top row started from one feature, middle row started with 5 unique
states per sequence) successfully explain each action with one primary state, while [4]?s sampler (bottom row)
started from 5 unique features remains stuck with multiple unique states for one true action.
Ballet
Walk
Squat
Sword
Lambada
Dribble Basketball
Box
Climb
Indian Dance
Tai Chi
Figure 4: Analysis of 124 mocap sequences, showing 10 of 33 recovered behaviors. Skeleton trajectories are
built from contiguous segments ? 1 sec long assigned to each behavior. Boxed groups contain segments from
distinct sequences assigned to the same state. Some states only appear in one sequence.
In Fig. 3, we compare a sampler which interleaves SM and DD moves with [4]?s original method.
We run 20 chains of each method for ten hours from two initializations: unique5, which assigns
5 unique features per sequence (as done in [4]), and one, using a single feature across all items.
In both log probability and Hamming distance, SM+DD methods are noticeably superior. Most
interestingly, SM+DD starting from a parsimonious one feature achieves best performance overall,
showing that clever initialization is not necessary with our algorithm. The best run of SM+DD from
one achieves Hamming distance of 0.22, compared to 0.3 reported in [4]. No Prior proposal run
from one created any additional states, indicating the importance of using our improved methods
of feature exploration even in moderate dimensions.
Our SM moves are critical for effectively creating and deleting features. Example segmentations of
arm-circles and jogging actions in Fig. 3 show that SM+DD consistently use one dominant behavior
across all segments where the action appears. In contrast, the Prior remains stuck with some unique
behaviors used in different sequences, yielding lower probability and larger Hamming distance.
Next, we study a larger dataset of 124 sequences, all ?Physical Activities & Sports? examples from
the CMU mocap dataset. Analyzing a dataset of this size is computationally infeasible using the
methods of [4]. Initializing from unique5 would create over 500 features, requiring a prohibitively
long sampling run to merge related behaviors. When initialized from one, the Prior sampler creates
no additional features. In contrast, starting from one, our SM+DD moves rapidly identify a diverse
set of 33 behaviors. A set of 10 behaviors representative of this dataset are shown in Fig. 4. Our
improved algorithm robustly explores the posterior space, enabling this large-scale analysis.
4.3
CMU Kitchen: Activity Discovery from Video
Finally, we apply our new inference methods to discover common motion patterns from videos of
recipe preparation in the CMU Kitchen dataset [23]. Each video is several minutes long and depicts
a single actor in the same kitchen preparing either a pizza, a sandwich, a salad, brownies, or eggs.
Our previous work [13] showed promising results in activity discovery with the BP-HMM using a
7
Eggs
Flip Omelette
Pizza Sandwich Salad
Grate Cheese
Brownie
Slice/Chop
Light Switch
Pour Bowl
Stir Bowl 2
Stir Bowl 1
Open Fridge
65 Others
0.25
0.5
0.75
1
Fraction of Time Elapsed
Figure 5: Activity discovery with 126 Kitchen videos, showing locations of select behaviors over time. Each
row summarizes zi for a single video, labeled at left by recipe type (label not provided to the BP-HMM). We
show only behaviors assigned to at least two timesteps in a local window.
small collection of 30 videos from this collection. We compare our new SM moves on this small
dataset, and then study a larger set of 126 Kitchen videos using our improved sampler.
Using only the 30 video subset, Fig. 6 compares the combined SM+DD sampler with just DD or
Prior moves, using fixed hyperparameter settings as in [13] and starting with just one feature. DD
proposals offer significant gains over the prior, and further interleaving DD and SM moves achieves
the best overall configuration, showing the benefits of proposing global changes to F, z.
5
6
x 10
log joint prob.
Finally, we run the SM+DD sampler on 126 Kitchen sequences, choosing the best of 4 chains after several days
of computer time (trace plots show convergence in half this
time). Fig. 5 maps behavior assignments over time across all
five recipes, using the last MCMC sample. Several intuitive
behavior sharing patterns exist: chopping happens with carrots (salad) and pepperoni (pizza), while stirring occurs when
preparing brownies and eggs. Non-uniform behavior usage
patterns within a category are due to differences in available
cooking equipment across videos. Please see the supplement
for more experimental details and results.
?7.05
?7.1
SM+DD
DD
Prior
?7.15
?7.2
0
1
2
3
4
5
cpu time (sec)
6
7
8
4
x 10
Figure 6: Joint log probability versus
computation time for various samplers
on the CMU Kitchen data [23] previously studied by [13].
Discussion
We have developed efficient MCMC inference methods for the BP-HMM. Our proposals do not require careful initialization or parameter tuning, and enable exploration of large datasets intractable
under previous methods. Our approach makes efficient use of data and applies to any choice of conjugate emission model. We expect the guiding principles of data-driven and sequentially-allocated
proposals to apply to many other models, enabling new applications of nonparametric analysis.
Acknowledgments
M. Hughes was supported in part by an NSF Graduate Research Fellowship under Grant No.
DGE0228243. E. Fox was funded in part by AFOSR Grant FA9550-12-1-0453.
8
References
[1] M. Beal, Z. Ghahramani, and C. Rasmussen. The infinite hidden Markov model. In NIPS, 2002.
[2] Y. W. Teh, M. I. Jordan, M. J. Beal, and D. M. Blei. Hierarchical Dirichlet processes. Journal of the
American Statistical Association, 101(476):1566?1581, 2006.
[3] E. B. Fox, E. B. Sudderth, M. I. Jordan, and A. S. Willsky. A sticky HDP-HMM with application to
speaker diarization. Annals of Applied Statistics, 5(2A):1020?1056, 2011.
[4] E. B. Fox, E. B. Sudderth, M. I. Jordan, and A. S. Willsky. Sharing features among dynamical systems
with beta processes. In NIPS, 2010.
[5] S. L. Scott. Bayesian methods for hidden Markov models: Recursive computing in the 21st century.
JASA, 97(457):337?351, 2002.
[6] J. Van Gael, Y. Saatci, Y. W. Teh, and Z. Ghahramani. Beam sampling for the infinite hidden Markov
model. In ICML, 2008.
[7] C. Yau, O. Papaspiliopoulos, G. O. Roberts, and C. Holmes. Bayesian non-parametric hidden Markov
models with applications in genomics. JRSS B, 73(1):37?57, 2011.
[8] S. Jain and R.M. Neal. A split-merge Markov chain Monte Carlo procedure for the Dirichlet process
mixture model. Journal of Computational and Graphical Statistics, 13(1):158?182, 2004.
[9] D. B. Dahl. Sequentially-allocated merge-split sampler for conjugate and nonconjugate Dirichlet process
mixture models. Submitted to Journal of Computational and Graphical Statistics, 2005.
[10] M. D. Escobar and M. West.
90(430):577?588, 1995.
Bayesian density estimation and inference using mixtures.
JASA,
[11] Z. Tu and S. C. Zhu. Image segmentation by data-driven Markov chain Monte Carlo. PAMI, 24(5):657?
673, 2002.
[12] P.J. Green. Reversible jump Markov chain Monte Carlo computation and Bayesian model determination.
Biometrika, 82(4):711?732, 1995.
[13] M. C. Hughes and E. B. Sudderth. Nonparametric discovery of activity patterns from video collections.
In CVPR Workshop on Perceptual Organization in Computer Vision, 2012.
[14] R. Thibaux and M. I. Jordan. Hierarchical beta processes and the Indian buffet process. In AISTATS,
2007.
[15] T. L. Griffiths and Z. Ghahramani. Infinite latent feature models and the Indian buffet process. In NIPS,
2007.
[16] S. Jain and R.M. Neal. Splitting and merging components of a nonconjugate Dirichlet process mixture
model (with invited discussion). Bayesian Analysis, 2(3):445?500, 2007.
[17] C. Wang and D. Blei. A split-merge MCMC algorithm for the hierarchical Dirichlet process.
arXiv:1201.1657v1 [stat.ML], 2012.
[18] E. Meeds, R. Neal, Z. Ghahramani, and S. Roweis. Modeling dyadic data with binary latent factors. In
NIPS, 2008.
[19] K. Miller, T. Griffiths, and M. Jordan. Nonparametric latent feature models for link prediction. In NIPS,
2009.
[20] M. M?rup, M. N. Schmidt, and L. K. Hansen. Infinite multiple membership relational modeling for
complex networks. In IEEE International Workshop on Machine Learning for Signal Processing, 2011.
[21] L. Tierney. Markov chains for exploring posterior distributions (with discussion). Annals of Statistics,
22:1701?1762, 1994.
[22] Carnegie Mellon University. Graphics lab motion capture database. http://mocap.cs.cmu.edu/.
[23] F. De la Torre et al. Guide to the Carnegie Mellon University Multimodal Activity (CMU-MMAC)
database. Technical Report CMU-RI-TR-08-22, Robotics Institute, Carnegie Mellon University, 2009.
9
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4,002 | 4,622 | Tensor Decomposition for Fast Parsing with
Latent-Variable PCFGs
Shay B. Cohen and Michael Collins
Department of Computer Science
Columbia University
New York, NY 10027
scohen,[email protected]
Abstract
We describe an approach to speed-up inference with latent-variable PCFGs, which
have been shown to be highly effective for natural language parsing. Our approach
is based on a tensor formulation recently introduced for spectral estimation of
latent-variable PCFGs coupled with a tensor decomposition algorithm well-known
in the multilinear algebra literature. We also describe an error bound for this
approximation, which gives guarantees showing that if the underlying tensors are
well approximated, then the probability distribution over trees will also be well
approximated. Empirical evaluation on real-world natural language parsing data
demonstrates a significant speed-up at minimal cost for parsing performance.
1
Introduction
Latent variable models have shown great success in various fields, including computational linguistics and machine learning. In computational linguistics, for example, latent-variable models are
widely used for natural language parsing using models called latent-variable PCFGs (L-PCFGs;
[14]).
The mainstay for estimation of L-PCFGs has been the expectation-maximization algorithm [14,
16], though other algorithms, such as spectral algorithms, have been devised [5]. A by-product
of the spectral algorithm presented in [5] is a tensor formulation for computing the inside-outside
probabilities of a L-PCFG. Tensor products (or matrix-vector products, in certain cases) are used as
the basic operation for marginalization over the latent annotations of the L-PCFG.
The computational complexity with the tensor formulation (or with plain CKY, for that matter) is
cubic in the number of latent states in the L-PCFG. This multiplicative factor can be prohibitive for
a large number of hidden states; various heuristics are used in practice to avoid this problem [16].
In this paper, we show that tensor decomposition can be used to significantly speed-up the parsing
performance with L-PCFGs. Our approach is also provided with a theoretical guarantee: given the
accuracy of the tensor decomposition, one can compute how accurate the approximate parser is.
The rest of this paper is organized as follows. We give notation and background in ?2?3, and then
present the main approach in ?4. We describe experimental results in ?5 and conclude in ?6.
2
Notation
Given a matrix A or a vector v, we write A> or v > for the associated transpose. For any integer
n ? 1, we use [n] to denote the set {1, 2, . . . n}. We will make use of tensors of rank 3:1
1
All PCFGs in this paper are assumed to be in Chomsky normal form. Our approach generalizes to arbitrary
PCFGs, which require tensors of higher rank.
1
Definition 1. A tensor C ? R(m?m?m) is a set of m3 parameters Ci,j,k for i, j, k ? [m]. Given
2
a tensor C, and vectors y 1 ? Rm and yP
? Rm , we define C(y 1 , y 2 ) to be the m-dimensional row
vector with components [C(y 1 , y 2 )]i = j?[m],k?[m] Ci,j,k yj1 yk2 . Hence C can be interpreted as a
function C : Rm ? Rm ? R1?m that maps vectors y 1 and y 2 to a row vector C(y 1 , y 2 ) ? R1?m .
In addition, we define the tensor C(1,2) ? R(m?m?m) for any tensor C ? R(m?m?m) to be the
P
1 2
function C(1,2) : Rm ? Rm ? Rm?1 defined as [C(1,2) (y 1 , y 2 )]k =
i?[m],j?[m] Ci,j,k yi yj .
Similarly, for any tensor C we define C(1,3) : Rm ? Rm ? Rm?1 as [C(1,3) (y 1 , y 2 )]j =
P
1 2
1 2
1 2
i?[m],k?[m] Ci,j,k yi yk . Note that C(1,2) (y , y ) and C(1,3) (y , y ) are both column vectors.
For two vectors x ? Rm and y ? Rm we denote by x y ? Rm the Hadamard product of x and y,
i.e. [x y]i = xi yi . Finally, for vectors x, y, z ? Rm , xy > z > is the tensor D ? Rm?m?m where
Di,j,k = xi yj zk (this is analogous to the outer product: [xy > ]i,j = xi yj ).
3
Latent-Variable Parsing
In this section we describe latent-variable PCFGs and their parsing algorithms.
3.1
Latent-Variable PCFGs
This section gives a definition of the L-PCFG formalism used in this paper; we follow the definitions
given in [5]. An L-PCFG is a 5-tuple (N, I, P, m, n) where:
? N is the set of non-terminal symbols in the grammar. I ? N is a finite set of in-terminals. P ? N
is a finite set of pre-terminals. We assume that N = I ? P, and I ? P = ?. Hence we have
partitioned the set of non-terminals into two subsets.
? [m] is the set of possible hidden states.
? [n] is the set of possible words.
? For all a ? I, b ? N, c ? N, h1 , h2 , h3 ? [m], we have a context-free rule a(h1 ) ? b(h2 ) c(h3 ).
? For all a ? P, h ? [m], x ? [n], we have a context-free rule a(h) ? x.
Note that for any binary rule, a ? b c, it holds that a ? I, and for any unary rule a ? x, it holds
that a ? P.
The set of ?skeletal rules? is defined as R = {a ? b c : a ? I, b ? N, c ? N}. The parameters of
the model are as follows:
? For each a ? b c ? R, and h1 , h2 , h3 ? [m], we have a parameter t(a ? b c, h2 , h3 |h1 , a).
? For each a ? P, x ? [n], and h ? [m], we have a parameter q(a ? x|h, a).
An L-PCFG corresponds to a regular PCFG with non-terminals annotated with latent states.
For each triplet of latent states and a rule a ? b c, we have a rule probability p(a(h1 ) ?
b(h2 ) c(h3 )|a(h1 )) = t(a ? b c, h2 , h3 |h1 , a). Similarly, we also have parameters p(a(h) ?
x|a(h)) = q(a ? x|h, a). In addition, there are initial probabilities of generating a non-terminal
with a latent at the top of the tree, denoted by ?(a, h).
L-PCFGs induce distributions over two type of trees: skeletal trees, i.e. trees without values for
latent states (these trees are observed in data), and full trees (trees with values for latent states). A
skeletal tree consists of a sequence of rules r1 . . . rN where ri ? R or ri = a ? x. See Figure 3.1
for an example.
We now turn to the problem of computing the probability of a skeletal tree, by marginalizing out
the latent states of full trees. Let r1 . . . rN be a derivation, and let ai be the non-terminal on the left
(2)
hand-side of rule ri . For any ri = a ? b c, define hi to be the latent state associated with the left
(3)
child of the rule ri and hi to be the hidden variable value associated with the right child.
The distribution over full trees is then:
p(r1 . . . rN , h1 . . . hN ) = ?(a1 , h1 ) ?
Y
i:ai ?I
2
(2)
(3)
t(ri , hi , hi |hi , ai ) ?
Y
i:ai ?P
q(ri |hi , ai )
S1
NP2
VP5
D3
N4
V6
P7
the
man
saw
him
r1
r2
r3
r4
r5
r6
r7
= S ? NP VP
= NP ? D N
= D ? the
= N ? man
= VP ? V P
= V ? saw
= P ? him
Figure 1: An s-tree with its sequence of rules.
(The nodes in the tree are indexed by the derivation order, which is canonicalized as top-down,
left-most derviation.)
Marginalizing out
P the latent states leads to the distribution over the skeletal tree r1 . . . rN :
p(r1 . . . rN ) = h1 ...hN p(r1 . . . rN , h1 . . . hN ).
It will be important for the rest of this paper to use of matrix form of parameters of an L-PCFG, as
follows:
? For each a ? b c ? R, we define T a?b c ? Rm?m?m to be the tensor with values
c
Tha?b
= t(a ? b c, h2 , h3 |a, h1 )
1 ,h2 ,h3
? For each a ? P, x ? [n], we define Qa?x ? R1?m to be the vector with values q(a ? x|h, a)
for h = 1, 2, . . . m.
? For each a ? I, we define the vector ? a ? Rm where [? a ]h = ?(a, h).
Parameter Estimation Several ways to estimate the parameters T a?b c , Qa?x and ? a have been
suggested in the literature. For example, vanilla EM has been used in [14], hierarchical state splitting
EM has been suggested in [16], and a spectral algorithm is proposed in [5].
In the rest of the paper, we assume that the parameters for these tensors have been identified, and
focus mostly on the problem of inference ? i.e. parsing unseen sentences. The reason for this is
two-fold: (a) in real-world applications, training can be done off-line to identify a set of parameters
once, and therefore its computational efficiency is of lesser interest; (b) our approach can speed-up
the inference problems existing in the EM algorithm, but the speed-up is of lesser interest, because
the inference problem in the EM algorithm is linear in the tree size (and not cubic, as in the case
of parsing). The reason for this linear complexity is that the skeletal trees are observed during EM
training. Still, EM stays cubic in the number of states.
3.2
Tensor Formulation for Inside-Outside
There are several ways to parse a sentence with latent-variable PCFGs. Most of these approaches are
taken by using an inside-outside algorithm [12] which computes marginals for various non-terminals
and spans in the sentence, and then eventually finding a parse tree which maximizes a score which
is the sum of the marginals of the spans that appear in the tree.
P
More formally, let ?(a, i, j) = ? ?T(x):(a,i,j)?? p(? ) for each non-terminal a ? N, for each (i, j)
such that 1 ? i ? j ? N . Here T(x) denotes the set of all possible s-trees for the sentence x, and
we write (a, i, j) ? ? if non-terminal a spans words xi . . . xj in the parse tree ? .
Then, the parsing
P algorithm seeks for a given sentence x = x1 . . . xN the skeletal tree
arg max? ?T(x) (a,i,j)?? ?(a, i, j).
Given the marginals ?(a, i, j), one can use the dynamic programming algorithm described in [7] in
order to find this highest scoring tree.
A key question is how to compute the marginals ?(a, i, j) using the inside-outside algorithm. Dynamic programming solutions are available for this end as well. The complexity of a na??ve implementation of the dynamic programming algorithm for this problem is cubic in the number of
latent states. This is where we suggest an alternative to the traditional dynamic programming solutions. Our alternative relies on an existing tensor formulation for the inside-outside algorithm [5],
which re-formalizes the dynamic programming algorithm using tensor, matrix and vector product
operations.
Algorithm 2 presents the re-formulation of the inside-outside algorithm using tensors. For more
details and proofs of correctness, refer to [5]. The re-formalized algorithm is still cubic in
3
the number of hidden states, and spends most of the time computing the tensor applications
b?c a
b?a c
T a?b c (?b,i,k , ?c,k+1,j ), T(1,2)
(? b,k,j , ?c,k,i?1 ) and T(1,3)
(? b,i,k , ?c,j+1,k ). This is the main
set of computations we aim to speed-up, as we show in the next section.
Inputs: Sentence x1 . . . xN , L-PCFG (N, I, P, m, n), parameters T a?b c ? R(m?m?m) for all a ? b c ? R,
Qa?x ? R(1?m) for all a ? P, x ? [n], ? a ? R(m?1) for all a ? I.
Data structures:
? Each ?a,i,j ? R1?m for a ? N, 1 ? i ? j ? N is a row vector of inside terms.
? Each ? a,i,j ? Rm?1 for a ? N, 1 ? i ? j ? N is a column vector of outside terms.
? Each ?(a, i, j) ? R for a ? N, 1 ? i ? j ? N is a marginal probability.
Algorithm:
(Inside base case) ?a ? P, i ? [N ], ?a,i,i = Qa?xi
(Inside recursion) ?a ? I, 1 ? i < j ? N,
?a,i,j =
j?1
X
X
T a?b c (?b,i,k , ?c,k+1,j )
k=i a?b c
(Outside base case) ?a ? I, ?
=?
(Outside recursion) ?a ? N, 1 ? i ? j ? N,
a,1,n
? a,i,j =
i?1 X
X
a
b?c a
T(1,2)
(? b,k,j , ?c,k,i?1 ) +
k=1 b?c a
N
X
X
b?a c
T(1,3)
(? b,i,k , ?c,j+1,k )
k=j+1 b?a c
(Marginals) ?a ? N, 1 ? i ? j ? N,
?(a, i, j) = ?a,i,j ? a,i,j =
X
?ha,i,j ?ha,i,j
h?[m]
Figure 2: The tensor form of the inside-outside algorithm, for calculation of marginal terms ?(a, i, j).
4
Tensor Decomposition
As mentioned earlier, most computation for the inside-outside algorithm is spent on the tensor calculation of T a?b c on the intermediate inside/outside quantities. These computations, appearing as
b?c a
b?a c
T a?b c (?b,i,k , ?c,k+1,j ), T(1,2)
(? b,k,j , ?c,k,i?1 ) and T(1,3)
(? b,i,k , ?c,j+1,k ) output a vector of
length m, where computation of each element in the vector is O(m2 ). Therefore, the inside-outside
has a multiplicative O(m3 ) factor in its computational complexity, which we aim to reduce.
For the rest of this section, fix a binary grammar rule a ? b c and consider the tensor T , T a?b c
associated with it. Consider a pair of two vectors y 1 , y 2 ? Rm , associated with the distributions
over latent-states for the left (y 1 ) and right child (y 2 ) of a given node in a parse tree. Our method
for improving the speed of this tensor computation relies on a simple observation. Given an integer
r ? 1,
that the tensor T had the following special form, which is also called ?Kruskal form?,
Passume
r
T = i=1 ui vi> wi> , i.e. it would be the sum of r tensors, each is the tensor product of three vectors.
In that case, the cost of computing T (y 1 , y 2 ) could be greatly reduced by computing:
"
1
2
T (y , y ) =
r
X
i=1
#
ui vi> wi>
(y 1 , y 2 ) =
r
X
ui (vi> y 1 )(wi> y 2 ) = U > (V y 1 W y 2 )
(1)
i=1
where U, V, W ? Rr?m with the ith row being ui , vi and wi respectively.
The total complexity of this computation is O(mr). We see later that our approach can be used
effectively for r as small as 2, turning the inside-outside algorithm for latent-variable PCFGs into a
linear algorithm in the number of hidden states.
4
We note that it is well-known that an exact tensor decomposition can be achieved by using r = m2
[11]. In that case, there is no computational gain. The minimal r required for an exact solution can
be smaller than m2 , but identifying that minimal r is NP-hard [9].
We focused on this section on the computation T a?b c (?b,i,k , ?c,k+1,j ), but the steps above can be
b?c a
b?a c
generalized easily for the cases of computing T(1,2)
(? b,k,j , ?c,k,i?1 ) and T(1,3)
(? b,i,k , ?c,j+1,k ).
4.1
CP Decomposition of Tensors
In the general case, for a fixed r, our latent-variable PCFG tensors will not have the exact decomposed form from the previous section. Still, by using decomposition algorithms from multilinear
algebra, we can approximate the latent-variable tensors, where the quality of approximation is measured according to some norm over the set of tensors Rm?m?m .
An example of such a decomposition is the canonical polyadic decomposition (CPD), also known
as CANDECOMP/PARAFAC decomposition [3, 8, 10]. Given an integer r, least squares CPD
aims to find the nearest tensor in Kruskal form according to the analogous norm (for tensors) to the
Frobenius norm (for matrices).
qP
2
More formally, for a given tensor D ? Rm?m?m , let ||D||F =
i,j,k Di,j,k . Let the set of
tensors in Kruskal form Cr be:
Cr = {C ? Rm?m?m | C =
r
X
ui vi> wi> s.t. ui , vi , wi ? Rm }.
i=1
? F.
||C ? C||
The least squares CPD of C is a tensor C? such that C? ? arg minC?C
?
r
There are various algorithms to perform CPD, such as alternating least squares, direct linear decomposition, alternating trilinear decomposition and pseudo alternating least squares [6]. Most of these
implementations treat the problem of identifying the approximate tensor as an optimization problem. These algorithms are not exact. Any of these implementations can be used in our approach.
We note that the decomposition optimization problem is hard, and often has multiple local maxima.
Therefore, the algorithms mentioned above are inexact.
In our experiments, we use the alternating least squares algorithm. This method works by iteratively
improving U , V and W from Eq. 1 (until convergence), each time solving a least squares problem.
4.2
Propagation of Errors
We next present a theoretical guarantee about the quality of the CP-approximated tensor formulation
of the inside-outside algorithm. We measure the propagation of errors in probability calculations
through a given parse tree. We derive a similar result for the marginals.
We denote by p? the distribution induced over trees (skeletal and full), where we approximate each
T a?b c using the tensor T?a?b c . Similarly, we denote by ?
?(a, i, j) the approximated marginals.
Lemma 4.1. Let C ? Rm?m?m and let y 1 , y 2 , y?1 , y?2 ? Rm . Then the following inequalities hold:
||C(y 1 , y 2 )||2 ? ||C||F ||y 1 ||2 ||y 2 ||2
1
2
1
2
(2)
1
2
1
1
2
2
||C(y , y ) ? C(?
y , y? )||2 ? ||C||F max{||y ||2 , ||?
y ||2 }(||y ? y? ||2 + ||y ? y? ||2 )
Proof. Eq. 2 is the result of applying Cauchy-Schwarz inequality twice:
?2
?
||C(y 1 , y 2 )||22 =
X
X
?
i
=
||C||2F
Ci,j,k yj1 yk2 ? ?
X
i
j,k
?
||y 1 ||22
?
||y 2 ||22
5
?
??
?
!
X
X
X
2
1
2
2
2
?
Ci,j,k ? ? (yj ) ?
(yk )
j,k
j
k
(3)
For Eq. 3, note that C(y 1 , y 2 ) ? C(?
y 1 , y?2 ) = C(y 1 , y 2 ) ? C(y 1 , y?2 ) + C(y 1 , y?2 ) ? C(?
y 1 , y?2 ), and
therefore from the triangle inequality and bi-linearity of C:
||C(y 1 , y 2 ) ? C(?
y 1 , y?2 )||2 ? ||C(y 1 , y 2 ? y?2 )||2 + ||C(y 1 ? y?1 , y?2 )||2
? ||C||F ||y 1 ||2 ||y 2 ? y?2 ||2 + ||y 1 ? y?1 ||2 ||?
y 2 ||2
? ||C||F max{||y 1 ||2 , ||?
y 2 ||2 }(||y 1 ? y?1 ||2 + ||y 2 ? y?2 ||2 )
Equipped with this Cauchy-Schwarz style lemma, we can prove the following theorem:
log( ?1 ) + 1
?
?
Theorem 4.2. Let d? =
where ? is the the ?tensor approximalog(2( m + 1)) + log(? + m)
tion error? defined as ? = maxa?b c ||T a?b c ? T?a?b c ||F , then:
? For a given skeletal tree r1 , . . . , rN , if the depth of the tree, denoted d, is such that
(
d ? min
log( ?1 ) ? log( m
)
?
? , d?
log(2( m + 1)) + log(? + m)
)
then |p(r1 , . . . , rN ) ? p?(r1 , . . . , rN )| ?
? For a given sentence x1 , . . . , xM , it holds that for all triplets (a, i, j), if
(
M ? min
log( ?1 ) ? log( m
)
?
? , d?
2 log(4|N|) + log(2( m + 1)) + log(? + m)
)
then |?(a, i, j)??
?(a, i, j)| ?
Proof. For the first part, the proof is using structural induction on the structure of the test tree.
Assume a fixed skeletal tree r1 , . . . , rN . The probability p(r1 , . . . , rN ) can be computed by using a
sequence of applications of T a?b c on distribution over latent states for left and right children. More
specifically, it can be shown that the vector of probabilities defined as [y i ]h = p(ti | ai , hi = h)
(ranging over [m]), where ti is the skeletal subtree rooted at node i can be defined recursively as:
? y i = Qa?xi if i is a leaf node with word xi and,
? y i = T a?b c (y j , y k ) if i is a non-leaf node with node j being the left child and node k being the
right child of node i.
Define the same quantities y?i , only using the approximate tensors T?a?b c . Let ?i = ||y i ? y?i ||. We
will prove inductively that if di is the depth of the subtree at node i, then:
(
?i ? min ?m
! )
?
?
d
(2( m + 1)(? + m)) i ? 1
?
?
,1
2( m + 1)(? + m) ? 1
For any leaf node (base case): ||y i ? y?i ||2 = 0. For a given non-leaf node i:
?i =||y i ? y?i ||2 = ||T a?b c (y j , y k ) ? T?a?b c (?
y j , y?k )||2
?||T a?b c (y j , y k ) ? T?a?b c (y j , y k )||2 + ||T?a?b c (y j , y k ) ? T?a?b c (?
y j , y?k )||2
?||T
a?b c
(4)
? T?a?b c ||F ||y j ||2 ||y k ||2
+ ||T?a?b c || max{||y j ||2 , ||?
y k ||2 }(||y j ? y?j ||2 + ||y k ? y?k ||2 )
?
?
??m + ( m + 1)(? + m)(?j + ?k )
!
?
?
?
? (2( m + 1)(? + m))di ?1 ? 1
?
?
??m 1 + 2( m + 1)(? + m)
2( m + 1)(? + m) ? 1
!
?
?
d
(2( m + 1)(? + m)) i ? 1
?
?
=?m
2( m + 1)(? + m) ? 1
6
(5)
(6)
?15
?10
?5
8
=
16
m
20
=
m
=
?20
?15
86.0
25
85.5
20
?
?
?10
?
?
?5
?
?
log threshold
threshold
seconds per sentence
F1
seconds per sentence
F1
seconds per sentence
F1
no approx.
4.6
85.72
14.6
85.59
23.7
85.20
84.5
?
?
?
?20
?15
F1
85.0
15
F1
85.0
?
?
?
log threshold
m
84.5
?
?
?
?
10
85.5
12
10
8
6
?
?
?
?
?
?10
?
?
?5
84.0
?20
?
?
?
?
5
?
?
?
?
?
?
84.0
3.5
?
?
?
?
?
?
4
?
?
84.5
?
seconds per sentence
85.5
4.0
?
?
F1
85.0
?
?
84.0
4.5
?
?
seconds per sentence
14
5.0
?
seconds per sentence
m = 20
86.0
m = 16
86.0
m=8
log threshold
10?8
4.2
85.72
9.8
85.58
15.6
85.21
10?5
3.4
85.60
3.5
85.46
3.6
85.15
0.001
3.5
85.61
3.2
85.49
3.2
85.14
Figure 3: Speed and performance of parsing with tensor decomposition for m ? {8, 16, 20} (left
plots, middle plots and right plots respectively). The left y axis is running time (red circles), the
right y axis is F1 performance of the parser (blue squares), the x axis corresponds to log t. Solid
lines describe decomposition with r = 2, dashed lines describe decomposition with r = 8. In
addition, we include the numerical results for various m for r = 8.
where Eq. 4 is the result of the triangle inequality, Eq. 5 comes from Lemma 4.1 and the fact that
?
?
||T?a?b c ||F ? ||T?a?b c ?T a?b c ||F +||T a?b c ||F ? ? + m and ||?
y k ||2 ? ?k +||y k ||2 ? 1+ m
for any node k (under ind. hyp.), and Eq. 6 is the result of applying the induction hypothesis. It can
also be verified that since di ? d ? d? we have ?i ? 1.
?
?
d
Since m ? 1, it holds that ?i ? ?m (2( m + 1)(? + m)) i . Consider |p(r1 , . . . , rN ) ?
1
a1
a1 1
p?(r1 , . . . , rN )| = |? (y ? y? )| ? ||? ||2 ?1 ? ?1 where a1 is the non-terminal at the root of
log( ?1 ) ? log( m
)
?
? , then ?1 ? , as needed.
the tree. It is easy to verify that if d1 ?
log(2( m + 1)) + log(? + m)
P
For the marginals, consider that: |?(a, i, j) ? ?
?(a, i, j)| ? ? ?T(x) |p(? ) ? p?(? )|.
We have d1 ? M . In addition, if
M?
m
log( ?1 ) ? log( /|T(x)|
)
log( ?1 ) ? 2M log(4|N|) ? log( m
)
?
? then d1 ?
?
?
log(2( m + 1)) + log(? + m)
log(2( m + 1)) + log(? + m)
(7)
because the number of labeled binary trees for a sentence of length M is at most (4|N|)2M (and
therefore |T(x)| ? (4|N|)2M ; 4l is a bound on the Catalan number, the number of binary trees over
l nodes), then |?(a, i, j) ? ?
?(a, i, j)| ? .
verified that the left hand-side of Eq. 7 is satisfied if M
?
log( ?1 ) ? log( m
)
?
? .
2 log(4|N|) + log(2( m + 1)) + log(? + m)
As expected, the longer a sentence is, or the deeper a parse tree is, the better we need the tensor
approximation to be (smaller ?) for the inside-outside to be more accurate.
It
can
be
5
Experiments
We devote this section to empirical evaluation of our approach. Our goal is to evaluate the trade-off
between the accuracy of the tensor decomposition and the speed-up in the parsing algorithm.
7
Experimental Setup We use sections 2?21 of the Penn treebank [13] to train a latent-variable
parsing model using the expectation-maximization algorithm (EM was run for 15 iterations) for
various number of latent states (m ? {8, 16, 20}), and then parse in various settings section 22 of
the same treebank (sentences of length ? 40). Whenever we report parsing accuracy, we use the
traditional F1 measure from the Parseval metric [2]. It computes the F1 measure of spans (a, i, j)
appearing in the gold standard and the hypothesized trees.
The total number of tensors extracted from the training data using EM was 7,236 (corresponding
to the number of grammar rules). Let ?a?b c = ||T a?b c ? T?a?b c ||F . In our experiments, we
vary a threshold t ? {0.1, 0.001, 10?5 , 10?6 , 10?8 , 0} ? an approximate tensor T?a?b c is used
instead of T a?b c only if ?a?b c ? t. The value t = 0 implies using vanilla inference, without any
approximate tensors. We describe experiments with r ? {2, 8}. For the tensor approximation, we
use the implementation provided in the Matlab tensor toolbox from [1]. The toolbox implements the
alternating least squares method.
As is common, we use a pruning technique to make the parser faster ? items in the dynamic programming chart are pruned if their value according to a base vanilla maximum likelihood model is
less than 0.00005 [4]. We report running times considering this pruning as part of the execution.
The parser was run on a single Intel Xeon 2.67GHz CPU.
We note that the performance of the parser improves as we add more latent states. The performance
of the parser with vanilla PCFG (m = 1) is 70.26 F1 measure.
Experimental Results Table 3 describes F1 performance and running time as we vary t. It is
interesting to note that the speed-up, for the same threshold t, seems to be larger when using r = 8
instead of r = 2. At first this may sound counter-intuitive. The reason for this happening is that
with r = 8, more of the tensors have an approximation error which is smaller than t, and therefore
more approximate tensors are used than in the case of r = 2.
Using t = 0.1, the speed-up is significant over non-approximate version of the parser. More specifically, for r = 8, it takes 72% of the time (without considering the pruning phase) of the nonapproximate parser to parse section 22 with m = 8, 24% of the time with m = 16 and 21% of the
time with m = 20. The larger m is, the more significant the speed-up is.
The loss in performance because of the approximation, on the other hand, is negligible. More
specifically, for r = 8, performance is decreased by 0.12% for m = 8, 0.11% for m = 16 and
0.08% for m = 20.
6
Conclusion
We described an approach to significantly improve the speed of inference with latent-variable
PCFGs. The approach approximates tensors which are used in the inside-outside algorithm. The
approximation comes with a minimal cost to the performance of the parser. Our algorithm can be
used in tandem with estimation algorithms such as EM or spectral algorithms [5]. We note that
tensor formulations are used with graphical models [15], for which our technique is also applicable.
Similarly, our technique can be applied to other dynamic programming algorithms which compute
marginals of a given statistical model.
References
[1] B. W. Bader and T. G. Kolda. Algorithm 862: MATLAB tensor classes for fast algorithm
prototyping. ACM Transactions on Mathematical Software, 32(4):635?653, 2006.
[2] E. Black, S. Abney, D. Flickenger, C. Gdaniec, R. Grishman, P Harrison, D. Hindle, R. Ingria,
F. Jelinek, J. Klavans, M. Liberman, M. Marcus, S. Roukos, B. Santorini, and T. Strzalkowski.
A procedure for quantitatively comparing the syntactic coverage of English grammars. In Proc.
of DARPA Workshop on Speech and Natural Language, 1991.
[3] J. D. Carroll and J. J. Chang. Analysis of individual differences in multidimensional scaling via
an N-way generalization of Eckart-Young decomposition. Psychometrika, 35:283?319, 1970.
8
[4] E. Charniak and M. Johnson. Coarse-to-fine n-best parsing and maxent discriminative reranking. In Proceedings of ACL, 2005.
[5] S. B. Cohen, K. Stratos, M. Collins, D. F. Foster, and L. Ungar. Spectral learning of latentvariable PCFGs. In Proceedings of ACL, 2012.
[6] N. M. Faber, R. Bro, and P. Hopke. Recent developments in CANDECOMP/PARAFAC algorithms: a critical review. Chemometrics and Intelligent Laboratory Systems, 65(1):119?137,
2003.
[7] J. Goodman. Parsing algorithms and metrics. In Proceedings of ACL, 1996.
[8] R. A. Harshman. Foundations of the PARAFAC procedure: Models and conditions for an
?explanatory? multi-modal factor analysis. UCLA working papers in phoentics, 16:1?84, 1970.
[9] J. H?astad. Tensor rank is NP-complete. Algorithms, 11:644?654, 1990.
[10] T. G. Kolda and B. W. Bader. Tensor decompositions and applications. SIAM Rev., 51:455?500,
2009.
[11] J. B. Kruskal. Rank, decomposition, and uniqueness for 3-way and N-way arrays. In R. Coppi
and S. Bolasco, editors, Multiway Data Analysis, pages 7?18, 1989.
[12] C.D. Manning and H. Sch?utze. Foundations of Statistical Natural Language Processing. MIT
Press, 1999.
[13] M. P. Marcus, B. Santorini, and M. A. Marcinkiewicz. Building a large annotated corpus of
English: The Penn treebank. Computational Linguistics, 19:313?330, 1993.
[14] T. Matsuzaki, Y. Miyao, and J. Tsujii. Probabilistic CFG with latent annotations. In Proceedings of ACL, 2005.
[15] A. Parikh, L. Song, and E. P. Xing. A spectral algorithm for latent tree graphical models. In
Proceedings of The 28th International Conference on Machine Learningy (ICML 2011), 2011.
[16] S. Petrov, L. Barrett, R. Thibaux, and D. Klein. Learning accurate, compact, and interpretable
tree annotation. In Proceedings of COLING-ACL, 2006.
9
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4,003 | 4,623 | Locating Changes in Highly Dependent Data
with Unknown Number of Change Points
Daniil Ryabko
SequeL-INRIA/LIFL-CNRS,
[email protected]
Azadeh Khaleghi
SequeL-INRIA/LIFL-CNRS,
Universit?e de Lille, France
[email protected]
Abstract
The problem of multiple change point estimation is considered for sequences with
unknown number of change points. A consistency framework is suggested that
is suitable for highly dependent time-series, and an asymptotically consistent algorithm is proposed. In order for the consistency to be established the only assumption required is that the data is generated by stationary ergodic time-series
distributions. No modeling, independence or parametric assumptions are made;
the data are allowed to be dependent and the dependence can be of arbitrary form.
The theoretical results are complemented with experimental evaluations.
1
Introduction
We are given a sequence x := X1 , X2 , . . . , Xn formed as the concatenation of an unknown number k + 1 of sequences, such that x = X1 . . . X?1 X?1 +1 . . . X?2 . . . X?k . . . Xn . The time-series
distributions that generate a pair of adjacent sequences separated by indices ?i , i = 1..k are different. (Non-adjacent sequences are allowed to be generated by the same distribution). The so-called
change points ?i , i = 1..k are unknown and to be estimated. Change point estimation is one of the
core problems in statistics, and as such, has been studied extensively under various formulations.
However, even nonparametric formulations of the problem typically assume that the data in each
segment are independent and identically distributed, and that the change necessarily affects singledimensional marginal distributions. In this paper we consider the most general nonparametric setting
where, the changes may be completely arbitrary (e.g., in the form of the long-range dependence).
We propose a change point estimation algorithm that is asymptotically consistent under such minimal assumptions.
Motivation. Change point analysis is an indispensable tool in a broad range of applications such
as market analysis, bioinformatics, network traffic, audio/video segmentation only to name a few.
Clearly, in these applications the data can be highly dependent and can not be easily modeled by
parametric families of distributions. From a machine learning perspective, change point estimation
is a difficult unsupervised learning problem: the objective is to estimate the change points in a given
sequence while no labeled examples are available. To better understand the challenging nature of
the problem, it is useful to compare it to time-series clustering. In time-series clustering, a set of
sequences is to be partitioned, whereas in change point estimation the partitioning is done on a sequence of sequences. While objectives are the same, in the latter, information about the individual
elements is no longer available, since only a single sequence formed by their concatenation is provided as input. This makes change point estimation a more challenging problem than time-series
clustering.
In the general setting of highly-dependent time-series correct estimation of the number of change
points is provably impossible, even in the weakest asymptotic sense, and even if there is at most
one change [23]. While a popular mitigation is to consider more restrictive settings, we are interested in intermediate formulations that can have asymptotically consistent solutions under the most
1
general assumptions. In light of the similarities between clustering and change point analysis, we
propose a formulation that is motivated by hierarchical clustering. When the number of clusters is
unknown, a hierarchical clustering algorithm produces a tree, such that some pruning of this tree
gives the ground-truth clustering (e.g., [3]). In change point estimation with an unknown number
k of change points, we suggest to aim for a sorted list of change points, whose first k elements are
some permutation of the true change points. An algorithm that achieves this goal is called consistent.
Related Work. Change point analysis is a classical problem in mathematical statistics [6, 4, 5,
17]. In a typical formulation, samples within each segment are assumed to be i.i.d. and the change
usually refers to the change in the mean. More general formulations are often considered as well,
however, it is usually assumed that the samples are i.i.d. in each of the segments [20, 8, 9, 21] or
that they belong to some specific model class (such as Hidden Markov processes) [15, 16, 27]. In
these frameworks the problem of estimating the number of change points is usually addressed with
penalized criteria, see, for example, [19, 18]. In nonparametric settings, the typical assumptions
usually impose restrictions on the form of the change or the nature of dependence (e.g., the timeseries are assumed strongly mixing) [6, 4, 10, 12]. Even when more general settings are considered,
it is almost exclusively assumed that the single-dimensional marginal distributions are different [7].
The framework considered in this paper is similar to that of [25] and of our recent paper [13], in
the sense that the only assumption made is that the distributions generating the data are stationary
ergodic. The particular case of k = 1 is considered in [25]. In [13] we provide a non-trivial extension
of [25] for the case where k > 1 is known and is provided to the algorithm. However, as mentioned
above, when the number k of change points is unknown, it is provably impossible to estimate it,
even under the assumption k ? {0, 1} [23]. In particular, if the input k is not the correct number of
change points, then the behavior of the algorithm proposed in [13] can be arbitrary bad.
Results. We present a nonparametric change point estimation algorithm for time-series data with
unknown number of change points. We consider the most general framework where the only assumption made is that the unknown distributions generating the data are stationary ergodic. This
means that we make no such assumptions as independence, finite memory or mixing. Moreover, we
do not need the finite-dimensional marginals of any fixed size before and after the change points to
be different. Also, the marginal distributions are not required to have densities.
We show that the proposed algorithm is asymptotically consistent in the sense that among the change
point estimates that it outputs, the first k converge to the true change points. Moreover, our algorithm
can be efficiently calculated; it has a computational complexity O(n2 polylog n) where n is the
length of the input sequence. To the best of our knowledge, this work is the first to address the
change point problem with an unknown number of change points in such general framework.
We further confirm our theoretical findings through experiments on synthetic data. Our experimental
setup is designed so as to demonstrate the generality of the suggested framework. To this end, we
generate our data by time-series distributions that, while being stationary ergodic, do not belong to
any ?simpler? class of processes. In particular they cannot be modeled as hidden Markov processes
with finite or countably infinite set of states. Through our experiments we show that the algorithm
is consistent in the sense that as the length of the input sequence grows, the produced change point
estimations converge to the actual change points.
Organization. In Section 2 we introduce some preliminary notation and definitions. We formulate
the problem in Section 3. Section 4 presents our main theoretical results, including the proposed
algorithm, and an informal description of how and why it works. In Section 5 we prove that the
proposed algorithm is asymptotically consistent under the general framework considered; we also
show that our algorithm can be computed efficiently. In Section 6 we present some experimental
results, and finally in Section 7 we provide some concluding remarks and future directions.
2
Notation and definitions
Let X be some measurable space (the domain); in this work we let X = R, but extensions to
more general spaces are straightforward. For a sequence X1 , . . . , Xn we use the abbreviation X1..n .
Consider the Borel ?-algebra B on X ? generated by the cylinders {B ?X ? : B ? B m,l , m, l ? N}
where, the sets B m,l , m, l ? N are obtained via the partitioning of X m into cubes of dimension m
and volume 2?ml (starting at the origin). Let also B m := ?l?N B m,l . Processes are probability
2
measures on the space (X ? , B). For x = X1..n ? X n and B ? B m let ?(x, B) denote the
frequency with which x falls in B, i.e.
?(x, B) :=
n?m+1
I{n ? m} X
I{Xi..i+m?1 ? B}
n ? m + 1 i=1
(1)
A process ? is stationary if for any i, j ? 1..n and B ? B m , m ? N, we have ?(X1..j ? B) =
?(Xi..i+j?1 ? B). A stationary process ? is called (stationary) ergodic if for all B ? B we have
limn?? ?(X1..n , B) = ?(B) with ?-probability 1. The distributional distance between a pair of
process distributions ?1 and ?2 is defined as follows
d(?1 , ?2 ) :=
?
X
X
wm wl
|?1 (B) ? ?2 (B)|
B?B m,l
m,l=1
where, wi := 2?i , i ? N. Note that any summable sequence of positive scores also works. It is
easy to see that d(?, ?) is a metric. For more on the distributional distance and its properties see [11].
In this work we use empirical estimates of this distance. Specifically, the empirical estimate of the
distance between a sequence x = X1..n ? X n , n ? N and a process distribution ? is defined as
?
X
? ?) :=
d(x,
X
wm wl
|?(x, B) ? ?(B)|
(2)
B?B m,l
m,l=1
and for a pair of sequences xi ? X ni ni ? N, i = 1, 2. it is defined as
? 1 , x2 ) :=
d(x
?
X
X
wm wl
|?(x1 , B) ? ?(x2 , B)|.
(3)
B?B m,l
m,l=1
Although expressions (2) and (3) involve infinite sums they can be easily calculated [22]. Moreover,
? ?) are asymptotically consistent [25]: for any pair of stationary ergodic distributions
the estimates d(?,
?1 , ?2 generating sequences xi ? X ni i = 1, 2 we have
lim
n1 ,n2 ??
? 1 , x2 ) = d(?1 , ?2 ), a.s., and
d(x
? i , ?j ) = d(?i , ?j ), i, j ? 1, 2, a.s.
lim d(x
ni ??
(4)
(5)
Moreover, a more general estimate of (.?, ?) may be obtained as
? 1 , x2 ) :=
d(x
mn X
ln
X
X
wm wl
m=1 l=1
|?(x1 , B) ? ?(x2 , B)|
(6)
B?B m,l
where, mn and ln are any sequences of integers that go to infinity with n. As shown in [22] the
? i.e. (2) and (3) equally hold for d? so long as mn , ln go to infinity with n.
consistency results for d,
Let x = X1..n be a sequence and consider a subsequence Xa..b of x with a < b ? 1..n. We define
the intra-subsequence distance of Xa..b as
?
?x (a, b) := d(X
a..b a+b c , Xd a+b e..b )
2
2
(7)
We further define the single-change point estimator of Xa..b , a < b as
? a?n?..t , Xt..b+n? ), ? ? (0, 1)
?x (a, b, ?) := argmax d(X
t?[a,b]
3
Problem Formulation
We formalize the multiple change point estimation problem as follows. We are given a sequence
x := X1 , . . . , Xn ? X n
which is the concatenation of an unknown number ? + 1 of sequences
X1..?1 , X?1 +1..?2 , . . . , X?? +1..n .
3
(8)
Each of these sequences is generated by an unknown stationary ergodic process distribution. Moreover, every two consecutive sequences are generated by two different process distributions. (A pair
of non-consecutive sequences may be generated by the same distribution.) The process distributions
are not required to be independent. The parameters ?k are unknown and have to be estimated; they
are called change points. Note that it is not required for the means, variances or single-dimensional
marginals of the distributions to be different. We are considering the most general scenario where
the process distributions are different.
Definition 1 (change point estimator). A change point estimator is a function that takes a sequence x
? := ?
and a parameter ? ? (0, 1) and outputs a sequence of change point estimates, ?
?1 , ?
?2 , . . . ?
?1/? .
(Note that the total number of estimated change points 1/? may be larger than the true number of
change points ?.)
To construct consistent algorithms, we assume that the change points ?k are linear in n i.e. ?k :=
n?k where ?k ? (0, 1) k = 1..? are unknown. We also define the minimum normalized distance
between the change points as
?min := min ?k ? ?k?1
(9)
k=1..?+1
where ?0 := 0 and ??+1 := 1, and assume ?min > 0. The reason why we impose these conditions is
that the consistency properties we are after are asymptotic in n. If the length of one of the sequences
is constant or sublinear in n then asymptotic consistency is impossible in this setting. We define the
consistency of a change point estimator as follows.
? := ?
Definition 2 (Consistency of a change point estimator). Let ?
?1 , ?
?2 , . . . ?
?1/? be a change point
1
1
?
?
?
estimator. Let ?(?) = (?1 , . . . , ?? ) := sort( n ?
?1 , . . . , n ?
?? ), where sort(?) orders the first ? elements
? with respect to their order of appearance in x. We call the change point estimator
?
?1 , . . . , ?
?? of ?
? asymptotically consistent if with probability 1 we have
?
lim sup |??k ? ?k | = 0.
n?? k=1..?
4
Theoretical Results
In this section we introduce a nonparametric multiple change point estimation algorithm for the
case where the number of change points is unknown. We also give an informal description of the
algorithm, and intuitively explain why it works. The main result is Theorem 1 which states that the
proposed algorithm is consistent under the most general assumptions. Moreover, the computational
complexity of the algorithm is O(n2 polylog n) where n denotes the length of the input sequence.
The main steps of the algorithm are as follows. Given ? ? (0, 1), a sequence of evenly-spaced
indices is formed. The index-sequence is used to partition x = X1..n into consecutive segments
of length n?, where ? := ?3 . The single-change point estimator ?(?, ?, ?) is used to generate a
candidate change point within every segment. Moreover, the intra-subsequence-distance ?(?, ?) of
each segment is used as its performance score s(?, ?). The change point candidates are ordered
according to the performance-scores of their corresponding segments. The algorithm assumes the
input parameter ? to be a lower-bound on the true normalized minimum distance ?min between
actual change points. Hence, the sorted list of estimated change points is filtered in such a way that
? of change point
its elements are at least ?n apart. The algorithm outputs an ordered sequence ?
estimates, where the ordering is done with respect to the performance scores s(?, ?). The length of
? may be larger than ?. However, as we show in Theorem 1, from some n on, the first ? elements
?
? converge to some permutation of the true change points, ?1 , ? ? ? , ?? .
?
?k , k = 1..? of the output ?
Theorem 1. Let x := X1..n ? X n , n ? N be a sequence with change points at least n?min apart,
for some ?min ? (0, 1). Then Alg1(x, ?) is asymptotically consistent for ? ? (0, ?min ].
? ?) involves taking infiRemark 2 (Computational complexity). While the definition (3) of d(?,
nite sums, the distance can be calculated efficiently. Indeed, in (3) all summands corresponding to m > maxi=1,2 ni equal 0; moreover, all summands corresponding to l > smin are
equal, where smin corresponds to the partition in which each cell has at most one point in it
smin := mini,j?1..n, Xi 6=Xj |Xi ? Xj |. Thus, even with a most naive implementation the computational complexity of the algorithm is at most polynomial in all arguments. A more efficient
? ?) given by (6), instead of d(?,
? ?), with m = log n,
implementation can be obtained if one uses d(?,
4
Algorithm 1 Estimating the change points
input: Sequence x = X1..n , Minimum Normalized Distance between the change points ?
? ? ()
initialize: Step size ? ? ?3 , Output change point Sequence ?
1. Generate 2 sets of index-sequences:
bti ? n?(i +
1
1
), i = 0.. , t = 1, 2
t+1
?
2. Calculate the intra-distance value (given by (7)) of every segment Xbti ..bti+1 , i = 1.. ?1 , t = 1, 2 as its
1
performance score:
t t
s(t, i) ? ?x (bi , bi+1 ), i = 1.. , t = 1, 2
?
3. Use the single-change point-estimator (given by (8)) to estimate a change point in every segment:
p?(t, i) := ?x (bti , bti+1 , ?), i = 1..
4. Remove duplicates and sort based on scores:
U ? {(t, i) : i ? 1..
1
? 1, t = 1, 2
?
1
? 1, t = 1, 2}
?
while U 6= ? do
?
i. Select an available change point estimate of highest score and add it to ?:
(?, l) ? argmax(t,i)?U s(t, i) - break the ties arbitrarily
? ??
? ? p?(?, l), i.e. append ?
? with p?(?, l)
?
ii. Remove the estimates within a radius of ?n/2 from ?
? (l):
U ? U \ {(t, i) : p?(t, i) ? (?
p(?, l) ? ?n/2, p?(?, l) + ?n/2)}
end while
? of change point estimates. Note: Elements of ?
? are at least n? apart and are
return: A sequence ?
sorted in decreasing order of their scores s(?, ?).
where n is the length of the samples; in this case, the consistency results are unaffected, and the
computational complexity of calculating the distance becomes n polylog n, making the complexity
of the algorithm n2 polylog n. The choice m = log n is further justified by the fact that the frequencies of cells in B m,l corresponding to higher values of m are not consistent estimates of their
probabilities (and thus only add to the error of the estimate); see [22, 14] for further discussion.
The proof of the theorem is given in the next section. Here we provide an intuition as to why the
consistency statement holds.
First, recall that the empirical distributional distance between a given pair of sequences converges
to the distributional distance between the corresponding process distributions. Consider a sequence
x = X1..n , and assume that a segment Xa..b , a, b ? 1..n does not contain any change points, so
that Xa.. a+b and X a+b ..b are generated by the same process. If the length of Xa..b is linear in n, so
2
2
that b ? a = ?n for some ? ? (0, 1), then its intra-subsequence distance ?x (a, b) (defined by (7))
converges to 0 with n going to infinity. On the other hand, if there is a single change point ? within
Xa..b whose distance from a and b is linear in n, then ?x (a, b) converges to a non-zero constant.
Now assume that Xa..b with its change point at ? ? a..b is contained within a larger segment
Xa?n?0 ..b+n?0 for some ?0 ? (0, 1). In this case, the single-change point estimator ?(a, b, ?0 )
(defined by (8)) produces an estimate that from some n on converges to ? provided that ? is the only
change point in Xa?n?0 ..b+n?0 . These observations are key to the consistency of the algorithm.
When ? ? ?min , each of the index-sequences generated with ? := ?3 partitions x in such a way
that every three consecutive segments of the partition contain at most one change point. Also, the
segments are of lengths linear in n. In this scenario, from some n on, the change point estimator
?(?, ?, ?) produces correct candidates within each of the segments that contains a true change point.
Moreover, from some n on, the performance scores s(?, ?) of the segments without change points
converge to 0, while those corresponding to the segments that encompass a change point converge
5
to a non-zero constant. Thus from some n on, the ? change point candidates of highest performance
score that are at least at a distance ?n from one another, each converge to a unique change point.
A problem occurs if the generated index-sequence is such that it includes some of the change points
as elements. As a mitigation strategy, we generate two index-sequences with the same gap ?n
between their consecutive elements but with distinct starting points: one starts at n?
2 and the other
at n?
.
Each
index-sequence
gives
a
different
partitioning
of
x
into
consecutive
segments.
This way,
3
every change point is fully encompassed by at least one segment from either of the two partitions.
We choose the appropriate segments based on their performance scores. From the above argument
we can see that segments with change points will have higher scores, and the change points within
will be estimated correctly; finally, this is used to prove the theorem in the next session.
5
Proof of Theorem 1
The proof relies on Lemma 1 and Lemma 2, which we borrow from [13] and state here without proof.
We also require the following additional notation.
Definition 3. For every change point ?k , k = 1..? and every fixed t = 1, 2 we denote by Lt (?k )
and by Rt (?k ) the elements of the index-sequence bti , i = 1.. ?1 that appear immediately to the left
and to the right of ?k respectively, i.e. Lt (?k ) :=
max 1 bti and Rt (?k ) :=
min 1 bti .
bti ??k , i=0.. ?
bti ??k , i=0.. ?
(Equality occurs when ?k for some k ? 1..? is exactly at the start or at the end of a segment.)
Lemma 1 ([13]). Let x = X1..n be generated by a stationary ergodic process ?. For all ? ? [0, 1)
and ? ? (0, 1) we have, lim
sup
?x (b1 , b2 ) = 0.
n?? b1 ??n, b2 ?b1 +?n
Lemma 2 ([13]). Let ? denote the minimum distance between the distinct distributions generating
the data. Denote by ? the ?unknown? number of change points and assume that for some ? ? (0, 1)
and some t = 1, 2 we have, inf k=1..? |bti ? ?k | ? ?n.
1
i=0.. ?
inf ?x (Lt (?k ), Rt (?k )) ? ??.
n?? k?1..?
[Lt (?k ) ? n?, Rt (?k ) + n?] ? [?k?1 , ?k+1 ]
(i) With probability one we have, lim
(ii) If additionally we have that
1
one we obtain, lim sup |?x (Lt (?k ), Rt (?k ), ?) ? ?k | = 0.
n?? k?1..? n
then with probability
Proof of Theorem 1. We first give an outline of the proof. In order for a change point ?k , k ? 1..?
to be estimated correctly through this algorithm, there needs to be at least one t = 1, 2 such that
1. ?k ? (Lt (?k ), Rt (?k )) and 2. [Lt (?k ) ? n?, Rt (?k ) + n?] ? [?k?1 , ?k ]
where ? := ?3 , as specified by the algorithm. We show that from some n on, for every change point
the algorithm selects an appropriate segment satisfying these conditions, and assigns it a performance score s(?, ?) that converges to a non-zero constant. Moreover, the performance scores of the
segments without change points converge to 0. Recall that, the change point candidates are finally
sorted according to their performance scores, and the sorted list is filtered to include only elements
that are at least ?n apart. For ? ? ?min , from some n on, the first ? elements of the output change
? are some permutation of the true change points. The proof follows.
point sequence ?
Fix an ? > 0. Recall that the algorithm specifies ? := ?3 and generates a sequence of evenly-spaced
1
indicies bti := n?(i + t+1
), i = 1.. ?1 , t = 1, 2. Observe that
1
bti ? bti?1 = n?, i = 1.. .
?
(10)
For every i ? 0.. ?1 and t ? 1, 2 we have that the index bti is either exactly equal to a change point or
1
has a linear distance from it. More formally, define ?(t, i) := min |?(i+ t+1
)??k |, i ? 0..1/? t ?
k?1..?
1..2. (Note that ?(t, i) can also be zero). For all i ? 0.. ?1 , t = 1, 2 and k ? 1..? we have
|bti ? ?k | ? n?(t, i).
6
(11)
For every t = 1, 2 and i = 0..1/?, a performance score s(t, i) is calculated as the intra-subsequence
distance ?x (bti , bti+1 ) of the segment Xbti ..bti+1 . Let I := {(t, i) : t ? 1, 2, i ? 1.. ?1 s.t. ?k ?
1..?, ?k ? (bti , bti+1 )}. Also define the complement set I 0 := {1, 2} ? {1.. ?1 } \ I. By (10), (11)
and Lemma 1, there exists some N1 such that for all n ? N1 we have,
sup s(t, i) ? ?.
(12)
(t,i)?I 0
Since ? ? ?min , we have ? ? (0, ?min /3]. Therefore, for every t = 1, 2 and every change point
?k , k ? 1..? we have
[Lt (?k ) ? n?, Rt (?k ) + n?] ? [?k?1 , ?k+1 ].
(13)
Define ?min := min(t,i)?I ?(t, i). It follows from the definition of I that
?min > 0.
(14)
By (10), (11), (13), (14) and Lemma 2.(i), there exists some N2 such that for all n ? N2 we have
inf s(t, i) ? ??min
(t,i)?I
(15)
where ? denotes the minimum distance between the distributions. Let ?(t, i), i ? 0..1/?, t = 1, 2
denote the change point that is contained within bti ..bti+1 , (t, i) ? I, i.e. ?(t, i) := ?k , k ?
1..? s.t. ?k ? (bti , bti+1 ). As specified in Step 3, the change point candidates are obtained as p?(t, i) :=
? (i) ? (i)
?x (bi , bi+1 , ?), i = 1..1/? ? 1. By (10), (11), (13), (14) and Lemma 2.(ii) there exists some N4
such that for all n ? N4 we have
sup
(t,i)?I
1
|?
p(t, i) ? ?(t, i)| ? ?.
n
(16)
Let N := maxi=1..4 Ni . Recall that (as specified in Step 4), the algorithm generates an output
? := ?
sequence ?
?1 , . . . , ?
?1/? by first sorting the change point candidates according to their performance scores, and then filtering the sorted list so that the remaining elements are at least n? apart.
?
It remains to see that the corresponding estimate of every change point appears exactly once in ?.
By (12) and (15) for all n ? N the segments bti ..bti+1 , (t, i) ? I are assigned higher scores than
bti ..bti+1 , (t, i) ? I 0 . Moreover, by construction for every change point ?k , k = 1..? there exists some (t, i) ? I such that ?k = ?(t, i) which, by (16) is estimated correctly for all n ? N .
? By (16) for all
Next we show that every estimate appears at most once in the output sequence ?.
(t, i), (t0 , i0 ) ? I such that ?(t, i) = ?(t0 , i0 ) and all n ? N we have
1
1
1
|?
p(t, i) ? p?(t0 , i0 )| ? |?
p(t, i) ? ?(t, i)| + |?
p(t0 , i0 ) ? ?(t0 , i0 )| ? 2?.
(17)
n
n
n
On the other hand, for all (t, i), (t0 , i0 ) ? I such that ?(t, i) 6= ?(t0 , i0 ) and all n ? N we have
1
1
1
1
|?
p(t, i) ? p?(t0 , i0 )| ? |?(t, i) ? ?(t0 , i0 )| ? |?
p(t, i) ? ?(t, i)| ? |?
p(t0 , i0 ) ? ?(t0 , i0 )|
n
n
n
n
1
? |?(t, i) ? ?(t0 , i0 )| ? 2? ? ?min ? 2?
(18)
n
where the last inequality follows from (16) and that the true change points are at least n?min apart.
By (17) and (18) the duplicate estimates of every change point are filtered, while estimates corresponding to different change points are left untouched. Finally, following the notation of Defi?
nition 2, let ?(?)
= (??1 , . . . , ??? ) := sort( n1 ?
?1 , ? ? ? , n1 ?
?? ), (sorted with respect to their order of
?
appearance in x). For n ? N we have, supk?1..? |?k ? ?k | ? ? and the statement follows.
6
Experimental Results
In this section we use synthetically generated time-series data to empirically evaluate our algorithm.
To generate the data we have selected distributions that while being stationary ergodic, do not belong to any ?simpler? class of time-series, and are difficult to approximate by finite-state models. In
particular they cannot be modeled by a hidden Markov process with a finite state-space. These distributions were used in [26] as examples of stationary ergodic processes which are not B-processes.
7
0.45
1
0.4
0.9
0.8
0.35
Error?rate
Error?rate
0.7
0.3
0.25
0.2
0.6
0.5
0.4
0.15
0.3
0.1
0.2
0.05
0
0.1
0
0.5
1
1.5
2
Length of the input sequence
0
2.5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Input parameter h
4
x 10
Figure 1: Left (Experiment 1): Average (over 20 runs) error as a function of the length of the input
sequence. Right (Experiment 2): Average (over 25 runs) error as a function the input parameter ?.
Time-series generation. To generate a sequence x = X1..n we proceed as follows. Fix some parameter ? ? (0, 1) and select r0 ? [0, 1]. For each i = 1..n let ri = ri?1 + ? ? bri?1 + ?c. The
samples Xi are obtained from ri by thresholding at 0.5, i.e. Xi := I{ri > 0.5}. We call this procedure DAS(?). If ? is irrational then x forms a stationary ergodic time-series. We simulate ? by
a longdouble with a long mantisa. For the purpose of our experiments we use four different process
distributions DAS(?i ), i = 1..4 with ?1 = 0.30..., ?2 = 0.35..., ?3 = 0.40... and ?4 = 0.45....
To generate an input sequence x = X1..n we fix some ?min = 0.23 and randomly generate ? = 3
change points at a minimum distance n?min . We use DAS(?i ), i = 1..4 to respectively generate
the four subsequences between every pair of consecutive change points.
Experiment 1: (Convergence with Sequence Length) In this experiment we demonstrate that the
estimation error converges to 0 as the sequence length grows. We iterate over n = 1000..20000; at
every iteration we generate an input sequence of length n as described above. We apply Algorithm 1
with ? = 0.18 to find the change points. Figure 1 (Left) shows the average error-rate as a function
of sequence length.
Experiment 2: (Dependence on ?) Algorithm 1 requires ? ? (0, 1) as a lower-bound on ?min .
In this experiment we show that this lower bound need not be tight. In particular, there is a rather
large range of ? ? ?min for which the estimation error is low. To demonstrate this, we fixed the
sequence length n = 20000 and observed the error-rate as we varied the input parameter ? between
0.01..0.35. Figure 1 (Right) shows the average error-rate as a function of ?.
7
Outlook
In this work we propose a consistency framework for multiple change points estimation in highly
dependent time-series, for the case where the number of change points is unknown. The notion of
consistency that we consider requires an algorithm to produce a list of change points such that the
first k change points approach the true unknown change points in asymptotic. While in the general
setting that we consider it is not possible to estimate the number of change points, other related
formulations may be of interest. For example, if the number of different time-series distributions is
known, but the number of change points is not, it may still be possible to estimate the latter. A simple
example of this scenario would be when two distributions generate many segments in alternation.
While the consistency result here (and in the previous works [14, 22, 25]) rely on the convergence of
frequencies, recent results of [1, 2] on uniform convergence can be used (see [24]) to solve related
statistical problems about time-series (e.g., clustering) and thus may also prove useful in change
point analysis.
Acknowledgements. This work is supported by the French Ministry of Higher Education and Research, NordPas-de-Calais Regional Council and FEDER through CPER 2007-2013, ANR projects EXPLO-RA (ANR-08COSI-004) and Lampada (ANR-09-EMER-007), by an INRIA Ph.D. grant to Azadeh Khaleghi, by the European Community?s Seventh Framework Programme (FP7/2007-2013) under grant agreement 231495 (project
CompLACS), and by Pascal-2.
8
References
[1] Terrence M. Adams and Andrew B. Nobel. Uniform convergence of Vapnik-Chervonenkis
classes under ergodic sampling. The Annals of Probability, 38:1345?1367, 2010.
[2] Terrence M. Adams and Andrew B. Nobel. Uniform approximation and bracketing properties
of VC classes. Bernoulli, to appear.
[3] M.F. Balcan and P. Gupta. Robust hierarchical clustering. In COLT, 2010.
[4] M. Basseville and I.V. Nikiforov. Detection of abrupt changes: theory and application. Prentice Hall information and system sciences series. Prentice Hall, 1993.
[5] P.K. Bhattacharya. Some aspects of change-point analysis. Lecture Notes-Monograph Series,
pages 28?56, 1994.
[6] B.E. Brodsky and B.S. Darkhovsky. Nonparametric methods in change-point problems. Mathematics and its applications. Kluwer Academic Publishers, 1993.
[7] E. Carlstein and S. Lele. Nonparametric change-point estimation for data from an ergodic
sequence. Teor. Veroyatnost. i Primenen., 38:910?917, 1993.
[8] L. Dumbgen. The asymptotic behavior of some nonparametric change-point estimators. The
Annals of Statistics, 19(3):pp. 1471?1495, 1991.
[9] D. Ferger. Exponential and polynomial tailbounds for change-point estimators. Journal of
statistical planning and inference, 92(1-2):73?109, 2001.
[10] L. Giraitis, R. Leipus, and D. Surgailis. The change-point problem for dependent observations.
Journal of Statistical Planning and Inference, 53(3), 1996.
[11] R. Gray. Prob. Random Processes, & Ergodic Properties. Springer Verlag, 1988.
[12] S. B. Hariz, J. J. Wylie, and Q. Zhang. Optimal rate of convergence for nonparametric changepoint estimators for nonstationary sequences. Annals of Statistics, 35(4):1802?1826, 2007.
[13] A. Khaleghi and D. Ryabko. Multiple change-point estimation in highly dependent time series.
Technical report, arXiv:1203.1515, 2012.
[14] A. Khaleghi, D. Ryabko, J. Mary, and P. Preux. Online clustering of processes. In AISTATS,
JMLR W&CP 22, pages 601?609, 2012.
[15] J. Kohlmorgen and S. Lemm. A dynamic hmm for on-line segmentation of sequential data.
Advances in Neural Inf. Proc. Systems, 14:793?800, 2001.
[16] John D. Lafferty, Andrew McCallum, and Fernando C. N. Pereira. Conditional random fields:
Probabilistic models for segmenting & labeling sequence data. In ICML, 2001.
[17] T.L. Lai. Sequential changepoint detection in quality control and dynamical systems. Journal
of the Royal Statistical Society, pages 613?658, 1995.
[18] Marc Lavielle. Using penalized contrasts for the change-point problem. Signal Processing,
85(8):1501 ? 1510, 2005.
[19] E. Lebarbier. Detecting multiple change-points in the mean of gaussian process by model
selection. Signal Processing, 85(4):717 ? 736, 2005.
[20] C.B. Lee. Nonparametric multiple change-point estimators. Statistics & probability letters,
27(4):295?304, 1996.
[21] Hidetoshi Murakami. A nonparametric locationscale statistic for detecting a change point. The
Inter. Journal of Advanced Manufacturing Technology, 2001.
[22] D. Ryabko. Clustering processes. In ICML, pages 919?926, Haifa, Israel, 2010.
[23] D. Ryabko. Discrimination between B-processes is impossible. Journal of Theoretical Probability, 23(2):565?575, 2010.
[24] D. Ryabko and J. Mary. Reducing statistical time-series problems to binary classification. In
NIPS, Lake Tahoe, USA, 2012.
[25] D. Ryabko and B. Ryabko. Nonparametric statistical inference for ergodic processes. IEEE
Transactions on Information Theory, 56(3), 2010.
[26] P. Shields. The Ergodic Theory of Discrete Sample Paths. AMS Bookstore, 1996.
[27] X. Xuan and K. Murphy. Modeling changing dependency structure in multivariate time series.
In ICML, pages 1055?1062. ACM, 2007.
9
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4,004 | 4,624 | Bayesian nonparametric models for ranked data
Yee Whye Teh
Department of Statistics
University of Oxford
Oxford, United Kingdom
[email protected]
Franc?ois Caron
INRIA
IMB - University of Bordeaux
Talence, France
[email protected]
Abstract
We develop a Bayesian nonparametric extension of the popular Plackett-Luce
choice model that can handle an infinite number of choice items. Our framework
is based on the theory of random atomic measures, with the prior specified by a
gamma process. We derive a posterior characterization and a simple and effective
Gibbs sampler for posterior simulation. We develop a time-varying extension of
our model, and apply it to the New York Times lists of weekly bestselling books.
1
Introduction
Data in the form of partial rankings, i.e. in terms of an ordered list of the top-m items, arise in many
contexts. For example, in this paper we consider datasets consisting of the top 20 bestselling books
as published each week by the New York Times. The Plackett-Luce model [1, 2] is a popular model
for modeling such partial rankings of a finite collection of M items. It has found many applications,
including choice modeling [3], sport ranking [4], and voting [5]. [6, Chap. 9] provides detailed
discussions on the statistical foundations of this model.
In the Plackett-Luce model, each item k ? [M ] = {1, . . . , M } is assigned a positive rating parameter wk , which represents the desirability or rating of a product in the case of choice modeling, or the
skill of a player in sport rankings. The Plackett-Luce model assumes the following generative story
for a top-m list ? = (?1 , . . . , ?m ) of items ?i ? [M ]: At each stage i = 1, . . . , m, an item is chosen
to be the ith item in the list from among the items that have not yet appeared, with the probability
that ?i is selected being proportional to its desirability w?i . The overall probability of a given partial
m
Y
ranking ? is then:
w?
P (?) =
(1)
i Pi?1
.
PM
k=1 wk ?
j=1 w?j
i=1
with the denominator in (1) being the sum over all items not yet selected at stage i.
In many situations the collection of available items can be very large and potentially unknown.
In this case, a nonparametric approach can be sensible, where the pool of items is assumed to
be infinite and the model allows for the possibility of items not observed in previous top-m lists
to appear in new ones. In this paper we propose such a Bayesian nonparametric Plackett-Luce
model. Our approach is built upon recent work on Bayesian inference for the (finite) Plackett-Luce
model and its extensions [7, 8, 9]. Our model assumes the existence of an infinite pool of items
?
{Xk }?
k=1 , each with its own rating parameter, {wk }k=1 . The probability of a top-m list of items,
say (X?1 , . . . , X?m ), is then a direct extension of the finite case (1):
m
Y
w?
(2)
P (X?1 , . . . , X?m ) =
i Pi?1
.
P?
k=1 wk ?
j=1 w?j
i=1
To formalize the framework, a natural representation to encapsulate the pool of items along with
?
their ratings is using an atomic measure:
X
G=
wk ?Xk
(3)
k=1
1
Using this representation, note that the top item X?1 in our list is simply a draw from the probability
measure obtained by normalizing G, while subsequent items in the top-m list are draws from probability measures obtained by first removing from G the atoms corresponding to previously picked
items and normalizing. Described this way, it is clear that the Plackett-Luce model is basically a partial size-biased permutation of the atoms in G [10], and the existing machinery of random measures
and exchangeable random partitions [11] can be brought to bear on our problem.
In particular, in Section 2 we will use a gamma process as the prior over the atomic measure G.
This is a completely random measure [12] with gamma marginals, such that the corresponding
normalized probability measure is a Dirichlet process. We will show that with the introduction of
a suitable set of auxiliary variables, we can characterize the posterior law of G given observations
of top-m lists distributed according to (2). A simple Gibbs sampler can then be derived to simulate
from the posterior distribution. In Section 3 we develop a time-varying extension of our model and
derive a simple and effective Gibbs sampler for posterior simulation. In Section 4 we apply our
time-varying Bayesian nonparametric Plackett- Luce model to the aforementioned New York Times
bestsellers datasets, and conclude in Section 5.
2
A Bayesian nonparametric model for partial ranking
We start this section by briefly describing a Bayesian approach to inference in finite Plackett-Luce
models [9], and taking the infinite limit to arrive at the nonparametric model. This will give good
intuitions for how the model operates, before we rederive the same nonparametric model more
formally using gamma processes. Throughout this paper we will suppose that our data consists
of L partial rankings, with ?` = (?`1 , . . . , ?`m ) for ` ? [L]. For notational simplicity we assume
that all the partial rankings are length m.
2.1
Finite Plackett-Luce model with gamma prior
Suppose we have M choice items, with item k ? [M ] having a positive desirability parameter wk .
A partial ranking ?` = (?`1 , . . . , ?`m ) can be constructed generatively by picking the ith item ?`i
at the ith stage for i = 1, . . . , m, with probability proportional to w?`i as in (1). An alternative
Thurstonian interpretation, which will be important in the following, is as follows: For each item k
let z`k ? Exp(wk ) be exponentially distributed with rate wk . Thinking of z`k as the arrival time of
item k in a race, let ?`i be the index of the ith item to arrive (the ith smallest value among (z`k )M
k=1 ).
The resulting probability of ?` can then be shown to still be (1). In this interpretation (z`k ) can be
understood as latent variables, and the EM algorithm can be applied to derive an algorithm to find
a ML parameter setting for (wk )M
k=1 given multiple partial rankings. Unfortunately the posterior
distribution of (z`k ) given ?` is difficult to compute directly, so we instead consider an alternative
parameterization: Let Z`i = z?`i ? z?` i?1 be the waiting time for the ith item to arrive after the
i ? 1th item (with z?`0 defined to be 0). Then it can be shown that the joint probability is:
L,m
M
P ((?` )L
`=1 , (Z`i )`=1,i=1 |(wk )k=1 ) =
L Y
m
Y
P
Pi?1
M
w?`i exp ?Z`i
k=1 wk ?
j=1 w?`j
(4)
`=1 i=1
PM
Note that the posterior of (Z`i )m
i=1 is simply factorized with Z`i |?, w ? Exp(
k=1 wk ?
Pi?1
w
),
and
the
ML
parameter
setting
can
be
easily
derived
as
well.
Taking
a
further
step,
?
`j
j=1
?
we note that a factorized gamma prior over (wk ) is conjugate to (4), say wk ? Gamma( M
, ?)
with hyperparameters ?, ? > 0. Now Bayesian inference can be carried out either with a VB EM
algorithm, or a Gibbs sampler. In this paper we shall consider only Gibbs sampling algorithms. In
this case the parameter updates are of the form
PL Pm
?
(5)
wk |(?` ), (Z`i ), (wk0 )k0 6=k ? Gamma M
+ nk , ? + `=1 i=1 ?`ik Z`i
where nk is the number of occurrences of item k among the observed partial rankings, and ?`ik = 0
if there is a j < i with ?`j = k and 1 otherwise. These terms arise by regrouping those in the
exponential in (4).
A nonparametric Plackett-Luce model can now be easily derived by taking the limit as the number
of choice items M ? ?. For those items k that have appeared among the observed partial rankings,
2
the limiting conditional distribution (5) is well defined since nk > 0. For items
P that did not appear
in the observations, (5) becomes degenerate at 0. Instead we can define w? = k:nk =0 wk to be the
total desirability among all infinitely many previously unobserved items, and show that
PL Pm
w? |(?` ), (Z`i ), (wk )k:nk >0 ? Gamma ?, ? + `=1 i=1 Z`i
(6)
The Gibbs sampler thus alternates between updating (Z`i ), and updating the ratings of the observed
items (wk )k:nk >0 and of the unobserved ones w? . This nonparametric model allows us to estimate
the probability of seeing new items appearing in future partial rankings in a consistent manner. While
intuitive, this derivation is ad hoc in the sense that it arises as the infinite limit of the Gibbs sampler
for finite models, and is unsatisfying as it did not directly capture the structure of the underlying
infinite dimensional object, which we will show in the next subsection to be a gamma process.
2.2
A Bayesian nonparametric Plackett-Luce model
Let X be a measurable space of choice items. A gamma process is a completely random measure
over X with gamma marginals. Specifically, it is a random atomic measure of the form (3), such
that for each measurable subset A, the (random) mass G(A) is gamma distributed. Assuming that
G has no fixed atoms (that is, for each element x ? X we have G({x}) = 0 with probability one)
and that the atom locations {Xk } are independent of their masses {wk }, it can be shown that such
a random measure can be constructed as follows: each Xk is iid according to a base distribution
H (which we assume is non-atomic with density h(x)), while the set of masses {wk } is distributed
according to a Poisson process over R+ with intensity ?(w) = ?w?1 e?w? where ? > 0 is the
concentration parameter and ? > 0 the inverse scale. We write this as G ? ?(?, ?, H). Under this
parametrization, we have that G(A) ? Gamma(?H(A), ? ).
Each atom Xk is a choice item, with its mass wk > 0 corresponding to the desirability parameter.
The Thurstonian view described in the finite model can be easily extended to the nonparametric one,
where a partial ranking (X?`1 . . . X?`m ) can be generated as the first m items to arrive in a race. In
particular, for each atom Xk let z`k ? Exp(wk ) be the time of arrival of Xk and X?`i the ith item to
arrive. The first m items to arrive (X?`1 . . . X?`m ) then constitutes our top-m list, with probability
as given in (2). Again reparametrizing using inter-arrival durations, let Z`i = z?`i ? z?`i?1 for
i = 1, 2, . . . (with z?0 = 0). Then the joint probability is:
m
P ((X?`i )m
i=1 , (Z`i )i=1 |G) = P ((z?`1 . . . z?`m ), and z`k > z?`m for all k 6? {?`1 , . . . , ?`m }) (7)
Y
Y
X
m
i?1
?
m
Y
X
=
w?`i e?w?`i z?`i
e?wk z?`m =
w?`i exp ? Z`i
w?`j
wk ?
i=1
k6?{?`i }m
i=1
i=1
k=1
j=1
m
Marginalizing out (Z`i )m
i=1 gives the probability of (X?`i )i=1 in (2). Further, conditional on ?`
it is seen that the inter-arrival durations Z`1 . . . Z`m are mutually independent and exponentially
distributed:
X
?
i?1
X
m
Z`i |(X?`i )i=1 , G ? Exp
wk ?
w?`j
(8)
k=1
j=1
The above construction is depicted on Figure 1(left). We visualize on right some top-m lists generated from the model, with ? = 1 and different values of ?.
2.3
Posterior characterization
Consider a number L of partial rankings, with the `th list denoted Y` = (Y`1 . . . Y`m` ) , for ` ? [L].
While previously our top-m list (X?1 . . . X?m ) consists of an ordered list of the atoms in G. Here G
is unobserved and (Y`1 . . . Y`m` ) is simply a list of observed choice items, which is why they were
not expressed as an ordered list of atoms in G. The task here is then to characterize the posterior law
of G under a gamma process prior and supposing that the observed partial rankings were drawn iid
from the nonparametric Plackett-Luce model given G. Re-expressing the conditional distribution
(2) of Y` given G, we have:
m`
Y
G({Y`i })
(9)
P (Y` |G) =
G(X\{Y
`1 . . . Y` i?1 })
i=1
3
G
2
2
2
4
4
4
6
6
8
8
8
10
10
10
12
12
12
14
14
14
16
16
16
18
18
20
20
5
u?
u?1
10
15
20
25
30
6
18
20
5
10
15
20
25
30
5
10
15
20
25
30
Figure 1: Bayesian
P nonparametric Plackett-Luce model.
Left: G and U = k uk ?Xk where uk = ? log(zk ). The
top-3 ranking is (?1 , ?2 , ?3 ). Right: Visualization of top-5
rankings with rows corresponding to different rankings and
columns to items sorted by size biased order. A lighter shade
corresponds to a higher rank. Each figure is for a different
G, with ? = .1, 1, 3.
E
2
u?
3
As before, for each `, we will also introduce a set of auxiliary variables Z` = (Z`1 . . . Z`m` ) (the
inter-arrival times) that are conditionally mutually independent given G and Y` , with:
Z`i |Y` , G ? Exp(G(X\{Y`1 , . . . , Y`i?1 }))
(10)
The joint probability of the item lists and auxiliary variables is then (c.f. (7)):
P ((Y` , Z` )L
`=1 |G) =
m`
L Y
Y
G({Y`i }) exp(?Z`i G(X\{Y`1 , . . . , Y` i?1 }))
(11)
`=1 i=1
Note that under the generative process described in Section 2.2, there is positive probability that an
item appearing in a list Y` appears in another list Y`0 with `0 6= `. Denote the unique items among all
?
, and for each k = 1, . . . , K let nk be the number of occurrences of Xk? among
L lists by X1? . . . XK
the item lists. Finally define occurrence indicators
0 if ?j < i with Y`j = Xk? ;
?`ik =
(12)
1 otherwise.
i.e. ?`ik is the indicator of the occurence that item Xk? does not appear at a rank lower than i in the
`th list. Then the joint probability under the nonparametric Plackett-Luce model is:
P ((Y` , Z` )L
`=1 |G) =
K
Y
G({Xk? })nk ?
m`
L Y
Y
!
= exp ?G(X)
exp(?Z`i G(X\{Y`1 , . . . , Y` i?1 }))
`=1 i=1
k=1
X
Z`i
`i
!
K
Y
G({Xk? })nk
exp
?G({Xk? })
k=1
X
(?`ik ? 1)Z`i
`i
(13)
Taking expectation of (13) with respect to G using the Palm formula gives:
Theorem 1 The marginal probability of the L partial rankings and auxiliary variables is:
K
Y
X
P
??( `i Z`i )
?
P ((Y` , Z` )L
)
=
e
h(X
)?
n
,
?
Z
k
`ik `i
`=1
k
k=1
(14)
`i
where ?(z) is the Laplace transform of ?,
h
i Z
?zG(X)
?(z) = ? log E e
=
z
?(w)(1 ? e?zw )dw = ? log 1 +
?
R+
and ?(n, z) is the nth moment of the exponentially tilted L?evy intensity ?(w)e?zw :
Z
?
?(n, z) =
?(w)wn e?zw dw =
?(n)
(z + ? )n
R+
(15)
(16)
Details are given in the supplementary material. Another application of the Palm formula now allows
us to derive a posterior characterisation of G:
4
Theorem 2 Given the observations and associated auxiliary variables (Y` , Z` )L
`=1 , the posterior
law of G is also a gamma process, but with atoms with both fixed and random locations. Specifically,
G|(Y` , Z` )L
`=1
?
=G +
K
X
wk? ?Xk?
(17)
k=1
?
where G? and w1? , . . . , wK
are mutually independent. The law of G? is still a gamma process,
X
?
G? |(X` , Z` )L
?? = ? +
Z`i
(18)
`=1 ? ?(?, ? , h)
`i
while the masses have distributions,
wk? |(Y` , Z` )L
`=1
? Gamma nk , ? +
X
?`ik Z`i
(19)
`i
2.4
Gibbs sampling
Given the results of the previous section, a simple Gibbs sampler can now be derived, where all the
conditionals are of known analytic form. In particular, we will integrate out all of G? except for its
total mass w?? = G? (X). This leaves the latent variables to consist of the masses w?? , (wk? ) and the
auxiliary variables (Z`i ). The update for Z`i is given by (10), while those for the masses are given
in Theorem 2:
P
Gibbs update for Z`i :
Z`i |rest ? Exp w?? + k ?`ik wk?
(20)
P
?
?
Gibbs update for wk :
wk |rest ? Gamma nk , ? + `i ?`ik Z`i
(21)
P
?
?
Gibbs update for w? :
w? |rest ? Gamma ?, ? + `i Z`i
(22)
Note that the auxiliary variables are conditionally independent given the masses and vice versa. Hyperparameters of the gamma process can be simply derived from the joint distribution in Theorem 1.
Since the marginal probability of the partial rankings is invariant to rescaling of the masses, it is
sufficient to keep ? fixed at 1. As for ?, if a Gamma(a, b) prior is placed on it, its conditional
distribution is still gamma:
P
Z
(23)
Gibbs update for ?:
?|rest ? Gamma a + K, b + log 1 + `i? `i
Note that this update was derived with w?? marginalized out, so after an update to ? it is necessary
to immediately update w?? via (22) before proceeding to update other variables.
3
Dynamic Bayesian nonparametric ranking models
In this section we develop an extension of the Bayesian nonparametric Plackett-Luce model to model
time-varying rankings, where the rating parameters of items may change smoothly over time and
reflected in a changing series of rankings. Given a series of times indexed by t = 1, 2, . . ., we
may model the rankings at time t using a gamma process distributed random measure Gt as in
Section 2.2, with Markov dependence among the sequence of measures (Gt ) enabling dependence
among the rankings over time.
3.1
Pitt-Walker dependence model
We will construct a dependent sequence (Gt ) which marginally follow a gamma process ?(?, ?, H)
using the construction of [13]. Suppose Gt ? ?(?, ?, H). Since Gt is atomic, we can write it in the
?
form:
X
Gt =
wtk ?Xtk
(24)
k=1
Define a random measure Ct with conditional law:
?
X
Ct |Gt =
ctk ?Xtk
ctk |Gt ? Poisson(?t wtk )
(25)
k=1
where ?t > 0 is a dependence parameter. Using the same method as in Section 2.3, we can show:
5
Proposition 3 Suppose the law of Gt is ?(?, ?, H). The conditional law of Gt given Ct is then:
Gt =
G?t
+
?
X
?
wtk
?Xtk
(26)
k=1
? ?
where G?t and (wtk
)k=1 are all mutually independent. The law of G?t is given by a gamma process,
while the masses are conditionally gamma,
G?t |Ct ? ?(?, ? + ?t , H)
?
wtk
|Ct ? Gamma(ctk , ? + ?t )
(27)
The idea of [13] is to define the conditional law of Gt+1 given Gt and Ct to coincide with the
conditional law of Gt given Ct as in Proposition 3. In other words, define
?
X
Gt+1 = G?t+1 +
wt+1,k ?Xtk
(28)
k=1
where G?t+1 ? ?(?, ? + ?t , H) and wt+1,k ? Gamma(ctk , ? + ?t ) are mutually independent. If
the prior law of Gt is ?(?, ?, H), the marginal law of Gt+1 will be ?(?, ?, H) as well when both
Gt and Ct are marginalized out, thus maintaining a form of stationarity. Further, although we have
described the process in order of increasing t, the joint law of Gt , Ct , Gt+1 can equivalently be
described in the reverse order with the same conditional laws as above. Note that if ctk = 0, the
conditional distribution of wt+1,k will be degenerate at 0. Hence Gt+1 has an atom at Xtk if and
only if Ct has an atom at Xtk , that is, if ctk > 0. In addition, it also has atoms (those in G?t+1 ) where
Ct does not (nor does Gt ). Finally, the parameter ?t can be interpreted as controlling the strength
of dependence between Gt+1 and Gt . Indeed it can be shown that
?
?t
Gt +
H.
(29)
E[Gt+1 |Gt ] =
?t + ?
?t + ?
Another measure of dependence can be gleaned by examining the ?lifetime? of an atom. Suppose
X is an atom in G1 with mass w > 0. The probability that X is an atom in C2 with positive mass
is 1 ? exp(??1 w), in which case it has positive mass in G2 as well. Conversely, once it is not an
atom, it will never be an atom in the future since the base distribution H is non-atomic. The lifetime
of the atom is then the smallest t such that it is no longer an atom. We can show by induction that:
(details in supplementary material)
Proposition 4 The probability that an atom X in G1 with mass w > 0 is dead at time t is given by
P (Gt ({X}) = 0|w) = exp(?yt|1 w)
where yt|1 can be obtained by the recurrence yt|t?1 = ?t?1 and yt|s?1 =
3.2
yt|s ?s?1
?s?1 +? +yt|s .
Posterior characterization and Gibbs sampling
Assume for simplicity that at each time step t = 1, . . . , T we observe one top-m list Yt =
(Yt1 , . . . , Ytm ) (it trivially extends to multiple partial rankings of differing sizes). We extend the
results of the previous section in characterizing the posterior and developing a Gibbs sampler for the
dynamical model.
Since each observed item at time t has to be an atom in its corresponding random measure Gt , and
atoms in Gt can propagate to neighboring random measures via the Pitt-Walker dependence model,
we conclude that the set of all observed items (through all times) has to include all fixed atoms in
the posterior of Gt . Thus let X ? = (Xk? ), k = 1, . . . , K be the set of unique items observed in
Y1 , . . . , YT , let ntk ? {0, 1} be the number of times the item Xk? appears at time t, and let ?t be
defined as Yt = (X??1 , . . . , X??m ). We write the masses of the fixed atoms as wtk = Gt ({Xk? }),
while the total mass of all other random atoms is denoted wt? = Gt (X\X ? ). Note that wtk has
to be positive on a random contiguous interval of time that includes all observations of Xk? ?it?s
lifetime?but is zero outside of the interval. We also write ctk = Ct ({Xk? }) and ct? = Ct (X\X ? ).
As before, we introduce, for t = 1, . . . , T and i = 1, . . . , m, latent variables
K
i?1
X
X
Zti ? Exp wt? +
wtk ?
wt?j
(30)
k=1
6
j=1
Figure 2: Sample path drawn from the Dawson-Watanabe superprocess. Each colour represents an
atom, with height being its (varying) mass. Left shows (Gt ) and right (Gt /Gt (X)), a Fleming-Viot
process.
Each iteration of the Gibbs sampler then proceeds as follows (details in supplementary material).
The latent variables (Zti ) are updated as above. Conditioned on the latent variables (Zti ), (ctk )
and (ct? ), we update the masses (wtk ), which are independent and gamma distributed since all
likelihoods are of gamma form. Note that the total masses (Gt (X)) are not likelihood identifiable, so
we introduce an extra step to improve mixing by sampling them from the prior (integrating out (ctk ),
(ct? )), scaling all masses along with it. Directly after this step we update (ctk ), (ct? ). We update
? along with the random masses (wt? ) and (ct? ) efficiently using a forward-backward recursion.
Finally, the dependence parameters (?t ) are updated.
3.3
Continuous time formulation using superprocesses
The dynamic model described in the previous section is formulated for discrete time data. When
the time interval between ranking observations is not constant, it is desirable to work with dynamic
models evolving over continuous-time instead, with the underlying random measures (Gt ) defined
over all t ? R, but with observations at a discrete set of times t1 < t2 < ? ? ? . Here we propose
a continuous-time model based on the Dawson-Watanabe superprocess [14, 15] (see also [16, 17,
18, 19]). This is a diffusion on the space of measures with the gamma process ?(?, ?, H) as its
equilibrium distribution. It is defined by a generator
Z
Z
Z
?2
?
?
L=?
G(dX)
+
?
H(dX)
?
?
G(dX)
?G(X)2
?G(X)
?G(X)
with ? parametrizing the rate of evolution. Figure 2 gives a sample path, where we see that it is
continuous but non-differentiable. For efficient inference, it is desirable to be able to integrate out all
Gt ?s except those Gt1 , Gt2 , . . . at observation times. An advantage to using the Dawson-Watanabe
superprocess is that, the conditional distribution of Gts given Gts?1 is remarkably simple [20]. In
particular it is simply given by the discrete-time process of the previous section with dependence
parameter ?ts |ts?1 = ? ?(ts ?t?s?1 ) . Thus the inference algorithm developed previously is directly
e
?1
applicable to the continuous-time model too.
4
Experiments
We apply the discrete-time dynamic Plackett-Luce model to the New York Times bestsellers data.
These consist of the weekly top-20 best-sellers list from June 2008 to April 2012 in various categories. We consider here the categories paperback nonfiction (PN) and hardcover fiction (HF), for
which respectively 249 and 916 books appear at least once in the top-20 lists over the 200 weeks.
We consider that the correlation parameter ?t = ? is constant over time, and assign flat improper
priors p(?) ? 1/? and p(?) ? 1/?. In order to take into account the publication date of a book,
we do not consider books in the likelihood before their first appearance in a list. We run the Gibbs
sampler with 10000 burn-in iterations followed by 10000 samples. Mean normalized weights for
the more popular books in both categories are shown in Figure 3.
The model is able to estimate the weights associated to each book that appeared at least once, as
well as the total weight associated to all other books, i.e. the probability that a new book enters at
the first rank in the list, represented by the black curve. Moreover, the Bayesian approach enables us
to have a measure of the uncertainty on the weights. The hardcover fiction category is characterized
by rapid changes in successive lists, compared to the paperback nonfiction. This is quantified by the
estimated value of the parameter ?, which are respectively 85 ? 20 and 140 ? 40 for PN and HF.
The estimated values of the shape parameter ? are 7 ? 1.5 and 2 ? 1 respectively.
7
Normalized weights
0.3
0.25
0.2
0.15
0.1
0.05
0
EAT, PRAY, LOVE
Date
Nov2008 Mar2009 Aug2009 Dec2009 May2010 Oct2010 Feb2011 Jul2011 Nov2011 Apr2012
THE AUDACITY OF HOPE
MARLEY AND ME
DREAMS FROM MY FATHER
THREE CUPS OF TEA
I HOPE THEY SERVE BEER IN HELL
GLENN BECK?S ?COMMON SENSE?
THE BLIND SIDE
THE LOST CITY OF Z
A PATRIOT?S HISTORY OF THE UNITED STATES
CONSERVATIVE VICTORY
MENNONITE IN A LITTLE BLACK DRESS
INSIDE OF A DOG
THE VOW
HEAVEN IS FOR REAL
Figure 3: Mean normalized weights for paperback nonfiction (left) and hardcover fiction (right).
The black lines represent the weight associated to all the books that have not appear in the top-20
lists.
5
Discussion
We have proposed a Bayesian nonparametric Plackett-Luce model for ranked data. Our approach
is based on the theory of atomic random measures, where we showed that the Plackett-Luce generative model corresponds exactly to a size-biased permutation of the atoms in the random measure.
We characterized the posterior distribution, and derived a simple MCMC sampling algorithm for
posterior simulation. Our approach can be see as a multi-stage generalization of posterior inference
in normalized random measures [21, 22, 23], and can be easily extended from gamma processes to
general completely random measures.
We also proposed dynamical extensions of our model for both discrete and continuous time data,
and applied it to modeling the bestsellers? lists on the New York Times. Our dynamic extension
may be useful for modeling time varying densities or clusterings as well. In our experiments we
found that our model is insufficient to capture the empirical observation that bestsellers often start
off high on the lists and tail off afterwards, since our model has continuous sample paths. We
adjusted for this by simply not including books in the model prior to their publication date. It may
be possible to model this better using models with discontinuous sample paths, for example, the
Orstein-Uhlenbeck approach of [24] where the process evolves via a series of discrete jump events
instead of continuously.
Acknowledgements
YWT thanks the Gatsby Charitable Foundation for generous funding.
8
References
[1] R.D. Luce. Individual choice behavior: A theoretical analysis. Wiley, 1959.
[2] R. Plackett. The analysis of permutations. Applied Statistics, 24:193?202, 1975.
[3] R.D. Luce. The choice axiom after twenty years. Journal of Mathematical Psychology,
15:215?233, 1977.
[4] D.R. Hunter. MM algorithms for generalized Bradley-Terry models. The Annals of Statistics,
32:384?406, 2004.
[5] I.C. Gormley and T.B. Murphy. Exploring voting blocs with the Irish electorate: a mixture
modeling approach. Journal of the American Statistical Association, 103:1014?1027, 2008.
[6] P. Diaconis. Group representations in probability and statistics, IMS Lecture Notes, volume 11.
Institute of Mathematical Statistics, 1988.
[7] I.C. Gormley and T.B. Murphy. A grade of membership model for rank data. Bayesian Analysis, 4:265?296, 2009.
[8] J. Guiver and E. Snelson. Bayesian inference for Plackett-Luce ranking models. In International Conference on Machine Learning, 2009.
[9] F. Caron and A. Doucet. Efficient Bayesian inference for generalized Bradley-Terry models.
Journal of Computational and Graphical Statistics, 21(1):174?196, 2012.
[10] G.P. Patil and C. Taillie. Diversity as a concept and its implications for random communities.
Bulletin of the International Statistical Institute, 47:497?515, 1977.
[11] J. Pitman. Combinatorial stochastic processes. Ecole d??et?e de Probabilit?es de Saint-Flour
XXXII - 2002, volume 1875 of Lecture Notes in Mathematics. Springer, 2006.
[12] J. F. C. Kingman. Completely random measures. Pacific Journal of Mathematics, 21(1):59?78,
1967.
[13] M.K. Pitt and S.G. Walker. Constructing stationary time series models using auxiliary variables
with applications. Journal of the American Statistical Association, 100(470):554?564, 2005.
[14] S. Watanabe. A limit theorem of branching processes and continuous state branching processes. Journal of Mathematics of Kyoto University, 8:141?167, 1968.
[15] D. A. Dawson. Stochastic evolution equations and related measure processes. Journal of
Multivariate Analysis, 5:1?52, 1975.
[16] S.N. Ethier and RC Griffiths. The transition function of a measure-valued branching diffusion
with immigration. Stochastic Processes. A Festschrift in Honour of Gopinath Kallianpur (S.
Cambanis, J. Ghosh, RL Karandikar and PK Sen, eds.), 71:79, 1993.
[17] R.H. Mena and S.G. Walker. On a construction of Markov models in continuous time. MetronInternational Journal of Statistics, 67(3):303?323, 2009.
[18] S. Feng. Poisson-Dirichlet Distribution and Related Topics. Springer, 2010.
[19] J.C. Cox, J.E. Ingersoll Jr, and S.A. Ross. A theory of the term structure of interest rates.
Econometrica: Journal of the Econometric Society, pages 385?407, 1985.
[20] S. N. Ethier and R. C. Griffiths. The transition function of a measure-valued branching diffusion with immigration. Stochastic Processes, 1993.
[21] L.F. James, A. Lijoi, and I. Pr?unster. Posterior analysis for normalized random measures with
independent increments. Scandinavian Journal of Statistics, 36(1):76?97, 2009.
[22] J.E. Griffin and S.G. Walker. Posterior simulation of normalized random measure mixtures.
Journal of Computational and Graphical Statistics, 20(1):241?259, 2011.
[23] S. Favaro and Y.W. Teh. MCMC for normalized random measure mixture models. Technical
report, University of Turin, 2012.
[24] J. E. Griffin. The Ornstein-Uhlenbeck Dirichlet process and other time-varying processes for
Bayesian nonparametric inference. Journal of Statistical Planning and Inference, 141:3648?
3664, 2011.
9
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4,005 | 4,625 | Value Pursuit Iteration
Amir-massoud Farahmand?
Doina Precup ?
School of Computer Science, McGill University, Montreal, Canada
Abstract
Value Pursuit Iteration (VPI) is an approximate value iteration algorithm that finds
a close to optimal policy for reinforcement learning problems with large state
spaces. VPI has two main features: First, it is a nonparametric algorithm that
finds a good sparse approximation of the optimal value function given a dictionary of features. The algorithm is almost insensitive to the number of irrelevant
features. Second, after each iteration of VPI, the algorithm adds a set of functions
based on the currently learned value function to the dictionary. This increases the
representation power of the dictionary in a way that is directly relevant to the goal
of having a good approximation of the optimal value function. We theoretically
study VPI and provide a finite-sample error upper bound for it.
1
Introduction
One often has to use function approximation to represent the near optimal value function of the
reinforcement learning (RL) and planning problems with large state spaces. Even though the conventional approach of using a parametric model for the value function has had successes in many
applications, it has one main weakness: Its success critically depends on whether the chosen function approximation method is suitable for the particular task in hand. Manually designing a suitable
function approximator, however, is difficult unless one has considerable domain knowledge about
the problem. To address this issue, the problem-dependent choice of function approximator and
nonparametric approaches to RL/Planning problems have gained considerable attention in the RL
community, e.g., feature generation methods of Mahadevan and Maggioni [1], Parr et al. [2], and
nonparametric regularization-based approaches of Farahmand et al. [3, 4], Taylor and Parr [5] .
One class of approaches that addresses the aforementioned problem is based on the idea of finding
a sparse representation of the value function in a large dictionary of features (or atoms). In this
approach, the designer does not necessarily know a priori whether or not a feature is relevant to the
representation of the value function. The feature, therefore, is simply added to the dictionary with
the hope that the algorithm itself figures out the necessary subset of features. The usual approach to
tackle irrelevant features is to use sparsity-inducing regularizers such as the l1 -norm of the weights
in the case of linear function approximators, e.g., Kolter and Ng [6], Johns et al. [7], Ghavamzadeh
et al. [8]. Another approach is based on greedily adding atoms to the representation of the target
function. Examples of these approaches in the supervised learning setting are Matching Pursuit and
Orthogonal Matching Pursuit (OMP) [9, 10]. These greedy algorithms have successfully been used
in the signal processing and statistics/supervised machine learning communities for years, but their
application in the RL/Planning problems has just recently attracted some attention. Johns [11] empirically investigated some greedy algorithms, including OMP, for the task of feature selection using
dictionary of proto-value functions [1]. A recent paper by Painter-Wakefield and Parr [12] considers
two algorithms (OMP-TD and OMP-BRM; OMP-TD is the same as one of the algorithms by [11])
in the context of policy evaluation and provides some conditions under which OMP-BRM can find
the minimally sparse solution.
?
?
Academic.SoloGen.net.
This research was funded in part by NSERC and ONR.
1
To address the problem of value function representation in RL when not much a priori knowledge
is available, we introduce the Value Pursuit Iteration (VPI) algorithm. VPI, which is an Approximate Value Iteration (AVI) algorithm (e.g., [13]), has two main features. The first is that it is a
nonparametric algorithm that finds a good sparse approximation of the optimal value function given
a set of features (dictionary), by using a modified version of OMP. The second is that after each
iteration, the VPI algorithm adds a set of functions based on the currently learned value function to
the dictionary. This potentially increases the representation power of the dictionary in a way that is
directly relevant to the goal of approximating the optimal value function.
At the core of VPI is the OMP algorithm equipped with a model selection procedure. Using OMP
allows VPI to find a sparse representation of the value function in large dictionaries, even countably
infinite ones1 . This property is very desirable for RL/Planning problems for which one usually does
not know the right representation of the value function, and so one wishes to add as many features
as possible and to let the algorithm automatically detect the best representation. A model selection
procedure ensures that OMP is adaptive to the actual difficulty of the problem.
The second main feature of VPI is that it increases the size of the dictionary by adding some basis
functions computed from previously learned value functions. To give an intuitive understanding of
how this might help, consider the dictionary B = {g1 , g2 , . . . }, in which each atom gi is a realvalued function defined on the state-action
P space. The goal is to learn the optimal value function by
a representation in the form of Q = i?1 wi gi .2 Suppose that we are lucky and the optimal value
function Q? belongs to the dictionary B, e.g., g1 = Q? . This is indeed an ideal atom to have in the
dictionary
P since one may have a sparse representation of the optimal value function in the form of
Q? = i?1 wi gi with w1 = 1 and wi = 0 for i ? 2. Algorithms such as OMP can find this sparse
representation quite effectively (details will be specified later). Of course, we are not usually lucky
enough to have the optimal value function in our dictionary, but we may still use approximation of
the optimal value function. In the exact Value Iteration, Qk ? Q? exponentially fast. This ensures
that Qk and Qk+1 = T ? Qk are close enough, so one may use Qk to explain a large part of Qk+1 and
use the other atoms of the dictionary to ?explain? the residual. In an AVI procedure, however, the
estimated value function sequence (Qk )k?1 does not necessarily converge to Q? , but one may hope
that it gets close to a region around the optimum. In that case, we may very well use the dictionary
of {Q1 , . . . , Qk } as the set of candidate atoms to be used in the representation of Qk+1 . We show
that adding these learned atoms does not hurt and may actually help.
To summarize, the algorithmic contribution of this paper is to introduce the VPI algorithm that finds
a sparse representation of the optimal value function in a huge function space and increases the
representation capacity of the dictionary problem-dependently. The theoretical contribution of this
work is to provide a finite-sample analysis of VPI and to show that the method is sound.
2
Definitions
We follow the standard notation and definitions of Markov Decision Processes (MDP) and Reinforcement Learning (RL) (cf. [14]). We also need some definitions regarding the function spaces
and norms, which are defined later in this section.
For a space ? with ?-algebra ?? , M(?) denotes the set of all probability measures over ?? . B(?)
denotes the space of bounded measurable functions w.r.t. (with respect to) ?? and B(?, L) denotes
the subset of B(?) with bound 0 < L < ?.
A finite-action discounted MDP is a 5-tuple (X , A, P, R, ?), where X is a measurable state space,
A is a finite set of actions, P : X ? A ? M(X ) is the transition probability kernel, R : X ? A ?
M(R) is the reward kernel, and ? ? [0, 1) is a discount factor. Let r(x, a) = E [R(?|x, a)], and
assume that r is uniformly bounded by Rmax . A measurable mapping ? : X ? A is called a
deterministic Markov stationary policy, or just a policy for short. A policy ? induces the m-step
transition probability kernels (P ? )m : X ? M(X ) and (P ? )m : X ? A ? M(X ? A) for m ? 1.
We use V ? and Q? to denote the value and action-value function of a policy ?. We also use V ?
and Q? for the optimal value and optimal action-value functions, with the corresponding optimal
1
2
From the statistical viewpoint and ignoring the computational difficulty of working with large dictionaries.
The notation will be defined precisely in Section 2.
2
policy ? ? . A policy ? is greedy w.r.t. an action-value function Q, denoted ? = ?
? (?; Q), if ?(x) =
argmaxa?A Q(x, a) holds for all x ? X (if there exist multiple maximizers, one of them is chosen in
an arbitrary deterministic manner). Define Qmax = Rmax /(1 ? ?). The Bellman optimality operator
is denoted by T ? . We use (P V )(x) to denote the expected value of V after the transition according
to a probability transition kernel P . Also for a probability measure ? ? M(X ), the symbol (?P )
represents the distribution over states when the initial state distribution is ? and we follow P for
a single step. A typical choice of P is (P ? )m for m ? 1 (similarly for ? ? M(X ? A) and
action-value functions).
2.1
Norms and Dictionaries
For a probability measure ? ? M(X ), and a measurable function V ? B(X ), we define the Lp (?)1/p
R
. The L? (X )-norm is defined as
norm (1 ? p < ?) of V as kV kp,? , X |V (x)|p d?(x)
kV k? , supx?X |V (x)|. Similarly for ? ? M(X ? A) and Q ? B(X ? A), we define k?kp,? as
R
p
kQkp,? , X ?A |Q(x, a)|p d?(x, a) and kQk? , sup(x,a)?X ?A |Q(x, a)|.
Let z1:n denote the Z-valued sequence (z1 , . . . , zn ). For Dn = z1:n , define the empirical norm of
Pn
p
p
function f : Z ? R as kf kp,z1:n = kf kp,Dn , n1 i=1 |f (zi )|p . Based on this definition, one
may define kV kDn (with Z = X ) and kQkDn (with Z = X ? A). Note that if Dn = Z1:n is
random
withi Zi ? ?, the empirical norm is random as well. For any fixed function f , we have
h
E kf kp,Dn = kf kp,? . The symbols k?k? and k?kDn refer to an L2 -norm. When we do not want to
emphasize the underlying measure, we use k?k to denote an L2 -norm.
Consider a Hilbert space H endowed with an inner product norm k?k. We call a family of functions
B = {g1 , g2 , . . . , } with atoms gi ? H a dictionary.
P The class L1 (B) = L1 (B; k?k) consists of those
functions f ? H that admits an expansion f = g?B cg g with (cg ) being an absolutely summable
sequence (these definitions are quoted from
The norm of a function f in this
P Barron et al. [15]).
P
space is defined as kf kL1 (B;k?k) , inf{ g?B |cg | : f = g?B cg g}. To avoid clutter, when the
norm is the empirical norm k?kDn , we may use L1 (B; Dn ) instead of L1 (B; k?kDn ), and when the
norm is k?k? , we may use L1 (B; ?). We denote a ball with radius r > 0 w.r.t. the norm of L1 (B; ?)
by Br (L1 (B; ?)).
For a dictionary B, we introduce a fixed exhaustion B1 ? B2 ? . . . ? B, with the number of atoms
|Bm | being m. If we index our dictionaries as Bk , the symbol Bk,m refers to the m-th element of
the exhaustion of Bk . For a real number ? > 0, the space L1,? (B; k?k) is defined as the set of
all functions f such that for all m = 1, 2, . . . , there exists a function h depending on m such that
khkL1 (Bm ;k?k) ? C and kf ? hk ? Cm?? . The smallest constant C such that these inequalities
hold defines a norm for L1,? (B; k?k). Finally, we define the truncation operator ?L : B(X ) ? B(X )
for some real number L > 0 as follows. For any function f ? B(X ), the truncated function of f at
the threshold level L is the function ?L f : B(X ) ? R such that for any x ? X , ?L f (x) is equal
to f (x) if ?L ? f (x) ? L, is equal to L if f (x) > L, and is equal to ?L if f (x) < ?L. We
overload ?L to be an operator from B(X ? A) to B(X ? A) by applying it component-wise, i.e.,
?L Q(x, ?) , [?L Q(x, a1 ), . . . , ?L Q(x, aA )]> .
3
VPI Algorithm
In this section, we first describe the behaviour of VPI in the ideal situation when the Bellman optimality operator T ? can be applied exactly in order to provide the intuitive understanding of why
VPI might work. Afterwards, we describe the algorithm that does not have access to the Bellman
optimality operator and only uses a finite sample of transitions.
VPI belongs to the family of AVI algorithms, which start with an initial action-value function Q0
and at each iteration k = 0, 1, . . . , approximately apply the Bellman optimality operator T ? to the
most recent estimate Qk to get a new estimate Qk+1 ? T ? Qk . The size of the error between Qk+1
and T ? Qk is a key factor in determining the performance of an AVI procedure.
3
Suppose that T ? Qk can be calculated, but it is not possible to represent it exactly. In this case, one
may use an approximant Qk+1 to represent T ? Qk . In this paper we would like to represent Qk+1
as a linear function of some atoms
P in a dictionary B = {g1 , g2 , . . . } (g ? H(X ? A) and kgk = 1
for all g ? B), that is Qk+1 = g?B cg g. Our goal is to find a representation that is as sparse as
possible, i.e., uses only a few atoms in B. From statistical viewpoint, the smallest representation
among all those that have the same function approximation error is desirable as it leads to smaller
estimation error. The goal of finding the sparsest representation, however, is computationally intractable. Nevertheless, it is possible to find a ?reasonable? suboptimal sparse approximation using
algorithms such as OMP, which is the focus of this paper.
? (0) = 0. For each i = 1, 2, . . . , define the
The OMP algorithm works as follows. Let Q
(i?1)
?
(i?1)
?
residual r
= T
Qk ? Q
. Define the new atom to be added to the representation as
g (i) ? Argmaxg?B r(i?1) , g , i.e., choose an element of the dictionary that has the maximum correlation with the residual. Here h ? , ? i is the inner product for a Hilbert space H(X ? A)
to which T ? Qk and atoms of the dictionary belong. Let ?(i) be the orthogonal projection onto
span(g (1) , . . . , g (i) ), i.e., ?(i) T ? Qk , argminQ?span(g(1) ,...,g(i) ) kQ ? T ? Qk k. We then have
? (i) = ?(i) T ? Qk . OMP continues iteratively.
Q
To quantify the approximation error at the i-th iteration, we use the L1 (B; k?k)-norm of the target
function of the OMP algorithm, which is T ? Qk in our case (with the norm being the one induced by
the inner product used in the
POMP procedure). Recall that this class consists of functions that admit
an expansion in the form g?B cg g and (cg ) being an absolutely summable sequence. If T ? Qk
belongs to the class of L1 (B; k?k), it can be shown (e.g., Theorem 2.1 of Barron et al. [15]) that
?
k kL (B;k?k)
1
? (i) is such that kQ
? (i) ? T ? Qk k ? kT Q?
after i iterations of OMP, the returned function Q
.
i+1
The problem with this result is that it requires T ? Qk to belong to L1 (B; k?k). This depends on how
expressive the dictionary B is. If it is not expressive enough, we still would like OMP to quickly
converge to the best approximation of T ? Qk ?
/ L1 (B; k?k) in L1 (B; k?k). Fortunately, such a result
exists (Theorem 2.3 by Barron et al. [15], quoted in the supplementary material) and we use it in the
proof of our main result.
We now turn to the more interesting case when we do not have access to T ? Qk . Instead we
(k)
(k)
(k)
(k)
(k)
are only given a set of transitions in the form of Dn = {(Xi , Ai , Ri , Xi0 )}ni=1 , where
(k)
(k)
(Xi , Ai ) are drawn from the sampling distribution ? ? M(X ? A), Xi0 ? P (?|Xi , Ai ), and
Ri ? R(?|Xi , Ai ). Instead of using T ? Qk , we use the empirical Bellman operator for the dataset
(k)
Dn . The operator is defined as follows.
Definition
1
(Empirical
Bellman
Optimality
Operator).
Let
Dn
=
{(X1 , A1 , R1 , X10 ), . . . , (Xn , An , Rn , Xn0 )}, defined similarly as above. Define the ordered multiset Sn = {(X1 , A1 ), . . . , (Xn , An )}. The empirical Bellman optimality operator T?? : Sn ? Rn is
defined as (T?? Q)(Xi , Ai ) , Ri + ? maxa0 Q(Xi0 , a0 ) for 1 ? i ? n.
i
h
(k)
(k)
(k)
(k)
(k)
(k)
= T ? Qk (Xi , Ai ), we can solve a regression
Since E T?? Qk (Xi , Ai ) Qk , Xi , Ai
problem and find an estimate for Qk+1 , which is close T ? Qk . This regression problem is the core of
the family of Fitted Q-Iteration (FQI) algorithms, e.g., [13, 4]. In this paper, the regression function
at each iteration is estimated using a modified OMP procedure introduced by Barron et al. [15].
We are now ready to describe the VPI algorithm (Algorithm 1). It gets as input a predefined dictionary B0 . This can be a dictionary of wavelets, proto-value functions, etc. The size of this dictionary
can be countably infinite. It also receives an integer m, which specifies how many atoms of B0
should be used by the algorithm. This defines the effective dictionary B0,m . This value can be set to
m = dna e for some finite a > 0, so it can actually be quite large. VPI also receives K, the number
of iterations, and ?, the sampling distribution. For the simplicity of analysis, we assume that the
sampling distribution is fixed, but in practice one may change this sampling distribution after each
iteration (e.g., sample new data according to the latest policy). Finally, VPI gets a set of m0 link
functions ?i : B(X ? A, Qmax ) ? B(X ? A, Qmax ) for some m0 that is smaller than m/K. We
describe the role of link functions shortly.
4
0
Algorithm 1 Value Pursuit Iteration(B0 , m, {?i }m
i=1 , ?, K)
0
Input: Initial dictionary B0 , Number of dictionary atoms used m, Link functions {?i }m
i=1 , Stateaction distribution ?, Number of iterations K.
Return: QK
Q0 ? 0.
B00 ? ?.
for k = 0, 1, . . . , K ? 1 do n
on
(k)
(k)
(k) i.i.d.
(k)
(k)
(k)
(k)
Construct a dataset Dn = (Xi , Ai , Ri , Xi0 )
, (Xi , Ai ) ? ?
i=1
? (0) ? 0
Q
k+1
// Orthogonal Matching Pursuit loop
Normalize elements of B0,m and Bk0 according to k?kD(k) and call them B?k and B?k0 .
n
for i = 1, 2, . . . , c1 n do
? (i?1)
r(i?1) ? T?? Qk ? Q
k+1
(i)
S
g ? Argmaxg?B?k B?0 r(i?1) , g D(k)
n
k
? (i) ? ?(i) T?? Qk
Q
{?(i) : Projection onto span(g (1) , . . . , g (i) )}
k+1
end for
2
(i)
i ln(n)
?
?
?
?
? T Qk
+ c2 (Qmax )
i ? argmin
{Complexity Regularization}
?Q Q
i?1
max
k+1
(k)
Dn
n
?
? (i )
Qk+1 ? Q
k+1
S
S
0
0
Bk+1 ? Bk0 {?i (?Qmax Qk+1 ; Bk Bk0 )}m
i=1 {Extending the dictionary}
end for
At the k-th iteration of the algorithm, we perform OMP for c1 n iterations (c1 > 0), similar to
what is described above with the difference that instead of using T ? Qk as the target, we use T?? Qk
over empirical samples.3 This means that we use the empirical inner product h Q1 , Q2 iD(k) ,
n
Pn
(k)
1
4
and
the
empirical
orthogonal
projection.
|Q
(X
,
A
)
?
Q
(X
,
A
)|
for
(X
,
A
)
?
D
n
1
i
i
2
i
i
i
i
i=1
n
? (i) )i?0 . Next, we perform a model selection procedure to choose
The result would be a sequence (Q
k+1
the best candidate. This can be done in different ways such as using a separate dataset as a validation
set. Here we use a complexity regularization technique that penalizes more complex estimates (those
? (i) in
that have more atoms in their representation). Note that we use the truncated estimate ?Qmax Q
k+1
the model selection procedure. This is required for the theoretical guarantees. The outcome of this
model selection procedure will determine Qk+1 .
0
0
Finally we use link functions {?i }m
i=1 to generate m new atoms, which are vector-valued Qmax |A|
bounded measurable functions from X ? A to R , to be added to the learned dictionary Bk0 .
The link functions
extract ?interesting? aspects of Qk+1 , potentially by considering the current
S
dictionary Bk Bk0 . VPI is quite flexible in how the new atoms are generated and how large m0 can
be. The theory allows m0 to be in the order of na (a > 0), so one may add many potentially useful
atoms without much deterioration in the performance. Regarding the choice of the link functions,
the theory requires that at least Qk+1 itself is being added to the dictionary, but it leaves other
possibilities open. For example, one might apply nonlinearities (e.g., sigmoid functions) to Qk+1 .
Or one might add atoms localized in parts of the state-action space with high residual errors ? a
heuristic which has been used previously in basis function construction. This procedure continues
for K iterations and the outcome will be QK . In the next section, we study the theoretical properties
of the greedy policy w.r.t. QK , i.e., ?K = ?
? (?; QK ).
Remark 1 (Comparison of VPI with FQI). Both VPI and FQI are indeed instances of AVI. If we
compare VPI with the conventional implementation of FQI that uses a fixed set of linear basis
3
The value of c1 depends only on Qmax and a. We do not explicitly specify it since the value that is
determined by the theory shall be quite conservative. One may instead find it by the trial and error. Moreover,
in practice we may stop much earlier than n iterations.
4
When the number of atoms is larger than the number of samples (i > n), one may use the Moore?Penrose
pseudoinverse to perform the orthogonal projection.
5
functions, we observe that FQI is the special case of VPI in which all atoms in the dictionary are
used in the estimation. As VPI has a model selection step, its chosen estimator is not worse than
FQI?s (up to a small extra risk) and is possibly much better if the target is sparse in the dictionary.
Moreover, extending the dictionary decreases the function approximation error with negligible effect
on the model selection error. The same arguments apply to many other FQI versions that use a fixed
data-independent set of basis functions and do not perform model selection.
4
Theoretical Analysis
In this section, we first study how the function approximation error propagates in VPI (Section 4.1)
and then provide a finite-sample error upper bound as Theorem 3 in Section 4.2. All the proofs are
in the supplementary material.
4.1
Propagation of Function Approximation Error
In this section, we present tools to upper bound the function approximation error at each iteration.
Definition 2 (Concentrability Coefficient of Function Approximation Error Propagation). (I) Let ?
be a distribution over the state-action pairs, (X, A) ? ?, ?X be the marginal distribution of X,
and ?b (?|?) be the conditional probability of A given X. Further, let P be a transition probability
kernel P : X ? A ? M(X) and Px,a = P (?|x, a). Define the concentrability coefficient of one
i 21
h
dP
step transitions w.r.t. ? by C??? = E sup(y,a0 )?X ?A ?b (a10 |y) d?X,A
(y)
, where C??? =
X
? if Px,a is not absolutely continuous w.r.t. ?X for some (x, a) ? X ? A, or if ?b (a0 |y) = 0
for some (y, a0 ) ? X ? A. (II) Furthermore, for an optimal policy ? ? and an integer m ? 0,
?
let ?(P ? )m ? M(X ? A) denote the future state-action distribution obtained after m-steps of
?? m
?
following ? ? . Define c? (m) , k d(?(Pd? ) ) k? . If the future state-action distribution ?(P ? )m is
not absolutely continuous w.r.t. ?, we let c? (m) = ?.
The constant C??? is large if after transition step, the future states can be highly concentrated at
some states where the probability of taking some action a0 is small or d?X is small. Hence, the name
?concentrability of one-step transitions?. The definition of C??? is from Chapter 5 of Farahmand
[16]. The constant c? (m) shows how much we deviate from ? whenever we follow an optimal policy
? ? . It is notable that if ? happens to be the stationary distribution of the optimal policy ? ? (e.g., the
samples are generated by an optimal expert), c? (m) = 1 for all m ? 0.
We now provide the following result that upper bounds the error caused by using Qk (which is the
newly added atom to the dictionary) to approximate T ? Qk . The proof is provided in the supplementary material.
Lemma 1. Let (Qi )ki=0 ? B(X ? A, Qmax ) be a Qmax -bounded sequence of measurable action2
value functions.
Define ?i , T ? Qi ? Qi+1 (0 ? i ? ki? 1). Then, kQk ? T ? Qk k? ?
h
2
(1+?C??? )2 Pk?1 k?1?i
c? (k ? 1 ? i) k?i k? + ? k (2Qmax )2 .
i=0 ?
1??
2
If there was no error at earlier iterations (i.e., k?i k? = 0 for 0 ? i ? k?1), the error kQk ? T ? Qk k?
would be O(? k Q2max ), which is decaying toward zero with a geometrical rate. This is similar to the
behaviour of the exact VI, i.e., kT ? Qk ? Qk k? ? (1 + ?)? k kQ? ? Q0 k? .
The following result is Theorem 5.3 of Farahmand [16]. For the sake of completeness, we provide
the proof in the supplementary material.
Theorem 2. Let (Qk )k?1
k=0 be a sequence of state-action value functions and define ?i ,
T ? Qi ? Qi+1 (0 ? i ? k). Let F |A| : X ? A ? R|A| be a
subset of vector-valued
measurable functions. Then, inf Q0 ?F |A| kQ0 ? T ? Qk k? ? inf Q0 ?F |A|
Q0 ? (T ? )(k+1) Q0
? +
Pk?1
k?i
k?i k? .
i=0 (? C??? )
This result quantifies the behaviour of the function approximation error and relates it to the function
approximation error of approximating (T ? )k+1 Q0 (which is a deterministic quantity depending only
on the MDP itself, the function space F |A| , and Q0 ) and the errors of earlier iterations. This allows
6
us to provide a tighter upper bound for the function approximation error compared to the so-called
inherent Bellman error supQ?F |A| inf Q0 ?F |A| kQ0 ? T ? Qk? introduced by Munos and Szepesv?ari
[17], whenever the errors at previous iterations are small.
4.2
Finite Sample Error Bound for VPI
In this section, we provide an upper bound on the performance loss kQ? ? Q?K k1,? . This performance loss indicates the regret of following the policy ?K instead of an optimal policy when the
initial state-action is distributed according to ?. We define the following concentrability coefficients
similar to Farahmand et al. [18].
Definition 3 (Expected Concentrability of the Future State-Action Distribution). Given
?, ? ? M(X ? A), m ? 0, and an arbitrary sequence of stationary policies
(?m )m?1 , let ?P ?1 P ?2 . . . P ?m ? M(X ? A) denote the future state-action distribution obtained after m transitions, when the first state-action pair is distributed according to ? and then we follow the sequence of policies (?k )m
For integers m1 , m2 ?
k=1 .
1, policy ? and the sequence of policies ?1 , . . . , ?k define the concentrability coefficients
"
2 #! 21
d ?(P ? )m1 (P ?? )m2
cVI1 ,?,? (m1 , m2 ; ?) , E
(X, A)
and cVI2 ,?,? (m1 ; ?1 , . . . , ?k ) ,
d?
2 12
d(?(P ?k )m1 P ?k?1 P ?k?2 ???P ?1 )
, where (X, A) ? ?. If the future state-action dis(X, A)
E
d?
?
tribution ?(P ? )m1 (P ? )m2 (similarly, if ?(P ?k )m1 P ?k?1 P ?k?2 ? ? ? P ?1 ) is not absolutely continuous w.r.t. ?, we let cVI1 ,?,? (m1 , m2 ; ?) = ? (similarly, cVI2 ,?,? (m1 ; ?1 , . . . , ?k ) = ?).
Assumption A1 We make the following assumptions:
? For all values of 0 ? k ? K ? 1, the dataset used by VPI at each iteration is
(k)
(k)
(k)
(k)
(k)
Dn = {(Xi , Ai , Ri , Xi0 )}ni=1 with independent and identically distributed (i.i.d.)
(k)
(k)
(k)
(k)
(k)
(k)
samples (Xi , Ai ) ? ? ? M(X ? A) and Xi0
? P (?|Xi , Ai ) and Ri ?
(k)
(k)
R(?, ?|Xi , Ai ) for i = 1, 2, . . . , n.
(k)
(k0 )
? For 1 ? k, k 0 ? K ? 1 and k 6= k 0 , the datasets Dn and Dn are independent.
? There exists a constant Qmax ? 1 such that for any Q ? B(X ? A; Qmax ), |T?? Q(X, A)| ?
Qmax almost surely (a.s).
? For all g ? B0 , kgk? ? L < ?.
? The number of atoms m used from the dictionary B0 is m = dna e for some finite a > 0.
The number of link functions m0 used at each iteration is at most m/K.
0
? At S
iteration k, each of the link functions {?i }m
i=1 maps ?Qmax Qk+1 and the dictionary
0
Bk Bk to an element of the space of vector-valued Qmax -bounded measurable functions
X ? A ? R|A| . At least one of the mappings returns ?Qmax Qk+1 .
Most of these assumptions are mild and some of them can be relaxed. The i.i.d. assumption can be
relaxed using the so called independent block technique [19], but it results in much more complicated
proofs. We conjecture that the independence of datasets at different iterations might be relaxed as
well under certain condition on the Bellman operator (cf. Section 4.2 of [17]). The condition on the
number of atoms m and the number of link functions being polynomial in n are indeed very mild.
In order to compactly present our result, we define ak =
(1??) ? K?k?1
1?? K+1
for 0 ? k < K. Note that
the behaviour of ak ? ? K?k?1 , so it gives more weight to later iterations. Also define C1 (k) ,
2
Pk?1 k?i 2(k?i)
??? )
C??? (k = 1, 2, . . . ) and C2 , (1+?C
. For 0 ? s ? 1, define
i=0 ?
1??
CVI,?,? (K; s) =
"
#2
K?1
X 2(1?s) X
1?? 2
m
0
0
0
,
(
) sup
ak
? cVI1 ,?,? (m, K ? k; ?K ) + cVI2 ,?,? (m + 1; ?k+1 , . . . , ?K )
0
2 ?10 ,...,?K
k=0
m?0
where in the last definition the supremum is taken over all policies. The following theorem is the
main theoretical result of this paper. Its proof is provided in the supplementary material.
7
Theorem 3. Consider the sequence (Qk )K
k=0 generated by VPI (Algorithm 1). Let Assumptions A1
hold. For any fixed 0 < ? < 1, recursively define the sequence (bi )K
i=0 as follows:
s
log nK
2
?
b20 , c1 Q3max
kQ0 ? T ? Q0 k? ,
+3 0
inf
n
Q ?BQmax (L1 (B0,m ;?))
s
log nK
2
3
?
bk , c2 Qmax
+
n
(
k?1
X
0
Q ? (T ? )k+1 Q0
2 + C1 (k)
c3 min
? k?i b2i ,
inf
?
Q0 ?BQmax (L1 (B0,m ;?))
C2
i=0
k?1
X
!)
?
k?1?i
c? (k ? 1 ?
i) b2i
k
+ ? (2Qmax )
2
,
(k ? 1)
i=0
for some c1 , c2 , c3 > 0 that are only functions of Qmax and L. Then, for any k = 0, 1, . . . , K ? 1,
2
it holds that kQk+1 ? T ? Qk k? ? b2k , with probability at least 1 ? k?
K . Furthermore, define the
PK?1 2s
discounted sum of errors as E(s) , k=0 ak bk (for s ? [0, 1]). Choose ? ? M(X ? A). The
?-weighted performance loss of ?K is upper bounded as
2?
1/2
1/2
K
kQ? ? Q?K k1,? ?
inf
C
(K;
s)E
(s)
+
2?
Q
max ,
(1 ? ?)2 s?[0,1] VI,?,?
with probability at least 1 ? ?.
The value of bk is a deterministic upper bound on the error kQk+1 ? T ? Qk k? of each iteration of
VPI. We would like bk to be close to zero, because the second part of the theorem implies that
kQ? ? Q?K k1,? would be small too. If we study b2k , we observe two main terms: The first term,
q
log(nK/?)
, is the estimation error. The second term describes the function
which behaves as
n
approximation error. For k ? 1, it consists of two terms from which the minimum is selected.
The first term inside min{?, ?} describes the behaviour of the function approximation error when
we only use the predefined dictionary B0,m to approximate T ? Qk (see Theorem 2). The second
term describes the behaviour of the function approximation error when we only consider Qk as
the approximant of T ? Qk (see Lemma 1). The error caused by this approximation depends on
the error made in earlier iterations. The current analysis only considers the atom Qk from the
learned dictionary, but VPI may actually use other atoms to represent T ? Qk . This might lead to
much smaller function approximation errors. Hence, our analysis shows that in terms of function
approximation error, our method is sound and superior to not increasing the size of the dictionary.
However, revealing the full power of VPI remains as future work. Just as an example, if B0 is
complete in L2 (?), by letting n, m ? ? both the estimation error and function approximation error
goes to zero and the method is consistent and converges to the optimal value function.
5
Conclusion
This work introduced VPI, an approximate value iteration algorithm that aims to find a close to
optimal policy using a dictionary of atoms (or features). The VPI algorithm uses a modified Orthogonal Matching Pursuit that is equipped with a model selection procedure. This allows VPI to find a
sparse representation of the value function in large, and potentially overcomplete, dictionaries. We
theoretically analyzed VPI and provided a finite-sample error upper bound for it. The error bound
shows the effect of the number of samples as well as the function approximation properties of the
predefined dictionary, and the effect of learned atoms.
This paper is a step forward to better understanding how overcomplete dictionaries and sparsity can
effectively be used in the RL/Planning context. A more complete theory describing the effect of
adding atoms to the dictionary remains to be established. We are also planning to study VPI?s empirical performance, and comparing with other feature construction methods. We note that our main
focus was on the statistical properties of the algorithm, not on computational efficiency; optimizing
computation speed will be an interesting topic for future investigation.
8
References
[1] Sridhar Mahadevan and Mauro Maggioni. Proto-value functions: A Laplacian framework for learning
representation and control in markov decision processes. Journal of Machine Learning Research, 8:
2169?2231, 2007. 1
[2] Ronald Parr, Christopher Painter-Wakefield, Lihong Li, and Michael Littman. Analyzing feature generation for value-function approximation. In ICML ?07: Proceedings of the 24th international conference
on Machine learning, pages 737 ? 744, New York, NY, USA, 2007. ACM. 1
[3] Amir-massoud Farahmand, Mohammad Ghavamzadeh, Csaba Szepesv?ari, and Shie Mannor. Regularized
policy iteration. In D. Koller, D. Schuurmans, Y. Bengio, and L. Bottou, editors, Advances in Neural
Information Processing Systems (NIPS - 21), pages 441?448. MIT Press, 2009. 1
[4] Amir-massoud Farahmand, Mohammad Ghavamzadeh, Csaba Szepesv?ari, and Shie Mannor. Regularized
fitted Q-iteration for planning in continuous-space Markovian Decision Problems. In Proceedings of
American Control Conference (ACC), pages 725?730, June 2009. 1, 4
[5] Gavin Taylor and Ronald Parr. Kernelized value function approximation for reinforcement learning. In
ICML ?09: Proceedings of the 26th Annual International Conference on Machine Learning, pages 1017?
1024, New York, NY, USA, 2009. ACM. 1
[6] J. Zico Kolter and Andrew Y. Ng. Regularization and feature selection in least-squares temporal difference
learning. In ICML ?09: Proceedings of the 26th Annual International Conference on Machine Learning,
pages 521?528. ACM, 2009. 1
[7] Jeff Johns, Christopher Painter-Wakefield, and Ronald Parr. Linear complementarity for regularized policy evaluation and improvement. In J. Lafferty, C. K. I. Williams, J. Shawe-Taylor, R.S. Zemel, and
A. Culotta, editors, Advances in Neural Information Processing Systems (NIPS - 23), pages 1009?1017.
2010. 1
[8] Mohammad Ghavamzadeh, Alessandro Lazaric, R?emi Munos, and Matthew Hoffman. Finite-sample
analysis of lasso-TD. In Lise Getoor and Tobias Scheffer, editors, Proceedings of the 28th International
Conference on Machine Learning (ICML-11), ICML ?11, pages 1177?1184, New York, NY, USA, June
2011. ACM. ISBN 978-1-4503-0619-5. 1
[9] Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad. Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition. In Proceedings of the 27th Annual Asilomar
Conference on Signals, Systems, and Computers, pages 40?44, 1993. 1
[10] Geoffrey M. Davis, St?ephane Mallat, and Marco Avellaneda. Adaptive greedy approximations. Journal
of Constructive Approximation, 13:57?98, 1997. 1
[11] Jeff Johns. Basis Construction and Utilization for Markov Decision Processes using Graphs. PhD thesis,
University of Massachusetts Amherst, 2010. 1
[12] Christopher Painter-Wakefield and Ronald Parr. Greedy algorithms for sparse reinforcement learning. In
Proceedings of the 29th International Conference on Machine Learning (ICML) (Accepted), 2012. 1
[13] Damien Ernst, Pierre Geurts, and Louis Wehenkel. Tree-based batch mode reinforcement learning. Journal of Machine Learning Research, 6:503?556, 2005. 2, 4
[14] Csaba Szepesv?ari. Algorithms for Reinforcement Learning. Morgan Claypool Publishers, 2010. 2
[15] Andrew R. Barron, Albert Cohen, Wolfgang Dahmen, and Ronald A. Devore. Approximation and learning
by greedy algorithms. The Annals of Statistics, 36(1):64?94, 2008. 3, 4
[16] Amir-massoud Farahmand. Regularization in Reinforcement Learning. PhD thesis, University of Alberta,
2011. 6
[17] R?emi Munos and Csaba Szepesv?ari. Finite-time bounds for fitted value iteration. Journal of Machine
Learning Research, 9:815?857, 2008. 7
[18] Amir-massoud Farahmand, R?emi Munos, and Csaba Szepesv?ari. Error propagation for approximate policy and value iteration. In J. Lafferty, C. K. I. Williams, J. Shawe-Taylor, R.S. Zemel, and A. Culotta,
editors, Advances in Neural Information Processing Systems (NIPS - 23), pages 568?576. 2010. 7
[19] Bin Yu. Rates of convergence for empirical processes of stationary mixing sequences. The Annals of
Probability, 22(1):94?116, January 1994. 7
9
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4,006 | 4,626 | Exact and Stable Recovery of Sequences of Signals
with Sparse Increments via Differential
?1-Minimization
Demba Ba1,2 , Behtash Babadi1,2 , Patrick Purdon2 and Emery Brown1,2
1
MIT Department of BCS, Cambridge, MA 02139
2
MGH Department of Anesthesia, Critical Care and Pain Medicine
55 Fruit st, GRJ 4, Boston, MA 02114
[email protected], {behtash,patrickp}@nmr.mgh.harvard.edu
[email protected]
Abstract
We consider the problem of recovering a sequence of vectors, (xk )K
k=0 , for which
the increments xk ? xk?1 are Sk -sparse (with Sk typically smaller than S1 ), based
on linear measurements (yk = Ak xk + ek )K
k=1 , where Ak and ek denote the measurement matrix and noise, respectively. Assuming each Ak obeys the restricted
isometry property (RIP) of a certain order?depending only on Sk ?we show that
in the absence of noise a convex program, which minimizes the weighted sum
of the ?1 -norm of successive differences subject to the linear measurement constraints, recovers the sequence (xk )K
k=1 exactly. This is an interesting result because this convex program is equivalent to a standard compressive sensing problem with a highly-structured aggregate measurement matrix which does not satisfy
the RIP requirements in the standard sense, and yet we can achieve exact recovery. In the presence of bounded noise, we propose a quadratically-constrained
convex program for recovery and derive bounds on the reconstruction error of the
sequence. We supplement our theoretical analysis with simulations and an application to real video data. These further support the validity of the proposed
approach for acquisition and recovery of signals with time-varying sparsity.
1 Introduction
In the field of theoretical signal processing, compressive sensing (CS) has arguably been one of the
major developments of the past decade. This claim is supported in part by the deluge of research
efforts (see for example Rice University?s CS repository [1]) which has followed the inception of
this field [2, 3, 4]. CS considers the problem of acquiring and recovering signals that are sparse
(or compressible) in a given basis using non-adaptive linear measurements, at a rate smaller than
what the Shannon-Nyquist theorem would require. The work [2, 4] derived conditions under which
a sparse signal can be recovered exactly from a small set of non-adaptive linear measurements.
In [3], the authors propose a recovery algorithm for the case of measurements contaminated by
bounded noise. They show that this algorithm is stable, that is, within a constant of the noise
tolerance. Recovery of these sparse or compressible signals is performed using convex optimization
techniques.
The classic CS setting does not take into account the structure, e.g. temporal or spatial, of the
underlying high-dimensional sparse signals of interest. In recent years, the attention has shifted to
formulations which incorporate the signal structure into the CS framework. A number of problems
and applications of interest deal with time-varying signals which may not only be sparse at any
given instant, but may also exhibit sparse changes from one instant to the next. For example, a video
1
of a natural scene consists of a sequence of natural images (compressible signals) which exhibits
sparse changes from one frame to the next. It is thus reasonable to hope that one would be able to
get away with far fewer measurements than prescribed by conventional CS theory to acquire and
recover such time-varying signals as videos. The problem of recovering signals with time-varying
sparsity has been referred to in the literature as dynamic CS. A number of empirically-motivated
algorithms to solve the dynamic CS problem have been proposed, e.g. [5, 6]. To our knowledge,
no recovery guarantees have been proved for these algorithms, which typically assume that the
support of the signal and/or the amplitudes of the coefficients change smoothly with time. In [5],
for instance, the authors propose message-passing algorithms for tracking and smoothing of signals
with time-varying sparsity. Simulation results show the superiority of the algorithms compared to
one based on applying conventional CS principles at each time instant. Dynamic CS algorithms
have potential applications to video processing [7], estimation of sources of brain activity from
MEG time-series [8], medical imaging [7], and estimation of time-varying networks [9].
To the best of our knowledge, the dynamic CS problem has not received rigorous, theoretical
scrutiny. In this paper, we develop rigorous results for dynamic CS both in the absence and in
the presence of noise. More specifically, in the absence of noise, we show that one can exactly recover a sequence (xk )K
k=0 of vectors, for which the increments xk ? xk?1 are Sk -sparse, based on
linear measurements yk = Ak xk and under certain regularity conditions on (Ak )K
k=1 , by solving a
convex program which minimizes the weighted sum of the ?1 -norms of successive differences. In
the presence of noise, we derive error bounds for a quadratically-constrained convex program for
recovery of the sequence (xk )K
k=0 .
In the following section, we formulate the problem of interest and introduce our notation. In Section 3, we present our main theoretical results, which we supplement with simulated experiments
and an application to real video data in Section 4. In this latter section, we introduce probability-ofrecovery surfaces for the dynamic CS problem, which generalize the traditional recovery curves of
CS. We give concluding remarks in Section 5.
2 Problem Formulation and Notation
We denote the support of a vector x ? Rp by supp(x) = {j : xj 6= 0}. We say that a vector
x ? Rp is S-sparse if ||x||0 ? S, where ||x||0 := |supp(x)|. We consider the problem of recovering
p
a sequence (xk )K
k=0 of R vectors such that xk ? xk?1 is Sk -sparse based on linear measurements
of the form yk = Ak xk + ek . Here, Ak ? Rnk ?p , ek ? Rnk and yk ? Rnk denote the measurement
matrix, measurement noise, and the observation vector, respectively. Typically, Sk < nk ? p,
which accounts for the compressive nature of the measurements. For convenience, we let x0 be the
Rp vector of all zeros.
For the rest of our treatment, it will be useful to introduce some notation. We will be dealing with
sequences (of sets, matrices, vectors), as such we let the index k denote the k th element of any such
sequence. Let J be the set of indices {1, 2, ? ? ? , p}. For each k, we denote by {akj : j ? J}, the
columns of the matrix Ak and by Hk the Hilbert space spanned by these vectors.
For two matrices A1 ? Rn1 ?p and A2 ? Rn2 ?p , n2 ? n1 , we say that A2 ? A1 if the rows of A2
are distinct and each row of A2 coincides with a row of A1 .
We say that the matrix A ? Rn?p satisfies the restricted isometry property (RIP) or order S if, for
all S-sparse x ? Rp , we have
2
2
2
(1 ? ?S ) ||x||2 ? ||Ax||2 ? (1 + ?S ) ||x||2 ,
(1)
where ?S ? (0, 1) is the smallest constant for which Equation 1 is satisfied [2].
Consider the following convex optimization programs
K
X
||xk ? xk?1 ||1
?
x1 ,x2 ,??? ,xK
Sk
k=1
min
K
X
kxk ? xk?1 k1
?
x1 ,x2 ,??? ,xK
Sk
k=1
min
s.t.
yk = Ak xk ,
s.t. kyk ? Ak xk k2 ? ?k ,
2
k = 1, 2, ? ? ? , K.
k = 1, 2, ? ? ? , K.
(P1)
(P2)
What theoretical guarantees can we provide on the performance of the above programs for recovery
of sequences of signals with sparse increments, respectively in the absence (P1) and in the presence
(P2) of noise?
3 Theoretical Results
We first present a lemma giving sufficient conditions for the uniqueness of sequences of vectors with
sparse increments given linear measurements in the absence of noise. Then, we prove a theorem
which shows that, by strengthening the conditions of this lemma, program (P1) can exactly recover
every sequence of vectors with sparse increments. Finally, we derive error bounds for program (P2)
in the context of recovery of sequences of vectors with sparse increments in the presence of noise.
Lemma 1 (Uniqueness of Sequences of Vectors with Sparse Increments).
Suppose (Sk )K
k=0 is such that S0 = 0, and for each k, Sk ? 1. Let Ak satisfy the RIP of order 2Sk .
Let xk ? Rp supported on Tk ? J be such that ||xk ? xk?1 ||0 ? Sk , for k = 1, 2, ? ? ? , K. Suppose
T0 = ? without loss of generality (w.l.o.g.). Then, given Ak and yk = Ak xk , the sequence of sets
K
(Tk )K
k=1 , and consequently the sequence of coefficients (xk )k=1 , can be reconstructed uniquely.
Proof. For brevity, and w.l.o.g., we prove the lemma for K = 2. We prove that there is a unique
choice of x1 and x2 such that ||x1 ? x0 ||0 ? S1 , ||x2 ? x1 ||0 ? S2 and obeying y1 = A1 x1 , y2 =
A2 x2 . We proceed by contradiction , and assume that there exist x?1 6= x1 and x?2 6= x2 supported
on T1? and T2? , respectively, such that y1 = A1 x1 = A1 x?1 , y2 = A2 x2 = A2 x?2 , ||x?1 ? x0 ||0 ? S1 ,
and ||x?2 ? x?1 ||0 ? S2 . Then ||A1 (x1 ? x?1 )||2 = 0. Using the lower bound in the RIP of A1 and the
2
fact that ?2S1 < 1, this leads to ||x1 ? x?1 ||2 = 0, i.e. x1 = x?1 , thus contradicting our assumption
?
that x1 6= x1 . Now consider the case of x2 and x?2 . We have
0 = A2 (x2 ? x?2 ) = A2 (x2 ? x1 + x1 ? x?2 ) = A2 (x2 ? x1 + x?1 ? x?2 ).
(2)
Using the lower bound in the RIP of A2 and the fact that ?2S2 < 1, this leads to
2
||x2 ? x1 + x?1 ? x?2 ||2 = 0, i.e. x2 ? x1 = x?2 ? x?1 , which implies x?2 = x2 , thus contradicting our assumption that x2 6= x?2 .
As in Cand`es and Tao?s work [2], this lemma only suggests what may be possible in terms of
recovery of (xk )K
k=1 through a combinatorial, brute-force approach. By imposing stricter conditions
K
on (?2Sk )K
k=1 , we can recover (xk )k=1 by solving a convex program. This is summarized in the
following theorem.
Theorem 2 (Exact Recovery in the Absence of Noise).
p
p
Let (?
xk )K
xk ? x?k?1 ||0 ? Sk for some
k=1 ? R be a sequence of R vectors such that, for each k, ||?
Sk < p/2. Suppose that the measurements yk = Ak x?k ? Rnk are given, such that nk < p, A1 ?
A2 , Ak = A2 for k = 3, ? ? ? , K and (Ak )K
k=1 satisfies ?Sk + ?2Sk + ?3Sk < 1 for k = 1, 2 ? ? ? , K.
Then, the sequence (?
xk )K
is
the
unique
minimizer
to the program (P1).
k=1
Proof. As before, we consider the case K = 2. The proof easily generalizes to the case of arbitrary
K. We can re-write the program as follows:
||x2 ? x1 ||1
||x1 ||
?
min ? 1 +
S1
S2
x1 ,x2
s.t.
A1 x1 = A1 x
?1 , A2 (x2 ? x1 ) = A2 (?
x2 ? x
?1 ),
(3)
where we have used the fact that A1 ? A2 : A2 x2 ? A1 x1 = A2 x
?2 ? A1 x?1 , which implies A2 (x2 ?
x1 ) = A2 (?
x2 ? x
?1 ).
Let x?1 and x?2 be the solutions to the above program. Let T1 = supp(?
x1 ) and ?T2 = supp(?
x2 ? x
?1 ).
Assume |T1 | ? S1 and |?T2 | ? S2 .
Key element of the proof: The key element of the proof is the existence of vectors u1 , u2 satisfying the exact reconstruction property (ERP) [10, 11]. It has been shown in [10] that given
?Sk + ?2Sk + ?3Sk < 1 for k = 1, 2:
3
1. hu1 , a1j i = sgn(x1,j ), for all j ? T1 , and hu2 , a2j i = sgn(x2,j ), for all j ? ?T2 .
2. |hu1 , a1j i| < 1, for all j ? T1c , and |hu2 , a2j i| < 1, for all j ? ?T2c .
Since x?1 and x?2 ? x?1 are feasible, we have
||x? ? x? ||
||?
x1 ||
||?
x2 ? x
?1 ||1
||x?1 ||1
?
?
+ 2? 1 1 ? ? 1 +
.
S1
S2
S1
S2
||x?1 ||1
?
S1
(4)
+
||x?2 ? x?1 ||1
1 X
1 X ?
?
=?
|?
x1,j + (x?1,j ? x?1,j )| + ?
|x |
S2
S1 j?T
S1 j?T c 1,j
+
1
?
S2
1
?
X
j??T2
+
x?1,j
?
? (?
x2,j
X
1
? x?1,j ) | + ?
|x?2,j ? x?1,j |
S2 j??T c
2
1 X ?
1 X
?
x1,j + (x?1,j ? x
?1,j )) + ?
sgn(?
x1,j )(?
x hu1 , a1j i
S1 j?T | {z }
S1 j?T c 1,j
1
+
|?
x2,j ? x
?1,j +
1
x?2,j
1
?
S2
1
hu1 ,a1j i
X
j??T2
sgn(?
x2,j ? x
?1,j )(?
x2,j ? x
?1,j + (x?2,j ? x?1,j ? (?
x2,j ? x
?1,j )))
|
{z
}
hu2 ,a2j i
X
1
?
(x? ? x?1,j )hu2 , a2j i
S2 j??T c 2,j
2
=
X
X
1 X
1
?
x
?1,j a1j i
x?1,j a1j ?
|?
x1,j | + ? hu1 ,
S1 j?T
S1
j?T1
j?J
1
{z
}
| {z } |
A1 x ?
1
+
A1 x
?1
X
X
1 X
1
?
(x?2,j ? x?1,j )a2j ?
|?
x2,j ? x
?1,j | + ? hu2 ,
(?
x2,j ? x?1,j )a2j i
S2 j??T
S2
j?J
j??T2
2
|
{z
} |
{z
}
?
A2 (x?
2 ?x1 )
=
A2 (?
x2 ??
x1 )
||?
x2 ? x?1 ||1
||?
x1 ||
? 1+
?
.
S1
S2
(5)
This implies that all of the inequalities in the derivation above must in fact be equalities. In particular,
X
1 X ?
1
?
|x1,j | + ?
|x? ? x?1,j |
S1 j?T c
S2 j??T c 2,j
1
2
=
X
1
1 X ?
?
x1,j hu1 , a1j i + ?
(x? ? x?1,j )hu2 , a2j i
S1 j?T c
S2 j??T c 2,j
?
X
1
1 X ?
?
|x1,j | |hu1 , a1j i| + ?
|x? ? x?1,j | |hu2 , a2j i| .
| {z }
| {z }
S1 j?T c
S2 j??T c 2,j
1
1
2
<1
2
<1
Therefore, x?1,j = 0 ?j ? T1c , and x?2,j ? x?1,j = 0 ?j ? ?T2c . Using the lower bounds in the RIP of
A1 and A2 leads to
0=
||A2 (x?2
0 = ||A1 (x?1 ? x
?1 )||2
? x?1 ? (?
x2 ? x?1 ))||2
?
?
?1 ||2
(1 ? ?2S1 ) ||x?1 ? x
x2 ? x
?1 )||2 ,
(1 ? ?2S2 ) ||x?2 ? x?1 ? (?
(6)
(7)
so that x?1 = x
?1 , and x?2 = x?2 . Uniqueness follows from simple convexity arguments.
A few remarks are in order. First, Theorem 2 effectively asserts that the program (P1) is equivalent
to sequentially solving (i.e. for k = 1, 2, ? ? ? , K) the following program, starting with x?0 the vector
of all zeros in Rp :
min xk ? x?k?1
s.t.
yk ? Ak x?k?1 = Ak (xk ? x?k?1 ), k = 1, 2, ? ? ? , K.
(8)
xk
1
4
Second, it is interesting and surprising that Theorem 2 would hold, if one naively applies standard
CS principles to our problem. To see this, if we let wk = xk ? xk?1 , then program (P1) becomes
K
X
||wk ||1
?
w1 ,??? ,wK
Sk
k=1
s.t.
min
y = Aw,
? ?
? ?
where w = (w1? , ? ? ? , wK
) ? RK?p , y = (y1? , ? ? ? , yK
) ?R
?
A1
? A2
A=?
? ..
.
AK
PK
0
A2
..
.
???
???
..
.
0
0
..
.
AK
???
AK
k=1
nk
(9)
and A is given by
?
?
?.
?
As K grows large, the columns of A become increasingly correlated or coherent, which intuitively
means that A would be far from satisfying RIP of any order. Yet, we get exact recovery. This is an
important reminder that the RIP is a sufficient, but not necessary condition for recovery.
Third, the assumption that A1 ? A2 , Ak = A2 for k = 3, ? ? ? , K makes practical sense as it
allows one to avoid the prohibitive storage and computational cost of generating several distinct
measurement matrices. Note that if a random A1 satisfies the RIP of some order and A1 ? A2 , then
A2 also satisfies the RIP (of lower order).
Lastly, the key advantage of dynamic CS recovery (P1) is the smaller number of measurements
required compared to the classical approach [2] which would solve K separate ?1 -minimization
problems. For each k = 1, ? ? ? , K, one would require nk ? CSk log(p/Sk ) measurements for
dynamic recovery, compared to nk ? CS1 log(p/S1 ) for classical recovery. Due to the hypothesis
of Sk ? S1 ? p, i.e., the sparse increments are small, we conclude that there are less number of
measurements required for dynamic CS.
We now move to the case where the measurements are perturbed by bounded noise. More specifically, we derive error bounds for a quadratically-constrained convex program for recovery of sequences of vectors with sparse increments in the presence of noise.
Theorem 3 (Conditionally Stable Recovery in Presence of Noise).
p
p
Let (?
xk )K
k=1 ? R be as stated in Theorem 2, and x0 be the vector of all zeros in R . Suppose that
nk
the measurements yk = Ak xk + ek ? R are given such that ||ek ||2 ? ?k and (Ak )K
k=1 satisfy
be
the
solution
to
the
program
(P2).
Finally,
let hk :=
?3Sk + 3?4Sk < 2, for each k. Let (x?k )K
k=1
(x?k ? x?k?1 ) ? (?
xk ? x?k?1 ), for k = 1, 2, ? ? ? , K, with the convention that x?0 := x?0 := 0 ? Rp .
Then, we have:
K
K
K
X
X
X
X
CSk Ak
2CSk ?k +
khk k2 ?
h?
(10)
k=1
k=2
k=1
?<k
2
where, for each k = 1, 2, ? ? ? , K, CSk is only a function of ?3Sk and ?4Sk .
Proof sketch. Cand`es et al.?s proof for stable recovery in the presence of bounded noise relies on
the so-called tube and cone constraints [3]. Our proof for Theorem 3 relies on generalization of
these two constraints. We omit some of the algebraic details of the proof as they can be filled in by
following the proof of [3] for the time-invariant case.
Generalized tube constraint: Let w
?k = x
?k ? x?k?1 , wk? = x?k ? x?k?1 , for k = 1, ? ? ? , K. The
generalized tube constraints are obtained using a simple application of the triangle inequality:
||A1 (w?1 ? w1? )||2
||A2 (w?2 ? w2? )||2
?
?
||Ak (w
?k ? wk? )||2
?
2?1
2?2 + ||A2 h1 ||2 and more generally,
X
2?k + Ak
h? , for k = 2, ? ? ? , K.
?<k
5
2
(11)
(12)
(13)
Generalized cone constraint: To obtain a generalization of the cone constraint in [3], we need
to account for the fact that the increments (xk ? xk?1 )K
k=1 (may) have different support sizes. The
resulting generalized cone constraint is as follows:
K
K
X
X
hk?Tkc 1
||hk?Tk ||1
?
?
?
,
(14)
S
Sk
k
k=1
k=1
where ?Tk = supp(?
?k?1 ). The proof proceeds along the lines of that presented in [3], with
? xk ? x
1+ 1/3
q
CSk = ?
.
1+?3S
1??4Sk ?
k
3
Equation (10) is an implicit bound: the second term in the inequality reflects the fact that, for a
given k, the error x?k ? x
?k depends on previous errors. Our bound proves a form of stability that
is conditional on the stability of previous estimates.The appeal of dynamic CS comes from the fact
that one may pick the constants CSk in the bound above to be much smaller that those from the
corresponding conventional CS bound [3] (Equation (10) without the second term). This ensures
that the errors do not propagate in an unbounded manner. One may obtain sharper bounds using
techniques as in [12]. In the next section, we use simulations to compare explicitly the average
mean-squared error (MSE) of conventional CS and our algorithm.
4 Experiments/Simulations
We ran a series of numerical experiments to assess the ability of the convex programs introduced
to recover signals with time-varying sparsity. In the absence of noise, the experiments result in
probability-of-recovery surfaces for the dynamic CS problem, which generalize the traditional recovery curves of CS. In the presence of noise, we compare dynamic CS to conventional CS in terms
of their reconstruction error as a function of signal-to-noise-ratio (SNR). We also show an application to real video data. All optimization problems were solved using CVX, a package for specifying
and solving convex programs [13, 14].
4.1 Simulated noiseless data
Experimental set-up:
1. Select nk , for k = 1, ? ? ? , K, and p, so that the Ak ?s are nk ? p matrices; sample Ak with
independent Gaussian entries, for k = 1, 2, ? ? ? , K.
2. Select S1 = ?s1 ? p?, s1 ? (0, 1), and Sk = ?s2 ? p?, s2 ? (0, 1), for k = 2, ? ? ? , K.
3. Select T1 of size S1 uniformly at random and set x
?1,j = 1 for all j ? T1 , and 0 otherwise;
for k = 2, ? ? ? , K, select ?Tk = supp(?
xk ? x
?k?1 ) of size Sk uniformly at random and set
x
?k,j ? x
?k?1,j = 1 for all j ? ?Tk , and 0 otherwise.
4. Make yk = Ak x
?k , for k = 1, 2, ? ? ? , K; solve the program (P1) to obtain (x?k )K
k=1 .
? K
5. Compare (?
xk )K
k=1 to (xk )k=1 .
6. Repeat 100 times for each (s1 , s2 ).
We compare dynamic CS to conventional CS applied independently at each k. Figure 1 shows
results for nk = 100, p = 200, and K = 2. We can infer the expected behavior for larger values of
K from the case K = 2 and from the theory developed above (see remarks below).
The probability of recovery for conventional CS is 1 on the set {(s1 , s2 ) : s1 + (K ? 1)s2 ?
s? }, and 0 on its complement, where s? is the sparsity level at which a phase transition occurs in
the conventional CS problem [2]. The figure shows that, when the measurement matrices Ak , for
k = 2, ? ? ? , K are derived from A1 as assumed in Theorem 1, dynamic CS (DCS 1) outperforms
conventional CS (CCS). However, when we used different measurement matrices (DCS 2), we see
that there is an asymmetry between s1 and s2 , which is not predicted by our Theorem 1. Intuitively,
this is because for small s2 , the program (P1) operates in a regime where we have not only one
but multiple measurements to recover a given sparse vector [15]. Program (P1) is equivalent to
sequential CS. Therefore, we expect the behavior of conventional CS to persist for larger K.
6
CCS
DCS 1
DCS 2
0.3
0.3
0.3
s1
0.4
s1
0.4
s1
0.4
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0.1
0.2
0.3
s2
0.4
0.1
0.2
0.3
s2
0.4
0
0.1
0.2
0.3
s2
0.4
Figure 1: Probability of recovery maps as a function of s1 and s2 .
4.2 Simulated noisy data
The experimental set-up differs slightly from the one of the noiseless case. In Step 2, we fix constant
values for S1 and Sk , k = 2, ? ? ? , K. Moreover, in Step 4, we form yk = Ak xk + ek , where the ek ?s
are drawn uniformly in (??, ?). In Step 6, we repeat the experiment 100 times for each ?. In our
experiments, we used n1 = 100, S1 = 5, n2 = 20, Sk = 1, for k = 2, ? ? ? , K, and p = 200. We
report results for K = 2 and K = 10, and choose values of ? resulting in SNRs in the range [5, 30]
dB, in increments of 5 dB.
2
1 PK
Figure 2 displays the average MSE given by 10 ? log10 ( K
xk ? x?k ||2 ) of conventional CS
k=1 ||?
and dynamic CS as a function of SNR. The Figure shows that the proposed algorithm outperforms
conventional CS, and is robust to noise.
Average MSE, K = 2
Average MSE, K = 10
4
6
Conventional CS
Dynamic CS
4
0
MSE (dB)
MSE (dB)
2
?2
?4
?6
?8
5
2
0
?2
?4
10
15
20
SNR (dB)
25
?6
5
30
10
15
20
SNR (dB)
25
30
Figure 2: Average MSE as a function of SNR.
4.3 Real video data
We consider the problem of recovering the first 10 frames of a real video using our dynamic CS
algorithm, and conventional CS applied to each frame separately. In both cases, we assume the
absence of noise. We use a video portraying a close-up of a woman engaged in a telephonic conversation [16]. The video has a frame rate of 12 Hz and a total of 150 frames, each of size 176 ? 144.
Due to computational constraints, we downsampled each frame by a factor of 3 in each dimension.
We obtained measurements in the wavelet domain by performing a two-level decomposition of each
frame using Daubechies-1 wavelet.
2
xk ?x?
1 P10 ||?
k ||2
In Table 1, we report the negative of the normalized MSE given by ?10 ? log10 ( 10
)
k=1
||?
xk ||22
in dB for various (n1 , n2 ) measurement pairs (nk = n2 , for k = 3, ? ? ? , 10). Larger numbers indicate better reconstruction accuracy. The table shows that, for all (n1 , n2 ) considered, dynamic CS
outperforms conventional CS. The average performance gap across (n1 , n2 ) pairs is approximately
7 dB. Interestingly, for sufficient number of measurements, dynamic CS improves as the video progresses. We observed this phenomenon in the small-s2 regime of the simulations. Figure 3 shows
the reconstructed frames highlighted in Table 1. The frames reconstructed using dynamic CS are
more appealing visually than their conventional CS counterparts.
7
Table 1: Normalized negated MSE in dB for frames 1, 5, 10, and average over all 10 frames. Each
frame consist of ? 3000 pixels. Each row of the table corresponds to a different (n1 , n2 ) pair (refer
to text). Larger numbers indicate better reconstruction accuracy.
Frame 1
Frame 5
Frame 10
Avg. (10 frames)
CCS DCS CCS DCS CCS DCS CCS
DCS
(2400,2400) 27.8 27.8
28.5
38
28
41.1
28.2
35
(2000,2000) 22.4 22.4
22.3 31.3
22.9 35.6
22.8
28.9
(2400,1200) 27.8 27.8
15.2 24.2
14.8 25.4
15.9
25.5
(1600,1600) 19.1 19.1
18.9
25
19.8 29.7
19.1
24.1
(1600,800)
19.2 19.2
8.4
17.6
9.3
16.7
8.4
17.8
Frame 5
Frame 10
DCS
CCS
Original
Frame 1
Figure 3: Comparison of frames reconstructed using dynamic CS and conventional CS, (n1 , n2 ) =
(2000, 2000).
5 Discussion
In this paper, we proved rigorous guarantees for convex programs for recovery of sequences of vectors with sparse increments, both in the absence and in the presence of noise. Our formulation of
the dynamic CS problem is more general than the empirically-motivated solutions proposed in the
literature, e.g. [5, 6]. Indeed, we only require that x1 is sparse, as well as the increments. Therefore,
there may exist values of k such that xk is not a sparse vector. We supplemented our theoretical
analysis with simulation experiments and an application to real video data. In the noiseless case, we
introduced probability-of-recovery surfaces which generalize traditional CS recovery curves. The
recovery surface showed that dynamic CS significantly outperforms conventional CS, especially for
large sequences (large K). In the noisy case, simulations showed that dynamic CS also outperforms
conventional CS for SNR values ranging from 5 to 30 dB. Our results on real video data demonstrated that dynamic CS outperforms conventional CS in terms of visual appeal of the reconstructed
frames, and by an average MSE gap of 7dB.
8
References
[1] Compressive
sensing
resources,
http://dsp.rice/edu/cs/.
rice
university.
Rice
University,
[2] E.J. Cand`es and T. Tao. Decoding by linear programming. Information Theory, IEEE Transactions on, 51(12):4203?4215, 2005.
[3] E.J. Cand`es, J.K. Romberg, and T. Tao. Stable signal recovery from incomplete and inaccurate
measurements. Communications on pure and applied mathematics, 59(8):1207?1223, 2006.
[4] D.L. Donoho. Compressed sensing. Information Theory, IEEE Transactions on, 52(4):1289?
1306, 2006.
[5] J. Ziniel, L.C. Potter, and P. Schniter. Tracking and smoothing of time-varying sparse signals
via approximate belief propagation. In Signals, Systems and Computers (ASILOMAR), 2010
Conference Record of the Forty Fourth Asilomar Conference on, pages 808?812. IEEE, 2010.
[6] M. Salman Asif and J. Romberg. Dynamic updating for ?1 -minimization. Selected Topics in
Signal Processing, IEEE Journal of, 4(2):421?434, 2010.
[7] H. Jung and J.C. Ye. Motion estimated and compensated compressed sensing dynamic magnetic resonance imaging: What we can learn from video compression techniques. International
Journal of Imaging Systems and Technology, 20(2):81?98, 2010.
[8] J.W. Phillips, R.M. Leahy, and J.C. Mosher. Meg-based imaging of focal neuronal current
sources. Medical Imaging, IEEE Transactions on, 16(3):338?348, 1997.
[9] M. Kolar, L. Song, A. Ahmed, and E.P. Xing. Estimating time-varying networks. The Annals
of Applied Statistics, 4(1):94?123, 2010.
[10] E. Cand`es, J. Romberg, and T. Tao. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory, June 2004.
Submitted.
[11] E. Cand`es and T. Tao. Near optimal signal recovery from random projections: Universal
encoding strategies? IEEE Trans. Inform. Theory, October 2004. Submitted.
[12] E.J. Cand`es. The restricted isometry property and its implications for compressed sensing.
Comptes Rendus Mathematique, 346(9):589?592, 2008.
[13] M. Grant and S. Boyd. CVX: Matlab software for disciplined convex programming, version
1.22. http://cvxr.com/cvx, May 2012.
[14] M. Grant and S. Boyd. Graph implementations for nonsmooth convex programs. In V. Blondel,
S. Boyd, and H. Kimura, editors, Recent Advances in Learning and Control, Lecture Notes in
Control and Information Sciences, pages 95?110. Springer-Verlag Limited, 2008.
[15] S.F. Cotter, B.D. Rao, K. Engan, and K. Kreutz-Delgado. Sparse solutions to linear inverse problems with multiple measurement vectors. Signal Processing, IEEE Transactions
on, 53(7):2477?2488, 2005.
[16] Softage
video
codec
demo
download
page.
Softage,
http:www.softage.ru/products/video-codec/uncompressed/suzie.avi.
9
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incorporate:1 phenomenon:1 correlated:1 |
4,007 | 4,627 | Variational Inference for Crowdsourcing
Qiang Liu
ICS, UC Irvine
[email protected]
Jian Peng
TTI-C & CSAIL, MIT
[email protected]
Alexander Ihler
ICS, UC Irvine
[email protected]
Abstract
Crowdsourcing has become a popular paradigm for labeling large datasets. However, it has given rise to the computational task of aggregating the crowdsourced
labels provided by a collection of unreliable annotators. We approach this problem by transforming it into a standard inference problem in graphical models,
and applying approximate variational methods, including belief propagation (BP)
and mean field (MF). We show that our BP algorithm generalizes both majority voting and a recent algorithm by Karger et al. [1], while our MF method is
closely related to a commonly used EM algorithm. In both cases, we find that the
performance of the algorithms critically depends on the choice of a prior distribution on the workers? reliability; by choosing the prior properly, both BP and MF
(and EM) perform surprisingly well on both simulated and real-world datasets,
competitive with state-of-the-art algorithms based on more complicated modeling
assumptions.
1
Introduction
Crowdsourcing has become an efficient and inexpensive way to label large datasets in many application domains, including computer vision and natural language processing. Resources such as
Amazon Mechanical Turk provide markets where the requestors can post tasks known as HITs (Human Intelligence Tasks) and collect large numbers of labels from hundreds of online workers (or
annotators) in a short time and with relatively low cost.
A major problem of crowdsoucing is that the qualities of the labels are often unreliable and diverse,
mainly since it is difficult to monitor the performance of a large collection of workers. In the extreme, there may exist ?spammers?, who submit random answers rather than good-faith attempts to
label, or even ?adversaries?, who may deliberately give wrong answers, either due to malice or to a
misinterpretation of the task. A common strategy to improve reliability is to add redundancy, such
as assigning each task to multiple workers, and aggregate the workers? labels. The baseline majority
voting heuristic, which simply assigns the label returned by the majority of the workers, is known to
be error-prone, because it counts all the annotators equally. In general, efficient aggregation methods
should take into account the differences in the workers? labeling abilities.
A principled way to address this problem is to build generative probabilistic models for the annotation processes, and assign labels using standard inference tools. A line of early work builds simple
models characterizing the annotators using confusion matrices, and infers the labels using the EM
algorithm [e.g., 2, 3, 4]. Recently however, significant efforts have been made to improve performance by incorporating more complicated generative models [e.g., 5, 6, 7, 8, 9]. However, EM is
widely criticized for having local optimality issues [e.g., 1]; this raises a potential tradeoff between
more dedicated exploitation of the simpler models, either by introducing new inference tools or fixing local optimality issues in EM, and the exploration of larger model space, usually with increased
computational cost and possibly the risk of over-fitting.
On the other hand, variational approaches, including the popular belief propagation (BP) and mean
field (MF) methods, provide powerful inference tools for probabilistic graphical models [10, 11].
1
These algorithms are efficient, and often have provably strong local optimality properties or even
globally optimal guarantees [e.g., 12]. To our knowledge, no previous attempts have taken advantage
of variational tools for the crowdsourcing problem. A closely related approach is a message-passingstyle algorithm in Karger et al. [1] (referred to as KOS in the sequel), which the authors asserted
to be motivated by but not equivalent to standard belief propagation. KOS was shown to have
strong theoretical guarantees on (locally tree-like) random assignment graphs, but does not have an
obvious interpretation as a standard inference method on a generative probabilistic model. As one
consequence, the lack of a generative model interpretation makes it difficult to either extend KOS to
more complicated models or adapt it to improve its performance on real-world datasets.
Contribution. In this work, we approach the crowdsourcing problems using tools and concepts from
variational inference methods for graphical models. First, we present a belief-propagation-based
method, which we show includes both KOS and majority voting as special cases, in which particular prior distributions are assumed on the workers? abilities. However, unlike KOS our method is
derived using generative principles, and can be easily extended to more complicated models. On
the other side, we propose a mean field method which we show closely connects to, and provides
an important perspective on, EM. For both our BP and MF algorithms (and consequently for EM
as well), we show that performance can be significantly improved by using more carefully chosen
priors. We test our algorithms on both simulated and real-world datasets, and show that both BP
and MF (or EM), with carefully chosen priors, is able to perform competitively with state-of-the-art
algorithms that are based on far more complicated models.
2
Background
Assume there are M workers and N tasks with binary labels {?1}. Denote by zi ? {?1}, i ? [N ]
the true label of task i, where [N ] represents the set of first N integers; Nj is the set of tasks labeled
by worker j, and Mi the workers labeling task i. The task assignment scheme can be represented by
a bipartite graph where an edge (i, j) denotes that the task i is labeled by the worker j. The labeling
results form a matrix L ? {0, ?1}N ?M , where Lij ? {?1} denotes the answer if worker j labels
task i, and Lij = 0 if otherwise. The goal is to find an optimal estimator
z? of the true labels z given
P
the observation L, minimizing the average bit-wise error rate N1 i?[N ] prob[?
zi 6= zi ].
We assume that all the tasks have the same level of difficulty, but that workers may have different
predictive abilities. Following Karger et al. [1], we initially assume that the ability of worker j is
measured by a single parameter qj , which corresponds to their probability of correctness: qj =
prob[Lij = zi ]. More generally, the workers? abilities can be measured by a confusion matrix, to
which our method can be easily extended (see Section 3.1.2).
The values of qj reflect the abilities of the workers: qj ? 1 correspond to experts that provide
reliable answers; qj ? 1/2 denote spammers that give random labels independent of the questions;
and qj < 1/2 denote adversaries that tend to provide opposite answers. Conceptually, the spammers
and adversaries should be treated differently: the spammers provide no useful information and only
degrade the results, while the adversaries actually carry useful information, and can be exploited to
improve the results if the algorithm can identify them and flip their labels. We assume the qj of all
workers are drawn independently from a common prior p(qj |?), where ? are the hyper-parameters.
To avoid the cases when adversaries and/or spammers overwhelm the system, it is reasonable to
require that E[qj |?] > 1/2. Typical priors include the Beta prior p(qj |?) ? qj??1 (1 ? qj )??1 and
discrete priors, e.g., the spammer-hammer model, where qj ? 0.5 or qj ? 1 with equal probability.
Majority Voting. The majority voting (MV) method aggregates the workers? labels by
X
z?imajority = sign[
Lij ].
j?Mi
The limitation of MV is that it weights all the workers equally, and performs poorly when the
qualities of the workers are diverse, especially when adversarial workers exist.
Expectation Maximization. Weighting the workers properly requires estimating their
P abilities qj ,
usually via a maximum a posteriori estimator, q? = arg max log p(q|L, ?) = log z p(q, z|L, ?).
This is commonly solved using an EM algorithm treating the z as hidden variables, [e.g., 2, 3, 4].
Assuming a Beta(?, ?) prior on qj , EM is formulated as
2
P
E-step: ?i (zi ) ?
Y
?
q?j ij (1
1??ij
? q?j )
,
M-step: q?j =
i?Nj
?i (Lij ) + ? ? 1
|Nj | + ? + ? ? 2
j?Mi
,
(1)
where ?ij = I[Lij = zi ]; the z?i is then estimated via z?i = arg maxzi ?i (zi ). Many approaches have
been proposed to improve this simple EM approach, mainly by building more complicated models.
Message Passing. A rather different algorithm in a message-passing style is proposed by Karger,
Oh and Shah [1] (referred to as KOS in the sequel). Let xi?j and yj?i be real-valued messages
0
from tasks to workers and from workers to tasks, respectively. Initializing yj?i
randomly from
0
Normal(1, 1) or deterministically by yj?i = 1, KOS updates the messages at t-th iteration via
X
X
t+1
Lij 0 yjt 0 ?i ,
Li0 j xt+1
(2)
xt+1
yj?i
=
i?j =
i0 ?j ,
j 0 ?Mi\j
i0 ?Nj\i
t
and the labels are estimated via s?ti = sign[?
xti ], where x
?ti =
j?Mi Lij yj?i . Note that the 0th
0
iteration of KOS reduces to majority voting when initialized with yj?i = 1. KOS has surprisingly
nice theoretical properties on locally tree-like assignment graphs: its error rate is shown to scale
in the same manner as an oracle lower bound that assumes the true qj are known. Unfortunately,
KOS is not derived using a generative model approach under either Bayesian or maximum likelihood principles, and hence is difficult to extend to more general cases, such as more sophisticated
worker-error models (Section 3.1.2) or other features and side information (see appendix). Given
that the assumptions made in Karger et al. [1] are restrictive in practice, it is unclear whether the
theoretical performance guarantees of KOS hold in real-world datasets. Additionally, an interesting phase transition phenomenon was observed in Karger et al. [1] ? the performance of KOS was
shown to degenerate, sometimes performing even worse than majority voting when the degrees of
the assignment graph (corresponding to the number of annotators per task) are small.
P
3
Crowdsourcing as Inference in a Graphical Model
We present our main framework in this section, transforming the labeling aggregation problem into
a standard inference problem on a graphical model, and proposing a set of efficient variational
methods, including a belief propagation method that includes KOS and majority voting as special
cases, and a mean field method, which connects closely to the commonly used EM approach.
To start, the joint posterior distribution of workers? abilities q = {qj : j ? [M ]} and the true labels
z = {zi : i ? [N ]} conditional on the observed labels L and hyper-parameter ? is
Y
Y
Y
c
p(z, q|L, ?) ?
p(qj |?)
p(Lij |zi , qj ) =
p(qj |?)qj j (1 ? qj )?j ?cj ,
j?[M ]
i?Nj
j?[M ]
P
where ?j = |Nj | is the number of predictions made by worker j and cj := i?Nj I[Lij = zi ] is
the number of j?s predictions that are correct. By standard Bayesian arguments, one can show that
the optimal estimator of z to minimize the bit-wise error rate is given by Z
X
z?i = arg max p(zi |L, ?)
where
p(zi |L, ?) =
p(z, q|L, ?)dq.
(3)
zi
z[N ]\i
q
Note that the EM algorithm (1), which maximizes rather than marginalizes qj , is not equivalent to
the Bayesian estimator (3), and hence is expected to be suboptimal in terms of error rate. However,
calculating the marginal p(zi |L, ?) in (3) requires integrating all q and summing over all the other zi ,
a challenging computational task. In this work we use belief propagation and mean field to address
this problem, and highlight their connections to KOS, majority voting and EM.
3.1
Belief Propagation, KOS and Majority Voting
It is difficult to directly apply belief propagation to the joint distribution p(z, q|L, ?), since it is
a mixed distribution of discrete variables z and continuous variables q. We bypass this issue by
directly integrating over qj , yielding a marginal posterior distribution over the discrete variables z,
Z
Y Z 1
def Y
c
p(z|L, ?) = p(z, q|L, ?)dq =
p(qj |?)qj j (1 ? qj )?j ?cj =
?j (zNj ), (4)
j?[M ]
0
j?[M ]
3
where ?j (zNj ) is the local factor contributed by worker j due to eliminating qj , which couples
all the tasks zNj labeled by j; here we suppress the dependency of ?j on ? and L for notational
simplicity. A key perspective is that we can treat p(z|L, ?) as a discrete Markov random field, and
re-interpret the bipartite assignment graph as a factor graph [13], with the tasks mapping to variable
nodes and workers to factor nodes. This interpretation motivates us to use a standard sum-product
belief propagation method, approximating p(zi |L, ?) with ?beliefs? bi (zi ) using messages mi?j
and mj?i between the variable nodes (tasks) and factor nodes (workers),
Y
mtj 0 ?i (zi ),
(5)
From tasks to workers:
mt+1
i?j (zi ) ?
j 0 ?Mi/j
mt+1
j?i (zi ) ?
From workers to tasks:
X
zNj/i
bt+1
(zi ) ?
i
Calculating the beliefs:
Y
Y
?j (zNj )
mt+1
i0 ?j (zi0 ),
(6)
i0 ?Nj
mt+1
j?i (zi ).
(7)
j?Mi
At the end of T iterations, the labels are estimated via z?it = arg maxzi bti (zi ). One immediate
difference between BP (5)-(7) and the KOS message passing (2) is that the messages and beliefs in
(5)-(7) are probability tables on zi , i.e., mi?j = [mi?j (+1), mi?j (?1)], while the messages in
(2) are real values. For binary labels, we will connect the two by rewriting the updates (5)-(7) in
terms of their (real-valued) log-odds, a standard transformation used in error-correcting codes.
The BP updates above appear computationally challenging, since step (6) requires eliminating a
high-order potential ?(zNj ), costing O(2?j ) in general. However, note that ?(zNj ) in (4) depends
on zNj only through cj , so that (with a slight abuse of notation) it can be rewritten as ?(cj , ?j ). This
structure enables us to rewrite the BP updates in a more efficient form (in terms of the log-odds):
Theorem 3.1.
Let
x
?i = log
bi (+1)
,
bi (?1)
xi?j = log
mi?j (+1)
,
mi?j (?1)
and
yj?i = Lij log
mj?i (+1)
.
mi?j (?1)
Then, sum-product BP (5)-(7) can be expressed as
P?j ?1
t+1
X
t+1
t
k=0 ?(k + 1, ?j ) ek
,
xt+1
=
L
y
,
y
=
log
0
P?j ?1
ij j ?i
i?j
j?i
t+1
k=0 ?(k, ?j ) ek
j 0 ?Mi\j
(8)
P
t+1
and x
?t+1
=
= 0, . . . , Nj ? 1, are the
i
j?Mi Lij yi?j , where the terms ek for k
0
0
elementary symmetric polynomials in variables {exp(Li j xi ?j )}i0 ?Nj\i , that is, ek =
P
Q
0
0
?it = sign[?
xti ].
s : |s|=k
i0 ?s exp(Li j xi ?j ). In the end, the true labels are decoded as z
The terms ek can be efficiently calculated by divide & conquer and the fast Fourier transform in
O(?j (log ?j )2 ) time (see appendix), making (8) much more efficient than (6) initially appears.
Similar to sum-product, one can also derive a max-product BP to find the joint maximum a posteriori
configuration, z? = arg maxz p(z|L, ?), which minimizes the block-wise error rate prob[?i : zi 6=
z?i ] instead of the bit-wise error rate. Max-product BP can be organized similarly to (8), with the
slightly lower computational cost of O(?j log ?j ); see appendix for details and Tarlow et al. [14]
for a general discussion on efficient max-product BP with structured high-order potentials. In this
work, we focus on sum-product since the bit-wise error rate is more commonly used in practice.
3.1.1
The Choice of Algorithmic Priors and connection to KOS and Majority Voting
Before further discussion, we should be careful to distinguish between the prior on qj used in our
algorithm (the algorithmic prior) and, assuming the model is correct, the true distribution of the qj
in the data generating process (the data prior); the algorithmic and data priors often do not match.
In this section, we discuss the form of ?(cj , ?j ) for different choices of algorithmic priors, and in
particular show that KOS and majority voting can be treated as special cases of our belief propagation (8) with the most ?uninformative? and most ?informative? algorithmic priors, respectively. For
more general priors that may not yield a closed form for ?(cj , ?j ), one can calculate ?(cj , ?j ) by
numerical integration and store them in a (? + 1) ? ? table for later use, where ? = maxj?[M ] ?j .
4
Beta Priors. If p(qj |?) ? qj??1 (1 ? qj )??1 , we have ?(cj , ?j ) ? B(? + cj , ? + ?j ? cj ), where
B(?, ?) is the Beta function. Note that ?(cj , ?j ) in this case equals (up to a constant) the likelihood
of a Beta-binomial distribution.
Discrete Priors. If p(qj |?) has non-zero probability mass on
Ponly finite points, that is, prob(qj =
qP
?k ) = pk , k ? [K], where 0 ? q?k ? 1, 0 ? pk ? 1 and k pk = 1, then we have ?(cj , ?j ) =
c
?kj (1 ? q?k )?j ?cj . One can show that log ?(cj , ?j ) in this case is a log-sum-exp function.
k pk q
Haldane Prior. The Haldane prior [15] is a special discrete prior that equals either 0 or 1 with equal
probability, that is, prob[qj = 0] = prob[qj = 1] = 1/2. One can show that in this case we have
?(0, ?j ) = ?(?j , ?j ) = 1 and ?(cj , ?j ) = 0 otherwise.
Claim 3.2. The BP update in (8) with Haldane prior is equivalent to KOS update in (2).
Proof. Just substitute the ?(cj , ?j ) of Haldane prior shown above into the BP update (8).
The Haldane prior can also be treated as a Beta(, ) prior with ? 0+ , or equivalently an improper
prior p(qj ) ? qj?1 (1 ? qj )?1 , whose normalization constant is infinite. One can show that the
Haldane prior is equivalent to putting a flat prior on the log-odds log[qj /(1 ? qj )]; also, it has
the largest variance (and hence is ?most uninformative?) among all the possible distributions of qj .
Therefore, although appearing to be extremely dichotomous, it is well known in Bayesian statistics
as an uninformative prior of binomial distributions. Other choices of objective priors include the
uniform prior Beta(1, 1) and Jeffery?s prior Beta(1/2, 1/2) [16], but these do not yield the same
simple linear message passing form as the Haldane prior.
Unfortunately, the use of Haldane prior in our problem suffers an important symmetry breaking issue: if the prior is symmetric, i.e., p(qj |?) = p(1 ? qj |?), the true marginal posterior distribution of
zj is also symmetric, i.e., p(zj |L, ?) = [1/2; 1/2], because jointly flipping the sign of any configuration does not change its likelihood. This makes it impossible to break the ties when decoding zj .
Indeed, it is not hard to observe that xi?j = yj?i = 0 (corresponding to symmetric probabilities)
is a fixed point of the KOS update (2). The mechanism of KOS for breaking the symmetry seems to
rely solely on initializing to points that bias towards majority voting, and the hope that the symmetric
distribution is an unstable fixed point. In experiments, we find that the use of symmetric priors usually leads to degraded performance when the degree of the assignment graph is low, corresponding
to the phase transition phenomenon discussed in Karger et al. [1]. This suggests that it is beneficial
to use asymmetric priors with E[qj |?] > 1/2, to incorporate the prior knowledge that the majority of
workers are non-adversarial. Interestingly, it turns out that majority voting uses such an asymmetric
prior, but unfortunately corresponding to another unrealistic extreme.
Deterministic Priors. A deterministic prior is a special discrete distribution that equals a single
point deterministically, i.e., prob[qj = q?|?] = 1, where 0 ? q? ? 1. One can show that log ? in this
case is a linear function, that is, log ?(cj , ?j ) = cj logit(?
q ) + const.
Claim 3.3. The BP update (8) with deterministic priors satisfying q? > 1/2 terminates at the first
iteration and finds the same solution as majority voting.
Proof. Just note that log ?(cj , ?j ) = cj logit(?
q ) + const, and logit(?
q ) > 0 in this case.
The deterministic priors above have the opposite properties to the Haldane prior: they can be also
treated as Beta(?, ?) priors, but with ? ? +? and ? > ?; these priors have the smallest variance
(equal to zero) among all the possible qj priors.
In this work, we propose to use priors that balance between KOS and majority voting. One reasonable choice is Beta(?, 1) prior with ? > 1 [17]. In experiments, we find that a typical choice of
Beta(2, 1) performs surprisingly well even when it is far from the true prior.
3.1.2
The Two-Coin Models and Further Extensions
We previously assumed that workers? abilities are parametrized by a single parameter qj . This is
likely to be restrictive in practice, since the error rate may depend on the true label value: false
positive and false negative rates are often not equal. Here we consider the more general case, where
the ability of worker j is specified by two parameters, the sensitivitiy sj and specificity tj [2, 4],
tj = prob[Lij = ?1|zi = ?1].
sj = prob[Lij = +1|zi = +1],
5
A typical prior on sj and tj are two independent Beta distributions. One can show that ?(zNj ) in
this case equals a product of two Beta functions, and depends on zNj only through two integers, the
true positive and true negative counts. An efficient BP algorithm similar to (8) can be derived for
the general case, by exploiting the special structure of ?(zNj ). See the Appendix for details.
One may also try to derive a two-coin version of KOS, by assigning two independent Haldane priors
on sj and tj ; it turns out that the extended version is exactly the same as the standard KOS in (2). In
this sense, the Haldane prior is too restrictive for the more general case. Several further extensions
of the BP algorithm that are not obvious for KOS, for example the case when known features of the
tasks or other side information are available, are discussed in the appendix due to space limitations.
3.2
Mean Field Method and Connection of EM
We next present a mean field method for computing the marginal p(zi |L, ?) in (3), and show its
close connection to EM. In contrast to the derivation of BP, here we directly work on the mixed joint
posterior p(z, q|L,
Q
Q ?). Let us approximate p(z, q|L, ?) with a fully factorized distribution b(z, q) =
i?[N ] ?i (zi )
j?[M ] ?j (qj ). The best b(z, q) should minimize the KL divergence,
X
X
KL[b(z, q) || p(z, q|L, ?)] = ?Eb [log p(z, q|L, ?)] ?
H(?i ) ?
H(?j ).
i?[N ]
j?[M ]
where Eb [?] denotes the expectation w.r.t. b(z, q), and H(?) the entropy functional. Assuming the
algorithmic prior of Beta(?, ?), one crucial property of the KL objective in this case is that the
optimal {?j? (qj )} is guaranteed to be a Beta distribution as well. Using a block coordinate descent
method that alternatively optimizes {?i (zi )} and {?j (qj )}, the mean field (MF) update is
Y ? 1??
Updating ?i : ?i (zi ) ?
aj ij bj ij ,
(9)
j?Mi
Updating ?j : ?j (qj ) ? Beta(
X
i?Nj
?i (Lij ) + ?,
X
?i (?Lij ) + ?),
(10)
i?Nj
where aj = exp(E?j [ln qj ]) and bj = exp(E?j [ln(1 ? qj )]). The aj and bj can be exactly calculated
by noting that E[ln x] = Digamma(?)?Digamma(?+?) if x ? Beta(?, ?). One can also instead
calculate the first-order approximation of aj and bj : by Taylor expansion, one have ln(1 + x) ? x;
taking x = (qj ? q?j )/?
qj , where q?j = E?j [qj ], and substituting it into the definition of aj and bj ,
one get aj ? q?j and bj ? 1 ? q?j ; it gives an approximate MF (AMF) update,
P
Y ?
i?Nj ?i (Lij ) + ?
ij
1??ij
Updating ?i : ?i (zi ) ?
q?j (1 ? q?j )
, Updating ?j : q?j =
. (11)
|Nj | + ? + ?
j?Mi
The update (11) differs from EM (1) only in replacing ??1 and ? ?1 with ? and ?, corresponding to
replacing the posterior mode of the Beta distribution with its posterior mean. This simple (perhaps
trivial) difference plays a role of Laplacian smoothing, and provides insights for improving the
performance of EM. For example, note that the q?j in the M-step of EM could be updated to 0 or 1 if
? = 1 or ? = 1, and once this happens, the q?j is locked at its current value, causing EM to trapped
at a local maximum. Update (11) can prevent q?j from becoming 0 or 1, avoiding the degenerate
case. One can of course interpret (11) as EM with prior parameters ?0 = ? + 1, and ? 0 = ? + 1;
under this interpretation, it is advisable to choose priors ?0 > 1 and ? 0 > 1 (corresponding to a less
common or intuitive ?informative? prior).
We should point out that it is widely known that EM can be interpreted as a coordinate descent on
variational objectives [18, 11]; our derivation differs in that we marginalize, instead of maximize,
over qj . Our first-order approximation scheme is also similar to the method by Asuncion [19]. One
can also extend this derivation to two-coin models with independent Beta priors, yielding the EM
update in Dawid and Skene [2]. On the other hand, discrete priors do not seem to lead to interesting
algorithms in this case.
4
Experiments
In this section, we present numerical experiments on both simulated and real-world Amazon Mechanical Turk datasets. We implement majority voting (MV), KOS in (2), BP in (8), EM in (1) and
6
?1
Log?error
?0.5
KOS
MV
Oracle
BP?True
BP?Beta(2,1)
BP?Beta(1,1)
EM?Beta(2,1)
AMF?Beta(2,1)
?0.5
?1.5
?1
?1
?2
?1.5
?1.5
?2.5
?2
?2
?3
?2.5
5
10 15 20 25
(?xed ? = 5)
2
5
10
? (?xed = 15)
?2.5
5
10
( = ?)
15
Figure 1: The performance of the algorithms as the degrees of the assignment graph vary; the left
degree ` denotes the number of workers per task, and the right degree ? denotes the number of tasks
per worker. The true data prior is prob[qj = 0.5] = prob[qj = 0.9] = 1/2.
Log?error
?0.5
?0.5
?0.5
?1.5
?1
?1
?1.5
?2
?2.5
KOS
MV
Oracle
BP?True
BP?Beta(2,1)
EM?Beta(2,1)
AMF?Beta(2,1)
(a) Beta prior (fixed ?/(? + ?) = 0.6)
?1
?1.5
?2.5
1 2 3 4 5 6 7 8 9 10
?+?
?0.5
?1.5
?2
0.55
0.6
0.65
?/(? + ? )
0.4
0.6
0.8
Percentage of Spammers
0.7
(b) Beta prior (fixed ? + ? = 1)
(c) Spammer-hammer prior
0
0.1
0.2
Percentage of Adversaries
(d) Adversary-spammer-hammer prior
Figure 2: The performance on data generated with different qj priors on (9,9)-regular random graphs.
?
(a) Beta prior with fixed ?+?
= 0.6. (b) Beta prior with fixed ? + ? = 1. (c) Spammer-hammer
prior, prob[qj = 0.5] = 1?prob[qj = 0.9] = p0 , with varying p0 . (d) Adversary-spammer-hammer
prior, prob[qj = 0.1] = p0 , prob[qj = 0.5] = prob[qj = 0.9] = (1 ? p0 )/2 with varying p0 .
its variant AMF in (11). The exact MF (9)-(10) was implemented, but is not reported because its
performance is mostly similar to AMF (11)
P in terms of error rates. We initialize BP (including KOS)
with yj?i = 1 and EM with ?i (zi ) = j?Mi I[Lij = zi ]/|Mi |, both of which reduce to majority voting at the 0-th iteration; for KOS, we also implemented another version that exactly follows
the setting of Karger et al. [1], which initializes yj?i by Normal(1, 1) and terminates at the 10-th
iteration; the best performance of the two versions was reported. For EM with algorithmic prior
Beta(?, ?), we add a small constant (0.001) on ? and ? to avoid possible numerical NaN values.
We also implemented a max-product version of BP, but found it performed similarly to sum-product
BP in terms of error rates. We terminate all the iterative algorithms at a maximum of 100 iterations
or with 10?6 message convergence tolerance. All results are averaged on 100 random trials.
Simulated Data. We generate simulated data by drawing the abilities qj from Beta priors or the
adversary-spammer-hammer priors, that equals 0.1, 0.5, or 0.9 with certain probabilities; the assignment graphs are randomly drawn from the set of (`, ?)-regular bipartite graphs with 1000 task
nodes using the configuration method [20]. For the simulated datasets, we also calculated the oracle
lower bound in Karger et al. [1] that assumes the true qj are known, as well as a BP equipped with
an algorithmic prior equal to the true prior used to generate the data, which sets a tighter (perhaps
approximate) ?Bayesian oracle? lower bound for all the algorithms that do not know qj . We find that
BP and AMF with a typical asymmetric prior Beta(2, 1) perform mostly as well as the ?Bayesian
oracle? bound, eliminating the necessity to search for more accurate algorithmic priors.
In Fig. 1, we show that the error rates of the algorithms generally decay exponentially w.r.t. the
degree ` and log(?) of the assignment graph on a spammer-hammer model. Perhaps surprisingly,
we find that the BP, EM and AMF with the asymmetric algorithmic prior beta(2, 1) scale similarly to
KOS, which has been theoretically shown to be order-optimal compared to the oracle lower bound.
On the other hand, BP with symmetric algorithmic priors, such as the Haldane prior Beta(0+ , 0+ ) of
KOS and the uniform prior Beta(1, 1), often result in degraded performance when the degrees of the
7
Error Rate
0.5
0.5
0.4
0.4
KOS
MV
BP-Beta(2,1)
EM-Beta(2,1)
AMF-Beta(2,1)
0.4
0.3
0.3
0.3
0.2
0.2
0.1
0.1
2
0.2
5
10
15
Number of Workers per Task
(a). The bluebird dataset
20
BP-Beta 2 (2,1)
0.1
4
6
8
Number of Workers per Task
(b) The rte dataset
10
2
EM-Beta 2 (2,1)
4
6
8
Number of Workers per Task
10
AMF-Beta 2 (2,1)
Welinder et al. 2010
(c). The temp dataset
Figure 3: The results on Amazon Mechanical Turk datasets. Averaged on 100 random subsamples.
assignment graphs are low, supporting our discussion in Section 3.1.1. Indeed, BP with symmetric
algorithmic priors often fails to converge in the low-degree setting.
Fig. 2 shows the performance of the algorithms when varying the true priors of the data. We find in
Fig. 2(b) and (d) that the performance of EM with Beta(2, 1) tends to degrade when the fraction of
adversaries increases, probably because the q?j is more likely to be incorrectly updated to and stuck
on 0 or 1 in these cases; see the discussion in Section 3.2. In all cases, we find that BP and AMF (and
MF) perform mostly equally well, perhaps indicating both Bethe and mean-field approximations are
reasonably good on the joint distribution p(z, q|L, ?) in terms of error rates. Our implementation
of EM (on both simulated data and the real data below) seems to perform better than previously
reported results, probably due to our careful choice on the prior and initialization.
Real Data. We tested our methods on three publicly available Amazon Mechanical Turk datasets.
The symmetric assumption of qj = sj = tj is likely to be violated in practice, especially on vision
datasets where a human?s perception decides on whether some object is present. Therefore we also
implemented the two-coin version of BP and AMF(EM) with the algorithmic priors of sj and tj
taken as two independent Beta(2, 1) (referred to as BP-Beta2 (2,1) and similar).
We first tested on the bluebird dataset of Welinder et al. [6], including 108 tasks and 39 workers
on a fully connected bipartite assignment graph, where the workers are asked whether the presented
images contain Indigo Bunting or Blue GrosBeak. Fig. 3(a) shows the performance when fixed
numbers of annotators are subsampled for each task. On this dataset, all methods, including KOS,
BP and AMF(EM), work poorly under the symmetric assumption, while the two-coin versions of
BP and AMF(EM) are significantly better, achieving equivalent performance to the algorithm by
Welinder et al. [6] based on an advanced high dimensional model. This suggests that the symmetric
assumption is badly violated on this dataset, probably caused by the non-expert workers with high
sensitivities but low specificities, having trouble identifying Indigo Bunting from Blue GrosBeak.
We then tested on two natural language processing datasets in [21], the rte dataset with 800 tasks and
164 workers, and the temp dataset with 462 tasks and 76 workers. As seen in Fig. 3(b)-(c), both the
symmetric and the two-coin versions of BP and AMF(EM) performed equally well, all achieving
almost the same performance as the SpEM algorithm reported in [4]. The KOS algorithm does
surprisingly poorly, probably due to the assignment graphs not having locally tree-like structures.
5
Conclusion
We have presented a spectrum of inference algorithms, in particular a novel and efficient BP algorithm, for crowdsourcing problems and clarified their connections to existing methods. Our exploration provides new insights into the existing KOS, MV and EM algorithms, and more importantly,
for separating the modeling factors and algorithmic factors in crowdsourcing problems, which provides guidance for both implementations of the current algorithms, and for designing even more
efficient algorithms in the future. Numerical experiments show that BP, EM and AMF, and exact
MF, when implemented carefully, all perform impressively in term of their error rate scaling. Further
directions include applying our methodology to more advanced models, e.g., incorporating variation
in task difficulties, and theoretical analysis of the error rates of BP, EM and MF that matches the
empirical behavior in Fig. 1.
Acknowledgements. Work supported in part by NSF IIS-1065618 and two Microsoft Research
Fellowships. We thank P. Welinder and S. Belongie for providing the data and code.
8
References
[1] D.R. Karger, S. Oh, and D. Shah. Iterative learning for reliable crowdsourcing systems. In
Neural Information Processing Systems (NIPS), 2011.
[2] A.P. Dawid and A.M. Skene. Maximum likelihood estimation of observer error-rates using the
em algorithm. Applied Statistics, pages 20?28, 1979.
[3] P. Smyth, U. Fayyad, M. Burl, P. Perona, and P. Baldi. Inferring ground truth from subjective
labelling of venus images. Advances in neural information processing systems, pages 1085?
1092, 1995.
[4] V.C. Raykar, S. Yu, L.H. Zhao, G.H. Valadez, C. Florin, L. Bogoni, and L. Moy. Learning
from crowds. The Journal of Machine Learning Research, 11:1297?1322, 2010.
[5] J Whitehill, P Ruvolo, T Wu, J Bergsma, and J Movellan. Whose vote should count more:
Optimal integration of labels from labelers of unknown expertise. In Advances in Neural
Information Processing Systems, pages 2035?2043. 2009.
[6] P. Welinder, S. Branson, S. Belongie, and P. Perona. The multidimensional wisdom of crowds.
In Neural Information Processing Systems Conference (NIPS), 2010.
[7] V.C. Raykar and S. Yu. Eliminating spammers and ranking annotators for crowdsourced labeling tasks. Journal of Machine Learning Research, 13:491?518, 2012.
[8] Fabian L. Wauthier and Michael I. Jordan. Bayesian bias mitigation for crowdsourcing. In
Advances in Neural Information Processing Systems 24, pages 1800?1808. 2011.
[9] B. Carpenter. Multilevel bayesian models of categorical data annotation. Unpublished
manuscript, 2008.
[10] D. Koller and N. Friedman. Probabilistic graphical models: principles and techniques. MIT
press, 2009.
[11] M. Wainwright and M. Jordan. Graphical models, exponential families, and variational inference. Found. Trends Mach. Learn., 1(1-2):1?305, 2008.
[12] Y. Weiss and W.T. Freeman. On the optimality of solutions of the max-product beliefpropagation algorithm in arbitrary graphs. Information Theory, IEEE Transactions on, 47
(2):736 ?744, Feb 2001.
[13] F.R. Kschischang, B.J. Frey, and H.A. Loeliger. Factor graphs and the sum-product algorithm.
Information Theory, IEEE Transactions on, 47(2):498?519, 2001.
[14] D. Tarlow, I.E. Givoni, and R.S. Zemel. Hopmap: Efficient message passing with high order
potentials. In Proc. of AISTATS, 2010.
[15] A. Zellner. An introduction to Bayesian inference in econometrics, volume 17. John Wiley and
sons, 1971.
[16] R.E. Kass and L. Wasserman. The selection of prior distributions by formal rules. Journal of
the American Statistical Association, pages 1343?1370, 1996.
[17] F. Tuyl, R. Gerlach, and K. Mengersen. A comparison of bayes-laplace, jeffreys, and other
priors. The American Statistician, 62(1):40?44, 2008.
[18] Radford Neal and Geoffrey E. Hinton. A view of the EM algorithm that justifies incremental,
sparse, and other variants. In M. Jordan, editor, Learning in Graphical Models, pages 355?368.
Kluwer, 1998.
[19] A. Asuncion. Approximate mean field for Dirichlet-based models. In ICML Workshop on
Topic Models, 2010.
[20] B. Bollob?as. Random graphs, volume 73. Cambridge Univ Pr, 2001.
[21] R. Snow, B. O?Connor, D. Jurafsky, and A.Y. Ng. Cheap and fast?but is it good?: evaluating
non-expert annotations for natural language tasks. In Proceedings of the Conference on Empirical Methods in Natural Language Processing, pages 254?263. Association for Computational
Linguistics, 2008.
9
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4,008 | 4,628 | Forward-Backward Activation Algorithm for
Hierarchical Hidden Markov Models
Kei Wakabayashi
Faculty of Library, Information and Media Science
University of Tsukuba, Japan
[email protected]
Takao Miura
Department of Engineering
Hosei University, Japan
[email protected]
Abstract
Hierarchical Hidden Markov Models (HHMMs) are sophisticated stochastic models that enable us to capture a hierarchical context characterization of sequence
data. However, existing HHMM parameter estimation methods require large computations of time complexity O(T N 2D ) at least for model inference, where D is
the depth of the hierarchy, N is the number of states in each level, and T is the
sequence length. In this paper, we propose a new inference method of HHMMs
for which the time complexity is O(T N D+1 ). A key idea of our algorithm is application of the forward-backward algorithm to state activation probabilities. The
notion of a state activation, which offers a simple formalization of the hierarchical
transition behavior of HHMMs, enables us to conduct model inference efficiently.
We present some experiments to demonstrate that our proposed method works
more efficiently to estimate HHMM parameters than do some existing methods
such as the flattening method and Gibbs sampling method.
1
Introduction
Latent structure analysis of sequence data is an important technique for many applications such
as speech recognition, bioinformatics, and natural language processing. Hidden Markov Models
(HMMs) play a key role in solving these problems. HMMs assume a single Markov chain of hidden
states as the latent structure of sequence data. Because of this simple assumption, HMMs tend to
capture only local context patterns of sequence data. Hierarchical Hidden Markov Models (HHMMs) are stochastic models which assume hierarchical Markov chains of hidden states as the latent
structure of sequence data [3]. HHMMs have a hierarchical state transition mechanism that yields
the capability of capturing global and local sequence patterns in various granularities. By their nature, HHMMs are applicable to problems of many kinds including handwritten letter recognition [3],
information extraction from documents [11], musical pitch structure modeling [12], video structure
modeling [13], and human activity modeling [8, 6].
For conventional HMMs, we can conduct unsupervised learning efficiently using the forwardbackward algorithm, which is a kind of dynamic programming [9]. In situations where few or
no supervised data are available, the existence of the efficient unsupervised learning algorithm is
a salient advantage of using HMMs. The unsupervised learning of HHMMs is an important technique, as it is for HMMs. In this paper, we discuss unsupervised learning techniques for HHMMs.
We introduce a key notion, activation probability, to formalize the hierarchical transition mechanism naturally. Using this notion, we propose a new exact inference algorithm which has less time
complexity than existing methods have.
The remainder of the paper is organized as follows. In section 2, we overview HHMMs. In section
3, we survey HHMM parameter estimation techniques proposed to date. In section 4, we introduce
our parameter estimation algorithm. Section 5 presents experiments to show the effectiveness of our
algorithm. We conclude our discussion in section 6.
1
Figure 1: (left) Dynamic Bayesian network of the HHMM. (top-right) Tree representation of the
HHMM state space. (bottom-right) State identification by the absolute path of the tree.
2
Hierarchical Hidden Markov Models
Let O = {O1 , ..., Ot , ..., OT } be a sequence of observations in which subscript t denotes the time
in the sequence. We designate time as an integer index of observation numbered from the beginning
of the sequence. HHMMs define Qdt for 1 ? t ? T, 1 ? d ? D as a hidden state at time t and
level d, where d = 1 represents the top level and d = D represents the bottom level. HHMMs also
define binary variables Ftd , called termination indicators. If Ftd = 1, then it is indicated that the
Markov chain of level d terminates at time t. In HHMMs, a state transition at level d is permitted
d+1
only when the Markov chain of level d + 1 terminates, i.e. Qdt = Qdt?1 if Ft?1
= 0. A terminated
Markov chain is initialized again at the next time. Figure 1 (left) presents a Dynamic Bayesian
Network (DBN) expression for an HHMM of hierarchical depth D = 3. The conditional probability
distribution of Q, F and O is defined as follows [7].
?
(if b = 0)
? ?(i, j)
d+1
d
d
A
(i,
j)
(if
b = 1, f = 0)
= f, Q1:d?1
= b, Ft?1
=
k)
=
p(Qdt = j|Qdt?1 = i, Ft?1
t
? ? dk(j)
(if
b
= 1, f = 1)
k
{
0
(if b = 0)
p(Ftd = 1|Qdt = i, Q1:d?1
= k, Ftd+1 = b) =
t
Adk (i, end) (if b = 1)
p(Ot = v|Q1:D
= k) = Bk (v)
t
}. Probabilities of the initializaWe use a notation Q1:d?1
as a combination of states {Q1t , ..., Qd?1
t
t
tion and the state transition of Markov chains at level d depend on all higher states Q1:d?1 . Adk (i, j)
is a model parameter of the transition probability at level d from state i to j when Q1:d?1
= k.
t
Adk (i, end) denotes a termination probability that state i terminates the Markov chain at level d
when Q1:d?1
= k. ?kd (j) is an initial state probability of state j at level d when Q1:d?1
= k. Bk (v)
t
t
is an output probability of observation v when Q1:D
=
k.
t
A state space of HHMM is expressed as a tree structure [3]. Figure 1 (top-right) presents a tree
expression of state space of an HHMM for which the depth D = 3 and the number of states in each
level N = 3. The level of the tree corresponds to the level of HHMM states. Each node at level d
corresponds to a combination of states Q1:d . Each node has N children because there are N possible
states for each level. The rectangles in the figure denote local HMMs in which nodes can mutually
transit directly using the transition probability A. For the analysis described herein, we assume the
balanced N-ary tree to simplify discussions of computational complexity. However, arbitrary state
space trees do not change the substance of what follows.
The behavior of Markov chain at level d depends on the combination of all higher-up states Q1:d?1 ,
not only on the individual Qd . In the tree structure, the absolute path which corresponds to Q1:d
is meaningful, rather than the relative path which corresponds to Qd . We refer to Q1:d as Z d and
call it absolute path state. Figure 1 (bottom-right) presents an absolute path state identification. The
set of values taken by an absolute path state at level d, denoted by ?d , contains N d elements in the
balanced N-ary tree state space. We define a function to obtain the parent absolute path state of Z d
as parent(Z d ). Similarly, we define a function to obtain the set of child absolute path states of Z d
as child(Z d ), and a function to obtain the set of siblings of Z d as sib(Z d ) = child(parent(Z d )).
2
Table 1: Notation for HHMMs.
D
N
?d
Ztd ? ?d
Ftd ? {0, 1}
Ot ? {1, ..., V }
Adij
AdiEnd
?di
Biv
Depth of hierarchy
Number of states in each level
Set of values taken by absolute path state at level d
Absolute path state at time t and level d
Termination indicator at time t and level d
Observation at time t
d
State transition probability from state Ztd = i to state Zt+1
= j at level d
Termination probability of Markov chain at level d from state Ztd = i
Initial state probability of state Z d = i at level d
Output probability of observation v with Z D = i
Table 1 presents the notation used for the HHMM description. We use the notation of the absolute
path state Z d rather than Qd throughout the paper. Therefore, we define compatible notations for
the model parameters. Whereas the conventional notation ?kd (j) denotes the initial state probability
of Qd = j when Q1:d?1 = k, we aggregate Qd and Q1:d?1 into Q1:d = Z d and define ?di as the
initial state probability ?
of Z d = i. Similarly, we
?define Adij as the state transition probability from
d
Z = i to j. Note that i0 ?sib(i) ?di0 = 1 and j 0 ?{sib(i)?End} Adij 0 = 1.
3
Existing Parameter Estimation Methods for HHMMs
The first work for HHMMs [3] proposed the generalized Baum-Welch algorithm. This algorithm is
based on an inside-outside algorithm used for inference of probabilistic context free grammars. This
method takes O(T 3 ) time complexity, which is not practical for long sequence data.
A more efficient approach is the flattening method [7]. The hierarchical state sequence can be
reduced to a single sequence of the bottom level absolute path states {Z1D , ..., ZTD }. If we regard
Z D as a flat HMM state, then we can conduct the inference by using the forward-backward algorithm
with O(T N 2D ) time complexity since |?D | = N D . Notice that the flat state Z D can transit to any
other flat state, and we cannot apply efficient algorithms for HMMs of sparse transition matrix. In
the flattening method, we must make a weak constraint on the HHMM parameters, say minimally
self-referential (MinSR) [12], which restricts the self-transition at higher levels i.e. Adii = 0 for 1 ?
d ? D ? 1. The MinSR constraint enables us to identify the path connecting two flat states uniquely.
This property is necessary for estimating HHMM parameters by using the flattening method.
We also discuss a sampling approach as an alternative parameter estimation technique. The Gibbs
sampling is often used for parameter estimation of probabilistic models including latent variables
[4]. We can estimate HMM parameters using a Gibbs sampler, which sample each hidden states
iteratively. This method is applicable to inference of HHMMs in a straightforward manner on the flat
HMM. This straightforward approach, called the Direct Gibbs Sampler (DGS), takes the O(T N D )
time complexity for a single iteration.
The convergence of a posterior distribution by the DGS method is said to be extremely slow for
HMMs [10] because the DGS ignores long time dependencies. Chib [2] introduced an alternative
method, called the Forward-Backward Gibbs Sampler (FBS), which calculates forward probabilities
in advance. FBS samples hidden states from the end of the sequence regarding the forward probabilities. FBS method requires larger computations for a single iteration than DGS does, but it can
bring a posterior of hidden states to its stationary distribution with fewer iterations [10].
Heller [5] proposed Infinite Hierarchical Hidden Markov Models (IHHMMs) which can have an
infinitely large depth by weakening the dependency between the states at different levels. They proposed the inference method for IHHMMs based on a blocked Gibbs sampler of which the sampling
unit is a state sequence from t = 1 to T at a single level. This inference takes only O(T D) time
for a single iteration. In HHMMs, the states in each level are strongly dependent, so resampling
a state at an intermediate level causes all lower states to alter into a state which has a completely
different behavior. Therefore, it is not practical to apply this Gibbs sampler to HHMMs in terms of
the convergence speed.
3
4
Forward-Backward Activation Algorithm
In this section, we introduce a new parameter estimation algorithm for HHMMs, which theoretically
has O(T N D+1 ) time complexity. The basic idea of our algorithm is a decomposition of the flat
D
transition probability distribution p(Zt+1
|ZtD ), which the flattening method calculates directly for
all pairs of the flat states. We can rewrite the flat transition probability distribution into a sum of two
cases that the Markov chain at level D terminates or not, as follows.
D
D
p(Zt+1
|ZtD ) = p(Zt+1
|ZtD , FtD = 0)p(FtD = 0|ZtD ) +
D?1
D?1
D
p(Zt+1
|Zt+1
, FtD = 1)p(Zt+1
|ZtD?1 , FtD = 1)p(FtD = 1|ZtD )
The first term corresponds to the direct transition without the Markov chain termination. The actual computational complexity for calculating this term is O(N D+1 ) because the direct transition
is permitted only between the sibling states, i.e. ADij = 0 if j ?
/ sib(i). The second term, corresponding to the case in which the Markov chain terminates at level D, contains two factors: The
D?1
upper level transition probability p(Zt+1
|ZtD?1 , FtD = 1) and the state initialization probability
D?1
D
for the terminated Markov chain p(Zt+1
|Zt+1
, FtD = 1). We attempt to compute these probability
distributions efficiently in a dynamic programming manner.
d
The transition probability at level d has the form p(Zt+1
|Ztd , Ftd+1 = 1). We define ending activad
tion et , as the condition of the transition probability from Ztd , formally:
?
? p(Ztd = i, Ftd+1 = 1) (if i 6= null and d < D)
d
p(et = i) =
p(Ztd = i)
(if i 6= null and d = D)
?
p(Ftd+1 = 0)
(if i = null)
The null value in edt indicates that the Markov chain at level d + 1 does not terminate at time t.
d+1
= 1). We define
The state initialization probability for level d + 1 has the form p(Ztd+1 |Ztd , Ft?1
d
beginning activation bt , as the condition of the state initialization probability from Ztd , formally, as
?
d+1
= 1) (if i 6= null and d < D and t > 1)
? p(Ztd = i, Ft?1
d
d
p(bt = i) =
p(Zt = i)
(if i 6= null and (d = D or t = 1))
?
d+1
p(Ft?1
= 0)
(if i = null)
The null value in bdt indicates that the Markov chain at level d + 1 does not terminate at time t ? 1.
Using these notations, we can represent the flat transition with propagations of activation probabilD
D
|ZtD ) = p(bD
ities as shown in figure 2 (left) because p(Zt+1
t+1 |et ). This representation naturally
describes the decomposition of the flat transition probability distribution discussed above, and it
enables us to apply the decomposition recursively for all levels. We can derive the conditional
probability distributions of edt and bdt+1 as
{ ?
d+1
= c)A(d+1)cEnd
(if i 6= null)
c?child(i) p(et
?
p(edt = i|ed+1
)
=
t
d+1
d+1
= c)(1?A(d+1)cEnd )+p(et = null) (if i = null)
c??d+1 p(et
{
?
d?1
p(bt+1 = parent(i))?di + j?sib(i) p(edt = j)Adji (if i 6= null)
d
d d?1
p(bt+1 = i|et , bt+1 ) =
p(edt = null)
(if i = null)
In the following subsections, we show the efficient inference algorithm and the parameter estimation
algorithm using the activation probabilities.
4.1
Inference using Forward and Backward Activation Probabilities
We can translate the DBN of HHMMs in figure 1 (left) equivalently into simpler DBN using activation probabilities. The translated DBN is portrayed in figure 2 (right). The inference algorithm
proposed herein is based on a forward-backward calculation over this DBN. We define forward
activation probability ? and backward activation probability ? as follows.
?edt (i) = p(edt = i, O1:t )
?bdt (i) = p(bdt = i, O1:t?1 )
?edt (i) = p(Ot+1:T , FT1 = 1|edt = i)
?bdt (i) = p(Ot:T , FT1 = 1|bdt = i)
4
Figure 2: (left) Propagation of activation probabilities for calculating the flat transition probability
from time t to t + 1. (right) Equivalent DBN of the HHMM using activation probabilities.
Algorithm 1 Calculate forward activation probabilities
1: for t = 1 to T do
2:
if t = 1 then
3:
?b11 (i ? ?1 ) = ?1i
4:
for d = 2 to D do
5:
?bd1 (i ? ?d ) = ?bd?1 (parent(i))?di
1
6:
end for
7:
else
?
8:
?b1t (i ? ?1 ) = j?sib(i) ?e1t?1 (j)A1ji
9:
for d = 2 to D do
?
10:
?bdt (i ? ?d ) = ?bd?1 (parent(i))?di + j?sib(i) ?edt?1 (j)Adji
t
11:
end for
12:
end if
13:
?eD
(i ? ?D ) = ?bD
(i)BiOt
t
t
14:
for d = D ? 1 to 1?do
15:
?edt (i ? ?d ) = c?child(i) ?ed+1 (c)A(d+1)cEnd
t
16:
end for
17: end for
These probabilities are efficiently calculable in a dynamic programming manner. Algorithm 1
presents the pseudocodes to calculate whole ?. ?bdt are derived downward from ?b1t to ?bD
by
t
summing up to the initialization probability from the parent and the transition probabilities from the
siblings (Line 8 to 11). ?edt are propagated upward from ?eD
to ?e1t by summing up to the probabilt
ities of the child Markov chain termination (Line 13 to 16). This algorithm includes the calculation
of |?d | = N d quantities involving the summation of |sib(i)| = N terms for d = 1 to D and for
?D
t = 1 to T . Therefore, the time complexity of algorithm 1 is O(T d=1 N d+1 ) = O(T N D+1 ).
Algorithm 2 propagates the backward activation probabilities similarly in backward order.
We can derive the conditional independence of O1:t and {Ot+1:T , FT1 = 1} given edt 6= null or
bdt+1 6= null, because both of edt 6= null and bdt+1 6= null indicates that the Markov chains at level
d + 1, ..., D terminates at time t. On the basis of this conditional independence, the exact inference
of a posterior of activation probabilities can be obtained using ? and ? as presented below.
p(edt = i|O1:T , FT1 = 1) ? p(edt = i, O1:t )p(Ot+1:T , FT1 = 1|edt = i) = ?edt (i)?edt (i)
p(bdt = i|O1:T , FT1 = 1) ? p(bdt = i, O1:t?1 )p(Ot:T , FT1 = 1|bdt = i) = ?bdt (i)?bdt (i)
The inference of the flat state p(ZtD |O1:T , FT1 = 1) is identical to of the bottom level activation
1
probability p(eD
t |O1:T , FT = 1). We can calculate the likelihood of the whole observation as follows.
?
?
p(O1:T , FT1 = 1) =
p(e1T = i, O1:T )p(FT1 = 1|e1T = i) =
?e1T (i)?e1T (i)
i??1
i??1
5
Algorithm 2 Calculate backward activation probabilities
1: for t = T to 1 do
2:
if t = T then
3:
?e1T (i ? ?1 ) = A1iEnd
4:
for d = 2 to D do
5:
?ed (i ? ?d ) = ?ed?1 (parent(i))AdiEnd
T
T
6:
end for
7:
else
?
8:
?e1t (i ? ?1 ) = j?sib(i) ?b1t+1 (j)A1ij
9:
for d = 2 to D do
?
10:
?edt (i ? ?d ) = ?ed?1 (parent(i))AdiEnd + j?sib(i) ?bdt+1 (j)Adij
t
11:
end for
12:
end if
13:
?bD
(i ? ?D ) = ?eD
(i)BiOt
t
t
14:
for d = D ? 1 to 1?do
15:
?bdt (i ? ?d ) = c?child(i) ?bd+1 (c)?(d+1)c
t
16:
end for
17: end for
4.2
Updating Parameters
Using the forward and backward activation probabilities, we can estimate HHMM parameters effi? is defined, where ? is a
ciently in the EM framework. In the EM algorithm, the function Q(?, ?)
?
parameter set before updating and ? is a parameter set after updating, as described below.
?
? =
Q(?, ?)
p? (Y |X) log p??(X, Y )
Y
In that equation, X represents a set of observed variables, and Y is a set of latent variables. The dif?
ference of log likelihood between the models of ? and ?? is known to be greater than Q(?, ?)?Q(?,
?)
[1]. For this reason, we can increase the likelihood monotonically by selecting a new parameter ?? to
maximize the function Q. For HHMMs, the set of parameters is ? = {A, ?, B}. The set of observed
1:D
1:D
variables is X = {O1:T , FT1 = 1}. The set of latent variables is Y = {Z1:T
, F1:T
?1 }. Therefore,
the function Q can be represented as shown below.
?
1:D
1:D
1
1:D
1:D
? ?
Q(?, ?)
p? (O1:T , FT1 = 1, Z1:T
, F1:T
(1)
?1 ) log p??(O1:T , FT = 1, Z1:T , F1:T ?1 )
1:D ,F 1:D
Z1:T
1:T ?1
The joint probability of observed variables and latent variables is given below.
1:D
1:D
, F1:T
p? (O1:T , FT1 = 1, Z1:T
?1 )
=
D
?
?dZ1d
d=1
T?
?1 ?
D
Fd
D
T
?
?
F d+1 (1?Ftd ) Ftd
?dZ d )
AdZ d End
BZtD Ot
T
t t+1
t+1
t=1
d=1
(AdZt d End AdZt d Z d
t
t=1 d=1
We substitute this equation for the joint probability in equation (1). We integrate out irrelevant
variables and organize around each parameter. Thereby, we obtain the following.
? ?
Q(?, ?)
D ?
?
g?di log ?
?di +
D ?
?
?
?iv
gBiv log B
i??D v=1
d=1 i??d j?{sib(i)?End}
d=1 i??d
V
? ?
gAdij log A?dij +
Therein, g?di , gAdij , gBiv are shown by equation (2)(3)(4)(5). They are calculable using forward
and backward activation probabilities.
g?di
= ?bd1 (i)?bd1 (i) +
T
?1
?
?bd?1 (parent(i))?di ?bdt+1 (i)
(2)
t+1
t=1
gAdiEnd
=
T
?1
?
?edt (i)AdiEnd ?ed?1 (parent(i)) + ?ed (i)?ed (i)
t
t=1
6
T
T
(3)
Table 2: Log-likelihood achieved at each iteration.
Iteration
FBA w/o MinSR
FBA with MinSR
FFB
1
-773.47
-773.89
-773.89
gAdij
=
2
-672.44
-672.47
-672.47
T
?1
?
3
-668.50
-670.40
-670.40
4
-631.30
-643.62
-643.62
5
-610.63
-614.98
-614.98
10
-577.33
-573.84
-573.84
50
-457.66
-453.09
-453.09
?edt (i)Adij ?bdt+1 (j)
100
-447.90
-448.52
-448.52
(4)
t=1
gBiv
?
=
?eD
(i)?eD
(i)
t
t
(5)
t:Ot =v
? B,
? which maximize the function Q
Using Lagrange multipliers,
we can obtain
? , A,
?
? parameters ?
? ?
under the constraint i0 ?sib(i) ?
?di0 = 1, j 0 ?{sib(i)?End} A?dij 0 = 1, v B
iv = 1 as shown below.
?
?di = ?
g?di
0
i0 ?sib(i) g?di
gAdij
, A?dij = ?
0
j 0 ?{sib(i)?End} gAdij
?iv = ?gBiv
,B
v gBiv
Consequently, we can calculate the update parameters using ? and ?. The time complexity for
computing a single EM iteration is O(T N D+1 ), which is identical to the calculation of forward and
backward activation probabilities.
5
Experiments
Firstly, we experimentally confirm that the forward-backward activation algorithm yields exactly
identical parameter estimation to the flattening method does. Remind that we must make the MinSR
constraint on the HHMM parameter set in the flattening method (see section 3). We compare
three parameter estimation algorithms: our forward-backward activation algorithm for a MinSR
HHMM (FBA with MinSR), for a HHMM without MinSR (FBA w/o MinSR), and the flattening
method(FFB). The dataset to learn includes 5 sequences of 10 length, which are artificially generated by a MinSR HHMM of biased parameter set. We execute three algorithms and examine the
log-likelihood achieved at each iteration.
Table 2 presents the result. The FBA with MinSR and the FFB achieve the identical log-likelihood
through the training. This result provides experimental evidence that our algorithm estimates
HHMM parameters exactly identically to the flattening method does. Furthermore, the FBA enables
us to conduct the parameter estimation of HHMMs which has non-zero self-transition parameters.
To evaluate the computational costs empirically, we compare four methods of HHMM parameter
estimation. Two are based on the EM algorithm with inference by the forward-backward activation
algorithm (FBA), and by the flattening forward-backward method (FFB). Another two are based on
a sampling approach: direct Gibbs sampling for the flat HMMs (DGS) and forward-backward activation sampling (FBAS). FBAS is a straightforward application of the forward-backward sampling
scheme to the translated DBN presented in figure 2. In FBAS, we first calculate forward activation
probabilities. Then we sample state activation variables from e1T to b11 in the backward order with
respect to forward activation probabilities. We evaluate four methods based on three aspects: execution time, convergence speed, and scalability of the state space size. We apply each method to four
different HHMMs of (D = 3,N = 3), (D = 3,N = 4), (D = 4,N = 3), and (D = 4,N = 4). We
examine the log-likelihood of the training dataset achieved at each iteration to ascertain the learning convergence. As a training dataset, we use 100 documents from the Reuters corpus as word
sequences. The dataset includes 36,262 words in all, with a 4,899 word vocabulary.
Figure 3 presents the log-likelihood of the training data. The horizontal axis shows the logarithmically scaled execution time. Table 2 presents the average execution time for a single iteration.
From these results, we can say primarily that FBA outperforms FFB in terms of execution time. The
improvement is remarkable, especially for the HHMMs of large state space size because FBA has
less time complexity for N and D than FFB has.
7
-180000
-190000
-190000
-200000
-200000
-210000
-210000
Log-Likelihood
Log-Likelihood
-180000
-220000
-230000
-240000
-230000
-240000
FB Activation
Flattening FB
FB Activation Sampling
Direct Gibbs Sampling
-250000
-260000
100
1000
10000
Execution Time (ms)
100000
FB Activation
Flattening FB
FB Activation Sampling
Direct Gibbs Sampling
-250000
-260000
100
1e+006
-180000
-180000
-190000
-190000
-200000
-200000
-210000
-210000
Log-Likelihood
Log-Likelihood
-220000
-220000
-230000
-240000
1000
10000
Execution Time (ms)
100000
1e+006
-220000
-230000
-240000
FB Activation
Flattening FB
FB Activation Sampling
Direct Gibbs Sampling
-250000
-260000
100
1000
10000
Execution Time (ms)
100000
FB Activation
Flattening FB
FB Activation Sampling
Direct Gibbs Sampling
-250000
-260000
1000
1e+006
10000
100000
1e+006
Execution Time (ms)
Figure 3: Convergence of log-likelihood for the training data on the Reuters corpus. Log-likelihood
(vertical) is shown against the log-scaled execution time (horizontal) to display the execution time
necessary to converge the learning of each algorithm. (top-left) HHMM of D = 3, N = 3. (topright) D = 3, N = 4. (bottom-left) D = 4, N = 3. (bottom-right) HHMM of D = 4, N = 4.
Table 3: Average execution time for a single iteration (ms).
Method
FBA
FFB
FBAS
DGS
D = 3, N = 3
(N D = 27)
186.65
1729.90
82.45
24.19
D = 3, N = 4
(N D = 64)
391.73
9242.35
142.20
37.50
D = 4, N = 3
(N D = 81)
476.92
19257.80
183.39
45.43
D = 4, N = 4
(N D = 256)
1652.03
220224.00
581.58
265.98
The results show that the likelihood convergence using DGS is much slower than that of other
methods.The execution time of DGS is less than that of other methods for a single iteration, but
this cannot compensate for the low convergence speed. However, FBAS achieves a competitive
likelihood in comparison to FBA. Results show that FBAS might be appropriate for some situations
because FBAS finds a better solution than that FBA do in some results.
6
Conclusion
In this work, we proposed a new inference algorithm for HHMMs based on the activation probability.
Results show that the performance of our proposed algorithm surpasses that of existing methods.
The forward-backward activation algorithm described herein enables us to conduct unsupervised
parameter learning with a practical computational cost for HHMMs of larger state space size.
References
[1] C. Bishop. Pattern Recognition and Machine Learning. Springer, 2007.
[2] S. Chib. Calculating posterior distributions and modal estimates in markov mixture models.
Journal of Econometrics, 1996.
[3] S. Fine, Y. Singer, and N. Tishby. The hierarchical hidden markov model: Analysis and applications. Machine Learning, 1998.
8
[4] T. Griffiths and M. Steyvers. Finding scientific topics. Proc. the National Academy of Sciences
of the United States of America, 2004.
[5] K. Heller, Y. Teh, and D. Gorur. Infinite hierarchical hidden markov models. In Proc. International Conference on Artificial Intelligence and Statistics, 2009.
[6] S. Luhr, H. Bui, S. Venkatesh, and G. West. Recognition of human activity through hierarchical
stochastic learning. In Proc. Pervasive Computing and Communication, 2003.
[7] K. Murphy and M. Paskin. Linear time inference in hierarchical hmms. In Proc. Neural
Information Processing Systems, 2001.
[8] N. Nguyen, D. Phung, and S. Venkatesh. Learning and detecting activities from movement trajectories using the hierarchical hidden markov models. In Proc. Computer Vision and Pattern
Recognition, 2005.
[9] L. Rabiner. A tutorial on hidden markov models and selected applications in speech recognition. Proc. IEEE, 1989.
[10] S. Scott. Bayesian methods for hidden markov models: Recursive computing in the 21st
century. Journal of the American Statistical Association, 2002.
[11] M. Skounakis, M. Craven, and S. Ray. Hierarchical hidden markov models for information
extraction. In Proc. International Joint Conference on Artificial Intelligence, 2003.
[12] M. Weiland, A. Smaill, and P. Nelson. Learning musical pitch structures with hierarchical
hidden markov models. In Proc. Journees Informatiques Musicales, 2005.
[13] L. Xie, S. Chang, A. Divakaran, and H. Sun. Learning hierarchical hidden markov models for
video structure discovery. Technical report, Columbia University, 2002.
9
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4,009 | 4,629 | Distributed Probabilistic Learning
for Camera Networks with Missing Data
Vladimir Pavlovic
Department of Computer Science
Rutgers University
[email protected]
Sejong Yoon
Department of Computer Science
Rutgers University
[email protected]
Abstract
Probabilistic approaches to computer vision typically assume a centralized setting,
with the algorithm granted access to all observed data points. However, many
problems in wide-area surveillance can benefit from distributed modeling, either
because of physical or computational constraints. Most distributed models to date
use algebraic approaches (such as distributed SVD) and as a result cannot explicitly deal with missing data. In this work we present an approach to estimation and
learning of generative probabilistic models in a distributed context where certain
sensor data can be missing. In particular, we show how traditional centralized
models, such as probabilistic PCA and missing-data PPCA, can be learned when
the data is distributed across a network of sensors. We demonstrate the utility
of this approach on the problem of distributed affine structure from motion. Our
experiments suggest that the accuracy of the learned probabilistic structure and
motion models rivals that of traditional centralized factorization methods while
being able to handle challenging situations such as missing or noisy observations.
1
Introduction
Traditional computer vision algorithms, particularly those that exploit various probabilistic and
learning-based approaches, are often formulated in centralized settings. A scene or an object is
observed by a single camera with all acquired information centrally processed and stored in a single
knowledge base (e.g., a classification model). Even if the problem setting relies on multiple cameras,
as may be the case in multi-view or structure from motion (SfM) tasks, all collected information is
still processed and organized in a centralized fashion. Increasingly modern computational settings
are becoming characterized by networks of peer-to-peer connected devices, with local data processing abilities. Nevertheless, the overall goal of such distributed device (camera) networks may still
be to exchange information and form a consensus interpretation of the visual scene. For instance,
even if a camera observes a limited set of object views, one would like its local computational model
to reflect a general 3D appearance of the object visible by other cameras in the network.
A number of distributed algorithms have been proposed to address the problems such as calibration,
pose estimation, tracking, object and activity recognition in large camera networks [1?3]. In order
to deal with high dimensionality of vision problems, distributed latent space search such as decentralized variants of PCA have been studied in [4, 5]. A more general framework using distributed
least squares [6] based on distributed averaging of sensor fusions [7] was introduced for PCA, triangulation, pose estimation and SfM. Similar approaches have been extended to settings such as the
distributed object tracking and activity interpretation [8,9]. Even though the methods such as PCA or
Kalman filtering have their well-known probabilistic counterparts, the aforementioned approaches
do not use probabilistic formulation when dealing with the distributed setting.
One critical challenge in distributed data analysis includes dealing with missing data. In camera
networks, different nodes will only have access to a partial set of data features because of varying
camera views or object movement. For instance, object points used for SfM may be visible only
1
in some cameras and only in particular object poses. As a consequence, different nodes will be
frequently exposed to missing data. However, most current distributed data analysis methods are
algebraic in nature and cannot seamlessly handle such missing data.
In this work we propose a distributed consensus learning approach for parametric probabilistic models with latent variables that can effectively deal with missing data. We assume that each node in
a network can observe only a fraction of the data (e.g., object views in camera networks). Furthermore, we assume that some of the data features may be missing across different nodes. The goal of
the network of sensors is to learn a single consensus probabilistic model (e.g., 3D object structure)
without ever resorting to a centralized data pooling and centralized computation. We will demonstrate that this task can be accomplished in a principled manner by local probabilistic models and
in-network information sharing, implemented as recursive distributed probabilistic learning.
In particular, we focus on probabilistic PCA (PPCA) as a prototypical example and derive its distributed version, the D-PPCA. We then suggest how missing data can be handled in this setting
using a missing-data PPCA and apply this model to solve the distributed SfM task in a camera network. Our model is inspired by the consensus-based distributed Expectation-Maximization (EM)
algorithm for Gaussian mixtures [10], which we extend to deal with generalized linear Gaussian
models [11]. Unlike other recently proposed decomposable Gaussian graphical models [4, 12], our
model does not depend on any specific type of graphs. Our network, of arbitrary topology, is assumed to be static with a single connected component. These assumptions are reasonably applicable
to many real world camera network settings.
In Section 2, we first explain the general distributed probabilistic model. Section 3 shows how DPPCA can be formulated as a special case of the probabilistic framework and propose the means for
handling missing data. We then explain how D-PPCA can be modified for the application in affine
SfM. In Section 5, we report experimental results of our model using both synthetic and real data.
Finally, we discuss our approach including its limitations and possible solutions in Section 6.
2
Distributed Probabilistic Model
We start our discussion by first considering a general parametric probabilistic model in a centralized
setting and then we show how to derive its distributed form.
2.1
Centralized Setting
Let X = {xn |xn ? RD } be a set of iid multivariate data points with the corresponding latent
variables Z = {zn |zn ? RM }, n = 1...N . Our model is a joint density defined on (xn , zn ) with a
global parameter ?
(xn , zn ) ? p(xn , zn |?),
Q
with p(X, Z|?) = n p(xn , zn |?), as depicted in Fig. 1a. In this general model, we can find an
optimal global parameter ?? (in a MAP sense) by applying standard EM learning. The EM follows a
recursive two-step procedure: (a) E-step, where the posterior density p(zn |xn , ?) is estimated, and
(b) M-step: parametric optimization ?? = arg max? EZ|X [log p(X, Z|?)]. It is important to point out
that each posterior density estimate at point n depends solely on the corresponding measurement xn
and does not depend on any other xk , k 6= n. This means that even if we partition independent measurements into arbitrary subsets, posterior density estimation is accomplished locally, within each
subset. However, in the M-step all measurements X affect the choice of ?? because of the depen? This is a typical characteristic of
dence of each term in the completed log likelihood on the same ?.
parametric models where the optimal parameters depend on summary data statistics.
2.2
Distributed Setting
Let G = (V, E) be an undirected connected graph with vertices i, j ? V and edges eij = (i, j) ? E
connecting the two vertices. Each i-th node is directly connected with 1-hop neighbors in Bi =
{j|eij ? E}. Suppose the set of data samples at i-th node is Xi = {xin |n = 1, ..., Ni }, where
xin ? RD is n-th measurement vector and Ni is the number of samples collected in i-th node.
Likewise, we define the latent variable set for node i as Zi = {zin |n = 1, ..., Ni }.
2
(a) Centralized
(b) Distributed
(c) Augmented
Figure 1: Centralized, distributed and augmented models for probabilistic PCA.
As observed previously, each posterior estimation is decentralized. Learning the model parameter
would be decentralized if each node had its own independent parameter ?i . Still, the centralized
model can be equivalently defined using the set of local parameters, with an additional constraint on
their consensus, ?1 = ?2 = ? ? ? = ?|V | . This is illustrated in Fig. 1b where the local node models
are constrained using ties defined on the underlying graph. The simple consensus tying can be more
conveniently defined using a set of auxiliary variables ?ij , one for each edge eij (Fig. 1c). This now
leads to the final distributed consensus learning formulation, similar to [10]:
?? = arg min ? log p(X|?, G) s.t. ?i = ?ij , ?ij = ?j , i ? V, j ? Bi
(1)
{?i :i?V }
where we marginalized on X. This is a constrained optimization task that can be solved in a principal manner using the Alternating Direction Method of Multipliers (ADMM) [13?15]. ADMM
iteratively, in a block-coordinate fashion, solves max? min? L(?) on the augmented Lagrangian
L(?, ?, ?) = ? log p(X|?1 , ?2 , ..., ?|V | , G) +
XXn
o
?Tij1 (?i ? ?ij ) + ?Tij2 (?ij ? ?j )
i?V j?Bi
?XX
+
||?i ? ?ij ||2 + ||?ij ? ?j ||2
2 i?V j?B
(2)
i
where ?ij1 , ?ij2 , i, j ? V are the Lagrange multipliers, ? is some positive scalar parameter and
|| ? || is induced norm. The last term (modulated by ?) is not strictly necessary for consensus but
introduces additional regularization. Further discussions on this term and the parameter can be found
in [15] and [16]. The auxiliary ?ij play a critical decoupling role and separate estimation of local
?i during block-coordinate ascent/descent. This classic (first introduced in 1970s) meta decompose
algorithm can be used to devise a distributed counterpart for any centralized problem that attempts
to maximize a global log likehood function over a connected network.
3
Distributed Probabilistic PCA (D-PPCA)
We now apply the general distributed probabilistic learning explained above to the specific case of
distributed PPCA. Traditional centralized formulation of probabilistic PCA (PPCA) [17] assumes
that latent variable zin ? N (zin |0, I), with a generative relation
xin = Wi zin + ?i + i ,
(3)
where i ? N (i |0, a?1
i I) and ai is the noise precision. Inference then yields
?1 ?1
T
p(zin |xin ) = N (zin |L?1
i Wi (xin ? ?i ), ai Li ),
(4)
a?1
i I.
where Li = WiT Wi +
We can find optimal parameters Wi , ?i , ai by finding the maximum
likelihood estimates of the marginal data likelihood or by applying the EM algorithm on expected
complete data log likelihood with respect to the posterior density p(Zi |Xi ).
3.1
Distributed Formulation
The distributed algorithm developed in Section 2 can be directly applied to this PPCA model. The
basic idea is to assign each subset of samples as evidence for the local generative models with
3
parameters Wi , ?i , a?1
i . The inference is accomplished locally in each node. The local parameter
estimates are then computed using the consensus updates that combine local summary data statistics
with the information about the model conveyed through neighboring network nodes. Below, we
outline specific details of this approach.
Let ?i = {Wi , ?i , ai } be the set of parameters for each node i. The global constrained consensus
optimization now becomes
Wi = ?ij , ?ij = Wj , i ? V, j ? Bi ,
?i = ?ij , ?ij = ?j , i ? V, j ? Bi ,
ai = ?ij , ?ij = aj , i ? V, j ? Bi
min{Wi ,?i ,ai :i?V } ?F (?i ) s.t.
where F (?i ) =
Ni
P
n=1
(5)
log p(xin |Wi , ?i , a?1
i ). The augmented Lagrangian is
L(?i ) = ?F (?i )+
XX
XX T
T
T
?ij1
(?i ? ?ij ) + ?ij2
(?ij ? ?j )
?ij1 (Wi ? ?ij ) + ?Tij2 (?ij ? Wj ) +
i?V j?Bi
i?V j?Bi
?XX
+
(?ij1 (ai ? ?ij ) + ?ij2 (?ij ? aj )) +
(||Wi ? ?ij ||2 + ||?ij ? Wj ||2 )
2
i?V j?Bi
i?V j?Bi
?XX
?XX
2
2
+
(||?i ? ?ij || + ||?ij ? ?j || ) +
((ai ? ?ij )2 + (?ij ? aj )2 )
(6)
2 i?V j?B
2 i?V j?B
XX
i
i
where ?i = {Wi , ?i , ai , ?ij , ?ij , ?ij ; i ? V, j ? Bi } and {?ijk }, {?ijk }, {?ijk } with k = 1, 2
are the Lagrange multipliers. The scalar value ? gives us control over the convergence speed of the
algorithm. With reasonably large positive ?, the overall optimization converges fairly quickly [10].
We will explore the converging behaviour with respect to various ? in synthetic data experiments.
Just like in a standard EM approach, we minimize the upper bound of L(?i ). Exploiting the posterior density in (4), we compute the expected mean and variance of latent variables in each node
as
T
E[zin ] = L?1
i Wi (xin ? ?i ),
?1
T
E[zin zTin ] = a?1
i Li + E[zin ]E[zin ] .
(7)
Maximization of the completed likelihood Lagrangian derived from (6) yields
(t+1)
Wi
=
ai
Ni
X
T
(xin ? ?i )E[zin ] ?
(t)
2?i
X (t)
(t)
?
+?
Wi + Wj
n=1
ai
Ni
X
!?1
E[zin zTin ]
+ 2?|Bi |I
n=1
j?Bi
(8)
(t+1)
?i
X
Ni
X (t)
(t)
(t)
? (Ni ai + 2?|Bi |)?1 ,
xin ? Wi E[zin ] ? 2?i + ?
?i + ?j )
= ai
n=1
(t+1)
?i
(t+1)
?i
(t+1)
?i
(9)
j?Bi
o
? X n (t+1)
(t+1)
Wi
? Wj
,
2 j?B
i
o
? X n (t+1)
(t)
(t+1)
= ?i +
?i
? ?j
,
2 j?B
i
o
? X n (t+1)
(t)
(t+1)
= ?i +
ai
? aj
.
2 j?B
(t)
= ?i +
(10)
(11)
(12)
i
For ai , we solve the quadratic equation
Ni D
(t+1) 2
(t+1)
0=?
+ 2?|Bi |ai
+ ai
?
2
(
(t)
2?i ? ?
X
j?Bi
(t)
(t)
ai + aj
?
Ni
X
E[zin ]T WiT (xin ? ?i )
n=1
Ni
)
h
io
1 Xn
2
T
T
+
||xin ? ?i || + tr E[zin zin ]Wi Wi
.
2 n=1
(13)
The overall distributed EM algorithm for D-PPCA is summarized in Algorithm 1. Detailed derivation can be found in the supplementary material.
4
,
Algorithm 1 Distributed Probabilistic PCA (D-PPCA)
(0)
(0)
(0)
(0)
Require: For every node i initialize Wi , ?i , ai randomly and set ?i
for t = 0, 1, 2, ... until convergence do
for all i ? V do
[E-step] Compute E[zin ] and E[zin zTin ] via (7).
(t+1)
(t+1)
(t+1)
[M-step] Compute Wi
, ?i
, ai
via (8,9,13).
end for
for all i ? V do
(t+1)
(t+1)
(t+1)
Broadcast Wi
, ?i
, and ai
to all neighbors of i ? Bi .
end for
for all i ? V do
(t+1)
(t+1)
(t+1)
Compute ?i
, ?i
, and ?i
via (10-12).
end for
end for
3.2
(0)
= 0, ?i
(0)
= 0, ?i
= 0.
Missing Data D-PPCA
Traditional PPCA is an effective tool for dealing with data missing-at-random (MAR) in traditional
PCA [18]. While more sophisticated methods including variational approximations, c.f., [18] are
possible direct use of PPCA is often sufficient in practice. Hence, we adopt D-PPCA as a method to
deal with missing data in a distributed consensus setting.
Generalization to missing data D-PPCA from D-PPCA is straightforward and follows [18]. From
the perspective of ADMM-based learning the only modifications comes in the form of adjusted
PNi
terms for local data summaries. For instance, in (9) the data summary term n=1
(xin ? Wi E[zin ])
becomes
X
T
xi,n,f ? wi,f
E[zin ],
(14)
n?Oi,f
where f = 1, . . . , D is the index of feature, Oi,f is the set of samples in node i that have the feature
T
f present, xi,n,f is the value of the present feature, and wi,f
is the f -th row of matrix Wi . Similar
expressions can be derived for other local parameters. Note that (10-12) incur no changes.
4
D-PPCA for Structure from Motion (SfM)
In this section, we consider a specific formulation of the modified distributed probabilistic PCA
for application in affine SfM. In SfM, our goal is to estimate the 3D location of N points on a
rigid object based on corresponding 2-D points observed from multiple cameras (or views). The
dimension D of our measurement matrix is thus twice the number of frames each camera observed.
A simple and effective way to solve this problem is the factorization method [19]. Given a 2D (image
coordinate) measurement matrix X, of size 2 ? #f rames ? #points, the matrix is factorized into a
2 ? #f rames ? 3 motion matrix M and the 3 ? #points 3D structure matrix S. In the centralized
setting this can be easily computed using SVD on X. Equivalently, the estimates of M and S can
be found using inference and learning in a centralized PPCA, where M is treated as the PPCA
parameter and S is the latent structure. There we obtain additional estimates of the variance of
structure S, which are not immediately available from the factorization approach (although, they
can be found).
However, the above defined (2 ? #f rames ? #points) data structure of X is not amenable to
distribution of different views (cameras, nodes), as considered in Section 3 of D-PPCA. Namely,
D-PPCA assumes that the distribution is accomplished by splitting the data matrix X into sets of
non-overlapping columns, one for each node. Here, however, we seek to distribute the rows of
matrix X, i.e., a set of (subsequent) frames is to be assigned to each node/camera.
Hence, to apply the D-PPCA framework to SfM we need to swap the role of rows and columns,
i.e., consider modeling of XT . This, subsequently, means that the 3D scene structure (which is to
be shared across all nodes in the network) will be treated as the D-PPCA parameter. The latent
D-PPCA variables will model the unknown and uncertain motion of each camera (and/or object in
its view).
5
Specifically, we will consider the model
XTi = W ? Zi + Ei
(15)
where XTi is the matrix of image coordinates of all points in node (camera) i of size #points ? 2 ?
#f rames in node i, W is the #points ? 3 3D structure (D-PPCA parameter) matrix and Zi is the
3 ? 2 ? #f rames motion matrix of node i.
One should note that we have implicitly assumed, in a standard D-PPCA manner, that each column
of Zi is iid and distributed as N (0, I). However, each pair of subsequent Zi columns represents
one 3 ? 2 affine motion matrix. While those columns are not truly independent our experiments (as
demonstrated in Section 5) show that this assumption is not detrimental in practice. Remaining task
is simply following the same process we did to derive D-PPCA.
Missing data in SfM will be handled using the formalism presented in Sec. 3.2. Strictly speaking,
the model of data missing-at-random is not always applicable to SfM. The reason is that occlusions,
the main source of missing data, cannot be treated as a random process. Instead, this setting corresponds to data missing-not-at-random [18] (MNAR). If treated blindly, this may introduce bias in the
estimated models. However, as we demonstrate in experiments this assumption does not adversely
affect SfM when the number of missing points is within a reasonable range.
5
Experiments
In our experiments we first study the general convergence properties of the D-PPCA algorithm in a
controlled synthetic setting. We then apply the D-PPCA to a set of SfM problems, both on synthetic
and on real data.
5.1
Empirical Convergence Analysis
Using synthetic data generated from Gaussian distribution, we observed that D-PPCA works well
regardless of the number of network nodes, topology, choice of the parameter ? or even with missing
values in both MAR and MNAR cases. Detailed results for the syntehtic data is provided in the
supplementary materials.
5.2
Affine Structure from Motion
We now show that the modified D-PPCA can be used as an effective framework for distributed affine
SfM. We first show results in a controlled environment with synthetic data and then report results
on data from real video sequences. We assume that correspondences across frames and cameras
are known. For the missing values of MNAR case, we either used the actual occlusions to induce
missing points or simulated consistently missing points over several frames.
5.2.1
Synthetic Data (Cube)
We first generated synthetic data with a rotating unit cube and 5 cameras facing the cube in a 3D
space, similar to synthetic experiments in [6]. The cube is centered at the origin of the space and
rotates 30? counterclockwise. We extracted 8 cube points projected on each camera view every 6? ,
i.e. each camera observed 5 frames. Cameras are placed on a skewed plane, making elevation along
z-axis as shown in Fig. 2a. For all synthetic and real SfM experiments, we picked ? = 10 and
initialized Wi matrix with feature point coordinates of the first frame visible in the i-th camera with
some small noise. The convergence criterion for D-PPCA for SfM was set as 10?3 relative error.
To measure the performance, we computed maximum subspace angle between the ground truth
3D coordinates and our estimated 3D structure matrix. For comparison, we conducted traditional
SVD-based SfM on the same data. In noise free case, D-PPCA for SfM always yielded the same
performance as SVD-based SfM with near 0? .
We also tested D-PPCA for SfM with noisy and missing-value cases. First, we generated 20 independent samples of all 25 frames with 10 different noise levels. Then we ran D-PPCA 20 times on
each of the independent sample and averaged the final structure estimates. As Fig. 2b shows, we
found that D-PPCA for SfM is fairly robust to noise and tends to stabilize even as the noise level
6
1.6
Subspace angle (degree)
1.4
2
1.5
1
0.5
0
?0.5
4
5
3
4
3
2
0
1
Centralized
SVD?based
SfM
0.8
0.6
0.4
D?PPCA
0.2
0
2
1
1.2
1
0
0
1
2
3
4
5
6
7
8
9
10
Noise level (%)
(a) Camera Setting
(b) Subspace Angle vs. Ground Truth
Figure 2: Rotating unit cube with multiple cameras. Red circles are camera locations and blue
arrows indicate each camera?s facing direction. Green and red crosses in the right plot are outliers
for centralized SVD-based SfM and D-PPCA for SfM, respectively.
increases. The mean subspace angle tends to be slightly larger than that estimated by the centralized SVD SfM, however both reside within the overlapping confidence intervals. Considering MAR
missing values, we obtained 1.66? for 20% missing points averaged over 10 different missing point
samples. In MNAR case with actual occlusions considered, D-PPCA yielded, relatively larger, 20?
error. Intuitively, this is because the missing points in the scene are naturally not random. However,
we argue that D-PPCA can still handle missing points given the evidence below.
5.2.2
Real Data
For real data experiement, we first applied D-PPCA for SfM on the Caltech 3D Objects on Turntable
dataset [20]. The dataset provides various objects rotating on a turntable under different lighting
conditions. The views of most objects were taken every 5? which make it challenging to extract
feature points with correspondence across frames. Instead, we used a subset of the dataset which
provides views taken every degree. This subset contains images of 5 objects. To simulate multiple
cameras, we adopted a setting similar to that of [6]. We first extracted first 30? images of each
object. We then used KLT [21] implementation in Voodoo Camera Tracker1 to extract feature points
with correspondence. Lastly, we sequentially and equally partitioned the 30 images into 5 nodes to
simulate 5 cameras. Thus, each camera observes 6 frames. Table 1 shows the 5 objects and statistics
of feature points we extracted from the objects. We used ? = 10 and convergence criterion 10?3 .
Due to the lack of the ground truth 3D coordinates, we compared the subspace angles between the
structure inferred using the traditional centralized SVD-based SfM and the D-PPCA-based SfM.
Results are shown in Table 1 as the mean and variance of 20 independent runs. 10% MAR and
MNAR results are also provided in the table.
Experimenal results indicate existance of differences between the reconstructions obtained by centralized factorization approach and that of D-PPCA. However, the differences are small, depend on
the object in question, and almost always include, within their confidence, the factorization result.
Qualitative examination reveals no noticable differences. Moreover, re-projecting back to the camera coordinate space resulted in close matching with the tracked feature points, as shown in videos
provided in supplementary materials.
We also tested the utility of D-PPCA for SfM on the Hopkins155 dataset [22]. We adopted virtually
identical experimental setting as in [6]. We collected 135 single-object sequences containing image
coordinates of points and we simulated multi-camera setting by partitioning the frames sequentially
and almost equally for 5 nodes and the network was connected using ring topology. Again, we
computed maximum subspace angle between centralized SVD-based SfM and distributed D-PPCA
for SfM. We chose the convergence criterion as 10?3 . Average maximum subspace angle between
1
http://www.digilab.uni-hannover.de/docs/manual.html
7
Table 1: Caltech 3D Objects on Turntable dataset statistics and quantitative results. Green dots indicate feature points tracked with correspondance across all 30 frames. All results ran 20 independent
initializations. MAR results provide variances over both various initializations and missing value
settings.
Object
BallSander
BoxStuff
Rooster
Standing
StorageBin
# Points
# Frames
62
67
189
310
102
30
30
30
30
30
Subspace angle b/w centralized SVD SfM and D-PPCA (degree)
Mean
1.4848
1.4397
1.4767
2.6221
0.4463
Variance
0.4159
0.4567
0.9448
1.6924
1.2002
Subspace angle b/w fully observable centralized PPCA SfM and D-PPCA with MAR (degree)
Mean
6.2991
2.1556
5.2506
7.6492
2.8358
Var.(init)
4.3562
0.1351
3.8810
6.6424
1.3591
Var.(miss)
0.5729
0.0161
0.1755
0.7603
0.0444
Subspace angle b/w fully observable centralized PPCA SfM and D-PPCA with MNAR (degree)
Mean
3.1405
6.4664
5.8027
9.2661
3.7965
Variance
0.0124
3.1955
2.4333
2.9720
0.0089
D-PPCA for SfM and SVD-based SfM for all objects was 3.97? with variance 7.06. However,
looking into the result more carefully, we found that even with substantially larger subspace angle,
3D structure estimates were similar to that of SVD-based SfM only with orthogonal ambiguity issue.
Moreover, more than 53% of all objects yielded the subspace angle below 1? , 77% of them below
5? and more than 94% were less than 15? . With 10% MAR, we obtained the mean 20.07? with
variance 27.94? with about 18% of them below 1? , 56% of them below 5? and more than 70% of
them less than the mean. We could not perform MNAR experiments on Hopkins as the ground truth
occlusion information is not provided with the dataset.
6
Discussion and Future Work
In this work we introduced a general approach for learning parameters of traditional centralized
probabilistic models, such as PPCA, in a distributed setting. Our synthetic data experiments showed
that the proposed algorithm is robust to choices of initial parameters and, more importantly, is not
adversely affected by variations in network size, topology or missing values. In the SfM problems,
the algorithm can be effectively used to distribute computation of 3D structure and motion in camera
networks, while retaining the probabilistic nature of the original model.
Despite its promising performance D-PPCA for SfM exhibits some limitations. In particular, we
assume the independence of the affine motion matrix parameters in (15). The assumption is clearly
inconsistent with the modeling of motion on the SE(3) manifold. However, our experiments demonstrate that, in practice, this violation is not crucial. This shortcoming can be amended in one of several possible ways. One can reduce the iid assumption of individual samples to that of subsequent
columns (i.e., full 3x2 motion matrices). Our additional experiments, not reported here, indicate
no discernable utility of this approach. A more principled approach would be to define priors for
motion matrices compatible with SE(3), using e.g., [23]. While appealing, the priors would render
the overall model non-linear and would require additional algorithmic considerations, perhaps in the
spirit of [1].
Acknowledgments
This work was supported in part by the National Science Foundation under Grant No. IIS 0916812.
8
References
[1] Roberto Tron and Rene Vidal. Distributed Computer Vision Algorithms. IEEE Signal Processing Magazine, 28:32?45, 2011.
[2] A.Y. Yang, S. Maji, C.M. Christoudias, T. Darrell, J. Malik, and S.S. Sastry. Multiple-view Object Recognition in Band-limited Distributed Camera Networks. In Distributed Smart Cameras, 2009. ICDSC 2009.
Third ACM/IEEE International Conference on, 30 2009-sept. 2 2009.
[3] Richard J. Radke. A Survey of Distributed Computer Vision Algorithms. In Hideyuki Nakashima, Hamid
Aghajan, and Juan Carlos Augusto, editors, Handbook of Ambient Intelligence and Smart Environments.
Springer Science+Business Media, LLC, 2010.
[4] A. Wiesel and A.O. Hero. Decomposable Principal Component Analysis. Signal Processing, IEEE
Transactions on, 57(11):4369?4377, 2009.
[5] Sergio V. Macua, Pavle Belanovic, and Santiago Zazo. Consensus-based Distributed Principal Component
Analysis in Wireless Sensor Networks. In Signal Processing Advances in Wireless Communications
(SPAWC), 2010 IEEE Eleventh International Workshop on, pages 1?5, June 2010.
[6] Roberto Tron and Rene Vidal. Distributed Computer Vision Algorithms Through Distributed Averaging.
In IEEE Conference on Computer Vision and Pattern Recognition, pages 57?63, 2011.
[7] Lin Xiao, Stephen Boyd, and Sanjay Lall. A Scheme for Robust Distributed Sensor Fusion Based on
Average Consensus. In International Conference on Information Processing in Sensor Networks, pages
63?70, April 2005.
[8] R. Olfati-Saber. Distributed Kalman Filtering for Sensor Networks. In Decision and Control, 2007 46th
IEEE Conference on, pages 5492 ?5498, dec. 2007.
[9] Bi Song, A.T. Kamal, C. Soto, Chong Ding, J.A. Farrell, and A.K. Roy-Chowdhury. Tracking and Activity
Recognition Through Consensus in Distributed Camera Networks. Image Processing, IEEE Transactions
on, 19(10):2564 ?2579, oct. 2010.
[10] P.A. Forero, A. Cano, and G.B. Giannakis. Distributed Clustering Using Wireless Sensor Networks.
Selected Topics in Signal Processing, IEEE Journal of, 5(4):707 ?724, aug. 2011.
[11] Sam Roweis and Zoubin Ghahramani. A Unifying Review of Linear Gaussian Models. Neural Computation, 11:305?345, 1999.
[12] Ami Wiesel, Yonina C. Eldar, and Alfred O. Hero. Covariance Estimation in Decomposable Gaussian
Graphical Models. IEEE Transactions on Signal Processing, 58(3):1482?1492, 2010.
[13] Andrew R. Conn, Nicholas I. M. Gould, and Philippe L. Toint. A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds. SIAM J. Numer. Anal.,
28:545?572, February 1991.
[14] Robert Michael Lewis and Virginia Torczon. A Globally Convergent Augmented Lagrangian Pattern
Search Algorithm for Optimization with General Constraints and Simple Bounds. SIAM J. on Optimization, 12:1075?1089, April 2002.
[15] Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato, and Jonathan Eckstein. Distributed Optimization
and Statistical Learning via the Alternating Direction Method of Multipliers. In Michael Jordan, editor,
Foundations and Trends in Machine Learning, volume 3, pages 1?122. Now Publishers, 2011.
[16] Pedro A. Forero, Alfonso Cano, and Geogios B. Giannakis. Consensus-Based Distributed Support Vector
Machines. Journal of Machine Learning Research, 11:1663?1707, 2010.
[17] Michael E. Tipping and Chris M. Bishop. Probabilistic Principal Component Analysis. Journal of the
Royal Statistical Society, Series B, 61:611?622, 1999.
[18] Alexander Ilin and Tapani Raiko. Practical Approaches to Principal Component Analysis in the Presence
of Missing Values. Journal of Machine Learning Research, 11:1957?2000, 2010.
[19] Carlo Tomasi and Takeo Kanade. Shape and motion from image streams under orthography: a factorization method. International Journal of Computer Vision, 9:137?154, 1992. 10.1007/BF00129684.
[20] Pierre Moreels and Pietro Perona. Evaluation of Features Detectors and Descriptors based on 3D Objects.
International Journal of Computer Vision, 73(3):263?284, July 2007.
[21] Carlo Tomasi and Takeo Kanade. Detection and Tracking of Point Features. Technical Report CMU-CS91-132, Carnegie Mellon University, April 1991.
[22] Roberto Tron and Rene Vidal. A Benchmark for the Comparison of 3-D Motion Segmentation Algorithms. In IEEE International Conference on Computer Vision and Pattern Recognition, 2007.
[23] Yasuko Chikuse. Statistics on Special Manifolds, volume 174 of Lecture Notes in Statistics. Springer, 1
edition, February 2003.
9
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4,010 | 463 | Illumination and View Position in 3D Visual
Recognition
Amnon Shashua
M.LT. Artificial Intelligence Lab., NE43-737
and Department of Brain and Cognitive Science
Cambridge, MA 02139
Abstract
It is shown that both changes in viewing position and illumination conditions can be compensated for, prior to recognition, using combinations
of images taken from different viewing positions and different illumination conditions. It is also shown that, in agreement with psychophysical
findings, the computation requires at least a sign-bit image as input contours alone are not sufficient.
1
Introduction
The task of visual recognition is natural and effortless for biological systems, yet
the problem of recognition has been proven to be very difficult to analyze from
a computational point of view. The fundamental reason is that novel images of
familiar objects are often not sufficiently similar to previously seen images of that
object. Assuming a rigid and isolated object in the scene, there are two major
sources for this variability: geometric and photometric. The geometric source of
variability comes from changes of view position. A 3D object can be viewed from a
variety of directions, each resulting with a different 2D projection. The difference is
significant, even for modest changes in viewing positions, and can be demonstrated
by superimposing those projections (see Fig. 4, first row second image). Much
attention has been given to this problem in the visual recognition literature ([9],
and references therein), and recent results show that one can compensate for changes
in viewing position by generating novel views from a small number of model views
of the object [10, 4, 8].
404
Illumination and View Position in 3D Visual Recognition
Figure 1: A 'Mooney' image. See text for details.
The photometric source of variability comes from changing illumination conditions
(positions and distribution of light sources in the scene). This has the effect of
changing the brightness distribution in the image, and the location of shadows
and specular reflections. The traditional approach to this problem is based on the
notion of edge detection. The idea is that discontinuities in image brightness remain
stable under changes of illumination conditions. This invariance is not complete
and furthermore it is an open question whether this kind of contour information is
sufficient, 01? even relevant, for purposes of visual recognition.
Consider the image in Fig. 1, adopted from Mooney's Closure Faces Test [6]. Most
observers show no difficulty in interpreting the shape of the object from the righthand image, but cannot identify the object when presented with only the contours.
Also, many of the contours are shadow contours and therefore critically rely on the
direction of light source. In Fig. 2 four frontal images of a doll from four different
illumination conditions are shown together with their intensity step edges. The
change in the contour image is significant and is not limited to shadow contours
- some object edges appear or disappear as a result of the change in brightness
distribution. Also shown in Fig. 4 is a sign-bit image of the intensity image followed
by a convolution with a Difference of Gaussians. As with the Mooney image, it is
considerably more difficult to interpret the image of a complex object with only the
zero-crossing (or level-crossing) contours than when the sign-bits are added.
It seems, therefore, that a successful recognition scheme should be able to cope
with changes in illumination conditions, as well as changes in viewing positions, by
working wit.h a richer source of information than just contours (for a different point
of view, see [1]). The minimal information that seems to be sufficient, at least for
coping with the photometric problem, is the sign-bit image.
The approach to visual recognition in this study is in line with the 'alignment'
approach [9] and is also inspired by the work of Ullman and Basri [10] who show that
the geometric source of variability can be handled by matching the novel projection
to a linear combination of a small number of previously seen projections of that
object. A recognition scheme that can handle both the geometric and photometric
sources of variability is suggested by introducing three new results: (i) any image of a
surface with a linear reflectance function (including Lambertian and Phong's model
without point specularities) can be expressed as a linear combination of a fixed
set of three images of that surface taken under different illumination conditions,
(ii) from a computational standpoint, the coefficients are better recovered using the
405
406
Shashua
sign-bit image rather than the contour image, and (iii) one can compensate for both
changes in viewing position and illumination conditions by using combinations of
images taken from different viewing positions and different illumination conditions.
2
Linear Combination of Images
We start by assuming that view position is fixed and the only parameter that is
allowed to change is the positions and distribution oflight sources. The more general
result that includes changes in viewing positions will be discussed in section 4.
Proposition 1 All possible images of a surface, with a linear reflectance function,
generated by all possible illumination conditions (positions and distribution of light
sources) are spanned by a linear combination of images of the 8urface taken from
independent illumination conditions.
Proof: Follows directly from the general result that if /j (x), x E Rk, j = 1, ... , k,
are k linear functions, which are also linearly independent, then for any linear
function f(x), we have that f(x) = Lj aj!i(x), for some constants aj.
0
The simplest case for which this result holds is the Lambertian reflectance model
under a point light source (observed independently by Yael Moses, personal communication). Let r be an object point projecting to p . Let nr represent the normal
and albedo at r (direction and magnitude), and s represent the light source and
its intensity. The brightness at p under the Lambertian model is I(p) = nr . 8,
and because 8 is fixed for all point p, we have I(p) = al II (p) + a2h(p) + a313(p)
where Ij(p) is the brightness under light source 8j and where 81,82,83 are linearly
independent. This result generalizes, in a straightforward manner, to the case of
multiple light sources as well.
The Lambertian model is suitable for matte surfaces, i.e. surfaces that diffusely
reflect incoming light rays. One can add a 'shininess' component to account for
the fact that for non-ideal Lambertian surfaces, more light is reflected in a direction making an equal angle of incidence with reflectance. In Phong's model of
reflectance [7] this takes the form of (n r . h)C where h is the bisector of 8 and
the viewer's direction v. The power constant c controls the degree of sharpness of
the point specularity, therefore outside that region one can use a linear version of
Phong's model by replacing the power constant with a multiplicative constant, to
get the following function: I(p) = nr . [8 + p( v + 8)]. As before, the bracketed vector
is fixed for all image points and therefore the linear combination result holds.
The linear combination result suggests therefore that changes in illumination can
be compensated for, prior to recognition, by selecting three points (that are visible
to 8,81,82,83) to solve for aI, a2, a3 and then match the novel image I with I'
Lj aj I j . The two images should match along all points p whose object points rare
visible to 81, S2, 83 (even if nr ?8 < 0, i.e. p is attached-shadowed); approximately
match along points for which nr . Sj < 0, for some j (Ij(p) is truncated to zero,
geometrically 8 is projected onto the subspace spanned by the remaining basis light
sources) and not match along points that are cast-shadowed in I (nr . 8 > but
r is not visible to 8 because of self occlusion). Coping with cast-shadows is an
important task, but is not in the scope of this paper .
=
?
Illumination and View Position in 3D Visual Recognition
Figure 2: Linear combination of model images taken from the same viewing positIOn
and under different illumination conditions. Row 1,2: Three model images taken under
a varying point light source, and the input image, and their brightness edges. Row 3:
The image generated by the linear combination of the model images, its edges, and the
difference edge image between the input and generated image.
The linear combination result also implies that, for the purposes of recognition, one
does not need to recover shape or light source direction in order to compensate for
changes in hrightness distribution and attached shadows. Experimental results, on
a non-ideal Lambertian surface, are shown in Fig. 2.
3
Coefficients fronl Contours and Sign-bits
Mooney pictures, such as in Fig. 1, demonstrate that humans can cope well with
situations of varying illumination by using only limited information from the input
image, namely the sign-bits, yet are not able to do so from contours alone. This
observation can be predicted from a computational standpoint, as shown below.
Proposition 2 The coejJiczents that span an image I by the basis of three other
images, as descnbed in proposition 1, can be solved, up to a common scale factor,
407
408
Shashua
Figure 3: Compensating for both changes in view and illumination. Row 1: Three model
images, one of which is taken from a different viewing direction (23 0 apart), and the input
image from a novel viewing direction (in between the model images) and illumination
condition. Row 2: difference image between the edges of the input image (shown separately
in Fig. 4) and the edges of the view transformed first model image (first row, lefthand),
the final generated image (linear combination of the three transformed model images), its
edges, and the difference image between edges of input and generated image.
from just the contours of I -
zero-crossings or level-crossings.
=
Proof: Let aj be the coefficients that span I by the basis images Ij, j
1,2,3, i.e.
I
Lj aj Ij. Let f, J; be the result of applying a Difference of Gaussians (DOG)
operator, with the same scale, on images I, I j , j
1,2,3. Since DOG is a linear
operator we have that f = Lj aj J;. Since J(p) = 0 along zero-crossing points p of
I, then by taking any three zero-crossing points, which are not on a cast-shadow
border, we get a homogeneous set of equations from which aj can be solved up to
a common scale factor.
=
=
Similarly, let k be an unknown threshold applied to I. Therefore, along level crossLj aj I j , hence 4 level-crossing points, that are visible to all
ings of I we have k
four light sources, are sufficient to solve for aj and k.
=
D
This result is in accordance with what is known from image compression literature
of reconstructing an image, up to a scale factor, from contours alone [2]. In both
cases, here and in image compression, this result may be difficult to apply in practice
because the contours are required to be given at sub-pixel accuracy. One can relax
the accuracy requirement by using the gradients along the contours - a technique
that works well in practice. Nevertheless, neither gradients nor contours at subpixel accuracy are provided by Mooney pictures, which leaves us with the sign- bits
as the source of information for solving for the coefficients.
Illumination and View Position in 3D Visual Recognition
Figure 4: Compensating for changes in viewing position and illumination from a single
view (model images are all from a single viewing position). Model images are the same
as in Fig. 2, input image the same as in Fig. 3. Row 1: edges of input image, overlay
of input edge image and edges of first model image, overlay with edges of the 2D affine
transformed first model image, sign-bit input image with marked 'example' locations (16
of them). Row 2: linear combination image of the 2D affine transformed model images,
the final generated image, its edges, overlay with edges of the input image.
Proposition 3 Solving for the coefficients from the sign- bit image of I is equtvalent to solving for a separating hyperplane in 3D in which image points serve as
'examples '.
=
=
Proof: Let z(p)
(II, 12, hf be a vector function and w (aI, a2, a3)T be the
unknown weight vector. Given the sign-bit image j of I, we have that for every
point p, excluding zero-crossings, the scalar product w T z(p) is either positive or
negative. In this respect , one can consider points in j as 'examples' in 3D space
and the coefficients aj as a vector norma) to the separating hyperplane.
0
A similar result can be obtained for the case of a thresholded image. The separating
hyperplane in that case is defined in 4D, rather than 3D. Many schemes for finding a
separating hyperplane have been described in Neural Network literature (see [5] for
review) and in Discriminant Analysis literature ([3], for example). Experimental
results shown in the next section show that 10-20 points, distributed over the
entire object, are sufficient to produce results that are indistinguishable from those
obtained from an exact solution.
By using the sign-bits instead of the zero-crossing contours we are trading a unique
(up to a scale factor), but unstable, solution for an approximate, but stable, one.
Also , by taking the sample points relatively far away from the contours (in order to
minimize the chance of error) the scheme can tolerate a certain degree of misalign-
409
410
Shashua
ment between the basis images and the novel image. This property will be used
in one of the schemes, described below, for combining changes of viewing positions
and illumination conditions.
4
Changing Illumination and Viewing Positions
In this section, the recognition scheme is generalized to cope with both changes in
illumination and viewing positions. Namely, given a set of images of an object as
a model and an input image viewed from a novel viewing position and taken under
a novel illumination condition we would like to generate an image, from the model,
that is similar to the input image.
Proposition 4 Any set of three images, satisfying conditions of proposition 1, of
an object can be used to compensate for both changes in view and illumination.
Proof: Any change in viewing position will induce both a change in the location
of points in the image, and a change in their brightness (because of change in
viewing angle and change in angle between light source and surface normal). From
proposition 1, the change in brightness can be compensated for provided all the
images are in alignment. What remains, therefore, is to bring the model images
and the input image into alignment.
Case 1: If each of the three model images is viewed from a different position, then
the remaining proof follows directly from the result of Ullman and Basri [10] who
show that any view of an object with smooth boundaries, undergoing any affine
transformat.ion in space, is spanned by three views of the object.
Case 2: If only two of the model images are viewed from different positions, then
given full correspondence between all points in the two model views and 4 corresponding points with the input image, we can transform all three model images
to align wit.h the input image in the following way. The 4 corresponding points
between the input image and one of the model images define three corresponding
vectors (taking one of the corresponding points, say 0, as an origin) from which a 2D
affine transformation, ma.trix A and vector w, can be recovered. The result, proved
in [8], is tha.t for every point p' in the input image who is in correspondence with p
in the model image we have that p' = [Ap + 0' - Ao] + apw. The parameter a p is
invariant to any affine transformation in space, therefore is also invariant to changes
in viewing position. One can, therefore, recover a p from the known correspondence
between two model images and use that to predict the location p'. It can be shown
that this scheme provides also a good approximation in the case of objects with
smooth boundaries (like an egg or a human head, for details see [8]).
Case 3: All three model images are from the same viewing position. The model
images are first brought into 'rough alignment' (term adopted from (10)) with the
input image by applying the transformation Ap + 0' - Ao + w to all points p in each
model image. The remaining displacement between the transformed model images
and the input image is (a p - l)w which can be shown to be bounded by the depth
variation of the surface [8]. (In case the object is not sufficiently fiat, more than
4 points may be used to define local transformations via a triangulation of those
points). The linear combination coefficients are then recovered using the sign-bit
Illumination and View Position in 3D Visual Recognition
scheme described in the previous section. The three transformed images are then
linearly combined to create a new image that is compensated for illumination but
is still displaced from the input image. The displacement can be recovered by using
a brightness correlation scheme along the direction w to find Q p - 1 for each point
p. (for details, see [B]).
0
Experimental results of the last two schemes are shown in Figs. 3 and 4. The four
corresponding points, required for view compensation, were chosen manually along
the tip of eyes, eye-brow and mouth of the doll. The full correspondence that is
required between the third model view and the other two in scheme 2 above, was established by first taking two pictures of the third view, one from a novel illumination
condition and the other from a similar illumination condition to one of the other
model images. Correspondence was then determined by using the scheme described
in [B]. The extra picture was then discarded. The sample points for the linear
combination were chosen automatically by selecting 10 points in smooth brightness
regions. The sample points using the sign-bit scheme were chosen manually.
5
Summary
It has been shown that the effects photometry and geometry in visual recognition
can be decoupled and compensated for prior to recognition. Three new results were
shown: (i) photometric effects can be compensated for using a linear combination
of images, (ii) from a computational standpoint, contours alone are not sufficient
for recognition, and (iii) geometrical effects can be compensated for from any set of
three images, from different illuminations, of the object.
Acknowledgments
I thank Shimon Ullman for his advice and support. Thanks to Ronen Basri, Tomaso
Poggio, Whitman Richards and Daphna Weinshall for many discussions. A.S. is
supported by NSF grant IRI-B900267.
References
[1] Cavana.gh,P. Proc. 19th ECVP, Andrei, G. (Ed.), 1990.
[2] Curtis,S.R and Oppenheim,A.V. in Whitman,R. and Ullman,S. (eds.) Image Understanding 1989. pp.92-110, Ablex, NJ 1990.
[3] Duda,R.O. and Hart,P.E. pattern classification and scene analysis. NY, Wiley 1973.
[4] Edelman,S. and Poggio,T. Massachusetts Institute of Technology, A.I. Memo 1181,
1990
[5] Lippmann,R.P. IEEE ASSP Magazine, pp.4-22, 1987.
[6] Mooney,C.M. Can. 1. Psychol. 11:219-226, 1957.
[7] Phong,B.T. Comm. A CM, 18, 6:311-317, 1975.
[8] Shashua,A. Massachusetts Institute of Technology, A.I. Memo 1927, 1991
[9] Ullman,S. Cognition,32:193-254, 1989.
[10] Ullman,s. and Basri,R. Massachusetts Institute of Technology, A.I. Memo 1052, 1989
411
| 463 |@word version:1 compression:2 seems:2 duda:1 open:1 closure:1 descnbed:1 brightness:10 selecting:2 recovered:4 incidence:1 yet:2 visible:4 shape:2 alone:4 intelligence:1 leaf:1 provides:1 location:4 along:8 edelman:1 ray:1 manner:1 tomaso:1 nor:1 brain:1 compensating:2 inspired:1 automatically:1 provided:2 bounded:1 what:2 weinshall:1 kind:1 cm:1 finding:2 transformation:4 nj:1 every:2 brow:1 control:1 grant:1 appear:1 before:1 positive:1 accordance:1 local:1 approximately:1 ap:2 therein:1 suggests:1 limited:2 unique:1 acknowledgment:1 practice:2 displacement:2 coping:2 projection:4 matching:1 induce:1 get:2 cannot:1 onto:1 operator:2 effortless:1 applying:2 demonstrated:1 compensated:7 straightforward:1 attention:1 iri:1 independently:1 sharpness:1 wit:2 spanned:3 his:1 handle:1 notion:1 variation:1 magazine:1 exact:1 homogeneous:1 origin:1 agreement:1 crossing:9 recognition:19 satisfying:1 richards:1 observed:1 solved:2 region:2 ne43:1 comm:1 personal:1 ings:1 solving:3 ablex:1 serve:1 basis:4 whitman:2 artificial:1 outside:1 whose:1 richer:1 solve:2 say:1 relax:1 transform:1 final:2 ment:1 product:1 relevant:1 combining:1 lefthand:1 requirement:1 produce:1 generating:1 object:20 transformat:1 ij:4 predicted:1 shadow:6 come:2 implies:1 trading:1 direction:9 norma:1 human:2 viewing:21 ao:2 proposition:7 biological:1 viewer:1 hold:2 sufficiently:2 normal:2 scope:1 predict:1 cognition:1 major:1 a2:2 albedo:1 purpose:2 proc:1 create:1 brought:1 rough:1 rather:2 varying:2 subpixel:1 rigid:1 lj:4 entire:1 transformed:6 pixel:1 classification:1 equal:1 manually:2 photometric:5 familiar:1 geometry:1 occlusion:1 detection:1 righthand:1 alignment:4 light:14 edge:16 poggio:2 modest:1 decoupled:1 isolated:1 minimal:1 introducing:1 rare:1 successful:1 considerably:1 combined:1 thanks:1 fundamental:1 tip:1 together:1 reflect:1 cognitive:1 ullman:6 account:1 fronl:1 includes:1 coefficient:7 bracketed:1 multiplicative:1 view:21 observer:1 lab:1 analyze:1 shashua:5 start:1 recover:2 hf:1 minimize:1 accuracy:3 who:3 identify:1 ronen:1 critically:1 mooney:6 oppenheim:1 ed:2 pp:2 proof:5 proved:1 massachusetts:3 fiat:1 tolerate:1 reflected:1 a2h:1 furthermore:1 just:2 correlation:1 working:1 replacing:1 aj:10 effect:4 hence:1 indistinguishable:1 self:1 generalized:1 complete:1 demonstrate:1 reflection:1 interpreting:1 bring:1 geometrical:1 gh:1 image:103 novel:9 common:2 diffusely:1 attached:2 phong:4 discussed:1 interpret:1 significant:2 cambridge:1 ai:2 similarly:1 stable:2 surface:9 add:1 align:1 recent:1 triangulation:1 apart:1 certain:1 seen:2 ii:4 multiple:1 full:2 smooth:3 match:4 compensate:4 hart:1 represent:2 ion:1 separately:1 source:20 standpoint:3 extra:1 ideal:2 iii:2 variety:1 specular:1 idea:1 amnon:1 whether:1 handled:1 simplest:1 generate:1 overlay:3 nsf:1 moses:1 sign:14 four:4 threshold:1 nevertheless:1 changing:3 neither:1 thresholded:1 geometrically:1 angle:3 bit:14 followed:1 correspondence:5 scene:3 span:2 relatively:1 department:1 combination:16 remain:1 reconstructing:1 making:1 projecting:1 invariant:2 taken:8 equation:1 previously:2 remains:1 adopted:2 generalizes:1 gaussians:2 yael:1 doll:2 apply:1 lambertian:6 away:1 remaining:3 daphna:1 reflectance:5 disappear:1 psychophysical:1 question:1 added:1 traditional:1 nr:6 gradient:2 subspace:1 thank:1 separating:4 discriminant:1 unstable:1 reason:1 assuming:2 difficult:3 memo:3 negative:1 unknown:2 convolution:1 observation:1 displaced:1 discarded:1 compensation:1 truncated:1 situation:1 variability:5 communication:1 excluding:1 head:1 assp:1 intensity:3 cast:3 namely:2 dog:2 required:3 photometry:1 established:1 discontinuity:1 able:2 suggested:1 below:2 pattern:1 including:1 shadowed:2 mouth:1 power:2 suitable:1 natural:1 difficulty:1 rely:1 scheme:13 technology:3 eye:2 picture:4 psychol:1 text:1 prior:3 geometric:4 literature:4 review:1 understanding:1 specularity:1 proven:1 degree:2 affine:5 sufficient:6 row:8 summary:1 supported:1 last:1 institute:3 face:1 taking:4 distributed:1 boundary:2 depth:1 contour:20 projected:1 far:1 cope:3 sj:1 approximate:1 lippmann:1 basri:4 incoming:1 curtis:1 complex:1 linearly:3 s2:1 border:1 allowed:1 fig:10 advice:1 egg:1 andrei:1 ny:1 wiley:1 sub:1 position:30 third:2 shimon:1 rk:1 undergoing:1 a3:2 magnitude:1 illumination:31 lt:1 specularities:1 visual:10 expressed:1 trix:1 scalar:1 chance:1 tha:1 ma:2 viewed:4 marked:1 change:26 determined:1 hyperplane:4 invariance:1 superimposing:1 experimental:3 bisector:1 support:1 frontal:1 |
4,011 | 4,630 | Compressive Sensing MRI with Wavelet Tree Sparsity
Chen Chen and Junzhou Huang
Department of Computer Science and Engineering
University of Texas at Arlington
[email protected]
[email protected]
Abstract
In Compressive Sensing Magnetic Resonance Imaging (CS-MRI), one can reconstruct a MR image with good quality from only a small number of measurements.
This can significantly reduce MR scanning time. According to structured sparsity
theory, the measurements can be further reduced to O(K + log n) for tree-sparse
data instead of O(K + K log n) for standard K-sparse data with length n. However, few of existing algorithms have utilized this for CS-MRI, while most of them
model the problem with total variation and wavelet sparse regularization. On the
other side, some algorithms have been proposed for tree sparse regularization, but
few of them have validated the benefit of wavelet tree structure in CS-MRI. In
this paper, we propose a fast convex optimization algorithm to improve CS-MRI.
Wavelet sparsity, gradient sparsity and tree sparsity are all considered in our model
for real MR images. The original complex problem is decomposed into three simpler subproblems then each of the subproblems can be efficiently solved with an
iterative scheme. Numerous experiments have been conducted and show that the
proposed algorithm outperforms the state-of-the-art CS-MRI algorithms, and gain
better reconstructions results on real MR images than general tree based solvers
or algorithms.
1
Introduction
Magnetic Resonance Imaging (MRI) is widely used for observing the tissue changes of the patients
within a non-invasive manner. One limitation of MRI is its imaging speed, including both scanning
speed and reconstruction speed. Long waiting time and slow scanning may result in patients? annoyance and blur on images due to local motion such as breathing, heart beating etc. According to
compressive sensing (CS) [1,2] theory, only a small number of measurements is enough to recover
an image with good quality. This is an extension of Nyquist-Shannon sampling theorem when data
is sparse or can be sparsely represented. Compressive Sensing Magnetic Resonance Imaging (CSMRI) becomes one of the most successful applications of compressive sensing, since MR scanning
time is directly related to the number of sampling measurements [3]. As most images can be transferred to some sparse domain (wavelet etc.), only O(K + K log n) samples are enough to obtain
robust MR image reconstruction.
Actually, this result can be better. Recent works on structured sparsity show that the required number
of sampling measurements could be further reduced to O(K + log n) by exploring the tree structure
[4-6]. A typical relationship in tree sparsity is that, if a parent coefficient has a large/small value,
its children also tend to be large/small. Some methods have been proposed to improve standard CS
reconstruction by utilizing this prior. Specially, two convex models are proposed to handle the treebased reconstruction problem [7]. They apply SpaRSA [11] to solve their models, with a relatively
slow convergence rate. In Bayesian compressive sensing, Markov Chain Monte Carlo (MCMC) and
variational Bayesian (VB) are used to solve the tree-based hierarchical models [8][9]. Turbo AMP
[10] also well exploits tree sparsity for compressive sensing with an iterative approximate message
1
passing approach. However, none of them has conducted numerous experiments on MR images to
validate their superiority.
In existing CS-MRI models, the linear combination of total variation and wavelet sparse regularization is very popular [3,12-15]. Classical conjugate gradient decent method is first used to solve
this problem [3]. TVCMRI [12] and RecPF [13] use an operator-splitting method and a variable
splitting method to solve this problem respectively. FCSA [14,15] decomposes the original problem
into two easy subproblems and separately solves each of them with FISTA [16,17]. They are the
state-of-the-art algorithms for CS-MRI, but none of them utilizes tree sparsity prior to enhance their
performance.
In this paper, we propose a new model for CS-MRI, which combines wavelet sparsity, gradient
sparsity and tree sparsity seamlessly. In tree structure modeling, we assign each pair of parentchild wavelet coefficients to one group, which forces them to be zeros or non-zeros simultaneously.
This is an overlapping group problem and hard to be solved directly. A new variable is introduced
to decompose this problem to three simpler subproblems. Then each of subproblems has closed
form solution or can be solved efficiently by existing techniques. We conduct extensive experiments
to compare the proposed algorithm with the state-of-the-art CS-MRI algorithms and several tree
sparsity algorithms. The proposed algorithm always achieves the best results in terms of SNR and
computational time.
Our contribution can be summarized as: (1) We introduce the wavelet tree sparsity to CS-MRI, and
provide a convex formulation to model the tree-based structure combining with total variation and
wavelet sparsity; (2) An efficient algorithm with fast convergence performance is proposed in this
paper to solve this model. Each iteration only cost O(n log n) time.(3) Numerous experiments have
been conducted to compare the proposed algorithm with the state-of-the-art CS-MRI algorithms
and several general tree-based algorithms or solvers. The results show that the proposed algorithm
outperforms all others on real MR images.
2
2.1
Related work
Tree based compressive sensing
If a signal is sparse or can be sparsely represented, the necessary samples to reconstruct it can
be significantly smaller than that needed in Nyquist-Shannon sampling theorem. Moreover, if we
know some prior about the structure of the original signal, such as group or graph structure, the
measurements can be further reduced [4,5]. Some previous algorithms have utilized the tree structure
of wavelet coefficients to improve CS reconstruction [7-10].
OGL [7] is a convex approach to model the tree structure:
n
X
1
1 XX
?? = arg min{F (?) = ||b ? A?T ?||22 + ?g
(?i ? ?ij )2 }
||?eg ||2 + ? 2
?
2
2 i=1
g?G
(1)
j?Ji
where ? is a set of the wavelet coefficients. A represents a partial Fourier transform for MR reconstruction problem and b is the measurements data. ?T denotes the inverse wavelet transform.
G denotes the set of all parent-child groups and g is one of such groups. ?e is an extended vector
of ? with replicates and the last term is a penalty to force the replicates to be the same. When
wavelet coefficients are recovered, they can be transformed to the recovered image by an inverse
wavelet transform.This method well explores the tree structure assumption, but may be slow in general as following reasons: a) the parent-child relationship in their model is hard to maintain; b) it
applies SpaRSA [11] to solve (1). Overall, their method can only achieve a convergence rate of
F (?k ) ? F (?? ) ' O(1/k) [16], where k is the iteration number and ?? is the optimal solution.
In statistical learning, AMP [10], MCMC [8], and VB [9] all solve (2) with probabilistic inference.
In (2), x is the original image to be reconstructed and w is Gaussian white noise. In these approaches, graphical models are used to represent the wavelet tree structure and the distribution of
each coefficient is decided by its parent?s value.
y = Ax + w = A?T ? + w
2
(2)
2.2
Efficient MR image reconstruction algorithms
In existing CS-MRI algorithms, the linear combination of total variation and wavelet sparsity constrains has shown good property for MR images. Recent fastest algorithms all attempt to solve
(3) in less computational
parameters, and ? denotes the wavelet
P time.
P p? and ? 2are two positive
2 , where ? and ? denote the forward fitransform. kxkT V =
(?
x
)
+
(?
x
)
1 ij
2 ij
1
2
i
j
nite difference operators on the first and second coordinates. TVCMRI [12] and RecPF [13] use an
operator-splitting method and a variable splitting method to solve this problem respectively. FCSA
[14,15] decomposes this problem into 2 simpler problems and solves them with FISTA respectively.
The convergence rate of FISTA is O(1/k 2 ). These approaches are very effective on real MR image
reconstruction, but none of them utilizes the wavelet tree structure to get further enhancement.
1
x
? = arg min{ kAx ? bk22 + ?kxkT V + ?k?xk1 }
x
2
2.3
(3)
Convex overlapped group sparsity solvers
SLEP [18] (Sparse Learning with Efficient Projections) has the package for tree structured group
lasso (4). Its main function is to iteratively solve the tree structured denoising problem. When it
comes to reconstruction problem, it applies FISTA to transfer the problem to denoising.
1
x
? = arg min{ kAx ? bk22 + ?k?xktree }
x
2
(4)
YALL1 [19] (Your Algorithms for L1) can solve the general overlapping group sparse problem efficiently. We put it in comparisons too. It first relaxes the constrained overlapping group minimization
to unconstrained problem by Lagrangian method. Then the minimization of the x and z subproblems
can be written as:
x
? = arg min{
x,z
s
X
?2
?1
wi ||zi ||2 }
kAx ? bk22 + ?T1 G?x + ||z ? G?x||22 ? ?T2 Ax +
2
2
i=1
(5)
where G indicates the grouping index with all its elements to be 1 or 0. s is the total number of
groups. ?1 , ?2 are multipliers and ?1 , ?2 are positive parameters.
3
Algorithm
Observations tell us that the wavelet coefficients of real MR images tend to be quadtree structured
[20], although not strictly. Moreover they are generally sparse in wavelet and gradient domain. So
we utilize all the sparse prior in our model. A new algorithm called Wavelet Tree Sparsity MRI
(WaTMRI) is proposed to efficiently solve this model. Tree-based MRI problem can be formulated
as follows:
min{F (x) =
x
X
1
kAx ? bk22 + ?kxkT V + ?(k?xk1 +
||?xg ||2 )}
2
(6)
g?G
The total variation and L1 term in fact have complemented the tree structure assumption, which
make our model more robust on real MR images. This is a main difference with previous tree
structured algorithms or solvers. However, this problem can not be solved efficiently. We introduce
3
a variable z to constrain x with overlapping structure. Then the problem becomes non-overlapping
convex optimization. Let G?x = z, (6) can be rewritten as:
min{F (x) =
x,z
s
X
1
?
kAx ? bk22 + ?kxkT V + ?(k?xk1 +
||zgi ||2 ) + ||z ? G?x||22 }
2
2
i=1
(7)
For the z subproblem, it has closed form solution by the group-wise soft thresholding. For the x
subproblem, we can combine the first and last quadratic penalty on the right side. Then the rest has
the similar form with FCSA and can be solved efficiently with an iterative scheme.
3.1
Solution
As mentioned above, z-subproblem in (7) can be written as:
zgi = arg min{?||zgi ||2 +
zgi
?
||zgi ? (G?x)gi ||22 }, i = 1, 2, ..., s
2
(8)
where gi is the i-th group and s is number of total groups. It has a closed form solution by soft
thresholding:
zgi = max(||ri ||2 ?
?
ri
, 0)
, i = 1, 2, ..., s
?
||ri ||2
(9)
where ri = (G?x)gi .
For the x-subproblem,
?
1
x = arg min{ kAx ? bk22 + ?kxkT V + ?k?xk1 + ||z ? G?x||22 }
x
2
2
(10)
Let f (x) = 21 kAx ? bk22 + ?2 ||z ? G?x||22 , which is a convex and smooth function with Lipschitz
Lf , and g1 (x) = ?kxkT V , g2 (x) = ?k?xk1 , which are convex but non-smooth functions. Then
this x problem can be solved efficiently by FCSA. For convenience, we denote (9) by
z = shrinkgroup(G?x, ?? ). Now, we can summarize our algorithm as in Algorithm 1:
Algorithm 1 WaTMRI
Input: ? = 1/Lf , r1 = x0 , t1 = 1, ?, ?, ?
for k = 1 to N do
1) z = shrinkgroup(G?xk?1 , ?/?)
2) xg = rk ? ??f (rk )
3) x1 = prox? (2?kxkT V )(xg )
4) x2 = prox? (2?(k?xk1 )(xg )
5) xk = (x1 + xp
2 )/2
6) tk+1 = [1 + 1 + 4(tk )2 ]/2
k
?1
7) rk+1 = xk + ttk+1
(xk ? xk?1 )
end for
where the proximal map is defined for any scaler ? > 0:
prox? (g)(x) := arg min{g(u) +
u
1
ku ? xk2 }
2?
(11)
and ?f (rk ) = AT (Ark ? b) + ??T GT (G?rk ? z) with AT denotes the inverse partial Fourier
transform.
4
3.2
Algorithm analysis
Suppose x represents an image with n pixels and z contains n0 elements. Although G is a n0 ? n
matrix, it is sparse with only n0 non-zero elements. So we can implement a multiplication by G
efficiently with O(n0 ) time. Step 1 shrinkgroup takes O(n0 + n log n) time. The total cost of step
2 takes O(n log n) time. Step 4 takes O(n log n) when the fast wavelet transform is applied. Steps
3,5 all cost O(n). Note that n0 ? 2n since we assign every parent-child coefficients to one group
and leave every wavelet scaling in one group. So the total computation complexity for each iteration
is O(n log n), the same complexity as that in TVCMRI, RecPF and FCSA. We introduce the wavelet
tree structure constrain in our model, without increasing the total computation complexity. The xsubproblem is accelerated by FISTA, and the whole algorithm shows a very fast convergence rate in
the following experiments.
4
4.1
Experiments
Experiments set up
Numerous experiments have been conducted to show the superiority of the proposed algorithm on
CS-MRI. In the MR imaging problem, A is partial Fourier transform with m rows and n columns.
We define the sampling ratio as m/n. The fewer measurements we samples, the less MR scanning
time is need. So MR imaging is always interested in low sampling ratio cases. We follow the
sampling strategy of previous works([12,14-15]), which randomly choose more Fourier coefficients
from low frequency and less on high frequency. All measurements are mixed with 0.01 Gaussian
white noise. Signal-to-Noise Ratio (SNR) is used for result evaluation.
All experiments are on a laptop with 2.4GHz Intel core i5 2430M CPU. Matlab version is 7.8(2009a).
We conduct experiments on four MR images : ?Cardiac?, ?Brain?, ?Chest? and ?Shoulder? ( Figure 1). We first compare our algorithm with the classical and fastest MR image reconstruction
algorithms: CG[3], TVCMRI[12], RecPF[13], FCSA[14,15], and then with general tree based algorithms or solvers: AMP[10], VB[9], YALL1[19], SLEP[18]. For fair comparisons, all codes are
downloaded from the authors? websites. We do not include MCMC[7] in experiements because it
has slow execution speed and untractable convergence [9][10]. OGL[7] solves its model by SpaRSA
[11] with only O(1/k) convergence rate, which can not be competitive with recent FISTA[16,17]
algorithms with O(1/k 2 ) convergence rate. The authors have not published the code yet. So we
do not include OGL for comparisons neither. We use the same setting ? = 0.001, ? = 0.035 in
previous works [12,14,15] for all convex models. ? = 0.2 ? ? is used for our model.
Figure 1: MR images: Cardiac; Brain; Chest; Shoulder and the sampling mask.
4.2
Comparisons with MR image reconstruction algorithms
We first compare our method with the state-of-the-art MR image reconstruction algorithms. For
convenience, all test images are resized to 256?256. Figure 2 shows the performance comparison on
?Brain? image. All algorithms terminate after 50 iterations. We decompose the wavelet coefficients
to 4 levels here since more levels would increase the computation cost and less levels will weaken
tree structure benefit. One could observe that the visual result recovered by the proposed algorithm
is the closest to the original with only 20% sampling ratio. Although tree structure definitely cost a
little more time to solve, it always achieves the best performance in terms of SNR and CPU time. We
have conducted experiments on other images and obtain similar results that the proposed algorithm
always has the best performance in terms of SNR and CPU time. This result is reasonable because
5
we exploit wavelet tree structure in our model, which can reduce requirement for the number of
measurements or increase the accuracy of the solution with the same measurements.
Figure 2: Brain image reconstruction with 20% sampling. Visual results from left to right, top to
bottom are original image, images reconstructed by CG [3], TVCMRI [12], RecPF [13], FCSA
[14,15], and the proposed algorithm. Their SNR are 10.26, 13.50, 14.29, 15.69 and 16.88. The right
side shows the average SNR to iterations and SNR to CPU time.
4.3
Comparisons with general algorithms of tree structure
We also compare our algorithm with existing algorithms for tree sparsity with statistical inference
and convex optimization. For statistical algorithms AMP [10] and VB [9], we use the default setting
in their code. For SLEP [18], we set the same parameters ? and ? as those in previous experiments.
For YALL1 [19], we set both ?1 and ?2 equal to ?. VB needs every column of A, which slows
down the whole algorithm. Due to the higher space requirement and time complexity of VB, we
resize all images to 128 ? 128. The wavelet decomposition level is set as 3. Figure 3 shows the
reconstruction results on ?Brain? image with only 20% measurements. All algorithm terminates
after 50 iterations. Due to high computational complexity of VB, we do not show the performance of
VB in the right bottom panel. As AMP and VB can converge with only a small number of iterations
and are much slower, we run them 10 iterations in all later experiments. The proposed algorithm
always achieves the highest SNR to CPU time among all tree-based algorithms or solvers. These
results are reasonable because none of the other algorithms uses the sparse prior of MR images in
wavelet and gradient domain simultaneously.
Table 1: Comparisons of SNR (db) on four MR images
Algorithms
Iterations
Cardiac
Brain
Chest
Shoulder
AMP [10]
VB [9]
SLEP [18]
YALL1 [19]
Proposed
10
10
50
50
50
11.36?0.95
9.62?1.82
12.24?1.08
9.56?0.13
14.80? 0.51
11.56?0.60
9.23?1.39
12.28?0.78
7.73?0.15
14.11? 0.41
11.00?0.30
8.93?0.79
12.34?0.28
7.76?0.56
12.90? 0.13
14.49?1.04
13.81?0.44
15.65?1.78
13.14?0.22
18.93? 0.73
Table 1 and 2 show the results on four MR images. Although statistical algorithms are slow in
general, they have the convenience without tuning parameters, as all parameters are learned from
data. Fortunately, good parameters for MR image reconstruction are easy to tune in our model.
Except the proposed algorithm, all other algorithms have a strong assumption of the tree structure.
However for real MR data, many images do not strictly follow this assumption. Due to this rea6
Table 2: Comparisons of execution time(sec) on four MR images
Algorithms
Iterations
Cardiac
Brain
Chest
Shoulder
AMP [10]
VB [9]
SLEP [18]
YALL1 [19]
Proposed
10
10
50
50
50
2.30?0.06
13.95?0.11
1.44?0.08
1.02?0.04
1.54?0.04
2.36?0.33
14.25?0.29
1.52?0.06
1.04?0.01
1.61?0.03
2.37?0.41
14.11?0.40
1.41?0.05
0.98?0.04
1.56?0.07
2.29?0.22
14.15?0.42
1.45?0.08
1.00?0.02
1.62?0.14
Figure 3: Brain image reconstruction with 20% sampling. Visual results from left to right, top to
bottom are original image, images reconstructed by AMP [10], VB [9], SLEP [18], YALL1 [19],
and the proposed algorithm. Their SNR are 11.56, 8.81, 12.28, 7.73 and 14.11. The right side shows
the average SNR to iterations and to CPU time. Note that the right bottom panel only shows the first
10 iterations time of AMP.
son, these tree-based algorithms can not do their best on real MR images. To show the benefit of
proposed model, we design another experiment on a toy MR image, which more strictly follow the
tree structure assumption. First we set the wavelet coefficients who have the smallest 0.1% energy
to zero. Then if a coefficient?s parent or child is zero, we set it to be zero. Hence coefficients in
the same group are both zeros or non-zeros. Figure 4 shows the original toy brain image and corresponding results of different algorithms. We found that all algorithms have improved a lot and the
performance of all algorithms becomes much closer. From Figure 4 and 3, we could find that the
proposed algorithm has great superiority on real MR image, because we combined TV and wavelet
sparsity, which ?soften? and complement the tree structure assumption for real MR data. Other tree
based algorithms depend on ?hard? tree structure only, which makes them hard to perform well on
CS-MRI.
Finally, we show the results at different sampling ratios at Figure 5. For the same algorithm, the
SNR of the solution tends to be higher when more measurements are used. On the same tested
image, the order of performance tends to be the same. It coincides the conclusion in previous paper
[14,15] that FCSA is better than TVCMRI and RecPF, and far better than classical method CG.
Comparing all these experiments, the proposed algorithm always achieves the highest SNR than all
other algorithms on real MR images.
7
Figure 4: Toy image reconstruction with 20% sampling. Visual results from left to right, top to
bottom are original image, images reconstructed by AMP [10], VB [9], SLEP [18], YALL1 [19],
and the proposed algorithm. Their SNR are 12.99, 10.12, 13.53, 13.19 and 15.29. The right side
shows the average SNR to iterations and to CPU time.
Figure 5: Average SNR with different sampling ratios on 4 MR images. All algorithms terminate
after 50 iterations except AMP [10] and VB [9] terminate after 10 iterations. From left to right,
results are on ?Cardiac?,?Brain?,?Chest? and ?Shoulder?.
5
Conclusions
Real MR images not only tend to be tree structured sparse, but also are sparse in wavelet and
gradient domain . In this paper, we consider all these priors in our model and all terms in this mode
are convex. To solve this model, we decompose the original problem into three simpler ones and
solve each of them very efficiently. Numerous experiments have been conducted to validate our
method. All experiments demonstrate that the proposed algorithm outperforms the state-of-the-art
ones in CS-MRI and general tree-based algorithms or solvers. Compared with the state-of-the-art
algorithms in CS-MRI, the tree structure in our model can help reduce the required measurements,
and leads to better performance. Compared with general tree sparsity algorithms, our algorithm can
obtain more robust results on real MR data. Future work will be combining the proposed algorithm
with the nonlocal total variation [22] for multi-contrast MRI [21].
8
References
[1] Donoho, D. (2006) Compressed sensing. IEEE Trans. on Information Theory 52(4):1289-1306.
[2] Candes, E., Romberg, J. & Tao, T. (2006) Robust uncertainty principles: Exact signal reconstruction from
highly incomplete frequency information. IEEE Trans. on Information Theory 52(2):489-509.
[3] Lustig, M., Donoho, D. & Pauly, J. (2007) Sparse MRI: The application of compressed sensing for rapid
MR imaging. Magnetic Resonance in Medicine 58(6):1182-1195.
[4] Huang, J., Zhang, T. & Metaxas, D. (2011) Learning With Structured Sparsity. Journal of Machine Learning
Research 12:3371-3412.
[5] Baraniuk, R.G., Cevher, V., Duarte, M.F. & Hegde, C. (2010) Model-based compressive sensing. IEEE
Trans. on Information Theory 56:1982-2001.
[6] Bach, F., Jenatton, R., Mairal, J. & Obozinski, G. (2012) Structured sparsity through convex optimization.
Technical report, HAL 00621245-v2, to appear in Statistical Science.
[7] Rao, N., Nowak, R., Wright, S. & Kingsbury, N. (2011) Convex approaches to model wavelet sparsity
patterns. In IEEE International Conference On Image Processing, ICIP?11
[8] He, L. & Carin, L. (2009) Exploiting Structure in Wavelet-Based Bayesian Compressive Sensing. IEEE
Trans. on Signal Processing 57(9):3488-3497.
[9] He, L., Chen, H. & Carin, L. (2010) Tree-Structured Compressive Sensing with Variational Bayesian Analysis. IEEE Signal Processing Letters 17(3):233-236
[10] Som, S., Potter, L.C. & Schniter, P. (2010) Compressive Imaging using Approximate Message Passing and
a Markov-Tree Prior. In Proceedings of Asilomar Conference on Signals, Systems, and Computers.
[11] Wright, S.J., Nowak, R.D. & Figueiredo, M.A.T. (2009) Sparse reconstruction by separable approximation.
IEEE Trans. on Signal Processing 57:2479-2493.
[12] Ma, S., Yin, W., Zhang, Y. & Chakraborty, A.(2008) An efficient algorithm for compressed MR imaging
using total variation and wavelets. In In Proc. of the IEEE Computer Society Conference on Computer Vision
and Pattern Recognition, CVPR?08.
[13] Yang, J., Zhang, Y. & Yin, W. (2010) A fast alternating direction method for tvl1-l2 signal reconstruction
from partial fourier data. IEEE Journal of Selected Topics in Signal Processing, Special Issue on Compressive
Sensing 4(2):288-297.
[14] Huang, J., Zhang, S. & Metaxas, D. (2011) Efficient MR Image Reconstruction for Compressed MR
Imaging. Medical Image Analysis 15(5):670-679.
[15] Huang, J., Zhang, S. & Metaxas, D. (2010) Efficient MR Image Reconstruction for Compressed MR
Imaging. In Proc. of the 13th Annual International Conference on Medical Image Computing and Computer
Assisted Intervention, MICCAI?10.
[16] Beck, A. & Teboulle, M. (2009) A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences 2(1):183-202.
[17] Beck, A. & Teboulle, M. (2009) Fast gradient-based algorithms for constrained total variation image
denoising and deblurring problems. IEEE Trans. on Image Processing 18(113):2419-2434
[18] Liu, J., Ji, S. & Ye, J. (2009) SLEP: Sparse Learning with Efficient Projections. Arizona State University.
http://www.public.asu.edu/ jye02/Software/SLEP.
[19] Deng, W., Yin, W. & Zhang, Y. (2011) Group Sparse Optimization by Alternating Direction Method. Rice
CAAM Report TR11-06.
[20] Manduca A., & Said A. (1996) Wavelet Compression of Medical Images with Set Partitioning in Hierarchical Trees. In Proceedings of International Conference IEEE Engineering in Medicine and Biology Society,
EMBS.
[21] Huang, J., Chen, C. & Axel, L. (2012) Fast Multi-contrast MRI Reconstruction. In Proc. of the 15th Annual
International Conference on Medical Image Computing and Computer Assisted Intervention, MICCAI?12.
[22] Huang, J., & Yang, F. (2012) Compressed Magnetic Resonace Imaging Based on Wavelet Sparsity and
Nonlocal Total Variation. IEEE International Symposium on Biomedical Imaging, ISBI?12.
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4,012 | 4,631 | On the connections between saliency and tracking
Nuno Vasconcelos
Statistical Visual Computing Laboratory
UC San Diego, La Jolla, CA 92092
[email protected]
Vijay Mahadevan
Yahoo! Labs
Bangalore, India
[email protected]
Abstract
A model connecting visual tracking and saliency has recently been proposed. This
model is based on the saliency hypothesis for tracking which postulates that tracking is achieved by the top-down tuning, based on target features, of discriminant
center-surround saliency mechanisms over time. In this work, we identify three
main predictions that must hold if the hypothesis were true: 1) tracking reliability should be larger for salient than for non-salient targets, 2) tracking reliability should have a dependence on the defining variables of saliency, namely feature contrast and distractor heterogeneity, and must replicate the dependence of
saliency on these variables, and 3) saliency and tracking can be implemented with
common low level neural mechanisms. We confirm that the first two predictions
hold by reporting results from a set of human behavior studies on the connection
between saliency and tracking. We also show that the third prediction holds by
constructing a common neurophysiologically plausible architecture that can computationally solve both saliency and tracking. This architecture is fully compliant
with the standard physiological models of V1 and MT, and with what is known
about attentional control in area LIP, while explaining the results of the human
behavior experiments.
1 Introduction
Biological vision systems have evolved sophisticated tracking mechanisms, capable of tracking
complex objects, undergoing complex motion, in challenging environments.These mechanisms have
been an area of active research in both neurophysiology [10, 34] and psychophysics [28], where research has been devoted to the study of object tracking by humans [29]. This effort has produced
several models of multi-object tracking, that account for the experimental evidence from human
psychometric data [28]. Prominent among these are the FINST model of Pylyshyn [29], and the
object file model of Kahnemman et al [18]. However, these models are not quantitative, and only
explain the psychophysics of tracking simple stimuli, such as dots or bars. They do not specify a set
of computations for the implementation of a general purpose tracking algorithm, and it is unclear
how they could be applied to natural scenes. While some computational models for multiple object
tracking (MOT) such as the oscillatory neural network model of Kazanovich et al. [19], and the
particle filter based model of Vul et al. [37], have been proposed, there have been no attempts to
demonstrate their applicability to real video scenes.
Visual tracking has also been widely studied in computer vision, where numerous tracking algorithms [38] have been proposed. Early solutions relied on simple object representations, and emphasized the prediction of object dynamics, typically using a Kalman filter. The prediction of these
dynamics turned out to be difficult, motivating the introduction of more sophisticated methods such
as particle filtering [15]. Nevertheless, because these approaches relied on simple target representations, they could not deal with complex scenes. This motivated research in the appearance-based
modeling techniques [17, 32, 9] where a model of object appearance is learned from the target location in the initial frame, and used to identify the target in the next. It is, however, difficult to learn
appearance models from complex scenes, where background detail tends to drift into the region used
to learn the model, corrupting the learning.
1
The best results among tracking algorithms have recently been demonstrated for a class of methods that pose object tracking as incremental target/background classification [22, 8, 2, 13]. These
discriminant trackers train a classifier to distinguish target from background at each frame. This
classifier is then used to determine the location of the target in the next frame. Target and background are extracted at this location, the classifier updated, and the process iterated.
Recent work in the computer vision literature [22] has postulated a connection between discriminant
tracking and one of the core processes of early biological vision - saliency, by suggesting that the
ability to track objects is a side-effect of the saliency mechanisms that are known to guide the deployment of attention. More precisely, [22] has hypothesized that tracking is a simple consequence
of object-based tuning, over time, of the mechanisms used by the attentional system to implement
bottom-up saliency. We refer to this as the saliency hypothesis for tracking. Working under this
hypothesis, [22] proposed a tracker based on the discriminant saliency principle of [12]. This is a
principle for bottom-up center-surround saliency, which poses saliency as discrimination between
a target (center) and a null (surround) hypothesis. Center-surround discriminant saliency has previously been shown to predict various psychophysical traits of human saliency and visual search
performance [11]. The extension proposed by [22], to the tracking problem, endows discriminant
saliency with a top-down feature selection mechanism. This mechanism enhances features that respond strongly to the target and weakly to the background, transforming the saliency operation from
a search for locations where center is distinct from the surround, to a search for locations where
target is present in the center but not in the surround. [22] has shown that this tracker has state-ofthe-art performance on a number of tracking benchmarks from the computer vision literature.
In this work, we evaluate the validity of the saliency hypothesis by identifying three main predictions that ensue from the saliency hypothesis: 1) tracking reliability should be larger for salient than
for non-salient targets, 2) tracking reliability should have a dependence on the defining variables of
saliency, namely feature contrast and distractor heterogeneity, and must replicate the dependence of
saliency on these variables and, 3) saliency and tracking can be implemented with common low level
neural mechanisms. We confirm that the first two of these predictions hold by performing several
human behavior experiments on the dependence between target saliency and human tracking performance. These experiments build on well understood properties of saliency, such as pop-out effects,
to show that tracking requires discrimination between target and background using a center-surround
mechanism. In addition, we characterize the dependence of tracking performance on the extent of
discrimination, by gradually varying feature contrast between target and distractors in the tracking
tasks. The results show that both tracking performance and saliency have highly similar patterns
of dependency on feature contrast and distractor heterogeneity. To confirm that the third prediction
holds, we show that both saliency and tracking can be implemented by a network compliant with
the widely accepted neurophysiological models of neurons in area V1 [5] and the middle temporal
area (MT) [36], and with the emerging view of attentional control in the lateral intra-parietal area
(LIP) [3]. This network extends the substantial connections between discriminant saliency and the
standard model that have already been shown [12] and is a biologically plausible optimal model for
both saliency and tracking.
2 Human Behavior Experiments on Saliency and Tracking
We start by reporting on human behavior experiments1 investigating the connections between the
psychophysics of tracking and saliency. To the best of our knowledge, this is the first report on
psychophysics experiments studying the relation between attentional tracking of a single target and
its saliency. Video stimuli were designed with the Psychtoolbox [4] on Matlab v7, running on
a Windows XP PC. A 27 inch LCD monitor of size 47.5? ? 30? visual angle and resolution of
1270 ? 1068 pixels was used to present the stimuli. Subjects were at a viewing distance of 57 cm.
The same apparatus was used for all experiments.
2.1 Experiment 1 : Saliency affects tracking performance
The experimental setting was inspired by the tracking paradigm of Pylyshyn [29]. Subjects viewed
displays containing a green target disk surrounded by 70 red distractor disks and a static fixation
square. Example displays are shown in [1]. At the start of each trial, the target disk was cued with
a bounding box. Subjects were instructed to track the target covertly, while their eyes fixated on a
black fixation square in the center. After a keystroke from the subject, all disks moved independently,
with random motion, for around 7 seconds. Then, the disks stopped moving and the colors of 3
1
IRB approved study, subjects provided informed consent and were compensated $8 per hour
2
disks were switched to 3 new colors - cyan, magenta and blue. Of these, one was the target and the
other two the spatially closest distractors. Subjects were asked to identify the target among the 3
highlighted disks.
Method 13 subjects (age 22-35, 9 male, 4 female) performed 4 trials each, organized into 2 versions of 2 conditions. First version: this version tested subject tracking performance under two
different stimulus conditions. In the first, denoted salient, the target remained green throughout the
presentation, changing randomly to one of the three highlight colors at the end of the 7 seconds. In
the second, denoted non-salient, the target remained green for the first half of this period, switched
to red for the remaining time, finally turning to a highlight color. While in the first condition the
target is salient throughout the presentation, the second makes the target non-salient throughout the
latter half of the trial. To eliminate potential effects of any other variables (e.g. target-distractor distances and motion patterns), the non-salient display was created by rotating each frame of a salient
display by 90? (and changing the green disk to red in the second half of the presentation).
Under the saliency hypothesis for tracking, the rate of successful target tracking should be much
higher for salient than for non-salient displays. However, this could be due to the fact that the target
was the only green disk in salient displays, and since it continuously popped-out subjects could be
acquiring the target at any time even after losing track. Second Version: The second version ruled
out this alternate hypothesis by using a different type of display for the salient condition. In this
case, the target was a red disk, and its 7 nearest spatial neighbors were green. All other distractors
were randomly assigned to either the red or green class. This eliminated the percept of pop-out.
As before, the display for the non-salient condition was created by rotation and color switch of
the target on the second half of the presentation. The video displays are available in the attached
supplement [1].
Results and Discussion Figure 2 (a) and (b) present the rate of successful tracking in the two versions. In both cases, this rate was much higher in the salient than in the non-salient condition. In the
latter, the tracking performance was almost at the chance level of 13 , suggesting complete tracking
failure. In fact, the similarity of detection rates in the two experiments suggests that target pop-out
does not aid human tracking performance at all. It only matters if the target is locally salient, i.e.
salient with respect to its immediate neighborhood. This is consistent with the saliency hypothesis,
since bottom-up saliency mechanisms are well known to have a center-surround structure [16, 12].
In fact, it suggests two new predictions. The first, motivated by the hypothesis that tracking requires
top-down biases of bottom-up saliency, is that center-surround organization also applies to tracking.
To address this prediction, we will investigate the spatial organization of tracking mechanisms in
greater detail in Experiment 3. The second, which follows from the fact that only target color varied between the two conditions, is that tracking performance depends on the discriminability of the
target. We study this prediction in Experiment 2. While the first experiment used color as a discriminant cue, the conclusion that saliency affect tracking performance applies even when other features
are salient. For example, studies on multiple object tracking with identical targets and distractors
have reported tracking failure when target and distractors are too close to each other [14]. This is
consistent with the discriminant hypothesis: when target and distractors are identical, the target must
be spatio-temporally salient (due to its trajectory or position) in an immediate neighborhood to be
tracked accurately.
2.2 Experiment 2: Tracking reliability as a function of feature contrast
Experiment 2 aimed to investigate the connection between the two phenomena in greater detail,
namely to quantify how tracking reliability depends on target saliency. Since saliency is not an independent variable, this quantification can only be done indirectly. One possibility is to resort to a
variable commonly used to manipulate saliency: the amount of feature contrast between target and
distractors. Several features can be used, as it is well known that targets which differ from distractors in terms of color, luminance, orientation or texture can be perceived as salient [27, 25]. In fact,
Nothdurft [26] has precisely quantified the dependence of saliency on orientation contrast in static
displays. His work has shown that perceived target saliency increases with the orientation contrast
between target and neighboring distractors. This increase is quite non-linear, exhibiting the threshold and saturation effects shown in Figure 1 (a), where we present curves of saliency as a function
of orientation contrast between target and distractors for three levels of distractor homogeneity [26].
The relationship between tracking reliability and target saliency can thus be characterized by manipulating orientation contrast and measuring the impact on tracking performance. If the saliency
3
hypothesis for tracking holds, saliency and tracking reliability should be equivalent functions of orientation contrast. In particular, increasing orientation contrast between target and distractors should
result in a non-linear increase of tracking reliability, with threshold and saturation effects similar to
those observed by Nothdurft.
Method 12 subjects (8 male and 4 female) in the age range 21-35 participated in the study. The
experimental setting was adapted from the work of Makovski and Jiang [23]. All displays had
size 26? ? 26? (700 ? 700 pixels) and consisted of 23 ellipses, all of color blue, against a black
background. Each ellipse had a major axis of ? 0.56? (15 pixels) and minor axis of ? 0.19? (5
pixels). The orientation of the ellipses depended on the condition from which the trial was drawn.
At the start of a trial, one of the ellipses was designated as target (cued with a white bounding box).
Subjects were instructed to track the target covertly, while fixating on a white square at the center of
the screen. On a keystroke, the ellipses started moving and continued to do so for ? 8-10 sec. At
the end of the trial, all ellipses were completely occluded by larger white disks and subjects asked to
click on the disk corresponding to the target. Each subject performed 30 trials under 7 conditions,
for a total of 210 trials, and no feedback was given on the accuracy of their selection.
The 7 conditions corresponded to different levels of orientation contrast between target and distractor
ellipses. Distractor orientation, defined by the major axis of the distractor ellipses, was always 0? .
Target orientation, determined by the major axis of the target ellipse, was selected from 7 values:
0? , 10? , 20? , 30? , 40? , 60? or 80? . This made orientation contrast equal to the target orientation.
Example displays are shown in the attached supplement [1]. To keep all other variables (e.g. distance
between items, motion patterns, distance from target to fixation square) identical, a trial was first
created for one condition (target orientation 0? ). The trials of all other conditions were obtained by
applying a transformation to each frame of this video clip. This consisted of an affine transformation
of the grid of ellipse centers, followed by the desired change in target orientation.
To study the effect of distractor heterogeneity [26], three versions of the experiment were conducted
with different numbers of ellipses in the target orientation. In the first version, only one ellipse (the
actual target) was in target orientation. In this case, there was no distractor heterogeneity. In the
second version, 18 of the 23 ellipses were in distractor orientation, and the remaining 5 in target
orientation. One of the latter was the actual target. Finally, in the third version, 13 ellipses were in
distractor and 10 in target orientation, for the largest degree of distractor heterogeneity.
Results and Discussion Figure 1 (b), shows the curves of tracking accuracy vs. orientation contrast obtained in all three versions of the experiment. These curves are remarkably similar to Nothdurft?s saliency curves, shown in (a). Again, there are 1) distinct threshold and saturation effects
for tracking, with tracking accuracy saturating for orientation contrasts beyond 40? , and 2) decreasing tracking accuracy as distractor heterogeneity increases. The co-variation of tracking accuracy
and saliency is illustrated in Figure 1 (c), where the two quantities are presented as a scatter plot
The correlation between the two variable is near perfect (r = 0.975). In summary, tracking has a
dependence on orientation contrast remarkably similar to that of saliency.
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Figure 1: (a) saliency vs. orientation contrast (adapted from [26]) (b) human tracking success rate vs. orientation contrast. (c) scatter plot of saliency values from (a) vs tracking accuracy from (b), r = 0.975. (d) model
prediction : tracking success rate vs. orientation contrast for the network of Figure 3.
2.3 Experiment 3: The spatial structure of tracking
It is well known that bottom-up saliency mechanisms are based on spatially localized centersurround processing [16, 6]. Hence, the saliency hypothesis for tracking predicts that tracking
performance depends only on distractors within a spatial neighborhood of the target. The results
4
of Experiment 2 provide some evidence in support of this prediction, by showing that tracking performance depends on distractor heterogeneity. This implies that the visual content of the background
affects human tracking performance. The open question is whether the effect of the background 1)
is limited to a localized neighborhood of the target, or 2) extends to the entire field of view. This
question motivated Experiment 3. In this experiment, the distance dcsd between the target and the
closest distractor of the same orientation, denoted the closest similar distractor (CSD), was controlled so that dcsd = R, where R is a parameter. This guaranteed that there were no distractors
with the target orientation inside a neighborhood of radius R around it. By jointly varying this parameter and the amount of distractor heterogeneity, it is possible to test three hypotheses: (a) no
surround region is involved in tracking: in this case, the rate of tracking success does not depend
on the distractor heterogeneity at all, (b) the entire visual field affects tracking performance: in this
case, for a fixed distractor heterogeneity, the rate of tracking success does not depend on R, (c) the
effect of the surround is spatially localized: in this case, there is a critical radius Rcritical beyond
which distractors have no influence in tracking performance. This implies that the rate of tracking
success does not depend on distractor heterogeneity for R > Rcritical . Experiment 2 already established that hypothesis (a) does not hold. Experiment 3 was designed to determine which of (b) and
(c) holds.
Method 9 subjects (7 male and 2 female) in the age range 21-35 participated in the study. The
target orientation was fixed at 40? for all stimuli. Two versions of the experiment were conducted,
with two levels of distractor heterogeneity. As in Experiment 2, the first version used 18 (5) of the
23 ellipses in distractor (target) orientation. In the second version, 13 ellipses were in distractor and
10 in target orientation. In both versions, the stimulus was produced with four values of average R
(average, over all frames in the video sequence, of the distance dcsd ). Across the 4 conditions, this
quantity was in the range 1.67? to 5.01? (about 45 pixels to 135 pixels).
Results and Discussion Figure 2(a) presents the rate of tracking success as a function of average
R, for the two versions of the experiment. The tracking accuracy for the case where there is no
distractor heterogeneity (no distractor with the target orientation) is also shown, as a flat line. Two
main observations are worth noting. First, for a fixed (non-zero) amount of distractor heterogeneity, tracking performance always increases with R. This implies that it is easier to track when the
CSD is farther from the target. Second, for large R tracking accuracy does not depend on distractor
heterogeneity (it is nearly the same under the two heterogeneity conditions), converging to the accuracy observed when there is no distractor heterogeneity (Experiment 3). These observations support
the conclusion that hypothesis (c) holds, i.e. tracking ability is influenced by a localized surround
region, of radius Rcritical ? 4? . When similar distractors are kept out of this region, the degree of
distractor heterogeneity has no effect in tracking performance.
In summary, results of the human behavior experiments show that the first two predictions made
by the saliency hypothesis for tracking hold. These predictions are that tracking reliability 1) is
larger for salient than for non-salient targets (Experiment 1), 2) depends on the defining variables
of saliency, namely feature contrast and distractor heterogeneity (Experiment 2), and replicates the
dependence of saliency on these variables. This includes the threshold and saturation effects of the
dependence of saliency on feature contrast (Experiment 2), and the spatially localized dependence
of saliency on distractor heterogeneity (Experiment 3). Overall, these experiments provide strong
evidence in support of the saliency hypothesis for tracking. We next consider the final prediction,
which is that saliency and tracking can be implemented with common neural mechanisms.
3 Joint neural architecture for saliency and tracking
To construct a saliency based neurally plausible computational model for tracking we start with the
neural model proposed by [12] to compute saliency and identify the mechanisms required to extend
it to perform tracking, and then show how these mechanisms can be implemented in a biologically
plausible manner.
In [12], saliency is equated to optimal decision-making between two classes of visual stimuli, with
label C ? {0, 1}, C = 1 for stimuli in a target class, and C = 0 for stimuli in a background
class. The classes are defined in a center-surround manner where, at each location l, the target
(background) class is that of stimuli in a center (surround) window. The stimuli are not observed
directly, but through projection onto a set of n features, of responses Y(l) = (Y1 (l), . . . , Yn (l)). The
saliency of location l is then equated to the expected accuracy of target/background classification,
5
1
(a)
1
(b)
tracking accuracy
tracking accuracy
0.95
0.9
0.85
0.8
0.75
0 similar distractors
4 similar distractors
9 similar distractors
0.7
0.9
0.8
0 similar distractors
4 similar distractors
9 similar distractors
0.7
0.6
2
3
4
5
average distance to nearest similar distractor (deg)
2
3
4
5
average distance to nearest similar distractor (deg)
(c)
(d)
Figure 2: (a) and (b) Experiment 1: successful target tracking rate for targets that are (a) globally salient
(pop-out), and (b) locally salient (do not pop-out). (c) and (d) Experiment 3: the effect of background on
tracking performance - (c) Tracking accuracy of human subjects for two versions of distractor homogeneities
are plotted as a function of the average target-similar distractor distance. Also shown, using a blue dashed line,
is the tracking accuracy for the version with no similar distractors at target orientation of 40? from Experiment
2. (d) model prediction for the same data using the saliency based model of Figure 3.
given the feature responses from the two classes and can be written as:
S(l) =
1X
Sk (l),
n k
Sk (l) = EY (l) {?[PC(l)|Yk (l) (1|y)]},
?(x) =
x?
0
1
2
x ? 0.5
x < 0.5
(1)
The saliency measure Sk (l) is the expected confidence with which the feature response Yk (l)
is assigned to the target class. ?(x) is a nonlinearity that thresholds the posterior probability
PC(l)|Yk (l) (1|y) to prevent locations assigned to the background class by the Bayes decision rule
(PC(l)|Yk (l) (1|y) ? 0.5), from contributing to the saliency. This tunes the saliency measure to respond only to the presence of target stimuli, not to its absence. This definition of saliency was
shown, in [12], to be computable using units that conform to the standard neurophysiological model
of cells in visual cortex area V1 [5], when the features are bandpass filters (e.g. Gabor filters) extracted from static natural images. However, for the tracking task, the feature set Y for representing
the target and background needs to contain spatiotemporal features that are tuned to the velocity
of moving patterns. It can be shown that saliency for such velocity tuned spatiotemporal features
can be computed by combining the outputs of a set of V1 like units of [12], akin to the widely
used approach for constructing models for MT cells from afferent V1 units [36, 33]. This enhanced
network, illustrated in Figure 3, is equivalent to an MT neuron tuned to a particular velocity (see
supplement [1]).
3.1 Neurophysiologically plausible feature selection
A key component of the saliency tracker of [22] is a feature selection procedure that continuously
adapts the saliency measure of (1) to the target. The basic idea is to select, at each time step, the
features in Y(l) that best discriminate between target (center) and background. This changes the
saliency from a bottom-up identification of locations where center and surround differ, to a topdown identification of locations containing the target in the center and background in the surround.
However, the procedure of [22] (based on feature ranking) is not biologically plausible. To derive
a biologically plausible feature selection mechanism, we replace the saliency measure of (1) with a
feature-weighted extension
X
X
?k = 1
?k Sk (l),
S(l) =
(2)
k
k
where ?k is the weight given to the saliency of the k th feature channel. We associate a binary
variable Fk with each feature Yk , such that Fk = 1 if and only if Yk is the most salient feature of the
target. We then assume that, given the knowledge of which feature is most salient, target presence
at location l is independent of the remaining feature responses, and so the posterior probability of
target presence given the observation of all features is written as:
PC(l)|Y(l),Fk (1|y, 1) = 2?[PC(l)|Yk (l) (1|y)],
(3)
This reflects a conservative strategy, where features cannot be considered salient unless they are individually discriminant for target presence. Given the location l? where the target has been detected,
the posterior probability of feature saliency can then be computed by Bayes rule
PFk |C(l? ) (1|1)
=
(1|1)PFk (1)
P ?
P C(l )|Fk
,
?
P
j C(l )|Fj (1|1)PFj (1)
6
where
(4)
PC(l? )|Fk (1|1)=
=
Z
PC(l? )|Y(l? ),Fk (1|y, 1)PY(l? )|Fk (y|1)dy
Z
2?[PC(l? )|Yk (l? ) (1|y)]PYk (l? ) (y)dy (using (3))
= 2EYk (l? ) {?[PC(l? )|Yk (l? ) (1|y)]} = 2Sk (l? ),
(5)
(6)
and the last equality follows from (1). Using (6) in (4), we get
PFk |C(l? ) (1|1)
S (l? )PFk (1)
Pk
.
?
j Sj (l )PFj (1)
=
(7)
Under reasonable assumptions of persistence of the dominant features in the target, this analysis can
be extended over time, by denoting the state of Fk and l? at time t by Fkt and lt? , and the sequence
?
of target locations till time t by l?t = (lt? , lt??
. . . l0? ), and we get the recursion (see [1]),
PF t |C(l?t ) (1|1)
k
=
Sk (lt? )PF t?? |C(l? ) (1|1)
t??
k
P
.
?
S
(l
)P
(1|1)
t??
j
t
j
F
|C(l?
)
j
(8)
t??
Hence, the posterior probability of feature k being the most salient at time t given that the target is
at lt? is computed by divisively normalizing a weighted version of Sk (lt? ), the bottom-up saliency of
the feature k at lt? , by the total saliency summed over all features. The weight applied to the saliency
of each feature (corresponding to ?k in (2)) is the posterior probability of the feature being the most
salient at time t ? ? . Therefore the posterior at time t ? ? is fed back with a delay, to become the
weight at time t. This enhances the most salient features, suppressing the non-salient ones, and is
equivalent to applying a soft-thresholding to select only the dominant features.
This feature selection mechanism involving selective enhancement and suppression of features, operating on the outputs of the MT stage bears a close resemblance to the phenomenon of feature-based
attention [24]. In fact, the proposed approach to feature selection is similar to previously proposed
biologically plausible models of feature-based attention that rely on a Bayesian formulation and
include divisive normalization [30, 31, 20, 7]. Further, neurophysiological studies have found evidence of feature-based attention in the lateral intraparietal (LIP) area [3]. LIP is also known to have
cortico-cortical connections to area MT [21], and attentional control is thought to be fed-back from
LIP to MT [35]. Studies also suggest that the LIP might be computing a priority map that combines
both bottom-up inputs and top-down signals, and the peak of this map response is used to guide
visual attention [3]. These findings are compatible with the feature selection approach of (8), and
therefore area LIP is a plausible candidate location for the feature selection stage of our model.
3.2 Neurophysiologically plausible discriminant tracker
A neurophysiologically plausible version of the discriminant tracker of [22] can be constructed with
the discriminant saliency measure of (1), and the feature selection mechanism of (8). As in [22],
in the absence of top-level information regarding the target, initialization and target acquisition can
be treated as discrimination between the visual stimulus contained in a pair of center (target) and
surround windows, at every location of the visual field. In this case, there is no explicit top-down
guidance about the object to recognize, and the saliency of location l is measured by the saliency
of all unmodulated feature responses. This consists of using the bottom-up saliency measure of (2)
with ?k = PFk0 (1), where PFk0 (1) is a uniform prior for feature selection, at time t = 0. The
outputs of all features or neurons are then summed with equal weights to produce a final saliency
map. The peak of this map represents the location which is most distinct from its surround, based
on the responses of the motion sensitive spatio-temporal features, and becomes the target. Spatial
attention is then shifted to the peak of this map.
Once the initial target location is attended, the feature selection mechanism modulates the saliency
response of the individual feature channels, using the weights of (8). The final saliency value at
that location also becomes the normalizing constant for the divisive normalization of (8). These
feature weights are fed back to MT neurons, where each feature map is enhanced or attenuated
depending on the corresponding feature weight. This enhances the features that are salient for
target detection, and suppresses the non-salient ones. LIP also feeds back the retinotopic weight
map corresponding to spatial attention, causing a suppression of feature responses in all areas other
than in a neighborhood of the current locus of attention.
7
Figure 3: The network for tracking using feature selection. The discriminant saliency network of [12] is used
to construct a model for an MT neuron. Feature selection, performed possibly in area LIP with weights being
fed-back to MT, is achieved by the modulation of the response of each feature channel by its saliency value
after divisive normalization across features.
After the latency due to feedback, say at time t + ? , the new feature weights and spatial weights,
modulate the feature maps, which are again fed forward to LIP, where the updated saliency map is
computed by simple summation. The top-down saliency of location l at time t + ? is then given by
S td (l) =
X
Sjtd (l) =
j
X
Sj (l)PF t |C(l?t ) (1|1).
j
(9)
j
where Sj (l) is the modulated saliency response of the j th feature.
Spatial attention suppresses all but a neighborhood of the last known target location lt? , and the
feature-based attention suppresses all features except those present in the target and discriminative
with respect to the background. Therefore, the peak of the new saliency map corresponds to the
position that best resembles the target at time t + ? , and attention is shifted to that position.
?
lt+?
= argmax S td (l)
(10)
l
The process is iterated, so as to track the target over time, as in [22]. The entire tracking network is
shown in Figure 3. The computation, in V1, of SZj (l) is implemented with the bottom-up network
of [12]. V1 outputs are then linearly combined with weights wjk (which are described in supplement [1]) to obtain the MT responses Sk (l). The remaining operations, possibly in LIP, compute the
probabilities of (8) and the top-down saliency map of (9).
4 Validation of joint architecture
We applied the network of Figure 3 to the sequences used in Experiment 1. Representative frames
of the result of tracking on the displays of the experiment and the videos are available from [1]. The
model replicates the trend observed in both versions of the experiment, accurately tracking the target
in the salient conditions, and losing track in the non-salient condition.
The results of applying the network to the stimuli in Experiments 2 and 3 are shown in Figures 1(d)
and 2(d) respectively. It is seen that the model predictions accurately match the trend observed in
all three versions of the Experiment 2. The model also predicts the effect of background seen in
Experiment 3.
5 Conclusion
We provide the first verifiable evidence for a connection between saliency and tracking that was
earlier only hypothesized [22]. In particular, we show that three main predictions of the hypothesis
hold. First, using psychophysics experiments we show that tracking requires discrimination between
target and background using a center-surround mechanism, and that tracking reliability and saliency
have a common dependence on feature contrast and distractor heterogeneity. Next we construct a
tracking model starting a neurally plausible architecture to compute saliency, and show that it can be
implemented with widely accepted models of cortical computation. Specifically, the model is based
on a feature selection mechanism akin to the well known phenomenon of feature-based attention in
MT. Finally, we show that the tracking model accurately replicates all our psychophysics results.
8
References
[1] See attached supplementary material.
[2] S. Avidan. Ensemble tracking. IEEE PAMI, 29(2):261?271, 2007.
[3] J. Bisley & M. Goldberg, ?Attention, intention, & priority in the parietal lobe,? Annu. Rev. Neurosci, 33, p.
1?21, 2010.
[4] D. H. Brainard. The psychophysics toolbox. Spatial Vision, 10:433?436, 1997.
[5] M. Carandini et al., Do we know what the early visual system does? J. Neuroscience, 25, 2005.
[6] J. Cavanaugh, W. Bair, & J. Movshon. Nature & interaction of signals from the receptive field center and
surround in macaque V1 neurons. J. Neurophysiol., 88:2530?2546, 2002.
[7] S. Chikkerur, et al., What & where: A Bayesian inference theory of attention. Vision Research, 2010.
[8] R. Collins, Y. Liu, & M. Leordeanu. On-line selection of discriminative tracking features. IEEE PAMI,
27(10):1631 ? 1643, October 2005.
[9] D. Comaniciu, V. Ramesh, & P. Meer. Kernel-based object tracking. IEEE PAMI, 25(5):564?577, 2003.
[10] J. C. Culham, et al., Cortical fmri activation produced by attentive tracking of moving targets. J. Neurophysiol, 80(5):2657?2670, 1998.
[11] D. Gao, V. Mahadevan, & N. Vasconcelos. On the plausibility of the discriminant center-surround hypothesis for visual saliency. Journal of Vision, 8(7):1?18, 6 2008.
[12] D. Gao & N. Vasconcelos. Decision-theoretic saliency: computational principle, biological plausibility,
& implications for neurophysiology & psychophysics. Neural Computation, 21:239?271, Jan 2009.
[13] H. Grabner & H. Bischof. On-line boosting & vision. IEEE CVPR, 1:260?267, 2006.
[14] J. Intriligator & P. Cavanagh. The spatial resolution of visual attention. Cog. Psych., 43:171?216, 1997.
[15] M. Isard & A. Blake. Condensation: conditional density propagation for visual tracking. IJCV, 29, 1998.
[16] L. Itti et al., A model of saliency-based visual attention for rapid scene analysis. IEEE PAMI,
20(11):1254?1259, 1998.
[17] A. D. Jepson et al., Robust online appearance models for visual tracking. IEEE PAMI, 25(10), 2003.
[18] D. Kahneman, A. Treisman, & B. J. Gibbs. The reviewing of object files: Object-specific integration of
information. Cognitive Psychology, 24(2):175?219, 1992.
[19] Y. Kazanovich & R. Borisyuk. An oscillatory neural model of multiple object tracking. Neural computation, 18(6):1413?1440, 2006.
[20] J. Lee & J. Maunsell. A normalization model of attentional modulation of single unit responses. PLoS
One, 4(2), 2009.
[21] J. Lewis & D. Van Essen, ?Corticocortical connections of visual, sensorimotor, & multimodal processing
areas in the parietal lobe of the macaque monkey,? J. Comparative Neurol., 428(1), p. 112?137, 2000.
[22] V. Mahadevan & N. Vasconcelos. Saliency-based discriminant tracking. CVPR, 2009.
[23] T. Makovski & Y. Jiang. Feature binding in attentive tracking of distinct objects. Visual cognition,
17(1):180?194, 2009.
[24] J. Maunsell & S. Treue. Feature-based attention in visual cortex. Trends in Neurosci., 29(6), 2006.
[25] H. C. Nothdurft. Texture segmentation & pop-out from orientation contrast. Vision Research, 31(6):1073?
1078, 1991.
[26] H. C. Nothdurft. The conspicuousness of orientation & motion contrast. Spatial Vision, 7:341?363, 1993.
[27] H. C. Nothdurft. Salience from feature contrast: additivity across dimensions. Vision Research, 40:1183?
1201, 2000.
[28] L. Oksama & J. Hyn. Is multiple object tracking carried out automatically by an early vision mechanism
independent of higher-order cognition? Visual Cognition, 11(5):631 ? 671, 2004.
[29] Z. W. Pylyshyn & R. W. Storm. Tracking multiple independent targets: evidence for a parallel tracking
mechanism. Spatial vision, 3(3):179?197, 1988.
[30] R. Rao. Bayesian inference & attentional modulation in the visual cortex. Neuroreport, 16(16), 2005.
[31] J. Reynolds & D. Heeger. The normalization model of attention. Neuron, 61(2):168?185, 2009.
[32] D. Ross et al., Incremental learning for robust visual tracking. IJCV, 77(1-3):125?141, 2008.
[33] N. Rust et al., How MT cells analyze the motion of visual patterns. Nat. Neurosci., 9(11), 2006.
[34] H. Sakata, H. Shibutani, & K. Kawano. Functional properties of visual tracking neurons in posterior
parietal association cortex of the monkey. J Neurophysiol, 49(6):1364?1380, 1983.
[35] Y. Saalmann, I. Pigarev, & T. Vidyasagar, ?Neural mechanisms of visual attention: how top-down feedback highlights relevant locations,? Science, 316(5831), p. 1612, 2007.
[36] E. Simoncelli & D. Heeger. A model of neuronal responses in visual area MT. Vision Research,
38(5):743?761, 1998.
[37] E. Vul et al., Explaining human multiple object tracking as resource-constrained approximate inference
in a dynamic probabilistic model. NIPS, 22:1955?1963, 2009.
[38] A. Yilmaz, O. Javed, & M. Shah. Object tracking: A survey. ACM Computing Surveys, 38(4):13, 2006.
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4,013 | 4,632 | Convex Multi-view Subspace Learning
Martha White, Yaoliang Yu, Xinhua Zhang? and Dale Schuurmans
Department of Computing Science, University of Alberta, Edmonton AB T6G 2E8, Canada
{whitem,yaoliang,xinhua2,dale}@cs.ualberta.ca
Abstract
Subspace learning seeks a low dimensional representation of data that enables
accurate reconstruction. However, in many applications, data is obtained from
multiple sources rather than a single source (e.g. an object might be viewed by
cameras at different angles, or a document might consist of text and images). The
conditional independence of separate sources imposes constraints on their shared
latent representation, which, if respected, can improve the quality of a learned
low dimensional representation. In this paper, we present a convex formulation
of multi-view subspace learning that enforces conditional independence while reducing dimensionality. For this formulation, we develop an efficient algorithm
that recovers an optimal data reconstruction by exploiting an implicit convex regularizer, then recovers the corresponding latent representation and reconstruction
model, jointly and optimally. Experiments illustrate that the proposed method
produces high quality results.
1
Introduction
Dimensionality reduction is one of the most important forms of unsupervised learning, with roots
dating to the origins of data analysis. Re-expressing high dimensional data in a low dimensional
representation has been used to discover important latent information about individual data items,
visualize entire data sets to uncover their global organization, and even improve subsequent clustering or supervised learning [1]. Modern data is increasingly complex, however, with descriptions
of increasing size and heterogeneity. For example, multimedia data analysis considers data objects
(e.g. documents or webpages) described by related text, image, video, and audio components. Multiview learning focuses on the analysis of such multi-modal data by exploiting its implicit conditional
independence structure. For example, given multiple camera views of a single object, the particular idiosyncrasies of each camera are generally independent, hence the images they capture will
be conditionally independent given the scene. Similarly, the idiosyncrasies of text and images are
generally conditionally independent given a topic. The goal of multi-view learning, therefore, is to
use known conditional independence structure to improve the quality of learning results.
In this paper we focus on the problem of multi-view subspace learning: reducing dimensionality
when data consists of multiple, conditionally independent sources. Classically, multi-view subspace
learning has been achieved by an application of canonical correlation analysis (CCA) [2, 3]. In
particular, many successes have been achieved in using CCA to recover meaningful latent representations in a multi-view setting [4?6]. Such work has been extended to probabilistic [7] and sparse
formulations [8]. However, a key limitation of CCA-based approaches is that they only admit efficient global solutions when using the squared-error loss (i.e. Gaussian models), while extensions to
robust models have had to settle for approximate solutions [9].
By contrast, in the single-view setting, recent work has developed new generalizations of subspace
learning that can accommodate arbitrary convex losses [10?12]. These papers replace the hard bound
on the dimension of the latent representation with a structured convex regularizer that still reduces
rank, but in a relaxed manner that admits greater flexibility while retaining tractable formulations.
?
Xinhua Zhang is now at the National ICT Australia (NICTA), Machine Learning Group.
1
Subspace learning can be achieved in this case by first recovering an optimal reduced rank response
matrix and then extracting the latent representation and reconstruction model. Such formulations
have recently been extended to the multi-view case [13, 14]. Unfortunately, the multi-view formulation of subspace learning does not have an obvious convex form, and current work has resorted
to local training methods based on alternating descent minimization (or approximating intractable
integrals). Consequently, there is no guarantee of recovering a globally optimal subspace.
In this paper we provide a formulation of multi-view subspace learning that can be solved optimally
and efficiently. We achieve this by adapting the new single-view training methods of [11, 12] to the
multi-view case. After deriving a new formulation of multi-view subspace learning that allows a
global solution, we also derive efficient new algorithms. The outcome is an efficient approach to
multi-view subspace discovery that can produce high quality repeatable results.
0
Notation: We use Ik for the
A, k ? k2 for the
p k ?k identity matrix, A for the transpose of matrix
P
0
Euclidean norm, kXkF = tr(X X) for the Frobenius norm and kXktr = i ?i (X) for the trace
norm, where ?i (X) is the ith singular value of X.
2
Background
nh io
x
Assume one is given t paired observations yjj
consisting of two views: an x-view and a y-view,
of lengths m and n respectively. The goal of multi-view subspace learning is to infer, for each pair,
a shared latent representation, hj , of dimension k < min(n, m), such that the original data can be
accurately modeled. We first consider a linear formulation. Given paired observations the goal is to
infer a set of latent representations, hj , and reconstruction models, A and B, such that Ahj ? xj
and Bhj ? yj for all j. hLetiX denote the n ? t matrix of x observations, Y the m ? t matrix of
X
Y
the concatenated (n + m) ? t data matrix. The problem can then be
h i
A , and a k ? t matrix
expressed as recovering a (n + m) ? k matrix of model parameters, C = B
of latent representations, H, such that Z ? CH.
y observations, and Z =
The key assumption of multi-view learning is that each of the two views, xj and yj , is conditionally independent given the shared latent representation, hj . Although multi-view data can always
be concatenated and treated as a single view, if the conditional independence assumption holds, explicitly representing multiple views enables more accurate identification of the latent representation
(as we will see). The classical formulation of multi-view subspace learning is given by canonical
correlation analysis (CCA), which is typically expressed as the problem of projecting two views so
that the correlation between them is maximized [2]. Assuming the data is centered (i.e. X1 = 0 and
Y 1 = 0), the sample covariance of X and Y is given by XX 0 /t and Y Y 0 /t respectively. CCA can
then be expressed as an optimization over matrix variables
max tr(U 0 XY 0 V ) s.t. U 0 XX 0 U = V 0 Y Y 0 V = I
U,V
(1)
for U ? Rn?k , V ? Rm?k [3]. Although this classical formulation (1) does not make the shared
latent representation explicit, CCA can be expressed by a generative model: given a latent representation, hj , the observations xj = Ahj +j and yj = Bhj +?j are generated by a linear mapping plus
independent zero mean Gaussian noise, ? N (0, ?x ), ? ? N (0, ?y ) [7]. In fact, one can show that
the classical CCA problem (1) is equivalent to the following multi-view subspace learning problem.
(XX 0 )?1/2 X
Proposition 1. Fix k, let Z? =
and
(Y Y 0 )?1/2 Y
(C, H) = arg min kZ? ? CHk2F ,
where C =
h i
A
B
C,H
? 21
. Then U = (XX 0 )
0
A and V = (Y Y 0 )
? 21
(2)
B provide an optimal solution to (1),
0
implying that A A = B B = I is satisfied in the solution to (2).
(The proof is given in Appendix A.) From Proposition 1, one can see how formulation (2) respects
the conditional independence of the separate views: given a latent representation hj , the reconstruction losses on the two views, xj and yj , cannot influence each other, since the reconstruction models
A and B are individually constrained. By contrast, in single-view subspace learning (i.e. principal
2
components analysis) A and B are concatenated in the larger variable C, where C as a whole is constrained but A and B are not. A and B must then compete against each other to acquire magnitude
to explain their respective ?views? given hj (i.e. conditional independence is not enforced). Such
sharing can be detrimental if the two views really are conditionally independent given hj .
Despite its elegance, a key limitation of CCA is its restriction to squared loss under a particular
normalization. Recently, subspace learning algorithms have been greatly generalized in the single
view case by relaxing the rank(H) = k constraint while imposing a structured regularizer that is
a convex relaxation of rank [10?12]. Such a relaxation allows one to incorporate arbitrary convex
losses, including robust losses [10], while maintaining tractability.
As mentioned, these relaxations of single-view subspace learning have only recently been proposed
for the multi-view setting [13, 14]. An extension of these proposals can be achieved by reformulating
(2) to first incorporate an arbitrary loss function L that is convex in its first argument (for examples,
see [15]), then relaxing the rank constraint by replacing it with a rank-reducing regularizer on H. In
particular, we consider the following training problem that extends [14]:
A
A:,i
min L
H; Z + ?kHk2,1 , s.t.
? C for all i,
B
B:,i
A,B,H
a
A
where C :=
: kak2 ? ?, kbk2 ? ? , C =
,
(3)
b
B
P
and kHk2,1 = i kHi,: k2 is a matrix block norm. The significance of using the (2, 1)-block norm
as a regularizer is that it encourages rows of H to become sparse, hence reducing the dimensionality
of the learned representation [16]. C must be constrained however, otherwise kHk2,1 can be pushed
arbitrarily close to zero simply by re-scaling H/s and Cs (s > 0) while preserving the same loss.
Unfortunately, (3) is not jointly convex in A, B and H. Thus, the algorithmic approaches proposed
by [13, 14] have been restricted to alternating block coordinate descent between components A, B
and H, which cannot guarantee a global solution. Our main result is to show that (3) can in fact be
solved globally and efficiently for A, B and H, improving on the previous local solutions [13, 14].
3
Reformulation
Our first main contribution is to derive an equivalent but tractable reformulation of (3), followed
by an efficient optimization algorithm. Note that (3) can in principle be tackled by a boosting
strategy; however, one would have to formulate a difficult weak learning oracle that considers both
views simultaneously [17]. Instead, we find that a direct matrix factorization approach of the form
developed in [11, 12] is more effective.
To derive our tractable reformulation, we first introduce the change of variable Z? = CH which
allows us to rewrite (3) equivalently as
n
o
? Z) + ? min
min L(Z;
min kHk2,1 .
(4)
?
{C:C:,i ?C} {H:CH=Z}
?
Z
A key step in the derivation is the following characterization of the inner minimization in (4).
? whose dual norm is
Proposition 2.
min
min kHk2,1 defines a norm ||| ? |||? (on Z)
?
{C:C:,i ?C} {H:CH=Z}
|||?||| :=
max
c?C,khk2 ?1
c0 ?h.
? i,: where
Proof. Let ?i = kHi,: k2 be the Euclidean norm of the i-th row of H. Then Hi,: = ?i H
?
?
Hi,: has unit length (if ?i = 0, then take Hi,: to be any unit vector). Therefore
min
min
?
{C:C:,i ?C} {H:CH=Z}
kHk2,1 =
min
? P ?i C:,i H
? i,: }
{C,?i :C:,i ?C,?i ?0, Z=
i
P
i
?i
=
min
?
{t?0:Z?tK}
t, (5)
where K is the convex hull of the set G := {ch0 : c ? C, khk2 = 1}. In other words, we seek a
? using only elements from G. Since the set K is convex and symmetric,
rank-one decomposition of Z,
3
(5) is known as a gauge function and defines a norm on Z? (see e.g. [18, Proposition V.3.2.1]). This
norm has a dual given by
|||?||| := max tr(?0 Z) =
Z?K
max
c?C,khk2 ?1
c0 ?h,
(6)
where the last equality follows because maximizing any linear function over the convex hull K of a
set G achieves the same value as maximizing over the set G itself.
Applying Proposition 2 to problem (3) leads to a simpler formulation of the optimization problem.
p
2
2
0
? Z)+? max kD??1 Zk
? tr , where D? = ? +? ? In p
.
Lemma 3. (3) = min L(Z;
??0
?
0
? 2 +? 2 /? Im
Z
Proof. The lemma is proved by first deriving an explicit form of the norm ||| ? ||| in (6), then deriving
its dual norm. The details are given in Appendix B.
Unfortunately the inner maximization problem in Lemma 3 is not concave in ?. However, it is
possible to re-parameterize D? to achieve a tractable formulation as follows. First, define a matrix
?
?/ ? In
? 0
E? := D ?2 (1??) =
, such that D? = E ?2 .
0
?/ 1 ? ? Im
? 2 ?+? 2
?2 ?
? with ? ? 0 corresponding to 0 ? ? ? 1. The
? = max0???1 kE??1 Zk,
Note that max??0 kD??1 Zk
following lemma proves that this re-parameterization yields an efficient computational approach.
? tr is concave in ? over [0, 1].
Lemma 4. h(?) := kE ?1 Zk
?
? q
?
? ?X
q
Z
2
?
? ?X 0 ?X
1?? ? Y 0 ? Y
?
? ?
Proof. Expand h(?) into
, where tr( ?)
? 2 (Z ) Z + ? 2 (Z ) Z
q 1?? ? Y
= tr
?2 Z
tr
means summing the square root of the eigenvalues (i.e. a spectral function). By [19], if a spectral
function f is concave on [0, ?), then tr(f (M )) must be concave on positive semidefinite matrices.
? Y 0 ? Y is positive semi-definite for ? ? [0, 1] and
The result follows since ??2 (Z? X )0 Z? X + 1??
? 2 (Z ) Z
?
f = ? is concave on [0, ?).
From Lemmas 3 and 4 we achieve the first main result.
Theorem 5.
? Z) + ? max kE??1 Zk
? tr = max min L(Z;
? Z) + ?kE??1 Zk
? tr . (7)
(3) = min L(Z;
?
Z
0???1
0???1
?
Z
Hence (3) is equivalent to a concave-convex maxi-min problem with no local maxima nor minima.
Thus we have achieved a new formulation for multi-view subspace learning that respects conditional
independence of the separate views (see discussion in Section 2) while allowing a globally solvable
formulation. To the best of our knowledge, this has not previously been achieved in the literature.
4
Efficient Training Procedure
This new formulation for multi-view subspace learning also allows for an efficient algorithmic approach. Before conducting an experimental comparison to other methods, we first develop an ef? = E??1 Z? in (7), which
ficient implementation. To do so we introduce a further transformation Q
leads to an equivalent but computationally more convenient formulation of (3):
? Z) + ?kQk
? tr .
(3) = max min L(E? Q;
0???1 Q
?
(8)
? Z) + ?kQk
? tr . The transformation does not affect the concavity of
Denote g(?) := minQ? L(E? Q;
the problem with respect to ? established in Lemma 4; therefore, (8) remains tractable. The training
? which allows Z? = E? Q
? to
procedure then consists of two stages: first, solve (8) to recover ? and Q,
?
be computed; then, recover the optimal factors H and C (i.e. A and B) from Z.
4
? The key to efficiently recovering Z? is to observe that (8) has a conveRecovering an optimal Z:
nient form. The concave outer maximization is defined over a scalar variable ?, hence simple line
search can be used to solve the problem, normally requiring at most a dozen evaluations to achieve
? is a standard trace-norm-regularized loss
a small tolerance. Crucially, the inner minimization in Q
minimization problem, which has been extensively studied in the matrix completion literature [20?
22]. By exploiting these algorithms, g(?) and its subgradient can both be computed efficiently.
? Once Z? is obtained, we need to recover a C and H that satisfy
Recovering C and H from Z:
?
? ? , and C:,i ? C for all i.
CH = Z,
kHk2,1 = |||Z|||
(9)
We exploit recent sparse approximation methods [23, 24] to solve this problem. First, note from
P
?
? ? = min
(5) that |||Z|||
? P ?i C:,i H
? i,: }
i ?i , where kHi,: k2 ? 1. Since we already
{C,?i :C:i ?C,?i ?0, Z=
i
? ? = kE ?1 Zk
? tr from the first stage, we can rescale the problem so that |||Z|||
? ? = 1 without
have |||Z|||
?
P
P
? i,: where ? ? 0 and
loss of generality. In such a case, Z? = i ?i C:,i H
i ?i = 1 (we restore the
?
?
proper scale to H afterward). So now, Z lies in the convex hull of the set G := {ch0 : c ? C, khk2 ?
1} and any expansion of Z? as a convex combination of the elements in G is a valid recovery. From
this connection, we can now exploit the recent greedy algorithms developed in [23, 24] to solve the
recovery problem. In particular, the recovery just needs to solve
min f (K), where f (K) := kZ? ? Kk2F .
(10)
K?convG
where conv denotes the convex hull. Note that the optimal value of (10) is 0. The greedy (boosting)
algorithm provided by [23, 24] produces a factorization of Z? into C and H and proceeds as follows:
1. Weak learning step: greedily pick Gt = ct h0t ? argminG?G h?f (Kt?1 ), Gi . Note that this step
can be computed efficiently with a form of power method iteration (see Appendix C.2).
P
t
t
P
(t)
?i Gi , then Kt =
f
?i Gi .
2. ?Totally corrective? step: ?(t) = argmin
P
??0,
i
i=1
?i =1
i=1
This procedure will find a Kt satisfying kZ? ? Kt k2F < within O(1/) iterations [23, 24].
Acceleration: In practice, this procedure can be considerably accelerated via more refined analysis.
? it is not hard to recover its dual
Recall Z? is penalized by the dual of the norm in (6). Given Z,
? Then given ?, the
variable ? by exploiting the dual norm relationship: ?= argmax?:|||?|||?1 tr(?0 Z).
following theorem guarantees many bases in C can be eliminated from the recovery problem (9).
0
Theorem 6. (C, H) satisfying Z? = CH is optimal iff k?0 C:,i k = 1 and Hi,: = kHi,: k2 C:,i
?, ?i.
Theorem 6 prunes many elements from G and the weak learning step only needs to consider a proper
subset. Interestingly this constrained search can be solved with no increase in the computational
complexity. The accelerated boosting generates ct in the weak learning step, giving the recovery
C = [c1 , . . . , ck ] and H = diag(?)C 0 ?. The rank, k, is implicitly determined by termination of the
boosting algorithm. The detailed algorithm and proof of Theorem 6 are given in Appendix C.
5
Comparisons
Below we compare the proposed global learning method, Multi-view Subspace Learning (MSL),
against a few benchmark competitors.
Local Multi-view Subspace Learning (LSL) An obvious competitor is to solve (3) by alternating
descent over the variables: optimize H with A and B fixed, optimize A with B and H fixed, etc.
This is the computational strategy employed by [13, 14]. Since A and B are both constrained and H
is regularized by the (2,1)-block norm which is not smooth, we optimized them using the proximal
gradient method [25].
Single-view Subspace Learning
h i (SSL) Single view learning can be cast as a relaxation of (3),
A are normalized as a whole, rather than individually for A and B:
where the columns of C = B
? Z) + ?(? 2 +? 2 )? 21 kHk
? 2,1 (11)
min ?
L(CH; Z) + ?kHk2,1 =
min
L(C? H;
{H,C:kC:,i k2 ?
? 2+? 2 }
? C:k
? C
?:,i k2 ?1}
{H,
? Z) + ?(? 2 + ? 2 )? 12 kZk
? tr .
= min L(Z;
?
Z
5
(12)
Equation
given in [10]. The equality in (11) is by change of variable
p (12) matches the formulation
p
? = ? 2 + ? 2 H. Equation (12) can be established from the basic results
C = ? 2 + ? 2 C? and H
of [11, 12] (or specializing Proposition 2 to the case where C is the unit Euclidean ball). To solve
(12), we used a variant of the boosting algorithm [22] when ? is large, due to its effectiveness
when the solution has low rank. When ? is small, we switch to the alternating direction augmented
Lagrangian method (ADAL) [26] which does not enforce low-rank at all iterations. This hybrid
? in (8) for MSL. Once an optimal Z? is
choice of solver is also applied to the optimization of Q
achieved, the corresponding C and H can be recovered by an SVD: for Z? = U ?V 0 , set C =
1
1
? tr , and so is an
(? 2 +? 2 ) 2 U and H = (? 2 +? 2 )? 2 ?V 0 which satisfies CH = Z? and kHk2,1 = kZk
optimal solution to (11).
6
Experimental results
Datasets We provide experimental results on two datasets: a synthetic dataset and a face-image
dataset. The synthetic dataset is generated as follows. First, we randomly generate a k-by-ttr matrix
Htr for training, a k-by-tte matrix Hte for testing, and two basis matrices, A (n-by-k) and B (mby-k), by (iid) sampling from a zero-mean unit-variance Gaussian distribution. The columns of A
and B are then normalized to ensure that the Euclidean norm of each is 1. Then we set
Xtr = AHtr , Ytr = BHtr , Xte = AHte , Yte = BHte .
? tr , Y?tr , X
? te , Y?te . Following [10], we use sparse
Next, we add noise to these matrices, to obtain X
non-Gaussian noise: 5% of the matrix entries were selected randomly and replaced with a value
drawn uniformly from [?M, M ], where M is 5 times the maximal absolute entry of the matrices.
The second dataset is based on the Extended Yale Face Database B [27]. It contains grey level
face images of 28 human subjects, each with 9 poses and 64 lighting conditions. To construct the
dataset, we set the x-view to a fixed lighting (+000E+00) and the y-view to a different fixed lighting
(+000E+20). We obtain a pair of views by randomly drawing a subject and a pose (under the two
fixed lightings). The underlying assumption is that each lighting has its own set of bases (A and B)
and each (person, pose) pair has the same latent representation for the two lighting conditions. All
images are down-sampled to 100-by-100, meaning n = m = 104 . We kept one view (x-view) clean
and added pixel errors to the second view (y-view). We randomly set 5% of the pixel values to 1,
mimicking the noise in practice, e.g. occlusions and loss of pixel information from image transfer.
The goal is to enable appropriate reconstruction of a noisy image using other views.
Model specification Due to the sparse noise model, we used L1,1 loss for L:
A
L
H, Z = kAH ? Xk1,1 + kBH ? Y k1,1 .
B
{z
} |
{z
}
|
:=L1 (AH,X)
(13)
:=L2 (BH,Y )
For computational reasons, we worked on a smoothed version of the L1,1 loss [26].
6.1
Comparing optimization quality
We first compare the optimization performance of MSL (global solver) versus LSL (local solver).
Figure 1(a) indicates that MSL consistently obtains a lower objective value, sometimes by a large
margin: more than two times lower for ? = 10?4 and 10?3 . Interestingly, as ? increases, the
difference shrinks. This result suggests that more local minima occur in the higher rank case (a large
? increases regularization and decreases the rank of the solution). In Section 6.2, we will see that
the lower optimization quality of LSL and the fact that SSL optimizes a less constrained objective
both lead to significantly worse denoising performance.
Second, we compare the runtimes of the three algorithms. Figure 1(b) presents runtimes for LSL
and MSL for an increasing number of samples. Again, the runtime of LSL is significantly worse
for smaller ?, as much as 4000x slower; as ? increases, the runtimes become similar. This result
is likely due to the fact that for small ?, the MSL inner optimization is much faster via the ADAL
solver (the slowest part of the optimization), whereas LSL still has to slowly iterate over the three
variables. They both appear to scale similarly with respect to the number of samples.
6
Objective LSL:MSL For Varying ?
?=1e-4
?=1e-3
1.5
?=1e-2
1
0.5
0
?=1e-1
?=1
200
400
600
800
Number of Samples
1000
3000
?=1e-3
2000
1000
?=1e-2
0
?=1e-1
?1000
0
1200
Runtime on Synthetic Data
?=1e-4
10
4000
(a) Objectives for LSL:MSL
Runtime (seconds)
2
Training Runtime LSL:MSL For Varying ?
5000
Runtime Ratio LSL:MSL
Objective Value Ratio LSL:MSL
2.5
8
6
SSL
MSL
MSL?R
4
2
?=1
200
400
600
800
Number of Samples
1000
1200
(b) Runtimes for LSL:MSL
0
0
200
400
600
800
Number of Samples/Features
1000
(c) Runtimes for SSL and MSL
Figure 1: Comparison between LSL and MSL on synthetic datasets with changing ?, n = m = 20 and 10
10
8
6
4
6
4
2
0
0
MSL
SSL
LSL
2
10
20
Run number
30
0
0
(a) tL = 100
4
4
10
10
MSL
SSL
LSL
Error of SSL
12
8
Error of LSL
10
SNR
SNR
repeats. (a) LSL often gets stuck in local minima, with a significantly higher objective than MSL. (b) For small
?, LSL is significantly slower than MSL. They scale similarly with the number of samples (c) Runtimes of SSL
and MSL for training and recovery with ? = 10?3 . For growing sample size, n = m = 20. MSL-R stands for
the recovery algorithm. The recovery time for SSL is almost 0, so it is not included.
2
10
0
10
0
10
20
Run number
30
2
(a) MSL vs LSL
(b) tL = 300
4
10
10
10
Error of MSL
2
10
0
10
0
2
4
10
10
10
Error of MSL
(b) MSL vs SSL
Figure 2: Signal-to-noise ratio of denoising algorithms on Figure 3: MSL versus SSL error in synthesynthetic data using recovered models on hold-out views. n =
m = 10. In (a), we used tL = 100 pairs of views for training
A and B and tested on 100 hold-out pairs, with 30 repeated
random draws of training and test data. In (b) we used tL =
300. Parameters were set to optimize respective methods.
sizing y-view, over 30 random runs. We set
n=m=200, tL =20, and ttest =80. In (a), LSL error is generally above the diagonal line, indicating higher error than MSL. In (b), SSL error is
considerably higher than MSL.
For SSL versus MSL, we expect SSL to be faster than MSL because it is a more straightforward
? (with a fixed ?) has the same form
optimization: in MSL, each inner optimization of (8) over Q
as the SSL objective. Figure 1(c), however, illustrates that this difference is not substantial for
increasing sample size. Interestingly, the recovery runtime seems independent of dataset size, and
is instead likely proportional to the rank of the data. The trend is similar for increasing features: for
tL = 200, at n = 200, MSL requires ? 20 seconds, and at n = 1000, it requires ? 60 seconds.
6.2
Comparing denoising quality
Next we compare the denoising capabilities of the algorithms on synthetic and face image datasets.
There are two denoising approaches. The simplest is to run the algorithm on the noisy Y?te , giving
the reconstructed Y?te as the denoised image. Another approach is to recover the models, A and B, in
s
s
? te = {?
? te and
a training phase. Given a new set of instances, X
xi }i=1 and Y?te = {?
yi }i=1 , noise in X
Y?te can be removed using A and B, without re-training. This approach requires first recovering the
? te = (h1 , . . . , hs ), for X
? te and Y?te . We use a batch approach for inference:
latent representation, H
?
?
Hte = argmin L1 (AH, Xte )+L2 (BH, Y?te )+?kHk2,1 .
(14)
H
? te = AH
? te and Y?te = B H
? te . We compared
The x-views and y-views are then reconstructed using X
these reconstructions with the clean data, Xte and Yte , in terms of the signal-to-noise ratio:
.
? te , Y?te ) = kXte k2 + kYte k2
? te k2F + kYte ? Y?te k2F .
SNR(X
kX
?
X
(15)
te
F
F
We present the recovery approach on synthetic data and the direct reconstruction approach on the
face dataset. We cross-validated over ? ? {10?4 , 10?3 , 10?2 , 10?1 , 0.5, 1} according to the highest
signal-to-noise ratio on the training data. We set ? = ? = 1 because the data is in the [0, 1] interval.
7
Clean
Noisy : 5%
SSL
LSL
MSL
20
20
20
20
20
40
40
40
40
40
60
60
60
60
60
80
80
80
80
80
100
20
40
60
80 100
100
20
40
60
80 100
100
20
Noisy : 10%
40
60
80 100
100
20
SSL
40
60
80 100
100
LSL
20
20
20
40
40
40
40
60
60
60
60
80
80
80
80
100
100
100
100
40
60
80 100
20
40
60
80 100
20
40
60
40
60
80 100
MSL
20
20
20
80 100
20
40
60
80 100
Figure 4: Reconstruction of a noisy image with 5% or 10% noise. LSL performs only slightly worse than
MSL for larger noise values: a larger regularization parameter is needed for more noise, resulting in fewer local
minima (as discussed in Figure 1). Conversely, SSL performs slightly worse than MSL for 5% noise, but as the
noise increases, the advantages of the MSL objective are apparent.
6.2.1
Using Recovered Models for Denoising
Figure 2 presents the signal-to-noise ratio for recovery on synthetic data. MSL produced the highest
value of signal-to-noise ratio. The performance of LSL is inferior to MSL, but still better than SSL,
corroborating the importance of modelling the data as two views.
6.2.2
Image Denoising
In Figure 4, we can see that MSL outperforms both SSL and LSL on the face image dataset for
two noise levels: 5% and 10%. Interestingly, in addition to having on average a 10x higher SNR
than SSL for these results, MSL also had significantly different objective values. SSL had larger
reconstruction error on the clean x-view (10x higher), lower reconstruction error on the noisy yview (3x lower) and a higher representation norm (3x higher). Likely, the noisy y-view skewed the
representation, due to the joint rather than separate constraint as in the MSL objective.
6.3
Comparing synthesis of views
In image synthesis, the latent representation is computed from only one view:
? te ) + ?kHk2,1 . The y-view is then synthesized: Y?te = B H
? te .
argminH L1 (AH, X
? te
H
=
Figure 3 shows the synthesis error, ||Y?te ?Yte ||2F , of MSL, LSL, and SSL over 30 random runs: MSL
generally incurs less error than LSL, and SSL incurs much higher error because it is not modelling
the conditional independence between views.
7
Conclusion
We provided a convex reformulation of multi-view subspace learning that enables global learning, as
opposed to previous local formulations. We also developed a new training procedure which reconstructs the data optimally and discovers the latent representations efficiently. Experimental results
on synthetic data and image data confirm the effectiveness of our method, which consistently outperformed other approaches in denoising quality. For future work, we are investigating extensions
to semi-supervised settings, such as global methods for co-training and co-regularization. It should
also be possible to extend our approach to more than two views and incorporate kernels.
Acknowledgements
We thank the reviewers for their helpful comments, in particular, an anonymous reviewer whose
suggestions greatly improved the presentation. Research supported by AICML and NSERC.
8
References
[1] J. Lee and M. Verleysen. Nonlinear Dimensionality Reduction. Springer, 2010.
[2] D. Hardoon, S. Szedmak, and J. Shawe-Taylor. Canonical correlation analysis: An overview
with application to learning methods. Neural Computation, 16:2639?2664, 2004.
[3] T. De Bie, N. Cristianini, and R. Rosipal. Eigenproblems in pattern recognition. In Handbook
of Geometric Computing, pages 129?170, 2005.
[4] P. Dhillon, D. Foster, and L. Ungar. Multi-view learning of word embeddings via CCA. In
NIPS, 2011.
[5] C. Lampert and O. Kr?omer. Weakly-paired maximum covariance analysis for multimodal
dimensionality reduction and transfer learning. In ECCV, 2010.
[6] L. Sigal, R. Memisevic, and D. Fleet. Shared kernel information embedding for discriminative
inference. In CVPR, 2009.
[7] F. Bach and M. Jordan. A probabilistic interpretation of canonical correlation analysis. Technical Report 688, Department of Statistics, University of California, Berkeley, 2006.
[8] C. Archambeau and F. Bach. Sparse probabilistic projections. In NIPS, 2008.
[9] J. Viinikanoja, A. Klami, and S. Kaski. Variational Bayesian mixture of robust CCA. In ECML,
2010.
[10] E. Cand`es, X. Li, Y. Ma, and J. Wright. Robust principal component analysis? J.ACM, 58(1):
1?37, 2011.
[11] X. Zhang, Y. Yu, M. White, R. Huang, and D. Schuurmans. Convex sparse coding, subspace
learning, and semi-supervised extensions. In AAAI, 2011.
[12] F. Bach, J. Mairal, and J. Ponce. Convex sparse matrix factorizations. arXiv:0812.1869v1,
2008.
[13] N. Quadrinto and C. Lampert. Learning multi-view neighborhood preserving projections. In
ICML, 2011.
[14] Y. Jia, M. Salzmann, and T. Darrell. Factorized latent spaces with structured sparsity. In NIPS,
pages 982?990, 2010.
[15] M. White and D. Schuurmans. Generalized optimal reverse prediction. In AISTATS, 2012.
[16] A. Argyriou, T. Evgeniou, and M. Pontil. Convex multi-task feature learning. Machine Learning, 73(3):243?272, 2008.
[17] D. Bradley and A. Bagnell. Convex coding. In UAI, 2009.
[18] J-B Hiriart-Urruty and C. Lemar?echal. Convex Analysis and Minimization Algorithms, I and
II, volume 305 and 306. Springer-Verlag, 1993.
[19] D. Petz. A survey of trace inequalities. In Functional Analysis and Operator Theory, pages
287?298. Banach Center, 2004.
[20] S. Ma, D. Goldfarb, and L. Chen. Fixed point and Bregman iterative methods for matrix rank
minimization. Mathematical Programming, 128:321?353, 2011.
[21] J. Cai, E. Cand`es, and Z. Shen. A singular value thresholding algorithm for matrix completion.
SIAM Journal on Optimization, 20(4):1956?1982, 2010.
[22] X. Zhang, Y. Yu, and D. Schuurmans. Accelerated training for matrix-norm regularization: A
boosting approach. In NIPS, 2012.
[23] A. Tewari, P. Ravikumar, and I. S. Dhillon. Greedy algorithms for structurally constrained high
dimensional problems. In NIPS, 2011.
[24] X. Yuan and S. Yan. Forward basis selection for sparse approximation over dictionary. In
AISTATS, 2012.
[25] A. Beck and M. Teboulle. A fast iterative shrinkage-thresholding algorithm for linear inverse
problems. SIAM Journal on Imaging Sciences, 2(1):183?202, 2009.
[26] D. Goldfarb, S. Ma, and K. Scheinberg. Fast alternating linearization methods for minimizing
the sum of two convex functions. Mathematical Programming, to appear.
[27] A. Georghiades, P. Belhumeur, and D. Kriegman. From few to many: Illumination cone models
for face recognition under variable lighting and pose. IEEE TPAMI, 23:643?660, 2001.
9
Supplementary Material
A
Proof of Proposition 1
To show that (1) and (2) have equivalent solutions we exploit some developments from [28]. Let
1
1
N = (XX 0 )? 2 and M = (Y Y 0 )? 2 , hence
I
N XY 0 M
0
?
?
ZZ =
.
M Y X 0N
I
First consider (1). Its solution can be characterized by the maximal solutions to the generalized
eigenvalue problem [3]:
0
XY 0
u
XX 0
0
u
,
=?
Y X0
0
v
0
YY0
v
which, under the change of variables u = N a and v = M b and then shifting the eigenvalues by 1, is
equivalent to
?1
0
XY 0 M
a
N
0
a
?
=
?
Y X 0N
0
b
b
0
M ?1
0
0
N XY M
a
I 0
a
?
=?
M Y X 0N
0
b
0 I
b
a
a
?
Z? Z? 0
= (? + 1)
b
b
h i
A to the top k eigenvectors of Z
? Z? 0 one can show that U = N A and V = M B provides
By setting B
an optimal solution to (1) [3].
? where C ? denotes pseudo-inverse.
By comparison, for (2), an optimal H is given by H = C ? Z,
Hence
? 2F
min kZ? ? CHk2F = min k(I ? CC ? )Zk
C,H
C
= tr(Z? Z? 0 ) ?
max
{C:C 0 C=I}
tr(C 0 Z? Z? 0 C).
Here again the solution is given by the top k eigenvectors of Z? Z? 0 [29].1
B
Proof for Lemma 3
First, observe that
(3) =
min
? Z) + ?
min L(CH; Z) + ?kHk2,1 = min L(Z;
{C:C:,i ?C} H
?
Z
min
min
?
{C:C:,i ?C} {H:CH=Z}
kHk2,1
? Z) + ?|||Z|||
? ?,
= min L(Z;
?
Z
where the last step follows from Proposition 2.
? ? = max??0 kD??1 Zk
? tr , which was established in [11]. We reproduce
It only remains to show |||Z|||
the proof in [11] for the convenience of the reader.
We will use two diagonal matrices, I X = diag([1n ; 0m ]) and I Y = diag([0n ; 1m ]) such that I X +
I Y = Im+n . Similarly, for c ? Rm+n , we use cX (respectively cY ) to denote c1:m (respectively
cm+1:m+n ).
The first stage is to prove that the dual norm is characterized by
|||?||| = min kD? ?ksp .
??0
1
[30] gave a similar but not equivalent formulation to (2), due to the lack of normalization.
10
(16)
where the spectral norm kXksp = ?max (X) is the dual of the trace norm, kXktr . To this end, recall
that
|||?||| =
max
c?C,khk2 ?1
c0 ?h = max kc0 ?k2 =
max
c?C
{c:kcX k2 =?, kcY k2 =?}
kc0 ?k2
giving
|||?|||2 =
max
{c:kcX k2 =?, kcY k2 =?}
c0 ??0 c =
max
{?:?0, tr(?I X )?? 2 , tr(?I Y )?? 2 }
tr(???0 ),
(17)
using the fact that when maximizing a convex function, one of the extreme points in the constraint
set {? : ?0, tr(?In )?? 2, tr(?Im )?? 2 } must be optimal. Furthermore, since the extreme points
have rank at most one in this case [31], the rank constraint rank(?) = 1 can be dropped.
Next, form the Lagrangian L(?; ?, ?, ?) = tr(???0 ) + tr(??) + ?(? 2 ? tr(?I X )) + ?(? 2 ?
tr(?I Y )) where ? ? 0, ? ? 0 and ? 0. Note that the primal variable ? can be eliminated
by formulating the equilibrium condition ?L/?? = ??0 + ? ? ?I X ? ?I Y = 0, which implies
??0 ? ?I X ? ?I Y 0. Therefore, we achieve the equivalent dual formulation
(17) =
min
? 2 ? + ? 2 ?.
(18)
{?,?:??0, ??0, ?I X +?I Y ??0 }
Now observe that for ? ? 0 and ? ? 0, the relation ??0 ?I X + ?I Y holds if and only if
D?/? ??0 D?/? D?/? (?I X +?I Y )D?/? = (? 2 ?+? 2 ?)In+m , hence
(18) =
min 2
? 2 ?+? 2 ?
(19)
2
2
{?,?:??0, ??0, kD?/? ?ksp ?? ?+? ?}
The third constraint must be met with equality at the optimum due to continuity, for otherwise we
would be able to further decrease the objective, a contradiction to optimality. Note that a standard
compactness argument would establish the existence of minimizers. So
(19) = min kD?/? ?k2sp = min kD? ?k2sp .
??0
??0,??0
Finally, for the second stage, we characterize the target norm by observing that
? ?
|||Z|||
=
?
max tr(?0 Z)
?:|||?|||?1
=
max
max
=
max
=
? tr .
max kD??1 Zk
??0 ?:kD? ?ksp ?1
max
? ?k
? sp ?1
??0 ?:k
?
tr(?0 Z)
(20)
? 0 D??1 Z)
?
tr(?
(21)
??0
where (20) uses (16), and (21) exploits the conjugacy of the spectral and trace norms. The lemma
follows.
C
Proof for Theorem 6 and Details of Recovery
Once an optimal reconstruction Z? is obtained, we need to recover the optimal factors C and H that
satisfy
? , kHk2,1 = |||Z|||
? ? , and C:,i ? C for all i.
CH = Z,
(22)
Note that by Proposition 2 and Lemma 3, the recovery problem (22) can be re-expressed as
min
?
{C,H:C:,i ?C ?i, CH=Z}
kHk2,1
=
max
{?:|||?|||?1}
?
tr(?0 Z).
(23)
? then use ? to recover H
Our strategy will be to first recover the optimal dual solution ? given Z,
and C.
?
First, to recover ? one can simply trace back from (21) to (20). Let U ?V 0 be the SVD of D??1 Z.
0
?1
0
? = U V and ? = D U V automatically satisfies |||?||| = 1 while achieving the optimal
Then ?
?
? 0 D??1 Z)
? = tr(?) = kD??1 Zk
? tr .
trace in (23) because tr(?
11
Given such an optimal ?, we are then able to characterize an optimal solution (C, H). Introduce the
set
a
C(?) := arg max k?0 ck = c =
: kak = ?, kbk = ?, k?0 ck = 1 .
(24)
b
c?C
Theorem 6. For a dual optimal ?, (C, H) solves recovery problem (22) if and only if C:,i ? C(?)
0
?
?, such that CH = Z.
and Hi,: = kHi,: k2 C:,i
Proof. By (23), if Z? = CH, then
? ? = tr(?0 Z)
? = tr(?0 CH) =
|||Z|||
X
Hi,: ?0 C:,i .
(25)
i
0
0
Note that ?C:,i ? C, k?0 C:,i k2 ? 1 since |||?||| ? 1 and Hi,: ?
PC:,i = kHi,: ? C:,i k2 ?
0
kHi,: k2 k? C:,i k2 ? kHi,: k2 . If (C, H) is optimal, then (25) =
i kHi,: k2 , hence implying
0
k?0 C:,i k2 = 1 and Hi,: = kHi,: k2 C:,i
?.
0
? ? = P kHi,: k2 ,
On the other hand, if k?0 C:,i k2 = 1 and Hi,: = kHi,: k2 C:,i
?, then we have |||Z|||
i
implying the optimality of (C, H).
Therefore, given ?, the recovery problem (22) has been reduced to finding a vector ? and matrix C
?
such that ? ? 0, C:,i ? C(?) for all i, and C diag(?)C 0 ? = Z.
Next we demonstrate how to incrementally recover ? and C. Denote the range of C diag(?)C 0 by
the set
P
S := { i ?i ci c0i : ci ? C(?), ? ? 0} .
Note that S is the conic hull of (possibly infinitely many) rank one matrices {cc0 : c ? C(?)}.
However, by Carath?eodory?s theorem [32, ?17], any matrix K ? S can be written as the conic combination of finitely many rank one matrices of the form {cc0 : c ? C(?)}. Therefore, conceptually,
the recovery problem has been reduced to finding a sparse set of non-negative weights, ?, over the
set of feasible basis vectors, c ? C(?).
To find these weights, we use a totally corrective ?boosting? procedure [22] that is guaranteed to
converge to a feasible solution. Consider the following objective function for boosting
? 2F , where K ? S.
f (K) = kK? ? Zk
Note that f is clearly a convex function in K with a Lipschitz continuous gradient. Theorem 6
implies that an optimal solution of (22) corresponds precisely to those K ? S such that f (K) = 0.
The idea behind totally corrective boosting [22] is to find a minimizer of f (hence optimal solution
of (22)) incrementally. After initializing K0 = 0, we iterate between two steps:
1. Weak learning step: find
ct ? argmin h?f (Kt?1 ), cc0 i = argmax c0 Qc,
c?C(?)
(26)
c?C(?)
where Q = ??f (Kt?1 ) = 2(Z? ? Kt?1 ?)?0 .
2. ?Totally corrective? step:
?(t)
=
argmin f
P
?:?i ?0
Kt
=
Pt
i=1
t
i=1
?i ci c0i ,
(27)
(t)
?i ci c0i .
Three key facts can be established about this boosting procedure: (i) each weak learning step can
be solved efficiently; (ii) each totally corrective weight update can be solved efficiently; and (iii)
f (Kt ) & 0, hence a feasible solution can be arbitrarily well approximated. (iii) has been proved in
[22], while (ii) is immediate because (27) is a standard quadratic program. Only (i) deserves some
explanation. We show in the next subsection that C(?), defined in (24), can be much simplified, and
consequently we give in the last subsection an efficient algorithm for the oracle problem (26) (the
idea is similar to the one inherent in the proof of Lemma 3).
12
C.1
Simplification of C(?)
Since C(?) is the set of optimal solutions to
max k?0 ck ,
(28)
c?C
our idea is to first obtain an optimal solution to its dual problem, and then use it to recover the
optimal c via the KKT conditions. In fact, its dual problem has been stated in (18). Once we obtain
the optimal ? in (21) by solving (8), it is straightforward to backtrack and recover the optimal ? and
? in (18). Then by KKT condition [32, ?28], c is an optimal solution to (28) if and only if
X
c
= ?,
cY
= ?,
(29)
hR, cc0 i = 0, where R = ?I X + ?I Y ? ??0 0.
(30)
Since (30) holds iff c is in the null space of R, we find an orthonormal basis {n1 , . . . , nk } for this
null space. Assume
X
N
c = N ?, where N = [n1 , . . . , nk ] =
, ? = (?1 , . . . , ?k )0 .
(31)
NY
By (29), we have
2
2
(32)
0 = ? 2
cX
? ? 2
cY
= ?0 ? 2 (N X )0 N X ? ? 2 (N Y )0 N Y ?.
The idea is to go through some linear transformations for simplification. Perform eigendecomposition U ?U 0 = ? 2 (N X )0 N X ? ? 2 (N Y )0 N Y , where ? = diag(?1 , . . . , ?k ), and U ?
Rk?k is orthonormal. Let v = U 0 ?. Then by (31),
c = N U v,
(33)
and (32) is satisfied if and only if
X
v0 ?v =
?i vi2 = 0.
(34)
i
Finally, (29) implies
2
? 2 + ? 2 = kck = v0 U 0 N 0 N U v = v0 v.
In summary, by (33) we have
C(?) = {N U v : v satisfies (34) and (35)}
n
o
2
= N U v : v0 ?v = 0, kvk = ? 2 + ? 2 .
C.2
(35)
(36)
Solving the weak oracle problem (26)
The weak oracle needs to solve
max c0 Qc,
c?C(?)
where Q = ??f (Kt?1 ) = 2(Z? ? Kt?1 ?)?0 . By (36), this optimization is equivalent to
max 2
v0 T v,
v:v0 ?v=0, kvk =? 2 +? 2
0
0
where T = U N QN U . Using the same technique as in the proof of Lemma 3, we have
max0
v0 T v
0
v:v v=1,v ?v=0
0
(let H = vv ) =
(Lagrange dual) =
max
tr(T H)
H0,tr(H)=1,tr(?H)=0
min
?,?:? ?+?I?T 0
?
= min ?max (T ? ? ?),
? ?R
where ?max stands for the maximum eigenvalue. Since ?max is a convex function over real symmetric matrices, the last line search problem is convex in ? , hence can be solved globally and efficiently.
Given the optimal ? and the optimal objective value ?, the optimal v can be recovered using a similar
? = {?
? s }.
trick as in Appendix C.1. Let the null space of ?I + ? ? ? T be spanned by N
n1 , . . . , n
2
s
2
2
0
?
? ? R such that v := N ?
? satisfies kvk = ? + ? and v ?v = 0.
Then find any ?
13
Auxiliary References
[28] L. Sun, S. Ji, and J. Ye. Canonical correlation analysis for multilabel classification: A leastsquares formulation, extensions, and analysis. IEEE TPAMI, 33(1):194?200, 2011.
[29] M. Overton and R. Womersley. Optimality conditions and duality theory for minimizing sums
of the largest eigenvalues of symmetric matrices. Mathematical Programming, 62:321?357,
1993.
[30] B. Long, P. Yu, and Z. Zhang. A general model for multiple view unsupervised learning. In
ICDM, 2008.
[31] G. Pataki. On the rank of extreme matrices in semidefinite programs and the multiplicity of
optimal eigenvalues. Mathematics of Operations Research, 23(2):339?358, 1998.
[32] R. Rockafellar. Convex Analysis. Princeton U. Press, 1970.
14
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4,014 | 4,633 | A Stochastic Gradient Method with an Exponential
Convergence Rate for Finite Training Sets
Nicolas Le Roux
SIERRA Project-Team
INRIA - ENS
Paris, France
[email protected]
Mark Schmidt
SIERRA Project-Team
INRIA - ENS
Paris, France
[email protected]
Francis Bach
SIERRA Project-Team
INRIA - ENS
Paris, France
[email protected]
Abstract
We propose a new stochastic gradient method for optimizing the sum of a finite set
of smooth functions, where the sum is strongly convex. While standard stochastic gradient methods converge at sublinear rates for this problem, the proposed
method incorporates a memory of previous gradient values in order to achieve a
linear convergence rate. In a machine learning context, numerical experiments
indicate that the new algorithm can dramatically outperform standard algorithms,
both in terms of optimizing the training error and reducing the test error quickly.
1
Introduction
A plethora of the problems arising in machine learning involve computing an approximate minimizer
of the sum of a loss function over a large number of training examples, where there is a large
amount of redundancy between examples. The most wildly successful class of algorithms for taking
advantage of this type of problem structure are stochastic gradient (SG) methods [1, 2]. Although
the theory behind SG methods allows them to be applied more generally, in the context of machine
learning SG methods are typically used to solve the problem of optimizing a sample average over a
finite training set, i.e.,
n
1X
:=
minimize
g(x)
fi (x).
(1)
x?Rp
n i=1
In this work, we focus on such finite training data problems where each fi is smooth and the average
function g is strongly-convex.
As an example, in the case of `2 -regularized logistic regression we have fi (x) := ?2 kxk2 + log(1 +
exp(?bi aTi x)), where ai ? Rp and bi ? {?1, 1} are the training examples associated with a
binary classification problem and ? is a regularization parameter. More generally, any `2 -regularized
empirical risk minimization problem of the form
n
?
1X
2
minimize
kxk
+
li (x),
(2)
x?Rp
2
n i=1
falls in the framework of (1) provided that the loss functions li are convex and smooth. An extensive
list of convex loss functions used in machine learning is given by [3], and we can even include
non-smooth loss functions (or regularizers) by using smooth approximations.
The standard full gradient (FG) method, which dates back to [4], uses iterations of the form
n
?k X 0 k
xk+1 = xk ? ?k g 0 (xk ) = xk ?
f (x ).
(3)
n i=1 i
Using x? to denote the unique minimizer of g, the FG method with a constant step size achieves a
linear convergence rate:
g(xk ) ? g(x? ) = O(?k ),
1
for some ? < 1 which depends on the condition number of g [5, Theorem 2.1.15]. Linear convergence is also known as geometric or exponential convergence, because the cost is cut by a fixed
fraction on each iteration. Despite the fast convergence rate of the FG method, it can be unappealing
when n is large because its iteration cost scales linearly in n. SG methods, on the other hand, have an
iteration cost which is independent of n, making them suited for that setting. The basic SG method
for optimizing (1) uses iterations of the form
xk+1 = xk ? ?k fi0k (xk ),
(4)
where ?k is a step-size and a training example ik is selected uniformly among the set {1, . . . , n}.
The randomly chosen gradient fi0k (xk ) yields an unbiased estimate of the true gradient g 0 (xk ), and
one can show under standard assumptions that, for a suitably chosen decreasing step-size sequence
{?k }, the SG iterations achieve the sublinear convergence rate
E[g(xk )] ? g(x? ) = O(1/k),
where the expectation is taken with respect to the selection of the ik variables. Under certain assumptions this convergence rate is optimal for strongly-convex optimization in a model of computation
where the algorithm only accesses the function through unbiased measurements of its objective and
gradient (see [6, 7, 8]). Thus, we cannot hope to obtain a better convergence rate if the algorithm
only relies on unbiased gradient measurements. Nevertheless, by using the stronger assumption
that the functions are sampled from a finite dataset, in this paper we show that we can achieve an
exponential converengence rate while preserving the iteration cost of SG methods.
The primay contribution of this work is the analysis of a new algorithm that we call the stochastic
average gradient (SAG) method, a randomized variant of the incremental aggregated gradient (IAG)
method of [9], which combines the low iteration cost of SG methods with a linear convergence rate
as in FG methods. The SAG method uses iterations of the
form
n
?k X k
k+1
k
x
=x ?
y ,
(5)
n i=1 i
where at each iteration a random training example ik is selected and we set
0 k
fi (x ) if i = ik ,
k
yi =
yik?1
otherwise.
That is, like the FG method, the step incorporates a gradient with respect to each training example.
But, like the SG method, each iteration only computes the gradient with respect to a single training
example and the cost of the iterations is independent of n. Despite the low cost of the SAG iterations,
in this paper we show that the SAG iterations have a linear convergence rate, like the FG method.
That is, by having access to ik and by keeping a memory of the most recent gradient value computed
for each training example i, this iteration achieves a faster convergence rate than is possible for
standard SG methods. Further, in terms of effective passes through the data, we also show that for
certain problems the convergence rate of SAG is faster than is possible for standard FG method.
In a machine learning context where g(x) is a training cost associated with a predictor parameterized
by x, we are often ultimately interested in the testing cost, the expected loss on unseen data points.
Note that a linear convergence rate for the training cost does not translate into a similar rate for the
testing cost, and an appealing propertly of SG methods is that they achieve the optimal O(1/k) rate
for the testing cost as long as every datapoint is seen only once. However, as is common in machine
learning, we assume that we are only given a finite training data set and thus that datapoints are
revisited multiple times. In this context, the analysis of SG methods only applies to the training cost
and, although our analysis also focuses on the training cost, in our experiments the SAG method
typically reached the optimal testing cost faster than both FG and SG methods.
The next section reviews closely-related algorithms from the literature, including previous attempts
to combine the appealing aspects of FG and SG methods. However, despite 60 years of extensive
research on SG methods, most of the applications focusing on finite datasets, we are not aware of
any other SG method that achieves a linear convergence rate while preserving the iteration cost of
standard SG methods. Section 3 states the (standard) assumptions underlying our analysis and gives
the main technical results; we first give a slow linear convergence rate that applies for any problem,
and then give a very fast linear convergence rate that applies when n is sufficiently large. Section 4
discusses practical implementation issues, including how to reduce the storage cost from O(np) to
O(n) when each fi only depends on a linear combination of x. Section 5 presents a numerical
comparison of an implementation based on SAG to SG and FG methods, indicating that the method
may be very useful for problems where we can only afford to do a few passes through a data set.
2
2
Related Work
There is a large variety of approaches available to accelerate the convergence of SG methods, and a
full review of this immense literature would be outside the scope of this work. Below, we comment
on the relationships between the new method and several of the most closely-related ideas.
Momentum: SG methods that incorporate a momentum term use iterations of the form
xk+1 = xk ? ?k fi0k (xk ) + ?k (xk ? xk?1 ),
see [10]. It is common to set all ?k = ? for some constant ?, and in this case we can rewrite the SG
with momentum method as
Pk
xk+1 = xk ? j=1 ?j ? k?j fi0j (xj ).
We can re-write the SAG updates (5) in a similar form as
Pk
xk+1 = xk ? j=1 ?k S(j, i1:k )fi0j (xj ),
(6)
where the selection function S(j, i1:k ) is equal to 1/n if j corresponds to the last iteration where
j = ik and is set to 0 otherwise. Thus, momentum uses a geometric weighting of previous gradients
while the SAG iterations select and average the most recent evaluation of each previous gradient.
While momentum can lead to improved practical performance, it still requires the use of a decreasing
sequence of step sizes and is not known to lead to a faster convergence rate.
Gradient Averaging: Closely related to momentum is using the sample average of all previous
gradients,
Pk
xk+1 = xk ? ?kk j=1 fi0j (xj ),
which is similar to the SAG iteration in the form (5) but where all previous gradients are used. This
approach is used in the dual averaging method [11], and while this averaging procedure leads to
convergence for a constant step size and can improve the constants in the convergence rate [12], it
does not improve on the O(1/k) rate.
Iterate Averaging: Rather than averaging the gradients, some authors use the basic SG iteration but
take an average over xk values. With a suitable choice of step-sizes, this gives the same asymptotic
efficiency as Newton-like second-order SG methods and also leads to increased robustness of the
convergence rate to the exact sequence of step sizes [13]. Baher?s method [14, ?1.3.4] combines
gradient averaging with online iterate averaging, and also displays appealing asymptotic properties.
The epoch SG method uses averaging to obtain the O(1/k) rate even for non-smooth objectives [15].
However, the convergence rates of these averaging methods remain sublinear.
Stochastic versions of FG methods: Various options are available to accelerate the convergence of
the FG method for smooth functions, such as the accelerated full gradient (AFG) method of Nesterov [16], as well as classical techniques based on quadratic approximations such as non-linear
conjugate gradient, quasi-Newton, and Hessian-free Newton methods. Several authors have analyzed stochastic variants of these algorithms [17, 18, 19, 20, 12]. Under certain conditions these
variants are convergent with an O(1/k) rate [18]. Alternately, if we split the convergence rate into
a deterministic and stochastic part, these methods can improve the dependency on the deterministic
part [19, 12]. However, as with all other methods we have discussed thus far in this section, we are
not aware of any existing method of this flavor that improves on the O(1/k) rate.
Constant step size: If the SG iterations are used with a constant step size (rather than a decreasing
sequence), then the convergence rate of the method can be split into two parts [21, Proposition 2.4],
where the first part depends on k and converges linearly to 0 and the second part is independent
of k but does not converge to 0. Thus, with a constant step size the SG iterations have a linear
convergence rate up to some tolerance, and in general after this point the iterations do not make
further progress. Indeed, convergence of the basic SG method with a constant step size has only been
shown under extremely strong assumptions about the relationship between the functions fi [22].
This contrasts with the method we present in this work which converges to the optimal solution
using a constant step size and does so with a linear rate (without additional assumptions).
Accelerated methods: Accelerated SG methods, which despite their name are not related to the
aforementioned AFG method, take advantage of the fast convergence rate of SG methods with a
constant step size. In particular, accelerated SG methods use a constant step size by default, and only
decrease the step size on iterations where the inner-product between successive gradient estimates
3
is negative [23, 24]. This leads to convergence of the method and allows it to potentially achieve
periods of linear convergence where the step size stays constant. However, the overall convergence
rate of the method remains sublinear.
Hybrid Methods: Some authors have proposed variants of the SG method for problems of the
form (1) that seek to gradually transform the iterates into the FG method in order to achieve a
linear convergence rate. Bertsekas proposes to go through the data cyclically with a specialized
weighting that allows the method to achieve a linear convergence rate for strongly-convex quadratic
functions [25]. However, the weighting is numerically unstable and the linear convergence rate treats
full passes through the data as iterations. A related strategy is to group the fi functions into ?batches?
of increasing size and perform SG iterations on the batches [26]. In both cases, the iterations that
achieve the linear rate have a cost that is not independent of n, as opposed to SAG.
Incremental Aggregated Gradient: Finally, Blatt et al. presents the most closely-related algorithm,
the IAG method [9]. This method is identical to the SAG iteration (5), but uses a cyclic choice of
ik rather than sampling the ik values. This distinction has several important consequences. In
particular, Blatt et al. are only able to show that the convergence rate is linear for strongly-convex
quadratic functions (without deriving an explicit rate), and their analysis treats full passes through
the data as iterations. Using a non-trivial extension of their analysis and a proof technique involving
bounding the gradients and iterates simultaneously by a Lyapunov potential function, in this work
we give an explicit linear convergence rate for general strongly-convex functions using the SAG
iterations that only examine a single training example. Further, as our analysis and experiments
show, when the number of training examples is sufficiently large, the SAG iterations achieve a linear
convergence rate under a much larger set of step sizes than the IAG method. This leads to more
robustness to the selection of the step size and also, if suitably chosen, leads to a faster convergence
rate and improved practical performance. We also emphasize that in our experiments IAG and
the basic FG method perform similarly, while SAG performs much better, showing that the simple
change (random selection vs. cycling) can dramatically improve optimization performance.
3
Convergence Analysis
In our analysis we assume that each function fi in (1) is differentiable and that each gradient fi0 is
Lipschitz-continuous with constant L, meaning that for all x and y in Rp we have
kfi0 (x) ? fi0 (y)k ? Lkx ? yk.
This is a fairly weak assumption on the fi functions, and in cases where the fi are twicedifferentiable it is equivalent to saying that the eigenvalues of the Hessians
Pn of each fi are bounded
above by L. In addition, we also assume that the average function g = n1 i=1 fi is strongly-convex
with constant ? > 0, meaning that the function x 7? g(x) ? ?2 kxk2 is convex. This is a stronger
assumption and is not satisfied by all machine learning models. However, note that in machine learning we are typically free to choose the regularizer, and we can always add an `2 -regularization term
as in Eq. (2) to transform any convex problem into a strongly-convex problem (in this case we have
? ? ?). Note that strong-convexity implies that the problem is solvable, meaning that there exists
some unique x? that achieves the optimal function value. Our convergence results assume that we
initialize yi0 to a zero vector for all i, and
P our results depend on the variance of the gradient norms
at the optimum x? , denoted by ? 2 = n1 i kfi0 (x? )k2 . Finally, all our convergence results consider
expectations with respect to the internal randomization of the algorithm, and not with respect to the
data (which are assumed to be deterministic and fixed).
1
We first consider the convergence rate of the method when using a constant step size of ?k = 2nL
,
which is similar to the step size needed for convergence of the IAG method in practice.
1
Proposition 1 With a constant step size of ?k = 2nL
, the SAG iterations satisfy for k ? 1:
k h
k
?
9? 2 i
E kx ? x? k2 6 1 ?
3kx0 ? x? k2 +
.
8Ln
4L2
The proof is given in the supplementary material. Note that the SAG iterations also trivially obtain
the O(1/k) rate achieved by SG methods, since
? k
k? 8Ln
1?
6 exp ?
6
= O(n/k),
8Ln
8Ln
k?
albeit with a constant which is proportional to n. Despite this constant, they are advantageous
over SG methods in later iterations because they obtain an exponential convergence rate as in FG
4
methods. We also note that an exponential convergence rate is obtained for any constant step size
1
smaller than 2nL
.
In terms of passes through the data, the rate in Proposition 1 is similar to that achieved by the basic
FG method. However, our next result shows that, if the number of training examples is slightly
larger than L/? (which will often be the case, as discussed in Section 6), then the SAG iterations
can use a larger step size and obtain a better convergence rate that is independent of ? and L (see
proof in the supplementary material).
8L
? ,
1
with a step size of ?k = 2n?
the SAG iterations satisfy for k > n:
1 k
E g(xk ) ? g(x? ) 6 C 1 ?
,
8n
16L 0
?n
4? 2
with C =
8 log 1 +
+1 .
kx ? x? k2 +
3n
3n?
4L
Proposition 2 If n >
We state this result for k > n because we assume that the first n iterations of the algorithm use an SG
method and that we initialize the subsequent SAG iterations with the average of the iterates, which
leads to an O((log n)/k) rate. In contrast, using the SAG iterations from the beginning gives the
same rate but with a constant proportional to n. Note that this bound is obtained when initializing
all yi to zero after the SG phase.1 However, in our experiments we do not use the SG initialization
but rather use a minor variant of SAG (discussed in the next section), which appears more difficult
to analyze but which gives better performance.
It is interesting to compare this convergence rate with the known convergence rates of first-order
methods [5, see ?2]. For example, if we take n = 100000, L = 100, and ? = 0.01 then the basic
2
FG method has
p a rate of ((L ? ?)/(L + ?)) = 0.9996 and the ?optimal? AFG method has a faster
rate of (1 ? ?/L) = 0.9900. In contrast, running n iterations of SAG has a much faster rate of
(1 ? 1/8n)n = 0.8825 using the same
of
of fi0 . Further, the lower-bound for
? number
? evaluations
?
? 2
a black-box first-order method is (( L ? ?)/( L + ?)) = 0.9608, indicating that SAG can
be substantially faster than any FG method that does not use the structure of the problem.2 In the
supplementary material, we compare Propositions 1 and 2 to the rates of primal and dual FG and
coordinate-wise methods for the special case of `2 -regularized leasts squares.
Even though n appears in the convergence rate, if we perform n iterations of SAG (i.e., one effective
pass through the data), the error is multiplied by (1 ? 1/8n)n ? exp(?1/8), which is independent
of n. Thus, each pass through the data reduces the excess cost by a constant multiplicative factor
that is independent of the problem, as long as n > 8L/?. Further, while the step size in Proposition
2 depends on ? and n, we can obtain the same convergence rate by using a step size as large as
?
1
. This is because the proposition is true for all values of ? satisfying L
> n8 , so we can
?k = 16L
8L
choose the smallest possible value of ? = n . We have observed in practice that the IAG method
1
with a step size of ?k = 2n?
may diverge, even under these assumptions. Thus, for certain problems
the SAG iterations can tolerate a much larger step size, which leads to increased robustness to the
selection of the step size. Further, as our analysis and experiments indicate, the ability to use a large
step size leads to improved performance of the SAG iterations.
While we have stated Proposition 1 in terms of the iterates and Proposition 2 in terms of the function
values, the rates obtained on iterates and function values are equivalent because, by the Lipschitz
and strong-convexity assumptions, we have ?2 kxk ? x? k2 6 g(xk ) ? g(x? ) 6 L2 kxk ? x? k2 .
4
Implementation Details
In this section we describe modifications that substantially reduce the SAG iteration?s memory requirements, as well as modifications that lead to better practical performance.
Structured gradients: For many problems the storage cost of O(np) for the yik vectors is prohibitive, but we can often use structure in the fi0 to reduce this cost. For example, many loss functions fi take the form fi (aTi x) for a vector ai . Since ai is constant, for these problems we only
1
While it may appear suboptimal to not use the gradients computed during the n iterations of stochastic
gradient descent, using them only improves the bound by a constant.
2
Note that L in the SAG rates is based on the fi0 functions, while in the FG methods it is based on g 0 which
can be much smaller.
5
need to store the scalar fi0k (uki ) for uki = aTik xk rather than the full gradient aTi fi0 (uki ), reducing the
storage cost to O(n). Further, because of the simple form of the SAG updates, if ai is sparse we can
use ?lazy updates? in order to reduce the iteration cost from O(p) down to the sparsity level of ai .
Mini-batches: To employ vectorization and parallelism, practical SG implementations often group
training examples into ?mini-batches? and perform SG iterations on the mini-batches. We can also
use mini-batches within the SAG iterations, and for problems with dense gradients this decreases
the storage requirements of the algorithm since we only need a yik for each mini-batch. Thus, for
example, using mini-batches of size 100 leads to a 100-fold reduction in the storage cost.
Step-size re-weighting: On early iterations of the SAG algorithm, when most yik are set to the
uninformative zero vector, rather than dividing ?k in (5) by n we found it was more effective to
divide by m, the number of unique ik values that we have sampled so far (which converges to n).
This modification appears more difficult to analyze, but with this modification we found that the
SAG algorithm outperformed the SG/SAG hybrid algorithm analyzed in Proposition 2.
Exact regularization: For regularized objectives like (2) we can use the exact gradient of the regularizer, rather than approximating it. For example, our experiments on `2 -regularized optimization
problems used the recursion
?
d ? d ? yi ,
yi ? li0 (xk ),
d ? d + yi ,
x ? 1 ? ?? x ? d .
(7)
m
This can be implemented efficiently for sparse data sets by using the representation x = ?z, where
? is a scalar and z is a vector, since the update based on the regularizer simply updates ?.
Large step sizes: Proposition 1 requires ?k 6 1/2Ln while under an additional assumption Proposition 2 allows ?k 6 1/16L. In practice we observed better performance using step sizes of
?k = 1/L and ?k = 2/(L + n?). These step sizes seem to work even when the additional assumption of Proposition 2 is not satisfied, and we conjecture that the convergence rates under these
step sizes are much faster than the rate obtained in Proposition 1 for the general case.
Line search: Since L is generally not known, we experimented with a basic line-search, where
we start with an initial estimate L0 , and we double this estimate whenever we do not satisfy the
instantiated Lipschitz inequality
fik (xk ? (1/Lk )fi0k (xk )) 6 fik (xk ) ?
1
kf 0 (xk )k2 .
2Lk ik
To avoid instability caused by comparing very small numbers, we only do this test when
kfi0k (xk )k2 > 10?8 . To allow the algorithm to potentially achieve a faster rate due to a higher
degree of local smoothness, we multiply Lk by 2(?1/n) after each iteration.
5
Experimental Results
Our experiments compared an extensive variety of competitive FG and SG methods. In the supplementary material we compare to the IAG method and an extensive variety of SG methods, and
we allow these competing methods to choose the best step-size in hindsight. However, our experiments in the main paper focus on the following methods, which we chose because they have no
dataset-dependent tuning parameters:
? Steepest: The full gradient method described by iteration (3), with a line-search that uses cubic
Hermite polynomial interpolation to find a step size satisfying the strong Wolfe conditions, and
where the parameters of the line-search were tuned for the problems at hand.
? AFG: Nesterov?s accelerated full gradient method [16], where iterations of (3) with a fixed step
size are interleaved with an extrapolation step, and we use an adaptive line-search based on [27].
? L-BFGS: A publicly-available limited-memory quasi-Newton method that has been tuned for
log-linear models.3 This method is by far the most complicated method we considered.
? Pegasos: The state-of-the-art SG method described by iteration (4) with a step size of ?k =
1/?k and a projection step onto a norm-ball known to contain the optimal solution [28].
? RDA: The regularized dual averaging method of [12], another recent state-of-the-art SG method.
? ESG: The epoch SG method of [15], which runs SG with a constant step size and averaging in
a series of epochs, and is optimal for non-smooth stochastic strongly-convex optimization.
3
http://www.di.ens.fr/?mschmidt/Software/minFunc.html
6
0
10
?2
10
?3
10
Steepest
AFG
L?BFGS
pegasos
RDA
ESG
NOSG
SAG?C
SAG?LS
?2
10
?4
10
Objective minus Optimum
?1
10
10
Steepest
AFG
L?BFGS
pegasos
RDA
ESG
NOSG
SAG?C
SAG?LS
Objective minus Optimum
Objective minus Optimum
0
0
10
?6
10
?8
10
?2
10
?3
10
?4
10
?5
?10
10
10
?4
Steepest
AFG
L?BFGS
pegasos
RDA
ESG
NOSG
SAG?C
SAG?LS
?1
10
10
0
5
10
15
20
25
0
5
Effective Passes
10
15
20
0
25
10
15
20
25
Effective Passes
5
x 10
7000
Steepest
AFG
L?BFGS
pegasos
RDA
ESG
NOSG
SAG?C
SAG?LS
3.5
3
6000
Test Logistic Loss
4
2.5
2
1.5
2
Steepest
AFG
L?BFGS
pegasos
RDA
ESG
NOSG
SAG?C
SAG?LS
6500
5500
5000
Steepest
AFG
L?BFGS
pegasos
RDA
ESG
NOSG
SAG?C
SAG?LS
1.95
1.9
Test Logistic Loss
5
4.5
Test Logistic Loss
5
Effective Passes
4
x 10
4500
4000
3500
1.85
1.8
1.75
1.7
1.65
3000
1
2500
0.5
2000
1.6
1.55
1.5
0
5
10
15
Effective Passes
20
25
0
5
10
15
20
25
0
5
Effective Passes
10
15
20
25
Effective Passes
Figure 1: Comparison of optimization strategies for `2 -regularized logistic regression. Top: training
excess cost. Bottom: testing cost. From left to right are the results on the protein, rcv1 and covertype
data sets. This figure is best viewed in colour.
? NOSG: The nearly-optimal SG method of [19], which combines ideas from SG and AFG methods to obtain a nearly-optimal dependency on a variety of problem-dependent constants.
? SAG: The proposed stochastic average gradient method described by iteration (5) using the
modifications discussed in the previous section. We used a step-size of ?k = 2/(Lk + n?)
where Lk is either set constant to the global Lipschitz constant (SAG-C) or set by adaptively
estimating the constant with respect to the logistic loss function using the line-search described
in the previous section (SAG-LS). The SAG-LS method was initialized with L0 = 1 .
In all the experiments, we measure the training and testing costs as a function of the number of
effective passes through the data, measured as the number of fi0 evaluations divided by n. These
results are thus independent of the practical implementation of the algorithms.
The theoretical convergence rates suggest the following strategies for deciding on whether to use an
FG or an SG method:
1. If we can only afford one pass through the data, then an SG method should be used.
2. If we can afford to do many passes through the data (say, several hundred), then an FG
method should be used.
We expect that the SAG iterations will be most useful between these two extremes, where we can
afford to do more than one pass through the data but cannot afford to do enough passes to warrant
using FG algorithms like L-BFGS. To test whether this is indeed the case on real data sets, we
performed experiments on a set of freely available benchmark binary classification data sets. The
protein (n = 145751, p = 74) data set was obtained from the KDD Cup 2004 website,4 while the
rcv1 (n = 20242, p = 47236) and covertype (n = 581012, p = 54) data sets were obtained from
the LIBSVM data website.5 Although our method can be applied to any differentiable function, on
these data sets we focus on an `2 -regularized logistic regression problem, with ? = 1/n. We split
each dataset in two, training on one half and testing on the other half. We added a (regularized)
bias term to all data sets, and for dense features we standardized so that they would have a mean of
zero and a variance of one. We plot the training and testing costs of the different methods for 30
effective passes through the data in Figure 1. In the supplementary material, we present additional
experimental results including the test classification accuracy and results on different data sets.
We can observe several trends across the experiments from both the main paper and the supplementary material.
4
5
http://osmot.cs.cornell.edu/kddcup
http://www.csie.ntu.edu.tw/?cjlin/libsvmtools/datasets
7
? FG vs. SG: Although the performance of SG methods can be catastrophic if the step size is not
chosen carefully (e.g., the covertype data), with a carefully-chosen step-size the SG methods
do substantially better than FG methods on the first few passes through the data (e.g., the rcv1
data). In contrast, FG methods are not sensitive to the step size and because of their steady
progress we also see that FG methods slowly catch up to the SG methods and eventually (or
will eventually) pass them (e.g., the protein data).
? (FG and SG) vs. SAG: The SAG iterations seem to achieve the best of both worlds. They
start out substantially better than FG methods, but continue to make steady (linear) progress
which leads to better performance than SG methods. In some cases (protein and covertype), the
significant speed-up observed for SAG in reaching low training costs also translates to reaching
the optimal testing cost more quickly than the other methods.
? IAG vs. SAG: Our experiments (in the supplementary material) show that the IAG method
performs similar to the regular FG method, and they also show the surprising result that the
randomized SAG method outperforms the closely-related deterministic IAG method by a very
large margin. This is due to the larger step sizes used by the SAG iterations, which would cause
IAG to diverge.
6
Discussion
Optimal regularization strength: One might wonder if the additional hypothesis in Proposition 2
is satisfied in practice. In a learning context, where each function fi is the loss associated to a
single data point, L is equal to the largest value of the loss second derivative ? (1 for the square
loss, 1/4 for the logistic loss) times R2 , where R is a the uniform bound on the norm of each data
2
?
point. Thus, the constraint L
> n8 is satisfied when ? > 8?R
n . In low-dimensional settings, the
optimal regularization parameter is of the form C/n [29] where C is a scalar constant, and may thus
violate the constraint.
However, the improvement with respect to regularization parameters of the
?
form ? = C/ n is known to be asymptotically negligible, and in any case in such low-dimensional
settings, regular stochastic or batch gradient descent may be efficient enough in practice. In the
more interesting high-dimensional settings where the dimension p of our covariates is not small
compared to the sample size n, then all theoretical analyses we are aware of advocate settings of ?
which satisfy this constraint. For example, [30] considers parameters of the form ? = ?Cn in the
parametric setting, while [31] considers ? = nC? with ? < 1 in a non-parametric setting.
Training cost vs. testing cost: The theoretical contribution of this work is limited to the convergence
rate of the training cost. Though there are several settings where this is the metric of interest (e.g.,
variational inference in graphical models), in many cases one will be interested in the convergence
speed of the testing cost. Since the O(1/k) convergence rate of the testing cost, achieved by SG
methods with decreasing step sizes (and a single pass through the data), is provably optimal when
the algorithm only accesses the function through unbiased measurements of the objective and its
gradient, it is unlikely that one can obtain a linear convergence rate for the testing cost with the SAG
iterations. However, as shown in our experiments, the testing cost of the SAG iterates often reaches
its minimum quicker than existing SG methods, and we could expect to improve the constant in the
O(1/k) convergence rate, as is the case with online second-order methods [32].
Step-size selection and termination criteria: The three major disadvantages of SG methods are: (i)
the slow convergence rate, (ii) deciding when to terminate the algorithm, and (iii) choosing the step
size while running the algorithm. This paper showed that the SAG iterations achieve a much faster
convergence rate, but the SAG iterations may also be advantageous in terms of tuning step sizes
and designing termination criteria. In particular, the SAG iterations suggest a natural termination
k
0 k
k
criterion; since the average
?xk?1 k converges to zero,
P k of the yi variables converges to g (x ) as kx
k
we can use (1/n)k i yi k as an approximation of the optimality of x . Further, while SG methods
require specifying a sequence of step sizes and mispecifying this sequence can have a disastrous
effect on the convergence rate [7, ?2.1], our theory shows that the SAG iterations iterations achieve
a linear convergence rate for any sufficiently small constant step size and our experiments indicate
that a simple line-search gives strong performance.
Acknowledgements
Nicolas Le Roux, Mark Schmidt, and Francis Bach are supported by the European Research Council
(SIERRA-ERC-239993). Mark Schmidt is also supported by a postdoctoral fellowship from the
Natural Sciences and Engineering Research Council of Canada (NSERC).
8
References
[1] H. Robbins and S. Monro. A stochastic approximation method. Annals of Mathematical Statistics,
22(3):400?407, 1951.
[2] L. Bottou and Y. LeCun. Large scale online learning. NIPS, 2003.
[3] C. H. Teo, Q. Le, A. J. Smola, and S. V. N. Vishwanathan. A scalable modular convex solver for regularized risk minimization. KDD, 2007.
[4] M. A. Cauchy. M?ethode g?en?erale pour la r?esolution des syst`emes d??equations simultan?ees. Comptes
rendus des s?eances de l?Acad?emie des sciences de Paris, 25:536?538, 1847.
[5] Y. Nesterov. Introductory lectures on convex optimization: A basic course. Springer, 2004.
[6] A. Nemirovski and D. B. Yudin. Problem complexity and method efficiency in optimization. Wiley, 1983.
[7] A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro. Robust stochastic approximation approach to
stochastic programming. SIAM Journal on Optimization, 19(4):1574?1609, 2009.
[8] A. Agarwal, P. L. Bartlett, P. Ravikumar, and M. J. Wainwright. Information-theoretic lower bounds
on the oracle complexity of stochastic convex optimization. IEEE Transactions on Information Theory,
58(5), 2012.
[9] D. Blatt, A. O. Hero, and H. Gauchman. A convergent incremental gradient method with a constant step
size. SIAM Journal on Optimization, 18(1):29?51, 2007.
[10] P. Tseng. An incremental gradient(-projection) method with momentum term and adaptive stepsize rule.
SIAM Journal on Optimization, 8(2):506?531, 1998.
[11] Y. Nesterov. Primal-dual subgradient methods for convex problems. Mathematical programming,
120(1):221?259, 2009.
[12] L. Xiao. Dual averaging methods for regularized stochastic learning and online optimization. Journal of
Machine Learning Research, 11:2543?2596, 2010.
[13] B. T. Polyak and A. B. Juditsky. Acceleration of stochastic approximation by averaging. SIAM Journal
on Control and Optimization, 30(4):838?855, 1992.
[14] H. J. Kushner and G. Yin. Stochastic approximation and recursive algorithms and applications. SpringerVerlag, Second edition, 2003.
[15] E. Hazan and S. Kale. Beyond the regret minimization barrier: an optimal algorithm for stochastic
strongly-convex optimization. COLT, 2011.
[16] Y. Nesterov. A method for unconstrained convex minimization problem with the rate of convergence
O(1/k2 ). Doklady AN SSSR, 269(3):543?547, 1983.
[17] N.N. Schraudolph. Local gain adaptation in stochastic gradient descent. ICANN, 1999.
[18] P. Sunehag, J. Trumpf, SVN Vishwanathan, and N. Schraudolph. Variable metric stochastic approximation
theory. International Conference on Artificial Intelligence and Statistics, 2009.
[19] S. Ghadimi and G. Lan. Optimal stochastic? approximation algorithms for strongly convex stochastic
composite optimization. Optimization Online, July, 2010.
[20] J. Martens. Deep learning via Hessian-free optimization. ICML, 2010.
[21] A. Nedic and D. Bertsekas. Convergence rate of incremental subgradient algorithms. In Stochastic
Optimization: Algorithms and Applications, pages 263?304. Kluwer Academic, 2000.
[22] M.V. Solodov. Incremental gradient algorithms with stepsizes bounded away from zero. Computational
Optimization and Applications, 11(1):23?35, 1998.
[23] H. Kesten. Accelerated stochastic approximation. Annals of Mathematical Statistics, 29(1):41?59, 1958.
[24] B. Delyon and A. Juditsky. Accelerated stochastic approximation. SIAM Journal on Optimization,
3(4):868?881, 1993.
[25] D. P. Bertsekas. A new class of incremental gradient methods for least squares problems. SIAM Journal
on Optimization, 7(4):913?926, 1997.
[26] M. P. Friedlander and M. Schmidt. Hybrid deterministic-stochastic methods for data fitting. SIAM Journal
of Scientific Computing, 34(3):A1351?A1379, 2012.
[27] J. Liu, J. Chen, and J. Ye. Large-scale sparse logistic regression. KDD, 2009.
[28] S. Shalev-Shwartz, Y. Singer, and N. Srebro. Pegasos: Primal estimated sub-gradient solver for svm.
ICML, 2007.
[29] P. Liang, F. Bach, and M. I. Jordan. Asymptotically optimal regularization in smooth parametric models.
NIPS, 2009.
[30] K. Sridharan, S. Shalev-Shwartz, and N. Srebro. Fast rates for regularized objectives. NIPS, 2008.
[31] M. Eberts and I. Steinwart. Optimal learning rates for least squares SVMs using Gaussian kernels. NIPS,
2011.
[32] L. Bottou and O. Bousquet. The tradeoffs of large scale learning. NIPS, 2007.
9
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4,015 | 4,634 | Latent Coincidence Analysis: A Hidden
Variable Model for Distance Metric Learning
Matthew Der and Lawrence K. Saul
Department of Computer Science and Engineering
University of California, San Diego
La Jolla, CA 92093
{mfder,saul}@cs.ucsd.edu
Abstract
We describe a latent variable model for supervised dimensionality reduction
and distance metric learning. The model discovers linear projections of high
dimensional data that shrink the distance between similarly labeled inputs
and expand the distance between differently labeled ones. The model?s
continuous latent variables locate pairs of examples in a latent space of
lower dimensionality. The model differs significantly from classical factor
analysis in that the posterior distribution over these latent variables is not
always multivariate Gaussian. Nevertheless we show that inference is completely tractable and derive an Expectation-Maximization (EM) algorithm
for parameter estimation. We also compare the model to other approaches
in distance metric learning. The model?s main advantage is its simplicity:
at each iteration of the EM algorithm, the distance metric is re-estimated
by solving an unconstrained least-squares problem. Experiments show that
these simple updates are highly effective.
1
Introduction
In this paper we propose a simple but new model to learn informative linear projections
of multivariate data. Our approach is rooted in the tradition of latent variable modeling,
a popular methodology for discovering low dimensional structure in high dimensional data.
Two well-known examples of latent variable models are factor analyzers (FAs), which recover
subspaces of high variance [1], and Gaussian mixture models (GMMs), which reveal clusters
of high density [2]. Here we describe a model that we call latent coincidence analysis (LCA).
The goal of LCA is to discover a latent space in which metric distances reflect meaningful
notions of similarity and difference.
We apply LCA to two problems in distance metric learning, where the goal is to improve
the performance of a classifier?typically, a k-nearest neighbor (kNN) classifier [3]?by a
linear transformation of its input space. Several previous methods have been proposed for
this problem, including neighborhood component analysis (NCA) [4], large margin neighbor
neighbor classification (LMNN) [5], and information-theoretic metric learning (ITML) [6].
These methods?all of them successful, all of them addressing the same problem?beg the
obvious question: why yet another?
One answer is suggested by the different lineages of previous approaches. NCA was conceived
as a supervised counterpart to stochastic neighborhood embedding [7], an unsupervised
method for dimensionality reduction. LMNN was conceived as a kNN variant of support
vector machines [8]. ITML evolved from earlier work in Bregman optimizations?that of
minimizing the LogDet divergence subject to linear constraints [9]. Perhaps it is due to
1
x
x
z
x
W,
x'
z
z
W,
y
y
x'
z'
z'
x'
z'
y
N
Figure 1: Bayesian network for latent coincidence analysis. The inputs x, x0 ? <d are
mapped into Gaussian latent variables z, z0 ? <p whose statistics are parameterized by the
linear transformation W ? <p?d and noise level ?. Coincidence in the latent space at length
scale ? is detected by the binary variable y ? {0, 1}. Observed nodes are shaded.
these different lineages that none of these methods completely dominates the others. They
all offer improvements in kNN classification, yet arguably their larger worth stems from
the related work they have inspired in other areas of pattern recognition. Distance metric
learning is a fundamental problem, and the more solutions we have, the better equipped we
are to solve its myriad variations.
It is in this spirit that we revisit the problem of distance metric learning in the venerable
tradition of latent variable modeling. We believe that LCA, like factor analysis and Gaussian
mixture modeling, is the simplest latent variable model that can be imagined for its purpose.
In particular, the inference in LCA (though not purely Gaussian) is tractable, and the
distance metric is re-estimated at each iteration of its EM algorithm by a simple leastsquares update. This update has stronger guarantees of convergence than the gradientbased methods in NCA; it also sidesteps the large number of linear inequality constraints
that appear in the optimizations for LMNN and ITML. For all these reasons, we believe
that LCA deserves to be widely known.
2
Model
We begin by describing the probabilistic model for LCA. Fig. 1 shows the model?s representation as a Bayesian network. There are three observed variables: the inputs x, x0 ? <d ,
which we always imagine to be observed in pairs, and the binary label y ? {0, 1}, which
indicates if the inputs map (or are desired to be mapped) to nearby locations in a latent
space of equal or reduced dimensionality p ? d. These locations are in turn represented by
the Gaussian latent variables z, z0 ? <p .
Each node in the Bayesian network is conditionally dependent on its parents. The conditional distributions P (z|x) and P (z0 |x0 ) are parameterized by a linear transformation
W ? <p?d (from the input space to the latent space) and a noise level ? 2 . They take the
simple Gaussian form:
2
1
P (z|x) = (2?? 2 )?p/2 e? 2?2 kz?Wxk ,
0
0
2 ?p/2 ? 2?12 kz0 ?Wx0 k2
P (z |x ) = (2?? )
e
(1)
.
(2)
Finally, the binary label y ? {0, 1} is used to detect the coincidence of the variables z, z0 in
the latent space. In particular, y follows a Bernoulli distribution with mean value:
1
0 2
P (y = 1|z, z0 ) = e? 2?2 kz?z k .
(3)
Eq. (3) states that y = 1 with certainty if z and z0 coincide at the exact same point in the
latent space; otherwise, the probability in eq. (3) falls off exponentially with their squared
distance. The length scale ? in eq. (3) governs the rate of this exponential decay.
2
2.1
Inference
Inference in this model requires averaging over the Gaussian latent variables z, z0 . The
required integrals take the form of simple Gaussian convolutions. For example:
Z
P (y = 1|x, x0 ) = dz dz0 P (y = 1|z, z0 ) P (z|x) P (z0 |x0 )
(4)
=
?2
?2 +2? 2
p/2
kW(x?x0 )k2
exp ?
.
2 (?2 +2? 2 )
(5)
Note that this marginal probability is invariant to uniform re-scalings of the model parameters W, ?, and ?; we will return to this observation later. For inputs (x, x0 ), we denote
the relative likelihood, or odds, of the event y = 1 by
?(x, x0 ) =
P (y = 1|x, x0 )
.
P (y = 0|x, x0 )
(6)
As we shall see, the odds appear in the calculations for many useful forms of inference. Note
that the odds ?(x, x0 ) has a complicated nonlinear dependence on the inputs (x, x0 ); the
numerator in eq. (6) is Gaussian, but the denominator (equal to one minus the numerator)
is not.
Of special importance for learning (as discussed in section 2.2) are the statistics of the
posterior distribution P (z, z0 |x, x0 , y). We obtain this distribution using Bayes rule:
P (z, z0 |x, x0 , y) =
P (y|z, z0 ) P (z|x) P (z0 |x0 )
.
P (y|x, x0 )
(7)
We note that the prior distribution P (z, z0 |x, x0 ) is multivariate Gaussian, as is the posterior
distribution P (z, z0 |x, x0 , y = 1) for positively labeled pairs of examples. However, this is
not true of the posterior distribution P (z, z0 |x, x0 , y = 0) for negatively labeled pairs. In
this respect, the model differs from classical factor analysis and other canonical models with
Gaussian latent variables (e.g., Kalman filters).
Despite the above wrinkle, it remains straightforward to compute the low-order moments
of the distribution in eq. (7) for both positively (y = 1) and negatively (y = 0) labeled pairs1
of examples. In particular, for the posterior means, we obtain:
?? 2
0
(x
?x)
,
(8)
E[z|x, x0 , y = 0] = W x ?
?2 + 2? 2
?2
0
E[z|x, x0 , y = 1] = W x +
(x
?x)
,
(9)
?2 + 2? 2
where the coefficient ? in eq. (8) is shorthand for the odds ?(x, x0 ) in eq. (6). Note how the
posterior means E[z|x, x0 , y] in eqs. (8?9) differ from the prior mean
E[z|x, x0 ] = Wx.
(10)
Analogous results hold for the prior and posterior means of the latent variable z0 . Intuitively,
these calculations show that the expected values of z and z0 move toward each other if the
observed label indicates a coincidence (y = 1) and away from each other if not (y = 0).
For learning it is also necessary to compute second-order statistics of the posterior distribution. For the posterior variances, straightforward calculations give:
?? 2
2
0
2
?k x, x , y = 0 = p? 1 + 2
,
(11)
E kz ? z
? + 2? 2
?2
?k2 x, x0 , y = 1 = p? 2 1 ? 2
E kz ? z
,
(12)
? + 2? 2
1
For the latter, the statistics can be expressed as the differences of Gaussian integrals.
3
? in these expressions denotes the posterior means in eqs. (8?9), and again the coefwhere z
ficient ? is shorthand for the odds ?(x, x0 ) in eq. (6). Note how the posterior variances in
eqs. (11?12) differ from the prior variance
2
(13)
E kz ? Wxk x, x0 = p? 2 .
Intuitively, we see that the posterior variance shrinks if the observed label indicates a coincidence (y = 1) and grows if not (y = 0). The expressions for the posterior variance of the
latent variable z0 are identical due to the model?s symmetry.
2.2
Learning
Next we consider how to learn the linear projection W, the noise level ? 2 , and the length
scale ?2 from data. We assume that the data comes in the form of paired inputs x, x0 ,
together with binary judgments y ? {0, 1} of similarity or difference. In particular, from
a training set {(xi , x0i , yi )}N
i=1 of N such examples, we wish to learn the parameters that
maximize the conditional log-likelihood
L(W, ? 2 , ?2 ) =
N
X
log P (yi |xi , x0i )
(14)
i=1
of observed coincidences (yi = 1) and non-coincidences (yi = 0). We say that the data is
incomplete or partially observed in the sense that the examples do not specify target values
for the latent variables z, z0 ; instead, such target values must be inferred from the model?s
posterior distribution.
Given values for the model parameters W, ? 2 , and ?2 , we can compute the right hand side of
eq. (14) from the result in eq. (5). However, the parameters that maximize eq. (14) cannot
be computed in closed form. In the absence of an analytical solution, we avail ourselves
of the EM algorithm, an iterative procedure for maximum likelihood estimation in latent
variable models [10]. The EM algorithm consists of two steps, an E-step which computes
statistics of the posterior distribution in eq. (7), and an M-step which uses these statistics
to re-estimate the model parameters. The two steps are iterated until convergence.
Intuitively, the EM algorithm uses the posterior means in eqs. (8?9) to ?fill in? the missing
values of the latent variables z, z0 . As shorthand, let
?i
z
?0i
z
= E[z|xi , x0i , yi ],
= E[z0 |xi , x0i , yi ]
(15)
(16)
denote these posterior means for the ith example in the training set, as computed from the
results in eq. (8?9). The M-step of the EM algorithm updates the linear transformation W
by minimizing the sum of squared errors
N
E(W) =
i
1 Xh
2
2
k?
zi ? Wxi k + k?
z0i ? Wx0i k ,
2 i=1
(17)
?i , z
?0i are computed with respect to the current model parameters
where the expected values z
(and thus treated as constants in the above minimization). Minimizing the sum of squared
errors in eq. (17) gives the update rule:
"N
#"X
#?1
N
X
>
>
0 0
? i x>
?0i x0i
W ?
z
xi x>
,
(18)
i +z
i + xi xi
i=1
i=1
where the product in eq. (18) is understood as a vector-matrix multiplication. The EM
update for the noise level ? 2 takes an equally simple form. As shorthand, let
2
2
0
?k xi , xi , yi
?i = E kz ? z
(19)
4
# train
# test
# classes
# features (D)
# inputs (d)
LCA dim (p)
Euclidean
PCA
LMNN
LCA
MNIST
60000
10000
10
784
164
40
2.83
2.12
1.34
1.61
BBC
1558
667
5
9635
200
4
30.10
11.90
3.40
3.41
Classic4
4257
1419
4
5896
200
32
7.89
9.74
3.19
3.54
Isolet
6238
1559
26
617
172
40
8.98
8.60
3.40
3.72
Letters
14000
6000
26
16
16
16
4.73
4.75
2.58
2.93
Seg
210
2100
7
19
19
18
13.71
13.71
8.57
8.57
Bal
438
187
3
4
4
4
18.82
18.07
8.98
4.06
Iris
105
45
3
4
4
4
5.33
5.33
5.33
2.22
Table 1: Summary of data sets for multiway classification. For efficiency we projected data
sets of high dimensionality D down to their leading d principal components. MNIST [11] is
a data set of handwritten digits; we deslanted the images to reduce variability. BBC [12]
and Classic4 [13] are text corpora with labeled topics. The last five data sets are from
the UCI repository [14]. The bottom four rows compare test error percentage using 3-NN
classification. For data sets without dedicated test sets, we averaged across multiple random
70/30 splits.
denote the posterior variance of z for the ith example in the training set, as computed from
the results in eqs. (11?12). Then the EM update for the noise level is given by:
"
#
N
X
1
?2 ?
min E(W) +
?2i .
(20)
pN W
i=1
The minimum of E(W) in this update is computed by substituting the right hand side of
eq. (18) into eq. (17).
The EM updates for W and ? 2 have the desirable property that they converge monotonically
to a stationary point of the log-likelihood: that is, at each iteration, they are guaranteed
to increase the right hand side of eq. (14) except at points in the parameter space with
vanishing gradient. A full derivation of the EM algorithm is omitted for brevity.
We have already noted that the log-likelihood in eq. (14) is invariant to uniform rescaling
of the model parameters W, ?, and ?. Thus without loss of generality we can set ?2 = 1 in
the simplest setting of the model, as described above. It does become necessary to estimate
the parameter ?2 , however, in slightly extended formulations of the model, as we consider
in section 3.2. Unlike the parameters W and ? 2 , the parameter ?2 does not have a simple
update rule for its re-estimation by EM. When necessary, however, this parameter can be
re-estimated by a simple line search. This approach also preserves the property of monotonic
convergence.
3
Applications
We explore two applications of LCA in which its linear transformation is used to preprocess
the data for different models of multiway classification. We assume that the original data
consists of labeled examples {(xi , ci )} of inputs xi ? <d and their class labels ci ? {1, 2, . . . , c}.
For each application, we show how to instantiate LCA by creating a particular data set of
labeled pairs, where the labels indicate whether the examples in each pair should be mapped
closer together (y = 1) or farther apart (y = 0) in LCA?s latent space of dimensionality p ? d.
In the first application, we use LCA to improve a parametric model of classification; in the
second application, a nonparametric one. The data sets in our experiments are summarized
in Table 1.
3.1
Gaussian mixture modeling
Gaussian mixture models (GMMs) offer perhaps the simplest parametric model of multiway
classification. In the most straightforward application of GMMs, the labeled examples in
5
each class c are modeled by a single multivariate Gaussian distribution with mean ?c and
covariance matrix ?c . Classification is also simple: for each unlabeled example, we use
Bayes rule to compute the class with the highest posterior probability.
Even in these simplest of GMMs, however, challenges arise when the data is very high
dimensional. In this case, it may be prohibitively expensive to estimate or store the covariance matrix for each class of the data. In this case two simple options are: (i) to reduce
the input?s dimensionality using principal component analysis (PCA) or linear discriminant
analysis (LDA) [15], or (ii) to model each multivariate Gaussian distribution using factor
analysis. In the latter, we learn distributions of the form:
P (x|c) ? N (?c , ?c + ?c ?>
c )
d?d
(21)
d?p
where the diagonal matrix ?c ? <
and loading matrix ?c ? <
are the model parameters of the factor analyzer belonging to the cth class. Factor analysis can be formulated as
a latent variable model, and its parameters estimated by an EM algorithm [1].
GMMs are generative models trained by maximum likelihood estimation. In this section,
we explore how LCA may yield classifiers of similar form but higher accuracy. To do so, we
learn one model of LCA for each class of the data. In particular, we use LCA to project
each example xi into a lower dimensional space where we hope for two properties: (i) that
it is closer to the mean of projected examples from the same class yi , and (ii) that it is
farther from the mean of projected examples from other classes c 6= yi .
More specifically, we instantiate the model of LCA for each class as follows. Let ?c ? <d
denote the mean of the labeled examples in class c. Then we create a training set of labeled
pairs {?c , xi , yic } over all examples xi where yic = 1 if yi = c and yic = 0 if yi 6= c. From
this training set, we use the EM algorithm in section 2.2 to learn a (class-specific) linear
projection Wc and variance ?c . Finally, to classify an unlabeled example x, we compute
the probabilities:
p/2
1
kWc (x??c )k2
P (yc = 1|x) =
exp ?
.
(22)
1+2?c2
2 (1+2?c2 )
We label the example x by the class c that maximizes the probability in eq. (22). As
we shall see, this decision rule for LCA often makes different predictions than Bayes rule
in maximum likelihood GMMs. Conveniently, we can train the LCA models for different
classes in parallel.
We evaluated the performance of LCA in this setting on the first four data sets in Table 1.
Over a range of reduced dimensionalities p < d, we compared the classification accuracy of
three approaches: (i) GMMs with full covariance matrices after projecting the data down
to p dimensions with PCA or LDA, (ii) GMMs with p-dimensional factor analyzers, and
(iii) p-dimensional models of LCA. Fig. 2 shows that LCA generally outperforms these
other methods; also, its largest gains occur in the regime of very aggressive dimensionality
reduction p d. To highlight the results in this regime, Fig. 3 contrasts the p = 2 dimensional
representations of the data discovered by PCA and LCA. Here it is visually apparent that
LCA leads to much better separation of the examples in different classes.
3.2
Distance metric learning
We can also apply LCA to learn a distance metric that improves kNN classification [4, 5, 6].
Our approach draws heavily on the ideas of LMNN [5], though differs in its execution. In
LMNN, each training example has k target neighbors, typically chosen as the k nearest
neighbors in Euclidean space with the same class label. LMNN learns a metric to shrink
the distances between examples and target neighbors while preserving (or increasing) the
distances between examples from different classes. Errors in kNN classification tend to
occur when differently labeled examples are closer together than pairs of target neighbors.
Thus LMNN seeks to minimize the number of differently labeled examples that invade the
perimeters established by target neighbors. These examples are known as impostors.
In LCA, we can view the matrix W> W as a Mahalanobis distance metric for kNN classification. The starting point of LCA is to create a training set of pairs of examples. Among
6
40
30
20
50
40
30
20
Isolet
80
30
PCA
FA
LDA
LCA
PCA
FA
LDA
LCA
25
% test error
% test error
50
Classic4
BBC
60
% test error
PCA
FA
LDA
LCA
20
15
10
PCA
FA
LDA
LCA
60
% test error
MNIST
60
40
20
10
5
10
0
2
4
8
16
p
32
64
128
0
2
4
8
16
p
32
64
128
0
2
4
8
16
p
32
64
128
0
2
4
8
16
p
32
64
128
Figure 2: Comparison of dimensionality reduction by principal components analysis (PCA),
factor analysis (FA), linear discriminant analysis (LDA), and latent coincidence analysis
(LCA). The plots show test set error versus dimensionality p.
BBC
Classic4
Isolet
LCA
PCA
MNIST
Figure 3: Comparison of two-dimensional (p = 2) representations of data discovered by PCA
and LCA. The examples are color-coded by class label.
these pairs, we wish the similarly labeled examples to coincide (y = 1) and the differently
labeled examples to diverge (y = 0). For the former, it is natural to choose all pairs of
examples and their target neighbors. For the latter, it is natural to choose all pairs of
differently labeled examples. Concretely, if there are c classes, each with m examples, then
this approach creates a training set of ckm pairs of similarly labeled examples (with y = 1)
and c(c ? 1)m2 pairs of differently labeled examples (with y = 0).
Unfortunately it is clear that this approach does not scale well with the number of examples.
We therefore adopt two pruning strategies in our implementation of LCA. First, we do
not include training examples without impostors. Second, among the pairs of differently
labeled examples, we only include each example with its current or previous impostors.
A complication of this approach is that every few iterations we must check to see if any
example has new impostors. If so, we add the example, its target neighbors, and impostors
into the training set. This strategy was used in all the experiments described below. Our
short-cuts are similar in spirit to the optimizations in LMNN [5] as well as more general
cutting plane strategies of constrained optimization [16].
The use of LCA for kNN classification also benefits from a slight but crucial extension of
the model in Fig. 1. Recall that the parameter ?2 determines the length scale at which
projected examples are judged to coincide in the latent space. For kNN classification, we
extend the model in Fig. 1 to learn a local parameter ?2 for each input in the training set.
These local parameters ?2 are needed to account for the fact that different inputs may reside
at very different distances from their target neighbors. In the graphical model of Fig. 1,
this extension amounts to drawing an additional plate that encloses the parameter ?2 and
the model?s random variables, but not the parameters W and ? 2 .
Note that the ? 2 and ?2 parameters of LCA, though important to estimate, are not ultimately used for kNN classification. In particular, after a model is trained, we simply
7
MNIST
BBC
10
8
Isolet
Classic4
30
PCA
LCA
25
60
8
PCA
LCA
PCA
LCA
20
% error
PCA
LCA
50
6
40
6
15
30
4
4
10
20
2
2
5
0
4
0
4
8
16
p
32
64
10
8
16
32
64
0
128 4
8
16
p
32
64
128
0
4
8
16
32
64
128
p
p
Figure 4: Comparison of dimensionality reduction by principal components analysis (PCA)
and latent coincidence analysis (LCA). The plots show kNN classification error (training
dotted, test solid) versus dimensionality p.
PCA
LMNN
LCA
20
15
% test error
% train error
15
10
5
0
PCA
LMNN
LCA
20
10
5
MNIST
BBC Classic4 Isolet Letters
Seg
Bal
0
Iris
MNIST
BBC Classic4 Isolet Letters
Seg
Bal
Iris
Figure 5: Comparison of kNN classification by PCA, LMNN, and LCA. We set k = 3 for all
experiments. Training error is computed using leave-one-out cross-validation. The values
of p used for LCA are given in Table 1.
perform kNN classification using the Mahalanobis distance metric parameterized by the
linear transformation W.
For comparison, we measure kNN classification accuracy using Euclidean distance, PCA,
and LMNN. We report all three along with LCA in Table 1, but we focus on PCA in Fig. 4 to
illustrate the effect of dimensionality reduction. LCA consistently outperforms PCA across
all dimensionalities. Additionally, we hold out a validation set to search for an optimal
dimensionality p. In Fig. 5, we compare LCA to PCA and LMNN. Again, LCA is clearly
superior to PCA and generally achieves comparable performance to LMNN. Advantageously,
we often obtain our best result with LCA using a lower dimensionality p < d.
4
Discussion
In this paper we have introduced Latent Coincidence Analysis (LCA), a latent variable model
for learning linear projections that map similar inputs closer together and different inputs
farther apart. Inference in LCA is entirely tractable, and we use an EM algorithm to learn
maximum likelihood estimates of the model parameters. Our approach values simplicity,
but not at the expense of efficacy. On problems in mixture modeling and distance metric
learning tasks, LCA performs competitively across a range of reduced dimensionalities.
There are many directions for future work. One challenge that we observed was slow convergence of the EM algorithm, an issue that may be ameliorated by the gradient or second-order
methods proposed in [17]. To handle larger data sets, we plan to explore online strategies
for distance metric learning [18], possibly based on Bayesian [19] or confidence-weighted
updates [20]. Finally, we will explore hybrid strategies between the mixture modeling in
section 3.1 and kNN classifiction in section 3.2, where multiple (but not all) examples in
each class are used as ?anchors? for distance-based classification. All these directions should
open the door to implementations on larger scales [21] than we have considered here.
8
References
[1] D. B. Rubin and D. T. Thayer. EM algorithms for ML factor analysis. Psychometrika, 47:69?
76, 1982.
[2] G. McLachlan and K. Basford. Mixture Models: Inference and Applications to Clustering.
Marcel Dekker, 1988.
[3] T. Cover and P. Hart. Nearest neighbor pattern classification. In IEEE Transactions in
Information Theory, IT-13, pages 21?27, 1967.
[4] J. Goldberger, S. Roweis, G. Hinton, and R. Salakhutdinov. Neighbourhood components
analysis. In L. K. Saul, Y. Weiss, and L. Bottou, editors, Advances in Neural Information
Processing Systems 17, pages 513?520, Cambridge, MA, 2005. MIT Press.
[5] K.Q. Weinberger and L.K. Saul. Distance metric learning for large margin nearest neighbor
classification. The Journal of Machine Learning Research, 10:207?244, 2009.
[6] J. V. Davis, B. Kulis, P. Jain, S. Sra, and I. S. Dhillon. Information-theoretic metric learning.
In ICML, pages 209?216, Corvalis, Oregon, USA, June 2007.
[7] G. Hinton and S. Roweis. Stochastic neighbor embedding. In S. Thrun S. Becker and K. Obermayer, editors, Advances in Neural Information Processing Systems 15, pages 833?840. MIT
Press, Cambridge, MA, 2003.
[8] C. Cortes and V. Vapnik. Support-vector networks. Machine Learning, 20:273?297, 1995.
[9] B. Kulis, M. A. Sustik, and I. S. Dhillon. Learning low-rank kernel matrices. In Proceedings
of the Twenty-Third International Conference on Machine Learning (ICML-06), 2006.
[10] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data
via the EM algorithm. Journal of the Royal Statistical Society B, 39:1?37, 1977.
[11] http://yann.lecun.com/exdb/mnist/.
[12] http://mlg.ucd.ie/datasets/bbc.html.
[13] http://www.dataminingresearch.com/index.php/2010/09/classic3-classic4-datasets/.
[14] http://archive.ics.uci.edu/ml/datasets.html.
[15] R A Fisher. The use of multiple measurements in taxonomic problems. Annals of Eugenics,
7(2):179?188, 1936.
[16] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, 2004.
[17] R. Salakhutdinov, S. T. Roweis, and Z. Ghahramani. On the convergence of bound optimization algorithms. In Proceedings of the Nineteenth Conference on Uncertainty in Artificial
Intelligence (UAI-03), pages 509?516, 2003.
[18] S. Shalev-Shwartz, Y. Singer, and A. Y. Ng. Online and batch learning of pseudo-metrics. In
Proceedings of the Twenty First International Conference on Machine Learning (ICML-04),
pages 94?101, Banff, Canada, 2004.
[19] T. Jaakkola and M. Jordan. A variational approach to bayesian logistic regression models and
their extensions. In Proceedings of the Sixth International Workshop on Artificial Intelligence
and Statistics, 1997.
[20] M. Dredze, K. Crammer, and F. Pereira. Confidence-weighted linear classification. In Andrew
McCallum and Sam Roweis, editors, Proceedings of the 25th Annual International Conference
on Machine Learning (ICML 2008), pages 264?271. Omnipress, 2008.
[21] G. Chechik, U. Shalit, V. Sharma, and S. Bengio. An online algorithm for large scale image
similarity learning. In Y. Bengio, D. Schuurmans, J. Lafferty, C. K. I. Williams, and A. Culotta,
editors, Advances in Neural Information Processing Systems 22, pages 306?314. 2009.
9
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support:2 latter:3 crammer:1 z0i:1 brevity:1 |
4,016 | 4,635 | Accuracy at the Top
Stephen Boyd
Stanford University
Packard 264
Stanford, CA 94305
Corinna Cortes
Google Research
76 Ninth Avenue
New York, NY 10011
[email protected]
[email protected]
Mehryar Mohri
Courant Institute and Google
251 Mercer Street
New York, NY 10012
Ana Radovanovic
Google Research
76 Ninth Avenue
New York, NY 10011
[email protected]
[email protected]
Abstract
We introduce a new notion of classification accuracy based on the top ? -quantile
values of a scoring function, a relevant criterion in a number of problems arising for search engines. We define an algorithm optimizing a convex surrogate of
the corresponding loss, and discuss its solution in terms of a set of convex optimization problems. We also present margin-based guarantees for this algorithm
based on the top ? -quantile value of the scores of the functions in the hypothesis
set. Finally, we report the results of several experiments in the bipartite setting
evaluating the performance of our solution and comparing the results to several
other algorithms seeking high precision at the top. In most examples, our solution
achieves a better performance in precision at the top.
1
Introduction
The accuracy of the items placed near the top is crucial for many information retrieval systems such
as search engines or recommendation systems, since most users of these systems browse or consider
only the first k items. Different criteria have been introduced in the past to measure this quality,
including the precision at k (Precision@k), the normalized discounted cumulative gain (NDCG)
and other variants of DCG, or the mean reciprocal rank (MRR) when the rank of the most relevant
document is critical. A somewhat different but also related criterion adopted by [1] is based on the
position of the top irrelevant item.
Several machine learning algorithms have been recently designed to optimize these criteria and other
related ones [6, 12, 11, 21, 7, 14, 13]. A general algorithm inspired by the structured prediction
technique SVMStruct [22] was incorporated in an algorithm by [15] which can be used to optimize
a convex upper bound on the number of errors among the top k items. The algorithm seeks to
solve a convex problem with exponentially many constraints via several rounds of optimization
with a smaller number of constraints, augmenting the set of constraints at each round with the
most violating one. Another algorithm, also based on structured prediction ideas, is proposed in
an unpublished manuscript of [19] and covers several criteria, including Precision@k and NDCG.
A regression-based solution is suggested by [10] for DCG in the case of large sample sizes. Some
other methods have also been proposed to optimize a smooth version of a non-convex cost function
in this context [8]. [1] discusses an optimization solution for an algorithm seeking to minimize the
position of the top irrelevant item.
1
However, one obvious shortcoming of all these algorithms is that the notion of top k does not generalize to new data. For what k should one train if the test data in some instances is half the size and
in other cases twice the size? In fact, no generalization guarantee is available for such precision@k
optimization or algorithm.
A more principled approach in all the applications already mentioned consists of designing algorithms that optimize accuracy in some top fraction of the scores returned by a real-valued hypothesis. This paper deals precisely with this problem. The desired objective is to learn a scoring function
that is as accurate as possible for the items whose scores are above the top ? -quantile. To be more
specific, when applied to a set of size n, the number of top items is k = ? n for a ? -quantile, while
for a different set of size n0 6= n, this would correspond to k 0 = ? n0 6= k.
The implementation of the Precision@k algorithm in [15] indirectly acknowledges the problem that
the notion of top k does not generalize since the command-line flag requires k to be specified as a
fraction of the positive samples. Nevertheless, the formulation of the problem as well as the solution
are still in terms of the top k items of the training set. A study of various statistical questions related
to the problem of accuracy at the top is discussed by [9]. The authors also present generalization
bounds for the specific case of empirical risk minimization (ERM) under some assumptions about
the hypothesis set and the distribution. But, to our knowledge, no previous publication has given
general learning guarantees for the problem of accuracy in the top quantile scoring items or carefully
addressed the corresponding algorithmic problem.
We discuss the formulation of this problem (Section 3.1) and define an algorithm optimizing a
convex surrogate of the corresponding loss in the case of linear scoring functions. We discuss the
solution of this problem in terms of several simple convex optimization problems and show that these
problems can be extended to the case where positive semi-definite kernels are used (Section 3.2).
In Section 4, we present a Rademacher complexity analysis of the problem and give margin-based
guarantees for our algorithm based on the ? -quantile value of the functions in the hypothesis set.
In Section 5, we also report the results of several experiments evaluating the performance of our
algorithm. In a comparison in a bipartite setting with several algorithms seeking high precision
at the top, our algorithm achieves a better performance in precision at the top. We start with a
presentation of notions and notation useful for the discussion in the following sections.
2
Preliminaries
Let X denote the input space and D a distribution over X ? X . We interpret the presence
of a pair (x, x0 ) in the support of D as the preference of x0 over x. We denote by S =
(x1 , x01 ), . . . , (xm , x0m ) 2 (X ? X )m a labeled sample of size m drawn i.i.d. according to D
b the corresponding empirical distribution. D induces a marginal distribution over
and denote by D
X that we denote by D0 , which in the discrete case can be defined via
1 X
D0 (x) =
D(x, x0 ) + D(x0 , x) .
2 0
x 2X
b 0 the empirical distribution associated to D0 based on the sample S.
We also denote by D
The learning problems we are studying are defined in terms of the top ? -quantile of the values taken
by a function h : X ! R, that is a score q such that Prx?D0 [h(x) > q] = ? (see Figure 1(a)). In
general, q is not unique and this equality may hold for all q in an interval [qmin , qmax ]. We will be
particularly interested in the properties of the set of points x whose scores are above a quantile, that
is sq = {x : h(x) > q}. Since for any (q, q 0 ) 2 [qmin , qmax ]2 , sq and sq0 differ only by a set of
measure zero, the particular choice of q in that interval has no significant consequence. Thus, in
what follows, when it is not unique, we will choose the quantile value to be the maximum, qmax .
For any ? 2 [0, 1], let ?? denote the function defined by
8u 2 R,
?? (u) =
? (u) + (1
? )(u)+ ,
where (u)+ = max(u, 0) and (u) = min(u, 0) (see Figure 1(b)). ?? is convex as a sum
of two convex functions since u 7! (u)+ is convex, u 7! (u) concave. We will denote by
argMinu f (u) the largest minimizer of function f . It is known (see for example [17]) that the
2
?-Quantile
Set U = {u1 , . . . , un }
R , ? 2 [0, 1] .
??
top ? fraction
of scores
u
0
page 5
Mehryar Mohri - Courant & Google
(a)
(b)
Figure 1: (a) Illustration of the ? -quantile. (b) Graph of function ?? for ? = .25.
(maximum) ? -quantile value qb of a sample of real numbers X = (u1 , . . . , un ) 2 Rn can be
given by qb P
= argMinu2R F? (u), where F? is the convex function defined for all u 2 R by
n
F? (u) = n1 i=1 ?? (ui u).
page 1
Mehryar Mohri - Courant & Google
3
3.1
Accuracy at the top (AATP)
Problem formulation and algorithm
The learning problem we consider is that of accuracy at the top (AATP) which consists of achieving
an ordering of all items so that items whose scores are among the top ? -quantile are as relevant as
possible. Ideally, all preferred items are ranked above the quantile and non-preferred ones ranked
below. Thus, the loss or generalization error of a hypothesis h : X ! R with top ? -quantile value
qh is the average number of non-preferred elements that h ranks above qh and preferred ones ranked
below:
?
?
1
R(h) =
E
1h(x)>qh + 1h(x0 )<qh .
2 (x,x0 )?D
qh can be defined as follows in terms of the distribution D0 : qh = argMinu2R Ex?D0 [?? (h(x) u)].
The quantile value qh depends on the true distribution D. To define the empirical error of h for a
sample S = (x1 , x01 ), . . . , (xm , x0m ) 2 (X ?X )m , we will use instead an empirical estimate qbh of
qh : qbh = argMinu2R Ex?Db 0 [?? (h(x) u)]. Thus, we define the empirical error of h for a labeled
sample as follows:
m
?
1 X?
b
R(h)
=
1h(xi )>bqh + 1h(x0i )<bqh .
2m i=1
We first assume that X is a subset of RN for some N 1 and consider a hypothesis set H of linear
functions h : x 7! w ? x. We will use a surrogate empirical loss taking into consideration how much
the score w?xi of a non-preferred item xi exceeds qbh , and similarly how much lower the score w?x0i
for a preferred point x0i is than qbh , and seek a solution w minimizing a trade-off of that surrogate
loss and the norm squared kwk2 . This leads to the following optimization problem for AATP:
m
hX
i
1
min
kwk2 + C
w ? xi qbw + 1 + + qbw w ? x0i + 1 +
(1)
w
2
i=1
subject to qbw = argMin Q? (w, u),
u2R
where C
0 is a regularization parameter and Q? the quantile function defined as follows for a
sample S, for any w 2 RN and u 2 R:
m
i
1 hX
?? (w ? xi ) u) + ?? (w ? x0i ) u) .
Q? (w, u) =
2m i=1
In the following, we will assume that ? is a multiple of 1/2m, otherwise it can be rounded to the
nearest such value.
3.2
Analysis of the optimization problem
Problem (1) is not a convex optimization problem since, while the objective function is convex, the
equality constraint is not affine. Here, we further analyze the problem and discuss a solution.
3
The equality constraint could be written as an infinite number of inequalities of Q? (w, qbw ) ?
Q? (w, u) for all u 2 R. Observe, however, that the quantile value qw must coincide with the score
of one of training points xk or x0k , that is w ? xk or w ? x0k . Thus, Problem (1) can be equivalently
written with a finite number of constraints as follows:
m
hX
i
1
min
kwk2 + C
w ? xi qbw + 1 + + qbw w ? x0i + 1 +
w
2
i=1
subject to qbw 2 {w ? xk , w ? x0k : k 2 [1, m]}
8k 2 [1, m], Q? (w, qbw ) ? Q? (w, w ? xk ), 8k 2 [1, m], Q? (w, qbw ) ? Q? (w, w ? x0k ).
The inequality constraints do not correspond to non-positivity constraints on convex functions.
Thus, the problem is not a standard convex optimization problem, but our analysis leads us
to a simple approximate solution for the problem. For convenience, let (z1 , . . . , z2m ) denote
(x1 , . . . , xm , x01 , . . . , x0m ). Our method consists of solving the convex quadratic programming (QP)
problem for each value of k 2 [1, 2m]:
hX
1
kwk2 + C
w ? xi
2
i=1
m
min
w
subject to qbw = w ? zk .
qbw + 1
+ qbw
+
w ? x0i + 1
+
i
(2)
Let wk be the solution of Problem (2). For each k 2 [1, 2m], we determine the ? -quantile value
of the scores {wk ?zi : i 2 [1, 2m]}. This can be checked straightforwardly in time O(m log m) by
sorting the scores. Then, the solution w? we return is the wk for which wk ? zk is closest to the
? -quantile value, the one for which the objective function is the smallest in the presence of ties. The
method for determining w? is thus based on the solution of 2m simple QPs. Our solution naturally
parallelizes so that on a distributed computing environment, the computational time for solving the
problem can be reduced to roughly the same as that of solving a single QP.
3.3
Kernelized formulation
For any i 2 [1, 2m], let yi = 1 if i ? m, yi = +1 otherwise. Then, Problem (2) admits the following
equivalent dual optimization problem similar to that of SVMs:
max
?
2m
X
?i
i=1
2m
1 X
?i ?j yi yj (zi
2 i,j=1
zk ) ? (zj
zk )
(3)
subject to: 8i 2 [1, 2m], 0 ? ?i ? C,
which depends only on inner products between points of the training set. The vector w can be
P2m
obtained from the solution via w = i=1 ?i yi (zi zk ). The algorithm can therefore be generalized
by using equivalently any positive semi-definite kernel symmetric (PDS) kernel K : X ? X ! R
instead of the inner product in the input space, thereby also extending it to the case of non-vectorial
input spaces X . The corresponding hypothesis set H is that of linear functions h : x 7! w ? (x)
where : X ! H is a feature mapping to a Hilbert space H associated to K and w an element of
H. In view of (3), for any k 2 [1, 2m], the dual problem of (2) can then be expressed as follows:
max
?
2m
X
i=1
?i
2m
1 X
?i ?j yi yj Kk (zi , zj )
2 i,j=1
subject to: 8i 2 [1, 2m], 0 ? ?i ? C,
(4)
where, for any k 2 [1, 2m], Kk is the PDS kernel defined by Kk : (z, z0 ) 7! K(z, z0 ) K(z, zk )
K(zk , z0 ) + K(zk , zk ). Our solution can therefore also be found in the dual by solving the 2m QPs
defined by (4).
4
Theoretical guarantees
We here present margin-based generalization bounds for the AATP learning problem.
4
Let ? : R ! [0; 1] be the function defined by ? : x 7! 1x?0 + (1 x/?)+ 1x>0 . For any ? > 0
b? (h, t), both with
and t 2 R, we define the generalization error R(h, t) and empirical margin loss R
respect to t, by
m
R(h, t) =
X?
?
?
1
b? (h, t) = 1
E
1h(x)>t + 1h(x0 )<t R
0
2 (x,x )?D
2m i=1
? (t
h(xi )) +
0
? (h(xi )
?
t) .
b? (h, qh ) to the empirical margin
In particular, R(h, qh ) corresponds to the generalization error and R
b? (h, t) is upper bounded
loss of a hypothesis h for AATP. For any t > 0, the empirical margin loss R
by the average of the fraction of non-preferred elements xi that h ranks above t or less than ? below
t, and the fraction of preferred ones x0i it ranks below t or less than ? above t:
m
X?
b? (h, t) ? 1
R
1t
2m i=1
h(xi )<?
+ 1h(x0i )
t<?
?
.
(5)
We denote by D1 the marginal distribution of the first element of the pairs in X ? X derived from
D, and by D2 the marginal distribution with respect to the second element. Similarly, S1 is the
sample derived from S by keeping only the first element of each pair: S1 = x1 , . . . , xm and
S2 the one obtained by keeping only the second element: S2 = x01 , . . . , x0m . We also denote
1
by RD
m (H) the Rademacher complexity of H with respect to the marginal distribution D1 , that is
D1
b S (H)], and RD2 (H) = E[R
b S (H)].
Rm (H) = E[R
m
1
2
Theorem 1 Let H be a set of real-valued functions taking values in [ M, +M ] for some M > 0.
Fix ? 2 [0, 1] and ? > 0, then, for any > 0, with probability at least 1
over the choice of a
sample S of size m, each of the following inequalities holds for all h 2 H and t 2 [ M, +M ]:
?
? r
1
2M
log 1/
D1
D2
b
R(h, t) ? R? (h, t)+
Rm (H) + Rm (H) + p
+
?
2m
m
?
? r
2M
log 2/
b S (H) + R
b S (H) + p
b? (h, t)+ 1 R
R(h, t) ? R
+3
.
1
2
?
2m
m
e be the family of hypotheses mapping (X ? X ) to R defined by H
e = {z = (x, x0 ) 7!
Proof. Let H
0
0
e
t h(x) : h 2 H, t 2 [ M, +M ]} and similarly H = {z = (x, x ) 7! h(x0 ) t : h 2 H, t 2
e and H
e 0 taking values in [0, 1] defined by
[ M, +M ]}. Consider the two families of functions H
e = { ? f : f 2 H}
e and H
e0 = { ? f : f 2 H
e 0 }. By the general Rademacher complexity
H
bounds for functions taking values in [0, 1] [18, 3, 20], with probability at least 1
,
r
?
?
?
1 ?
b? (h, t) + 2Rm 1 (H
e+H
e 0 ) + log 1/
E ? (t h(x)) + ? (h(x0 ) t) ? R
2
2
2m
r
log
1/
b? (h, t) + Rm (H)
e + Rm (H
e0 +
?R
,
2m
for all h 2 H. Since 1u<0 ? ? (u)
? for all u 2 R, the generalization
? error R(h, t) is a lower bound
on left-hand side: R(h, t) ? 12 E ? (t h(x)) + ? (h(x0 ) t) , we obtain
r
0
b? (h, t) + Rm (H)
e + Rm (H
e + log 1/ .
R(h, t) ? R
2m
e ? (1/?)Rm (H)
e
Since ? is 1/?-Lipschitz, by Talagrand?s contraction lemma, we have Rm H
e 0 ? (1/?)Rm (H
e 0 ). By definition of the Rademacher complexity,
and Rm H
"
#
"
#
m
m
m
X
X
X
1
1
e =
Rm (H)
E
sup
h(xi )) =
E sup
i (t
i t + sup
i h(xi )
m S?Dm , h2H,t i=1
m S,
t
h2H i=1
i=1
m
m
i 1 ?
X
X
1 h
= E
sup
t
E sup
i +
i h(xi ) .
m
m
h2H i=1
t2[ M,+M ] i=1
5
Since the random variables i and
i follow the same distribution, the second term coincides with
1
RD
(H).
The
first
term
can
be
rewritten
and upper bounded as follows using Jensen?s inequality:
m
"
#
m
m
m
X
X
X
X
X
1
M
M
E
sup
Pr[ ]
Pr[ ]
it =
i
i
m
m Pm
m Pm
M ?t?M i=1
i=1
i=1
>0
<0
i
i
i=1
i=1
" m
#
m
m
h
i1
X
X
M
M
M h X 2 i 12
M
2 2
E
?
E
E
=
=
=p .
i
i
i
m
m
m
m
i=1
i=1
i=1
Note
the last upper bound used is tight modulo a constant
pthat, by the Kahane-Khintchine inequality,
p
2
e 0 ) ? RD
(1/ 2). Similarly, we can show that Rm (H
m (H)+M/ m. This proves the first inequality
of the theorem; the second inequality can be derived from the first one using the standard bound
relating the empirical and true Rademacher complexity. 2
Since the bounds of the theorem hold uniformly for all t 2 [ M, +M ], they hold in particular for
any quantile value qh .
Corollary 1 (Margin bounds for AATP) Let H be a set of real-valued functions taking values in
[ M, +M ] for some M > 0. Fix ? 2 [0, 1] and ? > 0, then, for any > 0, with probability at least
1
over the choice of a sample S of size m, for all h 2 H it holds that:
?
? r
1
log 1/
2M
D1
D2
b
R(h) ? R? (h, qh )+
Rm (H) + Rm (H) + p
+
?
2m
m
?
? r
2M
log 2/
b S (H) + p
b S (H) + R
b? (h, qh )+ 1 R
R(h) ? R
+3
.
2
1
?
2m
m
A more explicit version of this corollary can be derived for kernel-based hypotheses (Appendix A).
In the results of the previous theorem and corollary, the right-hand side of the generalization bounds
is expressed in terms of the empirical margin loss with respect to the true quantile value qh , which
is upper bounded (see (5)) by half the fraction of non-preferred points in the sample whose score is
above qh ? and half the fraction of the preferred points whose score is less than qh + ?. These
fractions are close to the same fractions with qh replaced with qbh since the probability
that a score
p
falls between qh and qbh can be shown to be uniformly bounded by a term in O(1/ m).1 Altogether,
this analysis provides a strong support for our algorithm which is precisely seeking to minimize the
sum of an empirical margin loss based on the quantile and a term that depends on the complexity, as
in the right-hand side of the learning guarantees above.
5
Experiments
This section reports the results of experiments with our AATP algorithm on several datasets. To
measure the effectiveness of our algorithm, we compare it to two other algorithms, the I NFINITE P USH algorithm [1] and the SVMP ERF algorithm [15], which are both algorithms seeking to emphasize the accuracy near the top. Our experiments are carried out using three data sets from the
UC Irvine Machine Learning Repository http://archive.ics.uci.edu/ml/datasets.html:
Ionosphere, Housing, and Spambase. (Results for Spambase can be found in Appendix C). In addition, we use the TREC 2003 (LETOR 2.0) data set which is available for download from the
following Microsoft Research URL: http://research.microsoft.com/letor.
All the UC Irvine data sets we experiment with are for two-group classification problems. From
these we construct bipartite ranking problems where a preference pair consists of one positive and
one negative example. To explicitly indicate the dependency on the quantile, we denote by q? the
value of the top ? -th quantile of the score distribution of a hypothesis. We will use N to denote the
number of instances in a particular data set, as well as si , i = 1, . . . , N , to denote the particular
score values. If n+ denotes the number of positive examples in the data set and n denotes the
number of negative examples, then N = n+ + n and the number of preferences is m = n+ n .
1
Note that the Bahadur-Kiefer representation is known to provide a uniform convergence bound on the
difference of the true and empirical quantiles when the distribution admits a density [2, 16], a stronger result
than what is needed in our context.
6
Table 1: Ionosphere data: for each top quantile ? and each evaluation metric, the three rows correspond to AATP (top), SVMP ERF(middle) and I NFINITE P USH (bottom). For the I NFINITE P USH
algorithm we only report mean values over the folds.
? (%)
P@?
19
0.89 ? 0.04
0.89 ? 0.06
0.85
14 0.91 ? 0.05
0.82 ? 0.11
0.87
9.50 0.93 ? 0.06
0.77 ? 0.18
0.90
5
0.91 ? 0.14
0.66 ? 0.27
0.86
1
0.85 ? 0.24
0.35 ? 0.41
0.85
AP
DCG@?
NDCG@?
Positives@top
0.86 ? 0.03
0.83 ? 0.04
0.80
0.84 ? 0.03
0.79 ? 0.04
0.80
0.84 ? 0.03
0.79 ? 0.04
0.80
0.84 ? 0.03
0.79 ? 0.04
0.81
0.84 ? 0.03
0.79 ? 0.04
0.80
29.21 ? 0.10
28.88 ? 1.37
27.83
28.15 ? 0.95
27.02 ? 1.37
27.91
28.15 ? 0.95
27.02 ? 1.35
27.90
28.15 ? 0.95
27.02 ? 1.36
27.90
28.15 ? 0.95
27.02 ? 1.36
27.91
0.92 ? 0.06
0.89 ? 0.11
0.85
0.91 ? 0.07
0.75 ? 0.16
0.87
0.91 ? 0.09
0.70 ? 0.21
0.89
0.89 ? 0.15
0.60 ? 0.30
0.87
0.88 ? 0.19
0.34 ? 0.41
0.86
12.1 ? 12.5
6.00 ? 11.1
10.32
13.31 ? 12.5
4.10 ? 11.1
11.51
13.31 ? 12.49
4.50 ? 10.9
11.51
13.31 ? 12.49
4.60 ? 11.0
11.59
13.30 ? 12.53
4.50 ? 11.0
11.50
5.1 Implementation
We solved the convex optimization problems (2) using the CVX solver http://cvxr.com/. As
already noted, the AATP problem can be solved efficiently using a distributed computing environment. The convex optimization problem of the I NFINITE P USH algorithm (see (3.9) of [1])
can also be solved using CVX. However, this optimization problem has as many variables as
the product of the numbers of positively and negatively labeled instances (n+ n ), which makes
it prohibitive to solve for large data sets within a runtime of a few days. Thus, we experimented with the I NFINITE P USH algorithm only on the Ionosphere data set. Finally, for SVMP ERF?s training and score prediction we used the binary executables downloaded from the URL
http://www.cs.cornell.edu/people/tj and used the SVMP ERF?s settings that are the closest to our optimization formulation. Thus, we used L1-norm for slack variables and allowed the
constraint cache and the tolerance for termination criterion to grow in order to control the algorithm?s convergence, especially for larger values of the regularization constant.
5.2 Evaluation measures
To evaluate and compare the AATP, I NFINITE P USH, and SVMP ERF algorithms, we used a number
of standard metrics: Precision at the top (P@? ), Average Precision (AP), Number of positives at the
absolute top (Positives@top), Discounted Cumulative Gain (DCG@? ), and Normalized Discounted
Cumulative Gain (NDCG@? ). Definitions are included in Appendix B.
5.3 Ionosphere data
The data set?s 351 instances represent radar signals collected from phased antennas, where ?good?
signals (225 positively labeled instances) are those that reflect back toward the antennas and ?bad?
signals (126 negatively labeled instances) are those that pass through the ionosphere. The data has
34 features. We split the data set into 10 independent sets of instances, say S1 , . . . , S10 . Then, we
ran 10 experiments, where we used 3 consecutive sets for learning and the rest (7 sets) for testing.
We evaluated and compared the algorithms for 5 different top quantiles ? 2 {19, 14, 9.5, 5, 1} (%),
which would correspond to the top 20, 15, 10, 5, 1 items, respectively. For each ? , the regularization parameter C was selected based on the average value of P@? . The performance of AATP is
significantly better than that of the other algorithms, particularly for the smallest top quantiles. The
two main criteria on which to evaluate the AATP algorithm are Precision at the top, (P@? ), and
Number of positive at the top, (Positives@top). For ? = 5% the AATP algorithm obtains a stellar
91% accuracy with an average of 13.3 positive elements at the top (Table 1).
7
Table 2: Housing data: for each quantile value ? and each evaluation metric, there are two rows
corresponding to AATP (top) and SVMP ERF(bottom).
? (%)
P@?
AP
DCG@?
NDCG@?
Positives@top
6
0.14 ? 0.05
0.13 ? 0.05
0.17 ? 0.07
0.12 ? 0.10
0.19 ? 0.13
0.14 ? 0.05
0.20 ? 0.12
0.17 ? 0.12
0.23 ? 0.10
0.25 ? 0.17
0.20 ? 0.27
0.30 ? 0.27
0.11 ? 0.03
0.10 ? 0.02
0.10 ? 0.03
0.09 ? 0.03
0.12 ? 0.03
0.10 ? 0.02
0.10 ? 0.03
0.09 ? 0.02
0.10 ? 0.03
0.10 ? 0.03
0.12 ? 0.03
0.09 ? 0.02
4.64 ? 0.40
4.81 ? 0.46
4.69 ? 0.26
4.76 ? 0.60
4.83 ? 0.45
4.66 ? 0.25
4.70 ? 0.26
4.65 ? 0.40
4.69 ? 0.26
4.89 ? 0.48
4.80 ? 0.45
4.74 ? 0.56
0.13 ? 0.08
0.16 ? 0.09
0.16 ? 0.07
0.16 ? 0.14
0.18 ? 0.15
0.13 ? 0.07
0.18 ? 0.11
0.18 ? 0.13
0.19 ? 0.11
0.27 ? 0.16
0.17 ? 0.23
0.29 ? 0.27
0.20 ? 0.45
0.21 ? 0.45
0.00 ? 0.00
0.20 ? 0.48
0.00 ? 0.00
0.00 ? 0.00
0.00 ? 0.00
0.00 ? 0.00
0.00 ? 0.00
0.20 ? 0.46
0.00 ? 0.00
0.20 ? 0.45
5
4
3
2
1
5.4
Housing data
The Boston Housing data set has 506 examples, 35 positive and 471 negative, described by 13
features. We used feature 4 as the binary target value. Two thirds of the data instances was randomly
selected and used for training, and the rest for testing. We created 10 experimental folds analogously
as in the case of the Ionosphere data. The Housing data is very unbalanced with less than 7%
positive examples. For this dataset we obtain results very comparable to SVMP ERF for the very top
quantiles, see Table 2. Naturally, the standard deviations are large as a result of the low percentage
of positive examples, so the results are not always significant. For higher top quantiles, e.g., top
4%, the AATP algorithm significantly outperforms SVMP ERF, obtaining 19% accuracy at the top
(P@? ). For the highest top quantiles the difference in performance between the two algorithms is
not significant.
5.5
LETOR 2.0
This data set corresponds to a relatively hard ranking problem, with an average of only 1% relevant
query-URL pairs per query. It consists of 5 folds. Our Matlab implementation (with CVX) of the
algorithms prevented us from trying our approach on larger data sets. Hence from each training fold
we randomly selected 500 items for training. For testing, we selected 1000 items at random from the
test fold. Here, we only report results for P@1%. SVMP ERF obtained an accuracy of 1.5% ? 1.5%
while the AATP algorithm obtained an accuracy of 4.6% ? 2.4%. This significantly better result
indicates the power of the algorithm proposed.
6
Conclusion
We presented a series of results for the problem of accuracy at the top quantile, including an AATP
algorithm, a margin-based theoretical analysis in support of that algorithm, and a series of experiments with several data sets demonstrating the effectiveness of our algorithm. These results are of
practical interest in applications where the accuracy among the top quantile is sought. The analysis
of problems based on other loss functions depending on the top ? -quantile scores is also likely to
benefit form the theoretical and algorithmic results we presented.
The optimization algorithm we discussed is highly parallelizable, since it is based on solving 2m
independent QPs. Our initial experiments reported here were carried out using Matlab with CVX,
which prevented us from evaluating our approach on larger data sets, such as the full LETOR 2.0
data set. However, we have now designed a solution for very large m based on the ADMM (Alternating Direction Method of Multipliers) framework [4]. We have implemented that solution and
will present and discuss it in future work.
8
References
[1] S. Agarwal. The infinite push: A new support vector ranking algorithm that directly optimizes
accuracy at the absolute top of the list. In Proceedings of the SIAM International Conference
on Data Mining, 2011.
[2] R. R. Bahadur. A note on quantiles in large samples. Annals of Mathematical Statistics, 37,
1966.
[3] P. L. Bartlett and S. Mendelson. Rademacher and Gaussian complexities: Risk bounds and
structural results. Journal of Machine Learning Research, 3:2002, 2002.
[4] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in
Machine Learning, 3(1):1?122, 2011.
[5] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, 2004.
[6] J. S. Breese, D. Heckerman, and C. M. Kadie. Empirical analysis of predictive algorithms for
collaborative filtering. In UAI ?98: Proceedings of the Fourteenth Conference on Uncertainty
in Artificial Intelligence. Morgan Kaufmann, 1998.
[7] C. Burges, T. Shaked, E. Renshaw, A. Lazier, M. Deeds, N. Hamilton, and G. Hullender.
Learning to rank using gradient descent. In Proceedings of the 22nd international conference
on Machine learning, ICML ?05, pages 89?96, New York, NY, USA, 2005. ACM.
[8] C. J. C. Burges, R. Ragno, and Q. V. Le. Learning to rank with nonsmooth cost functions. In
NIPS, pages 193?200, 2006.
[9] S. Cl?emenc?on and N. Vayatis. Ranking the best instances. Journal of Machine Learning
Research, 8:2671?2699, 2007.
[10] D. Cossock and T. Zhang. Statistical analysis of Bayes optimal subset ranking. IEEE Transactions on Information Theory, 54(11):5140?5154, 2008.
[11] K. Crammer and Y. Singer. PRanking with ranking. In Neural Information Processing Systems
(NIPS 2001). MIT Press, 2001.
[12] Y. Freund, R. Iyer, R. E. Schapire, and Y. Singer. An efficient boosting algorithm for combining
preferences. J. Mach. Learn. Res., 4, December 2003.
[13] R. Herbrich, K. Obermayer, and T. Graepel. Advances in Large Margin Classifiers, chapter
Large Margin Rank Boundaries for Ordinal Regression. MIT Press, 2000.
[14] T. Joachims. Optimizing search engines using clickthrough data. In Proceedings of the eighth
ACM SIGKDD international conference on Knowledge discovery and data mining, KDD ?02,
pages 133?142, New York, NY, USA, 2002. ACM.
[15] T. Joachims. A support vector method for multivariate performance measures. In ICML, pages
377?384, 2005.
[16] J. Kiefer. On Bahadur?s representation of sample quantiles. Annals of Mathematical Statistics,
38, 1967.
[17] R. Koenker. Quantile Regression. Cambridge University Press, 2005.
[18] V. Koltchinskii and D. Panchenko. Empirical margin distributions and bounding the generalization error of combined classifiers. Annals of Statistics, 30, 2002.
[19] Q. V. Le, A. Smola, O. Chapelle, and C. H. Teo. Optimization of ranking measures. Unpublished, 2009.
[20] M. Mohri, A. Rostamizadeh, and A. Talwalkar. Foundations of Machine Learning. The MIT
Press, 2012.
[21] C. Rudin, C. Cortes, M. Mohri, and R. E. Schapire. Margin-based ranking meets boosting in
the middle. In COLT, pages 63?78, 2005.
[22] I. Tsochantaridis, T. Joachims, T. Hofmann, and Y. Altun. Large margin methods for structured
and interdependent output variables. Journal of Machine Learning Research, 6:1453?1484,
2005.
9
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4,017 | 4,636 | Learning the Dependency Structure of Latent Factors
Yunlong He?
Georgia Institute of Technology
[email protected]
Yanjun Qi
NEC Labs America
[email protected]
Haesun Park?
Georgia Institute of Technology
[email protected]
Koray Kavukcuoglu
NEC Labs America
[email protected]
Abstract
In this paper, we study latent factor models with dependency structure in the latent space. We propose a general learning framework which induces sparsity on
the undirected graphical model imposed on the vector of latent factors. A novel
latent factor model SLFA is then proposed as a matrix factorization problem with
a special regularization term that encourages collaborative reconstruction. The
main benefit (novelty) of the model is that we can simultaneously learn the lowerdimensional representation for data and model the pairwise relationships between
latent factors explicitly. An on-line learning algorithm is devised to make the
model feasible for large-scale learning problems. Experimental results on two
synthetic data and two real-world data sets demonstrate that pairwise relationships
and latent factors learned by our model provide a more structured way of exploring
high-dimensional data, and the learned representations achieve the state-of-the-art
classification performance.
1
Introduction
Data samples described in high-dimensional feature spaces are encountered in many important areas.
To enable the efficient processing of large data collections, latent factor models (LFMs) have been
proposed to find concise descriptions of the members of a data collection. A random vector x ? RM
is assumed to be generated by a linear combination of a set of basis vectors, i.e.,
x = Bs + = B1 s1 + B2 s2 + ? ? ? + BK sK +
(1)
where B = [B1 , . . . , BK ] stores the set of unknown basis vectors and describes noise. The i-th
?factor? si (i ? {1, ..., K}) denotes the i-th variable in the vector s.
In this paper, we consider the problem of learning hidden dependency structure of latent factors
in complex data sets. Our goal includes two main aspects: (1) to learn the interpretable lowerdimensional representations hidden in a set of data samples, and (2) to simultaneously model the
pairwise interaction of latent factors. It is difficult to achieve both aspects at the same time using
existing models. The statistical structure captured by LFM methods, such as Principal Component
Analysis (PCA) are limited in interpretability, due to their anti-correlation assumption on the latent
factors. For example, when a face image is represented as a linear super-position of PCA bases
with uncorrelated coefficients learned by PCA, there exist complex cancellations between the basis
images [14]. Methods that theoretically assume independence of components like ICA [10] or sparse
coding [15] fail to generate independent representations in practice. Notable results in [13, 17] have
shown that the coefficients of linear features for natural images are never independent.
?
The work of these authors was supported in part by the National Science Foundation grant CCF-0808863.
Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors
and do not necessarily reect the views of the National Science Foundation.
1
Instead of imposing this unrealistic assumption, more recent works [18, 25, 27] propose to allow correlated latent factors, which shows to be helpful in obtaining better performance on various tasks.
However, the graphical structure of latent factors (i.e., conditional dependence/independence) is not
considered in these works. Particularly, the sparse structure of the latent factor network is often
preferred but has been never been explicitly explored in the learning process [2, 8, 23]. For example,
when mining the enormous on-line news-text documents, a method discovering semantically meaningful latent topics and a concise graph connecting the topics will greatly assist intelligent browsing,
organizing and accessing of these documents.
The main contribution in this paper is a general LFM method that models the pairwise relationships between latent factors by sparse graphical models. By introducing a generalized Tikhonov
regularization, we enforce the interaction of latent factors to have an influence on learning latent
factors and basis vectors. As a result, we learn meaningful latent factors and simultaneously obtain
a graph where the nodes represent hidden groups and the edges represent their pairwise relationships. This graphical representation helps us analyze collections of complex data samples in a much
more structured and organized way. The latent representations of data samples obtained from our
model capture deeper signals hidden in the data which produce the useful features for discriminative
task and in-depth analysis, e.g. our model achieves a state-of-the-art performance on classifying
cancer samples in our experiment.
2
2.1
Methods
Sparse Undirected Graphical Model of Latent Factors: A General Formulation
Following [4, 16], our framework considers data samples drawn from the exponential family of
distributions, i.e.,
p(x|?) = h(x)exp(? | T (x) ? A(?)),
(2)
where sufficient statitic T (x) ? RM , ? ? RM represents the natural parameter for the model, T (x),
h(x) and A(?) are known functions defining a particular member of the exponential family. This
family includes most of the common distributions, like normal, Dirichlet, multinomial, Poisson, and
many others.
To learn the hidden factors for generating x, the natural parameter ? is assumed to be represented
by a linear combination of basis vectors, i.e.,
? = Bs,
(3)
where B = [B1 , . . . , BK ] is the basis matrix. To model the pairwise interaction between latent
factors, we introduce a pairwise Markov Random Field (MRF) prior on the vector of factors s ? RK :
K
K K
X
1
1 XX
p(s|?, ?) =
?ij si sj )
exp(?
?i si ?
Z(?, ?)
2 i=1 j=1
i=1
(4)
with parameter ? = [?i ], symmetric ? = [?ij ], and partition function Z(?, ?) which normalizes
the distribution. The classic Ising model and Gaussian graphical model are two special cases of the
above MRF. Let G = (V, E) denote a graph with K nodes, corresponding to the K latent factors
{s1 , . . . , sK }, and with edge set
E = {(i, j) ? V ? V : ?ij 6= 0}.
(5)
Since ?ij = 0 indicates that latent factor si and latent factor sj are conditionally independent given
other latent factors, the graph G presents an illustrative view of the statistical dependencies between
latent factors.
With such a hierarchical and flexible model, there would be significant risk of over-fitting, especially
when we consider all possible interactions between K latent factors. Therefore, regularization has
to be introduced for better generalization property of the model. As we will see in subsection 3,
regularization is also necessary from the perspective of avoiding ill-posed optimization problem.
The regularization technique we use is to introduce a sparsity-inducing prior for ?:
1
p(?) ? exp(? ?k?k1 ),
2
2
(6)
P P
where ? is a positive hyper-parameter and k?k1 := i j |?ij |. We aim to achieve two goals when
designing such a prior distribution: (1) in practice irrelevant latent factors are not supposed to be
conditionally dependent and hence a concise graphical structure between latent factors is preferred
in many applications such as topic mining and image feature learning, and (2) in contrast to L0
regularization which is the number of non-zero components, we obtain a convex subproblem of ?,
that can be efficiently solved by utilizing the recently developed convex optimization techniques.
2.2
Learning Algorithm
We consider the posterior distribution of parameters, which is proportional to the product of data
likelihood and the prior distributions:
h(x)exp{s| B| T (x) ? A(Bs)} ?
1
1
1
exp(??| s ? s| ?s) ? exp(? ?k?k1 ).
Z(?, ?)
2
2
(7)
Given a set of data observations {x(1) , . . . , x(N ) }, the Maximum a Posteriori (MAP) estimates of
the basis matrix B, the latent factors in S = [s(1) , . . . , s(N ) ] and the parameters {?, ?} of the latent
factor network are therefore the solution of the following problem:
1 X
min
{? log h(x(i) ) + A(Bs(i) ) ? s(i)| B| T (x(i) )}
B,S,? N
i
1 |
1
1
? S1N +
tr(S| ?S) + ?k?k1
N
2N
2
s.t. B ? 0, kBk k2 ? 1, k = 1, . . . , K,
+ log Z(?, ?) +
(8)
where additional constrains B ? 0 and kBk k2 ? 1 are introduced for the identifiability of the
model.
The objective function in Eq. (8) is not convex with respect to all three unknowns (B, S and ?)
together. Therefore, a good algorithm in general exhibits convergence behavior to a stationary point
and we can use Block Coordinate Descent algorithm [1] to iteratively update B, S and ? as follows:
while not convergent do
For i = 1, . . . , N , solve
Solve
1
min ? log h(x(i) ) + A(Bs(i) ) ? s(i)T B| T (x(i) ) + ?| s(i) + s(i)T ?s(i)
2
s(i)
X
min
{? log h(x(i) ) + A(Bs(i) ) ? s(i)T B| T (x(i) )}
B?0,kBk k2 ?1
Solve
(10)
i
min log Z(?, ?) +
?,?
(9)
1
1
1 |
? S1N +
tr(S| ?S) + ?k?k1
N
2N
2
(11)
end do
Since p(x|?) is in the exponential family, the subproblem (10) with respect to B is convex and
smooth with simple constraints, for which quasi-Newton methods such as projected L-BFGS [22]
are among the most efficient methods. Subproblem (9) is easy to solve for real-valued s(i) but
generally hard when the latent factors only admit discrete values. For example for s ? {0, 1}K and
Gaussian p(x|?), subproblem (9) is a 0-1 quadratic programming problem and we can resort to SDP
based Branch and Bound algorithms [20] to solve it in a reasonable time. The subproblem (11) is
minimizing the sum of a differentiable convex function and an L1 regularization term, for which
a few recently developed methods can be very efficient, such as variants of ADMM [6]. For the
cases of discrete s with large K (usually K << M ), evaluation of the partition function Z(?, ?)
during the iterations is ]P-hard and Schmidt [21] discusses methods to solve the pseudo-likelihood
approximation of (11).
3
A Special Case: Structured Latent Factor Analysis
From this section on, we consider a special case of the learning problem in Eq. (8) when x follows
a multivariate normal distribution and s follows a sparse Gaussian graphical model (SGGM). We
name our model under this default setting as ?structured latent factor analysis? (SLFA) and compare
it to related works. Assume p(x|?) = (2?)?M/2 exp(? 2?1 2 kx ? ?k2 ) and s ? N (?, ??1 ), with
3
sparse precision matrix ? (inverse covariance). For simplicity we assume the given data matrix
X = [x(1) , . . . , x(N ) ] is centered and set ? = 0. Then the objective function in Eq. (8) becomes
1
1
kX ? BSk2F + ? 2 ( tr(S| ?S) ? log det(?) + ?k?k1 )
N
N
s.t. B ? 0, kBk k2 ? 1, k = 1, . . . , K, ? < 0.
min
B,S,?
(12)
If ? is fixed, the problem in Eq. (12) is a matrix factorization method with generalized Tikhonov
regularization: trace(S| ?S). If ?i,j > 0, minimizing the objective function will avoid si and
sj to be simultaneously large, and we say the i-th factor and the j-th factor are negatively related.
If ?i,j < 0, the solution is likely to have si and sj of the same sign, and we say the i-th factor
and the j-th factor are positively related. If ?i,j = 0, the regularization doesn?t induce interaction
between si and sj in the objective function. Therefore, this regularization term makes SLFA produce
a collaborative reconstruction based on the conditional dependencies between latent factors. On one
hand, the collaborative nature makes SLFA capture deeper statistical structure hidden in the data
set, compared to the matrix factorization problem with the Tikhonov regularization kSk2F or sparse
coding with the sparsity-inducing regularization such as kSk1 . On the other hand, SLFA encourages
sparse interactions which is very different from previous works such as correlated topic Model [2]
and latent Gaussian model [18], where the latent factors are densely related.
An On-line Algorithm For Learning SLFA: The convex subproblem
1
tr(S| ?S) ? log det(?) + ?k?k1
?<0 N
min
(13)
can be efficiently solved by a recent quadratic approximation method in [9]. For subproblem of S
we have closed-form solution
S = (B| B + ? 2 ?)?1 X.
Moreover, considering that many modern high-dimensional data sets include a large number of data
observations (e.g. text articles from web-news), we propose an online algorithm for learning SLFA
on larger data sets. As summarized in Algorithm 1, at each iteration, we randomly fetch a mini-batch
of observations simultaneously, compute their latent factor vector s. Then the latent factor vectors
are used to update the basis matrix B in stochastic gradient descent fashion with projections on the
constraint set. Lastly we update the precision matrix ?.
Algorithm 1 An on-line algorithm for learning SLFA.
Input: X = [x(1) , . . . , x(N ) ], initial guess of basis matrix B, initial precision matrix ? = I,
number of iterations T , parameters ? 2 and ?, step-size ?, mini-batch size N 0 .
? for t = 1 to T
? Draw N 0 observations randomly from X = [x(1) , . . . , x(N ) ] to form the matrix
Xbatch .
? Compute the latent factor vectors Sbatch = (B| B + ? 2 ?)?1 Xbatch .
? Update the basis matrix B using a gradient descent step:
B ? B ? N? 0 [BSbatch ? Xbatch ]S|batch .
? Project columns of B to the first orthant and the unit ball, i.e., B ? 0 and kBi k ? 1.
? Solve the subproblem (13) to update the sparse inverse covariance matrix ? using all
available latent factor vectors in S.
? end for
Parameter Selection: The hyper-parameter ? controls the sparsity of ?. A large ? will result in
a diagonal precision matrix ?, indicating that the latent factors are conditionally independent. As
? ? 0, ? becomes denser. However, if we set ? = 0, the subproblem with respect to ? has a
closed form solution ? = ( N1 SS| )?1 , i.e., inverse sample covariance matrix. Plugging it back to
the Eq. (12), we have
1
1
min kX ? BSk2F + ? 2 log det( SS| ),
B,S N
N
which doesn?t have a lower bound. Therefore the regularization is necessary and we choose positive
values for ? in the experiments. For supervised tasks, we use cross-validation to choose the proper
4
value of ? that optimizes the evaluation rule on validation set. For unsupervised applications, we
combine the BIC criterion in [28], with our model to obtain the following criterion:
1
1
log N
?? = min kX ? B(?)S(?)k2F + ? 2
tr(S(?)| ?(?)S(?)) ? log det(?(?)) +
k?(?)k0 ,
? N
N
N
where B(?), S(?) and ?(?) and learned from (12) with parameter ?. Alternatively, for visual
analysis of latent factors, we can select multiple values of ? to obtain ? with desired sparsity.
Relationship to Sparse Gaussian Graphical Model: We can also see SLFA as a generalization
of sparse Gaussian graphical model. In fact, if the reduced dimension K = M , the problem (12)
has trivial solution B = I and S = X, and the problem becomes the same as (13). When K < M ,
the subproblem with respect to s has solution s = (B| B + ? 2 ?)?1 x. Therefore, lower dimensional
random vector s has less variables among which each variable is a linear combination of the original
variables of x with the combination weights stored in W = (B| B + ? 2 ?)?1 . In this sense, SLFA
could be seen as the sparse Gaussian graphical model of s = Wx, i.e. it generalizes the concept
from the original (totally N ) variables to the merged (totally K) group variables.
A few recent efforts [3, 24] also combined the model of SGGM and with latent factor models. For
example, ?Kronecker GLasso? in [24] performs a joint learning of row and column covariances
for matrix-variate Gaussian models. Different from our SLFA, these methods still aim at modeling
the interaction between the original features and doesn?t consider interaction in the latent factor
space. Instead, SLFA is a hierarchical model and the learned pairwise relationships are on the
latent factor level. If we apply both SLFA and Kronecker GLasso on a text corpus where each
document is represented by a 50, 000 sparse vector and number of latent factors (topics) are fixed as
50, then Kronecker GLasso will produce a precision matrix of dimension 50, 000 ? 50, 000 and a
corresponding sparse graph of 50, 000 nodes. SLFA, however, can dramatically reduce the problem
to learning a 50 ? 50 sparse precision matrix and the corresponding graph of 50 nodes.
Relationship to other works: Sparse coding [19] can be modeled as:
1
min kX ? BSk2F + ?kSk1 .
B,S 2
(14)
For many high-dimensional data sets such as text in natural languages, the input data is already very
sparse or high dimensional. Thus, sparse coding is not easily applicable. Intuitively, sparse coding
based works (such as [7]) try to remove the redundancy in the representation of data while SLFA
encourages a (sparse) collaborative reconstruction of the data from the latent bases.
Recently, Jenatton et al. [12] proposed a method that can learn latent factors with given tree structure.
The optimization problem in Jenatton et al., 2010 is a penalized matrix factorization problem similar
to our Eq. (12) and Eq. (14), but uses a different regularization term which imposes the overlapped
group sparsity of factors. Differently, SLFA can learn a more general graphical structure among
latent factors and doesn?t assume that data sample maps to a sparse combination of basis vectors.
The model of SLFA has similar hierarchy with correlated topic model [2] and latent Gaussian
model [18]. Besides the key difference of sparsity, SLFA directly use precision matrix to learn
latent factor networks while the other two works learn the covariance matrix by Bayesian methods.
4
Experiments
In this section, we conduct experiments on both synthetic and real world data sets to show that:
(1) SLFA recovers latent basis vectors and finds the pairwise relationships of latent factors, (2)
SLFA generates useful features for various tasks such as images analysis, topic visualization and
microarray analysis.
4.1 Synthetic Data I: Four Different Graphical Relationships
The first experiment uses randomly generated synthetic data with different graphical structures of
latent factors. It aims to test if SLFA can find true latent factors and the true relationships among
latent factors and to study the effect of the parameter ? on the results. We use four special cases
of Sparse Gaussian Graphical Model to generate the latent factors. The underlying graph is either
a ring, a grid, a tree or a random sparse graph, which are shown in Figure 1. A sparse positive
5
0.4
0.2
0
?2
0.6
0.4
0.2
0
2
4
?log2(rho)
6
(e) F-score (ring)
8
0
?2
(d) Random
1
SLFA
NMF+SGGM
True Basis
L2 version
L2+SGGM
Scaled?BIC
0.8
F?score
0.6
(c) Tree
1
SLFA
NMF+SGGM
True Basis
L2 version
L2+SGGM
Scaled?BIC
0.8
F?score
0.8
F?score
(b) Grid
1
SLFA
NMF+SGGM
True Basis
L2 version
L2+SGGM
Scaled?BIC
0.6
0.4
0.2
0
2
4
?log2(rho)
6
8
(f) F-score (grid)
0
?2
SLFA
NMF+SGGM
True Basis
L2 version
L2+SGGM
Scaled?BIC
0.8
F?score
(a) Ring
1
0.6
0.4
0.2
0
2
4
?log2(rho)
6
(g) F-score (tree)
8
0
?2
0
2
4
?log2(rho)
6
8
(h) F-score (random)
Figure 1: Recovering structured latent factors from data. On the upper row are four different underlying graphical model of latent factors. Red edge means the two latent factors are positively related
(??ij < 0), blue edge implies the two latent factors are negatively related (??ij > 0). On the lower
row are the plots of F-score vs. ? for four settings. We can observe that SLFA (red lines) is as good
as an oracle method (True Basis, green lines). The pink dash lines of BIC score (scaled to [0, 1])
demonstrate that the parameter selection method works well.
definite matrix ?? ? R10?10 is constructed based on the graph of SGGM. Then we sample 200
Gaussian random vectors, s(1) , . . . , s(200) ? R10 , with precision matrix ?? . A set of vectors B? ?
R500?10 is randomly generated with normal distribution and then filtered by a sigmoid function
1
f (b) = 1+e?100b
such that most components of B? are close to either 0 or 1. B1 , B2 , . . . , B10
are then normalized as basis vectors. Finally,the synthetic data points are generated by x(i) =
Bs(i) + 0.1i , i = 1, . . . , 200, where i ? N (0, I).
We compare SLFA to other four methods for learning the basis matrix B and the precision matrix
? from the data. The first one is NMF, where we learn nonnegative basis B from the data and then
learn the sparse precision matrix ? for the corresponding factor vectors (non nonnegative constraint
on factors) by SGGM. The second one is an ideal case where we have the ?oracle? of the true basis
B? , then after fit the data to be true basis we learn the sparse precision matrix ? by SGGM. The third
one is named L2 version of SLFA as we replace the L1 regularization of ? by a Frobenius norm
regularization. The fourth method first applies L2 version of SLFA and then learns ? by SGGM.
In all cases except the oracle method, we have a non-convex problem so that after we obtain the
learned basis vectors we use Hungarian algorithm to align them to with the true basis vectors based
on the cosine similarity. We compute the precision and recall rates for recovering the relationship
between latent factors by comparing the learned ? with the true precision matrix ?? .
We plot F-score based on the precision and recall rates averaged over 10 experiments. According
to Figure 1, when ? is large, the estimated ? is diagonal so that recall rate is 0. As ? becomes
smaller, more nonzero elements appear in the estimated ? and both the recall and precision rate of
?positive/negative relationship? get increased. When ? is small enough, the recovered ? becomes
denser and may not even recover the ?positive/negative relationship? correctly. We can see that
for all four cases, our proposed method SLFA is as good as the ?oracle? method at recovering the
pairwise relationship between latent factors. NMF most probably fails to find the right basis since
it does consider any higher level information about the interactions between basis elements, hence
SGGM can?t find meaningful relationship between the factors obtained from NMF. L2 version of
SLFA also has poor F-score since it can?t recover the sparse structure. Since latent factors have dense
interactions in L2 version of SLFA, combining it with a postprocessing by SGGM improves the
performance significantly, however it still performs worse compared to SLFA. This experiment also
confirms that the idea of performing an integrated learning of the bases together with a regularized
precision matrix is essential for recovering the true structure in the data.
4.2
Synthetic Data II: Parts-based Images
The second experiment also utilizes a simulated data set based on images to compare SLFA with
popular latent factor models. We set up an experiment by generating 15000 images of ?bugs?, each
6
?i,j (?) rel. ?i,j (+) rel.
0.030
0.020
0.015
0.015
0.014
0.013
(a) True Bases
(b) Creation
(c) Samples
?0.016
?0.015
?0.013
?0.012
?0.011
?0.011
(d) Precision Matrix
(e) SLFA Basis
Figure 2: Table (e) shows the ?(i, j) values and corresponding Bi and Bj elements learned by
SLFA for the six highest and and six lowest entries in ?. For ?(i, j) > 0, Bi and Bj are negatively
related (exclusive), for ?(i, j) < 0, Bi and Bj are positively related (supportive).
of which is essentially a linear combination of five latent parts shown in Figure 2a. Given 37 basis
images, we first randomly select one of the five big circles as the body of the ?bugs?. Each shape of
body is associated with four positions where the legs of the bug is located. We then randomly pick 4
legs from its associated set of 4 small circles and 4 small squares. However, for each leg, circle and
square are exclusive of each other. We combine the selected five latent parts with random coefficients
that are sampled from the uniform distribution and multiplied by ?1 with probability 0.5. Finally,
we add a randomly selected basis with small random coefficients plus Gaussian random noise to
the image to introduce the noise and confusion in the data set. A few examples of the bug image
samples created by the above strategy are shown in Figure 2c. The generating process (Figure 2b)
indicates positive relationship between one type of body and its associates legs, as well as negative
relationship between the pair of circle and square that is located at the same position.
Using SLFA and other two baseline algorithms, PCA and NMF, we learn a set of latent bases and
compare the result of three methods in Figures 2e. We can see that the basis images generated by
SLFA is almost exactly same as the true latent bases. This is due to the fact that SLFA accounts for
the sparse interaction between factors in the joint optimization problem and encourages collaborative
reconstruction. NMF basis (shown in supplementary material due to space considerations) in this
case also turns out to be similar to true basis, however, one can still observe that many components
contain mixed structures since it can not capture the true data generation process. The bases learned
by PCA (also shown in supp. material) is not interpretable as expected.
More importantly, SLFA provides the convenience of analyzing the relationship between the bases
using the precision matrix ?. In Figure 2d, we analyze the relational structure learned in the precision matrix ?. The most negatively related (exclusive) pairs (the i and j entries with highest positive
entries in ?) are circular and square legs which conforms fully to the generation process, since only
one of them is chosen for any given location. Accordingly, the most positively related pairs are
a body shape and one of its associated legs since every bug has a body and four legs with fixed
positions.
4.3
Real Data I: NIPS Documents
In this section, we apply SLFA to the NIPS corpus1 which contains 1740 abstracts from the NIPS
Conferences 1?12 for the purpose of topic/content modeling. SLFA is used to organize and visualize
the relationship between the structured topics. SLFA is applied on the 13649 dimensional tf-idf
feature vector which is normalized to have the unit norm. We fix the number of topics to be 40 and
tune the parameters ? and ? to obtain ? with a proper sparsity for the visualization task. In figure 3,
we plot a graph of topics (standing-alone topics removed) with positive interaction between each
other and present the top 5 keywords for each topic. For example, the topic at the top is about general
notions in many learning algorithms and acts as the hub point of the graph. more specific words that
are relevant to a particular learning algorithm or a more specialized topic of interest. It is obvious
that SLFA not only extracts the underlying topics, but is also able to capture the (de)correlations
between topics. For example, on the far left, the topic related to cells is connected to ?motion,
velocity, ...?, ?objects, image,...? and ?spike, neurons, ...? nodes. This subgraph clearly represents
a few topics in computer vision and neuroscience. The node on the far right containing ?robot,
planning, ...? is connected to the node with ?controller, control, ...? which represents a robotics
related topic-cluster. It is also interesting to note that SLFA can obtain a graph of negatively related
topics(shown in supplementary material). One can see that closely related topics tend to exclude
each other.
1
http://cs.nyu.edu/ roweis/data.html
7
units
hidden
unit
layer
boltzmann
motor
arm
trajectory
movement
movements
spike
firing
spikes
neuron
neurons
motion
velocity
visual
direction
flow
cells
cell
orientation
receptive
cortex
memory
capacity
associative
hopfield
memories
object
objects
views
image
visual
hmm
hmms
speech
markov
mlp
face
faces
facial
images
pca
convergence
gradient
descent
stochastic
matrix
obs
risk
pruning
obd
validation
bayesian
posterior
gaussian
hyperparameters
carlo
mixture
em
experts
expert
likelihood
classifier
classifiers
rbf
classification
class
image
images
texture
wavelet
pixel
controller
control
plant
critic
forward
student
teacher
generalization
committee
overlaps
pca
obs
pruning
obd
adaboost
robot
planning
navigation
reinforcement
action
Figure 3: Positively related topics (learned by SLFA) discovered from NIPS text corpus. Each edge
corresponds to a negative element in the sparse precision matrix ?.
SLFA
34.22 ? 2.58
Lasso-overlapped-group
35.31 ? 2.05
Lasso
36.42 ? 2.50
SVM
36.93 ? 2.54
PCA
36.85 ? 3.02
Table 1: Cross-validation error rate (average and standard deviation) by different methods on Gene
Micro-array data. SLFA performs best and even better than Lasso-overlapped-group (t-test at significance level 0.02), which takes advantage of external information (42, 594 known edges between
gene variables from another biological resource).
4.4 Real Data II: Gene Microarray Data for Cancer Classification
Next, we test our model on a classification task which uses breast cancer microarray data set obtained
from [11]. This data set contains the gene expression values of 8, 141 genes for 295 breast cancer
tumor samples. The task is to classify the tumor samples into two classes (with 78 metastatic and
217 non-metastatic).
Using the classification error rates as the metric, we compare totally five methods, including
Lasso [26], Lasso-overlapped-group [11], linear SVM classifier [5], PCA with linear SVM classifier and SLFA with linear SVM classifier. Lasso-overlapped-group, which is a logistic regression
approach with the graph-guided sparsity enforced, uses a known biological network as the graphical
(overlapped group) regularization on the lasso regression. The other methods, including SLFA, do
not use this extra supervised information. We run 10-fold cross validation and use the averaged error
rate to indicate the predictive performance of different methods. The test is repeated 50 times and
each time all methods use the same split of training and validation sets.
The averaged cross-validation error rate is shown in Table 1. We can observe that SLFA (K = 100)
has lower error rates than other methods, including Lasso, SVM and PCA. Compared to the
method of Lasso-overlapped-group [11] which constructs the regularization from external information (42, 594 known edges as prior knowledge), our method based on SLFA performs better, even
though it does not utilize any extra evidence. This is a strong evidence which indicates that SLFA
can extract deeper structural information hidden in the data. Indeed, genes naturally act in the form
of functional modules (gene groups) to carry out specific functions. Gene groups that usually correspond to biological processes or pathways, exhibit diverse pairwise dependency relationships among
each other. SLFA discovers these relationships while learning the latent representation of each data
sample at the same time. That is why its learned lower-dimensional representation captures more
fundamental and strong signals, and achieves the state-of-art classification performance. The learned
structural information and latent gene groups also get confirmed by the biological function analysis
in supplementary document.
5
Conclusion
In this paper we have introduced a novel structured latent factor model that simultaneously learns
latent factors and their pairwise relationships. The model is formulated to represent data drawn
from the general exponential family of distributions. The learned sparse interaction between latent
factors is crucial for understanding complex data sets and to visually analyze them. SLFA model is
also a hierarchical extension of Sparse Gaussian Graphical Model by generalizing the application
of precision matrix from the original variable space to the latent factor space and optimizing the
bases together with the precision matrix simultaneously. We have also provided an efficient online
learning algorithm that can scale SLFA training to large-scale datasets and showed that SLFA not
only can predict the true basis and structured relationshop between bases, but also it can achieve
state-of-the-art results in challenging biological classification task.
8
References
[1] Bertsekas, D.: Nonlinear programming. Athena Scientific Belmont, MA (1999)
[2] Blei, D., Lafferty, J.: Correlated topic models. Advances in Neural Information Processing Systems
(2006)
[3] Chandrasekaran, V., Parrilo, P., Willsky, A.: Latent variable graphical model selection via convex optimization. Arxiv preprint arXiv:1008.1290 (2010)
[4] Collins, M., Dasgupta, S., Schapire, R.: A generalization of principal component analysis to the exponential family. Advances in neural information processing systems (2002)
[5] Fan, R., Chang, K., Hsieh, C., Wang, X., Lin, C.: Liblinear: A library for large linear classification. JMLR
(2008)
[6] Goldfarb, D., Ma, S., Scheinberg, K.: Fast alternating linearization methods for minimizing the sum of
two convex functions. Arxiv preprint arXiv:0912.4571 (2009)
[7] Gregor, K., Szlam, A., LeCun, Y.: Structured sparse coding via lateral inhibition. Advances in Neural
Information Processing Systems 24 (2011)
[8] Hinton, G., Osindero, S., Bao, K.: Learning causally linked markov random fields. In: AI & Statistics
(2005)
[9] Hsieh, C., Sustik, M., Ravikumar, P., Dhillon, I.: Sparse inverse covariance matrix estimation using
quadratic approximation. Advances in Neural Information Processing Systems (NIPS) 24 (2011)
[10] Hyv?arinen, A., Hurri, J., Hoyer, P.: Independent component analysis. Natural Image Statistics (2009)
[11] Jacob, L., Obozinski, G., Vert, J.: Group lasso with overlap and graph lasso. Proceedings of the 26th
Annual International Conference on Machine Learning (2009)
[12] Jenatton, R., Mairal, J., Obozinski, G., Bach, F.: Proximal methods for sparse hierarchical dictionary
learning. Proceedings of the International Conference on Machine Learning (2010)
[13] Karklin, Y., Lewicki, M.S.: Emergence of complex cell properties by learning to generalize in natural
scenes. Nature (2009)
[14] Lee, D., Seung, H.: Learning the parts of objects by non-negative matrix factorization. Nature (1999)
[15] Lee, H., Battle, A., Raina, R., Ng, A.: Efficient sparse coding algorithms. Advances in neural information
processing systems (2007)
[16] Lee, H., Raina, R., Teichman, A., Ng, A.: Exponential family sparse coding with applications to selftaught learning. Proceedings of the 21st international jont conference on Artifical intelligence (2009)
[17] Lyu, S., Simoncelli, E.: Nonlinear extraction of independent components of natural images using radial
gaussianization. Neural computation (2009)
[18] Murray, I., Adams, R.: Slice sampling covariance hyperparameters of latent gaussian models. Arxiv
preprint arXiv:1006.0868 (2010)
[19] Olshausen, B., et al.: Emergence of simple-cell receptive field properties by learning a sparse code for
natural images. Nature (1996)
[20] Rendl, F., Rinaldi, G., Wiegele, A.: Solving Max-Cut to optimality by intersecting semidefinite and
polyhedral relaxations. Math. Programming 121(2), 307 (2010)
[21] Schmidt, M.: Graphical model structure learning with l1-regularization. Ph.D. thesis, UNIVERSITY OF
BRITISH COLUMBIA (2010)
[22] Schmidt, M., Van Den Berg, E., Friedlander, M., Murphy, K.: Optimizing costly functions with simple
constraints: A limited-memory projected quasi-newton algorithm. In: AI & Statistics (2009)
[23] Silva, R., Scheine, R., Glymour, C., Spirtes, P.: Learning the structure of linear latent variable models.
The Journal of Machine Learning Research 7, 191?246 (2006)
[24] Stegle, O., Lippert, C., Mooij, J., Lawrence, N., Borgwardt, K.: Efficient inference in matrix-variate
gaussian models with iid observation noise. Advances in Neural Information Processing Systems (2011)
[25] Teh, Y., Seeger, M., Jordan, M.: Semiparametric latent factor models. In: AI & Statistics (2005)
[26] Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society.
Series B (Methodological) (1996)
[27] Wainwright, M., Simoncelli, E.: Scale mixtures of gaussians and the statistics of natural images. Advances
in neural information processing systems (2000)
[28] Yuan, M., Lin, Y.: Model selection and estimation in the gaussian graphical model. Biometrika (2007)
9
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4,018 | 4,637 | A Spectral Algorithm for Latent Dirichlet Allocation
Anima Anandkumar
University of California
Irvine, CA
[email protected]
Dean P. Foster
University of Pennsylvania
Philadelphia, PA
[email protected]
Sham M. Kakade
Microsoft Research
Cambridge, MA
[email protected]
Daniel Hsu
Microsoft Research
Cambridge, MA
[email protected]
Yi-Kai Liu
National Institute of Standards and Technology?
Gaithersburg, MD
[email protected]
Abstract
Topic modeling is a generalization of clustering that posits that observations
(words in a document) are generated by multiple latent factors (topics), as opposed to just one. This increased representational power comes at the cost of
a more challenging unsupervised learning problem of estimating the topic-word
distributions when only words are observed, and the topics are hidden.
This work provides a simple and efficient learning procedure that is guaranteed to
recover the parameters for a wide class of topic models, including Latent Dirichlet
Allocation (LDA). For LDA, the procedure correctly recovers both the topic-word
distributions and the parameters of the Dirichlet prior over the topic mixtures,
using only trigram statistics (i.e., third order moments, which may be estimated
with documents containing just three words). The method, called Excess Correlation Analysis, is based on a spectral decomposition of low-order moments via
two singular value decompositions (SVDs). Moreover, the algorithm is scalable,
since the SVDs are carried out only on k ? k matrices, where k is the number
of latent factors (topics) and is typically much smaller than the dimension of the
observation (word) space.
1
Introduction
Topic models use latent variables to explain the observed (co-)occurrences of words in documents.
They posit that each document is associated with a (possibly sparse) mixture of active topics, and
that each word in the document is accounted for (in fact, generated) by one of these active topics.
In Latent Dirichlet Allocation (LDA) [1], a Dirichlet prior gives the distribution of active topics
in documents. LDA and related models possess a rich representational power because they allow
for documents to be comprised of words from several topics, rather than just a single topic. This
increased representational power comes at the cost of a more challenging unsupervised estimation
problem, when only the words are observed and the corresponding topics are hidden.
In practice, the most common unsupervised estimation procedures for topic models are based on
finding maximum likelihood estimates, through either local search or sampling based methods, e.g.,
Expectation-Maximization [2], Gibbs sampling [3], and variational approaches [4]. Another body
of tools is based on matrix factorization [5, 6]. For document modeling, a typical goal is to form
a sparse decomposition of a term by document matrix (which represents the word counts in each
?
Contributions to this work by NIST, an agency of the US government, are not subject to copyright laws.
1
document) into two parts: one which specifies the active topics in each document and the other
which specifies the distributions of words under each topic.
This work provides an alternative approach to parameter recovery based on the method of moments [7], which attempts to match the observed moments with those posited by the model. Our
approach does this efficiently through a particular decomposition of the low-order observable moments, which can be extracted using singular value decompositions (SVDs). This method is simple
and efficient to implement, and is guaranteed to recover the parameters of a wide class of topic
models, including the LDA model. We exploit exchangeability of the observed variables and, more
generally, the availability of multiple views drawn independently from the same hidden component.
1.1
Summary of contributions
We present an approach called Excess Correlation Analysis (ECA) based on the low-order (cross)
moments of observed variables. These observed variables are assumed to be exchangeable (and,
more generally, drawn from a multi-view model). ECA differs from Principal Component Analysis
and Canonical Correlation Analysis in that it is based on two singular value decompositions: the
first SVD whitens the data (based on the correlation between two observed variables) and the second
SVD uses higher-order moments (third- or fourth-order moments) to find directions which exhibit
non-Gaussianity, i.e., directions where the moments are in excess of those suggested by a Gaussian
distribution. The SVDs are performed only on k?k matrices, where k is the number of latent factors;
note that the number of latent factors (topics) k is typically much smaller than the dimension of the
observed space d (number of words).
The method is applicable to a wide class of latent variable models including exchangeable and multiview models. We first consider the class of exchangeable variables with independent latent factors.
We show that the (exact) low-order moments permit a decomposition that recovers the parameters for
model class, and that this decomposition can be computed using two SVD computations. We then
consider LDA and show that the same decomposition of a modified third-order moment correctly
recovers both the probability distribution of words under each topic, as well as the parameters of the
Dirichlet prior. We note that in order to estimate third-order moments in the LDA model, it suffices
for each document to contain at least three words.
While the methods described assume exact moments, it is straightforward to write down the analogue ?plug-in? estimators based on empirical moments from sampled data. We provide a simple
sample complexity analysis that shows that estimating the third-order moments is not as difficult as
it might na??vely seem since we only need a k ? k matrix to be accurate.
Finally, we remark that the moment decomposition can also be obtained using other techniques,
including tensor decomposition methods and simultaneous matrix diagonalization methods. Some
preliminary experiments illustrating the efficacy of one such method is given in the appendix.
Omitted proofs, and additional results and discussion are provided in the full version of the paper [8].
1.2
Related work
Under the assumption that a single active topic occurs in each document, the work of [9] provides the
first provable guarantees for recovering the topic distributions (i.e., the distribution of words under
each topic), albeit with a rather stringent separation condition (where the words in each topic are
essentially non-overlapping). Understanding what separation conditions permit efficient learning
is a natural question; in the clustering literature, a line of work has focussed on understanding the
relationship between the separation of the mixture components and the complexity of learning. For
clustering, the first provable learnability result [10] was under a rather strong separation condition;
subsequent results relaxed [11?18] or removed these conditions [19?21]; roughly speaking, learning
under a weaker separation condition is more challenging, both computationally and statistically.
For the topic modeling problem in which only a single active topic is present per document, [22]
provides an algorithm for learning topics with no separation requirement, but under a certain full
rank assumption on the topic probability matrix.
For the case of LDA (where each document may be about multiple topics), the recent work of [23]
provides the first provable result under a natural separation condition. The condition requires that
2
each topic be associated with ?anchor words? that only occur in documents about that topic. This
is a significantly milder assumption than the one in [9]. Under this assumption, [23] provide the
first provably correct algorithm for learning the topic distributions. Their work also justifies the
use of non-negative matrix (NMF) as a provable procedure for this problem (the original motivation
for NMF was as a topic modeling algorithm, though, prior to this work, formal guarantees as such
were rather limited). Furthermore, [23] provides results for certain correlated topic models. Our
approach makes further progress on this problem by relaxing the need for this separation condition
and establishing a much simpler procedure for parameter estimation.
The underlying approach we take is a certain diagonalization technique of the observed moments.
We know of at least three different settings which use this idea for parameter estimation.
The work in [24] uses eigenvector methods for parameter estimation in discrete Markov models
involving multinomial distributions. The idea has been extended to other discrete mixture models
such as discrete hidden Markov models (HMMs) and mixture models with a single active topic
in each document (see [22, 25, 26]). For such single topic models, the work in [22] demonstrates
the generality of the eigenvector method and the irrelevance of the noise model for the observations,
making it applicable to both discrete models like HMMs as well as certain Gaussian mixture models.
Another set of related techniques is the body of algebraic methods used for the problem of blind
source separation [27]. These approaches are tailored for independent source separation with additive noise (usually Gaussian) [28]. Much of the literature focuses on understanding the effects of
measurement noise, which often requires more sophisticated algebraic tools (typically, knowledge
of noise statistics or the availability of multiple views of the latent factors is not assumed). These
algebraic ideas are also used by [29, 30] for learning a linear transformation (in a noiseless setting)
and provides a different provably correct algorithm, based on a certain ascent algorithm (rather than
joint diagonalization approach, as in [27]), and a provably correct algorithm for the noisy case was
recently obtained by [31].
The underlying insight exploited by our method is the presence of exchangeable (or multi-view)
variables (e.g., multiple words in a document), which are drawn independently conditioned on the
same hidden state. This allows us to exploit ideas both from [24] and from [27]. In particular, we
show that the ?topic? modeling problem exhibits a rather simple algebraic solution, where only two
SVDs suffice for parameter estimation.
Furthermore, the exchangeability assumption permits us to have an arbitrary noise model (rather
than an additive Gaussian noise, which is not appropriate for multinomial and other discrete distributions). A key technical contribution is that we show how the basic diagonalization approach can
be adapted for Dirichlet models, through a rather careful construction. This construction bridges the
gap between the single topic models (as in [22, 24]) and the independent latent factors model.
More generally, the multi-view approach has been exploited in previous works for semi-supervised
learning and for learning mixtures of well-separated distributions (e.g., [16,18,32,33]). These previous works essentially use variants of canonical correlation analysis [34] between the two views. This
work follows [22] in showing that having a third view of the data permits rather simple estimation
procedures with guaranteed parameter recovery.
2
The independent latent factors and LDA models
Let h = (h1 , h2 , . . . , hk ) ? Rk be a random vector specifying the latent factors (i.e., the hidden
state) of a model, where hi is the value of the i-th factor. Consider a sequence of exchangeable random vectors x1 , x2 , x3 , x4 , . . . ? Rd , which we take to be the observed variables. Assume throughout that d ? k; that x1 , x2 , x3 , x4 , . . . ? Rd are conditionally independent given h. Furthermore,
assume there exists a matrix O ? Rd?k such that
E[xv |h] = Oh
for each v ? {1, 2, 3, . . . }. Throughout, we assume the following condition.
Condition 2.1. O has full column rank.
This is a mild assumption, which allows for identifiability of the columns of O. The goal is to
estimate the matrix O, sometimes referred to as the topic matrix. Note that at this stage, we have not
made any assumptions on the noise model; it need not be additive nor even independent of h.
3
2.1
Independent latent factors model
In the independent latent factors model, we assume h has a product distribution, i.e., h1 , h2 , . . . , hk
are independent. Two important examples of this setting are as follows.
Multiple mixtures of Gaussians: Suppose xv = Oh + ?, where ? is Gaussian noise and h is a
binary vector (under a product distribution). Here, the i-th column Oi can be considered to be the
mean of the i-th Gaussian component. This generalizes the classic mixture of k Gaussians, as the
model now permits any number of Gaussians to be responsible for generating the hidden state (i.e.,
h is permitted to be any of the 2k vectors on the hypercube, while in the classic mixture problem,
only one component is responsible). We may also allow ? to be heteroskedastic (i.e., the noise may
depend on h, provided the linearity assumption E[xv |h] = Oh holds).
Multiple mixtures of Poissons: Suppose [Oh]j specifies the Poisson rate of counts for [xv ]j . For
example, xv could be a vector of word counts in the v-th sentence of a document. Here, O would be
a matrix with positive entries, and hi would scale the rate at which topic i generates words in a sentence (as specified by the i-th column of O). The linearity assumption is satisfied as E[xv |h] = Oh
(note the noise is not additive in this case). Here, multiple topics may be responsible for generating
the words in each sentence. This model provides a natural variant of LDA, where the distribution
over h is a product distribution (while in LDA, h is a probability vector).
2.2
The Dirichlet model
Now suppose the hidden state h is a distribution itself, with a density specified by the Dirichlet
distribution with parameter ? ? Rk>0 (? is a strictly positive real vector). We often think of h as a
distribution over topics. Precisely, the density of h ? ?k?1 (where the probability simplex ?k?1
denotes the set of possible distributions over k outcomes) is specified by:
k
1 Y ?i ?1
p? (h) :=
h
Z(?) i=1 i
Qk
?(? )
i
where Z(?) := i=1
and ?0 := ?1 + ?2 + ? ? ? + ?k . Intuitively, ?0 (the sum of the ?pseudo?(?0 )
counts?) characterizes the concentration of the distribution. As ?0 ? 0, the distribution degenerates
to one over pure topics (i.e., the limiting density is one in which, almost surely, exactly one coordinate of h is 1, and the rest are 0).
Latent Dirichlet Allocation: LDA makes the further assumption that each random variable
x1 , x2 , x3 , . . . takes on discrete values out of d outcomes (e.g., xv represents what the v-th word
in a document is, so d represents the number of words in the language). The i-th column Oi of
O is a probability vector representing the distribution over words for the i-th topic. The sampling
process for a document is as follows. First, the topic mixture h is drawn from the Dirichlet distribution. Then, the v-th word in the document (for v = 1, 2, . . . ) is generated by: (i) drawing
t ? [k] := {1, 2, . . . k} according to the discrete distribution specified by h, then (ii) drawing xv
according to the discrete distribution specified by Ot (the t-th column of O). Note that xv is independent of h given t. For this model to fit in our setting, we use the ?one-hot? encoding for xv
from [22]: xv ? {0, 1}d with [xv ]j = 1 iff the v-th word in the document is the j-th word in the
vocabulary. Observe that
E[xv |h] =
k
X
Pr[t = i|h] ? E[xv |t = i, h] =
i=1
k
X
hi ? Oi = Oh
i=1
as required. Again, note that the noise model is not additive.
3
Excess Correlation Analysis (ECA)
We now present efficient algorithms for exactly recovering O from low-order moments of the observed variables. The algorithm is based on two singular value decompositions: the first SVD
whitens the data (based on the correlation between two variables), and the second SVD is carried
4
Algorithm 1 ECA, with skewed factors
Input: vector ? ? Rk ; the moments Pairs and Triples.
1. Dimensionality reduction: Find a matrix U ? Rd?k such that
range(U ) = range(Pairs).
(See Remark 1 for a fast procedure.)
2. Whiten: Find V ? Rk?k so V > (U > Pairs U )V is the k ? k identity matrix. Set:
W = U V.
3. SVD: Let ? be the set of left singular vectors of
W > Triples(W ?)W
corresponding to non-repeated singular values (i.e., singluar values with multiplicity
one).
4. Reconstruct: Return the set
b := {(W + )> ? : ? ? ?}.
O
out on higher-order moments. We start with the case of independent factors, as these algorithms
make the basic diagonalization approach clear.
Throughout, we use A+ to denote the Moore-Penrose pseudo-inverse.
3.1
Independent and skewed latent factors
Define the following moments:
? := E[x1 ], Pairs := E[(x1 ? ?) ? (x2 ? ?)], Triples := E[(x1 ? ?) ? (x2 ? ?) ? (x3 ? ?)]
(here ? denotes the tensor product, so ? ? Rd , Pairs ? Rd?d , and Triples ? Rd?d?d ). It is
convenient to project Triples to matrices as follows:
Triples(?) := E[(x1 ? ?)(x2 ? ?)> h?, x3 ? ?i].
Roughly speaking, we can think of Triples(?) as a re-weighting of a cross covariance (by h?, x3 ?
?i).
Note that the matrix O is only identifiable up to permutation and scaling of columns. To see the
latter, observe the distribution of any xv is unaltered if, for any i ? [k], we multiply the i-th column
of O by a scalar c 6= 0 and divide the variable hi by the same scalar c. Without further assumptions,
we can only hope to recover a certain canonical form of O, defined as follows.
Definition 1 (Canonical form). We say O is in a canonical form (relative to h) if, for each i ? [k],
?i2 := E[(hi ? E[hi ])2 ] = 1.
The transformation O ? O diag(?1 , ?2 , . . . , ?k ) (and a rescaling of h) places O in canonical form
relative to h, and the distribution over x1 , x2 , x3 , . . . is unaltered. In canonical form, O is unique up
to a signed column permutation.
Let ?i,p := E[(hi ? E[hi ])p ] denote the p-th central moment of hi , so the variance and skewness of
hi are given by ?i2 := ?i,2 and ?i := ?i,3 /?i3 . The first result considers the case when the skewness
is non-zero.
Theorem 3.1 (Independent and skewed factors). Assume Condition 2.1 and ?i2 > 0 for each i ? [k].
Under the independent latent factor model, the following hold.
? No False Positives: For all ? ? Rk , Algorithm 1 returns a subset of the columns of O, in
canonical form up to sign.
? Exact Recovery: Assume ?i 6= 0 for each i ? [k]. If ? ? Rk is drawn uniformly at random
from the unit sphere S k?1 , then with probability 1, Algorithm 1 returns all columns of O,
in canonical form up to sign.
5
The proof of this theorem relies on the following lemma.
Lemma 3.1 (Independent latent factors moments). Under the independent latent factor model,
k
X
Pairs =
?i2 Oi ? Oi = O diag(?12 , ?22 , . . . , ?k2 )O> ,
i=1
Triples
=
k
X
?i,3 Oi ? Oi ? Oi ,
Triples(?) = O diag(O> ?) diag(?1,3 , ?2,3 , . . . , ?k,3 )O> .
i=1
Proof. The model assumption E[xv |h] = Oh implies ? = OE[h]. Therefore E[(xv ? ?)|h] =
O(h ? E[h]). Using the conditional independence of x1 and x2 given h, and the fact that h has a
product distribution,
Pairs = E[(x1 ? ?) ? (x2 ? ?)] = E[E[(x1 ? ?)|h] ? E[(x2 ? ?)|h]]
= OE[(h ? E[h]) ? (h ? E[h])]O> = O diag(?12 , ?22 , . . . , ?k2 )O> .
An analogous argument gives the claims for Triples and Triples(?).
Proof of Theorem 3.1. Assume O is in canonical form with respect to h. By Condition 2.1,
U > Pairs U ? Rk?k is full rank and hence positive definite. Thus the whitening step is possible, and M := W > O is orthogonal. Observe that W > Triples(W ?)W = M DM > , where
D := diag(M > ?) diag(?1 , ?2 , . . . , ?k ). Since M is orthogonal, the above is an eigendecomposition of W > Triples(W ?)W , and hence the set of left singular vectors corresponding to nonrepeated singular values are uniquely defined up to sign. Each such singular vector ? is of the
form si M ei = si W > Oei = si W > Oi for some i ? [k] and si ? {?1}, so (W + )> ? =
si W (W > W )?1 W > Oi = si Oi (because range(W ) = range(U ) = range(O)).
If ? is drawn uniformly at random from S k?1 , then so is M > ?. In this case, almost surely, the
diagonal entries of D are unique (provided that each ?i 6= 0), and hence every singular value of
W > Triples(W ?)W is non-repeated.
Remark 1 (Finding range(Pairs) efficiently). Let ? ? Rd?k be a random matrix with entries sampled independently from the standard normal distribution, and set U := Pairs ?. Then with probability 1, range(U ) = range(Pairs).
It is easy to extend Algorithm 1 to kurtotic sources where ?i := (?i,4 /?i4 ) ? 3 6= 0 for each i ? [k],
simply by using fourth-order cumulants in places of Triples(?). The details are given in the full
version of the paper.
3.2
Latent Dirichlet Allocation
Now we turn to LDA where h has a Dirichlet density. Even though the distribution on h is propor1 ?1 ?2 ?1
k ?1
h2
? ? ? h?
tional to the product h?
, the hi are not independent because h is constrained to
1
k
live in the simplex. These mild dependencies suggest using a certain correction of the moments with
ECA.
We assume ?0 is known. Knowledge of ?0 = ?1 + ?2 + ? ? ? + ?k is significantly weaker than
having full knowledge of the entire parameter vector ? = (?1 , ?2 , . . . , ?k ). A common practice
is to specify the entire parameter vector ? in a homogeneous manner, with each component being
identical (see [35]). Here, we need only specify the sum, which allows for arbitrary inhomogeneity
in the prior.
Denote the mean and a modified second moment by
? = E[x1 ],
Pairs?0 := E[x1 x>
2]?
?0
??> ,
?0 + 1
and a modified third moment as
Triples?0 (?) := E[x1 x>
2 h?, x3 i] ?
?0
>
>
>
>
E[x1 x>
2 ]?? + ?? E[x1 x2 ] + h?, ?iE[x1 x2 ]
?0 + 2
2?02
+
h?, ?i??> .
(?0 + 2)(?0 + 1)
6
Algorithm 2 ECA for Latent Dirichlet Allocation
Input: vector ? ? Rk ; the modified moments Pairs?0 and Triples?0 .
1?3. Execute steps 1?3 of Algorithm 1 with Pairs?0 and Triples?0 in place of Pairs and
Triples.
4. Reconstruct and normalize: Return the set
(W + )> ?
b
O :=
:???
~1> (W + )> ?
where ~1 ? Rd is a vector of all ones.
Remark 2 (Central vs. non-central moments). In the limit as ?0 ? 0, the Dirichlet model degenerates so that, with probability 1, only one coordinate of h equals 1 and the rest are 0 (i.e., each
document is about just one topic). In this case, the modified moments tend to the raw (cross) moments:
lim Pairs?0 = E[x1 ? x2 ],
lim Triples?0 = E[x1 ? x2 ? x3 ].
?0 ?0
?0 ?0
Note that the one-hot encoding of words in xv implies that
X
X
E[x1 ?x2 ] =
Pr[x1 = ei , x2 = ej ] ei ?ej =
Pr[1st word = i, 2nd word = j] ei ?ej ,
1?i,j?d
1?i,j?d
(and a similar expression holds for E[x1 ? x2 ? x3 ]), so these raw moments in the limit ?0 ? 0 are
precisely the joint probabilitiy tables of words across all documents.
At the other extreme ?0 ? ?, the modified moments tend to the central moments:
lim Pairs?0 = E[(x1 ? ?) ? (x2 ? ?)],
?0 ??
lim Triples?0 = E[(x1 ? ?) ? (x2 ? ?) ? (x3 ? ?)]
?0 ??
>
>
(to see this, expand the central moment and use exchangeability: E[x1 x>
2 ] = E[x2 x3 ] = E[x1 x3 ]).
Our main result here shows that ECA recovers both the topic matrix O, up to a permutation of the
columns (where each column represents a probability distribution over words for a given topic) and
the parameter vector ?, using only knowledge of ?0 (which, as discussed earlier, is a significantly
less restrictive assumption than tuning the entire parameter vector).
Theorem 3.2 (Latent Dirichlet Allocation). Assume Condition 2.1 holds. Under the LDA model,
the following hold.
? No False Positives: For all ? ? Rk , Algorithm 2 returns a subset of the columns of O.
? Topic Recovery: If ? ? Rk is drawn uniformly at random from the unit sphere S k?1 , then
with probability 1, Algorithm 2 returns all columns of O.
? Parameter Recovery:
The Dirichlet parameter ? satisfies ?
+
+ >~
1)O Pairs?0 (O ) 1, where ~1 ? Rk is a vector of all ones.
=
?0 (?0 +
The proof relies on the following lemma.
Lemma 3.2 (LDA moments). Under the LDA model,
1
Pairs?0 =
O diag(?)O> ,
(?0 + 1)?0
2
Triples?0 (?) =
O diag(O> ?) diag(?)O> .
(?0 + 2)(?0 + 1)?0
The proof of Lemma 3.2 is similar to that of Lemma 3.1, except here we must use the specific
properties of the Dirichlet distribution to show that the corrections to the raw (cross) moments have
the desired effect.
? := ?
Proof of Theorem 3.2. Note that with the rescaling O
?
?
?
1
O diag( ?1 , ?2 , . . . , ?k ),
(?0 +1)?0
?O
? > . This is akin to O
? being in canonical form as per the skewed factor
we have that Pairs?0 = O
7
model of Theorem 3.1. Now the proof of the first two claims is the same as that of Theorem 3.1; the
only modification is that we simply normalize the output of Algorithm 1. Finally, observe that claim
for estimating ? holds due to the functional form of Pairs?0 .
Remark 3 (Limiting behaviors). ECA seamlessly interpolates between the single topic model (?0 ?
0) of [22] and the skewness-based ECA, Algorithm 1 (?0 ? ?).
4
4.1
Discussion
Sample complexity
It is straightforward to derive a ?plug-in? variant of Algorithm 2 based on empirical moments rather
than exact population moments. The empirical moments are formed using the word co-occurrence
statistics for documents in a corpus. The following theorem shows that the empirical version of ECA
returns accurate estimates of the topics. The details and proof are left to the full version of the paper.
Theorem 4.1 (Sample complexity for LDA). There exist universal constants C1 , C2 > 0 such that
the following hold. Let pmin = mini ??0i and let ?k (O) denote the smallest (non-zero) singular
value of O. Suppose that we obtain N ? C1 ? ((?0 + 1)/(pmin ?k (O)2 ))2 independent samples
\? .
[ ?0 and Triples
of x1 , x2 , x3 in the LDA model, which are used to form empirical moments Pairs
0
?
?
?
With high probability, the plug-in variant of Algorithm 2 returns a set {O1 , O2 , . . . Ok } such that,
for some permutation ? of [k],
? ?(i) k2 ? C2 ?
kOi ? O
4.2
(?0 + 1)2 k 3
? ,
p2min ?k (O)3 N
?i ? [k].
Alternative decomposition methods
Algorithm 1 is a theoretically efficient and simple-to-state method for obtaining the desired decomPk
position of the tensor Triples = i=1 ?i,3 Oi ? Oi ? Oi (a similar tensor form for Triples?0 in
the case of LDA can also be given). However, in practice the method is not particularly stable, due
to the use of internal randomization to guarantee strict separation of singular values. It should be
noted that there are other methods in the literature for obtaining these decompositions, for instance,
methods based on simultaneous diagonalizations of matrices [36] as well as direct tensor decomposition methods [37]; and that these methods can be significantly more stable than Algorithm 1.
In particular, very recent work in [37] shows that the structure revealed in Lemmas 3.1 and 3.2 can
be exploited to derive very efficient estimation algorithms for all the models considered here (and
others) based on a tensor power iteration. We have used a simplified version of this tensor power iteration in preliminary experiments for estimating topic models, and found the results (Appendix A)
to be very encouraging, especially due to the speed and robustness of the algorithm.
Acknowledgements
We thank Kamalika Chaudhuri, Adam Kalai, Percy Liang, Chris Meek, David Sontag, and Tong
Zhang for many invaluable insights. We also give warm thanks to Rong Ge for sharing preliminary
results (in [23]) and early insights into this problem with us. Part of this work was completed while
all authors were at Microsoft Research New England. AA is supported in part by the NSF Award
CCF-1219234, AFOSR Award FA9550-10-1-0310 and the ARO Award W911NF-12-1-0404.
References
[1] David M. Blei, Andrew Ng, and Michael Jordan. Latent dirichlet allocation. JMLR, 3:993?1022, 2003.
[2] R. A. Redner and H. F. Walker. Mixture densities, maximum likelihood and the EM algorithm. SIAM
Review, 26(2):195?239, 1984.
[3] A. Asuncion, P. Smyth, M. Welling, D. Newman, I. Porteous, and S. Triglia. Distributed gibbs sampling for latent variable models. In Scaling Up Machine Learning: Parallel and Distributed Approaches.
Cambridge Univ Pr, 2011.
[4] M.D. Hoffman, D.M. Blei, and F. Bach. Online learning for latent dirichlet allocation. In NIPS, 2010.
8
[5] Thomas Hofmann. Probilistic latent semantic analysis. In UAI, 1999.
[6] Daniel D. Lee and H. Sebastian Seung. Learning the parts of objects by nonnegative matrix factorization.
Nature, 401, 1999.
[7] K. Pearson. Contributions to the mathematical theory of evolution. Phil. Trans. of the Royal Society,
London, A., 1894.
[8] A. Anandkumar, D. P. Foster, D. Hsu, S. M. Kakade, and Y.-K. Liu. Two svds suffice: spectral decompositions for probabilistic topic models and latent dirichlet allocation, 2012. arXiv:1204.6703.
[9] Christos H. Papadimitriou, Prabhakar Raghavan, Hisao Tamaki, and Santosh Vempala. Latent semantic
indexing: A probabilistic analysis. J. Comput. Syst. Sci., 61(2), 2000.
[10] S. Dasgupta. Learning mixutres of Gaussians. In FOCS, 1999.
[11] S. Dasgupta and L. Schulman. A two-round variant of em for gaussian mixtures. In UAI, 2000.
[12] S. Arora and R. Kannan. Learning mixtures of arbitrary Gaussians. In STOC, 2001.
[13] S. Vempala and G. Wang. A spectral algorithm for learning mixtures of distributions. In FOCS, 2002.
[14] R. Kannan, H. Salmasian, and S. Vempala. The spectral method for general mixture models. In COLT,
2005.
[15] D. Achlioptas and F. McSherry. On spectral learning of mixtures of distributions. In COLT, 2005.
[16] K. Chaudhuri and S. Rao. Learning mixtures of product distributions using correlations and independence.
In COLT, 2008.
[17] S. C. Brubaker and S. Vempala. Isotropic PCA and affine-invariant clustering. In FOCS, 2008.
[18] K. Chaudhuri, S. M. Kakade, K. Livescu, and K. Sridharan. Multi-view clustering via canonical correlation analysis. In ICML, 2009.
[19] A. T. Kalai, A. Moitra, and G. Valiant. Efficiently learning mixtures of two Gaussians. In STOC, 2010.
[20] M. Belkin and K. Sinha. Polynomial learning of distribution families. In FOCS, 2010.
[21] A. Moitra and G. Valiant. Settling the polynomial learnability of mixtures of Gaussians. In FOCS, 2010.
[22] A. Anandkumar, D. Hsu, and S. M. Kakade. A method of moments for mixture models and hidden
markov models. In COLT, 2012.
[23] S. Arora, , R. Ge, and A. Moitra. Learning topic models ? going beyond svd. In FOCS, 2012.
[24] J. T. Chang. Full reconstruction of Markov models on evolutionary trees: Identifiability and consistency.
Mathematical Biosciences, 137:51?73, 1996.
[25] E. Mossel and S. Roch. Learning nonsingular phylogenies and hidden Markov models. Annals of Applied
Probability, 16(2):583?614, 2006.
[26] D. Hsu, S. M. Kakade, and T. Zhang. A spectral algorithm for learning hidden Markov models. In COLT,
2009.
[27] Jean-Franois Cardoso and Pierre Comon. Independent component analysis, a survey of some algebraic
methods. In IEEE International Symposium on Circuits and Systems, pages 93?96, 1996.
[28] P. Comon and C. Jutten. Handbook of Blind Source Separation: Independent Component Analysis and
Applications. Academic Press. Elsevier, 2010.
[29] Alan M. Frieze, Mark Jerrum, and Ravi Kannan. Learning linear transformations. In FOCS, 1996.
[30] P. Q. Nguyen and O. Regev. Learning a parallelepiped: Cryptanalysis of GGH and NTRU signatures.
Journal of Cryptology, 22(2):139?160, 2009.
[31] S. Arora, R. Ge, A. Moitra, and S. Sachdeva. Provable ICA with unknown Gaussian noise, and implications for Gaussian mixtures and autoencoders. In NIPS, 2012.
[32] R. Ando and T. Zhang. Two-view feature generation model for semi-supervised learning. In ICML, 2007.
[33] Sham M. Kakade and Dean P. Foster. Multi-view regression via canonical correlation analysis. In COLT,
2007.
[34] H. Hotelling. The most predictable criterion. Journal of Educational Psychology, 26(2):139?142, 1935.
[35] Mark Steyvers and Tom Griffiths. Probabilistic topic models. In T. Landauer, D. Mcnamara, S. Dennis,
and W. Kintsch, editors, Latent Semantic Analysis: A Road to Meaning. Laurence Erlbaum, 2006.
[36] A. Bunse-Gerstner, R. Byers, and V. Mehrmann. Numerical methods for simultaneous diagonalization.
SIAM Journal on Matrix Analysis and Applications, 14(4):927?949, 1993.
[37] A. Anandkumar, R. Ge, D. Hsu, S. M. Kakade, and T. Telgarsky. Tensor decompositions for learning
latent variable models, 2012. arXiv:1210.7559.
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4,019 | 4,638 | Relax and Randomize: From Value to Algorithms
Alexander Rakhlin
University of Pennsylvania
Ohad Shamir
Microsoft Research
Karthik Sridharan
University of Pennsylvania
Abstract
We show a principled way of deriving online learning algorithms from a minimax
analysis. Various upper bounds on the minimax value, previously thought to be
non-constructive, are shown to yield algorithms. This allows us to seamlessly recover known methods and to derive new ones, also capturing such ?unorthodox?
methods as Follow the Perturbed Leader and the R2 forecaster. Understanding
the inherent complexity of the learning problem thus leads to the development of
algorithms. To illustrate our approach, we present several new algorithms, including a family of randomized methods that use the idea of a ?random playout?. New
versions of the Follow-the-Perturbed-Leader algorithms are presented, as well as
methods based on the Littlestone?s dimension, efficient methods for matrix completion with trace norm, and algorithms for the problems of transductive learning
and prediction with static experts.
1
Introduction
This paper studies the online learning framework, where the goal of the player is to incur small
regret while observing a sequence of data on which we place no distributional assumptions. Within
this framework, many algorithms have been developed over the past two decades [6]. More recently,
a non-algorithmic minimax approach has been developed to study the inherent complexities of sequential problems [2, 1, 15, 20]. It was shown that a theory in parallel to Statistical Learning can be
developed, with random averages, combinatorial parameters, covering numbers, and other measures
of complexity. Just as the classical learning theory is concerned with the study of the supremum of
empirical or Rademacher process, online learning is concerned with the study of the supremum of
martingale processes. While the tools introduced in [15, 17, 16] provide ways of studying the minimax value, no algorithms have been exhibited to achieve these non-constructive bounds in general.
In this paper, we show that algorithms can, in fact, be extracted from the minimax analysis. This
observation leads to a unifying view of many of the methods known in the literature, and also gives
a general recipe for developing new algorithms. We show that the potential method, which has
been studied in various forms, naturally arises from the study of the minimax value as a certain
relaxation. We further show that the sequential complexity tools introduced in [15] are, in fact,
relaxations and can be used for constructing algorithms that enjoy the corresponding bounds. By
choosing appropriate relaxations, we recover many known methods, improved variants of some
known methods, and new algorithms. One can view our framework as one for converting a nonconstructive proof of an upper bound on the value of the game into an algorithm. Surprisingly,
this allows us to also study such ?unorthodox? methods as Follow the Perturbed Leader [10], and
the recent method of [7] under the same umbrella with others. We show that the idea of a random
playout has a solid theoretical basis, and that Follow the Perturbed Leader algorithm is an example
of such a method. Based on these developments, we exhibit an efficient method for the trace norm
matrix completion problem, novel Follow the Perturbed Leader algorithms, and efficient methods
for the problems of online transductive learning. The framework of this paper gives a recipe for
developing algorithms. Throughout the paper, we stress that the notion of a relaxation, introduced
below, is not appearing out of thin air but rather as an upper bound on the sequential Rademacher
complexity. The understanding of inherent complexity thus leads to the development of algorithms.
1
Let us introduce some notation. The sequence x1 , . . . , xt is often denoted by x1?t , and the set of all
distributions on some set A by (A). Unless specified otherwise, ? denotes a vector (?1 , . . . , ?T )
of i.i.d. Rademacher random variables. An X -valued tree x of depth d is defined as a sequence
(x1 , . . . , xd ) of mappings xt ? {?1}t?1 ? X (see [15]). We often write xt (?) instead of xt (?1?t?1 ).
2
Value, The Minimax Algorithm, and Relaxations
Let F be the set of learner?s moves and X the set of moves of Nature. The online protocol dictates
that on every round t = 1, . . . , T the learner and Nature simultaneously choose ft ? F, xt ? X ,
and observe each other?s actions. The learner aims to minimize regret RegT ? ?Tt=1 `(ft , xt ) ?
inf f ?F ?Tt=1 `(f, xt ) where ` ? F ? X ? R is a known loss function. Our aim is to study this
online learning problem at an abstract level without assuming convexity or other such properties of
`, F and X . We do assume, however, that `, F, and X are such that the minimax theorem in the
space of distributions over F and X holds. By studying the abstract setting, we are able to develop
general algorithmic and non-algorithmic ideas that are common across various application areas.
The starting point of our development is the minimax value of the associated online learning game:
VT (F) =
inf
sup E . . .
q1 ? (F ) x1 ?X f1 ?q1
T
inf
sup
T
E ?? `(ft , xt ) ? inf ? `(f, xt )?
qT ? (F ) xT ?X fT ?qT
t=1
f ?F t=1
(1)
where (F) is the set of distributions on F. The minimax formulation immediately gives rise to
the optimal algorithm that solves the minimax expression at every round t and returns
T
T
argmin ?sup E ?`(ft , xt ) + inf sup E . . . inf sup E ? ? `(fi , xi ) ? inf ? `(f, xi )???
q? (F )
xt ft ?q
qt+1 xt+1 ft+1
qT xT fT
i=t+1
f ?F i=1
Henceforth, if the quantification in inf and sup is omitted, it will be understood that xt , ft , pt , qt
range over X , F, (X ), (F), respectively. Moreover, Ext is with respect to pt while Eft is with
respect to qt . We now notice a recursive form for the value of the game. Define for any t ? [T ? 1]
and any given prefix x1 , . . . , xt ? X the conditional value
VT (F?x1 , . . . , xt ) ? inf sup ? E [`(f, x)] + VT (F?x1 , . . . , xt , x)?
q? (F ) x?X
f ?q
with VT (F?x1 , . . . , xT ) ? ? inf f ?F ?Tt=1 `(f, xt ) and VT (F) = VT (F?{}). The minimax optimal
algorithm specifying the mixed strategy of the player can be written succinctly as
qt = argmin sup ? E [`(f, x)] + VT (F?x1 , . . . , xt?1 , x)? .
q? (F ) x?X
f ?q
(2)
Similar recursive formulations have appeared in the literature [8, 13, 19, 3], but now we have
tools to study the conditional value of the game. We will show that various upper bounds on
VT (F?x1 , . . . , xt?1 , x) yield an array of algorithms. In this way, the non-constructive approaches of
[15, 16, 17] to upper bound the value of the game directly translate into algorithms. We note that
the minimax algorithm in (2) can be interpreted as choosing the best decision that takes into account
the present loss and the worst-case future. The first step in our analysis is to appeal to the minimax
theorem and perform the same manipulation as in [1, 15], but only on the conditional values:
T
VT (F?x1 , . . . , xt ) = sup E . . . sup E ? ? inf
pt+1 xt+1
pT xT
T
E `(fi , xi ) ? inf ? `(f, xi )? .
i=t+1 fi ?F xi ?pi
f ?F i=1
(3)
The idea now is to come up with more manageable, yet tight, upper bounds on the conditional value.
A relaxation RelT is a sequence of real-valued functions RelT (F?x1 , . . . , xt ) for each t ? [T ]. We
call a relaxation admissible if for any x1 , . . . , xT ? X ,
RelT (F?x1 , . . . , xt ) ? inf sup ? E [`(f, x)] + RelT (F?x1 , . . . , xt , x)?
q? (F ) x?X
f ?q
(4)
for all t ? [T ? 1], and
RelT (F?x1 , . . . , xT ) ? ? inf f ?F ?Tt=1 `(f, xt ). We use the notation
RelT (F) for RelT (F?{}). A strategy q that minimizes the expression in (4) defines an optimal
Meta-Algorithm for an admissible relaxation RelT :
on round t, compute
qt = arg min sup ? E [`(f, x)] + RelT (F?x1 , . . . , xt?1 , x)? , (5)
q? (F ) x?X
2
f ?q
play ft ? qt and receive xt from the opponent. Importantly, minimization need not be exact: any qt
that satisfies the admissibility condition (4) is a valid method, and we will say that such an algorithm
is admissible with respect to the relaxation RelT .
Proposition 1. Let RelT be an admissible relaxation. For any admissible algorithm with respect
to RelT , (including the Meta-Algorithm), irrespective of the strategy of the adversary,
T
T
? Eft ?qt `(ft , xt ) ? inf ? `(f, xt ) ? RelT (F) ,
(6)
f ?F t=1
t=1
and therefore, E[RegT ] ? RelT (F) . If `(?, ?) is bounded, the Hoeffding-Azuma inequality yields a
high-probability bound on RegT . We also have that VT (F) ? RelT (F) . Further, if for all t ? [T ],
the admissible strategies qt are deterministic, RegT ? RelT (F) .
The reader might recognize RelT as a potential function. It is known that one can derive regret
bounds by coming up with a potential such that the current loss of the player is related to the difference in the potentials at successive steps, and that the regret can be extracted from the final potential.
The origin/recipe for ?good? potential functions has always been a mystery (at least to the authors).
One of the key contributions of this paper is to show that they naturally arise as relaxations on the
conditional value, and the conditional value is itself the tightest possible relaxation. In particular,
for many problems a tight relaxation is achieved through symmetrization applied to the expression
in (3). Define the conditional Sequential Rademacher complexity
T
t
s=t+1
s=1
RT (F?x1 , . . . , xt ) = sup E?t+1?T sup ?2 ? ?s `(f, xs?t (?t+1?s?1 )) ? ? `(f, xs )? .
x
f ?F
(7)
Here the supremum is over all X -valued binary trees of depth T ? t. One may view this complexity
as a partially symmetrized version of the sequential Rademacher complexity
T
RT (F) ? RT (F ? {}) = sup E?1?T sup ?2 ? ?s `(f, xs (?1?s?1 ))?
f ?F
x
(8)
s=1
defined in [15]. We shall refer to the term involving the tree x as the ?future? and the term being
subtracted off in (7) ? as the ?past?. This indeed corresponds to the fact that the quantity is conditioned on the already observed x1 , . . . , xt , while for the future we have the worst possible binary
tree.1
Proposition 2. The conditional Sequential Rademacher complexity is admissible.
We now show that several well-known methods arise as further relaxations on RT .
Exponential Weights [12, 21] Suppose F is a finite class and ?`(f, x)? ? 1. In this case, a (tight)
upper bound on sequential Rademacher complexity leads to the following relaxation:
?
?
t
?
?
?
?
?
?1
RelT (F ?x1 , . . . , xt ) = inf ? log ? exp ?? ? `(f, xi )? + 2 (T ? t)?
?
?f ?F
?
>0 ?
?
?
i=1
?
?
(9)
Proposition 3. The relaxation (9) is admissible and RT (F?x1 , . . . , xt ) ? RelT (F?x1 , . . . , xt ) .
Furthermore, it leads to a parameter-free version of the Exponential Weights algorithm, defined on
round t + 1 by the mixed strategy qt+1 (f ) ? exp ?? ?t ?ts=1 `(f, xs )? with ?t the optimal value in
?
(9). The algorithm?s regret is bounded by RelT (F) ? 2 2T log ?F? .
We point out that the exponential-weights algorithm arising from the relaxation (9) is a parameterfree algorithm. The learning rate ? can be optimized (via 1D line search) at each iteration.
Mirror Descent [4, 14] In the setting of online linear optimization [22], the loss is `(f, x) = ?f, x?.
Suppose F is a unit ball in some Banach space and X is the dual. Let ? ? ? be some (2, C)-smooth
norm on X (in the Euclidean case, C = 2). Using the notation x
?t?1 = ?t?1
s=1 xs , a straightforward
upper bound on sequential Rademacher complexity is the following relaxation:
RelT (F ?x1 , . . . , xt ) =
?
??
xt?1 ?2 + ?? ??
xt?1 ?2 , xt ? + C(T ? t + 1)
(10)
It is cumbersome to write out the indices on xs?t (?t+1?s?1 ) in (7), so we will instead use xs (?) whenever
this doesn?t cause confusion.
1
3
Proposition 4. The relaxation (10) is admissible and RT (F?x1 , . . . , xt ) ? RelT (F?x1 , . . . , xt ) .
?
xt?1 ?2
It yields the update ft = ? ????
with regret bound RelT (F) ? 2CT .
2
??
xt?1 ? +C(T ?t+1)
2
We would like to remark that the traditional mirror descent update can be shown to arise out of an
appropriate relaxation. The algorithms proposed are parameter free as the step size is tuned automatically. We chose the popular methods of Exponential Weights and Mirror Descent for illustration.
In the remainder of the paper, we develop new algorithms to show universality of our approach.
3
Classification
We start by considering the problem of supervised learning, where X is the space of instances and
Y the space of responses (labels). There are two closely related protocols for the online interaction
between the learner and Nature, so let us outline them. The ?proper? version of supervised learning
follows the protocol presented in Section 2: at time t, the learner selects ft ? F, Nature simultaneously selects (xt , yt ) ? X ? Y, and the learner suffers the loss `(f (xt ), yt ). The ?improper? version
is as follows: at time t, Nature chooses xt ? X and presents it to the learner as ?side information?,
the learner then picks y?t ? Y and Nature simultaneously chooses yt ? Y. In the improper version, the
loss of the learner is `(?
yt , yt ), and it is easy to see that we may equivalently state this protocol as the
learner choosing any function ft ? Y X (not necessarily in F), and Nature simultaneously choosing
(xt , yt ). We mostly focus on the ?improper? version of supervised learning in this section. For the
improper version, we may write the value in (1) as
VT (F ) = sup
inf
E . . . sup
sup
x1 ?X q1 ? (Y) y1 ?X y
?1 ?q1
inf
sup
E
xT ?X qT ? (Y) yT ?X y
?T ?qT
?? `(?
yt , yt ) ? inf ? `(f (xt ), yt )?
T
T
f ?F t=1
t=1
and a relaxation RelT is admissible if for any (x1 , y1 ) . . . , (xT , yT ) ? X ? Y,
sup inf sup ? E `(?
y , y) + RelT ?F ?{(xi , yi )}ti=1 , (x, y)?? ? RelT ?F ?{(xi , yi )}ti=1 ?
x?X q? (Y) y?Y
y
??q
(11)
Let us now focus on binary prediction, i.e. Y = {?1}. In this case, the supremum over y in (11)
becomes a maximum over two values. Let us now take the absolute loss `(?
y , y) = ??
y ? y? = 1 ? y?y.
We can see2 that the optimal randomized strategy, given the side information x, is given by (11) as
qt = argmin max ?1 ? q + RelT ?F ?{(xi , yi )}ti=1 , (x, 1)? , 1 + q + RelT ?F ?{(xi , yi )}ti=1 , (x, ?1)??
q? (Y)
1
or equivalently as : qt = ?RelT ?F ?{(xi , yi )}ti=1 , (x, 1)? ? RelT ?F ?{(xi , yi )}ti=1 , (x, ?1)?? (12)
2
We now assume that F has a finite Littlestone?s dimension Ldim(F) [11, 5]. Suppose the loss
function is `(?
y , y) = ??
y ? y?, and consider the ?mixed? conditional Rademacher complexity
T ?t
sup E? sup ?2 ? ?i f (xi (?)) ? ? ?f (xi ) ? yi ??
x
f ?F
t
i=1
(13)
i=1
as a possible relaxation. The admissibility condition (11) with the conditional sequential
Rademacher (13) as a relaxation would require us to upper bound
sup
inf
T ?t
max ? E ??
yt ? yt ? + sup E? sup ?2 ? ?i f (xi (?)) ? ? ?f (xi ) ? yi ???
xt qt ?[?1,1] yt ?{?1}
y
?t ?qt
f ?F
x
i=1
t
(14)
i=1
However, the supremum over x is preventing us from obtaining a concise algorithm. We need to
further ?relax? this supremum, and the idea is to pass to a finite cover of F on the given tree x and
then proceed as in the Exponential Weights example for a finite collection of experts. This leads to
an upper bound on (13) and gives rise to algorithms similar in spirit to those developed in [5], but
with more attractive computational properties and defined more concisely.
Define the function g(d, t) = ?di=0 ?ti?, which is shown in [15] to be the maximum size of
an exact (zero) cover for a function class with the Littlestone?s dimension Ldim = d. Given
{(x1 , yt ), . . . , (xt , yt )} and = ( 1 , . . . , t ) ? {?1}t , let Ft ( ) = {f ? F ? f (xi ) = i ?i ?
t}, the subset of functions that agree with the signs given by
on the ?past? data and let
F?x1 ,...,xt ? F?xt ? {(f (x1 ), . . . , f (xt )) ? f ? F} be the projection of F onto x1 , . . . , xt . Det
note Lt (f ) = ?ti=1 ?f (xi ) ? yi ? and Lt ( ) = ?ti=1 ? i ? yi ? for
? ? {?1} . The following proposition
gives a relaxation and an algorithm which achieves the O( Ldim(F)T log T ) regret bound. Unlike the algorithm of [5], we do not need to run an exponential number of experts in parallel and
only require access to an oracle that computes the Littlestone?s dimension.
2
The extension to k-class prediction is immediate.
4
Proposition 5. The relaxation
RelT ?F ?(xt , y t )? =
1
log
?
?
? g(Ldim(Ft ( )), T ? t) exp {? Lt ( )} + 2 (T ? t) .
? ?F ? t
?
x
is admissible and leads to an admissible algorithm which uses weights qt (?1) = 1 ? qt (+1) and
qt (+1) =
?(
?(
,+1)?F ?xt
, t )?F ?xt
g(Ldim(Ft ( , +1)), T ? t) exp {? Lt?1 ( )}
g(Ldim(Ft ( ,
t )), T
? t) exp {? Lt?1 ( )}
,
(15)
There is a very close correspondence between the proof of Proposition 5 and the proof of the combinatorial lemma of [15], the analogue of the Vapnik-Chervonenkis-Sauer-Shelah result.
4
Randomized Algorithms and Follow the Perturbed Leader
We now develop a class of admissible randomized methods that arise through sampling. Consider
the objective (5) given by a relaxation RelT . If RelT is the sequential (or classical) Rademacher
complexity, it involves an expectation over sequences of coin flips, and this computation (coupled
with optimization for each sequence) can be prohibitively expensive. More generally, RelT might
involve an expectation over possible ways in which the future might be realized. In such cases,
we may consider a rather simple ?random playout? strategy: draw the random sequence and solve
only one optimization problem for that random sequence. The ideas of random playout have been
discussed in previous literature for estimating the utility of a move in a game (see also [3]). We show
that random playout strategy has a solid basis: for the examples we consider, it satisfies admissibility.
In many learning problems the sequential and the classical Rademacher complexities are within a
constant factor of each other. This holds true, for instance, for linear functions in finite-dimensional
spaces. In such cases, the relaxation RelT does not involve the supremum over a tree, and the
randomized method only needs to draw a sequence of coin flips and compute a solution to an optimization problem slightly more complicated than ERM. We show that Follow the Perturbed Leader
(FPL) algorithms [10] arise in this way. We note that FPL has been previously considered as a rather
unorthodox algorithm providing some kind of regularization via randomization. Our analysis shows
that it arises through a natural relaxation based on the sequential (and thus the classical) Rademacher
complexity, coupled with the random playout idea. As a new algorithmic contribution, we provide
a version of the FPL algorithm for the case of the decision sets being `2 balls, with a regret bound
that is independent of the dimension. We also provide an FPL-style method for the combination of
`1 and `? balls. To the best of our knowledge, these results are novel.
The assumption below implies that the sequential and classical Rademacher complexities are within
constant factor C of each other. We later verify that it holds in the examples we consider.
Assumption 1. There exists a distribution D ? (X ) and constant C ? 2 such that for any t ? [T ]
and given any x1 , . . . , xt?1 , xt+1 , . . . , xT ? X and any ?t+1 , . . . , ?T ? {?1},
sup
E sup ? CAt+1 (f ) ? Lt?1 (f ) + E [`(f, x)] ? `(f, xt )? ?
p? (X ) xt ?p f ?F
x?p
E
sup [ CAt (f ) ? Lt?1 (f )]
?t ,xt ?D f ?F
T
where ?t ?s are i.i.d. Rademacher, Lt?1 (f ) = ?t?1
i=1 `(f, xi ), and At (f ) = ?i=t ?i `(f, xi ).
Under the above assumption one can use the following relaxation
RelT (F ?x1 , . . . , xt ) =
E
xt+1 ,...xT ?D
E? sup ?C ? ?i `(f, xi ) ? ? `(f, xi )?
f ?F
T
t
i=t+1
i=1
(16)
which is a partially symmetrized version of the classical Rademacher averages.
The proof of admissibility for the randomized methods is quite curious ? the forecaster can be seen as
mimicking the sequential Rademacher complexity by sampling from the ?equivalently bad? classical
Rademacher complexity under the specific distribution D specified by the above assumption.
Lemma 6. Under Assumption 1, the relaxation in Eq. (16) is admissible and a randomized strategy
that ensures admissibility is given by: at time t, draw xt+1 , . . . , xT ? D and ?t+1 , . . . , ?T and then:
(a) In the case the loss ` is convex in its first argument and set F is convex and compact, define
5
ft = argmin sup ?`(g, x) + sup ?C ? ?i `(f, xi ) ? ? `(f, xi ) ? `(f, x)??
g?F
f ?F
x?X
T
t?1
i=t+1
i=1
(17)
(b) In the case of non-convex loss, sample ft from the distribution
q?t = argmin sup ? E [`(f, x)] + sup ?C ? ?i `(f, xi ) ? ? `(f, xi ) ? `(f, x)??
f ??
q
q?? (F ) x?X
f ?F
T
t?1
i=t+1
i=1
(18)
The expected regret for the method is bounded by the classical Rademacher complexity:
E [RegT ] ? C Ex1?T ?D E ?sup ? ?t `(f, xt )? ,
T
?
f ?F t=1
Of particular interest are the settings of static experts and transductive learning, which we consider
in Section 5. In the transductive case, the xt ?s are pre-specified before the game, and in the static
expert case ? effectively absent. In these cases, as we show below, there is no explicit distribution
D and we only need to sample the random signs ??s. We easily see that in these cases, the expected
regret bound is simply two times the transductive Rademacher complexity.
The idea of sampling from a fixed distribution is particularly appealing in the case of linear loss,
`(f, x) = ?f, x?. Suppose X is a unit ball in some norm ? ? ? in a vector space B, and F is a unit ball
in the dual norm ? ? ?? . A sufficient condition implying Assumption 1 is then
Assumption 2. There exists a distribution D ? (X ) and constant C ? 2 such that for any w ? B,
sup E ?w + 2?t xt ? ? E E ?w + C?t xt ?
x?X xt ?p
(19)
xt ?D ?t
At round t, the generic algorithm specified by Lemma 18 draws fresh Rademacher random variables
? and xt+1 , . . . , xT ? D and picks
ft = argmin sup ??f, x? + ?C ? ?i xi ? ? xi ? x??
f ?F
x?X
T
t?1
i=t+1
i=1
(20)
We now look at `2 ?`2 and `1 ?`? cases and provide corresponding randomized algorithms.
Example : `1 ?`? Follow the Perturbed Leader
Here, we consider the setting similar to that
in [10]. Let F ? RN be the `1 unit ball and X the (dual) `? unit ball in RN . In [10], F is the
probability simplex and X = [0, 1]N but these are subsumed by the `1 ?`? case. Next we show that
any symmetric distribution satisfies Assumption 2.
Lemma 7. If D is any symmetric distribution over R, then Assumption 2 is satisfied by using the
product distribution DN and any C ? 6?Ex?D ?x?. In particular, Assumption 2 is satisfied with a
distribution D that is uniform on the vertices of the cube {?1}N and C = 6.
The above lemma is especially attractive with Gaussian perturbations as sum of normal random
variables is again normal. Hence, instead of drawing xt+1 , . . . , xT ? N (0, 1) on round t, one can
simply draw one vector Xt ? N (0, T ? t) as the perturbation. In this case, C ? 8.
The form of update in Equation (20), however, is not in a convenient form, and the following
lemma shows a simple Follow the Perturbed Leader type algorithm with the associated regret bound.
N
Lemma 8. Suppose F is the `N
1 unit ball and X is the dual `? unit ball, and let D be
any symmetric distribution. Consider the randomized algorithm that at each round t, freshly
draws Rademacher random variables ?t+1 , . . . , ?T and xt+1 , . . . , xT ? DN and picks ft =
T
argmin ?f, ?t?1
i=1 xi ? C ?i=t+1 ?i xi ? where C = 6?Ex?D ?x?. The expected regret is bounded as :
f ?F
E [RegT ] ? C
E
x1?T ?D N
E? ?? ?t xt ? + 4 ? Pyt+1?T ?D ?C ? ? yi ? ? 4?
T
T
T
?
For instance, for the case
? of coin flips (with C = 6) or the Gaussian distribution (with C = 3 2?)
the bound above is 4C T log N , as the second term is bounded by a constant.
Example : `2 ?`2 Follow the Perturbed Leader
We now consider the case when F and X
are both the unit `2 ball. We can use as perturbation the uniform distribution on the surface of unit
sphere, as the following lemma shows. This result was hinted at in [2], as in high dimensional case,
the random draw from the unit sphere is likely to produce orthogonal directions. However, we do
not require dimensionality to be high for our result.
Lemma 9. Let X and F be unit balls in Euclidean norm. Then Assumption
? 2 is satisfied with a
uniform distribution D on the surface of the unit sphere with constant C = 4 2.
t=1
?
6
t=1
i=t+1
As in the previous example the update in (20) is not in a convenient form and this is addressed below.
Lemma 10. Let X and F be unit balls in Euclidean norm, and D be the uniform distribution on
the surface of the unit sphere. Consider the randomized algorithm that at each round (say round
?
T
t) freshly draws xt+1 , . . . , xT ? D and picks ft = ?? ?t?1
i=1 xi + C ?i=t+1 xi ? ?L where C = 4 2
T
and scaling factor L = ??? ?t?1
i=1 xi + C ?i=t+1 ?i xi ?2 + 1?
1?2
2
. The randomized algorithm enjoys a
?
bound on the expected regret given by E [RegT ] ? C Ex1 ,...,xT ?D ??Tt=1 xt ?2 ? 4 2T .
Importantly, the bound does not depend on the dimensionality of the space. To the best of our
knowledge, this is the first such result for Follow the Perturbed Leader style algorithms. Further,
unlike [10, 6], we directly deal with the adaptive adversary.
5
Static Experts with Convex Losses and Transductive Online Learning
We show how to recover a variant of the R2 forecaster of [7], for static experts and transductive
online learning. At each round, the learner makes a prediction qt ? [?1, 1], observes the outcome
yt ? [?1, 1], and suffers convex L-Lipschitz loss `(qt , yt ). Regret is defined as the difference between learner?s cumulative loss and inf f ?F ?Tt=1 `(f [t], yt ), where F ? [?1, 1]T can be seen as a
set of static experts. The transductive setting is equivalent to this: the sequence of xt ?s is known
before the game starts, and hence the effective function class is once again a subset of [?1, 1]T . It
turns out that in these cases, sequential Rademacher complexity becomes the classical Rademacher
complexity (see [17]), which can thus be taken as a relaxation. This is also the reason that an efficient implementation by sampling is possible. For general convex loss, one possible admissible
relaxation is just a conditional version of the classical Rademacher averages:
RelT (F ?y1 , . . . , yt ) = E?t+1?T sup ?2L ? ?s f [s] ? Lt (f )?
T
f ?F
(21)
s=t+1
where Lt (f ) = ?ts=1 `(f [s], ys ). If (21) is used as a relaxation, the calculation of prediction y?t
involves a supremum over f ? F with (potentially nonlinear) loss functions of instances seen so
far. In some cases this optimization might be hard and it might be preferable if the supremum only
involves terms linear in f . To this end we start by noting that by convexity
yt , yt ) ? inf ? `(f (xt ), yt ) ? ? @`(?
yt , yt ) ? y?t ? inf ? @`(?
yt , yt ) ? f [t]
? `(?
T
T
f ?F t=1
t=1
T
T
f ?F t=1
t=1
(22)
One can now consider an alternative online learning problem which, if we solve, also solves the
original problem. More precisely, the new loss is `? (?
y , r) = r ? y?; we first pick prediction y?t (deterministically) and the adversary picks rt (corresponding to rt = @`(?
yt , yt ) for choice of yt picked
by adversary). Now note that `? is indeed convex in its first argument and is L Lipschitz because
?@`(?
yt , yt )? ? L. This is a one dimensional convex learning game where we pick y?t and regret is
given by the right hand side of (22). Hence, we can consider the relaxation
RelT (F ?@`(?
y1 , y1 ), . . . , @`(?
yt , yt )) = E?t+1?T sup ?2L ? ?i f [t] ? ? @`(?
yi , yi ) ? f [i]?
f ?F
T
t
i=t+1
i=1
(23)
as a linearized form of (21). At round t, the prediction of the algorithm is then
y?t = E ?sup ? ? ?i f [i] ?
T
?
f ?F
i=t+1
1
2L
yi , yi )f [i] + 12 f [t]?? sup ? ? ?i f [i] ?
? @`(?
t?1
T
f ?F
i=1
i=t+1
1
2L
yi , yi )f [i] ? 21 f [t]??
? @`(?
t?1
i=1
(24)
Lemma 11. The relaxation in Eq. (23) is admissible w.r.t. the prediction strategy specified in Equation (24). Further the regret of the strategy is bounded as RegT ? 2L E? ?supf ?F ?Tt=1 ?t f [t]? .
This algorithm is similar to R2 , with the main difference that R2 computes the infima over a sum
of absolute losses, while here we have a more manageable linearized objective. While we need
to evaluate the expectation over ??s on each round, we can estimate y?t by sampling ??s and using
McDiarmid?s inequality argue that the estimate is close to y?t with high probability. The randomized
prediction is now given simply as: on round t, draw ?t+1 , . . . , ?T and predict
y?t (?) = inf ?? ? ?i f [i] +
T
f ?F
i=t+1
1
2L
? `(f [i], yi ) + 12 f [t]? ? inf ?? ? ?i f [i] +
t?1
T
f ?F
i=1
i=t+1
1
2L
? `(f [i], yi ) ? 12 f [t]?
t?1
i=1
(25)
We now show that this predictor enjoys regret bound of the transductive Rademacher complexity :
7
Lemma 12. The relaxation specified in Equation (21) is admissible w.r.t. the randomized prediction
strategy specified in Equation (25), and enjoys bound E [RegT ] ? 2L E? ?supf ?F ?Tt=1 ?t f [t]? .
6
Matrix Completion
Consider the problem of predicting unknown entries in a matrix, in an online fashion. At each round
t the adversary picks an entry in an m ? n matrix and a value yt for that entry. The learner then
chooses a predicted value y?t , and suffers loss `(yt , y?t ), assumed to be ?-Lipschitz. We define our
regret with respect to the class F of all matrices whose trace-norm is at most B (namely, we can
use any such
? matrix to predict just by returning its relevant entry at each round). Usually, one has
B = ?( mn). Consider a transductive version, where we know in advance the location of all
entries we need to predict. We show how to develop an algorithm whose regret is?bounded by the
(transductive) Rademacher complexity of F, which by Theorem 6 of [18], is O(B n) independent
of T . Moreover, in [7], it was shown how one can convert algorithms with such guarantees to obtain
the same regret even in a ?fully? online case, where the set of entry locations is unknown in advance.
In this section we use the two alternatives provided for transductive learning problem in the previous
subsection, and provide two alternatives for the matrix completion problem. Both variants proposed
here improve on the one provided by the R2 forecaster in [7], since that algorithm competes against
the smaller class F ? of matrices with bounded trace-norm and bounded individual entries, and our
variants are also computationally more efficient. Our first variant also improves on the recently
proposed method in [9] in terms of memory requirements, and each iteration is simpler: Whereas
that method requires storing and optimizing full m ? n matrices every iteration, our algorithm only
requires computing spectral norms of sparse matrices (assuming T ? mn, which is usually the
case). This can be done very efficiently, e.g. with power iterations or the Lanczos method.
Our first algorithm follows from Eq. (24), which for our setting gives the following prediction rule:
y?t = B E ??? ? ?i xi ?
T
?
i=t+1
1
2?
yi , yi )xi + 12 xt ? ? ? ? ?i xi ?
? @`(?
t?1
T
i=1
i=t+1
1
2?
yi , yi )xi ? 12 xt ? ?? (26)
? @`(?
t?1
i=1
In the above ??? stands for the spectral norm and each xi is a matrix with a 1 at some specific position and 0 elsewhere. Notice that the algorithm only involves calculation of spectral norms on each
round, which can be done efficiently. As mentioned in previous subsection, one can approximately
evaluate the expectation by sampling several ??s on each round and averaging. The second algorithm
follows (25), and is given by first drawing ? at random and then predicting
y?t (?)= sup ? ? ?i f [xi ] ?
T
?f ?? ?B i=t+1
1
2?
? `(f [xi ], yi ) + 12 f [xt ]? ? sup ? ? ?i f [xi ] ?
t?1
T
?f ?? ?B i=t+1
i=1
1
2?
? `(f [xi ], yi ) ? 12 f [xt ]?
t?1
i=1
where ?f ?? is the trace norm of the m?n f , and f [xi ] is the entry of the matrix f at the position xi .
Notice that the above involves solving two trace norm constrained convex optimization problems per
round. As a simple corollary of Lemma 12, together with the bound on the Rademacher
?
? complexity
mentioned earlier, we get that the expected regret of either variant is O ?B ? ( m + n)?.
7
Conclusion
In [2, 1, 15, 20] the minimax value of the online learning game has been analyzed and nonconstructive bounds on the value have been provided. In this paper, we provide a general constructive recipe for deriving new (and old) online learning algorithms, using techniques from the
apparently non-constructive minimax analysis. The recipe is rather simple: we start with the notion
of conditional sequential Rademacher complexity, and find an ?admissible? relaxation which upper
bounds it. This relaxation immediately leads to an online learning algorithm, as well as to an associated regret guarantee. In addition to the development of a unified algorithmic framework, our
contributions include (1) a new algorithm for online binary classification whenever the Littlestone
dimension of the class is finite; (2) a family of randomized online learning algorithms based on the
idea of a random playout, with new Follow the Perturbed Leader style algorithms arising as special
cases; and (3) efficient algorithms for trace norm based online matrix completion problem which
improve over currently known methods.
Acknowledgements: We gratefully acknowledge the support of NSF under grants CAREER DMS0954737 and CCF-1116928.
8
References
[1] J. Abernethy, A. Agarwal, P. L. Bartlett, and A. Rakhlin. A stochastic view of optimal regret
through minimax duality. In COLT, 2009.
[2] J. Abernethy, P. L. Bartlett, A. Rakhlin, and A. Tewari. Optimal strategies and minimax lower
bounds for online convex games. In COLT, 2008.
[3] J. Abernethy, M.K. Warmuth, and J. Yellin. Optimal strategies from random walks. In COLT,
pages 437?445, 2008.
[4] A. Beck and M. Teboulle. Mirror descent and nonlinear projected subgradient methods for
convex optimization. Operations Research Letters, 31(3):167?175, 2003.
[5] S. Ben-David, D. P?al, and S. Shalev-Shwartz. Agnostic online learning. In COLT, 2009.
[6] N. Cesa-Bianchi and G. Lugosi. Prediction, Learning, and Games. Cambridge University
Press, 2006.
[7] N. Cesa-Bianchi and O. Shamir. Efficient online learning via randomized rounding. In NIPS,
2011.
[8] J. Hannan. Approximation to Bayes risk in repeated play. Contributions to the Theory of
Games, 3:97?139, 1957.
[9] E. Hazan, S. Kale, and S. Shalev-Shwartz. Near-optimal algorithms for online matrix prediction. In COLT, 2012.
[10] A. Kalai and S. Vempala. Efficient algorithms for online decision problems. J. Comput. Syst.
Sci., 71(3):291?307, 2005.
[11] N. Littlestone. Learning quickly when irrelevant attributes abound: A new linear-threshold
algorithm. Machine Learning, 2(4):285?318, 04 1988.
[12] N. Littlestone and M. K. Warmuth. The weighted majority algorithm. Information and Computation, 108(2):212?261, 1994.
[13] J.F. Mertens, S. Sorin, and S. Zamir. Repeated games. Univ. Catholique de Louvain, Center
for Operations Research & Econometrics, 1994.
[14] A.S. Nemirovsky and D.B. Yudin. Problem complexity and method efficiency in optimization.
1983.
[15] A. Rakhlin, K. Sridharan, and A. Tewari. Online learning: Random averages, combinatorial
parameters, and learnability. In NIPS, 2010. Available at http://arxiv.org/abs/1006.1138.
[16] A. Rakhlin, K. Sridharan, and A. Tewari. Online learning: Beyond regret. In COLT, 2011.
Available at http://arxiv.org/abs/1011.3168.
[17] A. Rakhlin, K. Sridharan, and A. Tewari. Online learning: Stochastic, constrained, and
smoothed adversaries. In NIPS, 2011. Available at http://arxiv.org/abs/1104.5070.
[18] O. Shamir and S. Shalev-Shwartz. Collaborative filtering with the trace norm: Learning,
bounding, and transducing. In COLT, 2011.
[19] S. Sorin. The operator approach to zero-sum stochastic games. Stochastic Games and Applications, NATO Science Series C, Mathematical and Physical Sciences, 570:417?426, 2003.
[20] K. Sridharan and A. Tewari. Convex games in banach spaces. In COLT, 2010.
[21] V.G. Vovk. Aggregating strategies. In Proc. Third Workshop on Computational Learning
Theory, pages 371?383. Morgan Kaufmann, 1990.
[22] M. Zinkevich. Online convex programming and generalized infinitesimal gradient ascent.
2003.
9
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4,020 | 4,639 | Learning curves for multi-task Gaussian process
regression
Simon R F Ashton
King?s College London
Department of Mathematics
Strand, London WC2R 2LS, U.K.
Peter Sollich
King?s College London
Department of Mathematics
Strand, London WC2R 2LS, U.K.
[email protected]
Abstract
We study the average case performance of multi-task Gaussian process (GP) regression as captured in the learning curve, i.e. the average Bayes error for a chosen
task versus the total number of examples n for all tasks. For GP covariances that
are the product of an input-dependent covariance function and a free-form intertask covariance matrix, we show that accurate approximations for the learning
curve can be obtained for an arbitrary number of tasks T . We use these to study
the asymptotic learning behaviour for large n. Surprisingly, multi-task learning
can be asymptotically essentially useless, in the sense that examples from other
tasks help only when the degree of inter-task correlation, ?, is near its maximal
value ? = 1. This effect is most extreme for learning of smooth target functions
as described by e.g. squared exponential kernels. We also demonstrate that when
learning many tasks, the learning curves separate into an initial phase, where the
Bayes error on each task is reduced down to a plateau value by ?collective learning? even though most tasks have not seen examples, and a final decay that occurs
once the number of examples is proportional to the number of tasks.
1
Introduction and motivation
Gaussian processes (GPs) [1] have been popular in the NIPS community for a number of years
now, as one of the key non-parametric Bayesian inference approaches. In the simplest case one can
use a GP prior when learning a function from data. In line with growing interest in multi-task or
transfer learning, where relatedness between tasks is used to aid learning of the individual tasks (see
e.g. [2, 3]), GPs have increasingly also been used in a multi-task setting. A number of different
choices of covariance functions have been proposed [4, 5, 6, 7, 8]. These differ e.g. in assumptions
on whether the functions to be learned are related to a smaller number of latent functions or have
free-form inter-task correlations; for a recent review see [9].
Given this interest in multi-task GPs, one would like to quantify the benefits that they bring compared
to single-task learning. PAC-style bounds for classification [2, 3, 10] in more general multi-task scenarios exist, but there has been little work on average case analysis. The basic question in this setting
is: how does the Bayes error on a given task depend on the number of training examples for all tasks,
when averaged over all data sets of the given size. For a single regression task, this learning curve
has become relatively well understood since the late 1990s, with a number of bounds and approximations available [11, 12, 13, 14, 15, 16, 17, 18, 19] as well as some exact predictions [20]. Already
two-task GP regression is much more difficult to analyse, and progress was made only very recently
at NIPS 2009 [21], where upper and lower bounds for learning curves were derived. The tightest of
these bounds, however, either required evaluation by Monte Carlo sampling, or assumed knowledge
of the corresponding single-task learning curves. Here our aim is to obtain accurate learning curve
approximations that apply to an arbitrary number T of tasks, and that can be evaluated explicitly
without recourse to sampling.
1
We begin (Sec. 2) by expressing the Bayes error for any single task in a multi-task GP regression
problem in a convenient feature space form, where individual training examples enter additively.
This requires the introduction of a non-trivial tensor structure combining feature space components
and tasks. Considering the change in error when adding an example for some task leads to partial
differential equations linking the Bayes errors for all tasks. Solving these using the method of
characteristics then gives, as our primary result, the desired learning curve approximation (Sec. 3). In
Sec. 4 we discuss some of its predictions. The approximation correctly delineates the limits of pure
transfer learning, when all examples are from tasks other than the one of interest. Next we compare
with numerical simulations for some two-task scenarios, finding good qualitative agreement. These
results also highlight a surprising feature, namely that asymptotically the relatedness between tasks
can become much less useful. We analyse this effect in some detail, showing that it is most extreme
for learning of smooth functions. Finally we discuss the case of many tasks, where there is an
unexpected separation of the learning curves into a fast initial error decay arising from ?collective
learning?, and a much slower final part where tasks are learned almost independently.
2
GP regression and Bayes error
We consider GP regression for T functions f? (x), ? = 1, 2, . . . , T . These functions have to be
learned from n training examples (x` , ?` , y` ), ` = 1, . . . , n. Here x` is the training input, ?` ?
{1, . . . , T } denotes which task the example relates to, and y` is the corresponding training output.
We assume that the latter is given by the target function value f?` (x` ) corrupted by i.i.d. additive
Gaussian noise with zero mean and variance ??2` . This setup allows the noise level ??2 to depend on
the task.
In GP regression the prior over the functions f? (x) is a Gaussian process. This means that for any
set of inputs x` and task labels ?` , the function values {f?` (x` )} have a joint Gaussian distribution.
As is common we assume this to have zero mean, so the multi-task GP is fully specified by the
covariances hf? (x)f? 0 (x0 )i = C(?, x, ? 0 , x0 ). For this covariance we take the flexible form from [5],
hf? (x)f? 0 (x0 )i = D? ? 0 C(x, x0 ). Here C(x, x0 ) determines the covariance between function values
at different input points, encoding ?spatial? behaviour such as smoothness and the lengthscale(s)
over which the functions vary, while the matrix D is a free-form inter-task covariance matrix.
One of the attractions of GPs for regression is that, even though they are non-parametric models
with (in general) an infinite number of degrees of freedom, predictions can be made in closed form,
see e.g. [1]. For a test point x for task ? , one would predict as output the mean of f? (x) over
the (Gaussian) posterior, which is y T K ?1 k? (x). Here K is the n ? n Gram matrix with entries
K`m = D?` ?m C(x` , xm ) + ??2` ?`m , while k? (x) is a vector with the n entries k?,` = D?` ? C(x` , x).
The error bar would be taken as the square root of the posterior variance of f? (x), which is
V? (x) = D? ? C(x, x) ? k?T (x)K ?1 k? (x)
(1)
The learning curve for task ? is defined as the mean-squared prediction error, averaged over the
location of test input x and over all data sets with a specified number of examples for each task, say
n1 for task 1 and so on. As is standard in learning curve analysis we consider a matched scenario
where the training outputs y` are generated from the same prior and noise model that we use for
inference. In this case the mean-squared prediction error ?? is the Bayes error, and is given by the
average posterior variance [1], i.e. ?? = hV? (x)ix . To obtain the learning curve this is averaged
over the location of the training inputs x` : ? = h?
? i. This average presents the main challenge
for learning curve prediction because the training inputs feature in a highly nonlinear way in V? (x).
Note that the training outputs, on the other hand, do not appear in the posterior variance V? (x) and
so do not need to be averaged over.
We now want to write the Bayes error ?? in a form convenient for performing, at least approximately, the averages required for the learning curve. Assume that all training inputs x` , and also the
test input x, are drawn from the same distribution P (x). One can decompose the input-dependent
part
function into eigenfunctions relative to P (x), according to C(x, x0 ) =
P of the covariance
0
0
0
0
i ?i ?i (x)?i (x ). The eigenfunctions are defined by the condition hC(x, x )?i (x )ix = ?i ?i (x)
and can be chosen to be orthonormal with respect to P (x), h?i (x)?j (x)ix = ?ij . The sum over i
here is in general infinite (unless the covariance function is degenerate, as e.g. for the dot product
kernel C(x, x0 ) = x ? x0 ). To make the algebra below as simple as possible, we let the eigenvalues
?i be arranged in decreasing order and truncate the sum to the finite range i = 1, . . . , M ; M is then
some large effective feature space dimension and can be taken to infinity at the end.
2
In terms of the above eigenfunction decomposition, the Gram matrix has elements
X
X
??` ,? ?i (x` )?i ?ij D? ? 0 ?j (xm )?? 0 ,?m +??2` ?`m
K`m = D?` ?m
?i ?i (x` )?i (xm )+??2` ?`m =
i,?,j,? 0
i
or in matrix form K = ?L?T + ? where ? is the diagonal matrix from the noise variances and
?`,i? = ??` ,? ?i (x` ),
Li?,j? 0 = ?i ?ij D? ? 0
(2)
Here ? has its second index ranging over M (number of kernel eigenvalues) times T (number of
tasks) values; L is a square matrix of this size. In Kronecker (tensor) product notation, L = D ? ?
if we define ? as the diagonal matrix with entries ?i ?ij . The Kronecker product is convenient for
the simplifications below; we will use that for generic square matrices, (A ? B)(A0 ? B 0 ) =
(AA0 ) ? (BB 0 ), (A ? B)?1 = A?1 ? B ?1 , and tr (A ? B) = (tr A)(tr B). In thinking about
the mathematical expressions, it is often easier to picture Kronecker products over feature spaces
and tasks as block matrices. For example, L can then be viewed as consisting of T ? T blocks, each
of which is proportional to ?.
To calculate the Bayes error, we need to average the posterior variance V? (x) over the test input x.
The first term in (1) then becomes D? ? hC(x, x)i = D? ? tr ?. In the second one, we need to average
hk?,` (x)k?,m ix
=
=
D? ?` hC(x` , x)C(x, xm )ix D?m ?
X
D? ?`
?i ?j ?i (x` )h?i (x)?j (x)ix ?j (xm )D?m ?
ij
=
X
D? ? 0 ?l,i? 0 ?i ?j ?ij ?m,j? 00 D? 00 ?
i,? 0 ,j,? 00
2
T
In matrix form this is hk? (x)k?T (x)ix = ?[(De? eT
= ?M? ?T Here the last
? D) ? ? ]?
equality defines M? , and we have denoted by e? the T -dimensional vector with ? -th component
equal to one and all others zero. Multiplying by the inverse Gram matrix K ?1 and taking the trace
gives the average of the second term in (1); combining with the first gives the Bayes error on task ?
?? = hV? (x)ix = D? ? tr ? ? tr ?M? ?T (?L?T + ?)?1
Applying the Woodbury identity and re-arranging yields
??
=
D? ? tr ? ? tr M? ?T ??1 ?(I + L?T ??1 ?)?1
=
D? ? tr ? ? tr M? L?1 [I ? (I + L?T ??1 ?)?1 ]
But
tr M? L?1
=
2
?1
tr {[(De? eT
}
? D) ? ? ][D ? ?]
=
T
tr {[De? eT
? ] ? ?} = e? De? tr ? = D? ? tr ?
so the first and second terms in the expression for ?? cancel and one has
??
=
tr M? L?1 (I + L?T ??1 ?)?1 = tr L?1 M? L?1 (L?1 + ?T ??1 ?)?1
=
2
?1
tr [D ? ?]?1 [(De? eT
(L?1 + ?T ??1 ?)?1
? D) ? ? ][D ? ?]
=
?1
tr [e? eT
+ ?T ??1 ?)?1
? ? I](L
The matrix in square brackets in the last line is just a projector P? onto task ? ; thought of as a matrix
of T ? T blocks (each of size M ? M ), this has an identity matrix in the (?, ? ) block while all other
blocks are zero. We can therefore write, finally, for the Bayes error on task ? ,
?? = tr P? (L?1 + ?T ??1 ?)?1
(3)
Because ? is diagonal and given the definition (2) of ?, the matrix ?T ??1 ? is a sum of contributions from the individual training examples ` = 1, . . . , n. This will be
Pimportant for deriving the
learning curve approximation
in passing that, because ? P? = I, the sum of the
P below. We note
Bayes errors on all tasks is ? ?? = tr (L?1 +?T ??1 ?)?1 , in close analogy to the corresponding
expression for the single-task case [13].
3
3
Learning curve prediction
To obtain the learning curve ? = h?
? i, we now need to carry out the average h. . .i over the training
inputs. To help with this, we can extend an approach forP
the single-task scenario [13] and define a
response or resolvent matrix G = (L?1 + ?T ??1 ? + ? v? P? )?1 with auxiliary parameters v?
that will be set back to zero at the end. One can then ask how G = hGi and hence ? 0 = tr P? 0 G
changes with the number n? of training points for task ? . Adding an example at position x for task
? increases ?T ??1 ? by ???2 ?? ?T
? , where ?? has elements (?? )i? 0 = ?i (x)?? ? 0 . Evaluating the
?1
difference (G ?1 + ???2 ?? ?T
? G with the help of the Woodbury identity and approximating it
?)
with a derivative gives
G?? ?T
?G
?G
=? 2
T
?n?
?? + ?? G??
This needs to be averaged over the new example and all previous ones. If we approximate by
averaging numerator and denominator separately we get
1
?G
?G
= 2
?n?
?? + tr P? G ?v?
(4)
0
00
Here we have exploited for the average over x that the matrix h?? ?T
? ix has (i, ? ), (j, ? )-entry
i
=
P
.
We
have
also
used
the
h?i (x)?j (x)ix ?? ? 0 ?? ? 00 = ?ij ?? ? 0 ?? ? 00 , hence simply h?? ?T
?
? x
auxiliary parameters to rewrite ?hGP? Gi = ?hGi/?v? = ?G/?v? . Finally, multiplying (4) by P? 0
and taking the trace gives the set of quasi-linear partial differential equations
?? 0
1
?? 0
= 2
?n?
?? + ? ?v?
(5)
The remaining task is now to find the functions ? (n1 , . . . , nT , v1 , . . . , vT ) by solving these differential equations. We initially attempted to do this by tracking the ? as examples are added one
task at a time, but the derivation is laborious already for T = 2 and becomes prohibitive beyond.
Far more elegant is to adapt the method of characteristics to the present case. We need to find a
2T -dimensional surface in the 3T -dimensional space (n1 , . . . , nT , v1 , . . . , vT , 1 , . . . , T ), which is
specified by the T functions ? (. . .). A small change (?n1 , . . . , ?nT , ?v1 , . . . , ?vT , ?1 , . . . , ?T ) in
all 3T coordinates is tangential to this surface if it obeys the T constraints (one for each ? 0 )
X ?? 0
?? 0
?? 0 =
?n? +
?v?
?n?
?v?
?
From (5), one sees that this condition is satisfied whenever ?? = 0 and ?n? = ??v? (??2 + ? )
It follows that all the characteristic curves given by ? (t) = ?,0 = const., v? (t) = v?,0 (1 ? t),
n? (t) = v?,0 (??2 + ?,0 ) t for t ? [0, 1] are tangential to the solution surface for all t, so lie within
this surface if the initial point at t = 0 does. Because at t = 0 there are no training examples
(n? (0) = 0), this initial condition is satisfied by setting
!?1
X
?1
?,0 = tr P? L +
v? 0 ,0 P? 0
?0
Because ? (t) is constant along the characteristic curve, we get by equating the values at t = 0 and
t=1
!?1
X
?,0 = tr P? L?1 +
v? 0 ,0 P? 0
= ? ({n? 0 = v? 0 ,0 (??20 + ? 0 ,0 )}, {v? 0 = 0})
?0
Expressing v? 0 ,0 in terms of n? 0 gives then
? = tr P?
L
?1
+
X
?0
n? 0
P? 0
??20 + ? 0
!?1
(6)
This is our main result: a closed set of T self-consistency equations for the average Bayes errors
? . Given L as defined by the eigenvalues ?i of the covariance function, the noise levels ??2 and the
4
number of examples n? for each task, it is straightforward to solve these equations numerically to
find the average Bayes error ? for each task.
The r.h.s. of (6) is easiest to evaluate if we view the matrix inside the brackets as consisting of
M ? M blocks of size T ? T (which is the reverse of the picture we have used so far). The matrix is
then block diagonal, with the blocks corresponding to different eigenvalues ?i . Explicitly, because
L?1 = D ?1 ? ??1 , one has
?1
X
n? 0
?1 ?1
? =
?i D + diag({ 2
})
(7)
?? 0 + ? 0
??
i
4
Results and discussion
We now consider the consequences of the approximate prediction (7) for multi-task learning curves
in GP regression. A trivial special case is the one of uncorrelated tasks, where D is diagonal. Here
one recovers T separate equations for the individual tasks as expected, which have the same form as
for single-task learning [13].
4.1
Pure transfer learning
Consider now the case of pure transfer learning, where one is learning a task of interest (say ? = 1)
purely from examples for other tasks. What is the lowest average Bayes error that can be obtained?
Somewhat more generally, suppose we have no examples for the first T0 tasks, n1 = . . . = nT0 = 0,
but a large number of examples for the remaining T1 = T ? T0 tasks. Denote E = D ?1 and write
this in block form as
E00 E01
E=
T
E01
E11
Now multiply by ??1
and add in the lower right block a diagonal matrix N = diag({n? /(??2 +
i
?1
?1
? )}? =T0 +1,...,T ). The matrix inverse in (7) then has top left block ?i [E00
+ E00
E01 (?i N +
?1
?1
T
?1 T
E11 ? E01 E00 E01 ) E01 E00 ]. As the number of examples for the last T1 tasks grows, so do all
?1
(diagonal) elements of N . In the limit only the term ?i E00
survives, and summing over i gives
?1
?1
1 = tr ?(E00 )11 = hC(x, x)i(E00 )11 . The Bayes error on task 1 cannot become lower than this,
placing a limit on the benefits of pure transfer learning. That this prediction of the approximation
(7) for such a lower limit is correct can also be checked directly: once the last T1 tasks f? (x)
(? = T0 + 1, . . . T ) have been learn perfectly, the posterior over the first T0 functions is, by standard
Gaussian conditioning, a GP with covariance C(x, x0 )(E00 )?1 . Averaging the posterior variance of
?1
f1 (x) then gives the Bayes error on task 1 as 1 = hC(x, x)i(E00
)11 , as found earlier.
This analysis can be extended to the case where there are some examples available also for the first
T0 tasks. One finds for the generalization errors on these tasks the prediction (7) with D ?1 replaced
by E00 . This is again in line with the above form of the GP posterior after perfect learning of the
remaining T1 tasks.
4.2
Two tasks
We next analyse how well the approxiation (7) does in predicting multi-task learning curves for
T = 2 tasks. Here we have the work of Chai [21] as a baseline, and as there we choose
1 ?
D=
? 1
The diagonal elements are fixed to unity, as in a practical application where one would scale both
task functions f1 (x) and f2 (x) to unit variance; the degree of correlation of the tasks is controlled
by ?. We fix ?2 = n2 /n and plot learning curves against n. In numerical simulations we ensure
integer values of n1 and n2 by setting n2 = bn?2 c, n1 = n ? n2 ; for evaluation of (7) we use
directly n2 = n?2 , n1 = n(1 ? ?2 ). For simplicity we consider equal noise levels ?12 = ?22 = ? 2 .
As regards the covariance function and input distribution, we analyse first the scenario studied
in [21]: a squared exponential (SE) kernel C(x, x0 ) = exp[?(x ? x0 )2 /(2l2 )] with lengthscale
l, and one-dimensional inputs x with a Gaussian distribution N (0, 1/12). The kernel eigenvalues ?i
5
1
1
1
1
?1
?1
0.8
?1
?1
0.8
0.001
1
1
1
?1
0.8
0.001
n 10000
1
0.01
1
n 10000
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
0
100
200
n 300
400
500
0
0
100
?1
200
n 300
400
500
0
0
100
200
n 1000
n 300
400
500
Figure 1: Average Bayes error for task 1 for two-task GP regression with kernel lengthscale l = 0.01,
noise level ? 2 = 0.05 and a fraction ?2 = 0.75 of examples for task 2. Solid lines: numerical
simulations; dashed lines: approximation (7). Task correlation ?2 = 0, 0.25, 0.5, 0.75, 1 from top to
bottom. Left: SE covariance function, Gaussian input distribution. Middle: SE covariance, uniform
inputs. Right: OU covariance, uniform inputs. Log-log plots (insets) show tendency of asymptotic
uselessness, i.e. bunching of the ? < 1 curves towards the one for ? = 0; this effect is strongest for
learning of smooth functions (left and middle).
are known explicitly from [22] and decay exponentially with i. Figure 1(left) compares numerically
simulated learning curves with the predictions for 1 , the average Bayes error on task 1, from (7).
Five pairs of curves are shown, for ?2 = 0, 0.25, 0.5, 0.75, 1. Note that the two extreme values
represent single-task limits, where examples from task 2 are either ignored (? = 0) or effectively
treated as being from task 1 (? = 1). Our predictions lie generally below the true learning curves,
but qualitatively represent the trends well, in particular the variation with ?2 . The curves for the different ?2 values are fairly evenly spaced vertically for small number of examples, n, corresponding
to a linear dependence on ?2 . As n increases, however, the learning curves for ? < 1 start to bunch
together and separate from the one for the fully correlated case (? = 1). The approximation (7)
correctly captures this behaviour, which is discussed in more detail below.
Figure 1(middle) has analogous results for the case of inputs x uniformly distributed on the interval
[0, 1]; the ?i here decay exponentially with i2 [17]. Quantitative agreement between simulations
and predictions is better for this case. The discussion in [17] suggests that this is because the
approximation method we have used implicitly neglects spatial variation of the dataset-averaged
posterior variance hV? (x)i; but for a uniform input distribution this variation will be weak except
near the ends of the input range [0, 1]. Figure 1(right) displays similar results for an OU kernel
C(x, x0 ) = exp(?|x ? x0 |/l), showing that our predictions also work well when learning rough
(nowhere differentiable) functions.
4.3
Asymptotic uselessness
The two-task results above suggest that multi-task learning is less useful asymptotically: when the
number of training examples n is large, the learning curves seem to bunch towards the curve for
? = 0, where task 2 examples are ignored, except when the two tasks are fully correlated (? = 1).
We now study this effect.
When the number of examples for all tasks becomes large, the Bayes errors ? will become small
and eventually be negligible compared to the noise variances ??2 in (7). One then has an explicit
prediction for each ? , without solving T self-consistency equations. If we write, for T tasks,
n? = n?? with ?? the fraction of examples for task ? , and set ?? = ?? /??2 , then for large n
?1 P
P
?1
1/2
? = i ??1
+ n? ? ? = i (??1/2 [??1
D?1/2 )?1 + nI]?1 ??1/2 )? ?
(8)
i D
i (?
where ? = diag(?1 , . . . , ?T ). Using an eigendecomposition of the symmetric matrix ?1/2 D?1/2 =
PT
T
a=1 ?a va va , one then shows in a few lines that (8) can be written as
P
? ? ???1 a (va,? )2 ?a g(n?a )
(9)
6
1
1
1
50000
?
5000
r
0.1
?
0.5
n=500
10
100
1000
n
0.1
0
0
0.2
0.4
?
2
0.6
0.8
1
1
10
100
1000
n
Figure 2: Left: Bayes error (parameters as in Fig. 1(left), with n = 500) vs ?2 . To focus on the
error reduction with ?, r = [1 (?) ? 1 (1)]/[1 (0) ? 1 (1)] is shown. Circles: simulations; solid
line: predictions from (7). Other lines: predictions for larger n, showing the approach to asymptotic
uselessness in multi-task learning of smooth functions. Inset: Analogous results for rough functions
(parameters as in Fig. 1(right)). Right: Learning curve for many-task learning (T = 200, parameters
otherwise as in Fig. 1(left) except ?2 = 0.8). Notice the bend around 1 = 1 ? ? = 0.106. Solid
line: simulations (steps arise because we chose to allocate examples to tasks in order ? = 1, . . . , T
rather than randomly); dashed line: predictions from (7). Inset: Predictions for T = 1000, with
asymptotic forms = 1 ? ? + ??
and = (1 ? ?)?
for the two learning stages shown as solid lines.
P ?1
where g(h) = tr (??1 + h)?1 =
+ h)?1 and va,? is the ? -th component of the a-th
i (?i
eigenvector va . This is the general asymptotic form of our prediction for the average Bayes error
for task ? .
To get a more explicit result, consider the case where sample functions from the GP prior have
(mean-square) derivatives up to order r. The kernel eigenvalues ?i then decay as1 i?(2r+2) for large
i, and using arguments from [17] one deduces that g(h) ? h?? for large h, with ? = (2r +1)/(2r +
2). In (9) we can then write, for large n, g(n?a ) ? (?a /?? )?? g(n?? ) and hence
P
? ? g(n?? ){ a (va,? )2 (?a /?? )1?? }
(10)
When there is only a single task, ?1 = ?1 and this expression reduces to 1 = g(n?1 ) = g(n1 /?12 ).
Thus g(n?? ) = g(n? /??2 ) is the error we would get by ignoring all examples from tasks other than ? ,
and the term in {. . .} in (10) gives the ?multi-task gain?, i.e. the factor by which the error is reduced
because of examples from other tasks. (The absolute error reduction always vanishes trivially for
n ? ?, along with the errors themselves.) One observation can be made directly. Learning of very
smooth functions, as defined e.g. by the SE kernel, corresponds to r ? ? and hence ? ? 1, so
the multi-task gain tends to unity: multi-task learning is asymptotically useless. The only exception
occurs when some of the tasks are fully correlated, because one or more of the eigenvalues ?a of
?1/2 D?1/2 will then be zero.
Fig. 2(left) shows this effect in action, plotting Bayes error against ?2 for the two-task setting of
Fig. 1(left) with n = 500. Our predictions capture the nonlinear dependence on ?2 quite well,
though the effect is somewhat weaker in the simulations. For larger n the predictions approach a
curve that is constant for ? < 1, signifying negligible improvement from multi-task learning except
at ? = 1. It is worth contrasting this with the lower bound from [21], which is linear in ?2 . While
this provides a very good approximation to the learning curves for moderate n [21], our results here
show that asymptotically this bound can become very loose.
When predicting rough functions, there is some asymptotic improvement to be had from multi-task
learning, though again the multi-task gain is nonlinear in ?2 : see Fig. 2(left, inset) for the OU case,
which has r = 1). A simple expression for the gain can be obtained in the limit of many tasks, to
which we turn next.
1
See the discussion of Sacks-Ylvisaker conditions in e.g. [1]; we consider one-dimensional inputs here
though the discussion can be generalized.
7
4.4
Many tasks
We assume as for the two-task case that all inter-task correlations, D?,? 0 with ? 6= ? 0 , are equal to
?, while D?,? = 1. This setup
? was used e.g. in [23], and can be interpreted as each task having a
component proportional to ? of a shared latent function, with an independent task-specific signal
in addition. We assume for simplicity that we have the same number n? = n/T of examples for
each task, and that all noise levels are the same, ??2 = ? 2 . Then also all Bayes errors ? = will be
the same. Carrying out the matrix inverses in (7) explicitly, one can then write this equation as
= gT (n/(? 2 + ), ?)
where gT (h, ?) is related to the single-task function g(h) from above by
1??
T ?1
(1 ? ?)g(h(1 ? ?)/T ) + ? +
g(h[? + (1 ? ?)/T ])
gT (h, ?) =
T
T
(11)
(12)
Now consider the limit T ? ? of many tasks. If n and hence h = n/(? 2 + ) is kept
fixed, gT (h, ?) ? (1 ? ?) + ?g(h?); here we have taken g(0) = 1 which corresponds to
tr ? = hC(x, x)ix = 1 as in the examples above. One can then deduce from (11) that the Bayes
error for any task will have the form = (1 ? ?) + ??
, where ? decays from one to zero with increasing n as for a single task, but with an effective noise level ?
? 2 = (1 ? ? + ? 2 )/?. Remarkably, then,
even though here n/T ? 0 so that for most tasks no examples have been seen, the Bayes error for
each task decreases by ?collective learning? to a plateau of height 1 ? ?. The remaining decay of to
zero happens only once n becomes of order T . Here one can show, by taking T ? ? at fixed h/T
in (12) and inserting into (11), that = (1 ? ?)?
where ? again decays as for a single task but with
an effective number of examples n
? = n/T and effective noise level ?
? 2 /(1 ? ?). This final stage of
learning therefore happens only when each task has seen a considerable number of exampes n/T .
Fig. 2(right) validates these predictions against simulations, for a number of tasks (T = 200) that
is in the same ballpark as in the many-tasks application example of [24]. The inset for T = 1000
shows clearly how the two learning curve stages separate as T becomes larger.
Finally we can come back to the multi-task gain in the asymptotic stage of learning. For GP priors
with sample functions with derivatives up to order r as before, the function ? from above will decay
as (?
n/?
? 2 )?? ; since = (1 ? ?)?
and ?
? 2 = ? 2 /(1 ? ?), the Bayes error is then proportional
to (1 ? ?)1?? . This multi-task gain again approaches unity for ? < 1 for smooth functions (? =
(2r + 1)/(2r + 2) ? 1). Interestingly,
for rough functions (? < 1), the multi-task gain decreases
p
for small ?2 as 1 ? (1 ? ?) ?2 and so always lies below a linear dependence on ?2 initially. This
shows that a linear-in-?2 lower error bound cannot generally apply to T > 2 tasks, and indeed one
can verify that the derivation in [21] does not extend to this case.
5
Conclusion
We have derived an approximate prediction (7) for learning curves in multi-task GP regression, valid
for arbitrary inter-task correlation matrices D. This can be evaluated explicitly knowing only the
kernel eigenvalues, without sampling or recourse to single-task learning curves. The approximation
shows that pure transfer learning has a simple lower error bound, and provides a good qualitative
account of numerically simulated learning curves. Because it can be used to study the asymptotic
behaviour for large training sets, it allowed us to show that multi-task learning can become asymptotically useless: when learning smooth functions it reduces the asymptotic Bayes error only if tasks
are fully correlated. For the limit of many tasks we found that, remarkably, some initial ?collective
learning? is possible even when most tasks have not seen examples. A much slower second learning
stage then requires many examples per task. The asymptotic regime of this also showed explicitly
that a lower error bound that is linear in ?2 , the square of the inter-task correlation, is applicable
only to the two-task setting T = 2.
In future work it would be interesting to use our general result to investigate in more detail the consequences of specific choices for the inter-task correlations D, e.g. to represent a lower-dimensional
latent factor structure. One could also try to deploy similar approximation methods to study the case
of model mismatch, where the inter-task correlations D would have to be learned from data. More
challenging, but worthwhile, would be an extension to multi-task covariance functions where task
and input-space correlations to not factorize.
8
References
[1] C K I Williams and C Rasmussen. Gaussian Processes for Machine Learning. MIT Press, Cambridge,
MA, 2006.
[2] J Baxter. A model of inductive bias learning. J. Artif. Intell. Res., 12:149?198, 2000.
[3] S Ben-David and R S Borbely. A notion of task relatedness yielding provable multiple-task learning
guarantees. Mach. Learn., 73(3):273?287, December 2008.
[4] Y W Teh, M Seeger, and M I Jordan. Semiparametric latent factor models. In Workshop on Artificial
Intelligence and Statistics 10, pages 333?340. Society for Artificial Intelligence and Statistics, 2005.
[5] E V Bonilla, F V Agakov, and C K I Williams. Kernel multi-task learning using task-specific features.
In Proceedings of the 11th International Conference on Artificial Intelligence and Statistics (AISTATS).
Omni Press, 2007.
[6] E V Bonilla, K M A Chai, and C K I Williams. Multi-task Gaussian process prediction. In J C Platt,
D Koller, Y Singer, and S Roweis, editors, NIPS 20, pages 153?160, Cambridge, MA, 2008. MIT Press.
[7] M Alvarez and N D Lawrence. Sparse convolved Gaussian processes for multi-output regression. In
D Koller, D Schuurmans, Y Bengio, and L Bottou, editors, NIPS 21, pages 57?64, Cambridge, MA,
2009. MIT Press.
[8] G Leen, J Peltonen, and S Kaski. Focused multi-task learning using Gaussian processes. In Dimitrios
Gunopulos, Thomas Hofmann, Donato Malerba, and Michalis Vazirgiannis, editors, Machine Learning
and Knowledge Discovery in Databases, volume 6912 of Lecture Notes in Computer Science, pages 310?
325. Springer Berlin, Heidelberg, 2011.
?
[9] M A Alvarez,
L Rosasco, and N D Lawrence. Kernels for vector-valued functions: a review. Foundations
and Trends in Machine Learning, 4:195?266, 2012.
[10] A Maurer. Bounds for linear multi-task learning. J. Mach. Learn. Res., 7:117?139, 2006.
[11] M Opper and F Vivarelli. General bounds on Bayes errors for regression with Gaussian processes. In
M Kearns, S A Solla, and D Cohn, editors, NIPS 11, pages 302?308, Cambridge, MA, 1999. MIT Press.
[12] G F Trecate, C K I Williams, and M Opper. Finite-dimensional approximation of Gaussian processes. In
M Kearns, S A Solla, and D Cohn, editors, NIPS 11, pages 218?224, Cambridge, MA, 1999. MIT Press.
[13] P Sollich. Learning curves for Gaussian processes. In M S Kearns, S A Solla, and D A Cohn, editors,
NIPS 11, pages 344?350, Cambridge, MA, 1999. MIT Press.
[14] D Malzahn and M Opper. Learning curves for Gaussian processes regression: A framework for good
approximations. In T K Leen, T G Dietterich, and V Tresp, editors, NIPS 13, pages 273?279, Cambridge,
MA, 2001. MIT Press.
[15] D Malzahn and M Opper. A variational approach to learning curves. In T G Dietterich, S Becker, and
Z Ghahramani, editors, NIPS 14, pages 463?469, Cambridge, MA, 2002. MIT Press.
[16] D Malzahn and M Opper. Statistical mechanics of learning: a variational approach for real data. Phys.
Rev. Lett., 89:108302, 2002.
[17] P Sollich and A Halees. Learning curves for Gaussian process regression: approximations and bounds.
Neural Comput., 14(6):1393?1428, 2002.
[18] P Sollich. Gaussian process regression with mismatched models. In T G Dietterich, S Becker, and
Z Ghahramani, editors, NIPS 14, pages 519?526, Cambridge, MA, 2002. MIT Press.
[19] P Sollich. Can Gaussian process regression be made robust against model mismatch? In Deterministic
and Statistical Methods in Machine Learning, volume 3635 of Lecture Notes in Artificial Intelligence,
pages 199?210. Springer Berlin, Heidelberg, 2005.
[20] M Urry and P Sollich. Exact larning curves for Gaussian process regression on large random graphs. In
J Lafferty, C K I Williams, J Shawe-Taylor, R S Zemel, and A Culotta, editors, NIPS 23, pages 2316?2324,
Cambridge, MA, 2010. MIT Press.
[21] K M A Chai. Generalization errors and learning curves for regression with multi-task Gaussian processes.
In Y Bengio, D Schuurmans, J Lafferty, C K I Williams, and A Culotta, editors, NIPS 22, pages 279?287,
2009.
[22] H Zhu, C K I Williams, R J Rohwer, and M Morciniec. Gaussian regression and optimal finite dimensional
linear models. In C M Bishop, editor, Neural Networks and Machine Learning. Springer, 1998.
[23] E Rodner and J Denzler. One-shot learning of object categories using dependent Gaussian processes.
In Michael Goesele, Stefan Roth, Arjan Kuijper, Bernt Schiele, and Konrad Schindler, editors, Pattern
Recognition, volume 6376 of Lecture Notes in Computer Science, pages 232?241. Springer Berlin, Heidelberg, 2010.
[24] T Heskes. Solving a huge number of similar tasks: a combination of multi-task learning and a hierarchical Bayesian approach. In Proceedings of the Fifteenth International Conference on Machine Learning
(ICML?98), pages 233?241. Morgan Kaufmann, 1998.
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evaluate:1 correlated:4 |
4,021 | 464 | Information Processing to Create Eye Movements
David A. Robinson
Departments of Ophthalmology
and Biomedical Engineering
The Johns Hopkins University
School of Medicine
Baltimore, MD 21205
ABSTRACT
Because eye muscles never cocontract and do not deal with external
loads, one can write an equation that relates motoneuron firing rate to
eye position and velocity - a very uncommon situation in the CNS.
The semicircular canals transduce head velocity in a linear manner by
using a high background discharge rate, imparting linearity to the
premotor circuits that generate eye movements. This has allowed
deducing some of the signal processing involved, including a neural
network that integrates. These ideas are often summarized by block
diagrams. Unfortunately, they are of little value in describing the
behavior of single neurons - a fmding supported by neural network
models.
1 INTRODUCTION
The neural networks in our studies are quite simple. They differ from other applications
in that they attempt to model real neural subdivisions of the oculomotor system which
have been extensively studied with microelectrodes. Thus, we can ask the extent to
which neural networks succeed in describing the behavior of hidden units that is already
known. A major benefit of using neural networks in the oculomotor system is to
illustrate clearly the shortcomings of block diagram models which tell one very little
about what one may expect if one pokes a microelectrode inside one of its boxes.
Conversely, single unit behavior is so loosely coupled to system behavior that, although
the simplicity of the oculomotor system allows the relationships to be understood, one
fears that, in a more complicated system, the behavior of single (hidden) units will give
351
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Robinson
little information about what a system is trying to do, never mind how.
2 SIMPLIFICATIONS IN OCUWMOTOR CONTROL
Because it is impossible to cocontract our eye muscles and because their viscoelastic load
never varies, it is possible to write an equation that uniquely relates the discharge rates
of their motoneurons and the position of the load (eye position). This cannot be done in
the case of, for example, limb muscles. Moreover, this system is well-approximated by
a first-order, linear differential equation. Linearity comes about from the design of the
semicircular canals, the origin of the vestibulo-ocular reflex (VOR). This reflex creates
eye movements that compensate for head movements to stabilize the eyes in space for
clear vision. The canals primarily transduce head velocity, neurally encoded into the
discharge rates of its afferents. These rates modulate above and below a high
background rate (typically 100 spikes/sec) that keeps them well away from cutoff and
provides a wide linear range. The core of this reflex is only three neurons long and the
canals impose their properties - linear modulation around a high background rate - onto
all down-stream neurons including the motoneurons.
In addition to linearity, the functions of the various oculomotor subsystems are clear.
There is no messy stretch reflex, the muscle fibers are straight and parallel, and there is
only one "joint." All these features combine to help us understand the premotor
organization of oculomotor signals in the caudal pons, a system that has enjoyed much
block-diagram modelling and now, neural network modelling.
3 DISTRIBUTION OF OCULOMOTOR SIGNAlS
The first application of neural networks to the oculomotor system was a study of
Anastasio and Robinson (1989). The problem addressed concerned the convergence of
diverse oculomotor signals in the caudal pons. There are three major oculomotor
subsystems: the VOR; the saccadic system that causes the eyes to jump rapidly from one
target to another; and the smooth pursuit system that allows the eyes to track a moving
target. Each appears in the caudal pons as a veloci.ty command. The canals, via the
vestibular nuclei, provide an eye-velocity command, Ev, for compensatory vestibular eye
'Povements. Burst neurons in the nearby pontine reticular formation provide a signal,
Eat for the desired eye velocity for a saccade. Purkinje cells in the cerebellum carry an
eye-velocity signal, Ep' for pursuit eye movements. Thus, three eye-velocity commands
converge in the region of the motoneurons.
When one records from cells in this region one fmds a discharge rate R of:
R.=.. Ro
.
.
.
+ rp Ep + rvEv + r.E.
(1)
where Ro is the high background rate previously described and rp, rv and r. are
coefficients that can assume any values, in a seemingly random way, for anyone neuron
(e.g. Tomlinson and Robinson, 1984). Now a block-diagram model need show only the
three velocity commands converging on the motoneurons and would not suggest the
existence of neurons carrying complicated signals like that of Equ. (1). On the other
hand, such behavior has a nice, messy, biological flavor. Somehow, it wOl;lld ~m oqd
if such signals did not exist. What is clearly happening is that the signals Ep, Ev and E.
Information Processing to Create Eye Movements
are being distributed over the intemeurons and then reassembled in the correct amount
on the motoneurons. This is just a simple, specific example of distributed parallel
processing in the nervous system.
A neural network model is merely an explicit statement of such a distribution. Initial
randomization of the synaptic weights followed by error-driven learning creates hidden
units that conform to Equ. (1). We concluded that a neural network model was entirely
appropriate for this neural system. This exercise also brought home, although in a simple
way, the obvious, but often overlooked, message that block-diagram models can be
misleading about how their conceptual functions are realized by neurons.
We next examined distribution of the spatial properties of the intemeurons of the VOR
(Anastasio and Robinson, 1990). We used only the vertical VOR to keep things simple.
The inputs were the primary afferents of the four vertical semicircular canals that sense
head rotations in all combinations of pitch and roll. The output layer was the four
motoneurons of the vertical recti and oblique muscles that move the eye vertically and
in cyclotorsion. The model was trained to perform compensatory eye movements in all
combinations of pitch and roll.
The sensitivity axis is that axis around which rotation of the head or eye produces
maximum modulation in discharge rate. The sensitivity axis of a canal unit is
perpendicular to the plane in which the canal lies. That of a motoneuron is that axis
around which its muscle will rotate the eye. What were the sensitivity axes of the hidden
units?
A block diagram of the spatial manipulations of the VOR consists of matrices. The
geometry of the canals can be described by a 3 x 3 matrix that converts a head-velocity
vector into its neurally encoded representation on canal nerves. The geometry of the
muscles can be described as another matrix that converts the neurally-encoded
motoneuron vector into a physical eye-rotation vector. The brain-stem matrix describes
how the canal neurons must project to the motoneurons (Robinson, 1982). In this
scheme, intemeurons would have only fixed sensitivity axes laying somewhere between
that of a canal unit and a motoneuron. In our model, however, sensitivity axes are
distributed in the network; those of the hidden units point in a variety of directions. This
has also been confirmed by microelectrode recordings (Fukushima et al., 1990). Thus,
spatial aspects of transformations, just like temporal aspects, are distributed over the
intemeurons.
Again, block-diagrams, in this case in the form of a matrix, are misleading about what
one will find with a microelectrode. Again, recording from single units tells one little
about what a network is trying to do. There is much talk in motor physiology about
coordinate systems and transformations from one to another. The question is asked
"What coordinate system is this neuron working in?" In this example, individual hidden
units do not behave as if they belonged to any coordinate system and this raises the
problem of whether this is really a meaningful question.
4 THE NEURAL INTEGRATOR
Muscles are largely position actuators; against a constant load, position is proportional
353
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Robinson
to innervation. The motoneurons of the extraocular muscles also need a signal
proportional to desired eye position as well as velocity. Since eye-movement commands
enter the caudal pons as eye-velocity commands. the necessary eye-position command is
obtained by integrating the velocity signals (see Robinson. 1989. for a review). The
location of the neural network has been discovered in the caudal pons and it is intriguing
to speculate how it might work. Hardwired networks. based on positive feedback. have
been proposed utilizing lateral inhibition (Cannon et a1.. 1983) and more recently a
learning neural network (dynamic) has been proposed for the VOR (Arnold and
Robinson. 1991). The hidden units are freely connected. the input is from two canal
units in push-pull. the output is two motoneurons also in push-pull. which operate on the
plant transfer function. lI(sTc + 1). (Tc is the plant time constant). to create an eye
position which should be the time integral of the input head velocity. The error is retinal
image slip (the difference between actual and ideal eye velocity). Its ems value over a
trial interval is used to change synaptic weights in a steepest descent method until the
error is negligible. To compensate the plant lag. the network must produce a
combination output of eye velocity plus its integral. eye position. and these two signals.
with various weights. are seen on all hidden units which. thus. look remarkably like the
integrator neurons that we record from.
This exercise raises several issues. The block-diagram model of this network is a box
marked lis in parallel with the direct velocity feedforward path given the gain Tc. The
parallel combination is (sTc + 1)/s. The zero cancels the pole of the plant leaving lis.
so that eye position is the perfect integral of head velocity. While such a diagram is
conceptually very useful in diagnosing disorders (Zee and Robinson. 1979). it contains
no hint of how neurons might effect integration and so is useless in this regard.
Moreover. Galiana and Outerbridge (1984) have pointed out. although in a more complex
context. that a direct feedforward path of gain Tc with a positive feedback path around
it containing a model of the plant. produces exactly the same transfer function. Should
we worry about which is correct - feedforward or feedback? Perhaps we should. but
note that the neural network model of the integrator just described contains both feedback
and feedforward pathways and relies on positive feedback. There is a suspicion that the
latter network may subsume both block diagrams making questions about which is correct
irrelevant. One thing is certain. at this level of organization. so close to the neuron level.
block-diagrams. while having conceptual value. are not only useless but can be
misleading if one is interested in describing real neural networks.
Finally. how does one test a model network such as that proposed for the neural
integrator? It involves the microcircuitry with which small sets of circumscribed cells
talk to each other and process signals. The technology is not yet available to allow us
to answer this question. I know of no real. successful examples. This. I believe. is a
true roadblock in neurophysiology. If we cannot solve it. we must forever be content to
describe what cell groups do but not how they do it.
Acknowledgements
This research is supported by Grant 5 R37 EYOO598 from the National Eye Institute of
the National Institutes of Health.
Information Processing to Create Eye Movements
References
T.J. Anastasio & D.A. Robinson. (l989) The distributed representation of vestibuloocular signals by brain-stem neurons. Bioi. Cybern., 61:79-88.
T.J. Anastasio & D.A. Robinson. (l990) Distributed parallel processing in the vertical
vestibulo-ocular reflex: Learning networks compared to tensor theory. Bioi. Cybern.,
63:161-167.
D.B. Arnold & D.A. Robinson. (1991) A learning network model of the neural integrator
of the oculomotor system. Bioi. Cybern., 64:447-454.
S.C. Cannon, D.A. Robinson & S. Shamma. (1983) A proposed neural network for the
integrator of the oculomotor system. Bioi. Cybern., 49: 127-136.
K. Fukushima, S.I. Perlmutter, J.F. Baker & B.W. Peterson. (1990) Spatial properties
of second-order vestibulo-ocular relay neurons in the alert cat. Exp. Brain Res., 81:462478.
H.L. Galiana & J.S. Outerbridge. (1984) A bilateral model for central neural pathways
in vestibuloocular reflex. J. Neurophysiol., 51:210-241.
D.A. Robinson. (1982) The use of matrices in analyzing the three-dimensional behavior
of the vestibulo-ocular reflex. Bioi. Cybern., 46:53-66.
D.A. Robinson. (1989) Integrating with neurons. Ann. Rev. Neurosci., 12:33-45.
R.D. Tomlinson & D.A. Robinson. (1984) Signals in vestibular nucleus mediating
vertical eye movements in the monkey. J. Neurophysiol., 51: 1121-1136.
D.S. Zee & D.A. Robinson. (l979) Clinical applications of oculomotor models. In H.S.
Thompson (ed.), Topics in Neuro-Ophthalmology, 266-285. Baltimore, MD: Williams
& Wilkins.
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4,022 | 4,640 | Best Arm Identification: A Unified Approach to Fixed
Budget and Fixed Confidence
Victor Gabillon
Mohammad Ghavamzadeh
Alessandro Lazaric
INRIA Lille - Nord Europe, Team SequeL
Victor Gabillon, Mohammad Ghavamzadeh & Alessandro Lazaric
Abstract
We study the problem of identifying the best arm(s) in the stochastic multi-armed
bandit setting. This problem has been studied in the literature from two different
perspectives: fixed budget and fixed confidence. We propose a unifying approach
that leads to a meta-algorithm called unified gap-based exploration (UGapE), with
a common structure and similar theoretical analysis for these two settings. We
prove a performance bound for the two versions of the algorithm showing that the
two problems are characterized by the same notion of complexity. We also show
how the UGapE algorithm as well as its theoretical analysis can be extended to
take into account the variance of the arms and to multiple bandits. Finally, we
evaluate the performance of UGapE and compare it with a number of existing
fixed budget and fixed confidence algorithms.
1
Introduction
The problem of best arm(s) identification [6, 3, 1] in the stochastic multi-armed bandit setting has
recently received much attention. In this problem, a forecaster repeatedly selects an arm and observes a sample drawn from its reward distribution during an exploration phase, and then is asked to
return the best arm(s). Unlike the standard multi-armed bandit problem, where the goal is to maximize the cumulative sum of rewards obtained by the forecaster (see e.g., [15, 2]), in this problem the
forecaster is evaluated on the quality of the arm(s) returned at the end of the exploration phase. This
abstract problem models a wide range of applications. For instance, let us consider a company that
has K different variants of a product and needs to identify the best one(s) before actually placing it
on the market. The company sets up a testing phase in which the products are tested by potential
customers. Each customer tests one product at the time and gives it a score (a reward). The objective
of the company is to return a product at the end of the test phase which is likely to be successful once
placed on the market (i.e., the best arm identification), and it is not interested in the scores collected
during the test phase (i.e., the cumulative reward).
The problem of best arm(s) identification has been studied in two distinct settings in the literature.
Fixed budget. In the fixed budget setting (see e.g., [3, 1]), the number of rounds of the exploration
phase is fixed and is known by the forecaster, and the objective is to maximize the probability of
returning the best arm(s). In the above example, the company fixes the length of the test phase before
hand (e.g., enrolls a fixed number of customers) and defines a strategy to choose which products to
show to the testers so that the final selected product is the best with the highest probability. Audibert
et al. [1] proposed two different strategies to solve this problem. They defined a strategy based
on upper confidence bounds, called UCB-E, whose optimal parameterization is strictly related to a
measure of the complexity of the problem. They also introduced an elimination algorithm, called
Successive Rejects, which divides the budget n in phases and discards one arm per phase. Both
algorithms were shown to have nearly optimal probability of returning the best arm. Deng et al. [5]
and Gabillon et al. [8] considered the extension of the best arm identification problem to the multi1
bandit setting, where the objective is to return the best arm for each bandit. Recently, Bubeck et
al. [4] extended the previous results to the problem of m-best arm identification and introduced a
new version of the Successive Rejects algorithm (with accept and reject) that is able to return the set
of the m-best arms with high probability.
Fixed confidence. In the fixed confidence setting (see e.g., [12, 6]), the forecaster tries to minimize the number of rounds needed to achieve a fixed confidence about the quality of the returned
arm(s). In the above example, the company keeps enrolling customers in the test until it is, e.g., 95%
confident that the best product has been identified. Maron & Moore [12] considered a slightly different setting where besides a fixed confidence also the maximum number of rounds is fixed. They
designed an elimination algorithm, called Hoeffding Races, based on progressively discarding the
arms that are suboptimal with enough confidence. Mnih et al. [14] introduced an improved algorithm, built on the Bernstein concentration inequality, which takes into account the empirical
variance of each arm. Even-Dar et al. [6] studied the fixed confidence setting without any budget
constraint and designed an elimination algorithm able to return an arm with a required accuracy
(i.e., whose performance is at least -close to the optimal arm). Kalyanakrishnan & Stone [10]
further extended this approach to the case where the m-best arms must be returned with a given
confidence. Finally, Kalyanakrishnan et al. [11] recently introduced an algorithm for the case of
m-best arm identification along with a thorough theoretical analysis showing the number of rounds
needed to achieve the desired confidence.
Although the fixed budget and fixed confidence problems have been studied separately, they display
several similarities. In this paper, we propose a unified approach to these two settings in the general
case of m-best arm identification with accuracy .1 The main contributions of the paper can be
summarized as follows:
Algorithm. In Section 3, we propose a novel meta-algorithm, called unified gap-based exploration
(UGapE), which uses the same arm selection and (arm) return strategies for the two settings. This
algorithm allows us to solve settings that have not been covered in the previous work (e.g., the case
of 6= 0 has not been studied in the fixed budget setting). Furthermore, we show in Appendix C of
[7] that UGapE outperforms existing algorithms in some settings (e.g., it improves the performance
of the algorithm by Mnih et al. [14] in the fixed confidence setting). We also provide a thorough
empirical evaluation of UGapE and compare it with a number of existing fixed budget and fixed
confidence algorithms in Appendix C of [7].
Theoretical analysis. Similar to the algorithmic contribution, in Section 4, we show that a large
portion of the theoretical analysis required to study the behavior of the two settings of the UGapE
algorithm can be unified in a series of lemmas. The final theoretical guarantees are thus a direct
consequence of these lemmas when used in the two specific settings.
Problem complexity. In Section 4.4, we show that the theoretical analysis indicates that the two
problems share exactly the same definition of complexity. In particular, we show that the probability
of success in the fixed budget setting as well as the sample complexity in the fixed confidence setting
strictly depend on the inverse of the gaps of the arms and the desired accuracy .
Extensions. Finally, in Appendix B of [7], we discuss how the proposed algorithm and analysis can
be extended to improved definitions of confidence interval (e.g., Bernstein-based bounds) and to
more complex settings, such as the multi-bandit best arm identification problem introduced in [8].
2
Problem Formulation
In this section, we introduce the notation used throughout the paper. Let A = {1, . . . , K} be the set
of arms such that each arm k ? A is characterized by a distribution ?k bounded in [0, b] with mean
?k and variance ?k2 . We define the m-max and m-argmax operators as2
m
?(m) = max ?k
k?A
and
m
(m) = arg max ?k ,
k?A
where (m) denotes the index of the m-th best arm in A and ?(m) is its corresponding mean so that
?(1) ? ?(2) ? . . . ? ?(K) . We denote by S m ? A any subset of m arms (i.e., |S m | = m < K) and
by S m,? the subset of the m best arms (i.e., k ? S m,? iif ?k ? ?(m) ). Without loss of generality, we
1
2
Note that when = 0 and m = 1 this reduces to the standard best arm identification problem.
Ties are broken in an arbitrary but consistent manner.
2
assume there exists a unique set S m,? . In the following we drop the superscript m and use S = S m
and S ? = S m,? whenever m is clear from the context. With a slight abuse of notation we further
extend the m-max operator to an operator returning a set of arms, such that
1..m
{?(1) , . . . , ?(m) } = max ?k
1..m
S ? = arg max ?k .
and
k?A
For each arm k ? A, we define the gap ?k as
?k ? ?(m+1)
?k =
?(m) ? ?k
k?A
if k ? S ?
.
if k ?
/ S?
This definition of gap indicates that if k ? S ? , ?k represents the ?advantage? of arm k over the
suboptimal arms, and if k ?
/ S ? , ?k denotes how suboptimal arm k is. Note that we can also write
m
the gap as ?k = | max ?i ? ?k |. Given an accuracy and a number of arms m, we say that an arm
i6=k
k is (,m)-optimal if ?k ? ?(m) ? . Thus, we define the (,m)-best arm identification problem as
the problem of finding a set S of m (,m)-optimal arms.
The (,m)-best arm identification problem can be formalized as a game between a stochastic bandit
environment and a forecaster. The distributions {?k } are unknown to the forecaster. At each round t,
the forecaster pulls an arm I(t) ? A and observes an independent sample drawn from the distribution
?I(t) . The forecaster estimates the expected value of each arm by computing the average of the
samples observed over time. Let Tk (t) be the number of times that arm k has been pulled by the end
PTk (t)
of round t, then the mean of this arm is estimated as ?
bk (t) = Tk1(t) s=1
Xk (s), where Xk (s) is
the s-th sample observed from ?k . For any arm k ? A, we define the notion of arm simple regret as
rk = ?(m) ? ?k ,
(1)
and for any set S ? A of m arms, we define the simple regret as
rS = max rk = ?(m) ? min ?k .
k?S
k?S
(2)
We denote by ?(t) ? A the set of m arms returned by the forecaster at the end of the exploration
phase (when the alg. stops after t rounds), and by r?(t) its corresponding simple regret. Returning
m (,m)-optimal arms is then equivalent to having r?(t) smaller than . Given an accuracy and a
number of arms m to return, we now formalize the two settings of fixed budget and fixed confidence.
Fixed budget. The objective is to design a forecaster capable of returning a set of m (,m)-optimal
arms with the largest possible confidence using a fixed budget of n rounds. More formally, given
a budget n, the performance of
is measured by the probability ?e of not meeting the
the forecaster
e the better the algorithm.
(,m) requirement, i.e., ?e = P r?(n) ? , the smaller ?,
Fixed confidence. The goal is to design a forecaster that stops as soon as possible and returns a set
of m (,m)-optimal arms with a fixed confidence. We denote by n
e the time when the algorithm stops
and
by
?(e
n
)
its
set
of
returned
arms.
Given
a
confidence
level
?,
the forecaster has to guarantee that
P r?(en) ? ? ?. The performance of the forecaster is then measured by the number of rounds n
e
either in expectation or high probability.
Although these settings have been considered as two distinct problems, in Section 3 we introduce
a unified arm selection strategy that can be used in both cases by simply changing the stopping
criteria. Moreover, we show in Section 4 that the bounds on the performance of the algorithm in the
two settings share the same notion of complexity and can be derived using very similar arguments.
3
Unified Gap-based Exploration Algorithm
In this section, we describe the unified gap-based exploration (UGapE) meta-algorithm and show
how it is implemented in the fixed-budget and fixed-confidence settings. As shown in Figure 1, both
fixed-budget (UGapEb) and fixed-confidence (UGapEc) instances of UGapE use the same armselection strategy, SELECT-ARM (described in Figure 2), and upon stopping, return the m-best
arms in the same manner (using ?). The two algorithms only differ in their stopping criteria. More
precisely, both algorithms receive as input the definition of the problem (, m), a constraint (the
3
budget n in UGapEb and the confidence level ? in UGapEc), and a parameter (a or c). While
UGapEb runs for n rounds and then returns the set of arms ?(n), UGapEc runs until it achieves the
desired accuracy with the requested confidence level ?. This difference is due to the two different
objectives targeted by the algorithms; while UGapEc optimizes its budget for a given confidence
level, UGapEb?s goal is to optimize the quality of its recommendation for a fixed budget.
UGapEc (, m, ?, c)
Parameters: accuracy , number of arms m,
confidence level ?, exploration parameter c
Initialize: Pull each arm k once, update
?
bk (K), set Tk (K) = 1 and t ? K + 1
SAMP
while BJ(t) (t) ? do
SELECT-ARM (t)
t?t+1
end while
SAMP
Return ?(t) = J(t)
UGapEb (, m, n, a)
Parameters: accuracy , number of arms m,
budget n, exploration parameter a
Initialize: Pull each arm k once, update ?
bk (K)
and set Tk (K) = 1
SAMP
for t = K + 1, . . . , n do
SELECT-ARM (t)
end for
SAMP
Return ?(n) = arg min BJ(t) (t)
J(t)
Figure 1: The pseudo-code for the UGapE algorithm in the fixed-budget (UGapEb) (left) and fixedconfidence (UGapEc) (right) settings.
Regardless of the final objective, how to select
an arm at each round (arm-selection strategy) is SELECT-ARM (t)
Compute Bk (t) for each arm k ? A
the key component of any multi-arm bandit al1..m
gorithm. One of the most important features of
Identify the set of m arms J(t) ? arg min Bk (t)
k?A
UGapE is having a unique arm-selection stratPull the arm I(t) = arg max ?k (t ? 1)
egy for the fixed-budget and fixed-confidence
k?{lt ,ut }
settings. We now describe the UGapE?s armObserve XI(t) TI(t) (t ? 1) + 1 ? ?I(t)
selection strategy, whose pseudo-code has been
Update ?
bI(t) (t) and TI(t) (t)
reported in Figure 2. At each time step t,
UGapE first uses the observations up to time t?
Figure 2: The pseudo-code for the UGapE?s armm
1 and computes an index Bk (t) = max Ui (t) ? selection strategy. This routine is used in both
i6=k
UGapEb and UGapEc instances of UGapE.
L (t) for each arm k ? A, where
k
?t, ?k ? A
Uk (t) = ?
bk (t ? 1) + ?k (t ? 1)
,
Lk (t) = ?
bk (t ? 1) ? ?k (t ? 1).
(3)
3
In Eq. 3, ?k (t ? 1) is a confidence interval, and Uk (t) and Lk (t) are high probability upper and
lower bounds on the mean of arm k, ?k , after t ? 1 rounds. Note that the parameters a and c are used
in the definition of the confidence interval ?k , whose shape strictly depends on the concentration
bound used by the algorithm. For example, we can derive ?k from the Chernoff-Hoeffding bound as
s
3
r
c log 4K(t?1)
a
?
UGapEb: ?k (t ? 1) = b
,
UGapEc: ?k (t ? 1) = b
.
(4)
Tk (t ? 1)
Tk (t ? 1)
In Sec. 4, we discuss how the parameters a and c can be tuned and we show that while a should be
tuned as a function of n and in UGapEb, c = 1/2 is always a good choice for UGapEc. Defining
the confidence interval in a general form ?k (t ? 1) allows us to easily extend the algorithm by taking
into account different (higher) moments of the arms (see Appendix B of [7] for the case of variance,
where ?k (t ? 1) is obtained from the Bernstein inequality). From Eq. 3, we may see that the index
Bk (t) is an upper-bound on the simple regret rk of the kth arm (see Eq. 1). We also define an index
for a set S as BS (t) = maxi?S Bi (t). Similar to the arm index, BS is also defined in order to
upper-bound the simple regret rS with high probability (see Lemma 1).
After computing the arm indices, UGapE finds a set of m arms J(t) with minimum upper-bound
1..m
on their simple regrets, i.e., J(t) = arg min Bk (t). From J(t), it computes two arm indices ut =
k?A
arg maxj ?J(t)
Uj (t) and lt = arg mini?J(t) Li (t), where in both cases the tie is broken in favor of
/
3
To be more precise, ?k (t ? 1) is the width of a confidence interval or a confidence radius.
4
the arm with the largest uncertainty ?(t ? 1). Arms lt and ut are the worst possible arm among those
in J(t) and the best possible arm left outside J(t), respectively, and together they represent how bad
the choice of J(t) could be. Finally, the algorithm selects and pulls the arm I(t)
as the arm with the
larger ?(t ? 1) among ut and lt , observes a sample XI(t) TI(t) (t ? 1) + 1 from the distribution
?I(t) , and updates the empirical mean ?
bI(t) (t) and the number of pulls TI(t) (t) of the selected arm
I(t).
There are two more points that need to be discussed about the UGapE algorithm. 1) While UGapEc
defines the set of returned arms as ?(t) = J(t), UGapEb returns the set of arms J(t) with the
smallest index, i.e., ?(n) = arg minJ(t) BJ(t) (t), t ? {1, . . . , n}. 2) UGapEc stops (we refer to
the number of rounds before stopping as n
e) when BJ(en+1) (e
n + 1) is less than the given accuracy
, i.e., when even the mth worst upper-bound on the arm simple regret among all the arms in the
selected set J(e
n + 1) is smaller than . This guarantees that the simple regret (see Eq. 2) of the set
returned by the algorithm, ?(e
n) = J(e
n + 1), to be smaller than with probability larger than 1 ? ?.
4
Theoretical Analysis
In this section, we provide high probability upper-bounds on the performance of the two instances
of the UGapE algorithm, UGapEb and UGapEc, introduced in Section 3. An important feature of
UGapE is that since its fixed-budget and fixed-confidence versions share the same arm-selection
strategy, a large part of their theoretical analysis can be unified. We first report this unified part of
the proof in Section 4.1, and then provide the final performance bound for each of the algorithms,
UGapEb and UGapEc, separately, in Sections 4.2 and 4.3, respectively.
Before moving to the main results, we define additional notation used in the analysis. We first define
event E as
E = ?k ? A, ?t ? {1, . . . , T }, ?
bk (t) ? ?k < ?k (t) ,
(5)
where the values of T and ?k are defined for each specific setting separately. Note that event E plays
an important role in the sequel, since it allows us to first derive a series of results which are directly
implied by the event E and to postpone the study of the stochastic nature of the problem (i.e., the
probability of E) in the two specific settings. In particular, when E holds, we have that for any arm
k ? A and at any time t, Lk (t) ? ?k ? Uk (t). Finally, we define the complexity of the problem as
H =
K
X
b2
i=1
max( ?i2+ , )2
.
(6)
Note that although the complexity has an explicit dependence on , it also depends on the number of
arms m through the definition of the gaps ?i , thus making it a complexity measure of the (, m) best
arm identification problem. In Section 4.4, we will discuss why the complexity of the two instances
of the problem is measured by this quantity.
4.1
Analysis of the Arm-Selection Strategy
Here we report lower (Lemma 1) and upper (Lemma 2) bounds for indices BS on the event E, which
show their connection with the regret and gaps. The technical lemmas used in the proofs (Lemmas 3
and 4 and Corollary 1) are reported in Appendix A of [7]. We first prove that for any set S 6= S ?
and any time t ? {1, . . . , T }, the index BS (t) is an upper-bound on the simple regret of this set rS .
Lemma 1. On event E, for any set S 6= S ? and any time t ? {1, . . . , T }, we have BS (t) ? rS .
Proof. On event E, for any arm i ?
/ S ? and each time t ? {1, . . . , T }, we may write
m
m
Bi (t) = max Uj (t) ? Li (t) = max ?
bj (t ? 1) + ?j (t ? 1) ? ?
bi (t ? 1) ? ?i (t ? 1)
j6=i
j6=i
m
? max ?j ? ?i = ?(m) ? ?i = ri .
(7)
j6=i
Using Eq. 7, we have
BS (t) = max Bi (t) ?
i?S
max
i?(S?S ? )
Bi (t) ?
max
i?(S?S ? )
ri = rS ,
where the last passage follows from the fact that ri ? 0 for any i ? S ? .
5
Lemma 2. On event E, if arm k ? {lt , ut } is pulled at time t ? {1, . . . , T }, we have
BJ(t) (t) ? min 0, ??k + 2?k (t ? 1) + 2?k (t ? 1).
(8)
Proof. We first prove the statement for B(t) = Uut (t) ? Llt (t), i.e.,
B(t) ? min 0, ??k + 2?k (t ? 1) + 2?k (t ? 1).
(9)
We consider the following cases:
Case 1. k = ut :
Case 1.1. ut ? S ? : Since by definition ut ?
/ J(t), there exists an arm j ?
/ S ? such that j ? J(t).
Now we may write
(a)
(b)
(c)
(d)
?(m+1) ? ?j ? Lj (t) ? Llt (t) ? Lut (t) = ?
bk (t ? 1) ? ?k (t ? 1) ? ?k ? 2?k (t ? 1)
(10)
(a) and (d) hold because of event E, (b) follows from the fact that j ? J(t) and from the definition
of lt , and (c) is the result of Lemma 4. From Eq. 10, we may deduce that ??k + 2?k (t ? 1) ? 0,
which together with Corollary 1 gives us the desired result (Eq. 9).
Case 1.2. ut ?
/ S?:
Case 1.2.1. lt ? S ? : In this case, we may write
(a)
B(t) = Uut (t) ? Llt (t) ? ?ut + 2?ut (t ? 1) ? ?lt + 2?lt (t ? 1)
(b)
(c)
? ?ut + 2?ut (t ? 1) ? ?(m) + 2?lt (t ? 1) ? ??ut + 4?ut (t ? 1)
(11)
?
(a) holds because of event E, (b) is from the fact that lt ? S , and (c) is because ut is pulled, and
thus, ?ut (t ? 1) ? ?lt (t ? 1). The final result follows from Eq. 11 and Corollary 1.
Case 1.2.2. lt ?
/ S ? : Since lt ?
/ S ? and the fact that by definition lt ? J(t), there exists an
?
arm j ? S such that j ?
/ J(t). Now we may write
(a)
(b)
(c)
(d)
?ut + 2?ut (t ? 1) ? Uut (t) ? Uj (t) ? ?j ? ?(m)
(12)
(a) and (c) hold because of event E, (b) is from the definition of ut and the fact that j ?
/ J(t), and
(d) holds because j ? S ? . From Eq. 12, we may deduce that ??ut + 2?ut (t ? 1) ? 0, which
together with Corollary 1 gives us the final result (Eq. 9).
With similar arguments and cases, we prove the result of Eq. 9 for k = lt . The final statement of the lemma (Eq. 8) follows directly from BJ(t) (t) ? B(t) as shown in Lemma 3.
Using Lemmas 1 and 2, we define an upper and a lower bounds on BJ(t) in terms of quantities
related to the regret of J(t). Lemma 1 confirms the intuition that the B-values upper-bound the
regret of the corresponding set of arms (with high probability). Unfortunately, this is not enough
to claim that selecting J(t) as the set of arms with smallest B-values actually correspond to arms
with small regret, since BJ(t) could be an arbitrary loose bound on the regret. Lemma 2 provides
this complementary guarantee specifically for the set J(t), in the form of an upper-bound on BJ(t)
w.r.t. the gap of k ? {ut , lt }. This implies that as the algorithm runs, the choice of J(t) becomes
more and more accurate since BJ(t) is constrained between rJ(t) and a quantity (Eq. 8) that gets
smaller and smaller, thus implying that selecting the arms with the smaller B-value, i.e., the set J(t),
corresponds to those which actually have the smallest regret, i.e., the arms in S ? . This argument will
be implicitly at the basis of the proofs of the two following theorems.
4.2
Regret Bound for the Fixed-Budget Setting
Here we prove an upper-bound on the simple-regret of UGapEb. Since the setting considered by the
algorithm is fixed-budget, we may set T = n. From the definition of the confidence interval ?i (t)
in Eq. 4 and a union bound, we have that P(E) ? 1 ? 2Kn exp(?2a).4 We now have all the tools
needed to prove the performance of UGapEb for the m (,m)-best arm identification problem.
4
The extension to a confidence interval that takes into account the variance of the arms is discussed in
Appendix B of [7].
6
Theorem 1. If we run UGapEb with parameter 0 < a ? n?K
4H , its simple regret r?(n) satisfies
?e = P r?(n) ? ? 2Kn exp(?2a),
and in particular this probability is minimized for a =
n?K
4H .
Proof. The proof is by contradiction. We assume that r?(n) > on event E and consider the
following two steps:
Step 1: Here we show that on event E, we have the following upper-bound on the number of pulls
of any arm i ? A:
4ab2
Ti (n) <
(13)
2 + 1.
max ?i2+ ,
Let ti be the last time that arm i is pulled. If arm i has been pulled only during the initialization
phase, Ti (n) = 1 and Eq. 13 trivially holds. If i has been selected by SELECT-ARM, then we have
(a)
(b)
(c)
(d)
min ? ?i + 2?i (ti ? 1), 0 + 2?i (ti ? 1) ? B(ti ) ? BJ(ti ) (ti ) ? B?(n) (t` ) > ,
(14)
where t` ? {1, . . . , n} is the time such that ?(n) = J(t` ). (a) and (b) are the results of Lemmas 2
and 3, (c) is by the definition of ?(n), and (d) holds because using Lemma 1, we know that if
the algorithm suffers a simple regret r?(n) > (as assumed at the beginning of the proof), then
?t = 1, . . . , n + 1, B?(n) (t) > . By the definition of ti , we know Ti (n) = Ti (ti ? 1) + 1. Using
this fact, the definition of ?i (ti ? 1), and Eq. 14, it is straightforward to show that Eq. 13 holds.
Step 2: We know that
PK
i=1
Ti (n) = n. Using Eq. 13, we have
4ab2
PK
i=1
max
+K > n
2
?i +
,
2
on event E. It is easy to see that by selecting a ? n?K
4H , the left-hand-side of this inequality will be
smaller than or equal to n, which is a contradiction. Thus, we conclude that r?(n) ? on event E.
The final result follows from the probability of event E defined at the beginning of this section.
4.3
Regret Bound for the Fixed-Confidence Setting
Here we prove an upper-bound on the simple-regret of UGapEc. Since the setting considered by the
algorithm is fixed-confidence, we may set T = +?. From the definition of the confidence interval
?i (t) in Eq. 4 and a union bound on Tk (t) ? {0, . . . , t}, t = 1, . . . , ?, we have that P(E) ? 1 ? ?.
Theorem 2. The UGapEc algorithm stops after n
e rounds and returns a set of m arms, ?(e
n), that
satisfies
P r?(en+1) ? ? n
e ? N ? 1 ? ?,
where N = K + O(H log H? ) and c has been set to its optimal value 1/2.
Proof. We first prove the bound on the simple regret of UGapEc. Using Lemma 1, we have that on
event E, the simple regret of UGapEc upon stopping satisfies BJ(en+1) (e
n + 1) = B?(en+1) (e
n + 1) ?
r?(en+1) . As a result, on event E, the regret of UGapEc cannot be bigger than , because then it
contradicts the stopping condition of the algorithm, i.e., BJ(en+1) (e
n + 1) < . Therefore, we have
P r?(en+1) ? ? 1 ? ?. Now we prove the bound for the sample complexity. Similar to the proof
of Theorem 1, we consider the following two steps:
Step 1: Here we show that on event E, we have the following upper-bound on the number of pulls
of any arm i ? A:
2b2 log(4K(e
n ? 1)3 /?)
Ti (e
n) ?
+ 1.
(15)
2
max ?i2+ ,
Let ti be the last time that arm i is pulled. If arm i has been pulled only during the initialization
phase, Ti (e
n) = 1 and Eq. 15 trivially holds. If i has been selected by SELECT-ARM, then we have
BJ(ti ) (ti ) ? . Now using Lemma 2, we may write
BJ(ti ) (ti ) ? min 0, ??i + 2?i (ti ? 1) + 2?i (ti ? 1).
(16)
We can prove Eq. 15 by plugging in the value of ?i (ti ? 1) from Eq. 4 and solving Eq. 16 for Ti (ti )
taking into account that Ti (ti ? 1) + 1 = Ti (ti ).
7
PK
Step 2:
We know that i=1 Ti (e
n) = n
e. Using Eq. 15, on event E, we have 2H log K(e
n?
3
1) /? + K ? n
e. Solving this inequality gives us n
e ? N.
4.4
Problem Complexity
Theorems 1 and 2 indicate that both the probability of success and sample complexity of UGapE are
directly related to the complexity H defined by Eq. 6. This implies that H captures the intrinsic
difficulty of the (,m)-best arm(s) identification problem independently from the specific setting
considered. Furthermore, note that this definition generalizes existing notions of complexity. For
example, for = 0 and m = 1 we recover the complexity used in the definition of UCB-E [1] for
the fixed budget setting and the one defined in [6] for the fixed accuracy problem. Let us analyze
H in the general case of > 0. We define the complexity of a single arm i ? A, H,i =
b2 / max( ?i2+ , )2 . When the gap ?i is smaller than the desired accuracy , i.e., ?i ? , then
the complexity reduces to H,i = 1/2 . In fact, the algorithm can stop as soon as the desired
accuracy is achieved, which means that there is no need to exactly discriminate between arm i and
the best arm. On the other hand, when ?i > , then the complexity becomes H,i = 4b2 /(?i + )2 .
This shows that when the desired accuracy is smaller than the gap, the complexity of the problem is
smaller than the case of = 0, for which we have H0,i = 4b2 /?2i .
More in general, the analysis reported in the paper suggests that the performance of a upper confidence bound based algorithm such as UGapE is characterized by the same notion of complexity
in both settings. Thus, whenever the complexity is known, it is possible to exploit the theoretical
analysis (bounds on the performance) to easily switch from one setting to the other. For instance, as
also suggested in Section 5.4 of [9], if the complexity H is known, an algorithm like UGapEc can
be adapted to run in the fixed budget setting by inverting the bound on its sample complexity. This
would lead to an algorithm similar to UGapEb with similar performance, although the parameter
tuning could be more difficult because of the intrinsic poor accuracy in the constants of the bound.
On the other hand, it is an open question whether it is possible to find an ?equivalence? between
algorithms for the two different settings when the complexity is not known. In particular, it would
be important to derive a distribution-dependent lower bound in the form of the one reported in [1]
for the general case of ? 0 and m ? 1 for both the fixed budget and fixed confidence settings.
5
Summary and Discussion
We proposed a meta-algorithm, called unified gap-based exploration (UGapE), that unifies the two
settings of the best arm(s) identification problem in stochastic multi-armed bandit: fixed budget and
fixed confidence. UGapE can be instantiated as two algorithms with a common structure (the same
arm-selection and arm-return strategies) corresponding to these two settings, whose performance
can be analyzed in a unified way, i.e., a large portion of their theoretical analysis can be unified in
a series of lemmas. We proved a performance bound for the UGapE algorithm in the two settings.
We also showed how UGapE and its theoretical analysis can be extended to take into account the
variance of the arms and to multiple bandits. Finally, we evaluated the performance of UGapE and
compare it with a number of existing fixed budget and fixed confidence algorithms.
This unification is important for both theoretical and algorithmic reasons. Despite their similarities,
fixed budget and fixed confidence settings have been treated differently in the literature. We believe
that this unification provides a better understanding of the intrinsic difficulties of the best arm(s)
identification problem. In particular, our analysis showed that the same complexity term characterizes the hardness of both settings. As mentioned in the introduction, there was no algorithm
available for several settings considered in this paper, e.g., (,m)-best arm identification with fixed
budget. With UGapE, we introduced an algorithm that can be easily adapted to all these settings.
Acknowledgments This work was supported by Ministry of Higher Education and Research, NordPas de Calais Regional Council and FEDER through the ?contrat de projets ?tat region 2007?2013",
French National Research Agency (ANR) under project LAMPADA n? ANR-09-EMER-007, European Community?s Seventh Framework Programme (FP7/2007-2013) under grant agreement n?
270327, and PASCAL2 European Network of Excellence.
8
References
[1] J.-Y. Audibert, S. Bubeck, and R. Munos. Best arm identification in multi-armed bandits. In
Proceedings of the Twenty-Third Annual Conference on Learning Theory, pages 41?53, 2010.
[2] P. Auer, N. Cesa-Bianchi, and P. Fischer. Finite-time analysis of the multi-armed bandit problem. Machine Learning, 47:235?256, 2002.
[3] S. Bubeck, R. Munos, and G. Stoltz. Pure exploration in multi-armed bandit problems. In
Proceedings of the Twentieth International Conference on Algorithmic Learning Theory, pages
23?37, 2009.
[4] S. Bubeck, T. Wang, and N. Viswanathan. Multiple identifications in multi-armed bandits.
CoRR, abs/1205.3181, 2012.
[5] K. Deng, J. Pineau, and S. Murphy. Active learning for developing personalized treatment.
In Proceedings of the Twenty-Seventh International Conference on Uncertainty in Artificial
Intelligence, pages 161?168, 2011.
[6] E. Even-Dar, S. Mannor, and Y. Mansour. Action elimination and stopping conditions for
the multi-armed bandit and reinforcement learning problems. Journal of Machine Learning
Research, 7:1079?1105, 2006.
[7] V. Gabillon, M. Ghavamzadeh, and A. Lazaric. Best Arm Identification: A Unified Approach
to Fixed Budget and Fixed Confidence. Technical report 00747005, October 2012.
[8] V. Gabillon, M. Ghavamzadeh, A. Lazaric, and S. Bubeck. Multi-bandit best arm identification.
In Proceedings of Advances in Neural Information Processing Systems 25, pages 2222?2230,
2011.
[9] S. Kalyanakrishnan. Learning Methods for Sequential Decision Making with Imperfect Representations. PhD thesis, Department of Computer Science, The University of Texas at Austin,
Austin, Texas, USA, December 2011. Published as UT Austin Computer Science Technical
Report TR-11-41.
[10] S. Kalyanakrishnan and P. Stone. Efficient selection of multiple bandit arms: Theory and practice. In Proceedings of the Twenty-Seventh International Conference on Machine Learning,
pages 511?518, 2010.
[11] S. Kalyanakrishnan, A. Tewari, P. Auer, and P. Stone. Pac subset selection in stochastic multiarmed bandits. In Proceedings of the Twentieth International Conference on Machine Learning, 2012.
[12] O. Maron and A. Moore. Hoeffding races: Accelerating model selection search for classification and function approximation. In Proceedings of Advances in Neural Information Processing Systems 6, pages 59?66, 1993.
[13] A. Maurer and M. Pontil. Empirical bernstein bounds and sample-variance penalization. In
22th annual conference on learning theory, 2009.
[14] V. Mnih, Cs. Szepesv?ri, and J.-Y. Audibert. Empirical Bernstein stopping. In Proceedings of
the Twenty-Fifth International Conference on Machine Learning, pages 672?679, 2008.
[15] H. Robbins. Some aspects of the sequential design of experiments. Bulletin of the American
Mathematics Society, 58:527?535, 1952.
9
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4,023 | 4,641 | Density-Difference Estimation
Masashi Sugiyama1 Takafumi Kanamori2 Taiji Suzuki3
Marthinus Christoffel du Plessis1 Song Liu1 Ichiro Takeuchi4
1
Tokyo Institute of Technology, Japan 2 Nagoya University, Japan
3
University of Tokyo, Japan 4 Nagoya Institute of Technology, Japan
Abstract
We address the problem of estimating the difference between two probability densities. A naive approach is a two-step procedure of first estimating two densities
separately and then computing their difference. However, such a two-step procedure does not necessarily work well because the first step is performed without regard to the second step and thus a small estimation error incurred in the first stage
can cause a big error in the second stage. In this paper, we propose a single-shot
procedure for directly estimating the density difference without separately estimating two densities. We derive a non-parametric finite-sample error bound for
the proposed single-shot density-difference estimator and show that it achieves the
optimal convergence rate. We then show how the proposed density-difference estimator can be utilized in L2 -distance approximation. Finally, we experimentally
demonstrate the usefulness of the proposed method in robust distribution comparison such as class-prior estimation and change-point detection.
1 Introduction
When estimating a quantity consisting of two elements, a two-stage approach of first estimating
the two elements separately and then approximating the target quantity based on the estimates of
the two elements often performs poorly, because the first stage is carried out without regard to the
second stage and thus a small estimation error incurred in the first stage can cause a big error in the
second stage. To cope with this problem, it would be more appropriate to directly estimate the target
quantity in a single-shot process without separately estimating the two elements.
A seminal example that follows this general idea is pattern recognition by the support vector machine [1]: Instead of separately estimating two probability distributions of patterns for positive and
negative classes, the support vector machine directly learns the boundary between the two classes
that is sufficient for pattern recognition. More recently, a problem of estimating the ratio of two
probability densities was tackled in a similar fashion [2, 3]: The ratio of two probability densities is
directly estimated without going through separate estimation of the two probability densities.
In this paper, we further explore this line of research, and propose a method for directly estimating
the difference between two probability densities in a single-shot process. Density differences would
be more desirable than density ratios because density ratios can diverge to infinity even under a
mild condition (e.g., two Gaussians [4]), whereas density differences are always finite as long as
each density is bounded. Density differences can be used for solving various machine learning tasks
such as class-balance estimation under class-prior change [5] and change-point detection in time
series [6].
For this density-difference estimation problem, we propose a single-shot method, called the leastsquares density-difference (LSDD) estimator, that directly estimates the density difference without
separately estimating two densities. LSDD is derived with in the framework of kernel regularized
least-squares estimation, and thus it inherits various useful properties: For example, the LSDD
1
solution can be computed analytically in a computationally efficient and stable manner, and all
tuning parameters such as the kernel width and the regularization parameter can be systematically
and objectively optimized via cross-validation. We derive a finite-sample error bound for the LSDD
estimator and show that it achieves the optimal convergence rate in a non-parametric setup.
We then apply LSDD to L2 -distance estimation and show that it is more accurate than the difference of KDEs, which tends to severely under-estimate the L2 -distance [7]. Because the L2 -distance
is more robust against outliers than the Kullback-Leibler divergence [8], the proposed L2 -distance
estimator can lead to the paradigm of robust distribution comparison. We experimentally demonstrate the usefulness of LSDD in semi-supervised class-prior estimation and unsupervised change
detection.
2
Density-Difference Estimation
In this section, we propose a single-shot method for estimating the difference between two probability densities from samples, and analyze its theoretical properties.
Problem Formulation and Naive Approach: First, we formulate the problem of densitydifference estimation. Suppose that we are given two sets of independent and identically distributed
samples X := {xi }ni=1 and X := {xi }ni =1 from probability distributions on Rd with densities
p(x) and p (x), respectively. Our goal is to estimate the density difference,
f (x) := p(x) ? p (x),
from the samples X and X .
A naive approach to density-difference estimation is to use kernel density estimators (KDEs). However, we argue that the KDE-based density-difference estimator is not the best approach because
of its two-step nature. Intuitively, good density estimators tend to be smooth and thus the difference between such smooth density estimators tends to be over-smoothed as a density-difference
estimator [9]. To overcome this weakness, we give a single-shot procedure of directly estimating the
density difference f (x) without separately estimating the densities p(x) and p (x).
Least-Squares Density-Difference Estimation: In our proposed approach, we fit a densitydifference model g(x) to the true density-difference function f (x) under the squared loss:
2
g(x) ? f (x) dx.
argmin
g
We use the following Gaussian kernel model as g(x):
n+n
x ? c 2
? exp ?
,
g(x) =
2? 2
(1)
=1
where (c1 , . . . , cn , cn+1 , . . . , cn+n ) := (x1 , . . . , xn , x1 , . . . , xn ) are Gaussian kernel centers. If
n + n is large, we may use only a subset of {x1 , . . . , xn , x1 , . . . , xn } as Gaussian kernel centers.
For the model (1), the optimal parameter ? ? is given by
2
?
? := argmin
g(x) ? f (x) dx = argmin ? H? ? 2h ? = H ?1 h,
?
?
where H is the (n + n ) ? (n + n ) matrix and h is the (n + n )-dimensional vector defined as
x ? c 2
c ? c 2
x ? c 2
2 d/2
)
exp
?
exp
?
dx
=
(??
,
H, := exp ?
2? 2
2? 2
4? 2
x ? c 2
x ? c 2
p(x)dx
?
exp
?
p (x )dx .
h := exp ?
2? 2
2? 2
Replacing the expectations in h by empirical estimators and adding an 2 -regularizer to the objective
function, we arrive at the following optimization problem:
? + ?? ? ,
:= argmin ? H? ? 2h
(2)
?
?
2
is the (n + n )-dimensional vector defined as
where ? (? 0) is the regularization parameter and h
n
n
1
xi ? c 2
xi ? c 2
1
exp ?
exp
?
?
.
h :=
n i=1
2? 2
n
2? 2
i =1
Taking the derivative of the objective function in Eq.(2) and equating it to zero, we can obtain the
solution analytically as
= (H + ?I)?1 h,
?
where I denotes the identity matrix.
Finally, a density-difference estimator f
(x), which we call the least-squares density-difference
(LSDD) estimator, is given as
f
(x) =
n+n
=1
x ? c 2
?
exp ?
.
2? 2
Non-Parametric Error Bound: Here, we theoretically analyze an estimation error of LSDD.
We assume n = n, and let H? be the reproducing kernel
Hilbert space
(RKHS) corresponding to
the Gaussian kernel with width ?: k? (x, x ) = exp ?x ? x 2 /? 2 . Let us consider a slightly
modified LSDD estimator that is more suitable for non-parametric error analysis1 :
n
n
1
1
2
2
g(xi ) ?
g(xi ) + ?gH? .
f := argmin gL2 (Rd ) ? 2
n i=1
n
g?H?
i =1
Then we have the following theorem:
Theorem 1. Suppose that there exists a constant M such that p? ? M and p ? ? M .
Suppose also that the density difference f = p ? p is a member of Besov space with regularity ?.
?
?
where B2,?
is the Besov space with regularity ?, and
That is, f ? B2,?
?
f B2,?
:= f L2 (Rd ) + sup(t?? ?r,L2 (Rd ) (f, t)) < c for r = ? + 1,
t>0
where ? denotes the largest integer less than or equal to ? and ?r,L2 (Rd ) is the r-th modulus of
smoothness (see [10] for the definitions). Then, for all
> 0 and p ? (0, 1), there exists a constant
K > 0 depending on M , c,
, and p such that for all n ? 1, ? ? 1, and ? > 0, the LSDD estimator
f
in H? satisfies
2(1?p)d
1?p
? ?(1?p)(1+)d ? ? 1+p (1++ 4 )
?
?
2
2
?d
2?
+
f ? f L2 (Rd ) +?f H? ? K ?? +? +
+ 2 +
3p?p2
2
?p n
n ? n
? 1+p n 1+p
with probability not less than 1 ? 4e?? .
2?+d
1
If we set ? = n? (2?+d)(1+p)+(?p+p) and ? = n? (2?+d)(1+p)+(?p+p) , and take
and p sufficiently
small, then we immediately have the following corollary.
Corollary 1. Suppose that the same assumptions as Theorem 1 hold. Then, for all ?, ? > 0, there
exists a constant K > 0 depending on M, c, ?, and ? such that, for all n ? 1 and ? ? 1, the
density-difference estimator f
with appropriate choice of ? and ? satisfies
2?
f
? f 2L2 (Rd ) + ?f
2H? ? K n? 2?+d +? + ? n?1+?
with probability not less than 1 ? 4e?? .
1
More specifically, the regularizer is replaced from the squared 2 -norm of parameters to the squared RKHSnorm of a learned function, which is necessary to establish consistency. Nevertheless, we use the squared
2 -norm of parameters in experiments because it is simpler and seems to perform well in practice.
3
2?
?
Note that n? 2?+d is the optimal learning rate to estimate a function in B2,?
. Therefore, the densitydifference estimator with a Gaussian kernel achieves the optimal learning rate by appropriately
choosing the regularization parameter and the Gaussian width. Because the learning rate depends
on ?, the LSDD estimator has adaptivity to the smoothness of the true function.
It is known that, if the naive KDE with a Gaussian kernel is used for estimating a probability density
with regularity ? > 2, the optimal learning rate cannot be achieved [11, 12]. To achieve the optimal
rate by KDE, we should choose a kernel function specifically tailored to each regularity ? [13].
However, such a kernel function is not non-negative and it is difficult to implement it in practice.
On the other hand, our LSDD estimator can always achieve the optimal learning rate for a Gaussian
kernel without regard to regularity ?.
Model Selection by Cross-Validation: The above theoretical analysis showed the superiority of
LSDD. However, in practice, the performance of LSDD depends on the choice of models (i.e.,
the kernel width ? and the regularization parameter ?). Here, we show that the model can be
optimized by cross-validation (CV). More specifically, we first divide the samples X = {xi }ni=1
and X = {xi }ni =1 into T disjoint subsets {Xt }Tt=1 and {Xt }Tt=1 , respectively. Then we obtain a
density-difference estimate f
t (x) from X \Xt and X \Xt (i.e., all samples without Xt and Xt ), and
compute its hold-out error for Xt and Xt as
2
2
ft (x) +
ft (x ),
CV(t) := f
t (x)2 dx ?
|Xt |
|Xt |
x?Xt
x ?Xt
where |X | denotes the number of elements in the set X . We repeat this hold-out validation procedure for t = 1, . . . , T , and compute the average hold-out error. Finally, we choose the model that
minimizes the average hold-out error.
3 L2 -Distance Estimation by LSDD
In this section, we consider the problem of approximating the L2 -distance between p(x) and p (x),
2
L2 (p, p ) := (p(x) ? p (x)) dx,
from their independent and identically distributed samples X := {xi }ni=1 and X := {xi }ni =1 .
For an equivalent expression L2 (p, p ) = f (x)p(x)dx ? f (x )p (x )dx , if we replace f (x)
with an LSDD estimator f
(x) and approximate the expectations by empirical averages, we obtain
?.
Similarly, for another expression L2 (p, p ) = f (x)2 dx, replacing f (x) with
L2 (p, p ) ? h
H ?.
an LSDD estimator f
(x) gives L2 (p, p ) ? ?
?
themselves give approximations to L2 (p, p ), we argue that the use of
and ?
H ?
Although h
their combination, defined by
??
H ?,
?
2 (X , X ) := 2h
L
(3)
is more sensible. To explain the reason, let us consider a generalized L2 -distance estimator of the
?
where ? is a real scalar. If the regularization parameter ? (? 0) is
+ (1 ? ?)?
H ?,
form ? h
small, this can be expressed as
=h
H ?1 h
+ (1 ? ?)?
H ?
? ?(2 ? ?)h
H ?2 h
+ op (?),
?
?h
(4)
where op denotes the probabilistic order. Thus, up to Op (?), the bias introduced by regularization
(i.e., the second term in the right-hand side of Eq.(4) that depends on ?) can be eliminated if ? = 2,
and ?
H ?
?
which yields Eq.(3). Note that, if no regularization is imposed (i.e., ? = 0), both h
H ?1 h,
the first term in the right-hand side of Eq.(4).
yield h
4
Eq.(3) is actually equivalent to the negative of the optimal objective value of the LSDD optimization
problem without regularization (i.e., Eq.(2) with ? = 0). This can be naturally interpreted through a
lower bound of L2 (p, p ) obtained by Legendre-Fenchel convex duality [14]:
2
2
L (p, p ) = sup 2
g(x)p(x)dx ? g(x )p (x )dx ? g(x) dx ,
g
where the supremum is attained at g = f . If the expectations are replaced by empirical estimators and the Gaussian kernel model (1) is used as g, the above optimization problem is reduced
to the LSDD objective function without regularization (see Eq.(2)). Thus, LSDD corresponds to
approximately maximizing the above lower bound and Eq.(3) is its maximum value.
? h
?
??
H ?
? ?
H ?.
?
Through eigenvalue decomposition of H, we can show that 2h
?
and ?
H ?.
Thus, our approximator (3) is not less than the plain approximators h
4
Experiments
R
In this section, we experimentally demonstrate the usefulness of LSDD. A MATLAB
implementation of LSDD used for experiments is available from
?http://sugiyama-www.cs.titech.ac.jp/?sugi/software/LSDD/?.
Illustration: Let N (x; ?, ?) be the multi-dimensional normal density with mean vector ? and
variance-covariance matrix ? with respect to x, and let
p(x) = N (x; (?, 0, . . . , 0) , (4?)?1 I d )
and p (x) = N (x; (0, 0, . . . , 0) , (4?)?1 I d ).
We first illustrate how LSDD behaves under d = 1 and n = n = 200. We compare LSDD with
KDEi (KDE with two Gaussian widths chosen independently by least-squares cross-validation [15])
and KDEj (KDE with two Gaussian widths chosen jointly to minimize the LSDD criterion [9]). The
number of folds in cross-validation is set to 5 for all methods.
Figure 1 depicts density-difference estimation results obtained by LSDD, KDEi, and KDEj for ? = 0
(i.e., f (x) = p(x) ? p (x) = 0). The figure shows that LSDD and KDEj give accurate estimates
of the density difference f (x) = 0. On the other hand, the estimate obtained by KDEi is rather
fluctuated, although both densities are reasonably well approximated by KDEs. This illustrates an
advantage of directly estimating the density difference without going through separate estimation of
each density. Figure 2 depicts the results for ? = 0.5 (i.e., f (x) = 0), showing again that LSDD
performs well. KDEi and KDEj give the same estimation result for this dataset, which slightly
underestimates the peaks.
Next, we compare the performance of L2 -distance approximation based on LSDD, KDEi, and KDEj.
For ? = 0, 0.2, 0.4, 0.6, 0.8 and d = 1, 5, we draw n = n = 200 samples from the above p(x)
and p (x). Figure 3 depicts the mean and standard error of estimated L2 -distances over 1000 runs
as functions of mean ?. When d = 1 (Figure 3(a)), the LSDD-based L2 -distance estimator gives
the most accurate estimates of the true L2 -distance, whereas the KDEi-based L2 -distance estimator
slightly underestimates the true L2 -distance when ? is large. This is caused by the fact that KDE
tends to provide smooth density estimates (see Figure 2(b) again): Such smooth density estimates
are accurate as density estimates, but the difference of smooth density estimates yields a small L2 distance estimate [7]. The KDEj-based L2 -distance estimator tends to improve this drawback of
KDEi, but it still slightly underestimates the true L2 -distance when ? is large.
When d = 5 (Figure 3(b)), the KDE-based L2 -distance estimators even severely underestimate
the true L2 -distance when ? is large. On the other hand, the LSDD-based L2 -distance estimator
still gives reasonably accurate estimates of the true L2 -distance even when d = 5. However, we
note that LSDD also slightly underestimates the true L2 -distance when ? is large, because slight
underestimation tends to yield smaller variance and thus such stabilized solutions are more accurate
in terms of the bias-variance trade-off.
Semi-Supervised Class-Balance Estimation: In real-world pattern recognition tasks, changes in
class balance between the training and test phases are often observed. In such cases, naive classifier
5
1.6
1.6
1.6
f(x)
f(x)
1.4
^
p(x)?p?(x)
^
p(x)?p?(x)
p(x)
^
p(x)
p?(x)
^
p?(x)
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?0.5
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?1
?0.5
(a) LSDD
0
x
0.5
f(x)
f(x)
^
1.4
^
1
0.8
?0.4 b
1
?1
?0.5
(b) KDEi
0
x
0.5
1
(c) KDEj
Figure 1: Estimation of density difference when ? = 0 (i.e., f (x) = p(x) ? p (x) = 0).
1.5
1.5
b
1.5
f(x)
f(x)
f(x)
f(x)
^
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1
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p(x)
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?1
?0.5
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(a) LSDD
(b) KDEi
(c) KDEj
Figure 2: Estimation of density difference when ? = 0.5 (i.e., f (x) = p(x) ? p (x) = 0).
True
LSDD
KDE i
KDE j
1.6
1.4
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L2 distance
L2 distance
1.2
1
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LSDD
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KDE j
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0b
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(a) d = 1
0
0.1
0.2
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?
0.5
0.6
0.7
0.8
(b) d = 5
Figure 3: L2 -distance estimation by LSDD, KDEi, and KDEj for n = n = 200 as functions of the
Gaussian mean ?. Means and standard errors over 1000 runs are plotted.
training produces significant estimation bias because the class balance in the training dataset does
not properly reflect that of the test dataset.
Here, we consider a binary pattern recognition task of classifying pattern x ? Rd to class y ?
{+1, ?1}. Our goal is to learn the class balance of a test dataset in a semi-supervised learning setup
where unlabeled test samples are provided in addition to labeled training samples [16]. The class
balance in the test set can be estimated by matching a mixture of class-wise training input densities,
qtest (x; ?) := ?ptrain (x|y = +1) + (1 ? ?)ptrain (x|y = ?1),
to the test input density ptest (x) [5], where ? ? [0, 1] is a mixing coefficient to learn. See Figure 4
for schematic illustration. Here, we use the L2 -distance estimated by LSDD and the difference of
KDEs for this distribution matching. Note that, when LSDD is used to estimate the L2 -distance,
separate estimation of ptrain (x|y = ?1) is not involved, but the difference between ptest (x) and
qtest (x; ?) is directly estimated.
We use four UCI benchmark datasets (http://archive.ics.uci.edu/ml/), where we randomly choose 10 labeled training samples from each class and 50 unlabeled test samples following
true class-prior ? ? = 0.1, 0.2, . . . , 0.9. Figure 6 plots the mean and standard error of the squared
difference between true and estimated class-balances ? and the misclassification error by a weighted
2 -regularized least-squares classifier [17] with weighted cross-validation [18] over 1000 runs. The
results show that LSDD tends to provide better class-balance estimates than the KDEi-based, the
KDEj-based, and the EM-based methods [5], which are translated into lower classification errors.
6
Unsupervised Change Detection: The objective of change detection is to discover abrupt property changes behind time-series data. Let y(t) ? Rm be an m-dimensional time-series sample at
time t, and let Y (t) := [y(t) , y(t + 1) , . . . , y(t + k ? 1) ] ? Rkm be a subsequence of time
series at time t with length k. We treat the subsequence Y (t) as a sample, instead of a single point
y(t), by which time-dependent information can be incorporated naturally [6]. Let Y(t) be a set of r
retrospective subsequence samples starting at time t: Y(t) := {Y (t), Y (t + 1), . . . , Y (t + r ? 1)}.
Our strategy is to compute a certain dissimilarity measure between two consecutive segments Y(t)
and Y(t+r), and use it as the plausibility of change points (see Figure 5). As a dissimilarity measure,
we use the L2 -distance estimated by LSDD and the Kullback-Leibler (KL) divergence estimated by
the KL importance estimation procedure (KLIEP) [2, 3]. We set k = 10 and r = 50.
First, we use the IPSJ SIG-SLP Corpora and Environments for Noisy Speech Recognition (CENSREC) dataset (http://research.nii.ac.jp/src/en/CENSREC-1-C.html). This
dataset is provided by the National Institute of Informatics, Japan that records human voice in a
noisy environment such as a restaurant. The top graphs in Figure 7(a) display the original timeseries (true change points were manually annotated) and change scores obtained by KLIEP and
LSDD. The graphs show that the LSDD-based change score indicates the existence of change points
more clearly than the KLIEP-based change score.
Next, we use a dataset taken from the Human Activity Sensing Consortium (HASC) challenge
2011 (http://hasc.jp/hc2011/), which provides human activity information collected by
portable three-axis accelerometers. Because the orientation of the accelerometers is not necessarily
fixed, we take the 2 -norm of the 3-dimensional data. The HASC dataset is relatively simple, so
we artificially added zero-mean Gaussian noise with standard deviation 5 at each time point with
probability 0.005. The top graphs in Figure 7(b) display the original time-series for a sequence of
actions ?jog?, ?stay?, ?stair down?, ?stay?, and ?stair up? (there exists 4 change points at time 540,
1110, 1728, and 2286) and the change scores obtained by KLIEP and LSDD. The graphs show that
the LSDD score is much more stable and interpretable than the KLIEP score.
Finally, we compare the change-detection performance more systematically using the receiver operating characteristic (ROC) curves (i.e., the false positive rate vs. the true positive rate) and the
area under the ROC curve (AUC) values. In addition to LSDD and KLIEP, we test the L2 -distance
estimated by KDEi and KDEj and native change detection methods based on autoregressive models
(AR) [19], subspace identification (SI) [20], singular spectrum transformation (SST) [21], one-class
support vector machine (SVM) [22], kernel Fisher discriminant analysis (KFD) [23], and kernel
change-point detection (KCP) [24]. Tuning parameters included in these methods were manually optimized. For 10 datasets taken from each of the CENSREC and HASC data collections, mean ROC
curves and AUC values are displayed at the bottom of Figure 7(b). The results show that LSDD tends
to outperform other methods and is comparable to state-of-the-art native change-detection methods.
5
Conclusions
In this paper, we proposed a method for directly estimating the difference between two probability
density functions without density estimation. The proposed method, called the least-squares densitydifference (LSDD), was derived within the framework of kernel least-squares estimation, and its
solution can be computed analytically in a computationally efficient and stable manner. Furthermore,
LSDD is equipped with cross-validation, and thus all tuning parameters such as the kernel width and
the regularization parameter can be systematically and objectively optimized. We derived a finitesample error bound for LSDD in a non-parametric setup, and showed that it achieves the optimal
convergence rate. We also proposed an L2 -distance estimator based on LSDD, which nicely cancels
a bias caused by regularization. Through experiments on class-prior estimation and change-point
detection, the usefulness of the proposed LSDD was demonstrated.
Acknowledgments: We would like to thank Wittawat Jitkrittum for his comments and Za??d Harchaoui for providing us a program code of kernel change-point detection. MS was supported by
MEXT KAKENHI 23300069 and AOARD, TK was supported by MEXT KAKENHI 24500340,
TS was supported by MEXT KAKENHI 22700289, the Aihara Project, the FIRST program from
JSPS initiated by CSTP, and the Global COE Program ?The research and training center for new
development in mathematics?, MEXT, Japan, MCdP was supported by MEXT Scholarship, SL was
supported by the JST PRESTO program, and IT was supported by MEXT KAKENHI 23700165.
7
ptrain(x|y = +1)
Time
a
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i
Figure 5: Change-point detection.
Class balance squared error
Class balance squared error
Class balance squared error
LSDD
KDEi
h
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Y(t)
Figure 4: Class-balance estimation.
0.25
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d
(
k
Y (t + r ? 1)
ptest (x)
e
Class balance squared error
ptrain(x|y = ?1)
(
y(t + r)
y(t)
0.5
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(b) Diabetes dataset
0.2
0.4 ?* 0.6
0.4
0.3
0.2
0.1
0.8
(c) German dataset
(d) Statlogheart dataset
Figure 6: Results of semi-supervised class-balance estimation. Top: Squared error of class balance
estimation. Bottom: Misclassification error by a weighted 2 -regularized least-squares classifier.
0.1
5
Original data
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data
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KCP
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AUC LSDD KDEi KDEj KLIEP AR SI SST SVM KFD KCP AUC LSDD KDEi KDEj KLIEP AR SI SST SVM KFD KCP
Mean .879 .755 .705 .635 .749 .756 .580 .773 .905 .913 Mean .843 .764 .751 .638 .799 .762 .764 .815 .856 .730
.013 .029 .036 .020 .026 .020 .016 .018 .023 .032
SE
.014 .016 .023 .030 .013 .012 .023 .032 .013 .024
SE
(b) Accelerometer data
(a) Speech data
Figure 7: Results of unsupervised change detection. From top to bottom: Original time-series,
change scores obtained by KLIEP and LSDD, mean ROC curves over 10 datasets, and AUC values
for 10 datasets. The best method and comparable ones in terms of mean AUC values by the t-test at
the significance level 5% are indicated with boldface. ?SE? stands for ?Standard error?.
8
References
[1] V. N. Vapnik. Statistical Learning Theory. Wiley, New York, NY, USA, 1998.
[2] M. Sugiyama, T. Suzuki, S. Nakajima, H. Kashima, P. von B?unau, and M. Kawanabe. Direct importance estimation for covariate shift adaptation. Annals of the Institute of Statistical
Mathematics, 60(4):699?746, 2008.
[3] X. Nguyen, M. J. Wainwright, and M. I. Jordan. Estimating divergence functionals and the
likelihood ratio by convex risk minimization. IEEE Transactions on Information Theory,
56(11):5847?5861, 2010.
[4] C. Cortes, Y. Mansour, and M. Mohri. Learning bounds for importance weighting. In Advances
in Neural Information Processing Systems 23, pages 442?450, 2010.
[5] M. Saerens, P. Latinne, and C. Decaestecker. Adjusting the outputs of a classifier to new a
priori probabilities: A simple procedure. Neural Computation, 14(1):21?41, 2002.
[6] Y. Kawahara and M. Sugiyama. Sequential change-point detection based on direct densityratio estimation. Statistical Analysis and Data Mining, 5(2):114?127, 2012.
[7] N. Anderson, P. Hall, and D. Titterington. Two-sample test statistics for measuring discrepancies between two multivariate probability density functions using kernel-based density estimates. Journal of Multivariate Analysis, 50(1):41?54, 1994.
[8] A. Basu, I. R. Harris, N. L. Hjort, and M. C. Jones. Robust and efficient estimation by minimising a density power divergence. Biometrika, 85(3):549?559, 1998.
[9] P. Hall and M. P. Wand. On nonparametric discrimination using density differences.
Biometrika, 75(3):541?547, 1988.
[10] M. Eberts and I. Steinwart. Optimal learning rates for least squares SVMs using Gaussian
kernels. In Advances in Neural Information Processing Systems 24, pages 1539?1547, 2011.
[11] R. H. Farrell. On the best obtainable asymptotic rates of convergence in estimation of a density
function at a point. The Annals of Mathematical Statistics, 43(1):170?180, 1972.
[12] B. W. Silverman. Density Estimation for Statistics and Data Analysis. Chapman and Hall,
London, UK, 1986.
[13] E. Parzen. On the estimation of a probability density function and mode. The Annals of
Mathematical Statistics, 33(3):1065?1076, 1962.
[14] R. T. Rockafellar. Convex Analysis. Princeton University Press, Princeton, NJ, USA, 1970.
[15] W. H?ardle, M. M?uller, S. Sperlich, and A. Werwatz. Nonparametric and Semiparametric
Models. Springer, Berlin, Germany, 2004.
[16] O. Chapelle, B. Sch?olkopf, and A. Zien, editors. Semi-Supervised Learning. MIT Press,
Cambridge, MA, USA, 2006.
[17] R. Rifkin, G. Yeo, and T. Poggio. Regularized least-squares classification. In Advances in
Learning Theory: Methods, Models and Applications, pages 131?154. IOS Press, Amsterdam,
the Netherlands, 2003.
[18] M. Sugiyama, M. Krauledat, and K.-R. M?uller. Covariate shift adaptation by importance
weighted cross validation. Journal of Machine Learning Research, 8:985?1005, May 2007.
[19] Y. Takeuchi and K. Yamanishi. A unifying framework for detecting outliers and change points
from non-stationary time series data. IEEE Transactions on Knowledge and Data Engineering,
18(4):482?489, 2006.
[20] Y. Kawahara, T. Yairi, and K. Machida. Change-point detection in time-series data based on
subspace identification. In Proceedings of the 7th IEEE International Conference on Data
Mining, pages 559?564, 2007.
[21] V. Moskvina and A. A. Zhigljavsky. An algorithm based on singular spectrum analysis for
change-point detection. Communication in Statistics: Simulation & Computation, 32(2):319?
352, 2003.
[22] F. Desobry, M. Davy, and C. Doncarli. An online kernel change detection algorithm. IEEE
Transactions on Signal Processing, 53(8):2961?2974, 2005.
[23] Z. Harchaoui, F. Bach, and E. Moulines. Kernel change-point analysis. In Advances in Neural
Information Processing Systems 21, pages 609?616, 2009.
[24] S. Arlot, A. Celisse, and Z. Harchaoui. Kernel change-point detection. Technical Report
1202.3878, arXiv, 2012.
9
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4,024 | 4,642 | Exploration in Model-based Reinforcement Learning
by Empirically Estimating Learning Progress
Manuel Lopes
INRIA
Bordeaux, France
Tobias Lang
FU Berlin
Germany
Marc Toussaint
FU Berlin
Germany
Pierre-Yves Oudeyer
INRIA
Bordeaux, France
Abstract
Formal exploration approaches in model-based reinforcement learning estimate
the accuracy of the currently learned model without consideration of the empirical prediction error. For example, PAC-MDP approaches such as R- MAX base
their model certainty on the amount of collected data, while Bayesian approaches
assume a prior over the transition dynamics. We propose extensions to such
approaches which drive exploration solely based on empirical estimates of the
learner?s accuracy and learning progress. We provide a ?sanity check? theoretical analysis, discussing the behavior of our extensions in the standard stationary
finite state-action case. We then provide experimental studies demonstrating the
robustness of these exploration measures in cases of non-stationary environments
or where original approaches are misled by wrong domain assumptions.
1
Introduction
Reinforcement learning (RL) agents need to solve the exploitation-exploration tradeoff. They have
to exploit their current model of the environment. At the same time they need to explore the environment sufficiently to learn more about its reward-relevant structure. Established model-based
approaches like E 3 [8] and R- MAX [4] take into account how often a state-action pair has been
visited. With an efficient model learner, the estimated transition model can be guaranteed to be
approximately correct after a sufficient, and efficient, number of visitations in a stationary domain.
An alternative approach to exploration is Bayesian RL [11]. Bayesian RL exploits prior knowledge
about the transition dynamics to reason explicitly about the uncertainty of the estimated model. Interestingly, these existing approaches estimate the accuracy of the currently learned model based
only on visitation counts. They do not consider the actual empirical prediction performance or
learning rate of the learner w.r.t. the data seen so far.
What happens if the fundamental presumption of R- MAX and Bayesian RL fails that each single
seen data-point will increase the agent?s certainty about its model? This is the case if the transition dynamics change over time (so there is no correct stationary prior), or if we want to be able
to ignore non-learnable, currently ?too difficult? parts of the state space, either to the level of noise
or limitations of a learning algorithm. For example, a household robot cannot learn how to repair
a cupboard until it has achieved a basic understanding for the handling of tools. Such scenarios
require the development of new methods and even new measures of success. Previous work into this
direction emphasizes the concept of intrinsic motivation [10, 13, 12] and has shown empirical success in developmental robotics [1]. An interesting aspect about this work is its reliance on empirical
measures of learning progress to drive exploration in reinforcement learning [17, 6]. However, to
our knowledge this has not been made rigorous or related to fundamental methods like R- MAX or
Bayesian RL.
In this paper, we aim to draw these relations and make the following contributions: (i) We propose
to drive exploration in model-based RL by the estimated progress in model learning. We estimate
this progress in terms of the loss over the training data used for model learning. (ii) We introduce
1
two algorithms based on modifications of R- MAX and the recent Bayesian exploration bonus (BEB)
approach [9]. In contrast to the existing approaches, our algorithms do not have to assume correct
prior knowledge or that the visitation counts translate directly to model certainty. Hence, they can
also cope with changing dynamics. (iii) While our extensions are targeted at scenarios that go
beyond the standard domain of stationary unstructured, finite state and action spaces, we provide
a kind of theoretical sanity check of our extensions exactly under these standard assumptions: We
discuss exploration guarantees under standard assumptions analogous to those of R- MAX and BEB.
In the next section, we review background work on exploration in Markov decision processes. Then
we present our approaches for exploration based on an empirical estimate of the future model learning progress. Thereafter, we discuss guarantees on their exploration efficiency. Finally, we present
an empirical evaluation before we conclude.
2
Background on Exploration
We model the interaction of an agent with its environment as a Markov decision process (MDP).
An MDP is a discrete-time stochastic control process where at each time-step the process is in one
of a fixed set S of states and the agent can choose an action from a set A. The transition model T
specifies the conditional transition distribution T (s0 | a, s) over successor states s0 when executing
an action a in a given state s. In unstructured finite state and action spaces T can be defined by
separate multinomial distributions T (s, a) over successor states for each state-action pair s, a. The
agent receives rewards in states according to a function R : S ? R. A policy ? : S ? A
specifies for each state the action to take. The goal of planning in an MDP is to find the optimal
policy ? ? which maximizes the expected future rewards E[R]. The future rewards R can be defined
PH
P t
for a fixed horizon H, R =
t=1 R(st ), or for an infinite horizon, R =
t ? R(st ), using a
discount factor 0 < ? < 1. The value of state s when acting according to policy ? is defined as
the expected future rewards when starting from s, V ? (s) = E[R | s1 = s, ?]. The optimal policy
? ? can be found by classical algorithms such as value iteration or policy iteration. In reinforcement
learning, the agent does not know the transition model T . In a model-based approach, the agent
estimates T? from its interaction trace with the environment ? = hs1 , r1 , a1 , . . . , sT , rT i. Based on
T? it computes (approximately) optimal policies. A simple approach to the exploitation-exploration
tradeoff is -greedy: the agent performs a random action for exploration with probability and
exploits otherwise by executing a greedy policy with respect to T? . If decreases over time towards
0, -greedy exploration converges to ? ? . However, this may take an inefficiently large number of
non-optimal actions which is exponential in |S| and |A|.
In the PAC-MDP (probably approximately correct) framework, the efficiency of an exploration
algorithm A is measured by its sample complexity [7]. This is the number of time-steps when folA
?
lowing A where its policy ?tA at time t is not -optimal, that is, where V ?t (st ) < V ? (st ) ? .
Given ? with 0 < ? < 1, A is said to be PAC-MDP efficient if with probability 1 ? ? its sample complexity scales polynomially in quantities describing T as well as in ? and (and ?). The model-based
RL algorithms E 3 [8] and R- MAX [4] are PAC-MDP efficient exploration methods for unstructured
finite state and action spaces: their sample complexity scales polynomially in |S| and |A|. E 3
and R- MAX share the central concept of known states and actions which have been observed sufficiently often. (Counts are also used in the theoretical analysis of alternative PAC-MDP efficient
algorithms like MBIE-EB [15].) If the visitation count c(s, a) of a state-action pair s, a is larger
than some threshold m, the estimate T? (s, a) is guaranteed to be with high probability -close to
the true model. Following a policy in these known states achieves approximately the same rewards
in the learned model T? and the true model T . To drive exploration, R- MAX is ?optimistic in the
face of uncertainty? and assumes maximum reward Rmax in unknown states. This gives the reward
function
R(s, a) c(s, a) ? m (s, a known)
RR- MAX (s, a) =
.
Rmax
c(s, a) < m (s, a unknown)
Typically, the threshold m is very large as E 3 and R- MAX need to account for all possible model
instantiations within the model class of T [14]. For instance, for conservative choices = 0.1,
? = 0.9, ? = 0.1, S = 10, A = 5, we get m > 106 , which is unfeasible in practice. PAC-MDP
approaches like R- MAX ignore the current empirical progress in learning T : the threshold m is fixed
2
a-priori and remains the same for all s, a independently of the agent?s experiences or its estimated
relevance of s, a for large rewards.
Bayesian reinforcement learning [11] is an alternative approach to exploration. Here, the agent
takes its uncertainty about the learned model T? explicitly into account. This allows to incorporate
prior knowledge. More formally, the agent maintains a posterior belief b over all possible transition
models T given its previous experiences ? and a prior. The value function for a deterministic policy
?(b, s) is defined in terms of the state s and the belief state b and fulfills
Z
V ? (b, s) = R(s, ?(b, s)) +
T (b0 , s0 | b, s, ?(b, s)) V ? (b0 , s0 ) .
(1)
b0 ,s0
?
The optimal Bayesian policy ? = argmax? V ? (b, s) solves the exploitation-exploration tradeoff
implicitly: ? ? considers how actions affect not only the state of the world, but also the agent?s internal belief about the world. In a Bayesian RL context for a finite horizon H, the sample complexity
of an algorithm A can be defined as the number of time-steps when following A where its policy
?A
?
?tA at time t is not -Bayesian-optimal, that is, where VH t (bt , st ) < VH? (bt , st ) ? . Due to
the complexity of the belief state, however, Bayesian RL is typically intractable in terms of both
planning and updating the belief after an action. A recent approximate solution to Bayesian RL
for unstructured finite representations is the Bayesian exploration bonus (BEB) [9] which resembles
closely MBIE-EB [15]. In this setting, the belief b over T can be expressed by means of a separate
0
)
Dirichlet distribution ?(s, a, s0 ) for each s, a with a mean estimator T?b (s0 | s, a) = P?(s,a,s
0) .
?(s,a,s
0
s
BEB avoids reasoning in the belief space: it solves an MDP built from the mean estimate T?b using
an additional exploration bonus ?/(1 + c(s, a)) to reward state-action pairs inversely according to
their visitation counts c(s, a). In the undiscounted, fixed horizon H formulation BEB has a Bayesian
sample complexity which with probability 1 ? ?P
is polynomial in |S| and |A| when setting ? = 2H 2
and updating the belief for s, a is stopped once s0 ?(s, a, s0 ) > 4H 3 / [9].
Bayesian RL approaches are vulnerable to incorrect priors. Fard and Pineau [5] have combined
Bayesian and PAC approaches to derive bounds on the approximation error in the value function of
policies regardless of the correctness of the prior. However, their bounds do not apply to changing
transition dynamics and it remains unanswered how to incorporate them in efficient exploration
algorithms.
In a wider context of RL and developmental robotics, many strategies for efficient exploration have
been subsumed by the concept of intrinsic motivation [10, 13] which is also termed fun or curiosity
[12]. Many of these approaches take empirical learning progress into account. This includes methods that estimate from the agent?s experience the amount of potential learning progress in different
regions of the state space. Thereby, exploration focuses on those areas where learning progress
can indeed be made: areas which are neither already well-understood nor currently too difficult to
learn. The resulting algorithms enable an agent to develop progressively more complex skills. For
instance, this has been demonstrated for learning in robot control [1] and options learning in hierarchical reinforcement learning [16] and has also been considered lately in machine learning under the
name of curriculum learning [3]. Under the RL formalism only very recently have the concept of intrinsic motivation been mixed with standard exploration-exploitation strategies [6]. So far, however,
guarantees about the sample complexity of intrinsic motivation based exploration approaches have
been missing. In this paper, we take up the ideas of intrinsic motivation to extend the theoretically
founded exploration approaches described above.
3
Exploration by Empirically Estimated Model Accuracy and Learning
Progress
PAC-MDP approaches like R- MAX and Bayesian RL approaches like BEB have been developed
in the context of stationary finite state-action domains. In those problems, we know that after a
fixed number of visits to a state its estimated transition model becomes approximately correct and
we can perform exact belief updates to guarantee Bayesian optimality. In the following, we present
extensions which rely instead on previous exploration and learning experience to estimate in which
parts of the state and action space further exploration is promising and where not. This is helpful
in situations where the basic assumptions of R- MAX and BEB about model improvement might be
violated: for example, when we have an incorrect prior (large misspecified priors may impair the
3
performance of BEB[9]), when we cannot come up with an a-priori threshold on the data number
for learning accurate models (e.g. in domains where a polynomial KWIK learner is not available), or
when the transition dynamics change over time. We can also see our approach as a method to adjust
the standard R- MAX threshold online, allowing to adapt to different levels of noise on different parts
of the state space.
3.1
Exploration Driven by Learning Progress
Let ? : S ? A 7? R denote a measure for the expected learning and exploration progress when
visiting a state-action pair s, a. We discuss concrete definitions of ? later. Clearly, ? is a nonstationary function which changes with new experiences. Hence, an exploration strategy based on ?
needs to re-estimate ? with each new experience. We use ? to define two exploration algorithms.
Our first approach ?-R- MAX is based on R- MAX [4]. ?-R- MAX acts greedily with respect to the
optimal policy for the reward function
R(s, a) ?(s, a) < m
R?-R- MAX (s, a) =
.
(2)
Rmax
else
Instead of rewarding arbitrary states with low visitation counts (considered unknown) directly as
in R- MAX, ?-R- MAX gets large reward for exploring such state-action pairs where the expected
learning progress is large.
Our second approach ?-EB is based on Bayesian Exploration Bonus (BEB) [9]. ?-EB acts greedily
with respect to the optimal policy for the reward function
R?-EB (s, a) = R(s, a) +
?
1+ ?
1
?(s,a)
(3)
for some constant ?. Instead of setting the exploration bonus directly proportional to visitation
counts as in BEB , ?-EB gets a bonus for exploring state-actions pairs where the expected learning
progress is large. The idea of using expected learning progress to drive exploration is that we can
estimate ? empirically from the interaction data ? = hs1 , a1 , r1 , s2 , . . . , sT , rT i.
3.2
Empirically Estimated Model Accuracy and Learning Progress
We start by considering an empirical estimate of the current model accuracy. In a classical learning
context, model accuracy is ideally tested on a held-out test data set. However, to exploit the full
available data for model selection and algorithm comparison, cross validation methods have become
a standard method. The work [2] discusses the importance of estimating the variance of a crossvalidation estimator of the model accuracy, for instance to include this uncertainty of model accuracy
in a model selection criterion. We base our following treatment of ? on this previous work.
Let T? denote the estimated transition model based on data ?, which approximates the true model
T . In general, we assume that learning T? (s, a) implies minimizing the loss L(T? (s, a); Ds,a ) where
ns,a
Ds,a = {s0i }i=1
are the successor states in the transitions (s, a, s0i ) from s, a in ?. T? may generalize
over states and actions, in which case the data for learning T? (s, a) and evaluating the loss may
include experience sets Ds0 ,a0 with s0 6= s, a0 6= a (this is for example important for relational or
continuous domains). A typical loss is the neg-log data-likelihood,
Y
1
log
T? (s0 | s, a) .
(4)
L(T? ; Ds,a ) = ?
|Ds,a |
0
s ?Ds,a
Given such a loss, the predictive error is defined as
P E(s, a) = Es0 ?T (s,a) L(T? ; {s0 }) .
(5)
An empirical estimator of the PE based on the available data Ds,a is the leave-one-out crossvalidation estimator
1 X loo
? ?s0 ; {s0 })
CV (Ds,a , s, a) =
`s0 , `loo
(6)
s0 := L(T
|Ds,a | 0
s ?D
4
0
0
?s
where T? ?s is the model learned from data Ds,a
= Ds,a \ {s0 }.
Putting an absolute threshold directly on the loss to decide whether
a state is known or unknown
is hard. Note that the predictive error P E(s, a) = KL T (s, a) T? (s, a) + H T (s, a) has the
entropy of the true distribution as a lower bound, which is unknown. Therefore, we propose to
drive exploration based on the learning progress instead of the current learner accuracy. Using the
change in loss we may gain robustness by becoming independent of the loss? absolute value and can
potentially detect time-varying conditions.
?k
We define ? in terms of the change in the (empirically estimated) loss as follows. Let Ds,a
= {s0i ?
?k
Ds,a | i < ns,a ? k} denote the experiences in Ds,a except the last k. T?
is the transition model
?k
?
learned from the reduced data-set Ds,a , in contrast to T which is learned from all data Ds,a . We
define
? a) := CV (D?k , s, a) ? CV (Ds,a , s, a) ? L(T? ?k ; Ds,a ) ? L(T? ; Ds,a ) .
?(s,
(7)
s,a
This estimates to which extent the last k experiences help to learn a better model as evaluated over
? a) is small, then the last k visitations in the data-set Ds,a did not
the complete data. Thus, if ?(s,
have a significant effect on improving T? and in turn s, a does not require further exploration.
? a) defined above is only a mean estimator of the learning progress. The ?(s, a) we
The estimator ?(s,
use in concrete exploration algorithms ?-EB and ?-R- MAX includes an additional variance margin,
p
? a) + ? ?(s, a) ,
?(s, a) := ?(s,
(8)
where ?(s, a) is an estimate of the CV variance (discussed in more detail below),
X
1
2
?(s, a) =
[`loo
s0 ? CV (Ds,a , s, a)] .
|Ds,a | 0
(9)
s ?Ds,a
The variance margin increases robustness and is motivated by the following analysis.
3.3
Guarantees on the Exploration Efficiency
As discussed in the introduction, we propose our extensions of R- MAX and BEB to target scenarios
which go beyond the standard setting of stationary unstructured, finite state and action spaces. In
this subsection, however, we go back and consider the behavior of our extensions exactly under
these classical assumptions?this is meant as a sanity check to ensure that our extensions inherit
the standard exploration efficiency properties under standard assumptions. We will directly relate a
threshold on the empirical ?(s, a) to a threshold on model accuracy.
D
E
? a)
under random data. First we find
We start by providing two properties of the mean ?(s,
Ds,a
? a) converges with 1/n2 :
that the expected ?(s,
Lemma 1. For a Dirichlet learner in a stationary environment, we have
D
E
1
?
?(s, a)
=O
.
n2s,a
Ds,a
D
E
? a)
A proof is given in the supplementary material. Similarly, a threshold on the mean ?(s,
Ds,a
implies a model accuracy threshold:
Lemma 2. Given an approximated model T? of a true model T , for any there exists an 0 such
that:
D
E
? a)
| ?(s,
| < 0 ? |T? (s, a) ? T (s, a)| < .
(10)
Ds,a
Sketch of proof. For the case of multinomial variables, we know that the maximum likelihood
estimator is consistent ?
and unbiased and is equal to the normalized visit counts. In this situation we
p ? p)
N (0, ?). Our measure of progress ? is the difference between
know that as n ? ?, n(?
5
two maximum likelihood estimators and so, as n approaches infinity we have the same limiting result
on the model quality, with a higher variance due to the subtraction between two different random
variables and the correlation between them.
D
E
? a)
With the previous results we know that if we had access to the mean ?(s,
under random
Ds,a
data we would be able to assess model error by looking at its value. Unfortunately, the agent only
has access to an empirically estimate. To ensure that we can define robust criteria for considering a
? a) under random data Ds,a .
state as known, we have to consider the variability of the estimator ?(s,
As discussed in [2], the estimator is unbiased, that is hCV (Ds,a , s, a)iD = P E(s, a) and, in the
limit |Ds,a | ? ?, the (T? ?T ) becomes Gaussian. Its variance can be described by first considering
the covariance matrix C of the vector (`loo
s0 )s0 ?Ds,a under random Ds,a . The diagonal entries are the
loo
0
variances VarD {`loo
}
of
each
single
`
0
0
s
s under random data, which are independent of s (assuming
i.i.d. data) and therefore equally estimated as ?(s, a) given in Eq. (9). The off-diagonal entries of C
loo
capture the correlations between different `loo
s0 and `s00 and are constant (due to i.i.d. data; see [2] for
details). By assuming these correlations to vanish we over-estimate the CV variance and therefore
have, from Eq. (6),
VarDs,a {CV (Ds,a , s, a)} ? ?(s, a)
(11)
Having an overestimation of the variance of the loss we will consider what is the variance of the
? a). Both terms L(T? ; Ds,a ) and L(T? ?k ; Ds,a ) are estimated using LOO-CV,
estimation of ?(s,
? variance under random data, from Eq. (11), as:
allowing us to bound the ??s
? a)} ? 2?(Ds,a , s, a) .
VarDs,a {?(s,
(12)
Now that we have a confidence measurepon the estimator we can show that a threshold
on the
D
E
? a)
? a) + ? ?(s, a) implies a threshold on the mean ?(s,
:
empirical estimator ?(s, a) = ?(s,
Ds,a
Lemma 3. For any given ? with 0 < ? < 1 and > 0 there exists an ? such that
D
E
p
? a)| + ? ?(s, a) < ? | ?(s,
? a)
|?(s,
| < with probability 1 ? ? .
Ds,a
(13)
?
Proof. For a Gaussian variable x with mean ? and variance
? ? we know that x < ? + ? ? with
probability given by the error function ? = 1/2 + erf(?/ 2)/2. By inverting this we get ? to fulfill
the above.
Finally, we show that our exploration method using the empirical ?(s, a) to drive exploration is
PAC-MDP efficient.
Lemma 4. There is a threshold m such that ?-R- MAX using a Dirichlet learner in the standard
setting of stationary unstructured, finite state and action spaces is PAC-MDP efficient.
Proof. From Lemma 3 we know that a threshold on our empirical
measure
implies a threshold on
D
E
? a)
the mean measure. From Lemma 2 we know that a small ?(s,
corresponds, with high
Ds,a
probability, to a low model error. Under these conditions a state is only marked as known if the
empirical measure is below a certain threshold; this ensures with high probability that the error in
the model for the state-action is low. From the standard conditions of R- MAX, ?-R- MAX is PACMDP efficient.
4
Evaluation
We compare the empirical performance of our exploration algorithms ?-R- MAX and ?-EB with
the performance of R- MAX, BEB and simple model-based -greedy exploration with optimistic
initialization in unstructured finite state and action spaces. We investigate different scenarios where
the assumptions of R- MAX and BEB are fulfilled or violated. We define these scenarios by varying
the level of stochasticity in state transitions. BEB assumes to have a correct a prior about this
stochasticity. R- MAX assumes to know correct thresholds m for the number of visits to states to
6
ensure accurate transition models. We simulate satisfied or violated assumptions of R- MAX by
setting individual thresholds for states: in general, states with higher noise require more samples to
achieve the same level of model accuracy as states with low noise. Setting individual thresholds is
equivalent to setting individual initials counts (instead of 0s) for states.
We investigate three questions: (a) How close do our algorithms ?-R- MAX and ?-EB get to the
performance of the original algorithms when the assumptions of the latter are correct? (b) How
much do ?-R- MAX and ?-EB gain if the assumptions of the original algorithms are violated? (c)
And are our approaches more robust to unexpected changes in the transition dynamics?
Our evaluation environment (shown in Fig. 1(a)) is a discrete MDP with 25 states and five actions:
up, down, left, right and stop. There is a single goal state with a reward of 1 (marked with ?G?) and
several states with negative rewards ?0.1 (darker states). The lighter states are noisy states: their
actions have less predictable effects. The transition probabilities of the noisy states are sampled
from a Dirichlet distribution with parameters ? = 0.1, while the probabilities for all other states are
sampled from a Dirichlet with ? = 1.0, in both cases eventually permuted to ensure that the highest
probability corresponds to the expected next state according to the name definition of the actions.
The shortest path from start to goal is not optimal due to the uncertainty in the transitions. Instead,
the optimal path avoids the lighter states and takes the route below. To find this optimal path and
avoid local minima, an exploration algorithm needs to explore sufficiently and estimate the state
values accurately. We evaluate the performance of the algorithms in terms of the reward collected
in the true model T using the optimal policy ?T?? for their learned model T? . In our results, we report
the policy value error defined as VT (sI ; ?T?? ) ? VT (sI ; ?T? ) in the value of the start state sI with
respect to the optimal policy ? ? . In our experiments, the agent is reset to sI every 30 steps. All
figures presented in the following show the means and standard deviations over 20 runs. For the ?
estimation we use k = 10.
Experiment 1: Correct Assumptions In our first experiment, the assumptions of BEB and RMAX are fulfilled: BEB is given the correct prior; R- MAX uses appropriate thresholds for states
(depending on the state stochasticity). ?-R- MAX, ?-EB and -greedy are not given any knowledge.
The results presented in Fig. 1(b) show that our exploration methods ?-R- MAX and ?-EB achieve
similar performance to the original methods, even without having a correct prior or state-dependent
thresholds. Both ?-R- MAX and ?-EB converge to the correct final policy, requiring only moderately more steps. In contrast, -greedy does not find the optimal policy in reasonable time. Clearly,
the original algorithms could also be executed based on likewise correct, but less informative assumptions: BEB with an uninformative prior, R- MAX with more conservative (larger) uniform
threeholds. Then, R- MAX would need longer learning time; BEB might not converge, see [9].
Experiment 2: Violated Assumptions Here, the assumptions of R- MAX and BEB are violated
(wrong thresholds/priors). This may well happen in practice where correct priors cannot always be
specified or the counts cannot be translated directly to model certainty. In each run, we initialize
R- MAX and BEB with a random uniform prior (in the interval of the minimum and maximum values
of the true prior, translated to counts for R- MAX). The results in Fig. 1(c) show that as expected
R- MAX and BEB do not converge any longer to the optimal policy: they explore states too long
whose dynamics are already well estimated, while neglecting states which require more samples for
an accurate model estimation. In contrast, ?-R- MAX and ?-EB do not rely on these assumptions and
again converge to the optimal policy.
Experiment 3: Change in Dynamics In our final experiment, the transition dynamics for a randomly chosen state along the optimal path get permuted after 900 steps. As Fig. 1(d) shows, R- MAX
and BEB with correct assumptions for the original problem (before time-step 900) cannot compensate for this as they base their estimate of the model certainty only on the visitation counts, but do
not look at the data itself. In contrast, ?-R- MAX and ?-EB detect such an unexpected event and can
refocus their exploration efforts.
5
Conclusions and Future Work
We have proposed to drive exploration in model-based reinforcement learning using the estimated
future progress in model learning. When estimating this learning progress empirically, exploration
7
I
G
(a) Evaluation Environment
(b) Experiment 1?Correct Assumptions
(c) Experiment 2?Violated Assumptions
(d) Experiment 3?Change in Dynamics
Figure 1: Experiments: (a) The agent starts at state ?I? and needs to get to goal ?G?. Grey (darker)
states incur a negative reward. Green (lighter) states have very noisy transition dynamics. (b) Like
R- MAX and BEB with correct assumptions, our algorithms ?-R- MAX and ?-EB based on an empirical estimation of the learning progress converge to the optimal policy without relying on these
assumptions, but take a small extra amount of time. (c) When their assumptions are violated, RMAX and BEB fail to converge, while ?-R- MAX and ?-EB don?t rely on these assumptions and
again find the optimal policy. (d) In contrast to existing methods, ?-R- MAX and ?-EB can cope with
the change in transition dynamics after 900 steps and refocus their exploration.
algorithms can be defined which do not rely on correct prior knowledge and can cope with changing
transition dynamics. As a theoretical ?sanity check? we have discussed efficiency guarantees of our
approaches similar to the ones of the established algorithms. Our novel problem settings provide
interesting opportunities for the development of RL algorithms and theoretical analyses for relevant
real-world scenarios, in particular for structured, continuous and non-stationary domains. It is also
worth to investigate in more depth the relation of our approach to the general concept of intrinsic
motivation as proposed in developmental robotics. In our view, a combination of methods which
trades off both strong prior assumptions together with empirical estimates of the learning progress
seems to be the most promising direction for future work on exploration in the real world.
Acknowledgments
Work supported by the Flowers Team (INRIA/ENSTA-Paristech), Conseil R?egional d?Aquitaine
and the ERC grant EXPLORERS 24007. Tobias Lang and Marc Toussaint were supported by the
German Research Foundation under grants TO 409/1-3 and TO 409/7-1.
8
References
[1] A. Baranes and P.Y. Oudeyer. Active learning of inverse models with intrinsically motivated
goal exploration in robots. Robotics and Autonomous Systems, 2012.
[2] Yoshua Bengio and Yves Grandvalet. No unbiased estimator of the variance of k-fold crossvalidation. Journal of Machine Learning Research (JMLR), 5:1089?1105, 2004.
[3] Yoshua Bengio, J?er?ome Louradour, Ronan Collobert, and Jason Weston. Curriculum learning.
In Int. Conf. on Machine Learning (ICML), pages 41?48, 2009.
[4] Ronen I. Brafman and Moshe Tennenholtz. R-max - a general polynomial time algorithm for
near-optimal reinforcement learning. Journal of Machine Learning Research (JMLR), 3:213?
231, 2002.
[5] Mahdi Milani Fard and Joelle Pineau. Pac-bayesian model selection for reinforcement learning. In Conf. on Neural Information Processing Systems (NIPS). 2010.
[6] Todd Hester and Peter Stone. Intrinsically motivated model learning for a developing curious
agent. In AAMAS Workshop on Adaptive Learning Agents (ALA), 2012.
[7] Sham Kakade. On the Sample Complexity of Reinforcement Learning. PhD thesis, Gatsby
Computational Neuroscience Unit, University College London, 2003.
[8] Michael Kearns and Satinder Singh. Near-optimal reinforcement learning in polynomial time.
Machine Learning Journal, 49(2-3):209?232, 2002.
[9] J. Zico Kolter and Andrew Ng. Near-Bayesian exploration in polynomial time. In Int. Conf.
on Machine Learning (ICML), pages 513?520, 2009.
[10] P.Y. Oudeyer, F. Kaplan, and V.V. Hafner. Intrinsic motivation systems for autonomous mental
development. IEEE Transactions on Evolutionary Computation, 11(2):265?286, 2007.
[11] Pascal Poupart, Nikos Vlassis, Jesse Hoey, and Kevin Regan. An analytic solution to discrete
Bayesian reinforcement learning. In Int. Conf. on Machine Learning (ICML), 2006.
[12] J?urgen Schmidhuber. Curious model-building control systems. In Proc. of Int. Joint Conf. on
Neural Networks, volume 2, pages 1458?1463, 1991.
[13] Satinder Singh, Andrew G. Barto, and Nuttapong Chentanez. Intrinsically motivated reinforcement learning. In Conf. on Neural Information Processing Systems (NIPS), pages 1281?1288.
2005.
[14] Alexander L. Strehl, Lihong Li, and Michael Littman. Reinforcement learning in finite MDPs:
PAC analysis. Journal of Machine Learning Research (JMLR), 2009.
[15] Alexander L. Strehl and Michael L. Littman. An analysis of model-based interval estimation
for markov decision processes. J. Comput. Syst. Sci., 74(8):1309?1331, 2008.
[16] Christopher M. Vigorito and Andrew G. Barto. Intrinsically motivated hierarchical skill learning in structured environments. IEEE Transactions on Autonomous Mental Development
(TAMD), 2(2), 2010.
[17] Marco Wiering and J?urgen Schmidhuber. Efficient model-based exploration. In International
Conference on Simulation of Adaptive Behavior: From Animals to Animats 6, 1998.
9
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4,025 | 4,643 | Minimax Multi-Task Learning and a Generalized
Loss-Compositional Paradigm for MTL
Nishant A. Mehta? , Dongryeol Lee?, Alexander G. Gray?
[email protected], [email protected], [email protected]
?
College of Computing, Georgia Institute of Technology, Atlanta, GA 30332, USA
?
GE Global Research, Niskayuna, NY 12309, USA
Abstract
Since its inception, the modus operandi of multi-task learning (MTL) has been to
minimize the task-wise mean of the empirical risks. We introduce a generalized
loss-compositional paradigm for MTL that includes a spectrum of formulations as
a subfamily. One endpoint of this spectrum is minimax MTL: a new MTL formulation that minimizes the maximum of the tasks? empirical risks. Via a certain relaxation of minimax MTL, we obtain a continuum of MTL formulations spanning
minimax MTL and classical MTL. The full paradigm itself is loss-compositional,
operating on the vector of empirical risks. It incorporates minimax MTL, its relaxations, and many new MTL formulations as special cases. We show theoretically
that minimax MTL tends to avoid worst case outcomes on newly drawn test tasks
in the learning to learn (LTL) test setting. The results of several MTL formulations
on synthetic and real problems in the MTL and LTL test settings are encouraging.
1
Introduction
The essence of machine learning is to exploit what we observe in order to form accurate predictors
of what we cannot. A multi-task learning (MTL) algorithm learns an inductive bias to learn several
tasks together. MTL is incredibly pervasive in machine learning: it has natural connections to random effects models [15]; user preference prediction (including collaborative filtering) can be framed
as MTL [16]; multi-class classification admits the popular one-vs-all and all-pairs MTL reductions;
and MTL admits provably good learning in settings where single-task learning is hopeless [4, 12].
But if we see examples from a random set of tasks today, which of these tasks will matter tomorrow?
Not knowing in the present what challenges nature has in store for the future, a sensible strategy is
to mitigate the worst case by ensuring some minimum proficiency on each task.
Consider a simple learning scenario: A music preference prediction company is in the business of
predicting what 5-star ratings different users would assign to songs. At training time, the company learns a shared representation for predicting the users? song ratings by pooling together the
company?s limited data on each user?s preferences. Given this learned representation, a separate
predictor for each user can be trained very quickly. At test time, the environment draws a user
according to some (possibly randomized) rule and solicits from the company a prediction of that
user?s preference for a particular song. The environment may also ask for predictions about new
users, described by a few ratings each, and so the company must leverage its existing representation
to rapidly learn new predictors and produce ratings for these new users.
Classically, multi-task learning has sought to minimize the (regularized) sum of the empirical risks
over a set of tasks. In this way, classical MTL implicitly assumes that once the learner has been
trained, it will be tested on test tasks drawn uniformly at random from the empirical task distribution
of the training tasks. Notably, there are several reasons why classical MTL may not be ideal:
?
Work completed while at Georgia Institute of Technology
1
? While at training time the usual flavor of MTL commits to a fixed distribution over users (typically either uniform or proportional to the number of ratings available for each user), at test time
there is no guarantee what user distribution we will encounter. In fact, there may not exist any
fixed user distribution: the sequence of users for which ratings are elicited could be adversarial.
? Even in the case when the distribution over tasks is not adversarial, it may be in the interest of
the music preference prediction company to guarantee some minimum level of accuracy per user
in order to minimize negative feedback and a potential loss of business, rather than maximizing
the mean level of accuracy over all users.
? Whereas minimizing the average prediction error is very much a teleological endeavor, typically
at the expense of some locally egregious outcomes, minimizing the worst-case prediction error
respects a notion of fairness to all tasks (or people).
This work introduces minimax multi-task learning as a response to the above scenario.1 In addition, we cast a spectrum of multi-task learning. At one end of the spectrum lies minimax MTL,
and departing from this point progressively relaxes the ?hardness? of the maximum until full relaxation reaches the second endpoint and recovers classical MTL. We further sculpt a generalized
loss-compositional paradigm for MTL which includes this spectrum and several other new MTL
formulations. This paradigm equally applies to the problem of learning to learn (LTL), in which the
goal is to learn a hypothesis space from a set of training tasks such that this representation admits
good hypotheses on future tasks. In truth, MTL and LTL typically are handled equivalently at training time ? this work will be no exception ? and they diverge only in their test settings and hence
the learning theoretic inquiries they inspire.
Contributions. The first contribution of this work is to introduce minimax MTL and a continuum
of relaxations. Second, we introduce a generalized loss-compositional paradigm for MTL which
admits a number of new MTL formulations and also includes classical MTL as a special case.
Third, we empirically evaluate the performance of several MTL formulations from this paradigm
in the multi-task learning and learning to learn settings, under the task-wise maximum test risk and
task-wise mean test risk criteria, on four datasets (one synthetic, three real). Finally, Theorem 1
is the core theoretical contribution of this work and shows the following: If it is possible to obtain
maximum empirical risk across a set of training tasks below some level ?, then it is likely that the
maximum true risk obtained by the learner on a new task is bounded by roughly ?. Hence, if the goal
is to minimize the worst case outcome over new tasks, the theory suggests minimizing the maximum
of the empirical risks across the training tasks rather than their mean.
In the next section, we recall the settings of multi-task learning and learning to learn, formally
introduce minimax MTL, and motivate it theoretically. In Section 3, we introduce a continuously parametrized family of minimax MTL relaxations and the new generalized loss-compositional
paradigm. Section 4 presents an empirical evaluation of various MTL/LTL formulations with different models on four datasets. Finally, we close with a discussion.
2
Minimax multi-task learning
We begin with a promenade through the basic MTL and LTL setups, with an effort to abide by the
notation introduced by Baxter [4]. Throughout the rest of the paper, each labeled example (x, y)
will live in X ? Y for input instance x and label y. Typical choices of X include Rn or a compact
subset thereof, while Y typically is a compact subset of R or the binary {?1, 1}. In addition,
define a loss function ` : R ? Y ? R+ . For simplicity, this work considers `2 loss (squared loss)
`(y 0 , y) = (y 0 ? y)2 for regression and hinge loss `(y 0 , y) = max{0, 1 ? y 0 y} for classification.
MTL and LTL often are framed as applying an inductive bias to learn a common hypothesis space,
selected from a fixed family of hypothesis spaces, and thereafter learning from this hypothesis space
a hypothesis for each task observed at training time. It will be useful to formalize the various sets
and elements present in the preceding statement. Let H be a family of hypothesis spaces. Any
hypothesis space H ? H itself is a set of hypotheses; each hypothesis h ? H is a map h : X ? R.
1
Note that minimax MTL does not refer to the minimax estimators of statistical decision theory.
2
Learning to learn. In learning to learn, the goal is to achieve inductive transfer to learn the best
H from H. Unlike in MTL, there is a notion of an environment of tasks: an unknown probability
measure Q over a space of task probability measures P. The goal is to find the optimal representation
via the objective
inf EP ?Q inf E(x,y)?P `(y, h(x)).
H?H
h?H
(1)
In practice, T (unobservable) training task probability measures P1 , . . . , PT ? P are drawn iid from
Q, and from each task t a set of m examples are drawn iid from Pt .
Multi-task learning. Whereas in learning to learn there is a distribution over tasks, in multi-task
learning there is a fixed, finite set of tasks indexed by [T ] := {1, . . . , T }. Each task t ? [T ]
is coupled with a fixed but unknown probability measure Pt . Classically, the goal of MTL is to
minimize the expected loss at test time under the uniform distribution on [T ]:
1 X
inf
(2)
inf E(x,y)?Pt `(y, h(x)).
H?H T
h?H
t?[T ]
Notably, this objective is equivalent to (1) when Q is the uniform distribution on {P1 , . . . , PT }. In
terms of the data generation model, MTL differs from LTL since the tasks are fixed; however, just
as in LTL, from each task t a set of m examples are drawn iid from Pt .
2.1
Minimax MTL
A natural generalization of classical MTL results by introducing a prior distribution ? over the index
set of tasks [T ]. Given ?, the (idealized) objective of this generalized MTL is
inf Et?? inf E(x,y)?Pt `(y, h(x)),
H?H
h?H
(3)
given only the training data {(xt,1 , yt,1 ), . . . , (xt,m , yt,m )}t?[T ] . The classical MTL objective (2)
equals (3) when ? is taken to be the uniform prior over [T ]. We argue that in many instances, that
which is most relevant to minimize is not the expected error under a uniform distribution over tasks,
or even any pre-specified ?, but rather the expected error for the worst ?. We propose to minimize
the maximum error over tasks under an adversarial choice of ?, yielding the objective:
inf sup Et?? inf E(x,y)?Pt `(y, h(x)),
H?H ?
h?H
where the supremum is taken over the T -dimensional simplex. As the supremum (assuming it is
attained) is attained at an extreme point of the simplex, this objective is equivalent to
inf max inf E(x,y)?Pt `(y, h(x)).
H?H t?[T ] h?H
In practice, we approximate the true objective via a regularized form of the empirical objective
inf max inf
H?H t?[T ] h?H
m
X
`(yt,i , h(xt,i )).
i=1
In the next section, we motivate minimax MTL theoretically by showing that the worst-case performance on future tasks likely will not be much higher than the maximum of the empirical risks for
the training tasks. In this short paper, we restrict attention to the case of finite H.
2.2
A learning to learn bound for the maximum risk
In this subsection, we use the following notation. Let P (1) , . . . , P (T ) be probability measures drawn
iid from Q, and for t ? [T ] let z(t) be an m-sample (a sample of m points) from P (t) with corre(t)
sponding empirical measure Pm . Also, if P is a probability P
measure then P ` ? h := E`(y, h(x));
m
1
similarly, if Pm is an empirical measure, then Pm ` ? h := m
i=1 `(yi , h(xi )).
Our focus is the learning to learn setting with a minimax lens: when one learns a representation
H ? H from multiple training tasks and observes maximum empirical risk ?, we would like to
3
guarantee that H?s true risk on a newly drawn test task will be bounded by roughly ?. Such a goal is
in striking contrast to the classical emphasis of learning to learn, where the goal is to obtain bounds
on H?s expected true risk. Using H?s expected true risk and Markov?s inequality, Baxter [4, the
display prior to (25) ] showed that the probability that H?s true risk on a newly drawn test task is
above some level ? decays as the expected true risk over ?:
P
(t)
1
t?[T ] Pm ` ? ht + ?
T
Pr inf P ` ? h ? ? ?
(4)
h?H
?
where the size of ? is controlled by T , m, and the complexities of certain spaces.
The expected true risk is not of primary interest for controlling the tail of the (random) true risk,
and a more direct approach yields a much better bound. In this short paper we restrict the space of
representations H to be finite with cardinality C; in this case, the analysis is particularly simple and
illuminates the idea for proving the general case. The next theorem is the main result of this section:
Theorem 1. Let |H| = C, and let the loss ` be L-Lipschitz in its second argument and bounded by
B. Suppose T tasks P (1) , . . . , P (T ) are drawn iid from Q and from each task P (t) an iid m-sample
(t)
z(t) is drawn. Suppose there exists H ? H such that all t ? [T ] satisfy minh?H Pm ` ? h ? ?. Let
? be the empirical risk minimizer over the test
P be newly drawn probability measure from Q. Let h
m-sample. With probability at least 1 ? ? with respect to the random draw of the T tasks and their
T corresponding m-samples:
?
?
s
?
4?
2C
? > ? + 1 + 2L max Rm (H) + 8 log ? ? log ? + logdBe + log(T + 1) . (5)
Pr P ` ? h
H?H
?
T
m ?
T
In the above, Rm (H) is the Rademacher complexity of H (cf. [3]). Critically, in (5) the probability
of observing a task with high true risk decays with T , whereas in (4) the decay is independent of T .
Hence, when the goal is to minimize the probability of bad performance on future tasks uniformly,
this theorem motivates minimizing the maximum of the empirical risks as opposed to their mean.
For the proof of Theorem 1, first consider the singleton case H = {H1 }. Suppose that for ? fixed a
(t)
priori, the maximum of the empirical risks is bounded by ?, i.e. maxt?[T ] minh?H1 Pm ` ? h ? ?.
Let a new probability measure P drawn from Q correspond to a new test task. Suppose the probability of the event [minh?H1 Pm ` ? h > ?] is at least ?. Then the probability that ? bounds all T
empirical risks is at most (1 ? ?)T ? e?T ? . Hence, with probability at least 1 ? e?T ? :
Pr {minh?H1 Pm ` ? h > ?} ? ?.
A simple application of the union bound extends this result for finite H:
Lemma 1. Under the same conditions as Theorem 1, with probability at least 1 ? ?/2 with respect
to the random draw of the T tasks and their T corresponding m-samples:
log 2C
?
Pr min Pm ` ? h > ? ?
.
h?H
T
The bound in the lemma states a 1/T rate of decay for the probability that the empirical risk obtained
by H on a new task exceeds ?. Next, we relate this empirical risk to the true risk obtained by the
empirical risk minimizer. Note that at test time H is fixed and hence independent of any test msample. Then, from by now standard learning theory results of Bartlett and Mendelson [3]:
Lemma 2. Take loss ` as in Theorem 1. With probability at least 1 ? ?/2, for all h ? H uniformly:
p
P ` ? h ? Pm ` ? h + 2LRm (H) + (8 log(4/?))/m.
In particular, with high probability the true risk of the empirical risk minimizer is not much larger
than its empirical risk. Theorem 1 now follows from Lemmas 1 and 2 and a union bound over
? ? ? := {0, 1/T, 2/T, . . . , dBe}; note that mapping the observed maximum empirical risk ? to
min{? 0 ? ? | ? ? ? 0 } picks up the additional T1 term in (5).
In the next section, we introduce a loss-compositional paradigm for multi-task learning which includes as special cases minimax MTL and classical MTL.
4
3
A generalized loss-compositional paradigm for MTL
The paradigm can benefit from a bit of notation. Given a set of T tasks, we represent the empirical
Pm
risk for hypothesis ht ? H (? H) on task t ? [T ] as `?t (ht ) := i=1 `(yt,i , ht (xt,i )). Additionally
define a set of hypotheses for multiple tasks h := (h1 , . . . , hT ) ? HT and the vector of empirical
?
risks `(h)
:= (`?1 (h1 ), . . . , `?T (hT )).
With this notation set, the proposed loss-compositional paradigm encompasses any regularized minimization of a (typically convex) function ? : RT+ ? R+ of the empirical risks:
?
inf inf ? `(h)
+ ? (H, h) ,
(6)
H?H h?HT
where ?(?) : H ? ?H?H HT ? R+ is a regularizer.
`p MTL. One notable specialization that is still quite general is the case when ? is an `p -norm,
yielding `p MTL. This subfamily encompasses classical MTL and many new MTL formulations:
? Classical MTL as `1 MTL:
1 X ?
inf inf
`(ht ) + ? (H, h)
H?H h?HT T
t?[T ]
? Minimax MTL as `? MTL:
? t ) + ? (H, h)
inf inf max `(h
H?H h?HT t?[T ]
1 ?
k`(h)k1 + ? (H, h) .
T
? inf
inf
? inf
?
inf k`(h)k
? + ? (H, h) .
H?H h?HT
H?H h?HT
? A new formulation, `2 MTL:
1 X
1/2
1 ?
? t) 2
`(h
+ ? (H, h) ? inf inf ? k`(h)k
inf inf
2 + ? (H, h) .
H?H h?HT
H?H h?HT T
T
t?[T ]
A natural question is why one might consider minimizing `p -norms of the empirical risks vector for
1 < p < ?, as in `2 MTL. The contour of the `1 -norm of the empirical risks evenly trades off
empirical risks between different tasks; however, it has been observed that overfitting often happens
near the end of learning, rather than the beginning [14]. More precisely, when the empirical risk is
high, the gradient of the empirical risk (taken with respect to the parameter (H, h)) is likely to have
positive inner product with the gradient of the true risk. Therefore, given a candidate solution with a
corresponding vector of empirical risks, a sensible strategy is to take a step in solution space which
places more emphasis on tasks with higher empirical risk. This strategy is particularly appropriate
when the class of learners has high capacity relative to the amount of available data. This observation
sets the foundation for an approach that minimizes norms of the empirical risks.
In this work, we also discuss an interesting subset of the loss-compositional paradigm which does
not fit into `p MTL; this subfamily embodies a continuum of relaxations of minimax MTL.
?-minimax MTL. In some cases, minimizing the maximum loss can exhibit certain disadvantages because the maximum loss is not robust to situations when a small fraction of the tasks are
fundamentally harder than the remaining tasks. Consider the case when the empirical risk for each
task in this small fraction can not be reduced below a level u. Rather than rigidly minimizing the
maximum loss, a more robust alternative is to minimize the maximize loss in a soft way. Intuitively, the idea is to ensure that most tasks have low empirical risk, but a small fraction of tasks are
permitted to have higher loss. We formalize this as ?-minimax MTL, via the relaxed objective:
n
o
1 X
minimize min b +
max{0, `?t (ht ) ? b} + ? (H, h) .
b?0
?
H?H,h?HT
t?[T ]
In the above, ? from the loss-compositional paradigm (6) is a variational function of the empirical
risks vector. The above optimization problem is equivalent to the perhaps more intuitive problem:
1 X
minimize
b+
?t + ? (H, h)
subject to `?t (ht ) ? b + ?t , t ? [T ].
T
?
H?H,h?H ,b?0,??0
t?[T ]
5
Here, b plays the role of the relaxed maximum, and each ?t ?s deviation from zero indicates the
deviation from the (loosely enforced) maximum. We expect ? to be sparse.
To help understand how ? affects the learning problem, let us consider a few cases:
(1) When ? > T , the optimal value of b is zero, and the problem is equivalent to classical MTL. To
see this, note that for a given candidate solution with b > 0 the objective always can be reduced
by reducing b by some ? and increasing each ?t by the same ?.
(2) Suppose one task is much harder than all the other tasks (e.g. an outlier task), and its empirical
risk is separated from the maximum empirical risk of the other tasks by ?. Let 1 < ? < 2; now,
at the optimal hard maximum solution (where ? = 0), the objective can be reduced by increasing
one of the ?t ?s by ? and decreasing b by ?. Thus, the objective can focus on minimizing the
maximum risk of the set of T ? 1 easier tasks. In this special setting, this argument can be
extended to the more general case k < ? < k + 1 and k outlier tasks, for k ? [T ].
(3) As ? approaches 0, we recover the hard maximum case of minimax MTL.
This work focuses on ?-minimax MTL with ? = 2/(d0.1T + 0.5e?1 + d0.1T + 1.5e?1 ) i.e. the
harmonic mean of d0.1T + 0.5e and d0.1T + 1.5e. The reason for this choice is that in the idealized
case (2) above, for large T this setting of ? makes the relaxed maximum consider all but the hardest
10% of the tasks. We also try the 20% level (i.e. 0.2T replacing 0.1T in the above).
Models. We now provide examples of how specific models fit into this framework. We consider
two convex multi-task learning formulations: Evgeniou and Pontil?s regularized multi-task learning
(the EP model) [5] and Argyriou, Evgeniou, and Pontil?s convex multi-task feature learning (the
AEP model) [1]. The EP model is a linear model with a shared parameter v0 ? Rd and task-specific
parameters vt ? Rd (for t ? [T ]). Evgeniou and Pontil presented this model as
P
Pm
P
minv0 ,{vt }t?[T ] t?[T ] i=1 `(yt,i , hv0 + vt , xt,i i) + ?0 kv0 k2 + ?T1 t?[T ] kvt k2 ,
for ` the hinge loss or squared loss. This can be set in the new paradigm via H = {Hv0 | v0 ? Rd },
Pm
1
Hv0 = {h : x 7? hv0 + vt , xi | vt ? Rd }, and `?t (ht ) = m
i=1 ` yt,i , hv0 + vt , xt,i i .
The AEP model minimizes the task-wise average loss with the trace norm (nuclear norm) penalty:
P Pm
minW t i=1 `(yt,i , hWt , xt,i i) + ?kW ktr ,
P
where k?ktr : W 7? i ?i (W ) is the trace norm. In the new paradigm, H is a set where each element
is a k-dimensional subspace of linear estimators (for k d). Each ht = Wt in some H ? H lives
Pm
1
in H?s corresponding low-dimensional subspace. Also, `?t (ht ) = m
i=1 ` yt,i , hht , xt,i i .
For easy empirical comparison between the various MTL formulations from the paradigm, at times
it will be convenient to use constrained formulations of the EP and AEP model. If the regularized
forms are used, a fair comparison of the methods warrants plotting results according to the size of
the optimal parameter found (i.e. kW ktr for AEP). For EP, the constrained form is:
P
Pm
minv0 ,{vt }t?[T ] t?[T ] i=1 `(yt,i , hv0 + vt , xt,i i) subject to kv0 k ? ?0 , kvt k ? ?1 for t ? [T ].
P Pm
For AEP, the constrained form is: minW t i=1 `(yt,i , hWt , xt,i i) subject to kW ktr ? r.
4
Empirical evaluation
We consider four learning problems; the first three involve regression (MTL model in parentheses):
? A synthetic dataset composed from two modes of tasks (EP model),
? The school dataset from the Inner London Education Authority (EP model),
? The conjoint analysis personal computer ratings dataset 2 [11] (AEP model).
The fourth problem is multi-class classification from the MNIST digits dataset [10] with a reduction
to multi-task learning using a tournament of pairwise (binary) classifiers. We use the AEP model.
Given data, each problem involved a choice of MTL formulation (e.g. minimax MTL), model (EP
or AEP), and choice of regularized versus constrained. All the problems were solved with just a few
lines of code using CVX [9, 8]. In this work, we considered convex multi-task learning formulations
in order to make clear statements about the optimal solutions attained for various learning problems.
2
This data, collected at the University of Michigan MBA program, generously was provided by Peter Lenk.
6
250
250
200
200
400
350
100
squared?loss risk
squared?loss risk
squared?loss risk
300
150
150
100
250
200
150
100
50
50
50
0
?1
0
1
?noise
2
3
0
?1
4
0
1
?noise
2
3
0
?0.5
4
0
0.5
1
?task
1.5
2
2.5
Figure 1: Max `2 -risk (Top two lines) and mean `2 -risk (Bottom two lines). At Left and Center: `2 -risk vs
noise level, for ?task = 0.1 and ?task = 0.5 respectively. At Right: `2 -risk vs task variation, for ?noise = 0.1.
Dashed red is `1 , dashed blue is minimax. Error bars indicate one standard deviation. MTL results (not shown)
were similar to LTL results (shown), with MTL-LTL relative difference below 6.8% for all points plotted.
Two modes. The two modes regression problem consists of 50 linear prediction tasks for the first
type of task and 5 linear prediction tasks for the second task type. The true parameter for the first
task type is a vector ? drawn uniformly from the sphere of radius 5; the true parameter for the
second task type is ?2?. Each task is drawn from an isotropic Gaussian with mean taken from the
task type and the standard deviation of all dimensions set to ?task . Each data point for each task is
drawn from a product of 10 standard normals (so xt,i ? R10 ). The targets are generated according
to hWt , xt,i i + ?t , where the ?t ?s are iid univariate centered normals with standard deviation ?noise .
We fixed ?0 to a large value (in this case, ?0 = 10 is sufficient since the mean for the largest task
fits into a ball of radius 10) and ?1 to a small value (?1 = 2). We compute the average mean and
maximum test error over 100 instances of the 55-task multi-task problem. Each task?s training set
and test set are 5 and 15 points respectively. The average maximum (mean) test error is the 100experiment-average of the task-wise maximum (mean) of the `2 risks. For each LTL experiment, 55
new test tasks were drawn using the same ? as from the training tasks.
Figure 1 shows a tradeoff: when each task group is fairly homogeneous (left and center plots),
minimax is better at minimizing the maximum of the test risks while `1 is better at minimizing the
mean of the test risks. As task homogeneity decreases (right plot), the gap in performance closes
with respect to the maximum of the test risks and remains roughly the same with respect to the mean.
0.9
1.5
0.88
1.45
0.86
1.4
0.84
1.35
0.82
0.8
1.3
0.78
0
0.2
0.4
0.6
?1
0
0.2
0.4
0.6
0
0.2
0.4
0.6
?
0
0.2
0.4
0.6
1
Figure 2: Maximum RMSE (Left) and normalized mean RMSE (Right) versus task-specific parameter bound
?1 , for shared parameter bound ?0 fixed. In each figure, Left section is ?0 is 0.2 and Right section is ?0 = 0.6.
Solid red is `1 , solid blue ? is minimax, dashed green N is (0.1T )-minimax, dashed black H is (0.2T )minimax. The results for `2 MTL were visually identical to `1 MTL and hence were not plotted.
School. The school dataset has appeared in many previous works [7, 2, 6]. For brevity we just say
the goal is to predict student test scores using certain student-level features. Each school is treated as
a separate task. We report both the task-wise maximum of the root mean square error (RMSE) and
the taskwise-mean of the RMSE (normalized by number of points per task, as in previous works).
The results (see Figure 2) demonstrate that when the learner has moderate shared capacity ?0 and
high task-specific capacity ?1 , minimax MTL outperforms `1 MTL for the max objective; additionally, for the max objective in almost all parameter settings (0.1T )-minimax and (0.2T )-minimax
MTL outperform `1 MTL, and they also outperform minimax MTL when the task-specific capacity
?1 is not too large. We hypothesize that minimax MTL performs the best in the high??1 regime because stopping learning once the maximum of the empirical risks cannot be improved invokes early
stopping and its built-in regularization properties (see e.g. [13]). Interestingly, for the normalized
mean RMSE objective, both minimax relaxations are competitive with `1 MTL; however, when the
7
shared capacity ?0 is high (right section, right plot), `1 MTL performs the best. For high task-specific
capacity ?1 , minimax MTL and its relaxations again seem to resist overfitting compared to `1 MTL.
CV?mean Mean squared loss
CV?mean Maximum squared loss
0.24
0.85
0.8
0.75
0.7
0.65
0
5
10
15
bound on trace norm of W
0.5
0.2
0.18
0.16
0.14
5
10
15
bound on trace norm of W
20
0.155
CV?mean Mean squared loss
0.48
0.47
0.46
0.45
0.15
0.145
0.14
0.135
0.44
0.43
0
5
10
15
bound on trace norm of W
20
0.13
0
5
10
15
bound on trace norm of W
20
Figure 3: MTL (Top) and LTL (Bottom). Maximum `2 risk (Left)
and Mean `2 risk (Right) vs bound on kW ktr . LTL used 10-fold
cross-validation (10% of tasks left out in each fold). Solid red is
`1 , solid blue ? is minimax, dashed green N is (0.1T )-minimax,
dashed black H is (0.2T )-minimax, solid gold is `2 .
MNIST. The MNIST task is a 10-class problem; we approach it via a reduction to a tournament of 45 binary classifiers trained via the AEP model. The dimensionality was
reduced to 50 using principal component analysis (computed on the full training set), and only the first 2% of
each class?s training points were used for training.
Intuitively, the performance of the tournament tree of binary classifiers can only be as accurate as its paths, and
the accuracy of each path depends on the accuracy of
the nodes. Hence, our hypothesis is that minimax MTL
should outperform `1 MTL. The results in Figure 4 confirm our hypothesis. Minimax MTL outperforms `1 MTL
when the capacity kW ktr is somewhat limited, with the
gap widening as the capacity decreases. Furthermore, at
every capacity minimax MTL is competitive with `1 MTL.
5
0.22
0.12
0
20
0.49
0.5
0.45
Test multiclass zero?one loss
The results are shown in Figure 3. In
the MTL setting, for both the maximum RMSE objective and the mean
RMSE objective, `1 MTL appears to
perform the best. When the trace
norm of W is high, minimax MTL
displays resistance to overfitting and
obtains the lowest mean RMSE. In
the LTL setting for the maximum
RMSE objective, `2 , minimax, and
(0.1T )-minimax MTL all outperform
`1 MTL. For the mean RMSE, `1
MTL obtains the lowest risk for almost all parameter settings.
0.9
CV?mean Maximum squared loss
Personal computer. The personal
computer dataset is composed of 189
human subjects each of which rated
on a 0-10 scale the same 20 computers (16 training, 4 test). Each computer has 13 binary features (amount
of memory, screen size, price, etc.).
0.4
0.35
0.3
0.25
0.2
0.15
0.1
40
60
80
100
trace norm of W
120
140
Figure 4: Test multiclass 0-1 loss vs
kW ktr . Solid red is `1 MTL, solid blue is
minimax, dashed green is (0.1T )-minimax,
dashed black is (0.2T )-minimax. Regularized AEP used for speed and trace norm of
W ?s computed, so samples differ per curve.
Discussion
We have established a continuum of formulations for MTL which recovers as special cases classical
MTL and the newly formulated minimax MTL. In between these extreme points lies a continuum of
relaxed minimax MTL formulations. More generally, we introduced a loss-compositional paradigm
that operates on the vector of empirical risks, inducing the additional `p MTL paradigms. The empirical evaluations indicate that ?-minimax MTL at either the 10% or 20% level often outperforms `1
MTL in terms of the maximum test risk objective and sometimes even in the mean test risk objective.
All the minimax or ?-minimax MTL formulations exhibit a built-in safeguard against overfitting in
the case of learning with a model that is very complex relative to the available data.
Although efficient algorithms may make the various new MTL learning formulations practical for
large problems, a proper effort to develop fast algorithms in this setting would have detracted from
the main point of this first study. A good direction for the future is to obtain efficient algorithms
for minimax and ?-minimax MTL. In fact, such algorithms might have applications beyond MTL
and even machine learning. Another area ripe for exploration is to establish more general learning
bounds for minimax MTL and to extend these bounds to ?-minimax MTL.
8
References
[1] Andreas Argyriou, Theodoros Evgeniou, and Massimiliano Pontil. Convex multi-task feature
learning. Machine Learning, 73(3):243?272, 2008.
[2] Bart Bakker and Tom Heskes. Task clustering and gating for bayesian multitask learning.
Journal of Machine Learning Research, 4:83?99, 2003.
[3] Peter L. Bartlett and Shahar Mendelson. Rademacher and Gaussian complexities: Risk bounds
and structural results. Journal of Machine Learning Research, 3:463?482, 2002.
[4] Jonathan Baxter. A model of inductive bias learning. Journal of Artificial Intelligence Research, 12(1):149?198, 2000.
[5] Theodoros Evgeniou and Massimiliano Pontil. Regularized multi-task learning. In Proceedings of the tenth ACM SIGKDD international conference on Knowledge discovery and data
mining, pages 109?117. ACM, 2004.
[6] Theodoros Evgeniou, Massimiliano Pontil, and Olivier Toubia. A convex optimization approach to modeling consumer heterogeneity in conjoint estimation. Marketing Science,
26(6):805?818, 2007.
[7] Harvey Goldstein. Multilevel modelling of survey data. Journal of the Royal Statistical Society.
Series D (The Statistician), 40(2):235?244, 1991.
[8] Michael C. Grant and Stephen P. Boyd. Graph implementations for nonsmooth convex programs. In V. Blondel, S. Boyd, and H. Kimura, editors, Recent Advances in Learning and
Control, Lecture Notes in Control and Information Sciences, pages 95?110. Springer-Verlag
Limited, 2008.
[9] Michael C. Grant and Stephen P. Boyd. CVX: Matlab software for disciplined convex programming, version 1.21, April 2011.
[10] Yann LeCun, L?eon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning
applied to document recognition. Proceedings of the IEEE, 86(11):2278?2324, 1998.
[11] Peter J. Lenk, Wayne S. DeSarbo, Paul E. Green, and Martin R. Young. Hierarchical bayes
conjoint analysis: Recovery of partworth heterogeneity from reduced experimental designs.
Marketing Science, pages 173?191, 1996.
[12] Andreas Maurer. Transfer bounds for linear feature learning. Machine learning, 75(3):327?
350, 2009.
[13] Noboru Murata and Shun-ichi Amari. Statistical analysis of learning dynamics. Signal Processing, 74(1):3?28, 1999.
[14] Nicolas Le Roux, Pierre-Antoine Manzagol, and Yoshua Bengio. Topmoumoute online natural
gradient algorithm. In J.C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in
Neural Information Processing Systems 20, pages 849?856. MIT Press, Cambridge, MA, 2008.
[15] Kai Yu, John Lafferty, Shenghuo Zhu, and Yihong Gong. Large-scale collaborative prediction
using a nonparametric random effects model. In Proceedings of the 26th Annual International
Conference on Machine Learning, pages 1185?1192. ACM, 2009.
[16] Liang Zhang, Deepak Agarwal, and Bee-Chung Chen. Generalizing matrix factorization
through flexible regression priors. In Proceedings of the fifth ACM conference on Recommender systems, pages 13?20. ACM, 2011.
9
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4,026 | 4,644 | Minimization of Continuous Bethe Approximations:
A Positive Variation
Jason L. Pacheco and Erik B. Sudderth
Department of Computer Science, Brown University, Providence, RI
{pachecoj,sudderth}@cs.brown.edu
Abstract
We develop convergent minimization algorithms for Bethe variational approximations which explicitly constrain marginal estimates to families of valid distributions. While existing message passing algorithms define fixed point iterations corresponding to stationary points of the Bethe free energy, their greedy dynamics do
not distinguish between local minima and maxima, and can fail to converge. For
continuous estimation problems, this instability is linked to the creation of invalid
marginal estimates, such as Gaussians with negative variance. Conversely, our
approach leverages multiplier methods with well-understood convergence properties, and uses bound projection methods to ensure that marginal approximations
are valid at all iterations. We derive general algorithms for discrete and Gaussian
pairwise Markov random fields, showing improvements over standard loopy belief propagation. We also apply our method to a hybrid model with both discrete
and continuous variables, showing improvements over expectation propagation.
1
Introduction
Variational inference algorithms pose probabilistic inference as an optimization over distributions.
Typically the optimization is formulated by minimizing an objective known as the Gibbs free energy [1]. Variational methods relax an otherwise intractable optimal inference problem by approximating the entropy-based objective, and considering appropriately simplified families of approximating distributions [2]. Local message passing algorithms offer a computationally efficient method
for extremizing variational free energies. Loopy belief propagation (LBP), for example, optimizes
a relaxed objective known as the Bethe free energy [1, 2], which we review in Sec. 2. Expectation
propagation (EP) [3] is a generalization of LBP which shares the same objective, but optimizes over
a relaxed set of constraints [4] applicable to a broader family of continuous inference problems.
In general, neither LBP nor EP are guaranteed to converge. Even in simple continuous models, both
methods may improperly estimate invalid or degenerate marginal distributions, such as Gaussians
with negative variance. Such degeneracy typically occurs in classes of models for which convergence properties are poor, and there is evidence that these problems are related [5, 6],
Extensive work has gone into developing algorithms which improve on LBP for models with discrete
variables, for example by bounding [7, 8] or convexifying [9] the free energy objective. Gradient
optimization methods have been applied successfully to binary Ising models [10], but when applied
to Gaussian models this approach suffers similar non-convergence and degeneracy issues as LBP.
Work on optimization of continuous variational free energies has primarily focused on addressing
convergence problems [11]. None of these approaches directly address degeneracy in the continuous
case, and computation may be prohibitively expensive for these direct minimization schemes.
By leveraging gradient projection methods from the extensive literature on constrained nonlinear
optimization, we develop an algorithm which ensures that marginal estimates remain valid and normalizable at all iterations. In doing so, we account for important constraints which have been ignored
1
by previous variational derivations of the expectation propagation algorithm [12, 6, 11]. Moreover,
by adapting the method of multipliers [13], we guarantee that our inference algorithm converges for
most models of practical interest.
We begin by introducing the Bethe variational problem (Sec. 2). We briefly review the correspondence between the Lagrangian formalism and message passing and discuss implicit normalizability
assumptions which, when violated, lead to degeneracy in message passing algorithms. We discuss
the method of multipliers, gradient projection, and convergence properties (Sec. 3). We then provide derivations (Sec. 4) for discrete MRFs, Gaussian MRFs, and hybrid models with potentials
defined by discrete mixtures of Gaussian distributions. Experimental results in Sec. 5 demonstrate
substantial improvements over baseline message passing algorithms.
2
Bethe Variational Problems
For simplicity, we restrict our attention to pairwise Markov random fields (MRF) [2], with graphs
G(V, E) defined by nodes V and undirected edges E. The joint distribution then factorizes as
Y
1 Y
p(x) =
?s (xs )
?st (xs , xt )
(1)
Zp
s?V
(s,t)?E
for some non-negative potential functions ?(?). Often this distribution is a posterior given fixed
observations y, but we suppress this dependence for notational simplicity. We are interested in
computing the log partition function log Zp , and/or the marginal distributions p(xs ), s ? V.
Let q(x; ?) denote an exponential family of densities with sufficient statistics ?(x) ? Rd :
q(x; ?) ? exp{?T ?(x)}, ? = Eq [?(x)].
(2)
To simplify subsequent algorithm development, we index distributions via their mean parameters ?.
We associate each node s ? V with an exponential family qs (xs ; ?s ), ?s (x) ? Rds , and each edge
(s, t) ? E with a family qst (xs , xt ; ?st ), ?st (x) ? Rdst . Because qs (xs ; ?s ) is a valid probability
distribution, ?s must lie in a set of realizable mean parameters, ?s ? Ms . Similarly, ?st ? Mst .
For example, Ms and Mst might require Gaussians to have positive semidefinite covariances.
We can express the log partition as the solution to an optimization problem,
? log Zp = min E? [? log p(x)] ? H[?] = min F(?),
??M(G)
??M(G)
(3)
where H[?] is the entropy of q(x; ?), E? [?] denotes expectation with respect to q(x; ?), and F(?) is
known as the variational free energy. Mean parameters ? lie in the marginal polytope M(G) if and
only if there exists some valid, joint probability distribution with those moments.
Exactly characterizing M(G) may require exponentially many constraints, so we relax the optimization to be over a set of locally consistent marginal distributions L(G), which are properly normalized
and satisfy expectation constraints associated with each edge of the graph:
Z
Cs (?) = 1 ? qs (xs ; ?s ) dxs , Cts (?) = ?s ? Eqst [?s (xs )].
(4)
This is a relaxation in the sense that M(G) ? L(G) with strict equality if G does not contain cycles.
We approximate the entropy H[?] with the entropy of a tree-structured distribution q(x; ?). Such an
approximation is tractable and consistent with L(G), and yields the Bethe free energy:
X
X
F B (?) =
Eqst [log qst (xs , xt ; ?st ) ? ?st (xs , xt )] ?
(ns ? 1) Eqs [log qs (xs ; ?s ) ? ?s (xs )]
s?V
(s,t)?E
(5)
Here, S
?st (?) =
?
(?)?
(?)?
(?),
and
the
mean
parameters
?
are
valid
within
the
constraint
set
st
s
t
S
M = s Ms st Mst . The resulting objective is the Bethe variational problem (BVP):
minimize FB (?)
?
subject to Cts (?) = 0, ?s ? V, t ? N (s)
Cs (?) = 0, ?s ? V,
{?s : s ? V} ? {?st : (s, t) ? E} ? M.
Here, N (s) denotes the set of neighbors of node s ? V.
2
(6)
2.1
Correspondence to Message Passing
We can optimize the BVP (6) by relaxing the normalization and local consistency constraints with
Lagrange multipliers. Constrained minima are characterized by stationary points of the Lagrangian,
X
X X
L(x, ?) = F B (q) +
?s Cs +
?ts Cts .
(7)
s
s
t?N (s)
Equivalence between LBP fixed points and stationary points of the Lagrangian for the discrete case
have been discussed extensively [1, 2]. Similar correspondence has been shown more generally for
EP fixed points [2, 4]. Since our focus is on the continuous case we briefly review the correspondence between Gaussian LBP fixed points and the Gaussian Bethe free energy. For simplicity we
focus on zero-mean p(x) = N (x | 0, J ?1 ), where diagonal precision entries Jss = As and
1
1
0 Jst
xs
?s (xs ) = exp ? x2s As ,
?st (xs , xt ) = exp ? (xs xt )
.
Jst 0
xt
2
2
Pst
est = As Jst .
, and B
Let q(xs ) = N (xs | 0, Vs ), q(xs , xt ) = N (( xxst ) | 0, ?st ), ?st = PVts
Jst At
ts Vst
The Gaussian Bethe free energy then equals
X n ? 1
1 X
s
est ) ? log |?st | ?
(Vs As ? log Vs ) . (8)
tr(?st B
FGB (V, ?) =
2
2
s?V
(s,t)?E
The locally consistent marginal polytope L(G) consists of the constraints Cts (V ) = Vs ? Vts for
all nodes s ? V and edges (s, t) ? E. The Lagrangian is given by,
X X
L(V, ?, ?) = FGB (V, ?) +
?ts [Vs ? Vts ] .
(9)
s
t?N (s)
?L
Taking the derivative with respect to the node marginal variance ?V
= 0 yields the stationary point
s
P
1
?1
Vs = As + ns ?1 t?N (s) ?ts . For a Gaussian LBP algorithm with messages parametrized as
mt?s (xs ) = exp ? 12 x2s ?t?s , fixed points of the node marginal precision are given by
X
? s = As +
?t?s
t?N (s)
P
1
Let ?ts = ? 2 a?N (s)\t ?a?s . Substituting back into the stationary point conditions yields
Vs?1 ? ?s . A similar construction holds for the pairwise marginals. Inverting the correspondence
between multipliers and message parameters yields the converse Vs?1 ? ?s (c.f. [4]).
2.2
Message Passing Non-Convergence and Degeneracy
While local message passing algorithms are convenient for many applications, their convergence
is not guaranteed in general. In particular, LBP often fails to converge for networks with tight
loops [1] such as the 3?3 lattice of Figure 1(a). For non-Gaussian models with continuous variables,
convergence of the EP algorithm can be even more problematic [11].
For continuous models message updates may yield degenerate, unnormalizable marginal distributions which do not correspond to stationary points of the Lagrangian. For example, for Gaussian
MRFs the Bethe free energy F B (?) in (5) is derived from expectations with respect to variational
distributions qs (xs ; ?s ), qst (xs , xt ; ?st ). If a set of hypothesized marginals are not normalizable
(positive variance), the Gaussian Bethe free energy F GB (?) is invalid and undefined.
Degenerate marginals arise because the constraint set M is not represented in the Lagrangian (7);
this issue is mentioned briefly in [2] but is not dealt with computationally. Figure 1(b) demonstrates
this issue for a simple, three-node Gaussian MRF. Here LBP produces marginal variances which
oscillate between impossibly large positive, and non-sensical negative, values. Such degeneracies
are arguably more problematic for EP since its moment matching steps require expected values with
respect to an augmented distribution [3], which may involve an unbounded integral.
3
5
10
300
Belief Propagation
MoM
1000
True
LBP
MoM
250
? ? {?0.9, ?0.87, ?0.85, ?0.7, ?0.5}
800
4
10
600
Gaussian Bethe Free Energy
1
Variance of Node x
Bethe Free Energy
200
3
10
2
10
150
100
50
400
200
0
?200
?400
0
1
10
?600
?50
0
10
?800
?100
0
50
100
150
200
250
Iteration
300
350
400
450
0
10
20
30
40
(a)
50
60
Iteration #
70
(b)
80
90
100
?1000
0
10
1
10
2
10
Variance (V)
3
10
(c)
Figure 1: (a) Bethe free energy versus iteration for 3x3 toroidal binary MRF. (b) Node marginal variance
estimates per iteration for a symmetric, single-cycle Gaussian MRF with three nodes (plot is of x1 , other nodes
are similar). (c) For the model from (b), the Gaussian Bethe free energy is unbounded on the constraint set.
2.3
Unboundedness of the Gaussian Bethe Free Energy
Conditions under which the simple LBP and EP updates are guaranteed to be accurate are of great
practical interest. For Gaussian MRFs, the class of pairwise normalizable models are sufficient to
guarantee LBP stability and convergence [5]. For non-pairwise normalizable models the Gaussian
Bethe free energy is unbounded below [6] on the set of local consistency constraints L(G).
We offer a small example consisting of a non-pairwise normalizable symmetric single cycle with 3
nodes. Diagonal precision elements are Jss = 1.0, and off-diagonal elements Jst =
0.6. We
embed
V ?V
marginalization constraints into a symmetric parametrization Vs = V and ?st = ?V V . Feasible solutions within the constraint set are characterized by V > 0 and ?1 < ? < 1. Substituting
this parametrization into the Gaussian free energy (8), and performing some simple algebra, yields
3
3
3
FGB (V, ?) = ? log V + V (1 + 1.2?) ? log(1 ? ?2 ).
2
2
2
1
For ? < ? 1.2
the free energy is unbounded below at rate O(?V ). Figure 1(c) illustrates the Bethe
free energy for this model as a function of V , and for several values of ?.
More generally, it has been shown that Gaussian EP messages are always normalizable (positive
variance) for models with log-concave potentials [14]. It has been conjectured, but not proven,
that EP is also guaranteed to converge for such models [15]. For Gaussian MRFs, we note that
the family of log-concave models coincides with the pairwise normalizability condition. Our work
seeks to improve inference for non-log-concave models with bounded Bethe free energies.
3
Method of Multipliers
Given our complete constrained formulation of the Bethe variational problem, we avoid convergence
and degeneracy problems via direct minimization using the method of multipliers (MoM) [13]. In
general terms, given some convex feasible region M, consider the equality constrained problem
minimize f (x) subject to h(x) = 0
x?M
With penalty parameter c > 0, we form the augmented Lagrangian function,
1
(10)
Lc (x, ?) = f (x) + ?T h(x) + c||h(x)||2
2
Given a multiplier vector ?k and penalty parameter ck we update the primal and dual variables as,
xk = arg min Lck (x, ?k ), ?k+1 = ?k + ck h(xk ).
x?M
The penalty multiplier can be updated as ck+1 ? ck according to some fixed update schedule, or
based on the results of the optimization step. An update rule that we find useful [13] is to increase
the penalty parameter by ? > 1 if the constraint violation is not improved by a factor 0 < ? < 1
over the previous iteration,
?ck
if kh(xk )k > ?kh(xk?1 )k,
ck+1 =
ck
if kh(xk )k ? ?kh(xk?1 )k.
4
4
10
3.1
Gradient Projection Methods
The augmented Lagrangian Lc (x, ?) is a partial one, where feasibility of mean parameters (x ? M)
is enforced explicitly by projection. A simple gradient projection method [13] defines a sequence
+
xk+1 = xk + ?k (?
xk ? xk ),
x
?k = [xk ? sk ?f (xk )] .
The notation [?]+ denotes a projection onto the constraint set M. After taking a step sk > 0 in the
direction of the negative gradient, we project the result onto the constraint set to obtain a feasible
direction x
?k . We then compute xk+1 by taking a step ?k ? (0, 1] in the direction of (?
xk ? xk ). If
xk ? sk ?f (xk ) is feasible, gradient projection reduces to unconstrained steepest descent.
There are multiple such projection steps in each inner-loop iteration of MoM (e.g. each xk update).
For our experiments we use a projected quasi-Newton method [16] and step-sizes ?k and sk are
chosen using an Armijo rule [13, Prop. 2.3.1].
3.2
Convergence of Multiplier Methods
Convergence and rate of convergence results have been proven [17, Proposition 2.4] for the Method
of Multipliers with a quadratic penalty and multiplier iteration ?k+1 = ?k + ck h(xk ). The main
regularity assumptions are that the sequence {?k } is bounded, and there is a local minimum for
which a Lagrange multiplier pair (x? , ?? ) exists satisfying second-order sufficiency conditions, so
that ?x L0 (x? , ?? ) = 0 and z T ?2xx L0 (x? , ?? )z > 0 for all z 6= 0. It then follows that there exists
some c? such that for all c ? c?, the augmented Lagrangian also contains a strict local minimum
z T ?2xx Lc (x? , ?? )z > 0.
For convergence, the initialization of the Lagrange multiplier ?0 and penalty parameter c0 must be
such that k?0 ? ?? k < ?c0 for some ? > 0 and c ? c? which depend on the objective and constraints.
In practice, a poor initialization of the multiplier ?0 can often be offset by a sufficiently high c0 . A
final technical note is that convergence proofs assume the sequence of unconstrained optimizations
which yield xk stays in the neighborhood of x? after some k. This does not hold in general, but can
be encouraged by warm-starting the unconstrained optimization with the previous xk?1 .
To invoke existing convergence results we must show that a local minimum x? exists for each of
the free energies we consider; a sufficient condition is then that the Bethe free energy is bounded
from below. This property has been previously established for general discrete MRFs [18], for pairwise normalizable Gaussian MRFs [6], and for the clutter model [3]. For non-pairwise normalizable
Gaussian MRFs, the example of Section 2.3 shows that the Bethe variational objective is unbounded
below, and further may not contain any local optima. While the method of multipliers does not converge in this situation, its non-convergence is due to fundamental flaws in the Bethe approximation.
4
MoM Algorithms for Probabilistic Inference
We derive MoM algorithms which minimize the Bethe free energy for three different families of
graphical models. For each model we define the form of the joint distribution, Bethe free energy (5),
local consistency constraints, augmented Lagrangian, and the gradient projection step. Gradients,
which can be notationally cumbersome, are given in the supplemental material.
4.1
Gaussian Markov Random Fields
We have already introduced the Lagrangian (9) for the Gaussian MRF. The Gaussian Bethe free
energy (8) is always unbounded below off of the constraint set in node marginal variances Vs . We
correct this by adding an additional fixed penalty in the augmented Lagrangian,
X X
Lc (V, ?, ?) = FGB (V ) +
?ts [Vs ? Vts ]
s
t?N (s)
?X X
cX X
2
2
+
[log Vs ? log Vts ] +
[Vs ? Vts ] .
2 s
2 s
t?N (s)
t?N (s)
We keep ? ? 1 fixed so that existing convergence theory remains applicable. The set of realizeable
mean parameters M is the set of symmetric positive semidefinite matrices Vs , ?st . We therefore
5
must solve a series of constrained optimizations of the form, minV,? Lck (V, ?, ?k ), subject to Vs ?
0, ?st 0. The gradient projection step is easily expressed in terms of correlation coefficients ?st ,
?
?st Vst Vts
?Vst
?st =
.
?st Vst Vts
Vts
Then, ?st 0 if and only if Vst ? 0, Vts ? 0, and ?1 ? ?st ? 1. The projection step is then,
Vst = max(0, Vst ),
Vts = max(0, Vts ),
?st = max(?1, min(1, ?st )).
The full MoM algorithm then follows from gradient derivations in the supplemental material.
Recall that in Section 2.3, we showed that the Gaussian Bethe free energy is unbounded on the
constraint set for non-pairwise normalizable models. We run MoM on the symmetric three-node
cycle from this discussion and find that MoM, correctly, identifies an unbounded direction, and
Figure 1(b) shows that the node marginal variances indeed diverge to infinity.
4.2
Discrete Markov Random Fields
Consider a discrete MRF where all variables xs ? Xs = {1, . . . , Ks }. The variational marginal
QKs
distributions are then qs (xs ; ? ) = k=1
? (xs )I(xs ,k) , and have mean parameters ? ? RKs . Let
? (xs ) denote element xs of vector ? . Pairwise marginals have mean parameters ?st ? RKs ?Kt
similarly indexed as ?st (xs , xt ). The discrete Bethe free energy is then
X XX
FB (? ; ?) =
?st (xs , xt )[log ?st (xs , xt ) ? log ?st (xs , xt )]
(s,t)?E
xs
xt
XX
?
(ns ? 1)?s (xs )[log ?s (xs ) ? log ?s (xs )].
s?V xs
For this discrete model, our expectation constraints reduce to the following normalization and
marginalization constraints:
X
X
Cs (? ) = 1 ?
?s (xs ), Cts (xs ; ? ) = ?s (xs ) ?
?st (xs , xt ).
xs
xt
The augmented Lagrangian is then,
Lc (?, ?, ?; ?) = FB (? ; ?) +
X
(s,t)?E
+
X
s?V
"
X
#
?ts (xs )Cts (xs ; ? ) +
xs
X
?st (xt )Cst (xt ; ? )
(11)
xt
cX
c X
?ss Cs (? ) +
Cs (? )2 +
2
2
s?V
(s,t)?E
"
#
X
Cts (xs ; ? )2 +
xs
X
Cst (xt ; ? )2 .
xt
Mean parameters must be non-negative to be valid, so M = {?s , ?st : ?s ? 0, ?st ? 0}. This
constraint is enforced by a bound projection ?s (xs ) = max(0, ?s (xs )), and similarly for the pairwise marginals. While these constraints are never active in BP fixed point iterations, they must be
enforced in gradient optimization. With these pieces and the gradient computations presented in the
supplement, implementation of MoM optimization for the discrete MRF is straightforward.
4.3
Discrete Mixtures of Gaussian Potentials
We are particularly interested in tractable inference in hybrid models with discrete and conditionally
Gaussian random variables. A simple example of such a model is the clutter problem [3], whose
joint distribution models N conditionally independent Gaussian observations {yi }N
i=1 . These observations may either be centered on a target scalar x ? R (zi = 1) or drawn from a background clutter
distribution (zi = 0). If target observations occur with frequency ?0 , we then have
x ? N (?0 , P0 ),
zi ? Ber(?0 ),
yi | x, zi ? N (0, ?02 )(1?zi ) N (x, ?12 )zi
The corresponding variational posterior distributions are,
q0 (x) = N (m0 , V0 ),
qi (x, zi ) = ((1 ? ?i )N (x | mi0 , Vi0 ))
6
(1?zi )
z
(?i N (x | mi1 , Vi1 )) i .
Assuming normalizable marginals with V0 ? 0, Vi0 ? 0, Vi1 ? 0, as always ensured by our
multiplier method, we define the Bethe free energy F CGB (m, V, ?) in terms of the mean parameters
in the supplemental material. Expectation constraints are given by,
Cimean = E0 [x] ? Ei [x],
Civar = Var0 [x] ? Vari [x],
where Ei [?] and Var i [?] denote the mean and variance of the Gaussian mixture qi (x, zi ). Combining the free energy, constraints, and additional positive semidefinite constraints on the marginal
variances we have the BVP for the clutter model,
minimize
m,V,?
FCGB (m, V, ?; ?)
subject to Cimean = 0, Civar = 0, for all i = 1, 2, . . . , N
V0 ? 0, Vi0 ? 0, Vi1 ? 0
(12)
Derivation of the free energy and augmented Lagrangian is somewhat lengthy, and so is deferred to
the supplement. Projection of the variances onto the constraint set is a simple thresholding operation.
5
5.1
Experimental Results
Discrete Markov Random Fields
We consider binary Ising models, with variables arranged in NxN lattices with toroidal boundary
conditions. Potentials are parametrized as in [19], so that
exp(hs )
exp(Jst ) exp(?Jst )
?s =
, ?st =
.
exp(?hs )
exp(?Jst ) exp(Jst )
We sample 500 instances at random from a 10x10 toroidal lattice with each Jst ? N (0, 1) and
hs ? N (0, 0.01). LBP is run for a maximum of 1000 iterations, and MoM is initialized with a
single iteration of LBP. We report average L1 error of the approximate marginals as compared to
the true marginals computed with the junction tree algorithm [20]. Marginal errors are reported in
Figure 2(a,top), and there is a clear improvement over LBP in the majority of cases.
Direct evaluation of the Bethe free energy does not take into account constraint violations for nonconvergent LBP runs. The augmented Lagrangian penalizes constraint violation, but requires a
penalty parameter which LBP does not provide. For an objective comparison, we construct a penalized Bethe free energy by evaluating the augmented Lagrangian with fixed penalty c = 1 and
multipliers ? = 0. We evaluate this objective at the final iteration of both algorithms. As we see in
Figure 2(a,bottom), MoM finds a lower free energy for most trials.
Our implementations of LBP and MoM are in Matlab, and emphasize correctness over efficiency.
Nevertheless, computation time for LBP exceeds that of MoM. Wall clock time is measured in
seconds across various trials, and the percentiles for LBP are 25%: 1040.46, 50%: 1042.57, and
75%: 1044.85. For MoM they are 25%: 290.25, 50%: 381.62, and 75%: 454.52.
5.2
Gaussian Markov Random Fields
For the Gaussian case we again sample 500 random instances from a 10x10 lattice with toroidal
boundary conditions. We randomly sample only pairwise normalizable instances and initialization
is provided with a single iteration of Gaussian LBP. We find that MoM is generally insensitive to
initialization in this model. True marginals are computed by explicitly inverting the model precision
matrix and average symmetric L1 error with respect to truth is reported in Figure 2(b,top).
For pairwise normalizable models, Gaussian LBP is guaranteed to converge to the unique fixed point
of the Bethe free energy, so it is reassuring that MoM optimization matches LBP performance. The
value of the augmented Lagrangian at the final iteration is shown in Figure 2(b,bottom) and again
shows that MoM matches Gaussian LBP on pairwise normalizable models. Computation time for
MoM is slightly faster with median wall clock time of 58.76 seconds as compared to 103.17 seconds
for LBP. The 25% and 75% percentiles are 37.81 and 92.10 seconds for MoM compared to 88.40
and 125.59 seconds for LBP.
7
Method of Multipliers
0.4
0.3
0.2
0.5
1
0.49
0.9
0.48
0.8
0.47
0.7
Method of Multipliers
0.5
Method of Multipliers
Avg. Marginal Error
Avg. Marginal Error
Avg. Marginal Error
0.46
0.45
0.44
0.6
0.5
0.4
0.43
0.3
0.42
0.2
0.1
0.1
0.41
0
0
0.1
0.2
0.3
0.4
Belief Propagation
0.5
0.4
0.4
0.6
0.42
0.44
0.46
Belief Propagation
0.48
0
0.5
Augmented Lagrange
Augmented Lagrange
0
0.2
0.4
0.6
Expectation Propagation
0.8
1
Augmented Lagrange
7
10
120
6
?120
10
100
5
10
?130
?140
?150
?160
Method of Multipliers
Method of Multipliers
Method of Multipliers
80
60
40
20
4
10
3
10
2
10
1
10
0
?170
0
10
?20
?180
?1
10
?180
?170
?160
?150
?140
Belief Propagation
(a)
?130
?120
?20
0
20
40
60
80
Belief Propagation
(b)
100
120
0
10
2
4
10
10
Expectation Propagation
6
10
(c)
Figure 2: Performance of MoM and LBP on randomly generated (a) discrete 10 ? 10 toroidal Ising MRFs,
(b) 10 ? 10 toroidal Gaussian MRfs, and (c) clutter models with N = 30 observations. Each point corresponds
to a single model instance. Top: L1 error between estimated and true marginal distributions, averaged over all
nodes. Bottom: Penalized Bethe free energy constructed by setting ? = 0, c = 1 in the augmented Lagrangian.
5.3
Discrete Mixtures of Gaussian Potentials
To test the benefits of avoiding degenerate marginals, we consider the clutter model of Sec. 4.3 with
?0 = 0, P0 = 100 and ?0 = 0.25. The variance of the clutter distribution is ?02 = 10, and of the
target distribution ?12 = 1. We sample N = 30 observations for each trial instance.
A good initialization of the multipliers is critical to performance of MoM. We generate 10 initializations by running 5 iterations of EP, each with a different random message update schedule, compute
the corresponding Lagrange multipliers for each, and use the one with the lowest value of the augmented Lagrangian. Similarly, we measure EP?s performance by the best performing of 10 longer
runs. Both methods are run for a maximum of 1000 iterations, and true marginals are computed
numerically by finely discretizing the scalar target x.
We sample 500 random instances and report average L1 error with respect to true marginals in
Figure 2(c,top). We see a significant improvement in the majority of runs. Similarly, the augmented
Lagrangian comparison is shown in Figure 2(c,bottom) and MoM often finds a better penalized free
energy. While MoM and EP can both suffer from local optima, MoM avoids non-convergence and
the output of invalid (negative variance) marginal distributions. Median wall clock time for EP is
0.59 seconds, and 9.80 seconds for MoM. The 25% and 75% percentiles are 0.42 and 0.84 seconds
for EP and 0.51 and 49.19 seconds for MoM.
6
Discussion
We have proposed an approach for directly minimizing the Bethe variational problem motivated by
successful methods in nonlinear programming. Our approach is unique in that we do not relax the
constraint on normalizability of the marginals, rather we explicitly enforce it at all points in the optimization. This method directly avoids the creation of degenerate distributions ? for example with
negative variance ? which frequently occur in more greedy approaches for minimizing the Bethe
free energy. In addition we obtain convergence guarantees under broadly applicable assumptions.
8
References
[1] J.S. Yedidia, W.T. Freeman, and Y. Weiss. Constructing free-energy approximations and generalized Belief Propagation algorithms. Information Theory, IEEE Transactions on, 51(7):2282?
2312, 2005.
[2] M. J. Wainwright and M. I. Jordan. Graphical models, exponential families, and variational
inference. Technical report, UC Berkeley, Dept. of Statistics, 2003.
[3] T. P. Minka. Expectation Propagation for approximate Bayesian inference. Uncertainty in
Artificial Intelligence, 17:362?369, 2001.
[4] Tom Heskes, Wim Wiegerinck, Ole Winther, and Onno Zoeter. Approximate inference techniques with expectation constraints. Journal of Statistical Mechanics: Theory and Experiment,
page 11015, 2005.
[5] Dmitry M. Malioutov, Jason K. Johnson, and Alan S. Willsky. Walk-sums and Belief Propagation in Gaussian graphical models. Journal of Machine Learning Research, 7:2031?2064,
2006.
[6] B. Cseke and T. Heskes. Properties of bethe free energies and message passing in Gaussian
models. Journal of Artificial Intelligence Research, 41(2):1?24, 2011.
[7] A. Yuille. CCCP algorithms to minimize the Bethe and Kikuchi free energies: Convergent
alternatives to Belief Propagation. Neural Computation, 14:1691?1722, 2002.
[8] B. Kappen T. Heskes, K. Albers. Approximate inference and constrained optimization. Uncertainty in Artificial Intelligence, 13:313?320, 2003.
[9] Martin J. Wainwright, Tommi S. Jaakkola, and Alan S. Willsky. Tree-reweighted Belief Propagation algorithms and approximate ML estimation by pseudo-moment matching. In In AISTATS, 2003.
[10] M. Welling and Y.W. Teh. Belief optimization for binary networks: A stable alternative to
Loopy Belief Propagation. In Uncertainty in Artificial Intelligence, 2001.
[11] T. Heskes and O. Zoeter. Expectation Propagation for approximate inference in dynamic
Bayesian networks. Uncertainty in Artificial Intelligence, 18:216?223, 2002.
[12] T. Minka. The EP energy function and minimization schemes. Technical report, MIT Media
Lab, 2001.
[13] D.P. Bertsekas. Nonlinear programming. Athena Scientific, 1999.
[14] M. Seeger. Bayesian inference and optimal design for the sparse linear model. Journal of
Machine Learning Research, 9:759?813, 2008.
[15] C. Rasmussen. Gaussian Processes for Machine Learning. MIT Press, 2006.
[16] M. Schmidt, E. Van Den Berg, M. Friedlander, and K. Murphy. Optimizing costly functions
with simple constraints: A limited-memory projected quasi-Newton algorithm. In AI & Statistics, 2009.
[17] D.P. Bertsekas. Constrained optimization and Lagrange multiplier methods. Computer Science
and Applied Mathematics, Boston: Academic Press, 1982, 1, 1982.
[18] T. Heskes. On the uniqueness of loopy belief propagation fixed points. Neural Computation,
16(11):2379?2413, 2004.
[19] J.S. Yedidia, W.T. Freeman, and Y. Weiss. Generalized Belief Propagation. Advances in neural
information processing systems, pages 689?695, 2001.
[20] Joris M. Mooij. libDAI: A free and open source C++ library for discrete approximate inference
in graphical models. Journal of Machine Learning Research, 11:2169?2173, August 2010.
9
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4,027 | 4,645 | Stochastic optimization and sparse statistical
recovery: Optimal algorithms for high dimensions
Martin J. Wainwright
Sahand N. Negahban
Alekh Agarwal
Dept. of EECS and Statistics
Dept. of EECS
Microsoft Research
UC Berkeley
MIT
New York NY
[email protected] [email protected] [email protected]
Abstract
We develop and analyze stochastic optimization algorithms for problems in which
the expected loss is strongly convex, and the optimum is (approximately) sparse.
Previous approaches are able to exploit only one of these two structures, yielding
a O(d/T
) convergence rate for strongly convex objectives in d dimensions and
p
O( s(log d)/T ) convergence rate when the optimum is s-sparse. Our algorithm
is based on successively solving a series of ?1 -regularized optimization problems
using Nesterov?s dual averaging algorithm. We establish that the error of our solution after T iterations is at most O(s(log d)/T ), with natural extensions to approximate sparsity. Our results apply to locally Lipschitz losses including the logistic,
exponential, hinge and least-squares losses. By recourse to statistical minimax
results, we show that our convergence rates are optimal up to constants. The effectiveness of our approach is also confirmed in numerical simulations where we
compare to several baselines on a least-squares regression problem.
1 Introduction
Stochastic optimization algorithms have many desirable features for large-scale machine learning,
and have been studied intensively in the last few years (e.g., [18, 4, 8, 22]). The empirical efficiency
of these methods is backed with strong theoretical guarantees on their convergence rates, which
depend on various structural properties of the objective function. More precisely, for an objective
function that is strongly convex, stochastic gradient descent enjoys a convergence rate ranging from
O(1/T ), when features vectors are extremely sparse, to O(d/T ), when feature vectors are dense [9,
14, 10]. This strong convexity condition is satisfied for many common machine learning problems,
including boosting, least squares regression, SVMs and generalized linear models among others.
A complementary condition is that of (approximate) sparsity in the optimal solution. Sparse models
have proven useful in many applications (see e.g., [6, 5] and references therein), and many statistical
?
procedures seek to exploit such sparsity. It has been shown [15, 19] that when the optimal
p solution ?
is s-sparse, appropriate versions of the mirror descent algorithm converge at a rate O(s (log d)/T ).
Srebro et al. [20]
p exploit the smoothness of common loss functions, and
? obtain improved rates of
the form O(? (s log d)/T ), where ? is the noise variance. While the log d scaling makes these
methods attractive in?high dimensions, their scaling with respect to the iterations T is relatively
slow?namely, O(1/ T ) as opposed to O(1/T ) for strongly convex problems.
Many optimization problems encountered in practice exhibit both features: the objective function is
strongly convex, and the optimum is (approximately) sparse. This fact leads to the natural question:
is it possible to design algorithms for stochastic optimization that enjoy the best features of both
types of structure? More specifically, an algorithm should have a O(1/T ) convergence rate, as well
as a logarithmic dependence on dimension. The main contribution of this paper is to answer this
question in the affirmative, and to analyze a new algorithm that has convergence rate O((s log d)/T )
1
for a strongly convex problem with an s-sparse optimum in d dimensions. This rate is unimprovable
(up to constants) in our setting, meaning that no algorithm can converge at a substantially faster rate.
Our analysis also yields optimal rates when the optimum is only approximately sparse.
The algorithm proposed in this paper builds off recent work on multi-step methods for strongly
convex problems [11, 10, 12], but involves some new ingredients so as to obtain optimal rates for
statistical problems with sparse optima. In particular, we form a sequence of objective functions
by decreasing the amount of regularization as the optimization algorithm proceeds which is quite
natural from a statistical viewpoint. Each step of our algorithm can be computed efficiently, with a
closed form update rule in many common examples. In summary, the outcome of our development
is an optimal one-pass algorithm for many structured statistical problems in high dimensions, and
with computational complexity linear in the sample size. Numerical simulations confirm our theoretical predictions regarding the convergence rate of the algorithm, and also establish its superiority
compared to regularized dual averaging [22] and stochastic gradient descent algorithms. They also
confirm that a direct application of the multi-step method of Juditsky and Nesterov [11] is inferior
to our algorithm, meaning that our gradual decrease of regularization is quite critical. More details
on our results and their proofs can be found in the full-length version of this paper [2].
2 Problem set-up and algorithm description
Given a subset ? ? Rd and a random variable Z taking values in a space Z, we consider an
optimization problem of the form
?? ? arg min E[L(?; Z)],
(1)
???
where L : ? ? Z ? R is a given loss function. As is standard in stochastic optimization, we do
not have direct access to the expected loss function L(?) := E[L(?; Z)], nor to its subgradients.
Rather, for a given query point ? ? ?, we observe a stochastic subgradient, meaning a random
vector g(?) ? Rd such that E[g(?)] ? ?L(?). The goal of this paper is to design algorithms that are
suitable for solving the problem (1) when the optimum ?? is (approximately) sparse.
Algorithm description: In order to solve a sparse version of the problem (1), our strategy is to
consider a sequence of regularized problems of the form
min? L(?) + ?k?k1 .
(2)
???
Our algorithm involves a sequence of KT different epochs, where the regularization parameter ? > 0
and the constraint set ?? ? ? change from epoch to epoch. The epochs are specified by:
th
T
epoch,
? a sequence of natural numbers {Ti }K
i=1 , where Ti specifies the length of the i
KT
? a sequence of positive regularization weights {?i }i=1 , and
KT
T
vectors {yi }K
? a sequence of positive
i=1 , which specify the con radii {Ri }i=1 and d-dimensional
straint set, ?(Ri ) := ? ? ? | k? ? yi kp ? Ri , that is used throughout the ith epoch.
We initialize the algorithm in the first epoch with y1 = 0, and with any radius R1 that is an upper bound on k?? k1 . The norm k ? kp used in defining the constraint set ?(Ri ) is specified by
p = 2 log d/(2 log d ? 1), a choice that will be clarified momentarily.
The goal of the ith epoch is to update yi 7? yi+1 , in such a way that we are guaranteed that
2
2
kyi+1 ? ?? k21 ? Ri+1
for each i = 1, 2, . . .. We choose the radii such that Ri+1
= Ri2 /2, so
that upon termination, kyKT ? ?? k21 ? R12 /2KT ?1 . In order to update yi 7? yi+1 , we run Ti rounds
of the stochastic dual averaging algorithm [17] (henceforth DA) on the regularized objective
(3)
min L(?) + ?i k?k1 .
???(Ri )
i
i
The DA method generates two sequences of vectors {?t }Tt=0
and {?t }Tt=0
initialized as ?0 = 0
0
t Ti
and ? = yi , using a sequence of step sizes {? }t=0 . At iteration t = 0, 1, . . . , Ti , we let g t be a
stochastic subgradient of L at ?t , and we let ? t be any element of the subdifferential of the ?1 -norm
k ? k1 at ?t . The DA update at time t maps (?t , ?t ) 7? (?t+1 , ?t+1 ) via the recursions
(4)
?t+1 = ?t + g t + ?i ? t , and ?t+1 = arg min ?t+1 h?t+1 , ?i + ?yi ,Ri (?) ,
???(Ri )
2
where the prox function ? is specified below (5). The pseudocode describing the overall procedure
is given in Algorithm 1. In the stochastic dual averaging updates (4), we use the prox function
2 log d
1
k? ? yi k2p , where p =
.
(5)
?yi ,Ri (?) =
2Ri2 (p ? 1)
2 log d ? 1
This particular choice of the prox-function and the specific value of p ensure that the function ?
is strongly convex with respect to the ?1 -norm, and has been previously used for sparse stochastic
optimization (see e.g. [15, 19, 7]). In most of our examples, ? = Rd and owing to our choice of the
prox-function and the feasible set in the update (4), we can compute ?t+1 from ?t+1 in closed form.
Some algebra yields that the update (4) with ? = Rd is equivalent to
?
t+1
Ri2 ?t+1
|?t+1 |(q?1) sign(?t+1 )
?t+1 k?t+1 kq Ri
= yi +
?1 .
, where ? = max 0,
(q?2)
(p ? 1)(1 + ?)
p?1
k?t+1 kq
Here |?t+1 |(q?1) refers to elementwise operations and q = p/(p ? 1) is the conjugate exponent to
p. We observe that our update (4) computes a subgradient of the ?1 -norm rather than computing an
exact prox-mapping as in some previous methods [16, 7, 22]. Computing such a prox-mapping for
yi 6= 0 requires O(d2 ) computation, which is why we adopt the update (4) with a complexity O(d).
Algorithm 1 Regularization Annealed epoch Dual AveRaging (RADAR)
T
Require: Epoch length schedule {Ti }K
i=1 , initial radius R1 , step-size multiplier ?, prox-function
?, initial prox-center y1 , regularization parameters ?i .
for Epoch i = 1, 2, . . . , KT do
Initialize ?0 = 0 and ?0 = yi .
for Iteration t = 0, 1, . . . , Ti ? 1 do
?
Update (?t , ?t ) 7? (?t+1 , ?t+1 ) according to rule (4) with step size ?t = ?/ t.
end for
PT i t
?
Set yi+1 = t=1
.
Ti
2
Update Ri+1
= Ri2 /2.
end for
Return yKT +1
Conditions: Having defined our algorithm, we now discuss the conditions on the objective function L(?) and stochastic gradients that underlie our analysis.
Assumption 1 (Locally Lipschitz). For each R > 0, there is a constant G = G(R) such that
? ? G k? ? ?k
? 1
|L(?) ? L(?)|
(6)
for all pairs ?, ?? ? ? such that k? ? ?? k1 ? R and k?? ? ?? k1 ? R.
We note that it suffices to have k?L(?)k? ? G(R) for the above condition. As mentioned, our
goal is to obtain fast rates for objectives satisfying a local strong convexity condition, defined below.
Assumption 2 (Local strong convexity (LSC)). The function L : ? ? R satisfies a R-local form
of strong convexity (LSC) if there is a non-negative constant ? = ?(R) such that
? 2 ??, ?? ? ? with k?k1 ? R and k?k
? 1 ? R. (7)
? ? L(?) + h?L(?), ?? ? ?i + ? k? ? ?k
L(?)
2
2
Some of our results regarding stochastic optimization from a finite sample will use a weaker form of
the assumption, called local RSC, exploited in our recent work on statistics and optimization [1, 13].
Our final assumption is a tail condition on the error in stochastic gradients: e(?) := g(?) ? E[g(?)].
Assumption 3 (Sub-Gaussian stochastic gradients). There is a constant ? = ?(R) such that
E exp(ke(?)k2? /? 2 ) ? exp(1) for all ? such that k? ? ?? k1 ? R.
(8)
Clearly, this condition holds whenever the error vector e(?) has bounded components. More generally, the bound (8) holds whenever each component of the error vector has sub-Gaussian tails.
3
Some illustrative examples: We now describe some examples that satisfy the above conditions to
illustrate how the various parameters of interest might be obtained in different scenarios.
Example 1 (Classification under Lipschitz losses). In binary classification, the samples consist of
pairs z = (x, y) ? Rd ? {?1, 1}. Common choices for the loss function L(?; z) are the hinge loss
max(0, 1 ? yh?, xi) or the logistic loss log(1 + exp(?yh?, xi). Given a distribution P over Z (either
the population or the empirical distribution), a common strategy is to draw (xt , yt ) ? P at iteration
t and use g t = ?L(?; (xt , yt )). We now illustrate how our conditions are satisfied in this setting.
? Locally Lipschitz: Both the above examples actually satisfy a stronger global Lipschitz condition
since we have the bound G ? k?L(?)k? ? Ekxk? . Often, the data satisfies the normalization
kxk? ? B, in which case we get G ? B. More
? generally, tail conditions on the marginal
distribution of each coordinate of x ensure G = O( log d)) is valid with high probability.
? LSC: When the expectation in the objective (1) is under the population distribution, the above
examples satisfy LSC. Here we focus on the example of the logistic loss, where we define the link
function ?(?) = exp(?)/(1 + exp(?))2 . We also define ? = E[xxT ] to be the covariance matrix
and let ?min (?) denote its minimum singular value. Then a second-order Taylor expansion yields
e
? ? L(?) ? h?L(?), ?? ? ?i = ?(h?, xi) k?1/2 (? ? ?)k
? 2 ? ?(BR)?min (?) k? ? ?k
? 2,
L(?)
2
2
2
2
where ?e = a? + (1 ? a)?? for some a ? (0, 1). Hence ? ? ?(BR)?min (?) in this example.
? Sub-Gaussian gradients: Assuming the bound Ekxk? ? B, this condition is easily verified. A
simple calculation yields ? = 2B, since
ke(?)k? = k?L(?; (x, y)) ? ?L(?)k? ? k?L(?; (x, y))k? + k?L(?)k? ? 2B.
Example 2 (Least-squares regression). In the regression setup, we are given samples of the form
z = (x, y) ? Rd ? R. The loss function of interest is L(?; (x, y)) = (y ? h?, xi)2 /2. To illustrate
the conditions more clearly, we assume that our samples are generated as y = hx, ?? i + w, where
w ? N (0, ? 2 ) and ExxT = ? so that EL(?; (x, y)) = k?1/2 (? ? ?? )k22 /2.
? Locally Lipschitz: For this example, the Lipschitz parameter G(R) depends on the bound R.
If we define ?(?) = maxi ?ii to be the largest variance of a coordinate of x, then a direct
calculation yields the bound G(R) ? ?(?)R.
? LSC: Again we focus on the case where the expectation is taken under the population distribution, where we have ? = ?min (?).
? Sub-Gaussian gradients: Once again we assume that kxk? ? B. It can be shown with some
work that Assumption 3 is satisfied with ? 2 (R) = 8?(?)2 R2 + 4B 4 R2 + 10B 2 ? 2 .
3 Main results and their consequences
In this section we state our main results, regarding the convergence of Algorithm 1. We focus on
the cases where Assumptions 1 and 3 hold over the entire set ?, and RSC holds uniformly for
all k?k1 ? R1 ; key examples being the hinge and logistic losses from Example 1. Extensions to
examples such as least-squares loss, which are not Lipschitz on all of ? require a more delicate
treatment and these results as well the proofs of our results can be found in the long version [2].
Formally, we assume that G(R) ? G and ?(R) ? ? in Assumptions 1 and 3. We also use ? to
denote ?(R1 ) in Assumption 2. For a constant ? > 0 governing the error probability in our results,
we also define ?i2 = ? 2 + 24 log i at epoch i. Our results assume that we run Algorithm 1 with
2
s
2
2
2 2
T i ? c1
(G
+
?
)
log
d
+
?
?
+
log
d
,
(9)
i
? 2 Ri2
where c1 is a universal constant. For a total of T iterations in Algorithm 1, we state our results for
the parameter ?bT = y(KT +1) where KT is the last epoch completed in T iterations.
3.1 Main theorem and some remarks
We start with our main result which shows an overall convergence rate of O(1/T ) after T iterations.
This O(1/T ) convergence is analogous to earlier work on multi-step methods for strongly convex
4
objectives [11, 12, 10]. For each subset S ? {1, 2, . . . , d} of cardinality s, we define
?2 (?? ; S) := k?S? c k21 /s.
(10)
?
2
?
This quantity captures the degree of sparsity in the optimum ? ; for instance, ? (? ; S) = 0 if and
only if ?? is supported on S. Given the probability parameter ? > 0, we also define the shorthand
? 2 R12 T
?T = log2 2
log d.
(11)
s ((G2 + ? 2 ) log d + ? 2 ? 2 )
Theorem 1. Suppose the expected loss L satisfies Assumptions 1? 3 with parameters G(R) ? G,
? and ?(R) ? ?, and we perform updates (4) with epoch lengths (9) and parameters
s
q
R
?
log d
i
.
(12)
?2i = ?
(G2 + ? 2 ) log d + ?i2 ? 2 and ?(t) = 5Ri
2
(G
+
?2i + ? 2 )t
s Ti
Then for any subset S ? {1, . . . , d} of cardinality s and any T ? 2?T , there is a universal constant
c0 such that with probability at least 1 ? 6 exp(?? 2 /12) we have
s
?T
2
2
2
2
2 ?
k?bT ? ?? k22 ? c3
((G
+
?
)
log
d
+
?
(?
+
log
))
+
?
(?
;
S)
.
(13)
?2T
log d
Consequently, the theorem predicts a convergence rate of O(1/? 2 T ) which is the best possible under
our assumptions. Under the setup of Example 1, the error bound of Theorem 1 further simplifies to
2
sB
2
2 ?
(log
d
+
?
)
+
?
(?
;
S)
.
(14)
k?bT ? ?? k22 = O
?2T
We note that for an approximately sparse ?? , Theorem 1 guarantees convergence only to a tolerance ?2 (?? ; S) due to the error terms arising out of the approximate sparsity. Overall, the theorem
provides a family of upper bounds, one for each choice of S. The best bound can be obtained by
optimizing this choice, trading off the competing contributions of s and k?S? c k1 .
At this point, we can compare the result of Theorem 1 to some of the previous work. One approach
to minimize the objective (1) is to perform stochastic gradient descent on the objective,which has
a
ke(?)k22
2
2
2
e
e
convergence rate of O((G + ?
e )/(? T )) [10, 14], where k?L(?)k2 ? G and E exp
?
?
e2
2
e
exp(1). In the setup of Example 1, G = Bd and similarly for ?
e ; giving an exponentially worse
scaling in the dimension d. An alternative is to perform mirror descent [15, 19] or regularized dual
averaging [22] using the same prox-function as Algorithm 1 but without breaking it up into epochs.
As mentioned in the introduction, this single-step method
p fails to exploit the strong convexity of our
problem and obtains inferior convergence rates of O(s log d/T ) [19, 22, 7].
A proposal closer to our approach is to minimize the regularized objective (3), but with a fixed
value of ? instead of the decreasing schedule of ?i used in Theorem 1. This amounts to using the
method of Juditsky and Nesterov [11] on the regularized problem,
and by using the proof techniques
p
developedin this paper, it can be shown that setting ? = ? log d/T leads to an overall convergence
2
2
e sB
rate of O
2 (log d + ? ) , which exhibits the same scaling as Theorem 1. However, with this
? T
fixed setting of ?, the initial epochs tend to be much longer than needed for halving the error. Indeed,
our setting of ?i is based on minimizing the upper bound at each epoch, and leads to an improved
performance in our numerical simulations. The benefits of slowly decreasing the regularization in
the context of deterministic optimization were also noted in the recent work of Xiao and Zhang [23].
3.2 Some illustrative corollaries
We now present some consequences of Theorem 1 by making specific assumptions regarding the
sparsity of ?? . The simplest situation is when ?? is supported on some subset S of size s. More
generally, Theorem 1 also applies to the case when the optimum ?? is only approximately sparse.
One natural form of approximate sparsity is to assume that ?? ? Bq (Rq ) for 0 < q ? 1, where
)
(
d
X
q
d
|?i | ? Rq .
Bq (Rq ) := ? ? R |
i=1
5
For 0 < q ? 1, membership in the set Bq (Rq ) enforces a decay rate on the components of the
vector ?. We now present a corollary of Theorem 1 under such an approximate sparsity condition.
To facilitate comparison with minimax lower bounds, we set ? 2 = ? log d in the corollaries.
Corollary 1. Under the conditions of Theorem 1, for all T > 2?T with probability at least 1 ?
6 exp(?? log d/12), there is a universal constant c0 such that
h 2 2
i
?
d
?T
s?2
?
c0 G +?? 2(1+?) s log
+
log
?? is s-sparse,
?
2
T
? T
log d
? 2
2?q
2?q
b
o 2
2 2
n 2 2
?
k ?T ? ? k 2 ?
log logTd
(G +? (1+?)) log d
?
? c0 Rq
+ ??2 T
?? ? Bq (Rq ).
q
?2T
((1+?) log d) 2
The first part of the corollary follows directly from Theorem 1 by noting that ?2 (?? ; S) = 0 under
our assumptions. Note that as q ranges over the interval [0, 1], reflecting the degree of sparsity,?the
e
e
convergence rate ranges from from O(1/T
) (for q = 0 corresponding to exact sparsity) to O(1/
T)
(for q = 1). This is a rather interesting trade-off, showing in a precise sense how convergence rates
vary quantitatively as a function of the underlying sparsity.
It is useful to note that the results on recovery for generalized linear models presented here exactly
match those that have been developed in the statistics literature [13, 21], which are optimal under our
assumptions on the design vectors. Concretely, ignoring factors of O(log T ), we get a parameter ?bT
having error at most O(s log d/(? 2 T ) with an error probability decyaing to zero with d. Moreover,
in doing so our algorithm only goes over at most T data samples, as each stochastic gradient can be
evaluated with one fresh data sample drawn from the underlying distribution. Since the statistical
minimax lower bounds [13, 21] demonstrate that this is the smallest possible error that any method
can attain from T samples, our method is statistically optimal in the scaling of the estimation error
with the number of samples. We also observe that it is easy to instead set the error probability to
? = ? 2 log T , if an error probability decaying with T is desired, incurring at most additional log T
factors in the error bound. Finally, we also remark that our techniques extend to handle examples
such as the least-squares loss that are not uniformly Lipschitz. The details of this extension are
deferred to the long version of this paper [2].
Stochastic optimization over finite pools: A common setting for the application of stochastic optimization methods in machine learning is when one has a finite pool of examples, say {Z1 , . . . , Zn },
and the objective (1) takes the form
n
?? = arg min
???
1X
L(?; Zi )
n i=1
(15)
In this setting, a stochastic gradient g(?) can be obtained by drawing a sample Zj at random with
replacement from the pool {Z1 , . . . , Zn }, and returning the gradient ?L(?; Zj ).
In high-dimensional problems where d ? n, the sample loss is not strongly convex. However, it has
been shown by many researchers [3, 13, 1] that under suitable conditions, this objective does satisfy
restricted forms of the LSC assumption, allowing us to appeal to a generalized form of Theorem 1.
We will present this corollary only for settings where ?? is exactly sparse and also specialize to the
logistic loss, L(?; (x, y)) = log(1 + exp(?yh?, xi)) to illustrate the key aspects of the result. We
recall the definition of the link function ?(?) = exp(?)/(1 + exp(?))2 . We will state the result
for sub-Gaussian data design with parameters (?, ?x2 ), meaning that the E[xi xTi ] = ? and hu, xi i is
?x -sub-Gaussian for any unit norm vector u ? Rd .
Corollary 2. Consider the finite-pool loss (15), based on n i.i.d. samples from a sub-Gaussian
design with parameters (?, ?x2 ). Suppose that Assumptions 1-3 are satisfied and the optimum ??
of (15) is s-sparse. Then there are universal constants (c0 , c1 , c2 , c3 ) such that for all T ? 2?T and
d
2
n ? c3 ?2log(?)
max(?min
(?), ?x4 ), we have
min
k?bT ? ?? k22 ?
c0
2 (?)
?min
2
o
s? 2
s log d n
1
?T
B (1 + ?) + c0 2
log
.
2
T
? (2BR1 )
?min (?)? 2 (2BR1 )T
log d
2
with probability at least 1 ? 2 exp(?c1 n min(?min
(?)/?x4 , 1)) ? 6 exp(?? log d/12).
6
We observe that the bound only holds when the number of samples n in the objective (15) is large
enough, which is necessary for the restricted form of the LSC condition to hold with non-trivial
parameters in the finite sample setting.
A modified method with constant epoch lengths: Algorithm 1 as described is efficient and simple to implement. However, the convergence results critically rely on the epoch length Ti to be
set appropriately in a doubling manner. This could be problematic in practice, where it might be
tricky to know when an epoch should be terminated. Following Juditsky and Nesterov [11], we
next demonstrate how a variant of our algorithm with constant epoch lengths enjoys similar rates of
convergence. The key challenge here is that unlike the previous set-up [11], our objective function
changes at each epoch which leads to significant technical difficulties. At a very coarse level, if
we have a total budget of T iterations, then this version of our algorithm allows us to set the epoch
lengths to O(log T ), and guarantees convergence rates that are O((log T )/T ).
Theorem 2. Suppose the expected loss satisfies Assumptions 1- 3 with parameters G, ?, and ?
resp. Let S be any subset of {1, . . . , d} of cardinality s. Suppose we run Algorithm 1 for a total of
T iterations with epoch length Ti ? T log d/?T and with parameters as in Equation 12. Assuming
that this setting ensures Ti = O(log d), for any set S, with probability at least 1 ? 3 exp(? 2 /12)
(G2 + ? 2 ) log d + (? 2 + log(?/ log d))? 2 log d
k?bT ? ?? k22 = O s
.
T
?
The theorem shows that up to logarithmic factors in T , setting the epoch lengths optimally is not
critical. A similar result can also be proved for the case of least-squares regression.
4 Simulations
In this section we will present numerical simulations that back our theoretical convergence results.
We focus on least-squares regression, discussed in Example 2. Specifically, we generate samples
(xt , yt ) with each coordinate of xt distributed as Unif[?B, B] and yt = h?? , xt i + wt . We pick ??
to be s-sparse vector with s = ?log d?, and wt ? N (0, ? 2 ) with ? 2 = 0.5. Given an iterate ?t , we
generate a stochastic gradient of the expected loss (1) at (xt , yt ). For the ?1 -norm, we pick the sign
vector of ?t , with 0 for any component that is zero, a member of the ?1 -sub-differential.
Our first set of results evaluate Algorithm 1 against other stochastic optimization baselines assuming
a complete knowledge of problem parameters. Specifically, we epoch i is terminated once
kyi+1 ? ?? k2p ? kyi ? ?? k2p /2. This ensures that ?? remains feasible throughout, and tests the performance of Algorithm 1 in the most favorable scenario. We compare the algorithm against two
baselines. The first baseline is the regularized
dual averaging (RDA) algorithm [22], applied to the
p
regularized objective (3) with ? = 4? log d/T , which is the statistically optimal regularization pak?k2
p
, so that the theory for RDA
rameter with T samples. We use the same prox-function ?(?) = 2(p?1)
p
predicts a convergence rate of O(s log d/T ) [22]. Our second baseline is the stochastic gradient
(SGD) algorithm which exploits the strong convexity but not the sparsity of the problem (1). Since
the squared loss is not uniformly Lipschitz, we impose an additional constraint k?k1 ? R1 , without
which the algorithm does not converge. The results of this comparison are shown in Figure 1(a),
where we present the error k?t ? ?? k22 averaged over 5 random trials. We observe that RADAR
comprehensively outperforms both the baselines, confirming the predictions of our theory.
The second set of results focuses on evaluating algorithms better tailored for our assumptions. Our
first baseline here is the approach that we described in our remarks following Theorem 1. In this
approach we use the same multi-step strategy as Algorithm 1 but keepp? fixed. We refer to this as
Epoch Dual Averaging (henceforth EDA), and again employ ? = 4? log d/T with this strategy.
Our epochs are again determined by halving of the squared ?p -error measured relative to ?? .
Finally, we also evaluate the version of our algorithm with constant epoch lengths that we analyzed in
Theorem 2 (henceforth RADAR-CONST), using epochs of length log(T ). As shown in Figure 1(b),
the RADAR-CONST has relatively large error during the initial epochs, before converging quite
7
rapidly, a phenomenon consistent with our theory.1 Even though the RADAR-CONST method does
not use the knowledge of ?? to set epochs, all three methods exhibit the same eventual convergence
rates, with RADAR (set with optimal epoch lengths) performing the best, as expected. Although
RADAR-CONST is very slow in initial iterations, its convergence rate remains competitive with
EDA (even though EDA does exploit knowledge of ?? ), but is worse than RADAR as expected.
Overall, our experiments demonstrate that RADAR and RADAR-CONST have practical performance consistent with our theoretical predictions. Although optimal epoch length setting is not
too critical for our approach, better data-dependent empirical rules for determining epoch lengths
remains an interesting question for future research. The relatively poorer performance of EDA
demonstrates the importance of our decreasing regularization schedule.
Error vs. iterations
Error vs. iterations
6
5
RADAR
SGD
RDA
5
RADAR
EDA
RADAR-CONST
4
k? t ? ? ? k22
k? t ? ? ? k22
4
3
3
2
2
1
1
0
0
0.5
1
Iterations
1.5
0
0
2
4
x 10
(a)
0.5
1
Iterations
1.5
2
4
x 10
(b)
Figure 1. A comparison of RADAR with other stochastic optimization algorithms for d = 40000 and
s = ?log d?. The left plot compares RADAR with the RDA and SGD algorithms, neither of which
exploits both the sparsity and the strong convexity structures simultaneously. The right one compares
RADAR with the EDA and RADAR-CONST algorithms, all of which exploit the problem structure
but with varying degrees of effectiveness. We plot k?t ? ?? k22 averaged over 5 random trials versus the
number of iterations.
5 Discussion
In this paper we present an algorithm that is able to take advantage of the strong convexity and sparsity conditions that are satsified by many common problems in machine learning. Our algorithm
is simple and efficient to implement, and for a d-dimensional objective with an s-sparse optima, it
achieves the minimax-optimal convergence rate O(s log d/T ). We also demonstrate optimal convergence rates for problems that have weakly sparse optima, with implications for problems such as
sparse linear regression and sparse logistic regression. While we focus our attention exclusively on
sparse vector recovery due to space constraints, the ideas naturally extend to other structures such as
group sparse vectors and low-rank matrices. It would be interesting to study similar developments
for other algorithms such as mirror descent or Nesterov?s accelerated gradient methods, leading to
multi-step variants of those methods with optimal convergence rates in our setting.
Acknowledgements
The work of all three authors was partially supported by ONR MURI grant N00014-11-1-0688 to
MJW. In addition, AA was partially supported by a Google Fellowship, and SNN was partially
supported by the Yahoo KSC award.
1
To clarify, the epoch lengths in RADAR-CONST are set large enough to guarantee that we can attain an overall error bound of O(1/T ),
meaning that the initial epochs for RADAR-CONST are much longer than for RADAR. Thus, after roughly 500 iterations, RADAR-CONST
?
has done only 2 epochs and operates with a crude constraint set ?(R1 /4). During epoch i, the step size scales proportionally to Ri / t,
where t is the iteration number within the epoch; hence the relatively large initial steps in an epoch can take us to a bad solution even when
we start with a good solution yi when Ri is large. As Ri decreases further with more epochs, this effect is mitigated and the error of
RADAR-CONST does rapidly decrease like our theory predicts.
8
References
[1] A. Agarwal, S. N. Negahban, and M. J. Wainwright. Fast global convergence rates of gradient methods
for high-dimensional statistical recovery. To appear in The Annals of Statistics, 2012. Full-length version
http://arxiv.org/pdf/1104.4824v2.
[2] A. Agarwal, S. N. Negahban, and M. J. Wainwright. Stochastic optimization and sparse statistical recovery: An optimal algorithm for high dimensions. 2012. URL http://arxiv.org/abs/1207.4421.
[3] P. J. Bickel, Y. Ritov, and A. B. Tsybakov. Simultaneous analysis of Lasso and Dantzig selector. Ann.
Stat., 37(4):1705?1732, 2009.
[4] L. Bottou and O. Bousquet. The tradeoffs of large scale learning. In NIPS, 2007.
[5] P. B?uhlmann and S. Van De Geer. Statistics for High-Dimensional Data: Methods, Theory and Applications. Springer Series in Statistics. Springer, 2011.
[6] D. L. Donoho. High-dimensional data analysis: The curses and blessings of dimensionality, 2000.
[7] J. Duchi, S. Shalev-Shwartz, Y. Singer, and A. Tewari. Composite objective mirror descent. In Proceedings of the 23rd Annual Conference on Learning Theory, pages 14?26. Omnipress, 2010.
[8] J. Duchi and Y. Singer. Efficient online and batch learning using forward-backward splitting. Journal of
Machine Learning Research, 10:2873?2898, 2009.
[9] E. Hazan, A. Kalai, S. Kale, and A. Agarwal. Logarithmic regret algorithms for online convex optimization. In Proceedings of the Nineteenth Annual Conference on Computational Learning Theory, 2006.
[10] E. Hazan and S. Kale. Beyond the regret minimization barrier: an optimal algorithm for stochastic
strongly-convex optimization. Journal of Machine Learning Research - Proceedings Track, 19:421?436,
2011.
[11] A. Juditsky and Y. Nesterov. Primal-dual subgradient methods for minimizing uniformly convex functions. Available online http://hal.archives-ouvertes.fr/docs/00/50/89/33/PDF/Strong-hal.pdf, 2010.
[12] G. Lan and S. Ghadimi. Optimal stochastic approximation algorithms for strongly convex stochastic
composite optimization, part ii: shrinking procedures and optimal algorithms. 2010.
[13] S. Negahban, P. Ravikumar, M. J. Wainwright, and B. Yu. A unified framework for high-dimensional
analysis of M-estimators with decomposable regularizers. In NIPS Conference, Vancouver, Canada, December 2009. Full length version arxiv:1010.2731v1.
[14] A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro. Robust stochastic approximation approach to
stochastic programming. SIAM Journal on Optimization, 19(4):1574?1609, 2009.
[15] A. Nemirovski and D. Yudin. Problem Complexity and Method Efficiency in Optimization. Wiley, New
York, 1983.
[16] Y. Nesterov. Gradient methods for minimizing composite objective function. Technical Report 76, Center
for Operations Research and Econometrics (CORE), Catholic University of Louvain (UCL), 2007.
[17] Y. Nesterov. Primal-dual subgradient methods for convex problems. Mathematical Programming A,
120(1):261?283, 2009.
[18] S. Shalev-Shwartz, Y. Singer, and N. Srebro. Pegasos: Primal estimated sub-gradient solver for SVM. In
Proceedings of the 24th International Conference on Machine Learning, 2007.
[19] S. Shalev-Shwartz and A. Tewari. Stochastic methods for l1 regularized loss minimization. Journal of
Machine Learning Research, 12:1865?1892, June 2011.
[20] N. Srebro, K. Sridharan, and A. Tewari. Smoothness, low noise, and fast rates. In Advances in Neural
Information Processing Systems 23, pages 2199?2207, 2010.
[21] S. A. van de Geer. High-dimensional generalized linear models and the lasso. The Annals of Statistics,
36:614?645, 2008.
[22] L. Xiao. Dual averaging methods for regularized stochastic learning and online optimization. Journal of
Machine Learning Research, 11:2543?2596, 2010.
[23] L. Xiao and T. Zhang. A proximal-gradient homotopy method for the sparse least-squares problem.
ICML, 2012. URL http://arxiv.org/abs/1203.3002.
9
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4,028 | 4,646 | On the (Non-)existence of Convex, Calibrated
Surrogate Losses for Ranking
Cl?ement Calauz`enes, Nicolas Usunier, Patrick Gallinari
LIP6 - UPMC
4 place Jussieu, 75005 Paris, France
[email protected]
Abstract
We study surrogate losses for learning to rank, in a framework where the rankings
are induced by scores and the task is to learn the scoring function. We focus on the
calibration of surrogate losses with respect to a ranking evaluation metric, where
the calibration is equivalent to the guarantee that near-optimal values of the surrogate risk imply near-optimal values of the risk defined by the evaluation metric.
We prove that if a surrogate loss is a convex function of the scores, then it is not
calibrated with respect to two evaluation metrics widely used for search engine
evaluation, namely the Average Precision and the Expected Reciprocal Rank. We
also show that such convex surrogate losses cannot be calibrated with respect to
the Pairwise Disagreement, an evaluation metric used when learning from pairwise preferences. Our results cast lights on the intrinsic difficulty of some ranking
problems, as well as on the limitations of learning-to-rank algorithms based on the
minimization of a convex surrogate risk.
1
Introduction
A surrogate loss is a loss function used as a substitute for the true quality measure during training in
order to ease the optimization of the empirical risk. The hinge loss or the exponential loss, which are
used in Support Vector Machines or AdaBoost as convex upper bounds of the classification error, are
well-known examples of surrogate losses for binary classification. In this paper, we study surrogate
losses for learning to rank, in a context where a set of items should be ranked given an input query
and where the ranking is obtained by sorting the items according to predicted numerical scores. This
work is motivated by the intensive research that has recently been carried out on machine learning
approaches to improve the quality of search engine results, and more specifically on the design of
surrogate losses that lead to high quality rankings (see [16] for a review).
Considering algorithms for learning to rank on the axis of scalability, there are first algorithms that
are designed for small-scale datasets only and that directly solve the NP-hard problem [5] without using any surrogate loss; after them come algorithms that use a surrogate loss chosen as a
non-convex but continuous and (almost everywhere) differentiable approximation of the evaluation
metric [3, 21, 10], and finally algorithms that use a convex surrogate loss. Most algorithms for
learning to rank fall into the latter category, including the reference algorithms RankBoost [12] and
Ranking SVMs [14, 4] or the regression approach of [8], because convex surrogate losses lead to
optimization problems that can be solved efficiently while non-convex approaches may require intensive computations to find a good local optimum. The disadvantage of convex surrogate losses is
that they cannot closely approximate the evaluation metrics on the whole prediction space. However,
as more examples are available and smaller values of the surrogate risk are achieved, the only region
of interest becomes that of near-optimal predictions. It is thus possible that the minimization of the
surrogate risk provably leads to optimal predictions according to the risk defined by the evaluation
measure. In that case, the surrogate loss is said to be calibrated with respect to the evaluation metric.
1
The calibration of surrogate losses has been extensively studied for various classification settings
[1, 26, 27, 18, 19] and for AUC optimization [7, 15]. For each of these tasks, many usual convex
losses are calibrated with respect to the natural evaluation metric. In the context of learning to rank
for search engines, several families of convex losses are calibrated with respect to the Discounted
Cumulative Gain (DCG) and its variants [8, 2, 17]. However, other metrics than the DCG are often
used as reference for the evaluation of ranked results, such as the Average Precision (AP), used in
past TREC competitions [22], the Expected Reciprocal Rank (ERR), used the Yahoo! Learning to
Rank Challenge [6], or the Pairwise Disagreement (PD), used when learning from pairwise preferences. And despite the multiplicity of convex losses that have been proposed for ranking, none
of them was proved to be calibrated with respect to any of these three metrics. This lead us to the
question of whether convex losses can be calibrated with respect to the AP, the ERR, or the PD.
Our main contribution is a definitive and negative answer to that question. We prove that if a surrogate loss is convex, then it cannot be calibrated with respect to any of the AP, the ERR or the
PD. Thus, if one of these metrics should be optimized, the price to pay for the computational advantage of convex losses is an inconsistent learning procedure, which may converge to non-optimal
predictions as the number of examples increases.
Our result generalizes previous works on non-calibration. First, Duchi et al. [11] showed that many
convex losses based on pairwise comparisons, such as those of RankBoost [12] or Ranking SVMs
[14, 4], are not calibrated with respect to the PD. Secondly, Buffoni et al. [2] showed that specific
convex losses, called order-preserving, are not calibrated with respect to the AP or the ERR, even
though these losses are calibrated with respect to (any variant of) the DCG. Our result is stronger
than those because we do not make any assumption on the exact structure of the loss; our approach as
a whole is also more general because it directly applies to the three evaluation metrics (AP, ERR and
PD). Finally, Duchi et al. conjectured that no convex loss can be calibrated with the PD in general
[11, Section 2.1] because it would provide a polynomial algorithm to solve an NP-hard problem.
Our approach thus leads to a direct proof of this conjecture.
In the next section, we describe our framework for learning to rank. We then present in Section 3
the general framework of calibration of [20], and give a new characterization of calibration for the
evaluation metrics we consider (Theorem 2), and the implications of the convexity of a surrogate
loss. Our main result is proved in Section 4. Section 5 concludes the paper, and Section 6 is a
technical part containing the full proof of Theorem 2.
Notation Let V, W be two sets. A set-valued function g from V to W maps all v ? V to a subset
of W (set-valued functions appear in the paper as the result of arg min operations). Given a subset
V of V, the image
S of V by g, denoted by g(V ), is the union of the images by g of all members of
V , i.e. g(V ) = v?V g(v). If n is a positive integer, [n] is the set {1, ..., n}, and Sn is the set of
permutations of [n]. Boldface characters are used for vectors of Rn . If x ? Rn , the i-th component
of x is denoted by xi (normal font and subscript). The cardinal of a finite set V is denoted by |V|.
2
Ranking Framework
We describe in this section the formal framework of ranking we consider. We first present the
prediction problem we address, and then define the two main objects of our study: evaluation metrics
for ranking and surrogate losses. We end the section with an outline of our technical contributions.
2.1
Framework and Definitions
We consider a framework similar to label ranking [9] or subset ranking [8]. Let X be a measurable
space (the instance space). An instance x ? X represents a query and its associated n items to rank,
for an integer n ? 3. The items are indexed from 1 to n, and the goal is to order the set of item
indexes [n] = {1, ..., n} given x. The ordering (or ranking) is predicted by a scoring function, which
is a measurable function f : X ? Rn . For any input instance x, the scoring function f predicts a
vector of n relevance scores (one score for each item) and the ranking is predicted by sorting the
item indexes by decreasing scores. We use permutations over [n] to represent rankings, with the
following conventions. First, given a permutation ? in Sn , k in [n] is the rank of the item ?(k);
second, low ranks are better, so that ?(1) is the top-ranked item.
2
Table 1: Formulas of r(y, ?) for some common ranking evaluation metrics
TYPE OF FEEDBACK
METRIC
FORMULA
n
P
Discounted Cumulative Gain
(higher values mean better performances)
n
y ? Y = {0, ..., Y } ,
Y ? N, Y ? 1
n
P
Expected Reciprocal Rank
(higher values mean better performances)
k=1
Average Precision
n
y ? Y = {0, 1}
k=1
k?1
Q
Pairwise Disagreement
q=1
P
(lower values mean better performances)
y?(k)
?1
2Y
(1 ? Rq ), Rk = 2
1
|{i:yi =1}|
(higher values mean better performances)
y ? Y = all DAGs over [n]
Rk
k
y
2 ?(k) ?1
log(1+k)
P
-1
?P
(i)
y?(k)
? -1 (i)
i:yi =1 k=1
I ? -1 (i) > ? -1 (j)
i?j?y
The quality of a ranking is measured by a ranking evaluation metric, relatively to a feedback. The
feedback space, denoted by Y, is a finite set, and an evaluation metric is a function r : Y ? Sn ? R.
We use the convention that lower values of r are preferable, and thus when we discuss existing
metrics for which higher values are better (e.g. the DCG, the AP or the ERR), we implicitly consider
their opposite. Table 1 gives the formula and feedback spaces of the evaluation metrics that we
discuss in the paper. The first three metrics ? the DCG, the ERR and the AP ? are commonly used
for search engine evaluation. The feedback they consider is a vector of relevance judgments (one
judgment per item). The last measure we consider is the PD, which is widely used when learning
from pairwise preferences. For the feedback space of the PD, we follow [11] and take Y as the set
of all directed acyclic graph (DAG) over [n]. For a DAG y ? Y, there is an edge from item i to j
(denoted i ? j ? y) when i is preferred to j, or, equivalently when i should have better rank than j.
In general, using a sorting algorithm, any ranking evaluation metric r induces a quality measure on
vectors of scores instead of rankings, considering that the sorting algorithm break ties randomly.
Thus, using the following set-valued function from Rn to Sn , called arg sort, which gives the set
of rankings induced by a vector of scores:
?s = (s1 , ..., sn ) ? Rn , arg sort(s) = ? ? Sn |?k ? [n ? 1] , s?(k) ? s?(k+1) ,
the evaluation metric on vectors of scores induced by r is defined by:
X
r (y, ?)
?y ? Y, ?s ? Rn ,
r0 (y, s) =
.
| arg sort(s)|
??arg sort(s)
For a fixed, but unknown, probability measure D on X ? Y, theR objective of a learning algorithm
is to find a scoring function f with low ranking risk R0(D, f ) = X ?Y r0 (y, f (x))dD(x, y) using a
training set of (instance, feedback) pairs (e.g. drawn i.i.d. according to D).
The optimization of the empirical ranking risk is usually intractable because the ranking loss is
discontinuous. To address this issue, algorithms optimize the empirical risk associated to a surrogate
loss instead. Throughout the paper, we assume that this loss is bounded below, so that all the infima
we take are well-defined. Without loss of generality, we assume that the surrogate loss has nonnegative values, and we define a surrogate loss as a measurable function
` : Y ? Rn ? R+ . The
R
surrogate risk of a scoring function f is then defined by L(D, f ) = X ?Y ` (y, f (x))dD(x, y).
2.2
Outline of the Analysis
Any learning algorithm that performs empirical or structural risk minimization on the surrogate risk
can, at most, be expected to reach low values of the surrogate risk. The question we address in
this paper is whether such an algorithm provably solves the real learning task, which is to achieve
low values of the ranking risk. More formally, the criterion under study is whether the following
implication holds for every sequence of scoring functions (fk )k?0 and every data distribution D:
L(D, fk ) ?? inf L(D, f )
k??
f
?
R0(D, fk ) ?? inf R0(D, f )
k??
f
(1)
where the infima are taken over all scoring functions. In particular, we show that if a surrogate loss
is convex in the sense that `(y, .) is convex for every y ? Y, and if the evaluation metric is the AP,
3
the ERR or the PD, then there are distributions and sequences of scoring functions for which (1)
does not hold. In other words, we show that learning-to-rank algorithms that define their objective
through a convex surrogate loss cannot provably optimize any of these evaluation metrics.
In order to perform a general analysis for all the three evaluation metrics, we consider Assumption
(A) below, which formalizes the common property of these metrics that is relevant to our study.
Intuitively, it means that for any given item, there is a feedback for which the performance only
depends on the rank of this item, with a strict improvement of performances when one improves the
rank of the item:
(A)
??1 < ?2 < ... < ?n such that ?i ? [n] , ?y ? Y : ?? ? Sn , r(y, ?) = ??-1 (i) .
Note that in the assumption, the values of ?k (i.e. the performance when item i is predicted at rank
k) are the same for all items. This is not a strong requirement because the metrics we consider do
not depend on how we index the elements. The DCG, the AP and the ERR satisfy (A): for each i,
we take the vector of relevance with a 1 for item i and 0 for all other items so that the values of the
metrics only depends on the rank of i (which should be ranked first). The PD satisfies Assumption
(A) as well: for each i, take y as the DAG containing the edges i ? j, ?j ? [n] \ {i} and only those
edges. For this feedback, i is preferred to all other items (and no preference is specified regarding
the other items) and thus the quality of a ranking only depends on the rank of i.
Our analysis is organized as follows. In the next section, we introduce the notion of a calibrated
surrogate loss defined by Steinwart [20], which is a criterion equivalent to (1). We then obtain a
new condition that is equivalent to calibration when (A) holds, and finally we restrict our attention
to evaluation metrics satisfying (A) and to convex surrogate losses. In that context, using our new
condition for calibration, we show that evaluation metrics with a calibrated surrogate loss necessarily
satisfy a specific property. Then, in Section 4, we prove that the AP, the ERR and the PD do not
satisfy this property. Since Assumption (A) holds for these three metrics, this latter result implies
that they do not have any convex and calibrated surrogate loss. Equivalently, it implies that (1) does
not hold in general for these metrics if the surrogate loss is convex.
3
A New Characterization of Calibration
We present in this section the notion of calibration as studied in [20], which is the basis of our
work. Then, we provide a characterization of calibration more specific to the evaluation metrics
we consider, that relates more closely calibrated surrogate losses and evaluation metrics. This more
specific characterization of calibration is the starting point of the analysis of convex and calibrated
surrogate losses carried out in the last subsection and that allows us to state the results of Section 4.
3.1
The Framework of Calibration
Applying the general results of [20] to our setting, the criterion defined by (1) can be studied by
restricting our attention to the contributions of a single instance to the surrogate and ranking risk.
These contributions are called the inner surrogate risk and the inner rankingP
risk respectively. Denoting the set of probability distributions over Y by P = p : Y ? [0, 1]| y?Y p(y) = 1 , the
inner risks are respectively defined for all p ? P and all s ? Rn by:
X
L (p, s) =
p(y)` (y, s)
y?Y
and R0 (p, s) =
X
??arg sort(s)
X
R (p, ?)
, where ?? ? Sn , R (p, ?) =
p(y)r (y, ?) .
| arg sort(s)|
y?Y
Their optimal values are denoted by L (p) = infn L (p, s) and R0 (p) = R (p) = min R (p, ?).
s?R
??Sn
More precisely, [20, Theorem 2.8] shows that (1) holds for any distribution D and any sequence of
scoring functions if and only if the surrogate loss is r-calibrated according to the definition below.
Similarly to (1), the calibration is an implication of two limits, but it involves the inner risks L and
R0 instead of the risks L and R0 . For convenience in the rest of the work, we write the implication
4
between the two limits of L and R0 as an inclusion of the sets of near-optimal vectors of scores. For
any ? > 0 and ? > 0, the latter sets are respectively denoted by
M` (p, ?) = {s ? Rn |L (p, s) ? L (p) < ?} and Mr (p, ?) = {s ? Rn |R0 (p, s) ? R0 (p) < ?} ,
so that the definition of an r-calibrated loss is the following:
Definition 1. [20, Definition 2.7] The surrogate loss ` is r-calibrated if
?p ? P, ?? > 0, ?? > 0 : M` (p, ?) ? Mr (p, ?) .
3.2
Calibration through Optimal Rankings
Definition 1 is the starting point of our analysis, and our goal is to show that if the evaluation metric
is the AP, the ERR or the PD, then no convex surrogate loss can satisfy it. The goal of this subsection
is to give a stronger characterization of r-calibrated surrogate losses when Assumption (A) holds.
The starting point of this characterization is to rewrite Definition 1 in terms of rankings induced by
the sets of near-optimal scores, from which we can deduce that ` is r-calibrated if and only if1 :
?p ? P, ?? > 0, ?? > 0 : arg sort(M` (p, ?)) ? arg sort(Mr (p, ?)) .
In contrast to this characterization of calibration, our result (Theorem 2 below), which is specific to
metrics that satisfy (A), replaces the inclusion (which can be strict in general) of sets of ranking by
an equality when ? tends to 0. More specifically, we define the set of optimal rankings for the inner
ranking risk with the following set-valued function from P to Sn :
?p ? P, Ar (p) = arg min R (p, ?) ,
??Sn
so that when Assumption (A) holds, the set of optimal rankings is equal to a set of rankings induced
by near-optimal scores of the inner surrogate risk:
Theorem 2. If Assumption (A) holds, then ` is r-calibrated if and only if
?p ? P, ?? > 0 s.t. arg sort(M` (p, ?)) = Ar (p) .
The proof of Theorem 2 is deferred to Section 6 at the end of the paper. This theorem enables us to
relate the surrogate loss and the evaluation metric so that the convexity of ` induces some constraints
on r that are not satisfied by all evaluation metrics.
3.3
The implication of Convexity on Sets of Optimal Rankings
If `(y, .) is convex for all y ? P, then the inner risk L(p, .) is also convex for every distribution
p ? P. This implies that M` (p, ?) is a convex subset of Rn . Thus, if ` is r-calibrated, then Theorem
2 implies that Ar (p) = arg sort(M` (p, ?)) is a set of rankings induced by a convex set of Rn .
The following theorem presents a condition that the set Ar (p) must satisfy if it is generated by a
convex set of scores: if there exists at least one pair of items (i, j) which are inverted in two rankings
of Ar (p), then i and j are ?indifferent? in Ar (p):
Theorem 3. Assume that for all y ? Y, the function s 7? ` (y, s) is convex. If Assumption (A) holds
and ` is r-calibrated, then r satisfies: ?p ? P, ?i, j ? [n] , ??, ? 0 ? Ar (p),
? -1 (i) < ? -1 (j) and ? 0-1 (i) > ? 0-1 (j) ? ?s ? Rn : si = sj and arg sort(s) ? Ar (p) .
(2)
Proof of Theorem 3. Assume that the conditions of the theorem are satisfied. From now on, we fix
some p ? P and two i and j in [n]. Take ? and ? 0 in Ar (p) and assume that ? -1 (i) < ? -1 (j)
and ? 0-1 (i) > ? 0-1 (j). Since Assumption (A) holds, there is a ? > 0 such that Ar (p) =
arg sort M` (p, ?) by Theorem 2. Thus, there are two score vectors u and v in M` (p, ?) such
that ui ? uj (u induces the ranking ?) and vi ? vj (v induces the ranking ? 0 ).
Moreover, since ` is convex, the function L (p, .) is convex for every p ? P, and thus M` (p, ?) is
convex. Consequently, for all t ? [0, 1], the vector ?(t) = (1 ? t)u + tv belongs to M` (p, ?). We
define g : t 7? ?i (t) ? ?j (t) for t ? [0, 1]. Then, g is continuous, with g(0) = ui ? uj ? 0 and
g(1) = vi ? vj ? 0. By the intermediate value theorem, there is t0 ? [0, 1] such that g(t0 ) = 0. The
consequence is that the score vector s, defined by s = ?(t0 ), satisfies s ? M` (p, ?) and si = sj .
1
We remind to the reader the notation arg sort(M` (p, ?)) =
5
S
s?M`(p,?)
arg sort(s).
Table 2: Examples for Corollary 4. There are three elements to rank. i j k represents the
permutation that ranks item i first, j second and k last. For the ERR and the AP, we consider binary
relevance judgments. p110 denotes a Dirac distribution at the feedback vector y = [1, 1, 0]. p001
is defined similarly. For the Pairwise Disagreement, p1 2 3 is the Dirac distribution at the DAG
containing the edges 1 ? 2, 2 ? 3 and 1 ? 3, i.e. the DAG corresponding to 1 2 3. The
Dirac distribution at the DAG containing only the edge 3 ? 1 is denoted by p3 1 . In all cases, p?(?)
is a mixture between two Dirac distributions. The sets Ar (?
p(?)) are obtained by direct calculations.
The set Ar (?
p(?)) is the same for all ?s in the range given in the third column.
DISTRIBUTION p(?)
?
METRIC
RANGE OF ?
Ar (p(?))
?
ERR
??
AP
?=
1 1
3, 2
5
13
(1 ? ?)p110 + ?p001
{(1 3 2), (2 3 1)}
{(1 2 3), (3 1 2),
(2 1 3), (3 2 1)}
PD
??
2
3, 1
{(2 3 1), (3 1 2)}
(1 ? ?)p1 2 3 + ?p3 1
The contrapositive of Theorem 3 is our technical tool to prove the nonexistence of convex and
calibrated losses. Indeed, for a given evaluation metric r, if we are able to exhibit a distribution
p ? P such that (2) is not satisfied, this evaluation metric cannot have a surrogate loss both convex
and calibrated. In the next subsection, we apply this argument to the AP, the ERR and the PD.
Remark 1. It has been proved by several authors that there exist convex surrogate losses that are
DCG-calibrated [8, 2, 17]. Thus, the DCG satisfies (2). It can be seen by observing that the optimal
rankings for the DCG are exactly
P those generated by sorting the items according to the vector of
score s? (p) defined by s?i (p) = y?Y p(y)2yi , i.e. Ar (p) = arg sort(s? (p)).
4
Nonexistence Results
We now present the main result of the nonexistence of convex, calibrated surrogate losses:
Corollary 4. No convex surrogate loss is calibrated with respect to the AP, the ERR or the PD.
Proof. We consider the case where there are three elements to rank, and we use the examples and
the notations of Table 2. Since all three metrics satisfy (A), Theorem 3 implies that if r (taken as
either the AP, the ERR or the PD) has a calibrated, convex surrogate loss, then, for any distribution
p?(?), we have: if item i is preferred to j according to a ranking in Ar (?
p(?)), and j is preferred to i
according to another ranking in Ar (?
p(?)), then one of the two assertions below must hold:
(a) (i j k), (j i k) ? Ar (?
p(?)) , (b) (k i j), (k j i) ? Ar (?
p(?))
because there exists s ? R3 such that arg sort(s) ? Ar (?
p(?)) for which either si = sj ? sk or
si = sj ? sk . Now, let us consider the case of the ERR. Taking an arbitrary ? ? 31 , 12 , we
see on the last column of Table 2 that Ar (?
p(?)) contains two rankings: one of them ranks item 1
before item 2, and the other one ranks 2 before 1. If the ERR had a convex calibrated surrogate loss,
then either (a) or (b) should hold. However, we see that neither (a) nor (b) holds. Thus their is no
convex, ERR-calibrated surrogate loss. For the AP, a similar argument with items 1 and 3 leads to
the conclusion. For the PD, taking any two items leads to the result.
A first consequence of Corollary 4 is that for ranking problems evaluated in terms of AP, ERR or
PD, surrogate losses defined as convex upper bounds on an evaluation metric as discussed in [24],
as well as convex surrogate losses proposed in the structured output framework such as SVMmap
[25] are not calibrated with respect to the evaluation metric they are designed for. The convex
surrogate losses used by most participants of the recent Yahoo! Learning to Rank Challenge [6] are
also not calibrated with respect to the ERR, the official evaluation metric of the challenge. The fact
that the minimization of a non-calibrated surrogate risk leads to suboptimal prediction functions on
some data distributions suggests that convex losses are not a definitive solution to learning to rank.
Significant improvements in performances may then be obtained by switching to other approaches
than the optimization of a convex risk.
6
5
Conclusion
We proved that convex surrogate losses cannot be calibrated with three major ranking evaluation
metrics. The result cast light on the intrinsic limitations of all algorithms based on (empirical)
convex risk minimization for ranking, even though most existing algorithms for learning to rank
follow this approach. A possible direction for future work is to study whether the calibration of
convex losses can be obtained under low noise conditions. Such studies was carried out for the
PD [11], and calibrated, convex surrogate losses were found for special cases of practical interest.
Nonetheless, in order to obtain algorithms that do not rely on low noise assumptions, our results
suggest to explore whether alternatives to convex surrogate approaches can lead to improvements in
terms of performances. A first possibility is to turn to non-convex losses for ranking as in [10, 3],
and to study the calibration of such losses. Another alternative is to use another surrogate approach
than scoring, such as directly learning pairwise preferences [13], even though the reconstruction of
an optimal ranking, given the pairwise predictions, that is optimal for evaluation metrics such as the
AP, the ERR or the PD is still mostly an open issue.
6
Proof of Theorem 2
We remind the statement of Theorem 2: if r satisfies (A), then ` is r-calibrated if and only if for all
p ? P, there exists ? > 0 such that Ar (p) = arg sort(M` (p, ?) . We prove the result using the
following set-valued function which defines the set of optimal rankings for the inner surrogate risk:
e (p, ?) where L
e (p, ?) = inf L (p, s) | s ? Rn s.t. ? ? arg sort(s) .
A` (p) = arg min L
??Sn
Then, Theorem 2 is a direct implication of the two following claims that we prove in this section:
(a) the assertion ?p ? P, ?? > 0, arg sort(M` (p, ?) = A` (p) is true in general;
(b) if Assumption (A) holds, then ` is r-calibrated if and only if ?p ? P, A` (p) = Ar (p).
The proof of these two claims is based on three lemmas that we present before the final proof. The
first lemma, which does not need any assumption on the evaluation metric, both proves equality
(a) and provides a general characterization of calibration in terms of optimal rankings. The second
lemma concerns the surrogate loss; it states that a slight perturbation in p does not affect ?too much?
A` (p). The third lemma concerns evaluation metrics and gives a simple consequence of Assumption
(A). The final proof of Theorem 2 connects all these pieces together to prove (b).
Lemma 5. The following claims are true:
(i) ?p ? P, ?? > 0, A` (p) ? arg sort M` (p, ?) .
(ii) ?p ? P, ??0 > 0 : A` (p) = arg sort M` (p, ?0 ) .
(iii) ` is r-calibrated if and only if: ?p ? P, A` (p) ? Ar (p).
e there is an s ? Rn such
Proof. (i) Fix p ? P and ? > 0. Let ? ? A` (p). By the definition of L,
e (p, ?) < ?. Since L
e (p, ?) = min?0 ?S L
e (p, ? 0 ) = L (p), we
that ? ? arg sort(s) and L (p, s) ? L
n
have L (p, s) ? L (p) < ? . This proves s ? M` (p, ?) and thus ? ? arg sort M` (p, ?) .
e (p, ?) ? L (p) > 0, with the convention min ? = +?.
(ii) Fix p ? P and take ?0 = min?6?A`(p) L
n
The choice of ?0 guarantees that ?s
? R , L (p, s) ? L (p) < ?0 ? arg sort(s) ? A` (p), which is
equivalent to arg sort M` (p, ?0 ) ? A` (p). The reverse inclusion is given by the first point.
(iii) Since r can only take a finite set of values, we can prove that ` is r-calibrated if and only if:
?p ? P, ?? > 0 : ?s ? Rn , L (p, s) ? L (p) < ? ? R0 (p, s) = R0 (p). Moreover, we
have R0 (p, s) = R0 (p) ? arg sort(s) ? Ar (p) since R0 (p, s) is the mean of R (p, ?) for
? ? arg sort(s). Thus, ` is r-calibrated if and only if for every p ? P, there exists ? > 0 such
that arg sort M` (p, ?) ? Ar (p). This characterization and the first two points give the result.
7
We now present a more technical result on A` , which shows the set of optimal rankings cannot
dramatically change under a slight perturbation in the distribution over the feedback space. From
now on, for any p ? P and any ? > 0, we denote by B(p, ?) the open ball of P (with respect to
k.k1 ) of radius ? centered at p, i.e. B(p, ?) = {p0 ? P|kp ? p0 k1 < ?}.
Lemma 6. ?p ? P, ?? > 0 such that A` (B(p, ?)) = A` (p).
Proof. Note that A` (p) ? A` (B(p, ?)) since p ? B(p, ?). We now prove A` (B(p, ?)) ? A` (p);
e (., ?) is continuous for every ? because Y is finite [23, Theorem 2].
the main argument is that L
e (p, ? 0 ) ? L (p) . For each ? ? Sn , since
Indeed, let us fix p ? P and define ? = 21 min?0 6?A`(p) L
e (., ?) is continuous, there exists ?? > 0 such that ?p0 ? B(p, ?? ), |L
e (p0 , ?) ? L
e (p, ?)| < ?.
L
Let ? = min??Sn ?? , and let p0 be an arbitrary member of B(p, ?). By the definition of ?, we have:
e (p0 , ? 0 ) = L
e (p0 , ? 0 ) ? L
e (p, ? 0 ) + L
e (p, ? 0 ) ? L (p) + L (p) > ?? + 2? + L (p) .
?? 0 6? A` (p) , L
e (p0 , ? 0 ) > L (p) + ?. Additionally, the definition of ? gives ?? ?
Thus, ?? 0 6? A` (p) , L
e (p0 , ?) < L (p) + ?. Thus, we have min?0 6?A (p) L
e (p0 , ? 0 ) > min??A (p) L
e (p0 , ?).
A` (p) , L
`
`
0
e
e
This proves that a ranking that is not optimal for L (p, .) cannot be optimal for L (p , .). Thus
A` (p0 ) ? A` (p) from which we conclude A` (B(p, ?)) ? A` (p).
Now that we have studied the properties of A` , we analyze in more depth the evaluation metrics. We
prove the following consequence of Assumption (A): for each possible ranking there is a distribution
over the feedback space for which this ranking is the unique optimal ranking.
Lemma 7. If Assumption (A) holds, then ?? ? Sn , ?p? ? P such that Ar (p? ) = {?}.
Proof. Assume (A) holds, and, for each item k, let us denote by y k the feedback corresponding
to item k in Assumption (A). Now, let us take some ? ? Sn and define p? as p? (y k ) = ??-1 (k)
Pn
0
with ?1 > ... > ?n > 0 and k=1
k = 1. Then, for any ? ? Sn , we have the equality
?P
P
n
n
?
0
k
0
R (p , ? ) = k=1 ??-1 (k) r y , ? = k=1 ??-1 (k) ??0-1 (k) . Since the ?s are non-negative, and
since there are ties neither the ?s nor in the ?s, the rearrangement inequality implies that the minimum value of R (p? , ? 0 ) is obtained for the single permutation ? 0 for which the ??0-1 (k) are in reverse
order relatively to the ??-1 (k) (i.e. smaller values ??0-1 (k) should be associated to greater values of
??-1 (k) ). Since the ?k s are decreasing with k and the ?k s are increasing, the minimum value of
Pn
? 0 7? R (p? , ? 0 ) = k=1 ??-1 (k) ??0-1 (k) is obtained if and only if ? -1 = ? 0-1 (i.e. ? 0 = ?).
Proof of Theorem 2. We remind to the reader that bythe second point of Lemma 5, for any p ? P,
there is ? > 0 such that A` (p) = arg sort(M` (p, ?) . What remains to show is that if Assumption
(A) holds, then ` is r-calibrated if and only if ?p ? P, A` (p) = Ar (p).
(?if? direction) If ?p ? P, A` (p) = Ar (p) then ` is r-calibrated by Lemma 5.
(?only if? direction) Assume that (A) holds and that ` is r-calibrated. Let p ? P. By Point (iii) of
Lemma 5, we know that A` (p) ? Ar (p). We now prove the reverse inclusion A` (p0 ) ? Ar (p0 ).
By Lemma 6, there exists some ? > 0 such that A` (B(p, ?)) = A` (p). Let ? ? Ar (p). The idea is
to use Lemma 7 to find some p0 ? B(p, ?) such that A` (p0 ) = {?} which would prove ? ? A` (p)
and thus the result. The rest of the proof consists in building p0 .
Using Lemma 7, let p? ? P such that Ar (p? ) = {?}. Now, let p0 = (1? ?4 )p+ ?4 p? . Then, we have
kp ? p0 k1 = ?4 kp ? p? k1 ? ?/2 and thus p0 ? B(p, ?). Moreover, Ar (p0 ) = {?} since ? is optimal
for both p and p? , and any other permutation is suboptimal for p? . We also have A` (p0 ) = {?}
because A` has non-empty values and calibration implies that A` (p0 ) ? Ar (p0 ) by Lemma 5.
Acknowledgements
This work was partially funded by the French DGA. The authors thank M. R. Amini, D. Buffoni, S.
Cl?emenc?on, L. Denoyer and G. Wisniewski for their comments and suggestions.
8
References
[1] P. L. Bartlett, M. I. Jordan, and J. D. McAuliffe. Convexity, classification, and risk bounds. J. of the
American Stat. Assoc., pages 1?36, 2006.
[2] D. Buffoni, C. Calauz`enes, P. Gallinari, and N. Usunier. Learning scoring functions with order-preserving
losses and standardized supervision. In Proc. of the Intl. Conf. on Mach. Learn., pages 825?832, 2011.
[3] C. J. Burges, R. Ragno, and Q. V. Le. Learning to rank with nonsmooth cost functions. In Proc. of Adv.
in Neural Info. Proc. Syst., pages 193?200, 2007.
[4] Y. Cao, J. Xu, T.-Y. Liu, H. Li, Y. Huang, and H.-W. Hon. Adapting ranking svm to document retrieval.
In Proc. of the ACM SIGIR Conf. on Res. and Dev. in Info. Retr., pages 186?193, 2006.
[5] A. Chang, C. Rudin, M. Cavaretta, R. Thomas, and G. Chou. How to reverse-engineer quality rankings.
Mach. Learn., 88(3):369?398, Sept. 2012.
[6] O. Chapelle and Y. Chang. Yahoo! learning to rank challenge overview. J. of Mach. Learn. Res., 14:1?24,
2011.
[7] S. Cl?emenc?on, G. Lugosi, and N. Vayatis. Ranking and scoring using empirical risk minimization. In
Proc. of the 18th Conf. on Learning Theory, COLT?05, pages 1?15, 2005.
[8] D. Cossock and T. Zhang. Statistical analysis of bayes optimal subset ranking. IEEE Trans. Info. Theory,
54:5140?5154, 2008.
[9] O. Dekel, C. D. Manning, and Y. Singer. Log-linear models for label ranking. In Proc. of Advances in
Neural Information Processing Systems (NIPS), 2003.
[10] C. B. Do, Q. Le, C. H. Teo, O. Chapelle, and A. Smola. Tighter bounds for structured estimation. In Proc.
of Adv. in Neural Inf. Processing Syst., pages 281?288, 2008.
[11] J. Duchi, L. W. Mackey, and M. I. Jordan. On the consistency of ranking algorithms. In Proc. of the Int.
Conf. on Mach. Learn., pages 327?334, 2010.
[12] Y. Freund, R. Iyer, R. E. Schapire, and Y. Singer. An efficient boosting algorithm for combining preferences. J. of Mach. Learn. Res., 4:933?969, 2003.
[13] E. Hullermeier, J. Furnkranz, W. Cheng, and K. Brinker. Label ranking by learning pairwise preferences.
Artificial Intelligence, 172(16-17):1897?1916, Nov. 2008.
[14] T. Joachims. Optimizing search engines using clickthrough data. In Proc. of Know. Disc. and Dat. Mining
(SIGKDD), pages 133?142, 2002.
[15] W. Kotlowski, K. Dembczynski, and E. Huellermeier. Bipartite ranking through minimization of univariate loss. In Proc. of the Intl. Conf. on Mach. Learn., pages 1113?1120, 2011.
[16] T.-Y. Liu. Learning to rank for information retrieval. Foundations and Trends in Information Retrieval,
3:225?331, March 2009.
[17] P. D. Ravikumar, A. Tewari, and E. Yang. On ndcg consistency of listwise ranking methods. J. of Mach.
Learn. Res. - Proc. Track, 15:618?626, 2011.
[18] M. D. Reid and R. C. Williamson. Surrogate Regret Bounds for Proper Losses. In Proc. of the Intl. Conf.
on Mach. Learn., pages 897?904, 2009.
[19] C. Scott. Surrogate losses and regret bounds for cost-sensitive classification with example-dependent
costs. Proc. of the Intl. Conf. on Mach. Learn., pages 153?160, 2011.
[20] I. Steinwart. How to compare different loss functions and their risks. Constructive Approximation,
26(2):225?287, 2007.
[21] M. Taylor, J. Guiver, S. Robertson, and T. Minka. Softrank: optimizing non-smooth rank metrics. In
Proceedings of the international conference on Web search and web data mining, WSDM ?08, pages
77?86, 2008.
[22] E. Voorhees, D. Harman, N. I. of Standards, and T. (U.S.). TREC: experiment and evaluation in information retrieval. Digital libraries and electronic publishing. MIT Press, 2005.
[23] R. A. Wijsman. Continuity of the bayes risk. The Annals of Math. Stat., 41(3):pp. 1083?1085, 1970.
[24] J. Xu, T.-Y. Liu, M. Lu, H. Li, and W.-Y. Ma. Directly optimizing evaluation measures in learning to rank.
In Proceedings of the 31st annual international ACM SIGIR conference on Research and development in
information retrieval, SIGIR ?08, pages 107?114, 2008.
[25] Y. Yue, T. Finley, F. Radlinski, and T. Joachims. A support vector method for optimizing average precision. In Proc. of the ACM SIGIR Intl. Conf. on Res. and Dev. in Info. Retr., pages 271?278, 2007.
[26] T. Zhang. Statistical analysis of some multi-category large margin classification methods. J. of Mach.
Learn. Res., 5:1225?1251, 2004.
[27] T. Zhang. Statistical behavior and consistency of classification methods based on convex risk minimization. The Annals of Stat., 32(1):pp. 56?85, 2004.
9
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4,029 | 4,647 | GenDeR: A Generic Diversified Ranking Algorithm
Hanghang Tong
IBM T.J. Watson Research
Yorktown Heights, NY 10598
[email protected]
Jingrui He
IBM T.J. Watson Research
Yorktown Heights, NY 10598
[email protected]
Boleslaw K. Szymanski
Rensselaer Polytechnic Institute
Troy, NY 12180
[email protected]
Qiaozhu Mei
University of Michigan
Ann Arbor, MI 48109
[email protected]
Abstract
Diversified ranking is a fundamental task in machine learning. It is broadly applicable in many real world problems, e.g., information retrieval, team assembling,
product search, etc. In this paper, we consider a generic setting where we aim
to diversify the top-k ranking list based on an arbitrary relevance function and
an arbitrary similarity function among all the examples. We formulate it as an
optimization problem and show that in general it is NP-hard. Then, we show that
for a large volume of the parameter space, the proposed objective function enjoys
the diminishing returns property, which enables us to design a scalable, greedy
algorithm to find the (1 ? 1/e) near-optimal solution. Experimental results on
real data sets demonstrate the effectiveness of the proposed algorithm.
1 Introduction
Many real applications can be reduced to a ranking problem. While traditional ranking tasks mainly
focus on relevance, it has been widely recognized that diversity is another highly desirable property.
It is not only a key factor to address the uncertainty and ambiguity in an information need, but also
an effective way to cover the different aspects of the information need [14]. Take team assembling
as an example. Given a task which typically requires a set of skills, we want to form a team of
experts to perform that task. On one hand, each team member should have some relevant skills.
On the other hand, the whole team should somehow be diversified, so that we can cover all the
required skills for the task and different team members can benefit from each other?s diversified,
complementary knowledge and social capital. More recent research discovers that diversity plays
a positive role in improving employees? performance within big organizations as well as their job
retention rate in face of lay-off [21]; in improving the human-centric sensing results [15, 17]; in the
decision of joining a new social media site (e.g., Facebook) [18], etc.
To date, many diversified ranking algorithms have been proposed. Early works mainly focus on
text data [5, 23] where the goal is to improve the coverage of (sub-)topics in the retrieval result. In
recently years, more attention has been paid to result diversification in web search [2, 20]. For example, if a query bears multiple meanings (such as the key word ?jaguar?, which could refer to either
cars or cats), we would like to have each meaning (e.g., ?cars? and ?cats? in the example of ?jaguar?)
covered by a subset of the top ranked web pages. Another recent trend is to diversify PageRank-type
of algorithms for graph data [24, 11, 16]. It is worth pointing out that almost all the existing diversified ranking algorithms hinge on the specific choice of the relevance function and/or the similarity
function. For example, in [2] and [20], both the relevance function and the similarity function implicitly depend on the categories/subtopics associated with the query and the documents; in [16], the
1
relevance function is obtained via personalized PageRank [8], and the similarity is measured based
on the so-called ?Google matrix?; etc.
In this paper, we shift the problem to a more generic setting and ask: given an arbitrary relevance
function wrt an implicit or explicit query, and an arbitrary similarity function among all the available
examples, how can we diversify the resulting top-k ranking list? We address this problem from
the optimization viewpoint. First, we propose an objective function that admits any non-negative
relevance function and any non-negative, symmetric similarity function. It naturally captures both
the relevance with regard to the query and the diversity of the ranking list, with a regularization
parameter that balances between them. Then, we show that while such an optimization problem
is NP-hard in general, for a large volume of the parameter space, the objective function exhibits
the diminishing returns property, including submodurality, monotonicity, etc. Finally, we propose a
scalable, greedy algorithm to find provably near-optimal solution.
The rest of the paper is organized as follows. We present our optimization formulation for diversified
ranking in Section 2, followed by the analysis of its hardness and properties. Section 3 presents our
greedy algorithm for solving the optimization problem. The performance of the proposed algorithm
is evaluated in Section 4. In Section 5, we briefly review the related work. Finally, we conclude the
paper in Section 6.
2 The Optimization Formulation
In this section, we present the optimization formulation for diversified ranking. We start by introducing the notation, and then present the objective function, followed by the analysis regarding its
hardness and properties.
2.1 Notation
In this paper: we use normal lower-case letters to denote scalers or functions, bold-face lower-case
letters to denote vectors, bold-face upper-case letters to denote matrices, and calligraphic upper-case
letters to denote sets. To be specific, for a set X of n examples {x1 , x2 , . . . , xn }, let S denote the
n ? n similarity matrix, which is both symmetric and non-negative. In other words, S i,j = S j,i
and S i,j ? 0, where S i,j is the element of S in the ith row and the j th column (i, j = 1, . . . , n).
For any ranking function r(?), which returns the non-negative relevance score for each example
in X with respect to an implicit or explicit query, our goal is to find a subset T of k examples,
which are relevant to the query and diversified among themselves. Here the positive integer k is the
budget of the ranking list size, and the ranking function r(?) generates an n ? 1 vector r, whose ith
element r i = r(xi ). When we describe the objective function as well as the proposed optimization
algorithm, it is convenient to introduce the following n ? 1 reference vector q = S ? r. Intuitively,
its ith element q i measures the importance of xi . To be specific, if xi is similar to many examples
(high S i,j (j = 1, 2, ...., )) that are relevant to the query (high r j (j = 1, 2, ...), it is more important
than the examples whose neighbors are not relevant. For example, if xi is close to the center of a
big cluster relevant to the query, the value of q i is large.
2.2 Objective Function
With the above notation, our goal is to find a subset T of k examples which are both relevant to
the query and diversified among themselves. To this end, we propose the following optimization
problem.
X
X
arg max g(T ) = w
q i ri ?
r i S i,j r j
(1)
|T |=k
i?T
i,j?T
where w is a positive regularization parameter that defines the trade-off between the two terms, and
T consists of the indices of the k examples that will be returned in the ranking list.
Intuitively, in the goodness function g(T ), the first term measures the weighted overall relevance
of T with respect to the query, and q i is the weight for xi . It favors relevant examples from big
clusters. In other words, if two examples are equally relevant to the query, one from a big cluster and
the other isolated, by using the weighted relevance, we prefer the former. The second term measures
2
the similarity among the examples within T . That is, it penalizes the selection of multiple relevant
examples that are very similar to each other. By including this term in the objective function, we seek
a set of examples which are relevant to the query, but also dissimilar to each other. For example, in
the human-centric sensing [15, 17], due to the homophily in social networks, reports of two friends
are likely correlated so that they are a lesser corroboration of events than reports of two socially
unrelated witnesses.
2.3 The Hardness of Equation (1)
In the optimization problem in Equation (1), we want to find a subset T of k examples that collectively maximize the goodness function g(T ). Unfortunately, by the following theorem, it is NP-hard
to find the optimal solution.
Theorem 2.1. The optimization problem in Equation (1) is NP-hard.
Proof. We will prove this from the reduction of the Densest k-Subgraph (DkS) problem, which is
known to be NP-hard [7].
To be specific, given an undirected graph G(V, E) with the connectivity matrix W , where V is the
set of vertices, and E is the set of edges. W is a |V| ? |V| symmetric matrix with elements being 0 or
1. Let |E| be the total number of the edges in the graph. The DkS problem is defined in Equation (2).
X
Q = arg max
W i,j
(2)
|Q|=k
i,j?Q
? as: W
? i,j = 1 ? W i,j . It is easy to see that
Define another |V| ? |V| matrix W
P
2
?
k ? i,j?Q W i,j . Therefore, Equation (2) is equivalent to
X
? i,j
Q = arg min
W
|Q|=k
Furthermore, notice that
is equivalent to
P|V|
i,j=1
|Q|=k
arg
i,j?Q
W i,j =
(3)
i,j?Q
? i,j = |V|2 ? |E| = constant. Let T = V \ Q, then Equation (3)
W
X
arg max
=
P
i?Q,j?T
max
|T |=|V|?k
2
X
? i,j +
W
? i,j +
W
i?T ,j?Q
X
? i,j +
W
i?Q,j?T
X
X
? i,j
W
i?T ,j?T
? i,j
W
(4)
i,j?T
Next, we will show that Equation (4) can be viewed as an instance of the optimization problem in
? , the ranking function r
Equation (1) with the following setting: let the similarity function S be W
be 1|V|?1 , the budget be |V| ? k, and the regularization parameter w be 2. Under such settings, the
objective function in Equation (1) becomes
X
X
? i,j r j
g(T ) = 2
q i ri ?
ri W
i?T
=
2
|V|
XX
i,j?T
i?T j=1
=
2
XX
2
XX
i?Q j?T
? i,j r j
ri W
(dfn. of q)
? i,j r j
ri W
?
(symmetry of W)
i,j?T
X
? ij r j +
ri W
i?Q j?T
=
X
? ij r j ?
ri W
? ij +
W
X
i,j?T
? i,j
W
(dfn. of r)
(5)
i,j?T
which is equivalent to the objective function in Equation (4). This completes the proof.
3
2.4 Diminish Returns Property of g(T )
Given that Equation (1) is NP-hard in general, we seek for a provably near-optimal solution instead
in the next section. Here, let us first answer the following question: under what condition (e.g., in
which range of the regularization parameter w), is it possible to find such a near-optimal solution
for Equation (1)?
To this end, we present the so-called diminishing returns property of the goodness function g(T )
defined in Equation (1), which is summarized in the following theorem. By Theorem 2.2, if we
add more examples into an existing top-k ranking list, the goodness of the overall ranking list is
non-decreasing (P2). However, the marginal benefit of adding additional examples into the ranking
list decreases wrt the size of the existing ranking list (P1).
Theorem 2.2. Diminish Returns Property of g(T ). The goodness function g(T ) defined in Equation (1) has the following properties:
(P1) submodularity. For any w > 0, the objective function g(T ) is submodular wrt T ;
(P2) monotonicity. For any w ? 2, The objective function g(T ) is monotonically nondecreasing wrt T .
Proof. We first prove (P1). For any T1 ? T2 and any given example x ?
/ T2 , we have
g(T1 ? x) ? g(T1 ) =
(w
X
q i ri ?
i?T1 ?x
=
wq x r x ? (
X
X
i,j?T1 ?x
ri S i,x r x +
i?T1
=
X
ri S i,j r j ) ? (w
X
X
qi ri ?
i?T1
r i S i,j rj )
i,j?T1
rx S x,j rj + rx S x,x rx )
j?T1
wq x r x ? S x,x r 2x ? 2rx
X
S x,j rj
(6)
j?T1
Similarly, we have g(T2 ? x) ? g(T2 ) = wq x r x ? S x,x r2x ? 2rx
Therefore, we have
(g(T1 ? x) ? g(T1 )) ? (g(T2 ? x) ? g(T2 )) =
2rx
X
P
j?T2
S x,j rj .
2rx
X
S x,j rj
j?T1
j?T2
=
X
S x,j rj ? 2rx
S x,j rj ? 0
(7)
j?T2 \T1
which completes the proof of (P1).
Next, we prove (P2). Given any T1 ? T2 = ?, where ? is the empty set, with w ? 2, we have
X
X
X
X
g(T2 ? T1 ) ? g(T2 ) = w
qiri ? (
ri S i,j r j +
r i S i,j rj +
r i S i,j rj )
i?T1
=
w
X
i?T1 ,j?T2
ri
i?T1
?
2
X
ri
2
X
=
2
X
i?T1
X
S i,j rj ?
X
X
ri S i,j rj )
i,j?T1
ri S i,j rj +
i?T1 ,j?T2
j=1
ri
X
i,j?T1
ri S i,j rj +
i?T1 ,j?T2
S i,j r j ? 2(
ri (
i?T1
i?T2 ,j?T1
X
S i,j r j ? (2
j=1
n
X
j=1
n
X
i?T1
=
n
X
X
ri S i,j rj )
i,j?T1
S i,j r j )
j?T1 ?T2
S i,j rj ? 0
(8)
j ?T
/ 1 ?T2
which completes the proof of (P2).
4
3 The Optimization Algorithm
In this section, we present our algorithm GenDeR for solving Equation (1), and analyze its performance with respect to its near-optimality and complexity.
3.1 Algorithm Description
Based on the diminishing returns property of the goodness function g(T ), we propose the following
greedy algorithm to find a diversified top-k ranking list. In Alg. 1, after we calculate the reference
vector q (Step 1) and initialize the ranking list T (Step 2), we try to expand the ranking list T
one-by-one (Step 4-8). At each iteration, we add one more example with the highest score si into
the current ranking list T (Step 5). Each time we expand the current ranking list, we update the
score vector s based on the newly added example i (Step 7). Notice that in Alg. 1, ??? means the
element-wise multiplication, and diag(S) returns an n ? 1 vector with the corresponding elements
being the diagonal elements in the similarity matrix S.
Algorithm 1 GenDeR
Input: The similarity matrix S n?n , the relevance vector rn?1 , the weight w ? 2, and the budget
k;
Output: A subset T of k nodes.
1: Compute the reference vector q: q = Sr;
2: Initialize T as an empty set;
3: Initialize the score vector s = w ? (q ? r) ? diag(S) ? r ? r;
4: for iter = 1 : k do
5:
Find i = argmaxj (sj |j = 1, ..., n; j ?
/ T );
6:
Add i to T ;
7:
Update the score vector s ? s ? 2ri S :,i ? r
8: end for
9: Return the subset T as the ranking list (earlier selected examples ranked higher).
3.2 Algorithm Analysis
The accuracy of the proposed GenDeR is summarized in Lemma 3.1, which says that for a large
volume of the parameter space (i.e., w ? 2), GenDeR leads to a (1 ? 1/e) near-optimal solution.
Lemma 3.1. Near-Optimality of GenDeR. Let T be the subset found by GenDeR, |T | = k, and
T ? = argmax|T |=k g(T ). We have that g(T ) ? (1 ? 1/e)g(T ? ), where e is the base of the natural
logarithm.
Proof. The key of the proof is to verify that for any example xj ?
/ T , sj = g(T ? xj ) ? g(T ),
where s is the score vector we calculate in Step 3 or update in Step 7, and T is the initial empty
ranking list or the current ranking list in Step 6. The remaining part of the proof directly follows the
diminishing returns property of the goodness function in Theorem 2.2, together with the fact that
g(?) = 0 [12]. We omit the detailed proof for brevity.
The complexity of the proposed GenDeR is summarized in Lemma 3.2. Notice that the quadratic
term in the time complexity comes from the matrix-vector multiplication in Step 1 (i.e., q = Sr);
and the quadratic term in the space complexity is the cost to store the similarity matrix S. If the
similarity matrix S is sparse, say we have m non-zero elements in S, we can reduce the time
complexity to O(m + nk); and reduce the space complexity to O(m + n + k).
Lemma 3.2. Complexity of GenDeR. The time complexity of GenDeR is O(n2 + nk); the space
complexity of GenDeR is O(n2 + k).
Proof. Omitted for Brevity.
5
4 Experimental Results
We compare the proposed GenDeR with several most recent diversified ranking algorithms, including DivRank based on reinforced random walks [11] (referred to as ?DR?), GCD via resistive graph
centers [6] (referred to as ?GCD?) and manifold ranking with stop points [25] (referred to as ?MF?).
As all these methods aim to improve the diversity of PageRank-type of algorithms, we also present
the results by PageRank [13] itself as the baseline. We use two real data sets, including an IMDB
actor professional network and an academic citation data set. In [11, 6], the authors provide detailed
experimental comparisons with some earlier methods (e.g., [24, 23, 5], etc) on the same data sets.
We omit the results by these methods for clarity.
4.1 Results on Actor Professional Network
The actor professional network is constructed from the Internet Movie Database (IMDB)1 , where
the nodes are the actors/actresses and the edges are the numbers of the co-stared movies between two
actors/actresses. For the inputs of GenDeR, we use the adjacency matrix of the co-stared network as
the similarity function S; and the ranking results by ?DR? as the relevance vector r. Given a top-k
ranking list, we use the density of the induced subgraph of S by the k nodes as the reverse measure
of the diversity (lower density means higher diversity). We also measure the diversity of the ranking
list by the so-called ?country coverage? as well as ?movie coverage? (higher coverage means higher
diversity), which are defined in [24]. Notice that for a good top-k diversified ranking list, it often
requires the balance between the diversity and the relevance in order to fulfill the user?s information
need. Therefore, we also present the relevance score (measured by PageRank) captured by the entire
top-k ranking list. In this application, such a relevance score measures the overall prestige of the
actors/actresses in the ranking list. Overall, we have 3,452 actors/actresses, 23,460 edges, 1,027
movies and 47 countries.
The results are presented in Fig. 1. First, let us compare GenDeR with the baseline method ?PageRank?. From Fig. 1(d), we can see that our GenDeR is as good as ?PageRank? in terms of capturing
the relevance of the entire top-k ranking list (notice that the two curves almost overlap with each
other). On the other hand, GenDeR outperforms ?PageRank? in terms of the diversity by all the
three measures (Fig. 1(a-c)). Since GenDeR uses the ranking results by ?DR? as its input, ?DR?
can be viewed as another baseline method. The two methods perform similarly in terms of density
(Fig. 1(c)). Regarding all the remaining measures, our GenDeR is always better than ?DR?. For
example, when k ? 300, GenDeR returns both higher ?country-coverage? (Fig. 1(a)) and higher
?movie-coverage? (Fig. 1(b)). In the entire range of the budget k (Fig. 1(d)), our GenDeR captures
higher relevance scores than ?DR?, indicating the actors/actresses in our ranking list might be more
prestigious than those by ?DR?. Based on these results, we conclude that our GenDeR indeed improves ?DR? in terms of both diversity and relevance. The most competitive method is ?MF?. We
can see that GenDeR and ?MF? perform similarly in terms of both density (Fig. 1(c)) and ?movie
coverage? (Fig. 1(b)). In terms of ?country coverage? (Fig. 1(a)), ?MF? performs slightly better than
our GenDeR when 300 ? k ? 400; and for the other values of k, the two methods mix with each
other. However, in terms of relevance (Fig. 1(d)), our GenDeR is much better than ?MF?. Therefore,
we conclude that ?MF? performs comparably with or slightly better than our GenDeR in terms of
diversity, at the cost of sacrificing the relevance of the entire ranking list. As for ?GCD?, although
it leads to the lowest density, it performs poorly in terms of balancing between the diversity and the
relevance (Fig. 1(d)), as well as the coverage of countries/movies (Fig. 1(a-b)).
4.2 Results on Academic Citation Networks
This data set is from ACL Anthology Network2. It consists of a paper citation network and a researcher citation network. Here, the nodes are papers or researchers; and the edges indicate the
citation relationship. Overall, we have 11,609 papers and 54,208 edges in the paper citation network; 9,641 researchers and 229,719 edges in the researcher citation network. For the inputs of
GenDeR, we use the symmetrized adjacency matrix as the similarity function S; and the ranking
results by ?DR? as the relevance vector r. We use the same measure as in [11] (referred to as ?coverage?), which is the total number of unique papers/researchers that cite the top-k papers/researchers in
the ranking list. As pointed out in [11], the ?coverage? might provide a better measure of the overall
quality of the top-k ranking list than those traditional measures (e.g., h-index) as they ignore the diversity of the ranking list. The results are presented in Fig. 2. We can see that the proposed GenDeR
1
2
http://www.imdb.com/
http://www.aclweb.org/anthology-new/
6
50
1000
PageRank
900
DR
45
MF
800
GCD
GenDeR
700
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country coverage
40
35
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25
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(a) Country Coverage (Higher is better)
(b) Movie Coverage (Higher is better)
0.09
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k
(c) Density (Lower is better)
(d) Relevance (Higher is better)
Figure 1: The evaluations on actor professional network. (a-c) are different diversity measures and
(d) measures the relevance of the entire ranking list.
performs better than all the alternative choices. For example, with k = 50, GenDeR improves the
?coverage? of the next best method by 416 and 157 on the two citation networks, respectively.
6000
8000
PageRank
7000
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5000
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GenDeR
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k
(a) Paper Citation Network
(b) Researcher Citation Network
Figure 2: The evaluations on academic citation networks. Higher is better.
5 Related Work
Carbonell et al [5] are among the first to study diversified ranking in the context of text retrieval and
summarization. To this end, they propose to use the Maximal Marginal Relevance (MMR) criterion
to reduce redundancy while maintaining query relevance, which is a linear combination of relevance
and novelty. In [23], Zhai et al address this problem from a different perspective by explicitly modeling the subtopics associated with a query, and proposing a framework to evaluate subtopic retrieval.
Recently, researchers leverage external information sources to help with diversified ranking. For example, in [2], Agrawal et al maximize the probability that the average user finds at least one useful
7
result within the top ranked results with the help of a taxonomy available through Open Directory
Project (ODP); in [4], Capannini et al mine the query log to find specializations of a given query,
and use the search results of the specializations to help evaluate the set of top ranked documents;
in [20], Welch et al model the expected number of hits based on the number of relevant documents
a user will visit, user intent in terms of the probability distribution over subtopics, and document
categorization, which are obtained from the query logs, WordNet or Wikipedia.
With the prevalence of graph data, such as social networks, author/paper citation networks, actor
professional networks, etc, researchers have started to study the problem of diversified ranking in
the presence of relationships among the examples. For instance, in [24], Zhu et al propose the
GRASSHOPPER algorithm by constructing random walks on the input graph, and iteratively turning
the ranked nodes into absorbing states. In [11], Mei et al propose the DivRank algorithm based on
a reinforced random walk defined on the input graph, which automatically balances the prestige
and the diversity among the top ranked nodes due to the fact that adjacent nodes compete for their
ranking scores. In [16], Tong et al propose a scalable algorithm to find the near-optimal solution to
diversify the top-k ranking list for PageRank. Due to the asymmetry in their formulation, it remains
unclear if the optimization problem in [16] is NP-hard. On a higher level, the method in [16]
can be roughly viewed as an instantiation of our proposed formulation with the specific choices
in the optimization problem (e.g, the relevance function, the similarity function, the regularization
parameter, etc). In [25], Zhu et al leverage the stopping points in the manifold ranking algorithms
to diversify the results. All these works aim to diversify the results of one specific type of ranking
function (i.e., PageRank and its variants).
Learning to rank [10, 1, 3] and metric learning [19, 22, 9] have been two very active areas in the
recent years. Most of these methods require some additional information (e.g., label, partial ordering, etc) for training. They are often tailored for other purposes (e.g., improving the F-score in the
ranking task, improving the classification accuracy in metric learning, etc) without the consideration
of diversity. Nonetheless, thanks to the generality of our formulation, the learned ranking functions
and metric functions from most of these works can be naturally admitted into our optimization objective function. In other words, our formulation brings the possibility to take advantage of these
existing research results in the diversified ranking setting.
Remarks. While generality is one of the major contributions of this paper, we do not disregard the
value of the domain-specific knowledge. The generality of our method is orthogonal to domainspecific knowledge. For example, such knowledge can be reflected in the (learnt) ranking function
and/or the (learnt) similarity function, which can in turn serve as the input of our method.
6 Conclusion
In this paper, we study the problem of diversified ranking. The key feature of our formulation lies
in its generality: it admits any non-negative relevance function and any non-negative, symmetric
similarity function as input, and outputs a top-k ranking list that enjoys both relevance and diversity.
Furthermore, we identify the regularization parameter space where our problem can be solved nearoptimally; and we analyze the hardness of the problem, the optimality as well as the complexity of
the proposed algorithm. Finally, we conduct experiments on several real data sets to demonstrate
the effectiveness of this algorithm. Future work includes extending our formulation to the on-line,
dynamic setting.
7 Acknowledgement
Research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-09-2-0053. This work was in part supported by the National
Science Foundation under grant numbers IIS-1054199 and CCF-1048168; and by DAPRA under
SMISC Program Agreement No. W911NF-12-C-0028. The views and conclusions contained in this
document are those of the authors and should not be interpreted as representing the official policies,
either expressed or implied, of the Army Research Laboratory, the National Science Foundation, or
the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for
Government purposes notwithstanding any copyright notation here on.
References
[1] A. Agarwal and S. Chakrabarti. Learning random walks to rank nodes in graphs. In ICML, pages 9?16,
2007.
8
[2] R. Agrawal, S. Gollapudi, A. Halverson, and S. Ieong. Diversifying search results. In WSDM, pages
5?14, 2009.
[3] C. J. C. Burges, K. M. Svore, P. N. Bennett, A. Pastusiak, and Q. Wu. Learning to rank using an ensemble
of lambda-gradient models. Journal of Machine Learning Research - Proceedings Track, 14:25?35, 2011.
[4] G. Capannini, F. M. Nardini, R. Perego, and F. Silvestri. Efficient diversification of search results using
query logs. In WWW (Companion Volume), pages 17?18, 2011.
[5] J. G. Carbonell and J. Goldstein. The use of mmr, diversity-based reranking for reordering documents
and producing summaries. In SIGIR, pages 335?336, 1998.
[6] A. Dubey, S. Chakrabarti, and C. Bhattacharyya. Diversity in ranking via resistive graph centers. In KDD,
pages 78?86, 2011.
[7] U. Feige, G. Kortsarz, and D. Peleg. The dense k-subgraph problem. Algorithmica, 29, 1999.
[8] T. H. Haveliwala. Topic-sensitive pagerank: A context-sensitive ranking algorithm for web search. IEEE
Trans. Knowl. Data Eng., 15(4):784?796, 2003.
[9] P. Jain, B. Kulis, and I. S. Dhillon. Inductive regularized learning of kernel functions. In NIPS, pages
946?954, 2010.
[10] T.-Y. Liu. Learning to rank for information retrieval. In SIGIR, page 904, 2010.
[11] Q. Mei, J. Guo, and D. R. Radev. Divrank: the interplay of prestige and diversity in information networks.
In KDD, pages 1009?1018, 2010.
[12] G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher. An analysis of approximations for maximizing submodular set functions?i. MATHEMATICAL PROGRAMMING, (1):265?294, 1973.
[13] L. Page, S. Brin, R. Motwani, and T. Winograd. The PageRank citation ranking: Bringing order to the
web. Technical report, Stanford Digital Library Technologies Project, 1998. Paper SIDL-WP-1999-0120
(version of 11/11/1999).
[14] F. Radlinski, P. N. Bennett, B. Carterette, and T. Joachims. Redundancy, diversity and interdependent
document relevance. SIGIR Forum, 43(2):46?52, 2009.
[15] M. Srivastava, T. Abdelzaher, and B. Szymanski. Human-centric sensing. Phil. Trans. R. Soc. 370 ser.
A(1958), pages 176?197, 2012.
[16] H. Tong, J. He, Z. Wen, R. Konuru, and C.-Y. Lin. Diversified ranking on large graphs: an optimization
viewpoint. In KDD, pages 1028?1036, 2011.
[17] M. Y. S. Uddin, M. T. A. Amin, H. Le, T. Abdelzaher, B. Szymanski, and T. Nguyen. On diversifying
source selection in social sensing. In INSS, 2012.
[18] J. Ugander, L. Backstrom, C. Marlow, and J. Kleinberg. Structural diversity in social contagion. PNAS,
109(16):596?5966, 2012.
[19] J. Wang, H. Do, A. Woznica, and A. Kalousis. Metric learning with multiple kernels. In NIPS, pages
1170?1178, 2011.
[20] M. J. Welch, J. Cho, and C. Olston. Search result diversity for informational queries. In WWW, pages
237?246, 2011.
[21] L. Wu. Social network effects on performance and layoffs: Evidence from the adoption of a social
networking tool. Job Market Paper, 2011.
[22] E. P. Xing, A. Y. Ng, M. I. Jordan, and S. J. Russell. Distance metric learning with application to clustering
with side-information. In NIPS, pages 505?512, 2002.
[23] C. Zhai, W. W. Cohen, and J. D. Lafferty. Beyond independent relevance: methods and evaluation metrics
for subtopic retrieval. In SIGIR, pages 10?17, 2003.
[24] X. Zhu, A. B. Goldberg, J. V. Gael, and D. Andrzejewski. Improving diversity in ranking using absorbing
random walks. In HLT-NAACL, pages 97?104, 2007.
[25] X. Zhu, J. Guo, X. Cheng, P. Du, and H. Shen. A unified framework for recommending diverse and
relevant queries. In WWW, pages 37?46, 2011.
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4,030 | 4,648 | Entangled Monte Carlo
Seong-Hwan Jun
Liangliang Wang
Alexandre Bouchard-C?ot?e
Department of Statistics
University of British Columbia
{seong.jun, l.wang, bouchard}@stat.ubc.ca
Abstract
We propose a novel method for scalable parallelization of SMC algorithms, Entangled Monte Carlo simulation (EMC). EMC avoids the transmission of particles
between nodes, and instead reconstructs them from the particle genealogy. In particular, we show that we can reduce the communication to the particle weights for
each machine while efficiently maintaining implicit global coherence of the parallel simulation. We explain methods to efficiently maintain a genealogy of particles
from which any particle can be reconstructed. We demonstrate using examples
from Bayesian phylogenetic that the computational gain from parallelization using EMC significantly outweighs the cost of particle reconstruction. The timing
experiments show that reconstruction of particles is indeed much more efficient as
compared to transmission of particles.
1
Introduction
In this paper, we focus on scalable parallelization of Monte Carlo simulation, a problem motivated by
the increasingly large inference problems occurring in a variety of fields in science and engineering.
Specifically, we assume that we are given a large scale inference problem involving an intractable
posterior expectation, for example a Bayes estimator, and that Monte Carlo simulation is to be used
to approximate the targeted expectation.
We are specifically interested in parallel Monte Carlo algorithms that scale not only in scientificcomputing clusters, where node communication is fast and cheap, but also in situations where communication between nodes is limited by a combination of latency, throughput, and cost. For example, severe communication constraints arise in peer-to-peer distributed computing projects such as
BOINC [1], and more generally in clusters assembled from commodity hardware.
Sequential Monte Carlo (SMC) is generally viewed as the leading candidate for massively parallel
simulation, but because of particle resampling, existing implementations require the network transfer
of a large number of particles and a central server with a global view on the weights carried by the
particles. As a consequence, the naive communication cost grows with the size of the inference
problem.
Our main contribution is a method, Entangled Monte Carlo simulation (EMC), for carrying out SMC
simulation in a cluster with a communication cost per particle independent of the problem size. Our
approach is fully generic and does not assume any known structure on the target distribution or the
proposal used in the simulation. These desirable characteristics are achieved by limiting the contents
of inter-node transmission to summary statistics on the particle weights. These summary statistics
are compact and of size independent in the size of the state space of the target integral. We show
that our summary statistics are sufficient, in the sense that they can be used in combination with the
particle genealogy to quickly reconstruct any particle in any node of the cluster.
We will illustrate the advantage of particle reconstruction versus network transmission in the context
of phylogenetic inference, a well known example where Monte Carlo simulation is both important
1
and challenging. In the case of the SMC sampler from [2], the cost of transmitting one particle is
proportional to the product of the number of species under study, times the number of sites in the
sequences, times the number of characters possible at each site.
We also introduce the algorithms needed to do these reconstructions efficiently while maintaining a
distributed representation of the particle genealogies. The main algorithm is based on an alternative
representation of simulation borrowed from the field of perfect simulation [3]. We demonstrate
that using our algorithms, the computational cost involved in these reconstructions is negligible
compared to the corresponding gains obtained from parallelization. While we describe EMC in the
context of SMC simulation, it can accommodate any MCMC proposal. This is done by using the
construction of artificial backward kernels [4, 5].
There is a large literature on parallelization of both MCMC and SMC algorithms. For SMC, most
of the work has been on parallelization of the proposal steps [6], which is sufficient in setups such
as GPU parallelization where communication between computing units is fast and cheap. However
in generic clusters or peer-to-peer architectures, we argue that our more efficient parallelization of
the resampling step is advantageous.
For MCMC, there is a large amount of literature on parallelization involving kernels that take the
form of local Gibbs update in a graphical model. These methods allow for several blocks of variables
to be updated in parallel. However, the communication cost can be high in a dense graphical model
as state information needs to be synchronized. Moreover, the method is restricted to certain kinds of
Gibbs kernel [7, 8, 9].
Another popular MCMC parallelization method is parallel tempering [10], where auxiliary chains
are added to enable faster exploration of the space by swapping states in different chains. While
parallel tempering has a low communication cost independent of the inference problem size, the
additional gain of parallelism can quickly decrease as more chains are added because many swaps
are needed to get from the most heated chain to the main chain.
2
Background
We will denote the target distribution by ?, which in a Bayesian problem would correspond to a
posterior distribution. The main goal is to compute the integral under ? of one or more test functions
h, which we denote by ?(h) for short. In a Bayesian problem, this arises as the posterior expectation
needed when computing a Bayes estimator. We will denote the state space by S, i.e. h : S ? R,
? : FS ? [0, 1], where (S, FS ) is a probability space.
2.1
Stochastic maps
An important concept used in the construction of our algorithms is the idea of a stochastic map. We
start by reviewing stochastic maps in the context of a Markov chain, where it was first introduced to
design perfect simulation algorithms.
Let T : S ? FS ? [0, 1] denote the transition kernel of a Markov chain (generally constructed
by first proposing and then deciding whether to move or not using a Metropolis-Hastings (MH)
ratio). A stochastic map is an equivalent view of this chain, pushing the randomness into a list
of random transition functions. Formally, it is a (S ? S)-valued random variable F such that
T (s, A) = P(F (s) ? A) for all state s ? S and event A ? FS . Concretely, these maps are
constructed by using the observation that T is typically defined as a transformation t(u, s) with
u ? [0, 1]. The most fundamental example is the case where t is based on the inverse cumulative
distribution method. We can then write F (s) = t(U, s) for a uniform random variable U on [0, 1].
With this notation, we get a non-standard, but useful way of formulating MCMC algorithms. First,
sample N stochastic maps F1 , F2 , . . . , FN , independently and identically. Second, to compute the
state of the chain after n transitions, simply return F1 (F2 (. . . (Fn (x0 )) . . . )) = F1 ?? ? ??Fn (x0 ), for
an arbitrary start state x0 ? S, n ? {1, 2, . . . , N }. This representation decouples the dependencies
induced by random number generation from the dependencies induced by operations on the state
space. In MCMC, the latter are still not readily amenable to parallelization, and this is the motivation
for using SMC as the foundation of our method. We will show in Section 3 that SMC algorithms
can also be rewritten using stochastic maps.
2
i
i
has |X| ? r trees.
The sets of partial states considered in this Section are assumed to satisfy
the following three conditions:
1. The sets of partial states of di?erent ranks should be disjoint, i.e. Sr ?Ss
for all r ?= s (in phylogenetics, this holds since a forest with r trees cannot
be a forest with s trees when r ?= s).
2. The set of partial state of smallest rank should have a single element
Resampling
s1,1
Proposal
s1,1
s2,1
~
w2,1 = 0.12
s2,2
s1,2
~
s2,2
w1,2 = 0.02
s1,3
on X).
s2,1
w1,1 = 0.03
s1,2
denoted by ?, S1 = {?} (in phylogenetics, ? is the disconnected graph
Weighting
w2,2 = 0.2
s2,3
s~1,3
s2,3
w1,3 = 0.08
w2,3 = 0.02
3. The set of partial states of rank R should coincide with the target space,
SR = X (in phylogenetics, at rank R = |X| ? 1, forests have a single tree
and are members of the target space X ).
These conditions will be subsumed by the more general framework of Section 4.5, but the more concrete conditions above help understanding the poset
framework.
In order to grow particles from one rank to the next, the user needs to
specify a proposal probability kernel ? + . Given an initial partial state s and
a set of destination partial states A, we denote the probability of proposing
(a)
(b)
an element in A from s by ?s+ (A). In the discrete case, we abuse the notation
14
Figure 1: (a) a graphical illustration of the SMC algorithm. (b) Particle genealogy.
2.2
SMC algorithms
Before going over our parallel version of SMC and to keep the exposition self-contained, we review
here the notation and description of standard, serial SMC algorithms from [11], which in turn is
based on the SMC framework of [12, 4, 5]. The samplers used in this paper are defined using
a proposal ? : S ? FS ? [0, 1]. Here, S can be an enlarged version of the target space, with
intermediate states added to ease sampling. We assume that ? has been correspondingly enlarged.
The technical conditions on ? and ? are explained in [11], but for the purpose of understanding our
method, only the algorithmic description of SMC given below is necessary.
SMC proceeds in a sequence of generations indexed by r. At each generation, the algorithm
keeps in memory a weighted list of K particles, sr,1 , . . . , sr,K ? S, with corresponding weights
wr,1 , . . . , wr,K (see Figure 1 (a)). The weighted particles induce a distribution on S defined by:
?r,K (A) ?
K
X
wr,k ?sr,k (A),
(1)
k=1
where A ? FS is an event, and ?x (A) = 1 if x ? A and 0 otherwise. We define the algorithm
recursively on the generation r. In the base case, we set w0,k = 1/K for all k ? {1, . . . , K}, and
the s0,k are initialized to a designated start state ?. Given the list of particles and weights from
the previous generation r ? 1, a new list is created in three steps. The first step can be understood
as a method for pruning unpromising particles. This is done by sampling independently K times
from the weighted particles distribution ?r,K defined above. The result of this step is that some
of the particles (mostly those of low weight) will be pruned. We denote the sampled particles by
s?r?1,1 , . . . , s?r?1,K . The second step is to create new particles, sr,1 , . . . , sr,K , by extending the
partial states of each of the sampled particles from the previous iteration. This is done by sampling
K times from the proposal distribution, sr,k ? ?s?r?1,k . The third step is to compute weights for the
new particles: wr,k = ?(?
sr?1,k , sr,k ), where the weight update function ? is an easy to evaluate
deterministic function ? : S 2 ? [0, ?). We give examples in Section 4.1.
Finally, the target integral ?(h) is approximated using the weighted distribution of the last generation
R, ?R,K (h). Note that using recent work on SMC, it is possible to convert any MCMC proposals
targeting a state space X into a valid SMC algorithm [4, 5]. This can be done for example by using
an expended state space S = X R and by constructing an auxiliary distribution on this new space.
See [4, 5] for details.
3
Entangled Monte Carlo Simulation
To parallelize SMC, we will view the applications of SMC proposals as a collection of stochastic
maps to be shared across machines. Note that there are K ? R proposal applications in total, which
we will index by I 3 i = i(r, k) = (r(i), k(i)) for convenience. Applying these stochastic maps,
denoted by F = {Fi : i ? I}, is often computationally intensive (for example because of RaoBlackwellization), and it is common to view this step as the computational bottleneck. At iteration
r, each machine, with index m ? {1, . . . , M }, will therefore be responsible of computing proposals
3
Algorithm 1 : EMC(?, ?, h, I0 )
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
1:
2:
3:
4:
5:
6:
(F , G , H ) ? entangle(?) {Section 3.3}
s ? empty-hashtable
? ? empty-genealogy
init(s, w)
for r ? {1, . . . , R} do
exchange(wr?1 )
resample(wr?1 , ?, Ir?1 , G )
{Supplementary Material}
Ir ? allocate(?, Ir?1 , H ) {Section 3.1}
for i ? Ir do
s(i) ? reconstruct(s, ?, i, F )
{Algorithm 2}
wr,k(i) ? ?(s(?(i)), s(i))
end for
end for
process(s, w, h)
=
F ?I
while (s(i) = nil) do
F ? F ? Fi
i ? ?(i)
end while
return F (s(i))
components decreases by one at every step. More precisely, we will build each
rooted X -tree t by proposing a sequence of X -forests s1 , s2 , . . . , sR = t, where
an X -forest sr = {(ti , Xi )} is a collection of rooted Xi -trees ti such that the
disjoint union of leaves of the trees in the forest is equal to the original set of
leaves, ?i Xi = X . Note that with this specific construction, a forest of rank r
has |X | ? r trees.
The sets of partial states considered in this Section are assumed to satisfy
the following three conditions:
1.
The sets of partial states of di?erent ranks should be disjoint, i.e. Sr ? Ss
for all r ?= s (in phylogenetics, this holds since a forest with r trees cannot
1
be a forest with s trees when r ?= s).
2.
The set of partial state of smallest rank should have a single element
2
denoted by ?, S1 = {?} (in phylogenetics, ? is the disconnected graph
on X ).
3.
11:
12:
13:
14:
reconstruct(s, ?, i, F
Algorithm 2 :
{Fi : i ? I})
3
The set of partial states of rank R should coincide with the target space,
SR = X (in phylogenetics, at rank R = |X | ? 1, forests have a single tree
and are members of the target space X ).
r
These conditions will be subsumed by the more general framework of Section 4.5, but the more concrete conditions above help understanding the poset
framework.
Figure 2: Illustration of compact particles (blue),
concrete particles (black), and discarded particles
(grey).
In order to grow particles from one rank to the next, the user needs to
specify a proposal probability kernel ? + .
Given an initial partial state s and
a set of destination partial states A, we denote the probability of proposing
an element in A from s by
+
?s
(A).
In the discrete case, we abuse the notation
14
for only a subset Ir of the particles indices {1, . . . , K}. We refer to machine m as the reference
machine. For brevity of notation, we omit notation m when it is clear that we refer to the reference
machine.
Parallelizing SMC is complicated by the resampling step. If roughly all particles were resampled
exactly once, we would be able to assign to each machine the same indices as the previous iteration,
avoiding communication. However, this rarely happens in practice. Instead, a small number of
particles is often resampled a large number of times while many others have no offspring. This
means that Ir can radically change across iterations. This raises an important question: how can a
machine compute a proposal if the particle from which to propose was itself computed by a different
machine?
The naive approach would consist in transmitting the ?missing? particles over the network. However,
even if basic optimizations are used (for example sending particles with multiplicities only once),
we show in Section 4 that this transfer can be slow in practice. Instead, our approach relies on a
combination of the stochastic maps with the particle genealogy to reconstruct the particle. Let us
see what this means in more detail, by going over the key steps of EMC, shown in Algorithm 1.
First, note that the resampling step in SMC algorithms induces a one-to-many relationship between
the particle in generation r and those in generation r ? 1. This relationship is called the particle
genealogy, illustrated in Figure 1 (b). Formally, a genealogy is a directed graph where nodes are
particles sr,k , r ? {1, . . . , R}, k ? {1, . . . , K}, and where node sr?1,k is deemed the parent of
node sr,k0 if the latter was obtained by resampling s?r?1,k0 = sr?1,k followed by proposing sr,k0
from s?r?1,k0 .
Suppose for now that each machine kept track of the full genealogy, in the form of a hashtable
? : I ? I of parent pointers. Each machine also maintains a hashtable holding the particles held in
memory in the reference machine s : I ? S ? {nil} (the value nil represent a particle not currently
represented explicitly in the reference machine). Algorithm 2 shows that this information, s, ?, F , is
sufficient to instantiate any query particle (indexed by i in the pseudo-code). Note that the procedure
reconstruct is guaranteed to terminate: in the procedure init, we set s(i(0, k)) to ?, and the weights
uniformly, hence ? is an ancestor of all particles.
This high-level description raises several questions. How can we efficiently store and retrieve the
stochastic maps? Can we maintain a sparse view of the genealogical information ?, s to keep space
requirements low? Finally, how can we do resampling and particle allocation in this distributed
framework? We will cover these issues in the remaining of this section, describing at the same time
how the procedures allocate, resample and exchange are implemented.
4
3.1
Allocation and resampling
In SMC algorithms, the weights are periodically used for resampling the particles, a step also known
as the bootstrapping stage and denoted by resample in Algorithm 1. This is the only stage where
EMC requires communication over the network to be done. With each machine having the full information of the weights in the current iteration, they can each perform a standard, global resampling
step without further communication.
In most cases of interest, each machine can transmit all the individual weights of its particles and to
communicate it with every other machine (either via a central server, or a decentralized scheme such
as [13]) without becoming the bottleneck. Extreme cases, where even the list of weights alone is too
large to transmit, can also be handled by transmitting only the sum of the weights of each machine,
and using a distributed hashtable [13] to represent the genealogy. The modifications needed to
implement this are discussed in Supplementary Material. We focus on the simpler case here.
Once the resampling step determines which particles survive to the next generation, the next step is to
determine allocation of particles to machines. Particle allocation is an optimization problem where
the objective is to reduce the reconstruction time with respect to the set of partition of particles.
Let {A1r , . . . , AM
r } be the set of partition of particles {1, . . . , K} at generation r and let cm denote
the maximum number of particles that can be processed by machine m. For i ? Am
r , let ?(i) be the
number of times the stochastic map needs to be applied. The objective function is defined as,
M X
X
min
m
{A1r ,...,AM
r s.t |Ar |?cm ?m}
?(i)
m=1 i?Am
r
Obtaining an exact solution to this optimization problem is infeasible in practice as it requires enumerating over the set of all possible partitions. We propose greedy methods where each machine
m
m
retains as many particles from Ir?1
as possible. Let I?rm ? Ir?1
be the set of particles resampled
m
?
from machine m. If |Ir | ? cm > 0, this machine is in surplus of particles. We propose variety of
0
greedy schemes to allocate the surplus of particles over to machines m0 6= m, where cm0 ?|I?rm | > 0.
FirstOpen: a deterministic scheme where a known list of preferred machines are known by all
machines. The surplus particles are allocated according to this list.
MostAvailable: attempts to allocate the surplus particles to machines with the most capacity as
0
defined by cm0 ? I?rm .
Random: samples a machine m0 at random with equal probability 1/M . The intention is that the
particles are mixed well over different machines so that the reconstruction algorithm rarely traces
back the genealogy to the root ancestor.
3.2
Genealogy
In this section, we argue that for the purpose of reconstruction, only a sparse subset of the genealogy
needs to be represented at any given iteration and machine. The key idea is that if a particle has no
descendant in the current generation, storing its parent is not necessary. In practice, we observed that
the vast majority of the ancestral particles have this property. We discuss at the end of this section
some intuition as of why this holds, using a coalescent model.
Let us first look at how we can efficiently exploit this property. First, it is useful to draw a distinction
between concrete particles, with s(i) 6= nil, and compact particles, which are particles implicitly
represented via an integer (the parent of the particle), and are therefore considerably more spaceefficient. For example, in the smallest phylogenetic example considered in Section 4, a compact
particle occupies about 50, 000 times less memory than a concrete particle. Whereas a concrete
particle can grow in size as the problem size increases, a compact particle size is fixed.
Particles, concrete or compact, can become obsolete, meaning that the algorithm can guarantee that
they will not be needed in subsequent iterations. This can happen for at least two reasons, each of
which is efficiently detected at a different stage of the algorithm.
Update after resampling: Any lineage (path in ?) that did not survive the resampling stage no
longer needs to be maintained. This is illustrated in Figure 2. The greyed out particles will never
5
be reconstructed in the future generation so they are no longer maintained. Note that it is easy to
harness a garbage collector to perform this update in practice.
Update after reconstruction: Once a particle is reconstructed, the lineage of the reconstructed
particle can be updated. Let j be the particle that is reconstructed at generation r. At any future
generation r0 > r, the reconstruction algorithm will only trace up to j (as s(j) 6= nil), and hence
all its parent can be discarded. Note that similar updates can be performed on s to keep s sparse as
well.
The coalescent [14] can provide a potential theoretical model for understanding why these strategies
are so effective in practice. If we assumed the weight function ? to be constant, the genealogy
induced by resampling can be viewed as a Wright-Fisher model [14, 15], which is well approximated
by the coalescent when the number of particles is large. For example, this means that (1?1/k)/(1?
1/K) is the expected time spent waiting for the last k copies to coalesce [15].
Note that the coalescent also gives an intuition for having Algorithm 2 terminating well before
reaching the initial symbol ?. Again, this reflects what we observed in our experiments.
3.3
Compact representation of the stochastic maps
The cardinality of the set of the stochastic maps F = {Fi : i ? I} grows proportionally to the
number of particles K time the number of generations R. To store these maps naively would require
the storage of O(KR) uniform random variables Ui . However, since in practice pseudo-random
numbers rather than true independent numbers are typically used, the sequence can be stored implicitly by maintaining only a random seed shared between machines. A drawback to this approach
is that it is not efficient to perform random access of the random numbers. Random access of random
numbers is an unusual requirement imposed by the genealogy reconstruction algorithm. Fortunately,
as we discuss in this section, it is not hard to modify pseudo-random generators to support random
access.
The simplest strategy to obtain faster random access is to cache intermediate internal states of the
pseudo-random generator. For example by doing so for every particle generation, we get a faster
access time of O(K) and a larger space requirement of O(R). More generally, this method can
provide tradeoff of O(n) space and O(m) time with mn = RK.
In Supplementary Material, we describe the details of an alternative that requires O(1) storage with
O(log(KR)) time for random access of any given map with index i ? I. This method could potentially change the quality of the pseudo-random sequences obtained, but as described in Section 4.2,
we have empirical evidence suggesting that the new pseudo-random scheme does not affect the
quality of the estimated posterior expectations.
4
Experiments
In this section, we demonstrate the empirical performance of our method on synthetic and real
datasets. As a first validation, we start by demonstrating that the behavior of our sampler equipped
with our stochastic map datastructure is indistinguishable from that of a sampler based on a standard pseudo-random generator. Then we show results on the task of Bayesian phylogenetic inference, a challenging domain where massively parallel simulation is likely to have an impact for
practitioners?running phylogenetic MCMC chains for months is not uncommon. To keep the exposition self-contained, we include a review of the phylogenetic SMC techniques we used.
4.1
Experimental setup
Given a collection of biological sequences for different species (taxa), Bayesian phylogenetics aims
to compute expectations under a posterior distribution over phylogenetic trees, which represent the
relationship among the species under study [16]. For intermediate to large numbers of species,
Bayesian phylogenetic inference via SMC requires a large number of particles to achieve an accurate
estimate. This is due to fact that the total number of distinct tree topologies increases at a superexponential rate as the number of species increases [16].
6
In the following section, we use the phylogenetic SMC algorithm described in [2], where particles
are proposed using a proposal with density ?(s ? s0 ). Starting from a fully disconnected forest over
the species, ? picks one pair of trees in the forest at random, and forms a new tree by connecting
their roots. Under weak conditions described in [11], the following weight update yields a consistent
estimator for the posterior over phylogenies:
?(?
sr?1 ? sr ) ?
?(sr )
1
?
,
?(?
sr?1 ) ?(?
sr?1 ? sr )
where ? is an unnormalized density over forests. In the experiments in Section 4.2 and Supplementary Material, where we wanted to run our SMC for more iterations, we use an alternation of
kernel: in a first phase, the kernel described above, and in the second phase, the MCMC kernel of
[17], transformed into a SMC kernel using the technique of [4, 5].
To generate synthetic datasets, we used a standard process [11]: we sampled trees from the coalescent, simulated data along the tree using a K2P likelihood model, discarded the values at internal
nodes to keep only the observations at the leaves and held out the tree.
For real datasets, we used the manually aligned ribosomal RNA (rRNA) dataset of [18]. We used a
subset of 28 sequences in the directory containing 5S rRNA sequences of Betaproteobacteria and a
larger subset of 4,510 sequences of 16S rRNA sequences from Actinobacteria. We did experiments
on two different numbers of subsampled species: 20 and 100.
4.2
Validation of the stochastic random maps datastructure
To check if the scheme described in Section 3.3 affects the quality of the SMC approximation of
the target distribution, we carried out experiments to compare the quality of the SMC approximation
based on pseudo-random numbers generated from uniform algorithm outlined in the Supplementary
Material against the standard pseudo-random number generator. The dataset is a synthetic phylogenetics data with 20 taxa and 1000 sites. We measured a tree metric, the Robinson Foulds metric,
on the consensus tree at every iteration, to detect potential biases in the estimator. We show random
examples of pairs of runs in the Supplementary Material.
4.3
Speed-up results
In this section, we show experimental results where we measure the speed-up of an EMC algorithm
on two sets of phylogenetics data by counting the number of times the maps Fi are applied. The
question we explore here is how deep the reconstruction algorithm has to trace back, or more precisely, how many times a parallelized version of our algorithm applies maps Fi compared to the
number of times the equivalent operation is performed in the serial version of SMC.
We denote N1 to be the number of times the proposal function is applied in serial SMC, and NM
to be the number of stochastic maps applied in our algorithm ran on M machines. We measure the
speedup as a ratio of M and RM , SM = RMM where RM = NNM1 .
We ran these experiments on the 16S and 5S subsets of the rRNA data described earlier. In both
subsets, we found a substantial speedup, suggesting that deep reconstruction was rarely needed in
practice. We also obtained the following empirical ranking across the performance of the allocation
methods: FirstOpen > MostAvailable > Random. We show the results on 100 taxa (species) for 5S
and 16S in Figure 3.
We also performed an experiment on synthetic data generated with 20 taxa and 1000 sites that
parallelization using EMC is beneficial in the corner case when the weights are all equal. For
the purposes of illustration, we included an extra allocation method, chaos. This is an allocation
method where particles are allocated at random, which disregards the greedy method suggested in
Section 3.1. We show the results in Figure 3, where it can be seen that the speedup is still substantial
in this context for all of the allocation methods.
4.4
Timing results
An SMC algorithm can easily be distributed over multiple machines by relying naively on particle
transmission between machines over the network. In this section, we compared the particle transmission time to reconstruction time of EMC on Amazon EC2 micro instances.
7
5S proteobacteria dataset with 100 taxa
Speedup
Speedup
Chaos experiment with 20 taxa
Speedup
16S actinobacteria dataset with 100 taxa
Number of Machines
Number of Machines
Number of Machines
(a)
(b)
(c)
Figure 3: The speedup factor for (a) the 16S actinobacteria dataset with 100 taxa, (b) the 5S actinobacteria
dataset with 100 taxa, and (c) the uniform weight synthetic experiment (see text).
Run time elapsed per generation
50000
40000
30000
Time (milliseconds)
20000
0
10000
200000 400000 600000 800000
0
Time (milliseconds)
1200000
60000
Total run time of EMC versus Particle transfer
500
1000
1500
2000
2500
3000
0
# of particles
20
40
60
80
100
Generation
(a)
(b)
Figure 4: (a) Total time for particle transfer (red), total time for EMC (blue). (b) Sample generation time
including reconstruction time (black), reconstruction time (blue), and particle transfer time (red) by generation.
The timing results in this section builds on the results from Section 4.3 where we showed that the
ratio of NM and N1 is small. Here, we ran SMC algorithm for 100 generations and measured
the total run time of the EMC algorithm and an SMC algorithm parallelized via explicit particle
transfer?see Figure 4 (a). We fixed the number of particles per machine at 100 and produced a
sequence of experiments by doubling the number of machines and hence the number of particles
at each step. In Figure 4 (b), we show the reconstruction time, the sample generation time (which
includes the reconstruction time), and the particle transmission time by generation. As expected,
the particle transmission is the bottleneck to the SMC algorithm whereas the reconstruction time is
stable, which verifies that the reconstruction algorithm rarely traced deep.
The total timing result in Figure 4 (a) shows that the overhead arising from increasing the number of particles (or increasing the number of machines) is much smaller compared to the particle
transmission method. The breakdown of time by generation in Figure 4 (b) shows that the particle
transmission time is volatile as it depends on the network latency and throughput. The reconstruction
time is stable as it relies only on the CPU cycles.
5
Conclusion
We have introduced EMC, a method to parallelize an SMC algorithm over multiple nodes. The new
method requires only a small amount of data communication over the network, of size per particle
independent of the scale of the inference problem. We have shown that the algorithm performs
very well in practice on a Bayesian phylogenetic example and our software can be downloaded at
stat.ubc.ca/?seong.jun/.
Acknowledgements
We thank Arnaud Doucet, Fabian Wauthier, and the anonymous reviewers for their helpful comments.
8
References
[1] D. P. Anderson. BOINC: A System for Public-Resource Computing and Storage. In GRID
?04: Proceedings of the 5th IEEE/ACM International Workshop on Grid Computing, pages
4?10, Washington, DC, USA, 2004. IEEE Computer Society.
[2] Y. W. Teh, H. Daum?e III, and D. M. Roy. Bayesian agglomerative clustering with coalescents.
In Advances in Neural Information Processing Systems (NIPS), 2008.
[3] J. G. Propp and D. B. Wilson. Coupling from the past: a user?s guide. Microsurveys in Discrete Probability. DIMACS Series in Discrete Mathematics and Theoretical Computer Science,
41:181?192, 1998.
[4] P. Del Moral, A. Doucet, and A. Jasra. Sequential Monte Carlo samplers. Journal of The Royal
Statistical Society Series B-statistical Methodology, 68(3):411?436, 2006.
[5] P. Del Moral, A. Doucet, and A. Jasra. Sequential Monte Carlo for Bayesian computation.
Bayesian Statistics, 8:1?34, 2007.
[6] A. Lee, C. Yau, M. B. Giles, A. Doucet, and C. C. Holmes. On the utility of graphics cards to
perform massively parallel simulation of advanced Monte Carlo methods. Journal of Computational and Graphical Statistics, 19(4):769?789, 2010.
[7] S. Singh and A. McCallum. Towards asynchronous distributed MCMC inference for large
graphical models. In Neural Information Processing Systems (NIPS), Big Learning Workshop
on Algorithms, Systems, and Tools for Learning at Scale, 2011.
[8] J. Gonzalez, Y. Low, A. Gretton, and C. Guestrin. Parallel Gibbs sampling: From colored fields
to thin junction trees. In In Artificial Intelligence and Statistics (AISTATS), Ft. Lauderdale, FL,
May 2011.
[9] S. Singh, A. Subramanya, F. Pereira, and A. McCallum. Large-scale cross-document coreference using distributed inference and hierarchical models. In Association for Computational
Linguistics: Human Language Technologies (ACL HLT), 2011.
[10] R. H. Swendsen and J-S. Wang. Replica Monte Carlo simulation of spin-glasses. Phys. Rev.
Lett., 57:2607?2609, Nov 1986.
[11] A. Bouchard-C?ot?e, S. Sankararaman, and M. I. Jordan. Phylogenetic inference via Sequential
Monte Carlo. Systematic Biology, 2011.
[12] A. Doucet, N. de Freitas, and N. Gordon. Sequential Monte Carlo methods in practice.
Springer, 2001.
[13] I. Stoica, R. Morris, D. Karger, M. F. Kaashoek, and H. Balakrishnan. Chord: A scalable
peer-to-peer lookup service for internet applications. ACM SIGCOMM 2001, pages 149?160,
2001.
[14] J. F. C. Kingman. On the Genealogy of Large Populations. Journal of Applied Probability,
19:27?43, 1982.
[15] J. Felsenstein. Inferring phylogenies. Sinauer Associates, 2003.
[16] C. Semple and M. Steel. Phylogenetics. Oxford, 2003.
[17] J. P. Huelsenbeck and F. Ronquist. MRBAYES: Bayesian inference of phylogenetic trees.
Bioinformatics, 17(8):754?755, August 2001.
[18] J.J. Cannone, S. Subramanian, M.N. Schnare, J.R. Collett, L.M. D?Souza, Y. Du, B. Feng,
N. Lin, L.V. Madabusi, K.M. Muller, N. Pande, Z. Shang, N. Yu, and R.R. Gutell. The comparative RNA web (CRW) site: An online database of comparative sequence and structure
information for ribosomal, intron, and other RNAs. BioMed Central Bioinformatics, 2002.
9
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4,031 | 4,649 | The Bethe Partition Function of Log-supermodular
Graphical Models
Nicholas Ruozzi
Communication Theory Laboratory
EPFL
Lausanne, Switzerland
[email protected]
Abstract
Sudderth, Wainwright, and Willsky conjectured that the Bethe approximation corresponding to any fixed point of the belief propagation algorithm over an attractive, pairwise binary graphical model provides a lower bound on the true partition
function. In this work, we resolve this conjecture in the affirmative by demonstrating that, for any graphical model with binary variables whose potential functions
(not necessarily pairwise) are all log-supermodular, the Bethe partition function
always lower bounds the true partition function. The proof of this result follows
from a new variant of the ?four functions? theorem that may be of independent
interest.
1
Introduction
Graphical models have proven to be a useful tool for performing approximate inference in a wide variety of application areas including computer vision, combinatorial optimization, statistical physics,
and wireless networking. Computing the partition function of a given graphical model, a typical inference problem, is an NP-hard problem in general. Because of this, the inference problem is often
replaced by a variational approximation that is, hopefully, easier to solve. The Bethe approximation,
one such standard approximation, is of great interest both because of its practical performance and
because of its relationship to the belief propagation (BP) algorithm: stationary points of the Bethe
free energy function correspond to fixed points of belief propagation [1]. However, the Bethe partition function is only an approximation to the true partition function and need not provide an upper
or lower bound.
In certain special cases, the Bethe approximation is conjectured to provide a lower bound on the true
partition function. One such example is the class of attractive pairwise graphical models: models in
which the interaction between any two neighboring variables places a greater weight on assignments
in which the two variables agree. Many applications in computer vision and statistical physics can
be expressed as attractive pairwise graphical models (e.g., the ferromagnetic Ising model). Sudderth,
Wainwright, and Willsky [2] used a loop series expansion of Chertkov and Chernyak [3, 4] in order
to study the fixed points of BP over attractive graphical models. They provided conditions on the
fixed points of BP under which the stationary points of the Bethe free energy function corresponding
to these fixed points are a lower bound on the true partition function. Empirically, they observed that,
even when their conditions were not satisfied, the Bethe partition function appeared to lower bound
the true partition function, and they conjectured that this is always the case for attractive pairwise
binary graphical models.
Recent work on the relationship between the Bethe partition function and the graph covers of a
given graphical model has suggested a new approach to resolving this conjecture. Vontobel [5]
demonstrated that the Bethe partition function can be precisely characterized by the average of the
1
true partition functions corresponding to graph covers of the base graphical model. The primary
contribution of the present work is to show that, for graphical models with log-supermodular potentials, the partition function associated with any graph cover of the base graph, appropriately
normalized, must lower bound the true partition function. As pairwise binary graphical models are
log-supermodular if and only if they are attractive, combining our result with the observations of [5]
resolves the conjecture of [2].
The key element in our proof, and the second contribution of this work, is a new variant of the
?four functions? theorem that is specific to log-supermodular functions. We state and prove this
variant in Section 3.1, and in Section 4.1, we use it to resolve the conjecture. As a final contribution, we demonstrate that our variant of the ?four functions? theorem has applications beyond
log-supermodular functions: as an example, we use it to show that the Bethe partition function can
also provide a lower bound on the number of independent sets in a bipartite graph.
2
Undirected Graphical Models
Let f : {0, 1}n ? R?0 be a non-negative function. We say that f factors with respect to a hypergraph G = (V, A) where A ? 2V , if there exist potential functions ?i : {0, 1} ? R?0 for each
i ? V and ?? : {0, 1}|?| ? R?0 for each ? ? A such that
Y
Y
f (x) =
?i (xi )
?? (x? )
i?V
??A
where x? is the subvector of the vector x indexed by the set ?.
We will express the hypergraph G as a bipartite graph that consists of a variable node for each i ? V ,
a factor node for each ? ? A, and an edge joining the factor node corresponding to ? to the variable
node representing i if i ? ?. This is typically referred to as the factor graph representation of G.
Definition 2.1. A function f : {0, 1}n ? R?0 is log-supermodular if for all x, y ? {0, 1}n
f (x)f (y) ? f (x ? y)f (x ? y)
where (x?y)i = min{xi , yi } and (x?y)i = max{xi , yi }. Similarly, a function f : {0, 1}n ? R?0
is log-submodular if for all x, y ? {0, 1}n
f (x)f (y) ? f (x ? y)f (x ? y).
Definition 2.2. A factorization of a function f : {0, 1}n ? R?0 over G = (V, A) is logsupermodular if for all ? ? A, ?? (x? ) is log-supermodular.
Every function that admits a log-supermodular factorization is necessarily log-supermodular, products of log-supermodular functions are easily seen to be log-supermodular, but the converse may not
be true outside of special cases. If |?| ? 2 for each ? ? A, then we call the factorization pairwise.
For any pairwise factorization, f is log-supermodular if and only if ?ij is log-supermodular for each
i and j.
Pairwise graphical models such that ?? (x? ) is log-supermodular for all ? ? A are referred to
as attractive graphical models. A generalization of attractive interactions to the non-pairwise case
is presented in [2]: for all ? ? A, ?? , when appropriately normalized, has non-negative central
moments. However, the relationship between this generalization and log-supermodularity remains
unclear.
2.1
Graph Covers
Graph covers have played an important role in our understanding of graphical models [5, 6].
Roughly, if a graph H covers a graph G, then H looks locally the same as G.
Definition 2.3. A graph H covers a graph G = (V, E) if there exists a graph homomorphism
h : H ? G such that for all vertices v ? G and all w ? h?1 (v), h maps the neighborhood ?w of w
in H bijectively to the neighborhood ?v of v in G. If h(w) = v, then we say that w ? H is a copy
of v ? G. Further, H is a k-cover of G if every vertex of G has exactly k copies in H.
2
1
2
1
2
3
4
4
3
1
2
3
4
(b) One possible cover of G.
(a) A graph,
G.
Figure 1: An example of a graph cover. The nodes in the cover are labeled for the node that they
copy in the base graph.
For an example of a graph cover, see Figure 1.
For the factor graph corresponding to G = (V, A), each k-cover consists of a variable node for
each of the k|V | variables, a factor node for each of the k|A| factors, and an edge joining each copy
of ? ? A to a distinct copy of each i ? ?. To any k-cover H = (VH , AH ) of G given by the
homomorphism h, we can associate a collection of potentials: the potential at node i ? VH is equal
to ?h(i) , the potential at node h(i) ? G, and for each ? ? AH , we associate the potential ?h(?) . In
this way, we can construct a function f H : {0, 1}kn ? R?0 such that f H factorizes over H.
Notice that if f G admits a log-supermodular factorization over G and H is a k-cover of G, then f H
admits a log-supermodular factorization over H.
2.2
Bethe Approximations
For a function f : {0, 1}n ? RP
?0 that factorizes over G = (V, A), we are interested computing
the partition function Z(G) = x f (x). In general, this is an NP-hard problem, but in practice,
algorithms, such as belief propagation, based on variational approximations produce reasonable estimates in many settings. One such variational approximation, the Bethe approximation at temperature
T = 1, is defined as follows:
XX
XX
log ZB (G, ? ) =
?i (xi ) log ?i (xi ) +
?? (x? ) log ?? (x? )
?
i?V xi
??A x?
XX
XX
?i (xi ) log ?i (xi ) ?
i?V xi
?? (x? )
i?? ?i (xi )
?? (x? ) log Q
??A x?
for ? in the local marginal polytope,
T , {? ? 0 | ?? ? A, i ? ?,
X
?? (x? ) = ?i (xi ) and ?i ? V,
x?\i
X
?i (xi ) = 1}.
xi
The fixed points of the belief propagation algorithm correspond to stationary points of log ZB (G, ? )
over T , the set of pseudomarginals [1], and the Bethe partition function is defined to be the maximum
value achieved by this approximation over T :
ZB (G) = max ZB (G, ? ).
? ?T
For a fixed factor graph G, we are interested in the relationship between the true partition function,
Z(G), and the Bethe approximation corresponding to G, ZB (G). While, in general, ZB (G) can be
either an upper or a lower bound on the true partition function, in this work, we address the following
conjecture of [2]:
Conjecture 2.4. If f : {0, 1}n ? R?0 admits a pairwise, log-supermodular factorization over
G = (V, A), then ZB (G) ? Z(G).
We resolve this conjecture in the affirmative, and show that it continues to hold for a larger class of
log-supermodular functions. Our results are based, primarily, on two observations: a variant of the
?four functions? theorem [7] and the following, recent theorem of Vontobel [5]:
3
Theorem 2.5.
ZB (G) = lim sup
s X
where C k (G) is the set of all k-covers of G.
Z(H)/|C k (G)|
k
k??
H?C k (G)
1
Proof. See Theorem 27 of [5].
Theorem 2.5 suggests that a reasonable strategy for proving that ZB (G) ? Z(G) would be to show
that Z(H) ? Z(G)k for any k-cover H of G. This is the strategy that we adopt in the remainder of
this work.
3
The ?Four Functions? Theorem and Related Results
The ?four functions? theorem [7] is a general result concerning nonnegative functions over distributive lattices. Many correlation inequalities from statistical physics, such as the FKG inequality, can
be seen as special cases of this theorem [8].
Theorem 3.1 (?Four Functions? Theorem). Let f1 , f2 , f3 , f4 : {0, 1}n ? R?0 be nonnegative
real-valued functions. If for all x, y ? {0, 1}n ,
f1 (x)f2 (y) ? f3 (x ? y)f4 (x ? y),
then
h
X
f1 (x)
ih
x?{0,1}n
X
i h
f2 (x) ?
x?{0,1}n
X
f3 (x)
ih
x?{0,1}n
X
i
f4 (x) .
x?{0,1}n
The following lemma is a direct consequence of the four functions theorem:
Lemma 3.2. If f : {0, 1}n ? R?0 is log-supermodular, then every marginal of f is also logsupermodular.
The four functions theorem can be extended to more than four functions, by generalizing ? and
?. For any collection of vectors x1 , . . . , xk ? Rn , let z i (x1 , . . . , xk ) be the vector whose j th
component is the ith largest element of x1j , . . . , xkj for each j ? {1, . . . , n}. As an example, for
Pk
vectors x1 , . . . , xk ? {0, 1}n , z i (x1 , . . . , xk )j = { a=1 xaj ? i} where {? ? ?} is one if the
inequality is satisfied and zero otherwise. The ?four functions? theorem is then a special case of the
more general ?2k functions? theorem [9, 10, 11]:
Theorem 3.3 (?2k Functions? Theorem). Let f1 , . . . , fk : {0, 1}n ? R?0 and g1 , . . . , gk :
{0, 1}n ? R?0 be nonnegative real-valued functions. If for all x1 , . . . , xk ? {0, 1}n ,
k
Y
gi (xi ) ?
i=1
k
Y
fi (z i (x1 , . . . , xk )),
(1)
i=1
then
k h
Y
i=1
3.1
X
k h
i Y
gi (x) ?
i=1
x?{0,1}n
X
i
fi (x) .
x?{0,1}n
A Variant of the ?Four Functions? Theorem
A natural generalization of Theorem 3.3 would be to replace the product of functions on the left-hand
side of Equation 1 with an arbitrary function over x1 , . . . , xk : we will show that we can replace this
product with an arbitrary log-supermodular function while preserving the conclusion of the theorem.
The key property of log-supermodular functions that makes this possible is the following lemma:
1
The proof of the theorem is demonstrated for ?normal? factor graphs, but it easily extends to the factor
graphs described above by replacing variable nodes with equality constraints.
4
Lemma 3.4. If g : {0, 1}n ? R?0 is log-supermodular, then for any integer k ? 1 and
Qk
Qk
x1 , . . . , xk ? {0, 1}n , i=1 g(xi ) ? i=1 g(z i (x1 , . . . , xk )).
Proof. This follows directly from the log-supermodularity of g.
The proof of our variant of the ?2k functions theorem? uses the properties of weak majorizations:
Definition
3.5. A vectorP
x ? Rn is weakly majorized by a vector y ? Rn , denoted x ?w y, if
Pt
t
i
i
i=1 z (x1 , . . . , xn ) ?
i=1 z (y1 , . . . , yn ) for all t ? {1, . . . , n}.
For the purposes of this paper, we will only need the following result concerning weak majorizations:
Pn
Pn
Theorem 3.6. For x, y ? Rn , x ?w y if and only if i=1 g(xi ) ? i=1 g(yi ) for all continuous,
increasing, and convex functions g : R ? R.
Proof. See 3.C.1.b and 4.B.2 of [12].
We now state and prove our variant of the 2k functions theorem in two pieces. First, we consider
the case where n = 1:
Lemma 3.7. Let f1 , . . . , fk : {0, 1} ? R?0 and g : {0, 1}k ? R?0 be nonnegative real-valued
functions such that g is log-supermodular. If for all x1 , . . . , xk ? {0, 1},
g(x1 , . . . , xk ) ?
k
Y
fi (z i (x1 , . . . , xk )),
i=1
then
X
g(x1 , . . . , xk ) ?
k h X
Y
i=1
x1 ,...,xk ?{0,1}
i
fi (x) .
x?{0,1}
k
Proof. For each c ? {0, . . . , k}, define X c = {(x1 , . . . , xk ) : x1 + . . . + xk = c}. Let Gc ? R( c )
c
c
be the vector obtained from by evaluating g at each element of X , and define F similarly for
Qk
f (x1 , . . . , xk ) , i=1 fi (xi ).
Our strategy will be to show that log Gc ?w log F c for each c or, equivalently,
that
QT
QT
k
t
c
c
t
c
c
z
(G
,
.
.
.
,
G
)
?
z
(F
,
.
.
.
,
F
)
for
all
c
?
{0,
.
.
.
,
k}
and
T
?
.
Then,
k
k
1
1
t=1
t=1
c
(c)
(c)
by Theorem 3.6 and the fact that 2x is convex and increasing, we will have
X
1
k
g(x , . . . , x ) =
(kc)
X
log Gct
2
?
t=1
(x1 ,...,xk )?X c
(kc)
X
c
X
2log Ft =
t=1
k
Y
fi (xi )
(x1 ,...,xk )?X c i=1
for all c. As the X c are disjoint, this will complete the proof. We note that, by continuity arguments,
this analysis holds even when some values of g and f are equal to zero.
Now, fix c ? {0, . . . , k} and T ? {1, . . . , kc }. Suppose v 1 , . . . , v T ? X c are T distinct vectors.
By Lemma 3.4, we must have
T
Y
g(v t ) ?
t=1
wjt =
T
j
t
1
T
Y
g(z t (v 1 , . . . , v T )) ?
t=1
T
t
T
Y
f (wt )
t=1
1
T
where
z (z (v , . . . , v )1 , . . . , z (v , . . . , v )k ) for each j ? {1, . . . , k}. Given any such
1
v , . . . , v ? X c , we will show how to construct distinct vectors v 1 , . . . , v T ? X c such that
QT
QT
t
t
t=1 f (w ) ?
t=1 f (v ). Consequently, we will have
T
Y
t=1
g(v t ) ?
T
Y
f (v t ) ?
t=1
T
Y
t=1
5
z t (F1c , . . . , F(ck) ).
c
As our construction will work for any choice of distinct vectors v 1 , . . . , v T ? X c , it will work,
QT
in particular, for the T distinct vectors in X c that maximize t=1 g(v t ), and the lemma will then
follow as a consequence of our previous arguments.
We now describe how to construct the vectors v 1 , . . . , v T from the vectors v 1 , . . . , v T . Let A ?
Rk?T be the matrix whose ith column is given by the vector v i .PConstruct P
A ? Rk?T from A
by swapping the rows of A so that for each i < j ? {1, . . . , k}, p Aip ? p Ajp . Intuitively,
the first row of A corresponds to the row of A with the most nonzero elements, the second row
of A corresponds to the row of A with the second largest number of nonzero elements, and so on.
Let v 1 , . . . , v T be the columns of A. Notice that v 1 , . . . , v T are distinct vectors in X c and that, by
construction, z j (z t (v 1 , . . . , v T )1 , . . . , z t (v 1 , . . . , v T )k ) = z t (v 1 , . . . , v T )j for each j ? {1, . . . , k}
and t ? {1, . . . , T }. Therefore, we must have
T
Y
g(v t ) ?
t=1
T
Y
g(z t (v 1 , . . . , v T )) ?
t=1
T
Y
f (z t (v 1 , . . . , v T )) =
t=1
T
Y
f (v t )
t=1
where the equality follows from the definition of f as a product of the fi . In addition, the vector
z t (v 1 , . . . , v T ) is simply a permuted version of the vector z t (v 1 , . . . , v T ) which means that their j th
largest elements must agree:
wjt = z j (z t (v 1 , . . . , v T )1 , . . . , z t (v 1 , . . . , v T )k )
= z j (z t (v 1 , . . . , v T )1 , . . . , z t (v 1 , . . . , v T )k )
= z t (v 1 , . . . , v T )j .
Therefore,
T
Y
t=1
T
Y
g(v t ) ?
f (wt ) =
t=1
T
Y
f (z t (v 1 , . . . , v T )) =
t=1
T
Y
f (v t )
t=1
and the lemma follows as a consequence .
Remark. In the case that n =Q
1 and k ? 1, this lemma is a more general result than the 2k functions
theorem: if g(x1 , . . . , xk ) = i gi (xi ) for g1 , . . . , gk : {0, 1} ? R?0 , then g is log-supermodular.
As in the proof of the 2k functions theorem, the general theorem for n ? 1 follows by induction on
n.This inductive proof closely follows the inductive argument in the proof of the ?four functions?
theorem described in [8] with the added observation that marginals of log-supermodular functions
continue to be log-supermodular.
Theorem 3.8. Let f1 , . . . , fk : {0, 1}n ? R?0 and g : {0, 1}kn ? R?0 be nonnegative real-valued
functions such that g is log-supermodular. If for all x1 , . . . , xk ? {0, 1}n ,
g(x1 , . . . , xk ) ?
k
Y
fi (z i (x1 , . . . , xk )),
i=1
then
X
g(x1 , . . . , xk ) ?
k h
Y
i=1
x1 ,...,xk ?{0,1}n
X
i
fi (x) .
x?{0,1}n
Proof. We will prove the result for general k and n by induction on n. The base case of n = 1
follows from Lemma 3.7. Now, for n ? 2, suppose that the result holds for k ? 1 and n ? 1, and
let f1 , . . . , fk : {0, 1}n ? R?0 and g : {0, 1}kn ? R?0 be nonnegative real-valued functions such
that g is log-supermodular.
Define f 0 : {0, 1}n?1 ? R?0 and g 0 : {0, 1}k(n?1) ? R?0 as
fi0 (y) = fi (y, 0) + fi (y, 1)
X
g 0 (y 1 , . . . , y k ) =
g(y 1 , s1 , . . . , y k , sk )
s1 ,...,sk ?{0,1}
6
Notice that g 0 is log-supermodular because it is the marginal of a log-supermodular function (see
Lemma 3.2). If we can show that
k
Y
g 0 (y 1 , . . . , y k ) ?
fi0 (z i (y 1 , . . . , y k ))
i=1
for all y 1 , . . . , y k ? {0, 1}n?1 , then the result will follow by induction on n. To show this, fix
y 1 , . . . , y k ? {0, 1}n?1 and define f : {0, 1} ? R?0 and g : {0, 1}k ? R?0 as
f i (s) = fi (z i (y 1 , . . . , y k ), s)
g(s1 , . . . , sk ) = g(y 1 , s1 , . . . , y k , sk ).
We can easily check that g(s1 , . . . , sk ) is log-supermodular and that g(s1 , . . . , sk )
Qk
i 1
k
1
k
i=1 f i (z (s , . . . , s )) for all s , . . . , s ? {0, 1}. Hence, by Lemma 3.7,
k
k
X
Y
X
Y
g(s1 , . . . , sk ) ?
f i (s) =
g 0 (y 1 , . . . , y k ) =
fi0 (z i (y 1 , . . . , y k )),
i=1 s?{0,1}
s1 ,...,sk
?
i=1
which completes the proof of the theorem.
4
Graph Covers and the Partition Function
In this section, we show how to apply Theorem 3.8 in order to resolve Conjecture 2.4. In addition, we
show that the theorem can be applied more generally to yield similar results for a class of functions
that can be converted into log-supermodular functions by a change of variables.
4.1
Log-supermodularity and Graph Covers
The following theorem follows easily from Theorem 3.8:
Theorem 4.1. If f G : {0, 1}n ? R?0 admits a log-supermodular factorization over G = (V, A),
then for any k-cover, H, of G, Z(H) ? Z(G)k .
Proof. Let H be a k-cover of G. Divide the vertices of H into k sets S1 , . . . , Sk such that each set
contains exactly one copy of each vertex i ? V . Let the assignments to the variables in the set Si be
denoted by the vector xi .
For each ? ? A, let y?i denote the assignment to the ith copy of ? by the elements of x1 , . . . , xk .
By Lemma 3.4,
k
k
k
k
Y
Y
Y
Y
?? (z i (x1 , . . . , xk )? ).
?? (z i (x1? , . . . , xk? )) =
?? (y?i ) ?
?? (z i (y?1 , . . . , y?k )) =
i=1
i=1
i=1
i=1
Qk
From this, we can conclude that f H (x1 , . . . , xk ) ? i=1 f G (z i (x1 , . . . , xk )). Now, by Theorem
3.8,
k hX
i
X
Y
f G (xi ) = Z(G)k .
Z(H) =
f H (x1 , . . . , xk ) ?
i=1
x1 ,...,xk
xi
This theorem settles the conjecture of [2] for any log-supermodular function that admits a pairwise
binary factorization, and the conjecture continues to hold for any graphical model that admits a
log-supermodular factorization.
Corollary 4.2. If f : {0, 1}n ? R?0 admits a log-supermodular factorization over G = (V, A),
then ZB (G) ? Z(G).
Proof. This follows directly from Theorem 4.1 and Theorem 2.5.
As the value of the Bethe approximation at any of the fixed points of BP is always a lower bound on
ZB (G), the conclusion of the corollary holds for any fixed point of the BP algorithm as well.
7
4.2
Beyond Log-supermodularity
While Theorem 4.1 is a statement only about log-supermodular functions, we can use it to infer
similar results even when the function under consideration is not log-supermodular. As an example
of such an application, we consider the problem of counting the number of independent sets in a
given graph, G = (V, E). An independent set, I ? V , in G is a subset of the vertices such that no
two adjacent vertices are in I. We define the following function:
Y
I G (x1 , . . . , x|V | ) =
(1 ? xi xj )
(i,j)?E
which is equal to one if the nonzero xi ?s define an independent set and zero otherwise. As every
potential function depends on at most two variables, I G factorizes over the graph G = (V, E).
Notice that f G is log-submodular, not log-supermodular.
In this section, we will focus on bipartite graphs: G = (V, E) is bipartite if we can partition the
vertex set into two sets A ? V and B = V \ A such that A and B are independent sets. Examples
of bipartite graphs include single cycles, trees, and grid graphs. We will denote bipartite graphs as
G = (A, B, E).
For any bipartite graph G = (A, B, E), I G can be converted into a log-supermodular graphical
model by a simple change of variables. Define ya = xa for all a ? A and yb = 1 ? xb for all b ? B.
We then have
Y
I G (x1 , . . . , x|V | ) =
(1 ? xi xj )
(i,j)?E
Y
=
(1 ? ya (1 ? yb ))
(a,b)?E,a?A,b?B
G
, I (y1 , . . . , y|V | ).
P G
P
G
I admits a log-supermodular factorization over G and y I (y) = x I G (x). Similarly, for any
P
P H
graph cover H of G, we have y I (y) = x I H (x). Consequently, by Theorem 4.1, we can
conclude that Z(G) ? ZB (G). Similar observations can be used to show that the Bethe partition
function provides a lower bound on the true partition function for other problems that factor over
pairwise bipartite graphical models (e.g., the antiferromagnetic Ising model on a grid).
5
Conclusions
While the results presented above were discussed in the case that the temperature parameter, T , was
equal to one, they easily extend to any T ? 0 (as exponentiation preserves log-supermodularity in
this case). Hence, all of the bounds discussed above can be extended to the problem of maximizing
a log-supermodular function. In particular, the inequality in Theorem 4.1 shows that the maximum
value of the objective function on any graph cover is achieved by a lift of a maximizing assignment
on the base graph.
This work also suggests a number of directions for future research. Related work on the Bethe
approximation for permanents suggests that conjectures similar to those discussed above can be
made for other classes of functions [13, 14]. While, like the ?four functions? theorem, many of the
above results can be extended to general distributive lattices, understanding when similar results may
hold for non-binary problems may be of interest for graphical models that arise in certain application
areas such as computer vision.
Acknowledgments
The author would like to thank Pascal Vontobel and Nicolas Macris for useful discussions and suggestions during the preparation of this work. This work was supported by EC grant FP7-265496,
?STAMINA?.
8
References
[1] J. S. Yedidia, W. T. Freeman, and Y. Weiss. Constructing free-energy approximations and generalized belief propagation algorithms. Information Theory, IEEE Transactions on, 51(7):2282
? 2312, July 2005.
[2] E. B. Sudderth, M. J. Wainwright, and A. S. Willsky. Loop series and Bethe variational bounds
in attractive graphical models. In Neural Information Processing Systems (NIPS), Vancouver,
BC, Canada, Dec. 2007.
[3] M. Chertkov and V. Y. Chernyak. Loop series for discrete statistical models on graphs. J. Stat.
Mech., 2006.
[4] V. G?omez, J. M. Mooij, and H. J. Kappen. Truncating the loop series expansion for BP. Journal
of Machine Learning Research (JMLR), 2007.
[5] P. O. Vontobel. Counting in graph covers: A combinatorial characterization of the Bethe
entropy function. CoRR, abs/1012.0065, 2010.
[6] P. O. Vontobel and R. Koetter. Graph-cover decoding and finite-length analysis of messagepassing iterative decoding of LDPC codes. CoRR, abs/cs/0512078, 2005.
[7] R. Ahlswede and D. E. Daykin. An inequality for the weights of two families of sets, their
unions and intersections. Probability Theory and Related Fields, 43:183?185, 1978.
[8] N. Alon and J.H. Spencer. The probabilistic method. Wiley-Interscience series in discrete
mathematics and optimization. Wiley, 2000.
[9] R. Aharoni and U. Keich. A generalization of the Ahlswede-Daykin inequality. Discrete
Mathematics, 152(13):1 ? 12, 1996.
[10] Y. Rinott and M. Saks. Correlation inequalities and a conjecture for permanents. Combinatorica, 13:269?277, 1993.
[11] Y. Rinott and M. Saks. On FKG-type and permanental inequalities. Lecture Notes-Monograph
Series, 22:pp. 332?342, 1992.
[12] A. W. Marshall and I. Olkin. Inequalities: Theory of Majorization and its Applications. Academic Press, New York, 1979.
[13] P. O. Vontobel. The Bethe permanent of a non-negative matrix. In Communication, Control,
and Computing (Allerton), 2010 48th Annual Allerton Conference on, pages 341 ?346, Oct.
2010.
[14] L. Gurvits. Unleashing the power of Schrijver?s permanental inequality with the help of the
Bethe approximation. ArXiv e-prints, June 2011.
9
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4,032 | 465 | Practical Issues in Temporal Difference Learning
Gerald Tesauro
IBM Thomas J. Watson Research Center
P. O. Box 704
Yorktown Heights, NY 10598
[email protected]
Abstract
This paper examines whether temporal difference methods for training
connectionist networks, such as Suttons's TO('\) algorithm, can be successfully applied to complex real-world problems. A number of important
practical issues are identified and discussed from a general theoretical perspective. These practical issues are then examined in the context of a case
study in which TO('\) is applied to learning the game of backgammon
from the outcome of self-play. This is apparently the first application of
this algorithm to a complex nontrivial task. It is found that, with zero
knowledge built in, the network is able to learn from scratch to play the
entire game at a fairly strong intermediate level of performance, which is
clearly better than conventional commercial programs, and which in fact
surpasses comparable networks trained on a massive human expert data
set. The hidden units in these network have apparently discovered useful
features, a longstanding goal of computer games research. Furthermore,
when a set of hand-crafted features is added to the input representation,
the resulting networks reach a near-expert level of performance, and have
achieved good results against world-class human play.
1
INTRODUCTION
We consider the prospects for applications of the TO('\) algorithm for delayed reinforcement learning, proposed in (Sutton, 1988), to complex real-world problems.
TO('\) is an algorithm for adjusting the weights in a connectionist network which
259
260
Tesauro
has the following form:
~Wt
= Q(PHI -
t
Pt )
L: At-I:VwPI:
(1)
1:=1
where Pt is the network's output upon observation of input pattern Zt at time t, W
is the vector of weights that parameterizes the network, and VwPI: is the gradient
of network output with respect to weights. Equation 1 basically couples a temporal
difference method for temporal credit assignment with a gradient-descent method
for structural credit assigment; thus it provides a way to adapt supervised learning
procedures such as back-propagation to solve temporal credit assignment problems.
The A parameter interpolates between two limiting cases: A 1 corresponds to an
explicit supervised pairing of each input pattern Zt with the final reward signal,
while A = 0 corresponds to an explicit pairing of Zt with the next prediction PHI.
=
Little theoretical guidance is available for practical uses of this algorithm. For example, one of the most important i88ues in applications of network learning procedures
is the choice of a good representation scheme. However, the existing theoretical
analysis of TD( A) applies primarily to look-up table representations in which the
network has enough adjustable parameters to explicitly store the value of every
p088ible state in the state space. This will clearly be intractable for real-world
problems, and the theoretical results may be completely inappropriate, as they indicate, for example, that every possible state in the state space has to be visited
infinitely many times in order to guarantee convergence.
Another important class of practical i88ues has to do with the nature of the task
being learned, e.g., whether it is noisy or deterministic. In volatile environments
with a high step-to-step variance in expected reward, TD learning is likely to be
difficult. This is because the value of Pt +1 , which is used as a heuristic teacher
signal for Pt , may have nothing to do with the true value of the state Zt. In such
cases it may be necessary to modify TD(A) by including a lookahead process which
averages over the step-to-step noise.
Additional difficulties must also be expected if the task is a combined predictioncontrol task, in which the predictor network is used to make control decisions, as
opposed to a prediction only task. As the network's predictions change, its control
strategies also change, and this changes the target predictions that the network is
trying to learn. In this case, theory does not say whether the combined learning
system would converge at all, and if so, whether it would converge to the optimal
predictor-controller. It might be possible for the system to get stuck in a selfconsistent but non-optimal predictor-controller.
A final set of practical i88ues are algorithmic in nature, such as convergence, scaling,
and the p088ibility of overtraining or overfitting. TD( A) has been proven to converge
only for a linear network and a linearly independent set of input patterns (Sutton,
1988; Dayan, 1992). In the more general case, the algorithm may not converge even
to a locally optimal solution, let alone to a globally optimal solution.
Regarding scaling, no results are available to indicate how the speed and quality
of TD learning will scale with the temporal length of sequences to be learned, the
dimensionality of the input space, the complexity of the task, or the size of the
network. Intuitively it seems likely that the required training time might increase
Practical Issues in Temporal Difference Learning
dramatically with the sequence length. The training time might also scale poorly
with the network or input space dimension, e.g., due to increased sensitivity to
noise in the teacher signal. Another potential problem is that the quality of solution
found by gradient-descent learning relative to the globally optimal solution might
get progressively worse with increasing network size.
Overtraining occurs when continued training of the network results in poorer performance. Overfitting occurs when a larger network does not do as well on a task
as a smaller network. In supervised learning, both of these problems are believed
to be due to a limited data set. In the TD approach, training takes place on-line
using patterns generated de novo, thus one might hope that these problems would
not occur. But both overtraining and overfitting may occur if the error function
minimized during training does not correspond to the performance function that
the user cares about. For example, in a combined prediction-control task, the user
may care only about the quality of control signals, not the absolute accuracy of the
predictions.
2
A CASE STUDY: TD LEARNING OF
BACKGAMMON STRATEGY
We have seen that existing theory provides little indication of how TD(A) will behave
in practical applications. In the absence of theory, we now examine empirically the
above-mentioned issues in the context of a specific application: learning to play the
game of backgammon from the outcome of self-play. This application was selected
because of its complexity and stochastic nature, and because a detailed comparison
can be made with the alternative approach of supervised learning from human
expert examples (Tesauro, 1989j Tesauro, 1990).
It seems reasonable that, by watching two fixed opponents play out a large number
of games, a network could learn by TD methods to predict the expected outcome of
any given board position. However, the experiments presented here study the more
interesting question of whether a network can learn from its own play. The learning
system is set up as follows: the network observes a sequence of board positions
Zl, Zl, ??? , Z J leading to a final reward signal z. In the simplest case, z = 1 if White
wins and z = 0 if Black wins. In this case the network's output P e is an estimate
of White's probability of winning from board position Ze. The sequence of board
positions is generated by setting up an initial configuration, and making plays for
both sides using the network's output as an evaluation function. In other words,
the move which is selected at each time step is the move which maximizes P e when
White is to play and minimizes Pe when Black is to play.
The representation scheme used here contained only a simple encoding of the "raw"
board description (explained in detail in figure 2), and did not utilize any additional
pre-computed "features" relevant to good play. Since the input encoding scheme
contains no built-in knowledge about useful features, and since the network only
observes its own play, we may say that this is a "knowledge-free" approach to
learning backgammon. While it's not clear that this approach can make any progress
beyond a random starting state, it at least provides a baseline for judging other
approaches using various forms of built-in knowledge.
261
262
Tesauro
The approach described above is similar in spirit to Samuel's scheme for learning
checkers from self-play (Samuel, 1959), but in several ways it is a more challenging
learning task. Unlike the raw board description used here, Samuel's board description used a number of hand-crafted features which were designed in consultation
with human checkers experts. The evaluation function learned in Samuel's study
was a linear function of the input variables, whereas multilayer networks learn more
complex nonlinear functions. Finally, Samuel found that it was necessary to give
the learning system at least one fixed intermediate goal, material advantage, as well
as the ultimate goal of the game. The proposed backgammon learning system has
no such intermediate goals.
The networks had a feedforward fully-connected architecture with either no hidden
units, or a single hidden layer with between 10 and 40 hidden units. The learning
algorithm parameters were set, after a certain amount of parameter tuning, at
Q = 0.1 and A = 0.7.
The average sequence length appeared to depend strongly on the quality of play.
With decent play on both sides, the average game length is about 50-60 time steps,
whereas for the random initial networks, games often last several hundred or even
several thousand time steps. This is one of the reasons why the proposed selflearning scheme appeared unlikely to work.
Learning was assessed primarily by testing the networks in actual game play against
Sun Microsystems' Gammontool program. Gammontool is representative of the
playing ability of typical commercial programs, and provides a decent benchmark
for measuring game-playing strength: human beginners can win about 40% of the
time against it, decent intermediate-level humans would win about 60%, and human
experts would win about 75%. (The random initial networks before training win
only about 1%.)
Networks were trained on the entire game, starting from the opening position and
going all the way to the end. This is an admittedly naive approach which was not
expected to yield any useful results other than a reference point for judging more
sensible approaches. However, the rather surprising result was that a significant
amount of learning actually took place. Results are shown in figure 1. For comparison purposes, networks with the same input coding scheme were also trained on
a massive human expert data base of over 15,000 engaged positions, following the
training procedure described in (Tesauro, 1989). These networks were also tested
in game play against Gammontool.
Given the complexity of the task, size of input space and length of typical sequences,
it seems remarkable that the TO nets can learn on their own to play at a level
substantially better than Gammontool. Perhaps even more remarkable is that the
TO nets surpass the EP nets trained on a massive human expert data base: the
best TO net won 66.2% against Gammontool, whereas the best EP net could only
manage 59.4%. This was confirmed in a head-to-head test in which the best TO
net played 10,000 games against the best EP net. The result was 55% to 45%
in favor of the TO net. This confirms that the Gammontool benchmark gives a
reasonably accurate measure of relative game-playing strength, and that the TO
net really is better than the EP net. In fact, the TO net with no features appears
to be as good as Neurogammon 1.0, backgammon champion of the 1989 Computer
Practical Issues in Temporal Difference Learning
c
0
.70
~
(/)
Q.)
E
.65
ctJ
en
......
.60
0
c
0
.1'""'1
+J
U
ctJ
'-
.55
.50
l.J....
.45
0
10
20
40
Number of hidden units
Figure 1: Plot of game performance against Gammontool vs. number of hidden
units for networks trained using TD learning from self-play (TD), and supervised
training on human expert preferences (EP). Each data point represents the result
of a 10,000 game test, and should be accurate to within one percentage point.
Olympiad, which does have features, and which wins 65% against Gammontool. A
10,000 game test of the best TD net against Neurogammon 1.0 yielded statistical
equality: 50% for the TD net and 50% for Neurogammon.
Ii is also of interest to examine the weights learned by the TD nets, shown in fig-
ure 2. One can see a great deal of spatially organized structure in the pattern of
weights, and some of this structure can be interpreted as useful features by a knowledgable backgammon player. For example, the first hidden unit in figure 2 appears
to be a race-oriented feature detector, while the second hidden unit appears to be
an attack-oriented feature detector. The TD net has apparently solved the longstanding "feature discovery" problem, which was recently stated in (Frey, 1986) as
follows: "Samuel was disappointed in his inability to develop a mechanical strategy
for defining features. He thought that true machine learning should include the
discovery and definition of features. Unfortunately, no one has found a practical
way to do this even though more than two and a half decades have passed."
The training times needed to reach the levels of performance shown in figure 1 were
on the order of 50,000 training games for the networks with 0 and 10 hidden units,
100,000 games for the 20-hidden unit net, and 200,000 games for the 40-hidden
unit net. Since the number of training games appears to scale roughly linearly with
the number of weights in the network, and the CPU simulation time per game on
a serial computer also scales linearly with the number of weights, the total CPU
time thus scales quadratically with the number of weights: on an IBM RS/6000
workstation, the smallest network was trained in several hours, while the largest
net required two weeks of simulation time.
In qualitative terms, the TD nets have developed a style of play emphasizing run-
263
264
Tesauro
24
23
22
21
20
19
18
17
16
15
1-1
13
12
11
10
9
a
7
e
S
4
3
2
1
2
3
Black pipces
4
234
White pieces
2
3
Black pi('ces
4
4
White
pit"t:~~
Figure 2: Weights from the input units to two hidden units in the best TO net.
Black squares represent negative weightsi white squares represent positive weightsi
size indicates magnitude of weights. Rows represent spatial locations 1-24, top row
represents no. of barmen, men off, and side to move. Columns represent number
of Black and White men as indicated. The first hidden unit has two noteworthy
features: a linearly increasing pattern of negative weights for Black blots and Black
points, and a negative weighting of White men off and a positive weighting of Black
men off. These contribute to an estimate of Black's probability of winning based
on his racing lead. The second hidden unit has the following noteworthy features:
strong positive weights for Black home board points, strong positive weights for
White men on bar, positive weights for White blots, and negative weights for White
points in Black's home board. These factors all contribute to the probability of a
successful Black attacking strategy.
ning and tactical play, whereas the EP nets favor more quiescent positional play
emphasizing blocking rather than racing. This is more in line with human expert
play, but it often leads to complex prime vs. prime and back-game situations that
are hard for the network to evaluate properly. This suggests one possible advantage of the TO approach over the EP approach: by imitating an expert teacher,
the learner may get itself into situations that it can't handle. With the alternative
approach of learning from experience, the learner may not reproduce the expert
strategies, but at least it will learn to handle whatever situations are brought about
by its own strategy.
It's also interesting that TO net plays well in early phases of play, whereas its play
becomes worse in the late phases of the game. This is contrary to the intuitive notion
that states far from the end of the sequence should be harder to learn than states
near the end. Apparently the inductive bias due to the representation scheme and
network architecture is more important than temporal distance to the final outcome.
Practical Issues in Temporal Difference Learning
3
TD LEARNING WITH BUILT-IN FEATURES
We have seen that TD networks with no built-in knowledge are able to reach computer championship levels of performance for this particular application. It is then
natural to wonder whether even greater levels of performance might be obtained
by adding hand-crafted features to the input representation. In a separate series of
experiments, TD nets containing all of Neurogammon's features were trained from
self-playas described in the previous section. Once again it was found that the performance improved monotonically by adding more hidden units to the network, and
training for longer training times. The best performance was obtained with a network containing 80 hidden units and over 25,000 weights. This network was trained
for over 300,000 training games, taking over a month of CPU time on an RS/6000
workstation. The resulting level of performance was 73% against Gammontool and
nearly 60% against Neurogammon. This is very close to a human expert level of
performance, and is the strongest program ever seen by this author.
The level of play of this network was also tested in an all-day match against twotime World Champion Bill Robertie, one of the world's best backgammon players.
At the end of the match, a total of 31 games had been played, of which Robertie won
18 and the TD net 13. This showed that the TD net was capable of a respectable
showing against world-class human play. In fact, Robertie thinks that the network's
level of play is equal to the average good human tournament player.
It's interesting to speculate about how far this approach can be carried. Further
substantial improvements might be obtained by training much larger networks on
a supercomputer or special-purpose hardware. On such a machine one could also
search beyond one ply, and there is some evidence that small-to-moderate improvements could be obtained by running the network in two-ply search mode. Finally,
the features in Berliner's BKG program (Berliner, 1980) or in some of the top commercial programs are probably more sophisticated than Neurogammon's relatively
simple features, and hence might give better performance. The combination of all
three improvements (bigger nets, two-ply search, better features) could conceivably
result in a network capable of playing at world-class level.
4
CONCLUSIONS
The experiments in this paper were designed to test whether TD(.\) could be successfully applied to complex, stochastic, nonlinear, real-world prediction and control
problems. This cannot be addressed within current theory because it cannot answer
such basic questions as whether the algorithm converges or how it would scale.
Given the lack of any theoretical guarantees, the results of these experiments are
encouraging. Empirically the algorithm always converges to at least a local
minimum, and the quality of solution generally improves with increasing network
size. Furthermore, the scaling of training time with the length of input sequences,
and with the size and complexity of the task, does not appear to be a serious problem. This was ascertained through studies of simplified endgame situations, which
took about as many training games to learn as the full-game situation (Tesauro,
1992). Finally, the network's move selection ability is better than one would expect based on its prediction accuracy. The absolute prediction accuracy is only at
ve~y
265
266
Tesauro
the 10% level, whereas the difference in expected outcome between optimal and
non-optimal moves is usually at the level of 1% or less.
The most encouraging finding, however, is a clear demonstration that TO nets with
zero built-in knowledge can outperform identical networks trained on a massive
data base of expert examples. It would be nice to understand exactly how this is
possible. The ability of TO nets to discover features on their own may also be of
some general importance in computer games research, and thus worthy of further
analysis.
Beyond this particular application area, however, the larger and more important
issue is whether learning from experience can be useful and practical for more
general complex problems. The quality of results obtained in this study indicates
that the approach may work well in practice. There may also be some intrinsic
advantages over supervised training on a fixed data set. At the very least, for tasks
in which the exemplars are hand-labeled by humans, it eliminates the laborious
and time-consuming process of labeling the data. Furthermore, the learning system
would not be fundamentally limited by the quantity of labeled data, or by errors
in the labeling process. Finally, preserving the intrinsic temporal nature of the
task, and informing the system of the consequences of its actions, may convey
important information about the task which is not necessarily contained in the
labeled exemplars. More theoretical and empirical work will be needed to establish
the relative advantages and disadvantages of the two approachesi this could result
in the development of hybrid algorithms combining the best of both approaches.
References
H. Berliner, "Computer backgammon." Sci. Am. 243:1, 64-72 (1980).
P. Dayan, "Temporal differences: TO{>') for general >.." Machine Learning, in press
(1992).
P. W. Frey, "Algorithmic strategies for improving the performance of game playing
programs." In: O. Farmer et a1. (Eds.), Evolution, Game6 and Learning. Amsterdam: North Holland (1986).
A. Samuel, "Some studies in machine learning using the game of checkers." IBM
J. of Re6earch and Development 3, 210-229 (1959).
R. S. Sutton, "Learning to predict by the methods of temporal differences. " Machine
Learning 3, 9-44 (1988).
G. Tesauro and T. J. Sejnowski, "A parallel network that learns to play backgammon." Artificial Intelligence 39, 357-390 (1989).
G. Tesauro, "Connectionist learning of expert preferences by comparison training."
In D. Touretzky (Ed.), Advance6 in Neural Information Proce66ing 1, 99-106 (1989).
G. Tesauro, "Neurogammon: a neural network backgammon program." IJCNN
Proceeding6 III, 33-39 (1990).
G. Tesauro, "Practical issues in temporal difference learning." Machine Learning,
in press (1992).
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4,033 | 4,650 | Nonparanormal Belief Propagation (NPNBP)
Gal Elidan
Department of Statistics
Hebrew University
[email protected]
Cobi Cario
School of Computer Science and Engineering
Hebrew University
[email protected]
Abstract
The empirical success of the belief propagation approximate inference algorithm
has inspired numerous theoretical and algorithmic advances. Yet, for continuous
non-Gaussian domains performing belief propagation remains a challenging task:
recent innovations such as nonparametric or kernel belief propagation, while useful, come with a substantial computational cost and offer little theoretical guarantees, even for tree structured models. In this work we present Nonparanormal BP
for performing efficient inference on distributions parameterized by a Gaussian
copulas network and any univariate marginals. For tree structured networks, our
approach is guaranteed to be exact for this powerful class of non-Gaussian models. Importantly, the method is as efficient as standard Gaussian BP, and its convergence properties do not depend on the complexity of the univariate marginals,
even when a nonparametric representation is used.
1
Introduction
Probabilistic graphical models [Pearl, 1988] are widely use to model and reason about phenomena
in a variety of domains such as medical diagnosis, communication, machine vision and bioinformatics. The usefulness of such models in complex domains, where exact computations are infeasible,
relies on our ability to perform efficient and reasonably accurate inference of marginal and conditional probabilities. Perhaps the most popular approximate inference algoritm for graphical models
is belief propagation (BP) [Pearl, 1988]. Guaranteed to be exact for trees, it is the surprising performance of the method when applied to general graphs (e.g., [McEliece et al., 1998, Murphy and
Weiss, 1999]) that has inspired numerous works ranging from attempts to shed theoretical light
on propagation-based algorithms (e.g., [Weiss and Freeman, 2001, Heskes, 2004, Mooij and Kappen, 2005]) to a wide range of algorithmic variants and generalizations (e.g., [Yedidia et al., 2001,
Wiegerinck and Heskes, 2003, Globerson and Jaakkola, 2007]).
In most works, the variables are either discrete or the distribution is assumed to be Gaussian [Weiss
and Freeman, 2001]. However, many continuous real-world phenomenon are far from Gaussian,
and can have a complex multi-modal structure. This has inspired several innovative and practically
useful methods specifically aimed at the continuous non-Gaussian case such as expectation propagation [Minka, 2001], particle BP [Ihler and McAllester, 2009], nonparametric BP [Sudderth et al.,
2010b], and kernel BP [Song et al., 2011]. Since these works are aimed at general unconstrained
distributions, they all come at a substantial computational price. Further, little can be said a-priori
about their expected performance even in tree structured models. Naturally, we would like an inference algorithm that is as general as possible while being as computationally convenient as simple
Gaussian BP [Weiss and Freeman, 2001].
In this work we present Nonparanormal BP (NPNBP), an inference method that strikes a balance
between these competing desiderata. In terms of generality, we focus on the flexible class of Copula
Bayesian Networks (CBNs) [Elidan, 2010] that are defined via local Gaussian copula functions and
any univariate densities (possible nonparametric). Utilizing the power of the copula framework
[Nelsen, 2007], these models can capture complex multi-modal and heavy-tailed phenomena.
1
Figure 1: Samples from the
bivariate Gaussian copula with
correlation ? = 0.25.
(left) with unit variance Gaussian and Gamma marginals;
(right) with a mixture of Gaussian and exponential marginals.
Algorithmically, our approach enjoys the benefits of Gaussian BP (GaBP). First, it is guaranteed to
converge and return exact results on tree structured models, regardless of the form of the univariate
densities. Second, it is computationally comparable to performing GaBP on a graph with the same
structure. Third, its convergence properties on general graphs are similar to that of GaBP and, quite
remarkably, do not depend on the complexity of the univariate marginals.
2
Background
In this section we provide a brief background on copulas in general, the Gaussian copula in particular, and the Copula Bayesian Network model of Elidan [2010].
2.1
The Gaussian Copula
A copula function [Sklar, 1959] links marginal distributions to form a multivariate one. Formally:
Definition 2.1: Let U1 , . . . , Un be real random variables marginally uniformly distributed on [0, 1].
A copula function C : [0, 1]n ? [0, 1] is a joint distribution
C? (u1 , . . . , un ) = P (U1 ? u1 , . . . , Un ? un ),
where ? are the parameters of the copula function.
Now consider an arbitrary set X = {X1 , . . . Xn } of real-valued random variables (typically not
marginally uniformly distributed). Sklar?s seminal theorem states that for any joint distribution
FX (x), there exists a copula function C such that FX (x) = C(FX1 (x1 ), . . . , FXn (xn )). When the
univariate marginals are continuous, C is uniquely defined.
The constructive converse, which is of central interest from a modeling perspective, is also true.
Since Ui ? Fi is itself a random variable that is always uniformly distributed in [0, 1], any copula
function taking any marginal distributions {Ui } as its arguments, defines a valid joint distribution
with marginals {Ui }. Thus, copulas are ?distribution generating? functions that allow us to separate
the choice of the univariate marginals and that of the dependence structure, encoded in the copula
function C. Importantly, this flexibility often results in a construction that is beneficial in practice.
Definition 2.2: The Gaussian copula distribution is defined by:
?1
?1
C? (u1 , . . . , un ) = ?? ? (u1 ), . . . , ? (un )) ,
(1)
?1
where ? is the inverse standard normal distribution, and ?? is a zero mean normal distribution
with correlation matrix ?.
Example 2.3: The standard Gaussian distribution is mathematically convenient but limited due
to its unimodal form and tail behavior. However, the Gaussian copula can give rise to complex
varied distribution and offers great flexibility. As an example, Figure 1 shows two bivariate distributions that are constructed using the Gaussian copula and two different sets of univariate marginals.
Generally, any univariate marginal, both parametric and nonparametric can be used.
Let ?? (x) denote the multivariate normal density with mean zero and covariance ?, and let ?(x)
denote the univariate standard normal density. Using the derivative chain rule and the derivative
2
inverse function theorem, the Gaussian copula density c(u1 , . . . , un ) =
c(u1 , . . . , un ) = ??
? n C? (u1 ,...,un )
?U1 ,...?Un
is
?1
?1
?1
?
?
(u
),
.
.
.
,
?
(u
)
Y
?
1
n
?1
?1
?? (ui )
Q
? (u1 ), . . . , ? (un )
.
=
?1
?U
i
i ?(? (ui ))
i
For a distribution defined by a Gaussian copula FX (x1 , . . . , xn ) = C? (F1 (x1 ), . . . , Fn (xn )), using
?Ui /?Xi = fi , we have
fX (x1 , . . . , xn ) =
?1
? n C? (F1 (x1 ), . . . , Fn (xn ))
?? (?
x ,...,x
?n ) Y
Q1
fi (xi ),
=
xi )
?X1 , . . . , ?Xn
i ?(?
i
(2)
?1
where x
?i ? ? (ui ) ? ? (Fi (xi )). We will use this compact notation in the rest of the paper.
2.2
Copula Bayesian Networks
Let G be a directed acyclic graph (DAG) whose nodes correspond to the random variables X =
{X1 , . . . , Xn }, and let Pai = {Pai1 , . . . , Paiki } be the parents of Xi in G. As for standard BNs, G
encodes the independence statements I(G) = {(Xi ? NonDescendantsi | Pai )}, where ? denotes
the independence relationship, and NonDescendantsi are nodes that are not descendants of Xi in G.
Definition 2.4: A Copula Bayesian Network (CBN) is a triplet C = (G, ?C , ?f ) that defines fX (x).
G encodes the independencies assumed to hold in fX (x). ?C is a set of local copula functions
Ci (ui , upai1 , . . . , upaiki ) that are associated with the nodes of G that have at least one parent. In
addition, ?f is the set of parameters representing the marginal densities fi (xi ) (and distributions
ui ? Fi (xi )). The joint density fX (x) then takes the form
fX (x) =
n
Y
ci (ui , upai1 , . . . , upaiki )
i=1
? K Ci (1,upai1 ,...,upaik )
fi (xi ) ?
i
?Upai1 ...?Upaik
n
Y
Rci (ui , upai1 , . . . , upaiki )fi (xi )
(3)
i=1
i
When Xi has no parents in G, Rci (?) ? 1.
Note that Rci (?)fi (xi ) is always a valid conditional density f (xi | pai ), and can be easily computed.
In particular, when the copula density c(?) in the numerator has an explicit form, so does Rci (?).
Elidan
[2010] showed that a CBN defines a valid joint density. When the model is tree-structured,
Q
R
(ui , upai1 , . . . , upaiki ) defines a valid copula so that the univariate marginals of the conc
i
i
structed density are fi (xi ). More generally, the marginals may be skewed. though in practice only
slightly so. In this case the CBN model can be viewed as striking a balance between the fixed
marginals and the unconstrained maximum likelihood objectives. Practically, the model leads to
substantial generalization advantages (see Elidan [2010] for more details).
3
Nonparanormal Belief Propagation
As exemplified in Figure 1, the Gaussian copula can give rise to complex multi-modal joint distributions. When local Gaussian copulas are combined in a high-dimensional Gaussian Copula BN
(GCBN), expressiveness is even greater. Yet, as we show in this section, tractable inference in this
highly non-Gaussian model is possible, regardless of the form of the univariate marginals.
3.1
Inference for a Single Gaussian Copula
We start by showing how inference can be carried out in closed form for a single Gaussian copula. While all that is involved is a simple change of variables, the details are instructive. Let
fX (x1 , . . . , xn ) be a density parameterized by a Gaussian copula. We start with the task of computing the multivariate marginal over a subset of variables Y ? X. For convenience and without loss
3
of generality, we assume that Y = {X1 , . . . , Xk } with k < n. From Eq. (2), we have
Z
fX1 ,...,XK (x1 , . . . , xk ) =
fX (x1 , . . . , xn )dxk+1 . . . dxn
Rn?k
k
Y
Z
n
Y
?1
?1
fi (xi )
fi (xi )
?? ? (F1 (x1 )), . . . , ? (Fn (xn ))
dxk+1 . . . dxn .
=
?(?
xi )
?(??1 (Fi (xi )))
i=1
i=k+1
?Ui
Changing the integral variables to Ui and using fi = ?X
so that fi (xi )dxi = dui , we have
i
"
#
Z
k
Y
?? ??1 (u1 ), . . . , ??1 (un )
fi (xi )
Qn
fX1 ,...,XK (x1 , . . . , xk ) =
duk+1 . . . dun .
?1 (u ))
?(?
xi ) [0,1]n?k
i
i=k+1 ?(?
i=1
?1
?1
? i /?Ui = ?(?
Changing variables once again to x
?i = ? (ui ), and using ? X
xi ) , we have
Z
k
Y
fi (xi )
fX1 ,...,XK (x1 , . . . , xk ) =
?? (?
x1 , . . . , x
?n ) d?
xk+1 . . . d?
xn .
?(?
xi ) Rn?k
i=1
The integral on the right hand side is now a standard marginalization of a multivariate Gaussian
(over x
?i ?s) and can be carried out in closed form.
Computation of densities conditioned on evidence Z = z can also be easily carried out. Letting
W = X \ {Z ? Y} denote non query or evidence variables, and plugging in the above, we have:
R
R
Q
?
f (x)dw
?? (?
x1 , . . . , x
? n ) dw
fi (xi )
RR
RR
fY|Z (y | z) =
= i?Y
.
? y
?(?
xi )
f (x)dwdy
?? (?
x1 , . . . , x
?n ) dwd?
The conditional density is now easy to compute since a ratio of normal distributions is also normal.
The final answer, of course, does involve fi (xi ). This is not only unavoidable but in fact desirable
since we would like to retain the complexity of the desired posterior.
3.2
Tractability of Inference in a Gaussian CBNs
We are now ready to consider inference in a Gaussian CBN (GCBN). In this case, the joint density
of Eq. (3) takes, after cancellation of terms, the following form:
fX (x1 , . . . , xn ) =
Y fi (xi ) Y ??i (?
xi , x
?pai1 , . . . , x
?paiki )
i
?(?
xi )
??? (?
xpai1 , . . . , x
?paiki )
i
,
i
where ??
i is used to denote the i?th local covariance matrix excluding the i?th row and column.
When Xi has no parents, the ratio reduces to ?(?
xi ). When the graph is tree structured, this density
is also a copula and its marginals are fi (xi ). In this case, the same change of variables as above
results in
Y ??i (?
xi , x
?pai1 , . . . , x
?paiki )
fXe (?
x1 , . . . , x
?n ) =
.
??? (?
xpai1 , . . . , x
?paiki )
i
i
Since a ratio of Gaussians is also a Gaussian, the entire density is Gaussian in x
?i space, and computation of any marginal fY
(?
y
)
is
easy.
The
required
marginal
in
x
space
is
then
recovered using
?
i
Y fi (xi )
fY (y) = fY
y)
(4)
? (?
?(?
xi )
i?Y
which essentially summarizes the detailed derivation of the previous section.
When we consider a non-tree structured CBN model, as noted in Section 2.2, the marginals may not
equal fi (xi ), and the above simplification is not applicable. However, for the Gaussian case, it is
always possible to estimate the local copulas in a topological order so that the univariate marginals
are equal to fi (xi ) (the model in this case is equivalent to the distribution-free continuous Bayesian
belief net model [Kurowicka and Cooke, 2005]). It follows that, for any structure,
Corollary 3.1: The complexity of inference in a Gaussian CBN model is the same as that of inference
in a multivariate Gaussian model of the same structure.
4
Algorithm 1: Nonparanormal Belief Propagation (NPNBP) for general CBNs.
Input: {fk (xk )} for all i, ?i for all nodes with parents. Output: belief bS (xS ) for each cluster S.
CG ? a valid cluster graph over the following potentials for all nodes i in the graph
? ??i (?
xi , x
?pai1 , . . . , x
?paiki )
? 1/??? (?
xpai1 , . . . , x
?paiki )
i
foreach cluster S in CG
bG (?
xS ) ? GaBP belief over cluster S.
foreach cluster S in CG
Q
bS (xS ) = bG (?
xS ) i?S
// use black-box GaBP in x
?i space
// change to xi space
fi (xi )
?(?
xi )
While mathematically this conclusion is quite straightforward, the implications are significant. A
GCBN model is the only general purpose non-Gaussian continuous graphical model for which exact
inference is tractable. At the same time, as is demonstrated in our experimental evaluation, the
model is able to capture complex distributions well both qualitatively and quantitatively.
A final note is worthwhile regarding the (possibly conditional) marginal density. As can be expected,
the result of Eq. (4) includes fi (xi ) terms for all variables that have not been marginalized out. As
noted, this is indeed desirable as we would like to preserve the complexity of the density in the
marginal computation. The marginal term, however, is now in low dimension so that quantities of
interest (e.g., expectation) can be readily computed using naive grid-based evaluation or, if needed,
using more sophisticated sampling schemes (see, for example, [Robert and Cassella, 2005]).
3.3
Belief Propagation for Gaussian CBNs
Given the above observations, performing inference in a Gaussian CBN (GCBN) appears to be a
solved problem. However, inference in large-scale models can be problematic even in the Gaussian
case. First, the large joint covariance matrix may be ill conditioned and inverting it may not be
possible. Second, matrix inversion can be slow when dealing with domains of sufficient dimension.
A possible alternative is to consider the popular belief propagation algorithm [Pearl, 1988]. For a
tree structured model represented as a product of singleton ?i and pairwise ?ij factors, the method
relies on the recursive computation of ?messages?
Z
Q
mi?j (xj ) ? ? [?ij (xi , xj )?i (xi ) k?N (i)\j mk?i (xi )]dxi ,
where ? is a normalization factor and N (i) are the indices of the neighbor nodes of Xi .
In the case of a GCBN model, performing belief propagation may seem difficult since ?i (xi ) ?
fi (xi ) can have a complex form. However, the change of variables used in the previous section
applies here as well. That is, one can perform inference in x
?i space using standard Gaussian BP
(GaBP) [Weiss and Freeman, 2001], and then perform the needed change of variables. In fact, this
is true regardless of the structure of the graph so that loopy GaBP can also be used to perform approximate computations for a general GCBN model in x
?i space. The approach is summarized in
Algorithm 1, where we assume access to a black-box GaBP procedure and a cluster graph construction algorithm. In our experiments we simply use a Bethe approximation construction (see [Koller
and Friedman, 2009] for details on BP, GaBP and the cluster graph construction).
Generally, little can be said about the convergence of loopy BP or its variants, particularly for nonGaussian domains. Appealingly, the form of our NPNBP algorithm implies that its convergence can
be phrased in terms of standard Gaussian BP convergence. In particular:
Q
? Observation 1: NPNBP converges whenever GaBP converges for the model defined by i Rci .
? Observation 2: Convergence of NPNBP depends only on the covariance matrices ?i that parameterize the local copula and does not depend on the univariate form.
It follows that convergence conditions identified for GaBP [Rusmevichientong and Roy, 2000, Weiss
and Freeman, 2001, Malioutov et al., 2006] carry over to NPNBP for CBN models.
5
Figure 2: Exact vs. Nonparametric BP marginals for the GCBN model learned from the wine quality
dataset. Shown are the marginal densities for the first four variables.
4
Experimental Evaluation
We now consider the merit of using our NPNBP algorithm for performing inference in a a Gaussian
CBN (GCBN) model. We learned a tree structured GCBN using a standard Chow-Liu approach
[Chow and Liu, 1968], and a model with up to two parents for each variable using standard greedy
structure search. In both cases we use the Bayesian Information Criterion (BIC) [Schwarz, 1978]
to guide the structure learning algorithm. For the univariate densities, we use a standard Gaussian
kernel density estimator (see, for example, [Bowman and Azzalini, 1997]). Using an identical procedure, we learn a linear Gaussian BN baseline where Xi ? N (?pai , ?i ) so that each variable Xi
is normally distributed around a linear combination of its parents Pai (see [Koller and Friedman,
2009] for details on this standard approach to structure learning).
For the GCBN model, we also compare to Nonparametric BP (NBP) [Sudderth et al., 2010a] using
D. Bickson?s code [Bickson, 2008] and A. Ihlers KDE Matlab package (http://www.ics.uci.edu/ ihler/code/kde.html), which relies on a mixture of Gaussians for message representation. In this case,
since our univariate densities are constructed using Gaussian kernels, there is no approximation
in the NBP representation and all approximations are due to message computations. To carry out
message products, we tried all 7 sampling-based methods available in the KDE package. In the experiments below we use only the multiresolution sequential Gibbs sampling method since all other
approaches resulted in numerical overflows even for small domains.
4.1
Qualitative Assessment
We start with a small domain where the qualitative nature of the inferred marginals is easily explored,
and consider performance and running time in more substantial domains in the next section. We
use the wine quality data set from the UCI repository which includes 1599 measurements of 11
physiochemical properties and a quality variable of red ?Vinho Verde? [Cortez et al., 2009].
We first examine a tree structured GCBN model where our NPNBP method allows us to perform
exact marginal computations. Figure 2 compares the first four marginals to the ones computed by the
NBP method. As can be clearly seen, although the NBP marginals are not nonsensical, they are far
from accurate (results for the other marginals in the domain are similar). Quantitatively, each NBP
marginal is 0.5 to 1.5 bits/instance less accurate than the exact ones. Thus, the accuracy of NPNBP
in this case is approximately twice that of NBP per variable, amounting to a substantial per sample
advantage. We also note that NBP was approximately an order of magnitude slower than NPNBP in
this domain. In the larger domains considered in the next section, NBP proved impractical.
Figure 3 demonstrates the quality of the bivariate marginals inferred by our NPNBP method relative
to the ones of a linear Gaussian BN model where inference can also be carried out efficiently. The
middle panel shows a Gaussian distribution constructed only over the two variables and is thus an
upper bound on the quality that we can expect from a linear Gaussian BN. Clearly, the Gaussian
representation is not sufficiently flexible to reasonably capture the distribution of the true samples
(left panel). In contrast, the bivariate marginals computed by our algorithm (right panel) demonstrate
the power of working with a copula-based construction and an effective inference procedure: in both
cases the inferred marginals capture the non-Gaussian distributions quite accurately. Results were
qualitatively similar for all other variable pairs (except for the few cases that are approximately
Gaussian in the original feature space and for which all models are equally beneficial).
6
Density vs. Alcohol
Free vs. Total Sulfur
(a) true samples
(b) optimal Gaussian
(c) CBN marginal
Figure 3: The bivariate density for two pairs of variables in a tree structured GCBN model learned
from the wine quality dataset. (a) empirical samples; (b) maximum likelihood Gaussian density; (c)
exact GCBN marginal computed using our NPNBP algorithm.
In Figure 4 we repeat the comparison for another pair of variables in a non-tree GCBN (as before,
results were qualitatively similar for all pairs of variables). In this setting, the bivariate marginal
computed by our algorithm (d) is approximate and we also compare to the exact marginal (c). As
in the case of the tree-structured model, the GCBN model captures the true density quite accurately,
even for this multi-modal example. NPNBP dampens some of this accuracy and results in marginal
densities that have the correct overall structure but with a reduced variance. This is not surprising
since it is well known that GaBP leads to reduced variances [Weiss and Freeman, 2001]. Nevertheless, the approximate result of NPNBP is clearly better than the exact Gaussian model, which
assigns very low probability to regions of high density (along the main vertical axis of the density).
Finally, Figure 5(left) shows the NPNBP vs. the exact expectations. As can be seen, the inferred
values are quite accurate and it is plausible that the differences are due to numerical round-offs.
Thus, it is possible that, similarly to the case of standard GaBP [Weiss and Freeman, 2001], the
inferred expectations are theoretically exact. The proof for the GaBP case, however, does not carry
over to the CBN setting and shedding theoretical light on this issue remains a future challenge.
4.2
Quantitative Assessment
We now consider several substantially larger domains with 100 to almost 2000 variables. For each
domain we learn a tree structured GCBN, and justify the need for the expressive copula-based model
by reporting its average generalization advantage in terms of log-loss/instance over a standard linear
Gaussian model. We justify the use of NPNBP for performing inference by comparing the running
time of NPNBP to exact computations carried out using matrix inversion. For all datasets, we
performed 10-fold cross-validation and report average results. We use the following datasets:
? Crime (UCI repository). 100 variables relating to crime ranging from household size to fraction
of children born outside of a marriage, for 1994 communities across the U.S.
? SP500. Daily changes of value of the 500 stocks comprising the Standard and Poor?s index (S&P
500) over a period of one year.
? Gene. A compendium of gene expression experiments used in [Marion et al., 2004]. We chose
genes that have only 1, 2, and 3 missing values and only use full observations. This resulted in
datasets with 765, 1400, and 1945 variables (genes), and 1088, 956, and 876 samples, respectively.
For the 100 variable Crime domain, average test advantage of the GCBN model over the linear
Gaussian one was 0.39 bits/instance per variable (as in [Elidan, 2010]). For the 765 variable Gene
expression domain the advantage was around 0.1 bits/instance/variable (results were similar for the
7
Sugar level vs. Density
(a) true samples
(b) optimal Gaussian
(c) exact CBN marginal
(d) inferred marginal
Figure 4: The bivariate density for a pair of variables in a non-tree GCBN model learned from the
wine quality dataset. (a) empirical samples; (b) maximum likelihood Gaussian density; (c) exact
CBN marginal; (d) marginal density computed by our NPNBP algorithm.
Figure 5: (left) exact vs.
NPNBP expected values.
(right) speedup relative to
matrix inversion for a tree
structured GCBN model.
765,1400,1945 correspond
to the three different datasets
extracted from the gene
expression compendium.
other gene expression datasets). In both cases, the differences are dramatic and each instance is
many orders of magnitude more likely given a GCBN model. For the SP500 domain, evaluation
of the linear Gaussian model resulted in numerical overflows (due to the scarcity of the training
data), and the advantage of he GCBN cannot be quantified. These generalization advantages make
it obvious that we would like to perform efficient inference in a GCBN model.
As discussed, a GCBN model is itself tractable in that inference can be carried out by first constructing the inverse covariance matrix over all variables and then inverting it so as to facilitate
marginalization. Thus, using our NPNBP algorithm can only be justified practically. Figure 5(right)
shows the speedup of NPNBP relative to inference based on matrix inversion for the different domains. Although NPNBP is somewhat slower for the small domains (in which inference is carried
out in less than a second), the speedup of NPNBP reaches an order of magnitude for the larger gene
expression domain. Appealingly, the advantage of NPNBP grows with the domain size due to the
growth in complexity of matrix inversion. Finally, we note that we used a Matlab implementation
where matrix inversion is highly optimized so that the gains reported are quite conservative.
5
Summary
We presented Nonparanormal Belief Propagation (NPNBP), a propagation-based algorithm for performing highly efficient inference in a powerful class of graphical models that are based on the
Gaussian copula. To our knowledge, ours is the first inference method for an expressive continuous
non-Gaussian representation that, like ordinary GaBP, is both highly efficient and provably correct
for tree structured models. Appealingly, the efficiency and convergence properties of our method do
not depend on the choice of univariate marginals, even when a nonparametric representation is used.
The Gaussian copula is a powerful model widely used to capture complex phenomenon in fields
ranging from mainstream economics (e.g., Embrechts et al. [2003]) to flood analysis [Zhang and
Singh, 2007]. Recent probabilistic graphical models that build on the Gaussian copula open the
door for new high-dimensional non-Gaussian applications [Kirshner, 2007, Liu et al., 2010, Elidan,
2010, Wilson and Ghahramani, 2010]. Our method offers the inference tools to make this practical.
8
Acknowledgements
G. Elidan and C. Cario were supported in part by an ISF center of research grant. G. Elidan was also supported
by a Google grant. Many thanks to O. Meshi and A. Globerson for their comments on an earlier draft.
References
D. Bickson. Gaussian Belief Propagation: Theory and Application. PhD thesis, The Hebrew University of
Jerusalem, Jerusalem, Israel, 2008.
A. Bowman and A. Azzalini. Applied Smoothing Techniques for Data Analysis. Oxford Press, 1997.
C. K. Chow and C. N. Liu. Approximating discrete probability distributions with dependence trees. IEEE
Trans. on Info. Theory, 14:462?467, 1968.
P. Cortez, A. Cerdeira, F. Almeida, T. Matos, and J. Reis. Modeling wine preferences by data mining from
physicochemical properties. Decision Support Systems, 47(4):547?553, 2009.
G. Elidan. Copula bayesian networks. In Neural Information Processing Systems (NIPS), 2010.
P. Embrechts, F. Lindskog, and A. McNeil. Modeling dependence with copulas and applications to risk management. Handbook of Heavy Tailed Distributions in Finance, 2003.
A. Globerson and T. Jaakkola. Fixing max-product: Convergent message passing algorithms for map lprelaxations. In Neural Information Processing Systems (NIPS), 2007.
T. Heskes. On the uniqueness of loopy belief propagation fixed points. Neural Comp, 16:2379?2413, 2004.
A. Ihler and D. McAllester. Particle belief propagation. In Conf on AI and Statistics (AISTATS), 2009.
S. Kirshner. Learning with tree-averaged densities and distributions. In Neural Info Proc Systems (NIPS), 2007.
D. Koller and N. Friedman. Probabilistic Graphical Models: Principles and Techniques. The MIT Press, 2009.
D. Kurowicka and R. M. Cooke. Distribution-free continuous bayesian belief nets. In Selected papers based
on the presentation at the international conference on mathematical methods in reliability (MMR), 2005.
H. Liu, J. Lafferty, and L. Wasserman. The nonparanormal: Semiparametric estimation of high dimensional
undirected graphs. Journal of Machine Learning Research, 2010.
D. Malioutov, J. Johnson, and A. Willsky. Walk-sums and belief propagation in gaussian graphical models.
Journal of Machine Learning Research, 7:2031?2064, 2006.
R. Marion, A. Regev, E. Segal, Y. Barash, D. Koller, N. Friedman, and E. O?Shea. Sfp1 is a stress- and nutrientsensitive regulator of ribosomal protein gene expression. Proc Natl Acad Sci U S A, 101(40):14315?22, 2004.
R. McEliece, D. McKay, and J. Cheng. Turbo decoding as an instance of pearl?s belief propagation algorithm.
IEEE Journal on Selected Areas in Communication, 16:140?152, 1998.
T. P. Minka. Expectation propagation for approximate Bayesian inference. In Proc. Conference on Uncertainty
in Artificial Intelligence (UAI), pages 362?369, 2001.
J. Mooij and B. Kappen. Sufficient conditions for convergence of loopy belief propagation. In Proc. Conference
on Uncertainty in Artificial Intelligence (UAI), 2005.
K. Murphy and Y. Weiss. Loopy belief propagation for approximate inference: An empirical study. In Proc.
Conference on Uncertainty in Artificial Intelligence (UAI), pages 467?475, 1999.
R. Nelsen. An Introduction to Copulas. Springer, 2007.
J. Pearl. Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann, 1988.
C. P. Robert and G. Casella. Monte Carlo Statistical Methods (Springer Texts in Statistics. Springer, 2005.
P. Rusmevichientong and B. Van Roy. An analysis of belief propagation on the turbo decoding graph with
gaussian densities. IEEE Transactions on Information Theory, 47:745?765, 2000.
G. Schwarz. Estimating the dimension of a model. Annals of Statistics, 6:461?464, 1978.
A. Sklar. Fonctions de repartition a n dimensions et leurs marges. Publications de l?Institut de Statistique de
L?Universite de Paris, 8:229?231, 1959.
L. Song, A. Gretton, D. Bickson, Y. Low, and C. Guestrin. Kernel belief propagation. In Conference on
Artificial Intelligence and Statistics (AIStats), 2011.
E.B. Sudderth, A.T. Ihler, M. Isard, W.T. Freeman, and A.S. Willsky. Nonparametric belief propagation. Communications of the ACM, 53(10):95?103, 2010a.
Erik Sudderth, Alexander Ihler, Michael Isard, William Freeman, and Alan Willsky. Nonparametric belief
propagation. Communications of the ACM, 53(10):95?103, October 2010b.
Y. Weiss and W. Freeman. Correctness of belief propagation in gaussian graphical models of arbitrary topology.
Neural Computation, 13:2173?2200, 2001.
W. Wiegerinck and T. Heskes. Fractional belief propagation. In Neural Information Processing Systems 15,
Cambridge, Mass., 2003. MIT Press.
A. Wilson and Z. Ghahramani. Copula processes. In Neural Information Processing Systems (NIPS), 2010.
J. S. Yedidia, W. T. Freeman, and Y. Weiss. Generalized belief propagation. In Neural Information Processing
Systems 13, pages 689?695, Cambridge, Mass., 2001. MIT Press.
L. Zhang and V. Singh. Trivariate flood frequency analysis using the Gumbel-Hougaard copula. Journal of
Hydrologic Engineering, 12, 2007.
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4,034 | 4,651 | Learning Probability Measures with Respect to
Optimal Transport Metrics
Guillermo D. Canas?,?
Lorenzo A. Rosasco?,?
? Laboratory for Computational and Statistical Learning - MIT-IIT
? CBCL, McGovern Institute - Massachusetts Institute of Technology
{guilledc,lrosasco}@mit.edu
Abstract
We study the problem of estimating, in the sense of optimal transport metrics, a
measure which is assumed supported on a manifold embedded in a Hilbert space.
By establishing a precise connection between optimal transport metrics, optimal
quantization, and learning theory, we derive new probabilistic bounds for the performance of a classic algorithm in unsupervised learning (k-means), when used to
produce a probability measure derived from the data. In the course of the analysis,
we arrive at new lower bounds, as well as probabilistic upper bounds on the convergence rate of empirical to population measures, which, unlike existing bounds,
are applicable to a wide class of measures.
1
Introduction and Motivation
In this paper we study the problem of learning from random samples a probability distribution
supported on a manifold, when the learning error is measured using transportation metrics.
The problem of learning a probability distribution is classic in statistics, and is typically analyzed
for distributions in X = Rd that have a density with respect to the Lebesgue measure, with total
variation, and L2 among the common distances used to measure closeness of two densities (see for
instance [10, 32] and references therein.) The setting in which the data distribution is supported on
a low dimensional manifold embedded in a high dimensional space has only been considered more
recently. In particular, kernel density estimators on manifolds have been described in [36], and their
pointwise consistency, as well as convergence rates, have been studied in [25, 23, 18]. A discussion
on several topics related to statistics on a Riemannian manifold can be found in [26].
Interestingly, the problem of approximating measures with respect to transportation distances has
deep connections with the fields of optimal quantization [14, 16], optimal transport [35] and, as
we point out in this work, with unsupervised learning (see Sec. 4.) In fact, as described in the
sequel, some of the most widely-used algorithms for unsupervised learning, such as k-means (but
also others such as PCA and k-flats), can be shown to be performing exactly the task of estimating
the data-generating measure in the sense of the 2-Wasserstein distance. This close relation between
learning theory, and optimal transport and quantization seems novel and of interest in its own right.
Indeed, in this work, techniques from the above three fields are used to derive the new probabilistic
bounds described below.
Our technical contribution can be summarized as follows:
(a) we prove uniform lower bounds for the distance between a measure and estimates based on
discrete sets (such as the empirical measure or measures derived from algorithms such as kmeans);
(b) we provide new probabilistic bounds for the rate of convergence of empirical to population
measures which, unlike existing probabilistic bounds, hold for a very large class of measures;
1
(c) we provide probabilistic bounds for the rate of convergence of measures derived from k-means
to the data measure.
The structure of the paper is described at the end of Section 2, where we discuss the exact formulation of the problem as well as related previous works.
2
Setup and Previous work
Consider the problem of learning a probability measure ? supported on a space M, from an i.i.d.
sample Xn = (x1 , . . . , xn ) ? ?n of size n. We assume M to be a compact, smooth d-dimensional
manifold of bounded curvature, with C 1 metric and volume measure ?M , embedded in the unit ball
of a separable Hilbert space X with inner product h?, ?i, induced norm k ? k, and distance d (for
instance M = B2d (1) the unit ball in X = Rd .) Following [35, p. 94], let Pp (M) denote the
Wasserstein space of order 1 ? p < ?:
Z
Pp (M) := ? ? P (M) :
kxkp d?(x) < ?
M
of probability measures P (M) supported on M, with finite p-th moment. The p-Wasserstein distance
n
o
1/p
Wp (?, ?) = inf [EkX ? Y kp ]
: Law(X) = ?, Law(Y ) = ?
(1)
X,Y
where the random variables X and Y are distributed according to ? and ? respectively, is the optimal
expected cost of transporting points generated from ? to those generated from ?, and is guaranteed to
be finite in Pp (M) [35, p. 95]. The space Pp (M) with the Wp metric is itself a complete separable
metric space [35]. We consider here the problem of learning probability measures ? ? P2 (M),
where the performance is measured by the distance W2 .
There are many possible choices of distances between probability measures [13]. Among them,
Wp metrizes weak convergence (see [35] theorem 6.9), that is, in Pp (M), a sequence (?i )i?N of
measures converges weakly to ? iff Wp (?i , ?) ? 0 and their p-th order moments converge to that of
?. There are other distances, such as the L?evy-Prokhorov, or the weak-* distance, that also metrize
weak convergence. However, as pointed out by Villani in his excellent monograph [35, p. 98],
1. ?Wasserstein distances are rather strong, [...]a definite advantage over the weak-* distance?.
2. ?It is not so difficult to combine information on convergence in Wasserstein distance with
some smoothness bound, in order to get convergence in stronger distances.?
Wasserstein distances have been used to study the mixing and convergence of Markov chains [22], as
well as concentration of measure phenomena [20]. To this list we would add the important fact that
existing and widely-used algorithms for unsupervised learning can be easily extended (see Sec. 4)
to compute a measure ?0 that minimizes the distance W2 (?
?n , ?0 ) to the empirical measure
n
??n :=
1X
?x ,
n i=1 i
a fact that will allow us to prove, in Sec. 5, bounds on the convergence of a measure induced by
k-means to the population measure ?.
The most useful versions of Wasserstein distance are p = 1, 2, with p = 1 being the weaker of the
two (by H?older?s inequality, p ? q ? Wp ? Wq .) In particular, ?results in W2 distance are usually
stronger, and more difficult to establish than results in W1 distance? [35, p. 95]. A discussion of
p = ? would take us out of topic, since its behavior is markedly different.
2.1
Closeness of Empirical and Population Measures
By the strong law of large numbers, the empirical measure converges almost surely to the population
measure: ??n ? ? in the sense of the weak topology [34]. Since weak convergence and convergence
in Wp plus convergence of p-th moments are equivalent in Pp (M), this means that, in the Wp sense,
the empirical measure ??n converges to ?, as n ? ?. A fundamental question is therefore how fast
the rate of convergence of ??n ? ? is.
2
2.1.1
Convergence in expectation
The rate of convergence of ??n ? ? in expectation has been widely studied in the past, resulting in upper bounds of order EW2 (?, ??n ) = O(n?1/(d+2) ) [19, 8], and lower bounds of order
EW2 (?, ??n ) = ?(n?1/d ) [29] (both assuming that the absolutely continuous part of ? is ?A 6= 0,
with possibly better rates otherwise).
More recently, an upper bound of order EWp (?, ??n ) = O(n?1/d ) has been proposed [2] by proving
a bound for the Optimal Bipartite Matching (OBM) problem [1], and relating this problem to the
expected distance EWp (?, ??n ). In particular, given two independent samplesPXn , Yn , the OBM
problem is that of finding a permutation ? that minimizes the matching cost n?1 kxi ?y?(i) kp [24,
30]. It is not hard to show that the optimal matching cost is Wp (?
?Xn , ??Yn )p , where ??Xn , ??Yn are
the empirical measures associated to Xn , Yn . By Jensen?s inequality, the triangle inequality, and
(a + b)p ? 2p?1 (ap + bp ), it holds
EWp (?, ??n )p ? EWp (?
?Xn , ??Yn )p ? 2p?1 EWp (?, ??n )p ,
and therefore a bound of order O(n?p/d ) for the OBM problem [2] implies a bound EWp (?, ??n ) =
O(n?1/d ). The matching lower bound is only known for a special case: ?A constant over a bounded
set of non-null measure [2] (e.g. ?A uniform.) Similar results, with matching lower bounds are found
for W1 in [11].
2.1.2
Convergence in probability
Results for convergence in probability, one of the main results of this work, appear to be considerably
harder to obtain. One fruitful avenue of analysis has been the use of so-called transportation, or
Talagrand inequalities Tp , which can be used to prove concentration inequalities on Wp [20]. In
particular, we say that ? satisfies a Tp (C) inequality with C > 0 iff Wp (?, ?)2 ? CH(?|?), ?? ?
Pp (M), where H(?|?) is the relative entropy [20]. As shown in [6, 5], it is possible to obtain
probabilistic upper bounds on Wp (?, ??n ), with p = 1, 2, if ? is known to satisfy a Tp inequality
of the same order, thereby reducing the problem of bounding Wp (?, ??n ) to that of obtaining a Tp
inequality. Note that, by Jensen?s inequality, and as expected from the behavior of Wp , the inequality
T2 is stronger than T1 [20].
While it has been shown that ? satisfies a T1 inequality iff it has a finite square-exponential moment
2
(E[e?kxk ] finite for some ? > 0) [4, 7], no such general conditions have been found for T2 . As
an example, consider that, if M is compact with diameter D then, by theorem 6.15 of [35], and the
celebrated Csisz?ar-Kullback-Pinsker inequality [27], for all ?, ? ? Pp (M), it is
Wp (?, ?)2p ? (2D)2p k? ? ?k2TV ? 22p?1 D2p H(?|?),
where k ? kTV is the total variation norm. Clearly, this implies a Tp=1 inequality, but for p ? 2 it
does not.
The T2 inequality has been shown by Talagrand to be satisfied by the Gaussian distribution [31], and
then slightly more generally by strictly log-concave measures (see [20, p. 123], and [3].) However, as
noted in [6], ?contrary to the T1 case, there is no hope to obtain T2 inequalities from just integrability
or decay estimates.?
Structure of this paper. In this work we obtain bounds in probability (learning rates) for the
problem of learning a probability measure in the sense of W2 . We begin by establishing (lower)
bounds for the convergence of empirical to population measures, which serve to set up the problem
and introduce the connection between quantization and measure learning (sec. 3.) We then describe
how existing unsupervised learning algorithms that compute a set (k-means, k-flats, PCA,. . . ) can
be easily extended to produce a measure (sec. 4.) Due to its simplicity and widespread use, we focus
here on k-means. Since the two measure estimates that we consider are the empirical measure, and
the measure induced by k-means, we next set out to prove upper bounds on their convergence to
the data-generating measure (sec. 5.) We arrive at these bounds by means of intermediate measures,
which are related to the problem of optimal quantization. The bounds apply in a very broad setting
(unlike existing bounds based on transportation inequalities, they are not restricted to log-concave
measures [20, 3].)
3
3
Learning probability measures, optimal transport and quantization
We address the problem of learning a probability measure ? when the only observations we have at
our disposal are n i.i.d. samples Xn = (x1 , . . . , xn ). We begin by establishing some notation and
useful intermediate results.
Given a closed set S ? X , let {Vq : q ? S} be a Borel Voronoi partition of X composed of sets
Vq closest to each q ? S, that is, such that each Vq ? {x ? X : kx ? qk = minr?S kx ? rk} is
measurable (see for instance [15].) Consider the projection function ?S : X ? S mapping each
x ? Vq to q. By virtue of {Vq }q?S being a Borel Voronoi partition, the map ?S is measurable [15],
and it is d (x, ?S (x)) = minq?S kx ? qk for all x ? X .
For any ? ? Pp (M), let ?S ? be the pushforward, or image measure of ? under the mapping ?S ,
which is defined to be (?S ?)(A) := ?(?S?1 (A)) for all Borel measurable sets A. From its definition,
it is clear that ?S ? is supported on S.
We now establish a connection between the expected distance to a set S, and the distance between ?
and the set?s induced pushforward measure. Notice that, for discrete sets S, the expected Lp distance
to S is exactly the expected quantization error
Ep,? (S) := Ex?? d(x, S)p = Ex?? kx ? ?S (x)kp
incurred when encoding points x drawn from ? by their closest point ?S (x) in S [14]. This close
connection between optimal quantization and Wasserstein distance has been pointed out in the past
in the statistics [28], optimal quantization [14, p. 33], and approximation theory [16] literatures.
The following two lemmas are key tools in the reminder of the paper. The first highlights the close
link between quantization and optimal transport.
Lemma 3.1. For closed S ? X , ? ? Pp (M), 1 ? p < ?, it holds Ex?? d(x, S)p = Wp (?, ?S ?)p .
Note that the key element in the above lemma is that the two measures in the expression Wp (?, ?S ?)
must match. When there is a mismatch, the distance can only increase. That is, Wp (?, ?S ?) ?
Wp (?, ?S ?) for all ? ? Pp (M). In fact, the following lemma shows that, among all the measures
with support in S, ?S ? is closest to ?.
Lemma 3.2. For closed S ? X , and all ? ? Pp (M) with supp(?) ? S, 1 ? p < ?, it holds
Wp (?, ?) ? Wp (?, ?S ?).
When combined, lemmas 3.1 and 3.2 indicate that the behavior of the measure learning problem is
limited by the performance of the optimal quantization problem. For instance, Wp (?, ??n ) can only
be, in the best-case, as low as the optimal quantization cost with codebook of size n. The following
section makes this claim precise.
3.1
Lower bounds
Consider the situation depicted in fig. 1, in which a sample X4 = {x1 , x2 , x3 , x4 } is drawn from
a distribution ? which we assume here to be absolutely continuous on its support. As shown, the
projection map ?X4 sends points x to their closest point in X4 . The resulting Voronoi decomposition
of supp(?) is drawn in shades of blue. By lemma 5.2 of [9], the pairwise intersections of Voronoi
regions have null ambient measure, and since ? is absolutely continuous, the pushforward measure
P4
can be written in this case as ?X4 ? =
j=1 ?(Vxj )?xj , where Vxj is the Voronoi region of xj .
Note that, even for finite sets S, this particular decomposition is not always possible if the {Vq }q?S
form a Borel Voronoi tiling, instead of a Borel Voronoi partition. If, for instance, ? has an atom
falling on two Voronoi regionsPin a tiling, then both regions would count the atom as theirs, and
double-counting would imply q ?(Vq ) > 1. The technicalities required to correctly define a Borel
Voronoi partition are such that, in general, it is simpler to write ?S ?, even though (if S is discrete)
this measure can clearly be written as a sum of deltas with appropriate masses.
By lemma 3.1, the distance Wp (?, ?X4 ?)p is the (expected) quantization cost of ? when using X4
as codebook. Clearly, this cost can never be lower than the optimal quantization cost of size 4. This
reasoning leads to the following lower bound between empirical and population measures.
4
Theorem 3.3. For ? ? Pp (M) with absolutely continuous part ?A 6= 0, and 1 ? p < ?, it holds
Wp (?, ??n ) = ?(n?1/d ) uniformly over ??n , where the constants depend on d and ?A only.
Proof: Let Vn,p (?) := inf S?M,|S|=n Ex?? d(x, S)p be the optimal quantization cost of ? of order
p with n centers. Since ?A 6= 0, and since ? has a finite (p + ?)-th order moment, for some ? > 0
(since it is supported on the unit ball), then it is Vn,p (?) = ?(n?p/d ), with constants depending on
d and ?A (see [14, p. 78] and [16].) Since supp(?
?n ) = Xn , it follows that
Wp (?, ??n )p
?
lemma 3.2
Wp (?, ?Xn ?)p
=
lemma 3.1
Ex?? d(x, Xn )p ? Vn,p (?) = ?(n?p/d )
Note that the bound of theorem 3.3 holds for ??n derived from any sample Xn , and is therefore
stronger than the existing lower bounds on the convergence rates of EWp (?, ??n ) ? 0. In particular,
it trivially induces the known lower bound ?(n?1/d ) on the rate of convergence in expectation.
4
Unsupervised learning algorithms for learning a probability measure
As described in [21], several of the most widely used unsupervised learning algorithms can be
interpreted to take as input a sample Xn and output a set S?k , where k is typically a free parameter
of the algorithm, such as the number of means in k-means1 , the dimension of affine spaces in PCA,
Pn
etc. Performance is measured by the empirical quantity n?1 i=1 d(xi , S?k )2 , which is minimized
among all sets in some class (e.g. sets of size k, affine spaces of dimension k,. . . ) This formulation is
general enough to encompass k-means and PCA, but also k-flats, non-negative matrix factorization,
and sparse coding (see [21] and references therein.)
Using the discussion of Sec. 3, we can establish a clear connection between unsupervised learning
and the problem of learning probability measures with respect to W2 . Consider as a running example
the k-means problem, though the argument is general. Given an input Xn , the k-means problem is
to find a set |S?k | = k minimizing its average distance from points in Xn . By associating to S?k the
pushforward measure ?S?k ??n , we find that
n
1X
d(xi , S?k )2 = Ex???n d(x, S?k )2
n i=1
=
lemma 3.1
W2 (?
?n , ?S?k ??n )2 .
(2)
Since k-means minimizes equation 2, it also finds the measure that is closest to ??n , among those
with support of size k. This connection between k-means and W2 measure approximation was, to
the best of the authors? knowledge, first suggested by Pollard [28] though, as mentioned earlier, the
argument carries over to many other unsupervised learning algorithms.
Unsupervised measure learning algorithms. We briefly clarify the steps involved in using an
existing unsupervised learning algorithm for probability measure learning. Let Uk be a parametrized
algorithm (e.g. k-means) that takes a sample Xn and outputs a set Uk (Xn ). The measure learning
algorithm Ak : Mn ? Pp (M) corresponding to Uk is defined as follows:
1. Ak takes a sample Xn and outputs the measure ?S?k ??n , supported on S?k = Uk (Xn );
Pn
2. since ??n is discrete, then so must ?S?k ??n be, and thus Ak (Xn ) = n1 i=1 ??S? (xi ) ;
k
h
i
3. in practice, we can simply store an n-vector ?S?k (x1 ), . . . , ?S?k (xn ) , from which Ak (Xn )
can be reconstructed by placing atoms of mass 1/n at each point.
In the case that Uk is the k-means algorithm, only k points and k masses need to be stored.
Note that any algorithm A0 that attempts to output a measure A0 (Xn ) close to ??n can be cast in the
above framework. Indeed, if S 0 is the support of A0 (Xn ) then, by lemma 3.2, ?S 0 ??n is the measure
closest to ??n with support in S 0 . This effectively reduces the problem of learning a measure to that of
1
In a slight abuse of notation, we refer to the k-means algorithm here as an ideal algorithm that solves the
k-means problem, even though in practice an approximation algorithm may be used.
5
finding a set, and is akin to how the fact that every optimal quantizer is a nearest-neighbor quantizer
(see [15], [12, p. 350], and [14, p. 37?38]) reduces the problem of finding an optimal quantizer to
that of finding an optimal quantizing set.
Clearly, the minimum of equation 2 over sets of size k (the output of k-means) is monotonically
non-increasing with k. In particular, since S?n = Xn and ?S?n ??n = ??n , it is Ex???n d(x, S?n )2 =
W2 (?
?n , ?S?n ??n )2 = 0. That is, we can always make the learned measure arbitrarily close to ??n
by increasing k. However, as pointed out in Sec. 2, the problem of measure learning is concerned
with minimizing the 2-Wasserstein distance W2 (?, ?S?k ??n ) to the data-generating measure. The
actual performance of k-means is thus not necessarily guaranteed to behave in the same way as the
empirical one, and the question of characterizing its behavior as a function of k and n naturally
arises.
Finally, we note that, while it is Ex???n d(x, S?k )2 = W2 (?
?n , ?S?k ??n )2 (the empirical performances
are the same in the optimal quantization, and measure learning problem formulations), the actual
performances satisfy
Ex?? d(x, S?k )2 = W2 (?, ? ? ?)2 ? W2 (?, ? ? ??n )2 , 1 ? k ? n.
lemma 3.1
Sk
lemma 3.2
Sk
Consequently, with the identification between sets S and measures ?S ??n , the measure learning
problem is, in general, harder than the set-approximation problem (for example, if M = Rd and ?
is absolutely continuous over a set of non-null volume, it is not hard to show that the inequality is
almost surely strict: Ex?? d(x, S?k )2 < W2 (?, ?S?k ??n )2 for 1 < k < n.)
In the remainder, we characterize the performance of k-means on the measure learning problem, for
varying k, n. Although other unsupervised learning algorithms could have been chosen as basis for
our analysis, k-means is one of the oldest and most widely used, and the one for which the deep
connection between optimal quantization and measure approximation is most clearly manifested.
Note that, by setting k = n, our analysis includes the problem of characterizing the behavior of
the distance W2 (?, ??n ) between empirical and population measures which, as indicated in Sec. 2.1,
is a fundamental question in statistics (i.e. the speed of convergence of empirical to population
measures.)
5
Learning rates
In order to analyze the performance of k-means as a measure learning algorithm, and the convergence of empirical to population measures, we propose the decomposition shown in fig. 2. The
diagram includes all the measures considered in the paper, and shows the two decompositions used
to prove upper bounds. The upper arrow (green), illustrates the decomposition used to bound the distance W2 (?, ??n ). This decomposition uses the measures ?Sk ? and ?Sk ??n as intermediates to arrive
at ??n , where Sk is a k-point optimal quantizer of ?, that is, a set Sk minimizing Ex?? d(x, S)2 over
all sets of size |S| = k. The lower arrow (blue) corresponds to the decomposition of W2 (?, ?S?k ??n )
(the performance of k-means), whereas the labelled black arrows correspond to individual terms in
the bounds. We begin with the (slightly) simpler of the two results.
5.1
Convergence rates for the empirical to population measures
Let Sk be the optimal k-point quantizer of ? of order two [14, p. 31]. By the triangle inequality and
the identity (a + b + c)2 ? 3(a2 + b2 + c2 ), it follows that
W2 (?, ??n )2 ? 3 W2 (?, ?Sk ?)2 + W2 (?Sk ?, ?Sk ??n )2 + W2 (?Sk ??n , ??n )2 .
(3)
This is the decomposition depicted in the upper arrow of fig. 2.
By lemma 3.1, the first term in the sum of equation 3 is the optimal k-point quantization error of
? over a d-manifold M which, using recent techniques from [16] (see also [17, p. 491]), is shown
in the proof of theorem 5.1 (part a) to be of order ?(k ?2/d ). The remaining terms, b) and c), are
slightly more technical and are bounded in the proof of theorem 5.1.
Since equation 3 holds for all 1 ? k ? n, the best bound on W2 (?, ??n ) can be obtained by optimizing the right-hand side over all possible values of k, resulting in the following probabilistic bound
for the rate of convergence of the empirical to population measures.
6
x1
x2
x
?{x1 ,x2 ,x3 ,x4 }
W2 (?, ??n )
supp ?
?
x3
a)
?Sk ?
b)
?Sk ??n
c)
d)
??n
?S?k ??n
W2 (?, ?S?k ??n )
x4
Figure 1: A sample {x1 , x2 , x3 , x4 } is
drawn from a distribution ? with support in
supp ?. The projection map ?{x1 ,x2 ,x3 ,x4 }
sends points x to their closest one in the sample. The induced Voronoi tiling is shown in
shades of blue.
Figure 2: The measures considered in this paper
are linked by arrows for which upper bounds for
their distance are derived. Bounds for the quantities of interest W2 (?, ??n )2 , and W2 (?, ?S?k ??n )2 ,
are decomposed by following the top and bottom
colored arrows.
Theorem 5.1. Given ? ? Pp (M) with absolutely continuous part ?A 6= 0, sufficiently large n, and
? > 0, it holds
2
W2 (?, ??n ) ? C ? m(?A ) ? n?1/(2d+4) ? ?, with probability 1 ? e?? .
R
where m(?A ) := M ?A (x)d/(d+2) d?M (x), and C depends only on d.
5.2
Learning rates of k-means
The key element in the proof of theorem 5.1 is that the distance between population and empirical
measures can be bounded by choosing an intermediate optimal quantizing measure of an appropriate
size k. In the analysis, the best bounds are obtained for k smaller than n. If the output of k-means
is close to an optimal quantizer (for instance if sufficient data is available), then we would similarly
expect that the best bounds for k-means correspond to a choice of k < n.
The decomposition of the bottom (blue) arrow in figure 2 leads to the following bound in probability.
Theorem 5.2. Given ? ? Pp (M) with absolutely continuous part ?A 6= 0, and ? > 0, then for all
sufficiently large n, and letting
k = C ? m(?A ) ? nd/(2d+4) ,
it holds
2
W2 (?, ?S?k ??n ) ? C ? m(?A ) ? n?1/(2d+4) ? ?, with probability 1 ? e?? .
R
where m(?A ) := M ?A (x)d/(d+2) d?M (x), and C depends only on d.
Note that the upper bounds in theorem 5.1 and 5.2 are exactly the same. Although this may appear
surprising, it stems from the following fact. Since S = S?k is a minimizer of W2 (?S ??n , ??n )2 , the
bound d) of figure 2 satisfies:
W2 (?S?k ??n , ??n )2 ? W2 (?Sk ??n , ??n )2
and therefore (by the definition of c), the term d) is of the same order as c). It follows then that
adding term d) to the bound only affects the constants, but otherwise leaves it unchanged. Since
d) is the term that takes the output measure of k-means to the empirical measure, this implies that
the rate of convergence of k-means (for suitably chosen k) cannot be worse than that of ??n ? ?.
Conversely, bounds for ??n ? ? are obtained from best rates of convergence of optimal quantizers,
whose convergence to ? cannot be slower than that of k-means (since the quantizers that k-means
produces are suboptimal.)
7
Since the bounds obtained for the convergence of ??n ? ? are the same as those for k-means with
k of order k = ?(nd/(2d+4) ), this suggests that estimates of ? that are as accurate as those derived
from an n point-mass measure ??n can be derived from k point-mass measures with k n.
Finally, we note that the introduced bounds are currently limited by the statistical bound
sup |W2 (?S ??n , ??n )2 ? W2 (?S ?, ?)2 |
|S|=k
=
sup |Ex???n d(x, S)2 ? Ex?? d(x, S)2 |
lemma 3.1 |S|=k
(4)
(see for instance [21]), for which non-matching lower bounds are known. This means that, if better
upper bounds can be obtained for equation 4, then both bounds in theorems 5.1 and 5.2 would
automatically improve (would become closer to the lower bound.)
References
[1] M. Ajtai, J. Komls, and G. Tusndy. On optimal matchings. Combinatorica, 4:259?264, 1984.
[2] Franck Barthe and Charles Bordenave. Combinatorial optimization over two random point sets. Technical
Report arXiv:1103.2734, Mar 2011.
[3] Gordon Blower. The Gaussian isoperimetric inequality and transportation. Positivity, 7:203?224, 2003.
[4] S. G. Bobkov and F. G?otze. Exponential integrability and transportation cost related to logarithmic
Sobolev inequalities. Journal of Functional Analysis, 163(1):1?28, April 1999.
[5] Emmanuel Boissard. Simple bounds for the convergence of empirical and occupation measures in 1wasserstein distance. Electron. J. Probab., 16(83):2296?2333, 2011.
[6] F. Bolley, A. Guillin, and C. Villani. Quantitative concentration inequalities for empirical measures on
non-compact spaces. Probability Theory and Related Fields, 137(3):541?593, 2007.
[7] F. Bolley and C. Villani. Weighted Csisz?ar-Kullback-Pinsker inequalities and applications to transportation inequalities. Annales de la Faculte des Sciences de Toulouse, 14(3):331?352, 2005.
[8] Claire Caillerie, Fr?ed?eric Chazal, J?er?ome Dedecker, and Bertrand Michel. Deconvolution for the Wasserstein metric and geometric inference. Rapport de recherche RR-7678, INRIA, July 2011.
[9] Kenneth L. Clarkson. Building triangulations using -nets. In Proceedings of the thirty-eighth annual
ACM symposium on Theory of computing, STOC ?06, pages 326?335, New York, NY, USA, 2006. ACM.
[10] Luc Devroye and G?abor Lugosi. Combinatorial methods in density estimation. Springer Series in Statistics. Springer-Verlag, New York, 2001.
[11] V. Dobri and J. Yukich. Asymptotics for transportation cost in high dimensions. Journal of Theoretical
Probability, 8:97?118, 1995.
[12] A. Gersho and R.M. Gray. Vector Quantization and Signal Compression. Kluwer International Series in
Engineering and Computer Science. Kluwer Academic Publishers, 1992.
[13] Alison L. Gibbs and Francis E. Su. On choosing and bounding probability metrics. International Statistical Review, 70:419?435, 2002.
[14] Siegfried Graf and Harald Luschgy. Foundations of quantization for probability distributions. SpringerVerlag New York, Inc., Secaucus, NJ, USA, 2000.
[15] Siegfried Graf, Harald Luschgy, and Gilles Page`s. Distortion mismatch in the quantization of probability
measures. Esaim: Probability and Statistics, 12:127?153, 2008.
[16] Peter M. Gruber. Optimum quantization and its applications. Adv. Math, 186:2004, 2002.
[17] P.M. Gruber. Convex and discrete geometry. Grundlehren der mathematischen Wissenschaften. Springer,
2007.
[18] Guillermo Henry and Daniela Rodriguez. Kernel density estimation on riemannian manifolds: Asymptotic results. J. Math. Imaging Vis., 34(3):235?239, July 2009.
[19] Joseph Horowitz and Rajeeva L. Karandikar. Mean rates of convergence of empirical measures in the
Wasserstein metric. J. Comput. Appl. Math., 55(3):261?273, November 1994.
[20] M. Ledoux. The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs. American Mathematical Society, 2001.
[21] A. Maurer and M. Pontil. K?dimensional coding schemes in Hilbert spaces. IEEE Transactions on
Information Theory, 56(11):5839 ?5846, nov. 2010.
[22] Yann Ollivier. Ricci curvature of markov chains on metric spaces. J. Funct. Anal., 256(3):810?864, 2009.
8
[23] Arkadas Ozakin and Alexander Gray. Submanifold density estimation. In Y. Bengio, D. Schuurmans,
J. Lafferty, C. K. I. Williams, and A. Culotta, editors, Advances in Neural Information Processing Systems
22, pages 1375?1382. 2009.
[24] C. Papadimitriou. The probabilistic analysis of matching heuristics. In Proc. of the 15th Allerton Conf.
on Communication, Control and Computing, pages 368?378, 1978.
[25] Bruno Pelletier. Kernel density estimation on Riemannian manifolds. Statist. Probab. Lett., 73(3):297?
304, 2005.
[26] Xavier Pennec. Intrinsic statistics on riemannian manifolds: Basic tools for geometric measurements. J.
Math. Imaging Vis., 25(1):127?154, July 2006.
[27] M. S. Pinsker. Information and information stability of random variables and processes. San Francisco:
Holden-Day, 1964.
[28] David Pollard. Quantization and the method of k-means. IEEE Transactions on Information Theory,
28(2):199?204, 1982.
[29] S.T. Rachev. Probability metrics and the stability of stochastic models. Wiley series in probability and
mathematical statistics: Applied probability and statistics. Wiley, 1991.
[30] J.M. Steele. Probability Theory and Combinatorial Optimization. Cbms-Nsf Regional Conference Series
in Applied Mathematics. Society for Industrial and Applied Mathematics, 1997.
[31] M. Talagrand. Transportation cost for Gaussian and other product measures. Geometric And Functional
Analysis, 6:587?600, 1996.
[32] Alexandre B. Tsybakov. Introduction to nonparametric estimation. Springer Series in Statistics. Springer,
New York, 2009. Revised and extended from the 2004 French original, Translated by Vladimir Zaiats.
[33] A.W. van der Vaart and J.A. Wellner. Weak Convergence and Empirical Processes. Springer Series in
Statistics. Springer, 1996.
[34] V. S. Varadarajan. On the convergence of sample probability distributions. Sankhy?a: The Indian Journal
of Statistics, 19(1/2):23?26, Feb. 1958.
[35] C. Villani. Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften. Springer,
2009.
[36] P. Vincent and Y. Bengio. Manifold Parzen Windows. In Advances in Neural Information Processing
Systems 22, pages 849?856. 2003.
9
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4,035 | 4,652 | Continuous Relaxations for Discrete
Hamiltonian Monte Carlo
Yichuan Zhang, Charles Sutton, Amos Storkey
School of Informatics
University of Edinburgh
United Kingdom
[email protected],
[email protected],
[email protected]
Zoubin Ghahramani
Department of Engineering
University of Cambridge
United Kingdom
[email protected]
Abstract
Continuous relaxations play an important role in discrete optimization, but have
not seen much use in approximate probabilistic inference. Here we show that a
general form of the Gaussian Integral Trick makes it possible to transform a wide
class of discrete variable undirected models into fully continuous systems. The
continuous representation allows the use of gradient-based Hamiltonian Monte
Carlo for inference, results in new ways of estimating normalization constants
(partition functions), and in general opens up a number of new avenues for inference in difficult discrete systems. We demonstrate some of these continuous
relaxation inference algorithms on a number of illustrative problems.
1
Introduction
Discrete undirected graphical models have seen wide use in natural language processing [11, 24] and
computer vision [19]. Although sophisticated inference algorithms exist for these models, including
both exact algorithms and variational approximations, it has proven more difficult to develop discrete
Markov chain Monte Carlo (MCMC) methods. Despite much work and many recent advances [3],
the most commonly used MCMC methods in practice for discrete models are based on MetropolisHastings, the effectiveness of which is strongly dependent on the choice of proposal distribution.
An appealing idea is to relax the constraint that the random variables of interest take integral values.
This is inspired by optimization methods such as linear program relaxation. Continuous problems
are appealing because the gradient is on your side: Unlike discrete probability mass functions, in
the continuous setting, densities have derivatives, contours, and curvature that can be used to inform
sampling algorithms [6, 16, 18, 20, 27]. For this reason, continuous relaxations are widespread in
combinatorial optimization, and likewise a major appeal of variational methods is that they convert discrete inference problems into continuous optimization problems. Comparatively speaking,
relaxations in an MCMC setting have been generally overlooked.
In this paper we provide a method for relaxing a discrete model into a continuous one, using a
technique from statistical physics that Hertz et al. [8] call the ?Gaussian integral trick,? and that we
present in a more general form than is typical. This trick is also known as the Hubbard-Stratonovich
transform [10]. Starting with a discrete Markov random field (MRF), the trick introduces an auxiliary Gaussian variable in such a way that the discrete dependencies cancel out. This allows the
discrete variables to be summed away, leaving a continuous problem.
The continuous representation allows the use of gradient-based Hamiltonian Monte Carlo for inference, highlights an equivalence between Boltzmann machines and the Gaussian-Bernoulli harmonium model [25], and in general opens up a number of new avenues for inference in difficult discrete
1
systems. On synthetic problems and a real world problem in text processing, we show that HMC in
the continuous relaxation can be much more accurate than standard MCMC methods on the discrete
distribution.
The only previous work of which we are aware that uses the Gaussian integral trick for inference
in graphical models is Martens and Sutskever [12]. They use the trick to transform an arbitrary
MRF into an equivalent restricted Boltzmann machine (RBM), on which they then do block Gibbs
sampling. They show that this transformation is useful when each block Gibbs step can be performed
in parallel. However, unlike the current work, they do not sum out the discrete variables, so they do
not perform a full continuous relaxation.
2
Background
Consider an undirected graphical model over random vectors t = (t1 , t2 , . . . tM ) where each
ti ? {0, 1, 2, . . . Ki ? 1}. We will employ a 1 of Ki representation for each non-binary ti and
concatenate the resulting binary variables into the vector s = (s1 . . . sN ). We will also focus on
pairwise models over a graph G = (V, E) where V = {1, 2, . . . N }. Every discrete undirected
model can be converted into a pairwise model at the cost of expanding the state space. The undirected pairwise graphical model can be written in the form
1 Y
p(s) =
exp(?Eij (si , sj ))
(1)
Z
(i,j)?G
where Z is a normalisation term, and is a sum over all valid states of (s1 , s2 , . . . , sN ) that comply
with the 1 of Ki constraints. Equivalently we can set Eij (si , sj ) to be very large when si and sj
are derived from the same variable tk (for some k and i 6= j, and expanding G to include (i, j)),
making the resulting product for the terms that break the 1 of Ki constraints to be exponentially
small. Henceforth, without loss of generality, we can consider binary pairwise models, and assume
E captures any additional constraints that might apply. Then this model takes the general form of a
Boltzmann machine or binary MRF, and can be conveniently rewritten as
1
1
(2)
p(s) = exp aT s + sT W s
Z
2
where a ? RN , and W , a real symmetric matrix, are the model parameters. The normalization
function is
X
1
Z=
exp aT s + sT W s .
(3)
2
s
3
Gaussian Integral Trick
Inference in Boltzmann machines (which is equivalent to inference in Ising models) has always been
a challenging problem. Typically Markov chain Monte Carlo procedures such as Gibbs sampling
have been used, but the high levels of connectivity in Boltzmann machines can cause trouble and
result in slow mixing in many situations. Furthermore for frustrated systems, such models are highly
multimodal [1], often with large potential barriers between the different modes.
In many situations, the Hamiltonian Monte Carlo method has provided a more efficient sampling
method for highly coupled systems [17], but is only appropriate in real valued problems. For this
reason, we choose to work with a real valued augmentation of the Boltzmann machine using the
Gaussian integral trick. The main idea is to introduce a real valued auxiliary vector x ? RN in such
a way that the sT W s term from (2) cancels out [8]. We generalise the standard form of the Gaussian
integral trick by using the following form for the conditional distribution of the auxiliary variable x:
p(x|s) = N (x; A(W + D)s, A(W + D)AT )
(4)
for any choice of invertible matrix A and any diagonal matrix D for which W + D is positive
definite. N (x; m, ?) denotes the Gaussian distribution in x with mean m and covariance ?. The
resulting joint distribution over x and s is
1
1
p(x, s) ? exp(? (x?A(W +D)s)T (A?1 )T (W +D)?1 A?1 (x?A(W +D)s)+ sT W s+aT s).
2
2
(5)
2
If d denotes a vector containing the diagonal elements of D, this simplifies to
1
1
p(x, s) ? exp ? xT (A?1 )T (W + D)?1 A?1 x + sT A?1 x + (a ? d)T s .
2
2
(6)
The key point is that the sT W s term has vanished. We can then marginalise out the s variables, as
they are decoupled from one another in the energy function, and can be summed over independently.
Define the vector ?x = A?1 x. Then the marginal density is
Y
1 T ?1
di
?1
?1 T
p(x) ? exp ? x A (W + D) (A ) x
1 + exp ?x;i + ai ?
.
(7)
2
2
i
The constant of proportionality in the above equation is Z ?1 |2?A(W + D)AT |?1/2 . The distribution p(x) is a mixture of 2N Gaussians, i.e., the Gaussians are p(x|s) with mixing proportion p(s)
for each possible assignment s.
We have now converted the discrete distribution p(s) into a corresponding continuous distribution
p(x). To understand the sense in which the two distributions ?correspond?, consider reconstructing
s using the conditional distribution p(s|x). First, all of the si are independent given x, because s
appears only log-linearly in (6). Using the sigmoid ?(z) = (1 + exp{?z})?1 , this is
1?si
s
di
di i
p(si |x) = ? ??x;i ? ai +
? ?x;i + ai ?
(8)
2
2
Two choices for A are of particular interest because they introduce additional independence rela1
tionships into the augmented model. First, if A = ?? 2 V T for the eigendecomposition W + D =
V ?V T , then the result is an undirected bipartite graphical model in the joint space of (x, s):
1
1
1
p(x, s) ? exp ? xT x + sT V ? 2 x + (a ? d)T s .
(9)
2
2
This is a Gaussian-Bernoulli form of exponential family harmonium [25]. Hence we see that the
Gaussian-Bernoulli harmonium is equivalent to a general Boltzmann machine over the discrete variables only. Second, if A = I we get
(
)
T
1
1
p(x, s) = Z ?1 |2?(W + D)|?1/2 exp
a+x? d
s ? xT (W + D)?1 x ,
(10)
2
2
which is of particular interest in that the coupling between s and x is one-to-one. A given xi
determines the Bernoulli probabilities for the variable si , independent of the states of any of the
other variables. This yields a marginal density
Y
di
1
1 + exp ai + xi ?
p(x) = Z ?1 |2?(W + D)|?1/2 exp ? xT (W + D)?1 x
2
2
i
(11)
and a particularly nice set of Bernoulli conditional probabilities
1?si
s
di
di i
p(si |x) = ? ?ai ? xi +
? ai + xi ?
(12)
2
2
In this model, the marginal of p(x) is a mixture of Gaussian distributions. Then, conditioned on x,
the log odds of si = 1 is a recentered version of xi , in particular, xi ? ai ? di /2.
The different versions of the Gaussian integral trick can be compactly summarized by the independence relations that they introduce. All versions of Gaussian integral trick give us that all si and
sj are independent given x. If we take A = ??1/2 V T , we additionally get that all xi and xj are
independent given s. Finally if we instead take A = I, we get that si and sj are independent given
only xi and xj . These independence relations are presented graphically in Figure 1.
3
x
x
x
s
s
s
p(s)
Original MRF
A = ? 1/2 V T
[MS10; HKP91]
General A
A=I
Current Approach
Figure 1: Graphical depiction of the different versions of the Gaussian integral trick. In all of the
models here si ? {0, 1} while xi ? R. Notice that when A = I the x have the same dependence
structure as the s did in the original MRF.
3.1
Convexity of Log Density
Because probabilistic inference is NP-hard, it is too much to expect that the continuous transformation will always help. Sometimes difficult discrete distributions will be converted into difficult
continuous ones. Experimentally we have noticed that highly frustrated systems typically result in
multimodal p(x).
The modes of p(x) are particularly easy to understand if A = ??1/2 V T , because p(x|s) =
N (x; ?1/2 V s; I), that is, the covariance does not depend on W + D. Without loss of generality assume that the diagonal of W is 0. Then write (W + D) = W + cD0 . Interpreting p(x) as a
mixture of Gaussians, one for each assignment s, as c ? ? the Gaussians become farther apart and
we get 2n modes, one each at ?1/2 V s for each assignment to binary vector s. If we take a small
c, however, we can sometimes get fewer modes, and as shown next, we can sometimes even get
log p(x) convex. This is a motivation to make sure that the elements of D are not too large.
In the following proposition we characterize the conditions on p(s) under which the resultant p(x)
is log-concave. For any N ? N matrix M , let ?1 (M ) ? . . . ? ?N (M ) denote the eigenvalues of
M . Recall that we have already required that D be chosen so that W + D is positive definite, i.e.,
?N (W + D) > 0. Then
Proposition 1. p(x) is log-concave if and only if W + D has a narrow spectrum, by which we mean
?1 (W + D) < 4.
Proof. The Hessian of log p(x) is easy to compute. It is
Hx := ?2x log p(x) = Cx ? (W + D)?1
(13)
where Cx is a diagonal matrix with elements cii = ?(?ai ? xi + d2i )(1 ? ?(?ai ? xi + d2i )). We
use the simple eigenvalue inequalities that ?1 (A) + ?N (B) ? ?1 (A + B) ? ?1 (A) + ?1 (B). If
?1 (W + D) ? 4, then
?1 (Hx ) ? ?1 (Cx ) ? [?1 (W + D)]?1 ? 0.25 ? [?1 (W + D)]?1 ? 0.
So p(x) is log-concave. Conversely suppose that p(x) is log-concave. Then
0.25 ? [?1 (W + D)]?1 = sup ?N (Cx ) ? [?1 (W + D)]?1 ? sup ?1 (Hx ) ? 0.
x
x
So ?1 (W + D) ? 4.
3.2
MCMC in the Continuous Relaxation
Now we discuss how to perform inference in the augmented distribution resulting from the trick.
One simple choice is to focus on the joint density p(x, s). It is straightforward to generate samples
from the conditional distributions p(x|s) and p(s|x). Therefore one can sample the joint distribution
p(x, s) in a block Gibbs style that switches sampling between p(x|s) and p(s|x). In spite of the simplicity of this method, it has the potential difficulty that it may generate highly correlated samples,
due to the coupling between discrete and continuous samples.
To overcome the drawbacks of block Gibbs sampling, we propose running MCMC directly on the
marginal p(x). We can efficiently evaluate the unnormalized density of p(x) from (11) up to a
4
constant, so we can approximately sample from p(x) using MCMC. The derivatives of log p(x)
have a simple form and can be computed at a linear cost of the number of dimension of x.
That suggests the use of Hamiltonian Monte Carlo, an advanced MCMC method that uses gradient
information to traverse the continuous space efficiently. We refer to the use of HMC on p(x) as
discrete Hamiltonian Monte Carlo (DHMC). An important benefit of HMC is that it is more likely
than many other Metropolis-Hastings methods to accept a proposed sample that has a large change
of log density compared with the current sample.
3.3
Estimating Marginal Probabilities
Given a set of samples that are approximately distributed from p(x) we can estimate the marginal
distribution over any subset Sq ? S of the discrete variables. This is possible because all of the
variables Sq decompose given x. There is no need to generate samples of s because p(s|x) is easy
to compute. The marginal probability p(sq ) can be estimated as
p(sq ) ?
M
M
1 X
1 X Y
p(si |x(m) )
p(Sq |x(m) ) =
M m=1
M m=1
x(m) ? p(x)
si ?Sq
This gives us a Rao-Blackwellized estimate of p(sq ) without needing to sample s directly. We will
not typically be able to obtain exact samples from p(x), as in general it may be multimodal, but we
can obtain approximate samples from an MCMC method.
3.4
Normalizing Constants
Because the normalizing factor Z ?1 of the Boltzmann machine is equal to the probability p(s = 0),
we can estimate the normalizing factor using the technique from the previous section
Z
?1
M
1 X
p(s = 0|x(m) ) x(m) ? p(x).
= p(s = 0) ?
M m=1
(14)
Although if we can sample exactly from p(x), this estimator is unbiased, it suffers from two problems. First, because it is an estimator of Z ?1 , as [15] explains, such an estimator may underestimate
Z and log Z in practice. Second, it can suffer a similar problem as the harmonic mean estimator. The
non-Gaussian term in p(x) corresponds to p(s = 0|x)?1 . In general it is difficult to approximate
the expectation of a function f (x) with respect to a distribution q(x) if f is large precisely where q
is not. This is potentially the situation with this estimator of Z.
We use an alternative estimator using a ?mirrored variant? of the importance trick. First, we introduce a distribution q(x).
Define p? (x) = Zp(x) to be an unnormalized version of p. Using the
R
?1
?1
dx q(x) we have
identity Z = Z
Z
Z
p? (x)
q(x)
Z ?1 = Z ?1 dx q(x) ?
= dx ?
p(x),
p (x)
p (x)
P q(x(m) )
1
(m)
A Monte Carlo estimate of this integral is Z? ?1 = M
? p(x). This estimator
m p? (x(m) ) for x
reduces to (14) if we take q(x) to be the Gaussian portion of p(x) i.e., q(x) = N (x; 0, A(W +
D)AT ). However in practice we have found other choices of q to be much better. Intuitively, the
variance of Z? depends on the ratio of q(x)/p? (x). If p(x) = q(x), the variance of this estimator is
asymptotically zero. This importance trick is well known in the statistics literature, e.g., [14].
4
Related Work
The use of Hamiltonian mechanics in Boltzmann machines and related models (e.g., Ising Models,
stochastic Hopfield models) has an interesting history. The Ising model was studied as a model
of physical spin systems, and so the dynamics used were typically representative of the physics,
with Glauber dynamics [7] being a common model, e.g., [2]. In the context of stochastic neural
models, though, there was the potential to examine other dynamics that did not match the standard
5
physics of spin systems. Hamiltonian dynamics were considered as a neural interaction model [13,
21, 26], but were not applied directly to the Ising model itself, or used for inference. Hamiltonian
dynamics were also considered for networks combining excitatory and inhibitory neurons [22]. All
these approaches involved developing Hamiltonian neural models, rather than Hamiltonian auxiliary
methods for existing models.
The Gaussian integral trick is also known as the Hubbard-Stratonovich transformation in
physics[10]. In the area of neural modelling, the Gaussian integral trick was also common for
theoretical reasons rather than as a practical augmentation strategy [8]. The Gaussian integral trick
formed a critical part of replica-theoretical analysis [8] for phase analysis of spin systems, as it
enabled ensemble averaging of the spin components, leading to saddle-point equations in the continuous domain. These theoretical analysis relied on the ensemble randomness of the interaction
matrix of the stochastic Hopfield model, and so were not directly relevant in the context of a learnt
Boltzmann machine, where the weight matrix has a specific structure.
The Gaussian integral trick relates the general Boltzmann machines to exponential family harmoniums [25], which generalise the restricted Boltzmann machines. The specific Gaussian-Bernoulli
harmonium is in common use, but where the real valued variables are visible units and the binary
variables are hidden variables [9]. This is quite distinct from the use here where the visible and
hidden units are all binary and the Gaussian variables are auxiliary variables.
The only work of which we are aware that uses the Gaussian integral trick for probabilistic inference
is that of Martens and Sutskever [12]. This work also considers inference in MRFs, using the special
1
case of the Gaussian integral trick in which A = ?? 2 V T . However, they do not use the full version
of the trick, as they do not integrate s out, so they never obtain a fully continuous problem. Instead
they perform inference directly in the resulting harmonium, using block Gibbs sampling alternating
between s and x. On serial computers, they do not find that this expanded representation offers much
benefit over performing single-site Gibbs in the discrete space. Indeed they find that the sampler in
the augmented model is actually slightly slower than the one in the original discrete space. This is
in sharp contrast to our work, because we use a Rao-Blackwellized sampler on the x.
5
Results
In this section we evaluate the accuracy of the relaxed sampling algorithms on both synthetic grids
and a real-world task. We evaluate both the estimation of node marginal and of the normalisation
factor estimation on two synthetic models.
We compare the accuracy of the discrete HMC sampler to Gibbs sampling in the original discrete
model p(s) and to block Gibbs sampling the augmented model p(x, s). We choose the number of
MCMC samples so that the total computational time for each method is roughly the same. The
Gibbs sampler resamples one node at a time from p(si |s?i ). The node marginal probability p(si )
is estimated by the empirical probability of the samples. The normalizing constant is estimated in
Chib-style using the Gibbs transition kernel, for more details see [15].
The block Gibbs sampler over p(x, s) we use is based on [12]. This comparison is designed to evaluate the benefits of summing away the discrete variables. To estimate the node marginals, we use
the block Gibbs sampler to generate samples of x and then apply the Rao-Blackwellized estimators
from Sections 3.3 and 3.4. We empirically choose the mirror distribution q as a Gaussian distribution, with mean and variance given by the empirical mean and covariance of the x samples from
MCMC. The samples of s are simply discarded at the estimation stage.
HMC can generate better samples while a large number of leapfrog steps is used, but this requires
much more computation time. For a fixed computational budget, using more leapfrog steps causes
fewer samples to be generated, which can also undermine the accuracy of the estimator. So, we
empirically pick 5 leapfrog steps and tuning the leapfrog step size so that acceptance rate is around
90%. We use A = I for DHMC. To estimate the marginals p(si ) and the partition function, we apply
the Rao-Blackwellized estimators from Sections 3.3 and 3.4 in the same way as for block Gibbs.
Synthetic Boltzmann Machines. We evaluate the performance of the samplers across different
types of weight matrices by using synthetically generated models. The idea is to characterize what
types of distributions are difficult for each sampler.
6
HMC on p(x)
Block Gibbs on p(x, s)
3
3
3
2.5
2.5
2.5
2
1.5
25
20
biases scale
biases scale
biases scale
Gibbs on p(s)
2
1.5
2
15
1.5
10
1
1
1
0.5
0.5
0.5
5
0.5
1
1.5
2
2.5
3
0.5
1
1.5
2
2.5
0.5
3
1
3
2.5
2
1.5
2
2.5
3
0.022
3
0.02
2.5
biases scale
3
2.5
1.5
weights scale
weights scale
biases scale
biases scale
weights scale
2
1.5
0.018
2
0.016
0.014
1.5
0.012
1
1
1
0.5
0.5
0.5
0.01
0.5
1
1.5
2
weights scale
2.5
3
0.5
1
1.5
2
weights scale
2.5
3
0.008
0.5
1
1.5
2
2.5
3
weights scale
Figure 2: Performance of samplers on synthetic grid-structured Boltzmann machines. The axes show
the standard deviations of the distributions used to select the synthetic models. The top row shows
error in the normalization constant, while the bottom row shows average error in the single-mode
marginal distributions.
We randomly generate 10 ? 10 grid models over binary variables. We use two different generating
processes, a ?standard? one and a ?frustrated? one. In the standard case, for each node si , the biases
are generated as ai ? c1 N (0, 4). The weights are generated as wij ? c2 N (0, 4). The parameters
c1 and c2 define the scales of the biases and weights and determine how hard the problem is. In
the frustrated case, we shift the weights to make the problem more difficult.
P We still generate the
weights as wij ? c2 N (0, 4) but now we generate the biases as ai ? c1 N ( i wij , 4). This shift in
the Gaussian distribution tends to encourage the multimodality of p(x).
We test all three samplers on 36 random graphs from each of the two generating processes, using
different values of c1 and c2 for each random graph. Each MCMC method is given 10 runs of 10000
samples with 2000 burn-in samples. We report the MSE of the node marginal estimate and the log
normalising constant estimate averaged over 10 runs.
The results are shown in Figure 2 and 3. The axes show c1 and c2 , which determine the difficulty
of the problem. The higher value in the heat maps means a larger error. On the standard graphs
(Figure 2), the DHMC method significantly outperforms both competitors. DHMC beats Gibbs on
p(s) at the normalization constant and beats block Gibbs on p(x, s) at marginal estimation.
The frustrated graphs (Figure 3) are significantly more difficult for DHMC, as expected. All three
samplers seem to have trouble in the same area of model space, although DHMC suffers somewhat
worse than the other methods in marginal error, while still beating Chib?s method for normalization
constants. It is noted that in worst scenarios, the weights of the model are very extreme. Examining a
few representative graphs seems to indicate that in the regime with large weights, the HMC sampler
becomes stuck in a bad mode. We observe that in both cases block Gibbs of p(x, s) performs roughly
the same at marginal estimation as Gibbs on p(s). This is consistent with the results in [12].
Text data. We also evaluate the relaxed inference methods on a small text labelling data set. The
data are a series of email messages that announce academic seminars [5]. We consider the binary
problem of determining whether or not each word in the message is part of the name of the seminar?s
speaker, so we have one random variable for each token in the message. We use a ?skip-chain CRF?
[4, 23] model which contains edges between adjacent tokens and also additional edges between any
pair of identical capitalized words.
We trained the CRF on a set of 485 messages using belief propagation. We evaluate the performance
of different inference methods on inferring the probability distribution over labels on a held out set
7
Gibbs on p(s)
HMC on p(x)
Block Gibbs on p(x, s)
3
3
3
2.5
2.5
2.5
100
2
1.5
biases scale
biases scale
biases scale
90
2
1.5
80
70
2
60
50
1.5
40
1
1
1
0.5
0.5
0.5
30
20
0.5
1
1.5
2
2.5
3
0.5
1
weights scale
1.5
2
2.5
3
10
0.5
1
weights scale
1.5
2
2.5
3
weights scale
3
3
3
2.5
2.5
2.5
0.45
2
1.5
0.35
biases scale
biases scale
biases scale
0.4
2
1.5
0.3
2
0.25
1.5
0.2
0.15
1
1
1
0.5
0.5
0.5
0.1
0.5
1
1.5
2
2.5
3
0.5
weights scale
1
1.5
2
2.5
3
0.05
0.5
weights scale
1
1.5
2
2.5
3
weights scale
Figure 3: Performance of samplers on a set of highly frustrated grid-structured Boltzmann machines.
The axes show the standard deviations of the distributions used to select the synthetic models. The
top row shows error in the normalization constant, while the bottom row shows average error in the
single-mode marginal distributions.
of messages. Our test set uses a subset of the messages which are small enough that we can run exact
inference. The test set contained 75 messages whose length ranged from 50 words to 628 words.
We evaluate whether the approximate inference methods match the solution from exact inference.
The accuracy of the three approximate inference methods are shown in Table 1. We see that the HMC
sampler is much more accurate than either of the other samplers at estimating single-node marginals.
Chib?s method and DHMC have roughly the same accuracy on the normalization constant. The block
Gibbs sampler yields both worse estimates of the marginals and a significantly worse estimate of the
normalization constant.
RMSE (node marginal)
RSME (log normalizing constant)
Gibbs p(s)
DHMC p(x)
Block Gibbs p(s, x)
0.2346
3.3041
0.1619
3.3171
0.2251
12.9685
Table 1: Root mean squared error of single site marginal and normalising constant against the ground
truth computed by the variable elimination algorithm
6
Conclusion
We have provided a general strategy for approximate inference based on relaxing discrete distributions into continuous ones using the classical Gaussian integral trick. We described a continuum of
different versions of the trick that have different properties. Although we illustrated the benefits of
the continuous setting by using Hamiltonian Monte Carlo, in future work other inference methods
such as elliptical slice sampling or more advanced HMC methods may prove superior. We hope that
this work might open the door to a larger space of interesting relaxations for approximate inference.
Acknowledgments
We thank Iain Murray for helpful comments, Max Welling for introducing ZG to the GIT and Peter
Sollich for bringing the paper by Hubbard to our attention. This work was supported by the Engineering and Physical Sciences Research Council [grant numbers EP/I036575/1 and EP/J00104X/1].
8
References
[1] D. J. Amit. Modeling Brain Function. Cambridge University Press, 1989.
[2] A. Coolen, S. Laughton, and D. Sherrington. Modern analytic techniques to solve the dynamics of recurrent neural networks. In Advances in Neural Information Processing Systems 8 (NIPS95), 1996.
[3] S. Ermon, C. P. Gomes, A. Sabharwal, and B. Selman. Accelerated adaptive Markov chain for partition
function computation. In J. Shawe-Taylor, R. Zemel, P. Bartlett, F. Pereira, and K. Weinberger, editors,
Advances in Neural Information Processing Systems 24, pages 2744?2752. 2011.
[4] J. Finkel, T. Grenager, and C. D. Manning. Incorporating non-local information into information extraction systems by Gibbs sampling. In Annual Meeting of the Association for Computational Linguistics
(ACL), 2005.
[5] D. Frietag and A. McCallum. Information extraction with HMMs and shrinkage. In AAAI Workshop on
Machine Learning for Information Extraction, 1999.
[6] M. Girolami, B. Calderhead, and S. A. Chin. Riemannian manifold Hamiltonian Monte Carlo. Journal
of the Royal Statistical Society, B, 73(2):1?37, 2011.
[7] R. J. Glauber. Time-dependent statistics of the Ising model. J. Math. Phys., 4:294?307, 1963.
[8] J. Hertz, A. Krogh, and R. G. Palmer. Introduction to the Theory of Neural Computation. Perseus Books,
1991.
[9] G. E. Hinton and R. R. Salakhutdinov. Reducing the dimensionality of data with neural networks. Science,
313:504?507, 2006.
[10] J. Hubbard. Calculation of partition functions. Phys. Rev. Lett., 3:77?78, Jul 1959. doi: 10.1103/
PhysRevLett.3.77. URL http://link.aps.org/doi/10.1103/PhysRevLett.3.77.
[11] M. Johnson, T. Griffiths, and S. Goldwater. Bayesian inference for PCFGs via Markov chain Monte Carlo.
In HLT/NAACL, 2007.
[12] J. Martens and I. Sutskever. Parallelizable sampling of Markov random fields. In Conference on Artificial
Intelligence and Statistics (AISTATS), 2010.
[13] R. V. Mendes and J. T. Duarte. Vector fields and neural networks. Complex Systems, 6:21?30, 1992.
[14] J. M?ller, A. N. Pettitt, R. Reeves, and K. K. Berthelsen. An efficient Markov chain Monte Carlo method
for distributions with intractable normalising constants. Biometrika, 93(2):pp. 451?458, 2006.
[15] I. Murray and R. Salakhutdinov. Evaluating probabilities under high-dimensional latent variable models. In D. Koller, D. Schuurmans, Y. Bengio, and L. Bottou, editors, Advances in Neural Information
Processing Systems 21, pages 1137?1144. 2009.
[16] I. Murray, R. P. Adams, and D. J. MacKay. Elliptical slice sampling. JMLR: W&CP, 9:541?548, 2010.
[17] R. Neal. Bayesian Learning for Neural Networks. PhD thesis, Computer Science, University of Toronto,
1995.
[18] R. M. Neal. MCMC using Hamiltonian dynamics. In S. Brooks, A. Gelman, G. Jones, and X.-L. Meng,
editors, Handbook of Markov Chain Monte Carlo. Chapman & Hall / CRC Press, 2010.
[19] S. Nowozin and C. H. Lampert. Structured prediction and learning in computer vision. Foundations and
Trends in Computer Graphics and Vision, 6(3-4), 2011.
[20] Y. Qi and T. P. Minka. Hessian-based Markov chain Monte-Carlo algorithms. In First Cape Cod Workshop
on Monte Carlo Methods, September 2002.
[21] U. Ramacher. The Hamiltonian approach to neural networks dynamics. In Proc. IEEE JCNN, volume 3,
1991.
[22] H. S. Seung, T. J. Richardson, J. C. Lagarias, and J. J. Hopfield. Minimax and Hamiltonian dynamics
of excitatory-inhibitory networks. In Advances in Neural Information Processing Systems 10 (NIPS97),
1998.
[23] C. Sutton and A. McCallum. Collective segmentation and labeling of distant entities in information
extraction. In ICML Workshop on Statistical Relational Learning and Its Connections to Other Fields,
2004.
[24] C. Sutton and A. McCallum. An Introduction to Conditional Random Fields. Foundations and Trends in
Machine Learning, 4(4), 2012.
[25] M. Welling, M. Rosen-Zvi, and G. Hinton. Exponential family harmoniums with an application to information retrieval. In Advances in Neural Information Processing 17, 2004.
[26] P. D. Wilde. Class of Hamiltonian neural networks. Physical Review E, 2:1392?1396, 47.
[27] Y. Zhang and C. Sutton. Quasi-Newton Markov chain Monte Carlo. In Advances in Neural Information
Processing Systems (NIPS), 2011.
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4,036 | 4,653 | Multiple Operator-valued Kernel Learning
Hachem Kadri
LIF - CNRS / INRIA Lille - Sequel Project
Universit?e Aix-Marseille
Marseille, France
[email protected]
Alain Rakotomamonjy
LITIS EA 4108
Universit?e de Rouen
St Etienne du Rouvray, France
[email protected]
Philippe Preux
INRIA Lille - Sequel Project
LIFL - CNRS, Universit?e de Lille
Villeneuve d?Ascq, France
[email protected]
Francis Bach
INRIA - Sierra Project
Ecole Normale Sup?erieure
Paris, France
[email protected]
Abstract
Positive definite operator-valued kernels generalize the well-known notion of
reproducing kernels, and are naturally adapted to multi-output learning situations. This paper addresses the problem of learning a finite linear combination
of infinite-dimensional operator-valued kernels which are suitable for extending
functional data analysis methods to nonlinear contexts. We study this problem in
the case of kernel ridge regression for functional responses with an `r -norm constraint on the combination coefficients (r ? 1). The resulting optimization problem is more involved than those of multiple scalar-valued kernel learning since
operator-valued kernels pose more technical and theoretical issues. We propose a
multiple operator-valued kernel learning algorithm based on solving a system of
linear operator equations by using a block coordinate-descent procedure. We experimentally validate our approach on a functional regression task in the context
of finger movement prediction in brain-computer interfaces.
1
Introduction
During the past decades, a large number of algorithms have been proposed to deal with learning
problems in the case of single-valued functions (e.g., binary-output function for classification or real
output for regression). Recently, there has been considerable interest in estimating vector-valued
functions [21, 5, 7]. Much of this interest has arisen from the need to learn tasks where the target is
a complex entity, not a scalar variable. Typical learning situations include multi-task learning [11],
functional regression [12], and structured output prediction [4].
In this paper, we are interested in the problem of functional regression with functional responses in
the context of brain-computer interface (BCI) design. More precisely, we are interested in finger
movement prediction from electrocorticographic signals [23]. Indeed, from a set of signals measuring brain surface electrical activity on d channels during a given period of time, we want to predict,
for any instant of that period whether a finger is moving or not and the amplitude of the finger flexion. Formally, the problem consists in learning a functional dependency between a set of d signals
and a sequence of labels (a step function indicating whether a finger is moving or not) and between
the same set of signals and vector of real values (the amplitude function). While, it is clear that this
problem can be formalized as functional regression problem, from our point of view, such problem
can benefit from the multiple operator-valued kernel learning framework. Indeed, for these problems, one of the difficulties arises from the unknown latency between the signal related to the finger
1
movement and the actual movement [23]. Hence, instead of fixing in advance some value for this
latency in the regression model, our framework allows to learn it from the data by means of several
operator-valued kernels.
If we wish to address functional regression problem in the principled framework of reproducing
kernel Hilbert spaces (RKHS), we have to consider RKHSs whose elements are operators that map
a function to another function space, possibly source and target function spaces being different.
Working in such RKHSs, we are able to draw on the important core of work that has been performed on scalar-valued and vector-valued RKHSs [28, 21]. Such a functional RKHS framework
and associated operator-valued kernels have been introduced recently [12, 13]. A basic question
with reproducing kernels is how to build these kernels and what is the optimal kernel choice for a
given application. In order to overcome the need for choosing a kernel before the learning process,
several works have tried to address the problem of learning the scalar-valued kernel jointly with
the decision function [18, 29]. Since these seminal works, many efforts have been carried out in
order to theoretically analyze the kernel learning framework [9, 3] or in order to provide efficient
algorithms [24, 1, 15]. While many works have been devoted to multiple scalar-valued kernel learning, this problem of kernel learning have been barely investigated for operator-valued kernels. One
motivation of this work is to bridge the gap between multiple kernel learning (MKL) and operatorvalued kernels by proposing a framework and an algorithm for learning a finite linear combination
of operator-valued kernels. While each step of the scalar-valued MKL framework can be extended
without major difficulties to operator-valued kernels, technical challenges arise at all stages because
we deal with infinite dimensional spaces. It should be pointed out that in a recent work [10], the
problem of learning the output kernel was formulated as an optimization problem over the cone
of positive semidefinite matrices, and a block-coordinate descent method was proposed to solve it.
However, they did not focus on learning the input kernel. In contrast, our multiple operator-valued
kernel learning formulation can be seen as a way of learning simultaneously input and output kernels, although we consider a linear combination of kernels that are fixed in advance.
In this paper, we make the following contributions: 1) we introduce a novel approach to infinitedimensional multiple operator-valued kernel learning (MovKL) suitable for learning the functional
dependencies and interactions between continuous data; 2) we extend the original formulation of
ridge regression in dual variables to the functional data analysis domain, showing how to perform
nonlinear functional regression with functional responses by constructing a linear regression operator in an operator-valued kernel feature space (Section 2); 3) we derive a dual form of the MovKL
problem with functional ridge regression, and show that a solution of the related optimization problem exists (Section 2); 4) we propose a block-coordinate descent algorithm to solve the MovKL
optimization problem which involves solving a challenging linear system with a sum of block operator matrices (Section 3); 5) we provide an empirical evaluation of MovKL performance which
demonstrates its effectiveness on a BCI dataset (Section 4).
2
Problem Setting
Before describing the multiple operator-valued kernel learning algorithm that we will study and experiment with in this paper, we first review notions and properties of reproducing kernel Hilbert
spaces with operator-valued kernels, show their connection to learning from multiple response
data (multiple outputs; see [21] for discrete data and [12] for continuous data), and describe the
optimization problem for learning kernels with functional response ridge regression.
2.1
Notations and Preliminaries
We start by some standard notations and definitions used all along the paper. Given a Hilbert
space H, h?, ?iH and k ? kH refer to its inner product and norm, respectively. We denote by Gx
and Gy the separable real Hilbert spaces of input and output functional data, respectively. In functional data analysis domain, continuous data are generally assumed to belong to the space of square
integrable functions L2 . In this work, we consider that Gx and Gy are the Hilbert space L2 (?) which
consists of all equivalence classes of square integrable functions on a finite set ?. ? being potentially different for Gx and Gy . We denote by F(Gx , Gy ) the vector space of functions f : Gx ?? Gy ,
and by L(G y ) the set of bounded linear operators from Gy to Gy .
2
We consider the problem of estimating a function f such that f (xi ) = yi when observed functional
data (xi , yi )i=1,...,n ? (Gx , Gy ). Since Gx and Gy are spaces of functions, the problem can be thought
of as an operator estimation problem, where the desired operator maps a Hilbert space of factors to
a Hilbert space of targets. We can define the regularized operator estimate of f ? F as:
n
1X
f? , arg min
kyi ? f (xi )k2Gy + ?kf k2F .
(1)
f ?F n
i=1
In this work, we are looking for a solution to this minimization problem in a function-valued reproducing kernel Hilbert space F. More precisely, we mainly focus on the RKHS F whose elements
are continuous linear operators on Gx with values in Gy . The continuity property is obtained by
considering a special class of reproducing kernels called Mercer kernels [7, Proposition 2.2]. Note
that in this case, F is separable since Gx and Gy are separable [6, Corollary 5.2].
We now introduce (function) Gy -valued reproducing kernel Hilbert spaces and show the correspondence between such spaces and positive definite (operator) L(G y )-valued kernels. These extend the
traditional properties of scalar-valued kernels.
Definition 1 (function-valued RKHS)
A Hilbert space F of functions from Gx to Gy is called a reproducing kernel Hilbert space if there is
a positive definite L(G y )-valued kernel KF (w, z) on Gx ? Gx such that:
i. the function z 7?? KF (w, z)g belongs to F, ?z ? Gx , w ? Gx , g ? Gy ,
ii. ?f ? F, w ? Gx , g ? Gy , hf, KF (w, ?)giF = hf (w), giGy (reproducing property).
Definition 2 (operator-valued kernel)
An L(G y )-valued kernel KF (w, z) on Gx is a function KF (?, ?) : Gx ? Gx ?? L(G y ); furthermore:
i. KF is Hermitian if KF (w, z) = KF (z, w)? , where ? denotes the adjoint operator,
ii. KF is positive definite on GxPif it is Hermitian and for every natural number r and all
{(wi , ui )i=1,...,r } ? Gx ? Gy , i,j hKF (wi , wj )uj , ui iGy ? 0.
Theorem 1 (bijection between function-valued RKHS and operator-valued kernel)
An L(G y )-valued kernel KF (w, z) on Gx is the reproducing kernel of some Hilbert space F, if and
only if it is positive definite.
The proof of Theorem 1 can be found in [21]. For further reading on operator-valued kernels and
their associated RKHSs, see, e.g., [5, 6, 7].
2.2
Functional Response Ridge Regression in Dual Variables
We can write the ridge regression with functional responses optimization problem (1) as follows:
n
1 X
1
min kf k2F +
k?i k2Gy
(2)
f ?F 2
2n? i=1
with ?i = yi ? f (xi ).
Now, we introduce the Lagrange multipliers ?i , i = 1, . . . , n which are functional variables since
the output space is the space of functions Gy . For the optimization problem (2), the Lagrangian
multipliers exist and the Lagrangian function is well defined. The method of Lagrange multipliers on
Banach spaces, which is a generalization of the classical (finite-dimensional) Lagrange multipliers
method suitable to solve certain infinite-dimensional constrained optimization problems, is applied
here. For more details, see [16]. Let ? = (?i )i=1,...,n ? Gyn the vector of functions containing the
Lagrange multipliers, the Lagrangian function is defined as
1
1
L(f, ?, ?) = kf k2F +
k?k2Gyn + h?, y ? f (x) ? ?iGyn ,
(3)
2
2n?
where ? = (?1 , . . . , ?n ) ? Gyn , y = (y1 , . . . , yn ) ? Gyn , f (x) = (f (x1 ), . . . , f (xn )) ? Gyn ,
n
P
? = (?1 , . . . , ?n ) ? Gyn , and ?a, b ? Gyn , ha, biGyn =
hai , bi iGy .
i=1
3
Differentiating (3) with respect to f ? F and setting to zero, we obtain
f (.) =
n
X
K(xi , .)?i ,
(4)
i=1
where K : Gx ? Gx ?? L(G y ) is the operator-valued kernel of F.
Substituting this into (3) and minimizing with respect to ?, we obtain the dual of the functional
response kernel ridge regression (KRR) problem
n?
1
k?k2Gyn ? hK?, ?iGyn + h?, yiGyn ,
(5)
2
2
where K = [K(xi , xj )]ni,j=1 is the block operator kernel matrix. The computational details regrading the dual formulation of functional KRR are derived in Appendix B of [14].
max ?
?
2.3
MovKL in Dual Variables
Let us now consider that the function f (?) is sum of M functions {fk (?)}M
k=1 where each fk belongs
to a Gy -valued RKHS with kernel Kk (?, ?). Similarly to scalar-valued multiple kernel learning, we
can cast the problem of learning these functions fk as
min min
M
X
kfk k2F
k
n
+
1 X
k?i k2Gy
2n? i=1
(6)
2dk
PM
with ?i = yi ? k=1 fk (xi ),
P
where d = [d1 , ? ? ? , dM ], D = {d : ?k, dk ? 0 and k drk ? 1} and 1 ? r ? ?. Note that this
problem can equivalently be rewritten as an unconstrained optimization problem. Before deriving
the dual of this problem, it can be shown by means of the generalized Weierstrass theorem [17] that
this problem admits a solution. We report the proof in Appendix A of [14].
d?D fk ?Fk
k=1
Now, following the lines of [24], a dualization of this problem leads to the following equivalent one
min max ?
d?D ??Gyn
where K =
M
P
1
n?
k?k2Gyn ? hK?, ?iGyn + h?, yiGyn ,
2
2
(7)
dk Kk and Kk is the block operator kernel matrix associated to the operator-valued
k=1
kernel Kk . The KKT conditions also state that at optimality we have fk (?) =
n
P
dk Kk (xi , ?)?i .
i=1
3
Solving the MovKL Problem
After having presented the framework, we now devise an algorithm for solving this MovKL problem.
3.1
Block-coordinate descent algorithm
Since the optimization problem (6) has the same structure as a multiple scalar-valued kernel learning
problem, we can build our MovKL algorithm upon the MKL literature. Hence, we propose to
borrow from [15], and consider a block-coordinate descent method. The convergence of a block
coordinate descent algorithm which is related closely to the Gauss-Seidel method was studied in
works of [30] and others. The difference here is that we have operators and block operator matrices
rather than matrices and block matrices, but this doesn?t increase the complexity if the inverse of
the operators are computable (typically analytically or by spectral decomposition). Our algorithm
iteratively solves the problem with respect to ? with d being fixed and then with respect to d with ?
being fixed (see Algorithm 1). After having initialized {dk } to non-zero values, this boils down to
the following steps :
1. with {dk } fixed, the resulting optimization problem with respect to ? has the following
form:
(K + ?I)? = y,
(8)
4
PM
where K = k=1 dk Kk . While the form of solution is rather simple, solving this linear
system is still challenging in the operator setting and we propose below an algorithm for its
resolution.
2. with {fk } fixed, according to problem (6), we can rewrite the problem as
min
d?D
M
X
kfk k2F
k
k=1
dk
(9)
which has a closed-form solution and for which optimality occurs at [20]:
2
kfk k r+1
.
dk = P
2r
( k kfk k r+1 )1/r
(10)
This algorithm is similar to that of [8] and [15] both being based on alternating optimization. The
difference here is that we have to solve a linear system involving a block-operator kernel matrix
which is a combination of basic kernel matrices associated to M operator-valued kernels. This
makes the system very challenging, and we present an algorithm for solving it in the next paragraph.
We also report in Appendix C of [14] a convergence proof of a modified version of the MovKL
algorithm that minimizes a perturbation of the objective function (6) with a small positive parameter
required to guarantee convergence [2].
3.2
Solving a linear system involving multiple operator-valued kernel matrices
One common way to construct operator-valued kernels is to build scalar-valued ones which are
carried over to the vector-valued (resp. function-valued) setting by a positive definite matrix (resp.
operator). In this setting an operator-valued kernel has the following form:
K(w, z) = G(w, z)T,
where G is a scalar-valued kernel and T is a positive operator in L(G y ). In multi-task learning,
T is a finite dimensional matrix that is expected to share information between tasks [11, 5]. More
recently and for supervised functional output learning problems, T is chosen to be a multiplication or an integral operator [12, 13]. This choice is motivated by the fact that functional linear
models for functional responses [25] are based on these operators and then such kernels provide
an interesting alternative to extend these models to nonlinear contexts. In addition, some works on
functional regression and structured-output learning consider operator-valued kernels constructed
from the identity operator as in [19] and [4]. In this work we adopt a functional data analysis point
of view and then we are interested in a finite combination of operator-valued kernels constructed
from the identity, multiplication and integral operators. A problem encountered when working with
operator-valued kernels in infinite-dimensional spaces is that of solving the system of linear operator
equations (8). In the following we show how to solve this problem for two cases of operator-valued
kernel combinations.
Case 1: multiple scalar-valued kernels and one operator. This is the simpler case where the
combination of operator-valued kernels has the following form
K(w, z) =
M
X
dk Gk (w, z)T,
(11)
k=1
where Gk is a scalar-valued kernel, T is a positive operator in L(G y ), and dk are the combination coefficients. In this setting, the block operator kernel matrix K can be expressed as a
PM
Kronecker product between the multiple scalar-valued kernel matrix G =
k=1 dk Gk , where
Gk = [Gk (xi , xj )]ni,j=1 , and the operator T . Thus we can compute an analytic solution of the
system of equations (8) by inverting K + ?I using the eigendecompositions of G and T as in [13].
Case 2: multiple scalar-valued kernels and multiple operators. This is the general case where
multiple operator-valued kernels are combined as follows
K(w, z) =
M
X
dk Gk (w, z)Tk ,
k=1
5
(12)
Algorithm 1 `r -norm MovKL
Algorithm 2 Gauss-Seidel Method
Input Kk for k = 1, . . . , M
1
for k = 1, . . . , M
d1k ??
M
? ?? 0
for t = 1, 2, . . . do
?0 ?? ?
P
K ?? k dtk Kk
choose an initial vector of functions ?(0)
repeat
for i = 1, 2, . . . , n
(t)
?? sol. of (13):
(t)
[K(xi , xi ) + ?I]?i = si
end for
until convergence
?i
? ?? solution of (K + ?I)? = y
if k? ? ?0 k < then
break
end if
2
kfk k r+1
for k = 1, . . . , M
dt+1
??
P
2r
k
( k kfk k r+1 )1/r
end for
where Gk is a scalar-valued kernel, Tk is a positive operator in L(G y ), and dk are the combination
coefficients. Inverting the associated block operator kernel matrix K is not feasible in this case,
that is why we propose a Gauss-Seidel iterative procedure (see Algorithm 2) to solve the system
of linear operator equations (8). Starting from an initial vector of functions ?(0) , the idea is to
iteratively compute, until a convergence condition is satisfied, the functions ?i according to the
following expression
(t)
[K(xi , xi ) + ?I]?i = yi ?
i?1
X
(t)
K(xi , xj )?j ?
j=1
n
X
(t?1)
K(xi , xj )?j
,
(13)
j=i+1
where t is the iteration index. This problem is still challenging because the kernel K(?, ?) still
involves a positive combination of operator-valued kernels. Our algorithm is based on the idea
that instead of inverting the finite combination of operator-valued kernels [K(xi , xi ) + ?I], we can
consider the following variational formulation of this system
M +1
1 X
(t)
(t)
(t)
Kk (xi , xi )?i , ?i iGy ? hsi , ?i iGy ,
min h
(t) 2
?i
k=1
where si = yi ?
i?1
P
j=1
(t)
K(xi , xj )?j ?
n
P
j=i+1
(t?1)
K(xi , xj )?j
, Kk = dk Gk Tk , ?k ? {1, . . . , M },
and KM +1 = ?I.
Now, by means of a variable-splitting approach, we are able to decouple the role of the different
kernels. Indeed, the above problem is equivalent to the following one :
1 ?
(t)
(t)
(t)
(t)
(t)
min hK(x
i , xi )?i , ?i iGyM ? hsi , ?i iGyM with ?i,1 = ?i,k for k = 2, . . . , M + 1,
(t) 2
?i
? i , xi ) is the (M + 1) ? (M + 1) diagonal matrix [Kk (xi , xi )]M +1 . ?(t) is the vector
where K(x
i
k=1
(t)
(t)
(?i,1 , . . . , ?i,M +1 ) and the (M + 1)-dimensional vector si = (si , 0, . . . , 0). We now have to deal
with a quadratic optimization problem with equality constraints. Writing down the Lagrangian
of this optimization problem and then deriving its first-order optimality conditions leads us to the
following set of linear equations
?
P
? K1 (xi , xi )?i,1 ? si + M
= 0
k=1 ?k
(14)
Kk (xi , xi )?i,k ? ?k
= 0
?
?i,1 ? ?i,k
= 0
where k = 2, . . . , M + 1 and {?k } are the Lagrange multipliers related to the M equality constraints. Finally, in this set of equations, the operator-valued kernels have been decoupled and thus,
if their inversion can be easily computed (which is the case in our experiments), one can solve the
problem (14) with respect to {?i,k } and ?k by means of another Gauss-Seidel algorithm after simple
reorganization of the linear system.
6
Ch. 1
20
1.5
6
0
?20
0
20
40
60
80
100
120
140
160
180
200
0
20
40
60
80
100
120
140
160
180
200
5
1
4
?10
Ch. 3
5
0
?5
0
20
40
60
80
100
120
140
160
180
200
Ch. 4
5
0
?0.5
3
2
1
0
?5
0
20
40
60
80
100
120
140
160
180
200
5
Ch. 5
0.5
Finger Movement
0
Finger Movement State
Ch. 2
10
0
?1
?1
0
?5
0
20
40
60
80
100
120
140
160
180
200
?1.5
0
50
100
Time samples
Time samples
150
200
0
50
100
Time samples
150
200
Figure 1: Example of a couple of input-output signals in our BCI task. (left) Amplitude modulation features extracted from ECoG signals over 5 pre-defined channels. (middle) Signal of labels
denoting whether the finger is moving or not. (right) Real amplitude movement of the finger.
4
Experiments
In order to highlight the benefit of our multiple operator-valued kernel learning approach, we have
considered a series of experiments on a real dataset, involving functional output prediction in a
brain-computer interface framework. The problem we addressed is a sub-problem related to finger movement decoding from Electrocorticographic (ECoG) signals. We focus on the problem of
estimating if a finger is moving or not and also on the direct estimation of the finger movement
amplitude from the ECoG signals. The development of the full BCI application is beyond the scope
of this paper and our objective here is to prove that this problem of predicting finger movement can
benefit from multiple kernel learning.
To this aim, the fourth dataset from the BCI Competition IV [22] was used. The subjects were 3
epileptic patients who had platinium electrode grids placed on the surface of their brains. The number of electrodes varies between 48 to 64 depending on the subject, and their position on the cortex
was unknown. ECoG signals of the subject were recorded at a 1KHz sampling using BCI2000 [27].
A band-pass filter from 0.15 to 200Hz was applied to the ECoG signals. The finger flexion of the
subject was recorded at 25Hz and up-sampled to 1KHz by means of a data glove which measures
the finger movement amplitude. Due to the acquisition process, a delay appears between the finger
movement and the measured ECoG signal [22]. One of our hopes is that this time-lag can be properly learnt by means of multiple operator-valued kernels. Features from the ECoG signals are built
by computing some band-specific amplitude modulation features, which is defined as the sum of the
square of the band-specific filtered ECoG signal samples during a fixed time window.
For our finger movement prediction task, we have kept 5 channels that have been manually selected
and split ECoG signals in portions of 200 samples. For each of these time segments, we have the
label of whether at each time sample, the finger is moving or not as well as the real movement
amplitudes. The dataset is composed of 487 couples of input-output signals, the output signals
being either the binary movement labels or the real amplitude movement. An example of inputoutput signals are depicted in Figure 1. In a nutshell, the problem boils down to be a functional
regression task with functional responses.
To evaluate the performance of the multiple operator-valued kernel learning approach, we use both:
(1) the percentage of labels correctly recognized (LCR) defined by (Wr /Tn ) ? 100%, where Wr
is the number of well-recognized labels and Tn the total number of labels to be recognized; (2) the
residual sum of squares error (RSSE) as evaluation criterion for curve prediction
Z X
RSSE =
{yi (t) ? ybi (t)}2 dt,
(15)
i
where ybi (t) is the prediction of the function yi (t) corresponding to real finger movement or the
finger movement state.
For the multiple operator-valued kernels having the form (12), we have used a Gaussian kernel
with 5 different bandwidths and a polynomial kernel of degree 1 to 3 combined with three oper2
ators T : identity T y(t) = y(t), multiplication operator associated with the function e?t defined
2
by T y(t) = e?t y(t),
and the integral Hilbert-Schmidt operator with the kernel e?|t?s| proposed
R ?|t?s|
in [13], T y(t) = e
y(s)ds. The inverses of these operators can be computed analytically.
7
Table 1: (Left) Results for the movement state prediction. Residual Sum of Squares Error (RSSE)
and the percentage number of Labels Correctly Recognized (LCR) of : (1) baseline KRR with the
Gaussian kernel, (2) functional response KRR with the integral operator-valued kernel, (3) MovKL
with `? , `1 and `2 -norm constraint. (Right) Residual Sum of Squares Error (RSSE) results for
finger movement prediction.
Algorithm
RSSE
LCR(%)
Algorithm
RSSE
KRR - scalar-valued KRR - functional response MovKL - `? norm MovKL - `1 norm MovKL - `2 norm -
68.32
49.40
45.44
48.12
39.36
72.91
80.20
81.34
80.66
84.72
KRR - scalar-valued KRR - functional response MovKL - `? norm MovKL - `1 norm MovKL - `2 norm -
88.21
79.86
76.52
78.24
75.15
While the inverses of the identity and the multiplication operators are easily and directly computable
from the analytic expressions of the operators, the inverse of the integral operator is computed from
its spectral decomposition as in [13]. The number of eigenfunctions as well as the regularization
parameter ? are fixed using ?one-curve-leave-out cross-validation? [26] with the aim of minimizing
the residual sum of squares error.
Empirical results on the BCI dataset are summarized in Table 1. The dataset was randomly partitioned into 65% training and 35% test sets. We compare our approach in the case of `1 and `2 -norm
constraint on the combination coefficients with: (1) the baseline scalar-valued kernel ridge regression algorithm by considering each output independently of the others, (2) functional response ridge
regression using an integral operator-valued kernel [13], (3) kernel ridge regression with an evenlyweighted sum of operator-valued kernels, which we denote by `? -norm MovKL.
As in the scalar case, using multiple operator-valued kernels leads to better results. By directly combining kernels constructed from identity, multiplication and integral operators we could reduce the
residual sum of squares error and enhance the label classification accuracy. Best results are obtained
using the MovKL algorithm with `2 -norm constraint on the combination coefficients. RSSE and
LCR of the baseline kernel ridge regression are significantly outperformed by the operator-valued
kernel based functional response regression. These results confirm that by taking into account the
relationship between outputs we can improve performance. This is due to the fact that an operatorvalued kernel induces a similarity measure between two pairs of input/output.
5
Conclusion
In this paper we have presented a new method for learning simultaneously an operator and a finite linear combination of operator-valued kernels. We have extended the MKL framework to deal
with functional response kernel ridge regression and we have proposed a block coordinate descent
algorithm to solve the resulting optimization problem. The method is applied on a BCI dataset to
predict finger movement in a functional regression setting. Experimental results show that our algorithm achieves good performance outperforming existing methods. It would be interesting for future
work to thoroughly compare the proposed MKL method for operator estimation with previous related methods for multi-class and multi-label MKL in the contexts of structured-output learning and
collaborative filtering.
Acknowledgments
We would like to thank the anonymous reviewers for their valuable comments. This research was
funded by the Ministry of Higher Education and Research, Nord-Pas-de-Calais Regional Council
and FEDER (Contrat de Projets Etat Region CPER 2007-2013), ANR projects LAMPADA (ANR09-EMER-007) and ASAP (ANR-09-EMER-001), and by the IST Program of the European Community under the PASCAL2 Network of Excellence (IST-216886). This publication only reflects the
authors? views. Francis Bach was partially supported by the European Research Council (SIERRA
Project).
8
References
[1] J. Aflalo, A. Ben-Tal, C. Bhattacharyya, J. Saketha Nath, and S. Raman. Variable sparsity kernel learning.
JMLR, 12:565?592, 2011.
[2] A. Argyriou, T. Evgeniou, and M. Pontil. Convex multi-task feature learning. Machine Learning,
73(3):243?272, 2008.
[3] F. Bach. Consistency of the group Lasso and multiple kernel learning. JMLR, 9:1179?1225, 2008.
[4] C. Brouard, F. d?Alch?e-Buc, and M. Szafranski. Semi-supervised penalized output kernel regression for
link prediction. In Proc. ICML, 2011.
[5] A. Caponnetto, C. A. Micchelli, M. Pontil, and Y. Ying. Universal multi-task kernels. JMLR, 68:1615?
1646, 2008.
[6] C. Carmeli, E. De Vito, and A. Toigo. Vector valued reproducing kernel Hilbert spaces of integrable
functions and mercer theorem. Analysis and Applications, 4:377?408, 2006.
[7] C. Carmeli, E. De Vito, and A. Toigo. Vector valued reproducing kernel Hilbert spaces and universality.
Analysis and Applications, 8:19?61, 2010.
[8] C. Cortes, M. Mohri, and A. Rostamizadeh. L2 regularization for learning kernels. In Proc. UAI, 2009.
[9] C. Cortes, M. Mohri, and A. Rostamizadeh. Generalization bounds for learning kernels. In ICML, 2010.
[10] F. Dinuzzo, C. S. Ong, P. Gehler, and G. Pillonetto. Learning output kernels with block coordinate descent.
In Proc. ICML, 2011.
[11] T. Evgeniou, C. A. Micchelli, and M. Pontil. Learning multiple tasks with kernel methods. JMLR,
6:615?637, 2005.
[12] H. Kadri, E. Duflos, P. Preux, S. Canu, and M. Davy. Nonlinear functional regression: a functional RKHS
approach. In Proc. AISTATS, pages 111?125, 2010.
[13] H. Kadri, A. Rabaoui, P. Preux, E. Duflos, and A. Rakotomamonjy. Functional regularized least squares
classification with operator-valued kernels. In Proc. ICML, 2011.
[14] H. Kadri, A. Rakotomamonjy, F. Bach, and P. Preux. Multiple operator-valued kernel learning. Technical
Report 00677012, INRIA, 2012.
[15] M. Kloft, U. Brefeld, S. Sonnenburg, and A. Zien. `p -norm multiple kernel learning. JMLR, 12:953?997,
2011.
[16] S. Kurcyusz. On the existence and nonexistence of lagrange multipliers in Banach spaces. Journal of
Optimization Theory and Applications, 20:81?110, 1976.
[17] A. Kurdila and M. Zabarankin. Convex Functional Analysis. Birkhauser Verlag, 2005.
[18] G. Lanckriet, N. Cristianini, L. El Ghaoui, P. Bartlett, and M. Jordan. Learning the kernel matrix with
semi-definite programming. JMLR, 5:27?72, 2004.
[19] H. Lian. Nonlinear functional models for functional responses in reproducing kernel Hilbert spaces. The
Canadian Journal of Statistics, 35:597?606, 2007.
[20] C. Micchelli and M. Pontil. Learning the kernel function via regularization. JMLR, 6:1099?1125, 2005.
[21] C. A. Micchelli and M. Pontil. On learning vector-valued functions. Neural Comput., 17:177?204, 2005.
[22] K. J. Miller and G. Schalk. Prediction of finger flexion: 4th brain-computer interface data competition.
BCI Competition IV, 2008.
[23] T. Pistohl, T. Ball, A. Schulze-Bonhage, A. Aertsen, and C. Mehring. Prediction of arm movement
trajectories from ECoG-recordings in humans. Journal of Neuroscience Methods, 167(1):105?114, 2008.
[24] A. Rakotomamonjy, F. Bach, Y. Grandvalet, and S. Canu. SimpleMKL. JMLR, 9:2491?2521, 2008.
[25] J. O. Ramsay and B. W. Silverman. Functional Data Analysis, 2nd ed. Springer Verlag, New York, 2005.
[26] John A. Rice and B. W. Silverman. Estimating the mean and covariance structure nonparametrically when
the data are curves. Journal of the Royal Statistical Society. Series B, 53(1):233?243, 1991.
[27] G. Schalk, D. J. McFarland, T. Hinterberger, N. Birbaumer, and J. R. Wolpaw. BCI2000: a generalpurpose brain-computer interface system. Biomedical Engineering, IEEE Trans. on, 51:1034?1043, 2004.
[28] B. Sch?olkopf and A. J. Smola. Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press, Cambridge, MA, USA, 2002.
[29] S. Sonnenburg, G. R?atsch, C. Sch?afer, and B. Sch?olkopf. Large scale multiple kernel learning. JMLR,
7:1531?1565, 2006.
[30] P. Tseng. Convergence of block coordinate descent method for nondifferentiable minimization. J. Optim.
Theory Appl., 109:475?494, 2001.
9
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4,037 | 4,654 | Augmented-SVM: Automatic space partitioning for
combining multiple non-linear dynamics
Ashwini Shukla
[email protected]
Aude Billard
[email protected]
Learning Algorithms and Systems Laboratory (LASA)
?
Ecole
Polytechnique F?ed?erale de Lausanne (EPFL)
Lausanne, Switzerland - 1015
Abstract
Non-linear dynamical systems (DS) have been used extensively for building generative models of human behavior. Their applications range from modeling brain
dynamics to encoding motor commands. Many schemes have been proposed for
encoding robot motions using dynamical systems with a single attractor placed
at a predefined target in state space. Although these enable the robots to react
against sudden perturbations without any re-planning, the motions are always directed towards a single target. In this work, we focus on combining several such
DS with distinct attractors, resulting in a multi-stable DS. We show its applicability in reach-to-grasp tasks where the attractors represent several grasping points
on the target object. While exploiting multiple attractors provides more flexibility in recovering from unseen perturbations, it also increases the complexity of
the underlying learning problem. Here we present the Augmented-SVM (A-SVM)
model which inherits region partitioning ability of the well known SVM classifier
and is augmented with novel constraints derived from the individual DS. The new
constraints modify the original SVM dual whose optimal solution then results in a
new class of support vectors (SV). These new SV ensure that the resulting multistable DS incurs minimum deviation from the original dynamics and is stable at
each of the attractors within a finite region of attraction. We show, via implementations on a simulated 10 degrees of freedom mobile robotic platform, that the
model is capable of real-time motion generation and is able to adapt on-the-fly to
perturbations.
1 Introduction
Dynamical systems (DS) have proved to be a promising framework for encoding and generating
complex motions. A major advantage of representing motion using DS based models [1, 2, 3, 4] is
the ability to counter perturbations by virtue of the fact that re-planning of trajectories is instantaneous. These are generative schemes that define the flow of trajectories in state space x ? RN by
means of a non-linear dynamical function x? = f (x). DS with single stable attractors have been used
in pick and place tasks to control for both the motion of the end-effector [5, 6, 7] and the placement
of the fingers on an object [8]. Assuming a single attractor, and hence a single grasping location on
the object, constrains considerably the applicability of these methods to realistic grasping problems.
A DS composed of multiple stable attractors provides an opportunity to encode different ways to
reach and grasp an object. Recent neuro-physiological results [9] have shown that a DS based modeling best explains the trajectories followed by humans while switching between several reaching
targets. From a robotics viewpoint, a robot controlled using a DS with multiple attractors would
1
2.5
2.5
2.5
2.5
2
2
2
2
1.5
1.5
1.5
1.5
1
1
1
1
0.5
0.5
0.5
0.5
0
0.5
1
1.5
2
(a) Motion 1
0
0.5
1
1.5
2
(b) Motion 2
0
0.5
1
1.5
0
2
(c) Crossing over
Training data
Streamlines
Attractors
0.5
1
1.5
2
(d) Fast switching
Figure 2: Combining motions using naive SVM classification based switching.
Y
be able to switch online across grasping strategies. This may be useful, e.g., when one grasping
point becomes no longer accessible due to a sudden change in the orientation of the object or the
appearance of an obstacle along the current trajectory. This paper presents a method by which
one can learn multiple dynamics directed toward different attractors in a single dynamical system.
The dynamical function f (x) is usually estimated us2
ing non-linear regression functions such as Gaussian Pro- 1.5
cess Regression (GPR) [10], Gaussian Mixture Regres1
sion (GMR) [7], Locally Weighted Projection Regression
(LWPR) [11] or Dynamical Movement Primitives (DMP) 0.5
[1]. However, all of these works modeled DS with a sin0
gle attractor. While [7, 12] ensure global stability at the
attractor, other approaches result in unstable DS with spu- ?0.5
rious attractors.
?1
Stability at multiple targets has been addressed to date ?1.5
?1.5
?1
?0.5
0
0.5
1
1.5
X
largely through neural networks approaches. The Hopfield network and variants offered a powerful means to
Figure 1: 8 attractor DS
encode several stable attractors in the same system to provide a form of content-addressable memory [13, 14]. The
dynamics to reach these attractors was however not controlled for, nor was the partitioning of the
state space that would send the trajectories to each attractor. Echo-state networks provide alternative
ways to encode various complex dynamics [15]. Although they have proved to be universal estimators, their ability to generalize in untrained regions of state space remains unverified. Also, the key
issue of global stability of the learned dynamics is achieved using heuristic rules. To our knowledge,
this is the first attempt at learning simultaneously a partitioning of the state space and an embedding
of multiple dynamical systems with separate regions of attractions and distinct attractors.
2 Preliminaries
A naive approach to building a multi-attractor DS would be to first partition the space and then learn
a DS in each partition separately. This would unfortunately rarely result in the desired compound
system. Consider, for instance, two DS with distinct attractors, as shown in Fig. 2(a)-(b). First, we
build a SVM classifier to separate data points of the first DS, labeled +1, from data points of the
other DS, labeled ?1. We then estimate each DS separately using any of the techniques reviewed in
the previous section. Let h : RN 7? R denote the classifier function that separates the state space
x ? RN into two regions with labels yi ? {+1, ?1}. Also, let the two DS be x? = fyi (x) with
stable attractors at x?yi . The combined DS is then given by x? = fsgn(h(x)) (x). Figure 2(c) shows
the trajectories resulting from this approach. Due to the non-linearity of the dynamics, trajectories
initialized in one region cross the boundary and converge to the attractor located in the opposite
region. In other words, each region partitioned by the SVM hyperplane is not a region of attraction
for its attractor. In a real-world scenario where the attractors represent grasping points on an object
and the trajectories are to be followed by robots, crossing over may take the trajectories towards
kinematically unreachable regions. Also, as shown in Fig. 2(d), trajectories that encounter the
boundary may switch rapidly between different dynamics leading to jittery motion.
To ensure that the trajectories do not cross the boundary and remain within the region of attraction of their respective attractors, one could adopt a more informed approach in which each of the
2
original DS is modulated such that the generated trajectories always move away from the classifier
boundary. Recall that by construction, the absolute value of the classifier function h(x) increases as
one moves away from the classification hyperplane. The gradient ?h(x) is hence positive, respectively negative, as one moves inside the region of the positive, respectively negative, class. We can
exploit this observation to deflect selective components of the velocity signal from the original DS
along, respectively opposite to, the direction ?h(x). Concretely, if x? O = fsgn(h(x)) (x) denotes the
velocity obtained from the original DS and
max ?, ?h(x)T x? O
if h(x) > 0
?(x) =
,
(1)
if h(x) < 0
min ??, ?h(x)T x? O
the modulated dynamical system is given by
x? = f?(x) = ?(x)?h(x) + x? ? .
(2)
?O
?h(x)T x
Here, ? is a small positive scalar and x? ? = x? O ?
?h(x) is the compok?h(xk2
nent of the original velocity perpendicular to ?h. This results in a vector field that flows
along increasing values of the classifier function in the regions of space where h(x) > 0
and along decreasing values for h(x) < 0. As a result, the trajectories move away from
the classification hyperplane and converge to a point located in the region where they were
initialized. Such modulated systems have been used extensively for estimating stability regions of interconnected power networks [16] and are known as quasi gradient systems [17].
If h(x) is upper bounded1, all trajectories converge to one of the stationary points {x :
?h(x) = 0} and h(x) is a Lyapunov function of the overall system (refer [17], proposition 1).
Figure 3 shows the result of applying the above modulation to our
pair of DS. As expected, it forces the trajectories to flow along the
gradient of the function h(x). Although this solves the problem of 2.5
?crossing-over? the boundary, the trajectories obtained are deficient
in two major ways. They depart heavily from the original dynamics
2
and do not terminate at the desired attractors. This is due to the fact
that the function h(x) used to modulate the DS was designed solely 1.5
for classification and contained no information about the dynamics
of the two original DS. In other words, the vector field given by
1
?h(x) was not aligned with the flow of the training trajectories and
the stationary points of the modulation function did not coincide 0.5
with the desired attractors.
0
0.5
1
1.5
2
In subsequent sections, we show how we can learn a new modulation function which takes into account the three issues we highFigure 3: Modulated trajs.
lighted in this preliminary discussion. We will seek a system that
a) ensures strict classification across regions of attraction (ROA)
for each DS, b) follows closely the dynamics of each DS in each ROA and c) ensures that all trajectories in each ROA reach the desired attractor. Satisfying requirements a) and b) above is equivalent
to performing classification and regression simultaneously. We take advantage of the fact that the
optimization in support vector classification and support vector regression have the same form to
phrase our problem in a single constrained optimization framework. In next sections, we show that
in addition to the usual SVM support vectors (SVs), the resulting modulation function is composed
of an additional class of SVs. We geometrically analyze the effect of these new support vectors on
the resulting dynamics. While this preliminary discussion considered solely binary classification,
we will now extend the problem to multi-class classification.
3 Problem Formulation
The N -dimensional state space of the system represented by x ? RN is partitioned into M different classes, one for each of the M motions to be combined. We collect trajectories in the state
space, yielding a set of P data points {xi ; x? i ; li }i=1...P where li ? {1, 2, ? ? ? , M } refers to the
class label of each point2 . To learn the set of modulation functions {hm (x)}m=1...M , we proceed recursively. We learn each modulation function in a one-vs-all classifier scheme and then
1
2
SVM classifier function is bounded if the Radial Basis Function (rbf) is used as kernel.
Bold faced fonts represent vectors. xi denotes the i-th vector and xi denotes the i-th element of vector x.
3
?
compute the final modulation function h(x)
=
max hm (x). In the multi-class setting, the be-
m=1???M
havior of avoiding boundaries is obtained if the trajectories move along increasing values of the
?
function h(x).
the deflection term ?(x) presented in the binary case 1 becomes
To this effect,
T ?
?
?(x) = max ?, ?h(x) xO ; ?x ? RN . Next, we describe the procedure for learning a single
hm (x) function.
We follow classical SVM formulation and lift the data into a higher dimensional feature space
through the mapping ? : RN 7? RF where F denotes the dimension of the feature space. We
also assume that each function hm (x) is linear in feature space, i.e., hm (x) = wT ?(x) + b where
w ? RF , b ? R. We label the current (m ? th) motion class as positive and all others negative such
that the set of labels for the current sub-problem is given by
+1 if li = m
yi =
;
i = 1 ? ? ? P.
?1 if li 6= m
Also, the set indexing the positive class is then defined as I+ = {i : i ? [1, P ]; li = m}. With this,
we formalize the three constraints explained in Section 2 as:
Region separation: Each point must be classified correctly yields P constraints:
yi wT ?(xi ) + b ? 1 ?i = 1...P.
(3)
Lyapunov constraint: To ensure that the modulated flow is aligned with the training trajectories,
the gradient of the modulation function must have a positive component along the velocities at the
data points. That is,
?? i = wT J(xi )x
?? i ? 0 ?i ? I+
?hm (xi )T x
(4)
where J ? RF ?N is the Jacobian matrix given by J = [ ??1 (x)??2 (x) ? ? ? ??F (x) ]
?? i = x? i /kx? i k is the normalized velocity at the i ? th data point.
x
T
and
Stability: Lastly, the gradient of the modulation function must vanish at the attractor of the positive
class x? . This constraint can be expressed as
?hm (x? )T ei = wT J(x? )ei = 0 ?i = 1...N
(5)
N
where the set of vectors {ei }i=1???N is the canonical basis of R .
3.1 Primal & Dual forms
As in the standard SVM [18], we optimize for maximal margin between the positive and negative
class, subject to constraints 3-5 above. This can be formulated as:
?
?i = 1 ? ? ? P ?
yi wT ?(xi ) + b ? 1
?
?
X
?? i + ?i > 0
1
wT J(xi )x
?i ? I+
min kwk2 + C
?i subject to
. (6)
?i > 0
?i ? I+ ?
w,?i 2
?
i?I+
?
wT J(x? )ei = 0
?i = 1 ? ? ? N
Here ?i ? R are slack variables that relax the Lyapunov constraint in Eq. 4. We retain these in
our formulation to accommodate noise in the data representing the dynamics. C ? R+ is a penalty
parameter for the slack variables. The Lagrangian for the above problem can be written as
P
X
X
X
1
L(w, b, ?, ?, ?) = kwk2 + C
?i yi (wT ?(xi ) + b) ? 1
?i ?
?i ?i ?
2
i=1
i?I+
i?I+
?
X
i?I+
N
X
?? i + ?i +
?i wT J(x? )ei
?i wT J(xi )x
(7)
i=1
where ?i , ?i , ?i , ?i are the Lagrange multipliers with ?i , ?i , ?i ? R+ and ?i ? R. Employing a
similar analysis as in the standard SVM, it can be shown that the corresponding dual is given by the
constrained quadratic program:
?
K
1 h T T Ti?
GT
min
? ? ?
?,?,? 2
?GT?
G
H
?HT?
??
?
?G?
?
?H? ? ? ? ???T 1
?
H??
4
0 ? ?i
subject to P0 ? ?i ? C
P
i=1 ?i yi = 0
?i = 1...P
?i ? I+
where 1 ? RP is a vector with all entries equal to one. Let k : RN ? RN 7? R represents the kernel
function such that k(x1 , x2 ) = ?T (x1 )?(x2 ). The matrices K ? RP ?P , G ? RP ?|I+ | , G? ?
RP ?N , H ? R|I+ |?|I+| , H? ? R|I+ |?N , H?? ? RN ?N can be expressed in terms of the kernel
function and its first and second order derivatives:
?
2
?? T ? k(xi ,xj ) x
?? j ?
(K)ij = yi yj k(xi , xj )
;
(H)ij = x
?
i
?xi ?xj
?
T
?
2
?
?k(xi ,xj )
?
k(x
,x
)
i
T
?? j
?
?
x
;
(H
)
=
x
e
(G)ij = yi
(8)
? ij
j
i
?xj
?xi ?x?
?
T
?
2
?
?
?
?
k(x ,x )
i ,x )
ej ?
ej
;
(H?? )ij = eTi ? ?x
(G? )ij = yi ?k(x
? ?x?
?x?
where (.)ij denotes the i, j ?th entry of the corresponding matrix. Due to space constraints, detailed
development of the dual and proof of the above relations are given in appendices A and B of the
supplement material.
Note that since the matrices K, H and H?? are symmetric, the overall Hessian matrix for the resulting
quadratic program is also symmetric. However, unlike the standard SVM dual, it may not be positive
definite resulting in multiple solutions to the above problem. In our implementation, we use the
interior point solver IPOPT [19] to find a local optimum. We initialize the iterations using the ?
found by running first a standard SVM classification problem. All entries of ? and ? are set to 03 .
The solution to the above problem yields a modulation function (see Eq. A.11 for proof) given by
hm (x) =
P
X
i=1
N
?i yi k(x, xi ) +
X
?i x??Ti
i?I+
?k(x, xi ) X
?k(x, x? )
?i eTi
?
+b
?xi
?x?
i=1
(9)
which can be further expanded depending on the choice of kernel.
Expansions for the Radial Basis Function (rbf) kernel are given in Appendix C.
0.1
0.
The modulation function 9 learned using the A0
75
57
0. 008
1.5
01
5
22 77
SVM has noticeable similarities with the stan- 1.5
0.1
71
75
1
1
57
dard SVM classifier function. The first sum0.5
0.5
mation term is composed of the ? support vec0
0
tors (?-SV) which act as support to the classi- ?0.5
?0.5
?0
.1
fication hyperplane. The second term entails a ?1
75
?0
?1
57
?0 .01
?0
22
.0
new class of support vectors that perform a lin- ?1.5 .17
00
7
?1.5
85 1
55
7
7
7
?? i
ear combination of the normalized velocity x
?2
?2
?2
?1
0
1
?2
?1
0
1
at the training data points xi . These ? sup(a) ? = 1
(b) ? = 0.5
port vectors (?-SVs) collectively contribute to
the fulfillment of the Lyapunov constraint by
??T ?k(x,xi )
introducing a positive slope in the modulation Figure 4: Isocurves of f (x) = xi ?xi at
?? i = [ ?1 ?1 ]T for the rbf kernel.
?? i . Figure 4 xi = [0 0]T , x
function value along the directions x
2 2
shows the influence of a ?-SV for the rbf kernel
2
2
k(xi , xj ) = e1/2? kxi ?xj k with xi at the origin
?? i = [ ?1 ?1 ]T . It can be seen that the smaller the kernel width ?, the steeper the slope. The
and x
2 2
third summation term is a non-linear bias, which does not depend on the chosen support vectors, and
performs a local modification around the desired attractor x? to ensure that the modulation function
has a local maximum at that point. b is the constant bias which normalizes the classification margins
as ?1 and +1. We calculate its value by making use of the fact that for all the ?-SV xi , we must
have yi hm (xi ) = 1. We use average of the values obtained from the different support vectors.
Figure 5 illustrates the effects of the support vectors in a 2D example by progressively adding them
and overlaying the resulting DS flow in each case. The value of the modulation function hm (x) is
shown by the color plot (white indicates high values). As the ?-SVs are added in Figs. 5(b)-(d),
they push the flow of trajectories along their associated directions. In Figs. 5(e)-(f), adding the two
? terms shifts the location of the maximum of the modulation function to coincide with the desired
attractor. Once all the SVs have been taken into account, the streamlines of the resulting DS achieve
the desired criteria, i.e., they follow the training trajectories and terminate at the desired attractor.
3
Source code for learning is available at http://asvm.epfl.ch
5
0.4
?0.2
?0.2
?0.2
?0.8
?0.5
0
X
?0.8
0.5
Y
0.4
Y
Y
0.4
?0.5
(a) ? only
0
X
?0.8
0.5
?0.5
(b) ? and ?
0.4
0
X
0.5
(c) ? and ?
0.4
Modulated Streamlines
Training Data
0.4
? SV
? SV
?0.8
?0.5
0
X
(d) ? and ?
0.5
?0.2
?0.8
Y
Y
Y
Obtained attractor
Desired attractor
?0.2
?0.5
0
X
0.5
(e) ?, ? and ?1
?0.2
?0.8
?0.5
0
X
0.5
(f) ?, ?, ?1 and ?2
Figure 5: Progressively adding support vectors to highlight their effect on shaping the dynamics
of the motion. (a) ?-SVs largely affect classification. (b)-(d) ?-SVs guide the flow of trajectories
?? i shown by arrows. (e)-(f) The 2 ? terms force the local
along their respective associated directions x
maximum of the modulation function to coincide with the desired attractor along the X and Y axes
respectively.
4 Results
In this section, we validate the presented A-SVM model on 2D (synthetic) data and on a robotic
simulated experiment using a 7 degrees of freedom (DOF) KUKA-LWR arm mounted on a 3-DOF
Omnirob base to catch falling objects. A video of the robotic experiment - simulated and real is provided in Annexes. Next, we present a cross-validation analysis of the error introduced by
the modulation in the original dynamics. A sensitivity analysis of the region of attraction of the
resulting dynamical system with respect to the model parameters is also presented. We used the
rbf kernel for all the results presented in this section. As discussed in Section 2, the RBF kernel is
advantageous as it ensures that the function hm (x) is bounded. To generate an initial estimate of
each individual dynamical system, we used the technique proposed in [7].
2D Example Figure 6(a) shows a synthetic example with 4 motion classes, each generated
from a different closed form dynamics and containing 160 data points. The color plot indicates the
value of the combined modulation function ?h(x) = max hm (x) where each of the functions
m=1???M
hm (x) are learned using the presented A-SVM technique. A total of 9 support vectors were
obtained which is < 10% of the number of training data points. The trajectories obtained after
modulating the original dynamical systems flow along increasing values of the modulation function,
thereby bifurcating towards different attractors at the region boundaries. Unlike the dynamical
system in Fig. 3, the flow here is aligned with the training trajectories and terminates at the desired
attractors. To recall, this is made possible thanks to the additional constraints (Eq. 4 and 5) in our
formulation.
In a second example, we tested the ability of our model to accommodate a higher density of
attractors. We created 8 synthetic dynamics by capturing motion data using a screen mouse. Figure
1 shows the resulting 8 attractor system.
Error Analysis As formulated in Eq. 6, the Lyapunov constraints admit some slack, which
allows the modulation to introduce slight deviations from the original dynamics. Here we
statistically analyze this error via 5-fold cross validation. In the 4 attractor problem presented
6
Training Data
Modulated Trajs.
40
Attractors
Class
Class
Class
Class
y
1
0
0.5
?1
0
30
1
2
3
4
1
Training
0.6
20
0.4
10
0.2
?2
?3
?2
?1
x
0
1
2
(a) Combined flow
Testing
0.8
% Error
1
% Testing Error
2
0
0
5
10
?
15
20
0
(b) Cross validation error
Class 1
Class 2
Class 3
Class 4
(c) Best case errors
Figure 6: Synthetic 2D case with 4-attractors.
above, we generate a total of 10 trajectories per motion class and use 2:3 training to testing ratio
for cross validation. We calculate the average percentage error between the original velocity
(read off
D from? the data) and
E the modulated velocity (calculated using 2) for the m ? th class as
kx? i ?f (xi )k
em =
? 100
where < . > denotes average over the indicated range. Figure
kx? i k
i:li =m
6(b) shows the cross validation error (mean and standard deviation over the 5 folds) for a range
of values of kernel width. The general trend revealed here is that for each class of motion, there
exists a band of optimum values of the kernel width for which the testing error is the smallest. The
region covered by this band of optimal values may vary depending on the relative location of the
attractors and other data points. In Fig. 6(a), motion classes 2 (upper left) and 4 (upper right) are
better fitted and show less sensitivity to the choice of kernel width than classes 1 (lower left) and
3 (lower right). We will show later in this section that this is correlated to the distance between
the attractors. A comparison of testing and training errors for the least error case is shown in Fig.
6(c). We see that the testing errors for all the classes in the best case scenario are less than 1%.
0.5
Sensitivity analysis The partitioning of space created by our
method results in M regions of attraction (ROA) for each of our
M attractors. To assess the size of these regions and the existence
0
of spurious attractors, we adopt an empirical approach. For each
class, we compute the isosurfaces of the corresponding modulation
?0.5
function hm (x) in the range [0, hm (x? )]. These hypersurfaces
h(x) = const
incrementally span the volume of the m ? th region around its
ROA boundary
?1
attractor. We mesh each of these test surfaces and compute trajecMeshed contour
Actual attractor
tories starting from the obtained mesh-points, looking for spurious
Spurious attractor
attractors. hROA is the isosurface of maximal value that encloses
?0.5
0
0.5
no spurious attractor and marks the ROA of the corresponding
motion dynamics. We use the example in Fig. 5 to illustrate this Figure 7: Test trajectories
process. Figure 7 shows a case where one spurious attractor is generated from several points
detected using a larger test surface (dotted line) whereas the actual on an isocurve (dotted line) to
ROA (solid line) is smaller. Once hROA is calculated, we define determine spurious attractors.
the size of ROA as rROA = (h(x? ) ? hROA )/h(x? ). rROA = 0
when no trajectory except those originating at the attractor itself, lead to the attractor. rROA = 1
when the ROA is bounded by the isosurface h(x) = 0. The size of the rROA is affected by both
the choice of kernel width and the distance across nearby attractors. This is illustrated in Fig. 9
using data points from class 1 of Fig. 6(a) and translating the attractors so that they are either very
far apart (left, distance datt = 1.0) or very close to one another (right, datt = 0.2). As expected,
rROA increases as we reach the optimal range of parameters. Furthermore, when the attractors are
farther apart, high values of rROA are obtained for a larger range of values of the kernel width, i.e.,
the model is less sensitive to the chosen kernel width. With smaller distance between the attractors
(Fig. 9(b)), only a small deviation from the optimum kernel width results in a considerable loss in
rROA , exhibiting high sensitivity to the model parameter.
3D Example We validated our method on a real world 3D problem. The attractors here represent manually labeled grasping points on a pitcher. The 3D model of the object was taken
from the ROS IKEA object library. We use the 7-DOF KUKA-LWR arm mounted on the 3-DOF
7
0.4
0.4
0.2
0.2
0
0
?0.2
?0.2
?0.2
?0.2
0
0.2
0.1
0
?0.1
0.4 ?0.2
(a) Training data
(b) hm (x) = 0
(c) Trajectory 1
(d) Trajectory 2
0
0.2
0.4
0 0.1
?0.1
?0.2
(e) Combined flow
Figure 8: 3D Experiment. (a) shows training trajectories for three manually chosen grasping points.
(b) shows the isosurfaces hm (x) = 0; m = 1, 2, 3 along with the locations of the corresponding attractors. In (c) and (d), the robot executes the generated trajectories starting from different positions
and hence converging to different grasping points. (e) shows the complete flow of motion.
KUKA-Omnirob base for executing the modulated Cartesian trajectories in simulation. We control
all 10 DOF of the robot using the damped least square inverse kinematics. Training data for this
implementation was obtained by recording the end-effector positions xi ? R3 from kinesthetic
demonstrations of reach-to-grasp motions directed towards these grasping points, yielding a 3-class
problem (see Fig. 8(a)). Each class was represented by 75 data points. Figure 8(b) shows the
isosurfaces hm (x) = 0; m ? {1, 2, 3} learned using the presented method. Figures 8(c)-(d) show
the robot executing two trajectories when started from two different locations and converging to a
different attractor (grasping point). Figure 8(e) shows the flow of motion around the object. Note
that the time required to generate each trajectory point is O(S) where S denotes the total number of
support vectors in the model. In this particular example with a total of 18 SVs, the trajectory points
were generated at 1000 Hz which is well suited for real-time control. Such a fast generative model
allows the robot to switch on-the-fly between the attractors and adapt to real-time perturbations
in the object or the end-effector pose, without any re-planning or re-learning. Results for another
object (champagne glass) are included in Appendix D (Fig. D.1). A video illustrating how the robot
exploits multiple attractors to catch one of the grasping points on the object as it falls down is also
provided in the supplementary material.
5 Conclusions
C
C
In this work, we presented the A1.0
1.0
1400
1400
SVM model for combining non0.8
0.8
1200
1200
linear dynamical systems through a
1000
1000
partitioning of the space. We refor0.6
0.6
800
800
mulated the optimization framework
0.4
0.4
600
600
of SVM to encapsulate constraints
400
400
0.2
0.2
that ensure accurate reproduction of
200
200
the dynamics of motion. The new
0
0
0.5
1
1.5
2
0.5
1
1.5
2
set of constraints result in a new class
?
?
of support vectors that exploit partial
(a) datt = 1.0
(b) datt = 0.2
derivatives of the kernel function to
align the flow of trajectories with the Figure 9: Variation of rROA with varying model parameters.
training data. The resulting model
behaves as a multi-stable DS with attractors at the desired locations. Each of the classified regions are forward invariant w.r.t the learned DS. This ensures that the trajectories do not cross over
region boundaries. We validated the presented method on synthetic motions in 2D and 3D grasping
motions on real objects. Results show that even though spurious attractors may occur, in practice
they can be avoided by a careful choice of model parameters through grid search. The applicability
of the method for real-time control of a 10-DOF robot was also demonstrated.
Acknowledgments
This work was supported by EU Project First-MM (FP7/2007-2013) under grant agreement number
248258. The authors would also like thank Prof. Franc?ois Margot for his insightful comments on
the technical material.
8
References
[1] Peter Pastor, Heiko Hoffmann, Tamim Asfour, and Stefan Schaal. Learning and generalization of motor
skills by learning from demonstration. In Robotics and Automation, 2009. ICRA ?09. IEEE International
Conference on, pages 763 ?768, may 2009.
[2] G. Sch?oner and M. Dose. A dynamical systems approach to task-level system integration used to plan
and control autonomous vehicle motion. Robotics and Autonomous Systems, 10(4):253?267, 1992.
[3] G. Sch?oner, M. Dose, and C. Engels. Dynamics of behavior: Theory and applications for autonomous
robot architectures. Robotics and Autonomous Systems, 16(2):213?245, 1995.
[4] L.P. Ellekilde and H.I. Christensen. Control of mobile manipulator using the dynamical systems approach.
In Robotics and Automation, 2009. ICRA?09. IEEE International Conference on, pages 1370?1376. IEEE,
2009.
[5] H. Reimann, I. Iossifidis, and G. Sch?oner. Autonomous movement generation for manipulators with
multiple simultaneous constraints using the attractor dynamics approach. In Robotics and Automation
(ICRA), 2011 IEEE International Conference on, pages 5470?5477. IEEE, 2011.
[6] K.R. Dixon and P.K. Khosla. Trajectory representation using sequenced linear dynamical systems. In
Robotics and Automation, 2004. Proceedings. ICRA?04. 2004 IEEE International Conference on, volume 4, pages 3925?3930. IEEE, 2004.
[7] S. M. Khansari-Zadeh and Aude Billard. Learning Stable Non-Linear Dynamical Systems with Gaussian
Mixture Models. IEEE Transaction on Robotics, 2011.
[8] A. Shukla and A. Billard. Coupled dynamical system based armhand grasping model for learning fast
adaptation strategies. Robotics and Autonomous Systems, 60(3):424 ? 440, 2012.
[9] H. Hoffmann. Target switching in curved human arm movements is predicted by changing a single control
parameter. Experimental brain research, 208(1):73?87, 2011.
[10] C. Rasmussen. Gaussian processes in machine learning. Advanced Lectures on Machine Learning, pages
63?71, 2004.
[11] S. Schaal, C.G. Atkeson, and S. Vijayakumar. Scalable techniques from nonparametric statistics for real
time robot learning. Applied Intelligence, 17(1):49?60, 2002.
[12] Auke Jan Ijspeert, Jun Nakanishi, and Stefan Schaal. Movement imitation with nonlinear dynamical systems in humanoid robots. In In IEEE International Conference on Robotics and Automation (ICRA2002,
pages 1398?1403, 2002.
[13] A. Fuchs and H. Haken. Pattern recognition and associative memory as dynamical processes in a synergetic system. i. translational invariance, selective attention, and decomposition ofscenes. Biol. Cybern.,
60:17?22, November 1988.
[14] A.N. Michel and J.A. Farrell. Associative memories via artificial neural networks. Control Systems
Magazine, IEEE, 10(3):6 ?17, apr 1990.
[15] H. Jaeger, M. Lukosevicius, D. Popovici, and U. Siewert. Optimization and applications of echo state
networks with leaky-integrator neurons. Neural Networks, 20(3):335?352, 2007.
[16] J. Lee. Dynamic gradient approaches to compute the closest unstable equilibrium point for stability region
estimate and their computational limitations. Automatic Control, IEEE Transactions on, 48(2):321?324,
2003.
[17] H.D. Chiang and C.C. Chu. A systematic search method for obtaining multiple local optimal solutions of
nonlinear programming problems. Circuits and Systems I: Fundamental Theory and Applications, IEEE
Transactions on, 43(2):99?109, 1996.
[18] B. Sch?olkopf and A.J. Smola. Learning with kernels: Support vector machines, regularization, optimization, and beyond. MIT press, 2001.
[19] Andreas Wchter and Lorenz T. Biegler. On the implementation of an interior-point filter line-search
algorithm for large-scale nonlinear programming. Mathematical Programming, 106:25?57, 2006.
9
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correlated:1 |
4,038 | 4,655 | Identifiability and Unmixing of Latent Parse Trees
Daniel Hsu
Microsoft Research
Sham M. Kakade
Microsoft Research
Percy Liang
Stanford University
Abstract
This paper explores unsupervised learning of parsing models along two directions.
First, which models are identifiable from infinite data? We use a general technique for numerically checking identifiability based on the rank of a Jacobian matrix, and apply it to several standard constituency and dependency parsing models.
Second, for identifiable models, how do we estimate the parameters efficiently?
EM suffers from local optima, while recent work using spectral methods [1] cannot be directly applied since the topology of the parse tree varies across sentences.
We develop a strategy, unmixing, which deals with this additional complexity for
restricted classes of parsing models.
1
Introduction
Generative parsing models, which define joint distributions over sentences and their parse trees, are
one of the core techniques in computational linguistics. We are interested in the unsupervised learning of these models [2?6], where the goal is to estimate the model parameters given only examples
of sentences. Unsupervised learning can fail for a number of reasons [7]: model misspecification,
non-identifiability, estimation error, and computation error. In this paper, we delve into two of these
issues: identifiability and computation. In doing so, we confront a central challenge of parsing
models?that the topology of the parse tree is unobserved and varies across sentences. This is in
contrast to standard phylogenetic models [8] and other latent tree models for which there is a single
fixed global tree across all examples [9].
A model is identifiable if there is enough information in the data to pinpoint the parameters (up
to some trivial equivalence class); establishing the identifiability of a model is often a highly nontrivial task. A classic result of Kruskal [10] has been employed to prove the identifiability of a wide
class of latent variable models, including hidden Markov models and certain restricted mixtures of
latent tree models [11?13]. However, these techniques cannot be directly applied to parsing models
since the tree topology varies over an exponential set of possible topologies. Instead, we turn to
techniques from algebraic geometry [14?17]; we show that a simple numerical procedure can be
used to check identifiability for a wide class of models in NLP. Using this tool, we discover that
probabilistic context-free grammars (PCFGs) are non-identifiable, but that simpler PCFG variants
and dependency models are identifiable.
The most common way to estimate unsupervised parsing models is by using local techniques such
as EM [18] or MCMC sampling [19], but these methods can suffer from local optima and slow
mixing. Meanwhile, recent work [1,20?23] has shown that spectral methods can be used to estimate
mixture models and HMMs with provable guarantees. These techniques express low-order moments
of the observable distribution as a product of matrix parameters and use eigenvalue decomposition
to recover these matrices. However, these methods are not directly applicable to parsing models
because the tree topology again varies non-trivially. To address this, we propose a new technique,
unmixing. The main idea is to express moments of the observable distribution as a mixture over
the possible topologies. For restricted parsing models, the moments for a fixed tree structure can
E-mail: [email protected], [email protected], [email protected]
1
?
?
z03
z03
T
T
z02
T
T
T
z23
z01
O
O
x3
x1
T
T
A
A
A
x2
x3
Topology(z) = 1
z13
?
T
z01
z12
z12
z23
O
O
O
O
x1
x2
x2
x3
Topology(z) = 1
x1
?
x1
A
x2
A
x3
Topology(z) = 2
x1
x2
A
?
x2
A
x3
A
?
x3
Topology(z) = 5
x1
A
x2
A
?
x3
Topology(z) = 6
A
A
Topology(z) = 3
x1
x2
A
?
Topology(z) = 2
x1
x1
A
x2
?
x3
Topology(z) = 7
x3
Topology(z) = 4
(a) Constituency (PCFG-IE)
(b) Dependency (DEP-IE)
Figure 1: The two constituency trees and seven dependency trees over L = 3 words, x1 , x2 , x3 . (a)
A constituency tree consists of a hierarchical grouping of the words with a latent state zv for each
node v. (b) A dependency tree consists of a collection of directed edges between the words. In both
cases, we have labeled each edge from i to j with the parameters used to generate the state of node
j given i.
be ?unmixed?, thereby reducing the problem to one with a fixed topology, which can be tackled
using standard techniques [1]. Importantly, our unmixing technique does not require the training
sentences be annotated with the tree topologies a priori, in contrast to recent extensions of [21] to
learning PCFGs [24] and dependency trees [25, 26], which work on a fixed topology.
2
Notation
def
For a positive integer n, define [n] = {1, . . . , n} and hni = {e1 , . . . , en }, where ei is the vector
which is 1 in component i and 0 elsewhere. For integers a, b ? [n], let a?n b = (a?1)n+b ? [n2 ] be
the integer encoding of the pair (a, b). For a pair of matrices, A, B ? Rm?n , define the columnwise
2
tensor product A ?C B ? Rm ?n to be such that (A ?C B)(i1 ?m i2 )j = Ai1 j Bi2 j . For a matrix
A ? Rm?n , let A? denote the Moore-Penrose pseudoinverse.
3
Parsing models
A sentence is a sequence of L words, x = (x1 , . . . , xL ), where each word xi ? hdi is one of
d possible word types. A (generative) parsing model defines a joint distribution P? (x, z) over a
sentence x and its parse tree z (to be made precise later), where ? are the model parameters (a
collection of multinomials). Each parse tree z has a topology Topology(z) ? Topologies, which
is both unobserved and varying across sentences. The learning problem is to recover ? given only
samples of x.
Two important classes of models of natural language syntax are constituency models, which represent a hierarchical grouping and labeling of the phrases of a sentence (e.g., Figure 1(a)), and
dependency models, which represent pairwise relationships between the words of a sentence (e.g.,
Figure 1(b)).
2
3.1
Constituency models
A constituency tree z = (V, s) consists of a set of nodes V and a collection of hidden states s =
{sv }v?V . Each state sv ? hki represents one of k possible syntactic categories. Each node v has
the form [i : j] for 0 ? i < j ? L corresponding to the phrase between positions i and j of the
sentence. These nodes form a binary tree as follows: the root node is [0 : L] ? V , and for each node
[i : j] ? V with j ? i > 1, there exists a unique m with i < m < j defining the two children nodes
[i : m] ? V and [m : j] ? V . Let Topology(z) be an integer encoding of V .
PCFG. Perhaps the most well-known constituency parsing model is the probabilistic context-free
grammar (PCFG). The parameters of a PCFG are ? = (?, B, O), where ? ? Rk specifies the initial
2
state distribution, B ? Rk ?k specifies the binary production distributions, and O ? Rd?k specifies
the emission distributions.
A PCFG corresponds to the following generative process (see Figure 1(a) for an example): choose a
topology Topology(z) uniformly at random; generate the state of the root node using ?; recursively
generate pairs of children states given their parents using B; and finally generate words xi given
their parents using O. This generative process defines a joint probability over a sentence x and a
parse tree z:
P? (x, z) = | Topologies |?1 ? > s[0:L]
Y
(s[i:m] ?k s[m:j] )> Bs[i:j]
L
Y
x>
i Os[i?1:i] , (1)
i=1
[i:m],[m:j]?V
We will also consider two variants of the PCFG with additional restrictions:
PCFG-I. The left and right children states are generated independently?that is, we have the following factorization: B = T1 ?C T2 for some T1 , T2 ? Rk?k .
PCFG-IE. The left and the right productions are independent and equal: B = T ?C T .
3.2
Dependency tree models
In contrast to constituency trees, which posit internal nodes with latent states, dependency trees
connect the words directly. A dependency tree z is a set of directed edges (i, j), where i, j ? [L]
are distinct positions in the sentence. Let Root(z) denote the position of the root node of z. We
consider only projective dependency trees [27]: z is projective if for every path from i to j to k in
z, we have that j and k are on the same side of i (that is, j ? i and k ? i have the same sign). Let
Topology(z) be an integer encoding of z.
DEP-I. We consider the simple dependency model of [4]. The parameters of this model are ? =
(?, A. , A& ), where ? ? Rd is the initial word distribution and A. , A& ? Rd?d are the left and
right argument distributions. The generative process is as follows: choose a topology Topology(z)
uniformly at random, generate the root word using ?, and recursively generate argument words to
the left to the right given the parent word using A. and A& , respectively. The corresponding joint
probability distribution is as follows:
Y
P? (x, z) = | Topologies |?1 ? > xRoot(z)
x>
(2)
j Adir(i,j) xi ,
(i,j)?z
where dir(i, j) = . if j < i and & if j > i.
We also consider the following two variants:
DEP-IE. The left and right argument distributions are equal: A = A. = A& .
DEP-IES. A = A. = A& and ? is the stationary distribution of A (that is, ? = A?).
Usually a PCFG induces a topology via a state-dependent probability of choosing a binary production
versus an emission. Our model is a restriction which corresponds to a state-independent probability.
3
4
Identifiability
Our goal is to estimate model parameters ?0 ? ? given only access to sentences x ? P?0 . Specifically, suppose we have an observation function ?(x) ? Rm , which is the only lens through which an
algorithm can view the data. We ask a basic question: in the limit of infinite data, is it informationdef
theoretically possible to identify ?0 from the observed moments ?(?0 ) = E?0 [?(x)]?
To be more precise, define the equivalence class of ?0 to be the set of parameters ? that yield the
same observed moments:
S? (?0 ) = {? ? ? : ?(?) = ?(?0 )}.
(3)
It is impossible for an algorithm to distinguish among the elements of S? (?0 ). Therefore, one might
want to ensure that |S? (?0 )| = 1 for all ?0 ? ?. However, this requirement is too strong for two reasons. First, models often have natural symmetries?e.g., the k states of any PCFG can be permuted
without changing ?(?), so |S? (?0 )| ? k!. Second, |S? (?0 )| = ? for some pathological ?0 ?s?e.g.,
PCFGs where all states have the same emission distribution O are indistinguishable regardless of
the production distributions B. The following definition of identifiability accommodates these two
exceptional cases:
Definition 1 (Identifiability). A model family with parameter space ? is (globally) identifiable from
? if there exists a measure zero set E such that |S? (?0 )| is finite for every ?0 ? ?\E. It is locally
identifiable from ? if there exists a measure zero set E such that, for every ?0 ? ?\E, there exists an
open neighborhood N (?0 ) around ?0 such that S? (?0 ) ? N (?0 ) = {?0 }.
Example of non-identifiability. Consider the DEP-IE model with L = 2 with the full observation
function ?(x) = x1 ? x2 . The corresponding observed moments are ?(?) = 0.5A diag(?) +
0.5 diag(?)A> . Note that A diag(?) is an arbitrary d ? d matrix whose entries sum to 1, which
has d2 ?1 degrees of freedom, whereas ?(?) is a symmetric matrix whose
entries sum to 1, which
d
has d+1
?
1
degrees
of
freedom.
Therefore,
S
(?)
has
dimension
?
2
2 and therefore the model is
non-identifiable.
Parameter counting. It is important to compute the degrees of freedom correctly?simple parameter counting is insufficient. For example, consider the PCFG-IE model with L = 2. The observed
moments with respect to ?(x) = x1 ? x2 is a d ? d matrix, which places d2 constraints on the
k 2 + (d ? 1)k parameters. When d ? 2k, there are more constraints than parameters, but the PCFGIE model with L = 2 is actually non-identifiable (as we will see later). The issue here is that the
number of constraints does not reveal the fact that some of these constraints are redundant.
4.1
Observation functions
An observation function ?(x) and its associated observed moments ?(?0 ) = E?0 [?(x)] reveals
aspects of the distribution P?0 (x). For example, ?(x) = x1 would only reveal the marginal distribution of the first word, whereas ?(x) = x1 ? ? ? ? ? xL reveals the entire distribution of x. There is a
tradeoff: Higher-order moments provide more information, but are harder to estimate reliably given
finite data, and are also computationally more expensive. In this paper, we consider the following
intermediate moments:
def
def
?12 (x) = x1 ? x2
??? (x) = xi ? xj : i, j ? [L]
def
def
?123 (x) = x1 ? x2 ? x3
???? (x) = xi ? xj ? xk : i, j, k ? [L]
def
def
?123? (x) = (x1 ? x2 )(? > x3 )
????? (x) = (xi ? xj )(? > xk ) : i, j, k ? [L]
def
?all (x) = x1 ? ? ? ? ? xL
Above, ? ? Rd denotes a unit vector in Rd (e.g., e1 ) which picks out a linear combination of matrix
slices from a third-order d ? d ? d tensor.
4.2
Automatically checking identifiability
One immediate goal is to determine which models in Section 3 are identifiable from which of the
observed moments (Section 4.1). A powerful analytic tool that has been succesfully applied in
4
previous work is Kruskal?s theorem [10, 11], but (i) it is does not immediately apply to models with
random topologies, and (ii) only gives sufficient conditions for identifiability, and cannot be used to
determine non-identifiability. Furthermore, since it is common practice to explore many different
models for a given problem in rapid succession, we would like to check identifiability quickly and
reliably. In this section, we develop an automatic procedure to do this.
To establish identifiability, let us examine the algebraic structure of S? (?0 ) for ?0 ? ?, where we
assume that the parameter space ? is an open subset of [0, 1]n . Recall that S? (?0 ) is defined by the
moment constraints ?(?) = ?(?0 ). We can write these constraints as h?0 (?) = 0, where
def
h?0 (?) = ?(?) ? ?(?0 )
is a vector of m polynomials in ?.
Let us now compute the number of degrees of freedom of h?0 around ?0 . The key quantity is
J(?) ? Rm?n , the Jacobian of h?0 at ? (note that the Jacobian of h?0 does not depend on ?0 ; it
is precisely the Jacobian of ?). This Jacobian criterion is well-established in algebraic geometry,
and has been adopted in the statistical literature for testing model identifiability and other related
properties [14?17].
Intuitively, each row of J(?0 ) corresponds to a direction of a constraint violation, and thus the row
space of J(?0 ) corresponds to all directions that would take us outside the equivalence class S? (?0 ).
If J(?0 ) has less than rank n, then there is a direction orthogonal to all the rows along which we
can move and still satisfy all the constraints?in other words, |S? (?0 )| is infinite, and therefore the
model is non-identifiable. This intuition leads to the following algorithm:
C HECK I DENTIFIABILITY:
?1. Choose a point ?? ? ? uniformly at random.
?
?2. Compute the Jacobian matrix J(?).
?
?3. Return ?yes? if the rank of J(?) = n and ?no? otherwise.
The following theorem asserts the correctness of C HECK I DENTIFIABILITY. It is largely based on
techniques in [16], although we have not seen it explicitly stated in this form.
Theorem 1 (Correctness of C HECK I DENTIFIABILITY). Assume the parameter space ? is a nonempty open connected subset of [0, 1]n ; and the observed moments ? : Rn ? Rm , with respect to
observation function ?, is a polynomial map. Then with probability 1, C HECK I DENTIFIABILITY
returns ?yes? iff the model family is locally identifiable from ?. Moreover, if it returns ?yes?, then
there exists E ? ? of measure zero such that the model family with parameter space ? \ E is
identifiable from ?.
The proof of Theorem 1 is given in Appendix A.
4.3
Implementation of C HECK I DENTIFIABILITY
Computing the Jacobian. The rows of J correspond to ?E? [?j (x)]/?? and can be computed efficiently by adapting dynamic programs used in the E-step of an EM algorithm for parsing models.
There are two main differences: (i) we must sum over possible values of x in addition to z, and (ii)
we are not computing moments, but rather gradients thereof. Specifically, we adapt the CKY algorithm for constituency models and the algorithm of [27] for dependency models. See Appendix C.1
for more details.
Numerical issues. Because we implemented C HECK I DENTIFIABILITY on a finite precision machine, the results are subject to numerical precision errors. However, we verified that our numerical
results are consistent with various analytically-derived identifiability results (e.g., from [11]).
While we initially defined ? to be a tuple of conditional probability matrices, we will now use its nonredundant vectorized form ? ? Rn .
5
Model \ Observation function
PCFG
PCFG-I / PCFG-IE
DEP-I
DEP-IE / DEP-IES
?12
No
No
???
?123e1 ?123 ????e1
No, even from ?all for L ? {3, 4, 5}
Yes iff L ? 4
Yes iff L ? 3
Yes iff L ? 3
Yes iff L ? 3
????
Figure 2:
Local identifiability of parsing models.
These findings are given by
C HECK I DENTIFIABILITY have the semantics from Theorem 1. These were checked for d ?
{2, 3, . . . , 8}, k ? {2, . . . , d} (applies only for PCFG models), L ? {2, 3, . . . , 9}.
4.4
Identifiability of constituency and dependency tree models
We checked the identifiability status of various constituency and dependency tree models using our
implementation of C HECK I DENTIFIABILITY. We focus on the regime where d ? k for PCFGs;
additional results for d < k are given in Appendix B.
The results are reported in Figure 2. First, we found that the PCFG is not identifiable from ?all (and
therefore not identifiable from any ?) for L ? {3, 4, 5}; we believe that the same holds for all L. This
negative result motivates exploring restricted subclasses of PCFGs, such as PCFG-I and PCFG-IE,
which factorize the binary productions. For these classes, we found that the sentence length L and
choice of observation function can influence identifiability: Both models are identifiable for large
enough L (e.g., L ? 3) and with a sufficiently rich observation function (e.g., ?123? ).
The dependency models, DEP-I and DEP-IE, were all found to be identifiable for L ? 3 from
second-order moments ??? . The conditions for identifiability are less stringent than their constituency counterparts (PCFG-I and PCFG-IE), which is natural since dependency models are simpler without the latent states. Note that in all identifiable models, second-order moments suffice to
determine the distribution?this is good news because low-order moments are easier to estimate.
5
Unmixing algorithms
Having established which parsing models are identifiable, we now turn to parameter estimation for
these models. We will consider algorithms based on moment matching?those that try to find a ?
satisfying ?(?) = u for some u. Typically, u is an empirical estimate of ?(?0 ) = E?0 [?(x)] based
on samples x ? P?0 .
In general, solving ?(?) = u corresponds to finding solutions to systems of multivariate polynomials, which is NP-hard [28]. However, ?(?) often has additional structure which we can exploit.
For instance, for an HMM, the sliced third-order moments ?123? (?) can be written as a product of
parameter matrices in ?, and each matrix can be recovered by decomposing the product [1].
For parsing models, the challenge is that the topology is random, so the moments is not a single product, but a mixture over products. To deal with this complication, we propose a new technique, which
we call unmixing: We ?unmix? the products from the mixtures, essentially reducing the problem to
one with a fixed topology.
We will first present the general idea of unmixing (Section 5.1) and then apply it to the PCFG-IE
model (Section 5.2) and the DEP-IES model (Section 5.3).
5.1
General case
We assume the observation function ?(x) consists of a collection of observation matrices
{?o (x)}o?O (e.g., for o = (i, j), ?o (x) = xi ? xj ). Given an observation matrix ?o (x) and a
topology t ? Topologies, consider the mapping that computes the observed moment conditioned on
Note that these subclasses occupy measure zero subsets of the PCFG parameter space, which is expected
given the non-identifiability of the general PCFG.
We will develop our algorithms assuming true moments (u = ?(?0 )). The empirical moments converge
1
to the true moments at Op (n? 2 ), and matrix perturbation arguments (e.g., [1]) can be used derive sample
complexity arguments for the parameter error.
6
that topology: ?o,t (?) = E? [?o (x) | Topology = t]. As we range o over O and t over Topologies,
we will enounter a finite number of such mappings. We call these mappings compound parameters,
denoted {?p }p?P .
Now write the observed moments as a weighted sum:
X
?o (?) =
P(?o,Topology = ?p ) ?p
|
{z
}
p?P
for all o ? O,
(4)
def
= Mop
where we have defined Mop to be the probability mass over tree topologies that yield compound
parameter ?p . We let {Mop }o?O,p?P be the mixing matrix. Note that (4) defines a system of
equations ? = M ?, where the variables are the compound parameters and the constraints are the
observed moments. In a sense, we have replaced the original system of polynomial equations (in ?)
with a system of linear equations (in ?).
The key to the utility of this technique is that the number of compound parameters can be polynomial
in L even when the number of possible topologies is exponential in L. Previous analytic techniques
[13] based on Kruskal?s theorem [10] cannot be applied here because the possible topologies are too
many and too varied.
Note that the mixing equation ? = M ? holds for each sentence length L, but many compound parameters p appear in the equations of multiple L. Therefore, we can combine the equations across all
observed sentence lengths, yielding a more constrained system than if we considered the equations
of each L separately.
The following proposition shows how we can recover ? by unmixing the observed moments ?:
Proposition 1 (Unmixing). Suppose that there exists an efficient base algorithm to recover ? from
some subset of compound parameters {?p (?) : p ? P0 }, and that e>
p is in the row space of M for
each p ? P0 . Then we can recover ? as follows:
U NMIX(?):
?1. Compute the mixing matrix M (4).
?2. Retrieve the compound parameters ?p (?) = (M ? ?)p for each p ? P0 .
?3. Call the base algorithm on {?p (?) : p ? P0 } to obtain ?.
For all our parsing models, M can be computed efficiently using dynamic programming (Appendix C.2). Note that M is data-independent, so this computation can be done once in advance.
5.2
Application to the PCFG-IE model
As a concrete example, consider the PCFG-IE model over L = 3 words. Write A = OT . For
any ? ? Rd , we can express the observed moments as a sum over the two possible topologies in
Figure 1(a):
def
?1;? = A diag(T diag(?)A> ?)A> ,
def
def
?2;? = A diag(?)T > diag(A> ?)A> ,
def
?3;? = A diag(A> ?)T diag(?)A> ,
?123? = E[x1 ? x2 (? > x3 )] = 0.5?1;? + 0.5?2;? ,
def
?132? = E[x1 ? x3 (? > x2 )] = 0.5?3;? + 0.5?2;? ,
def
?231? = E[x2 ? x3 (? > x1 )] = 0.5?3;? + 0.5?1;? ,
or compactly in matrix form:
!
?123?
?132?
?231?
|
{z
}
observed moments ??
0.5I
0
0.5I
=
|
0.5I
0.5I
0
{z
0
0.5I
0.5I
mixing matrix M
!
}
!
?1;?
?2;?
?3;?
| {z }
.
compound parameters ??
Let us observe ?? at two different values of ?, say at ? = 1 and ? = ? for some random ? . Since
the mixing matrix M is invertible, we can obtain the compound parameters ?2;1 = (M ?1 ?1 )2 and
?2;? = (M ?1 ?? )2 .
7
Now we will recover ? from ?2;1 and ?2;? by first extracting A = OT via an eigenvalue decomposition, and then recovering ?, T , and O in turn (all up to the same unknown permutation) via
elementary matrix operations.
For the first step, we will use the following tool (adapted from Algorithm A of [1]), which allow us
to decompose two related matrix products:
Lemma 1 (Spectral decomposition). Let M1 , M2 ? Rd?k have full column rank and D be a diagonal matrix with distinct diagonal entries. Suppose we observe X = M1 M2> and Y = M1 DM2> .
Then D ECOMPOSE(X, Y ) recovers M1 up to a permutation and scaling of the columns.
D ECOMPOSE(X, Y ):
?1. Find U1 , U2 ? Rd?k such that range(U1 ) = range(X) and range(U2 ) = range(X > ).
?2. Perform an eigenvalue decomposition of (U1> Y U2 )(U1> XU2 )?1 = V SV ?1 .
?3. Return (U1> )? V .
>
First, run D ECOMPOSE(X = ?>
2;1 , Y = ?2;? ) (Lemma 1), which corresponds to M1 = A and
M2 = A diag(?)T > . This produces A?S for some permutation matrix ? and diagonal scaling S.
Since we know that the columns of A sum to one, we can identify A?.
To recover the initial distribution ? (up to permutation), take ?2;1 1 = A? and left-multiply by
(A?)? to get ??1 ?. For T , put the entries of ? in a diagonal matrix: ??1 diag(?)?. Take ?>
2;1 =
AT diag(?)A> and multiply by (A?)? on the left and ((A?)> )? (??1 diag(?)?)?1 on the right,
which yields ??1 T ?. (Note that ? is orthogonal, so ??1 = ?> .) Finally, multiply A? = OT ?
and (??1 T ?)?1 , which yields O?.
The above algorithm identifies the PCFG-IE from only length 3 sentences. To exploit sentences of
different lengths, we can compute a mixing matrix M which includes constraints from sentences
of length 1 ? L ? Lmax up to some upper bound Lmax . For example, Lmax = 10 results in a
990 ? 2376 mixing matrix. We can retrieve the same compound parameters (?2;1 and ?2;? ) from
the pseudoinverse of M and as proceed as before.
5.3
Application to the DEP-IES model
We now turn to the DEP-IES model over L = 3 words. Our goal is to recover the parameters
? = (?, A). Let D = diag(?) = diag(A?), where the second equality is due to stationarity of ?.
def
?1 = E[x1 ] = ?,
def
?12 = E[x1 ? x2 ] = 7?1 (DA> + DA> + DA> A> + AD + ADA> + AD + DA> ),
def
?13 = E[x1 ? x3 ] = 7?1 (DA> + DA> A> + DA> + ADA> + AD + AAD + AD),
def ?
?1
?
?12 = E[x
(DA> + AD),
1 ? x2 ] = 2
? is taken with respect to length 2 sentences. Having recovered ? from ?1 , it remains to
where E[?]
recover A. By selectively combining the moments above, we can compute AA + A = [7(?13 ?
?12 ) + 2?
?12 ] diag(?1 )?1 . Assuming A is generic position, it is diagonalizable: A = Q?Q?1 for
some diagonal matrix ? = diag(?1 , . . . , ?d ), possibly with complex entries. Therefore, we can
recover ?2 + ? = Q?1 (AA + A)Q. Since ? is diagonal, we simply have d independent quadratic
equations in ?i , which can be solved in closed form. After obtaining ?, we retrieve A = Q?Q?1 .
6
Discussion
In this work, we have shed some light on the identifiability of standard generative parsing models using our numerical identifiability checker. Given the ease with which this checker can be applied, we
believe it should be a useful tool for analyzing more sophisticated models [6], as well as developing
new ones which are expressive yet identifiable.
There is still a large gap between showing identifiability and developing explicit algorithms. We
have made some progress on closing it with our unmixing technique, which can deal with models
where the tree topology varies non-trivially.
8
References
[1] A. Anandkumar, D. Hsu, and S. M. Kakade. A method of moments for mixture models and hidden
Markov models. In COLT, 2012.
[2] F. Pereira and Y. Shabes. Inside-outside reestimation from partially bracketed corpora. In ACL, 1992.
[3] G. Carroll and E. Charniak. Two experiments on learning probabilistic dependency grammars from corpora. In Workshop Notes for Statistically-Based NLP Techniques, AAAI, pages 1?13, 1992.
[4] M. A. Paskin. Grammatical bigrams. In NIPS, 2002.
[5] D. Klein and C. D. Manning. Conditional structure versus conditional estimation in NLP models. In
EMNLP, 2002.
[6] D. Klein and C. D. Manning. Corpus-based induction of syntactic structure: Models of dependency and
constituency. In ACL, 2004.
[7] P. Liang and D. Klein. Analyzing the errors of unsupervised learning. In HLT/ACL, 2008.
[8] J. T. Chang. Full reconstruction of Markov models on evolutionary trees: Identifiability and consistency.
Mathematical Biosciences, 137:51?73, 1996.
[9] A. Anandkumar, K. Chaudhuri, D. Hsu, S. M. Kakade, L. Song, and T. Zhang. Spectral methods for
learning multivariate latent tree structure. In NIPS, 2011.
[10] J. B. Kruskal. Three-way arrays: Rank and uniqueness of trilinear decompositions, with application to
arithmetic complexity and statistics. Linear Algebra and Applications, 18:95?138, 1977.
[11] E. S. Allman, C. Matias, and J. A. Rhodes. Identifiability of parameters in latent structure models with
many observed variables. Annals of Statistics, 37:3099?3132, 2009.
[12] E. S. Allman, S. Petrovi, J. A. Rhodes, and S. Sullivant. Identifiability of 2-tree mixtures for group-based
models. Transactions on Computational Biology and Bioinformatics, 8:710?722, 2011.
[13] J. A. Rhodes and S. Sullivant. Identifiability of large phylogenetic mixture models. Bulletin of Mathematical Biology, 74(1):212?231, 2012.
[14] T. J. Rothenberg. Identification in parameteric models. Econometrica, 39:577?591, 1971.
[15] L. A. Goodman. Exploratory latent structure analysis using both identifiabile and unidentifiable models.
Biometrika, 61(2):215?231, 1974.
[16] D. Bamber and J. P. H. van Santen. How many parameters can a model have and still be testable? Journal
of Mathematical Psychology, 29:443?473, 1985.
[17] D. Geiger, D. Heckerman, H. King, and C. Meek. Stratified exponential families: graphical models and
model selection. Annals of Statistics, 29:505?529, 2001.
[18] K. Lari and S. J. Young. The estimation of stochastic context-free grammars using the inside-outside
algorithm. Computer Speech and Language, 4:35?56, 1990.
[19] M. Johnson, T. Griffiths, and S. Goldwater. Bayesian inference for PCFGs via Markov chain Monte Carlo.
In HLT/NAACL, 2007.
[20] E. Mossel and S. Roch. Learning nonsingular phylogenies and hidden Markov models. Annals of Applied
Probability, 16(2):583?614, 2006.
[21] D. Hsu, S. M. Kakade, and T. Zhang. A spectral algorithm for learning hidden Markov models. In COLT,
2009.
[22] S. M. Siddiqi, B. Boots, and G. J. Gordon. Reduced-rank hidden Markov models. In AISTATS, 2010.
[23] A. Parikh, L. Song, and E. P. Xing. A spectral algorithm for latent tree graphical models. In ICML, 2011.
[24] S. B. Cohen, K. Stratos, M. Collins, D. P. Foster, and L. Ungar. Spectral learning of latent-variable
PCFGs. In ACL, 2012.
[25] F. M. Luque, A. Quattoni, B. Balle, and X. Carreras. Spectral learning for non-deterministic dependency
parsing. In EACL, 2012.
[26] P. Dhillon, J. Rodue, M. Collins, D. P. Foster, and L. Ungar. Spectral dependency parsing with latent
variables. In EMNLP-CoNLL, 2012.
[27] J. Eisner. Three new probabilistic models for dependency parsing: An exploration. In COLING, 1996.
[28] S. Sahni. Computationally related problems. SIAM Journal on Computing, 3:262?279, 1974.
[29] J. Eisner. Bilexical grammars and their cubic-time parsing algorithms. In Advances in Probabilistic and
Other Parsing Technologies, pages 29?62, 2000.
9
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4,039 | 4,656 | Recognizing Activities by Attribute Dynamics
Weixin Li
Nuno Vasconcelos
Department of Electrical and Computer Engineering
University of California, San Diego
La Jolla, CA 92093, United States
{wel017, nvasconcelos}@ucsd.edu
Abstract
In this work, we consider the problem of modeling the dynamic structure of human activities in the attributes space. A video sequence is first represented in a
semantic feature space, where each feature encodes the probability of occurrence
of an activity attribute at a given time. A generative model, denoted the binary
dynamic system (BDS), is proposed to learn both the distribution and dynamics
of different activities in this space. The BDS is a non-linear dynamic system,
which extends both the binary principal component analysis (PCA) and classical
linear dynamic systems (LDS), by combining binary observation variables with
a hidden Gauss-Markov state process. In this way, it integrates the representation power of semantic modeling with the ability of dynamic systems to capture
the temporal structure of time-varying processes. An algorithm for learning BDS
parameters, inspired by a popular LDS learning method from dynamic textures,
is proposed. A similarity measure between BDSs, which generalizes the BinetCauchy kernel for LDS, is then introduced and used to design activity classifiers.
The proposed method is shown to outperform similar classifiers derived from the
kernel dynamic system (KDS) and state-of-the-art approaches for dynamics-based
or attribute-based action recognition.
1
Introduction
Human activity understanding has been a research topic of substantial interest in computer vision [1].
Inspired by the success of the popular bag-of-features (BoF) representation on image classification
problems, it is frequently based on the characterization of video as a collection of orderless spatiotemporal features [2, 3]. Recently, there have been attempts to extend this representation along
two dimensions that we explore in this work. The first is to introduce richer models for the temporal
structure, also known as dynamics, of human actions [4, 5, 6, 7]. This aims to exploit the fact that
actions are usually defined as sequences of poses, gestures, or other events over time. While desirable, modeling action dynamics can be a complex proposition, and this can sometimes compromise
the robustness of recognition algorithms, or sacrifice their generality, e.g., it is not uncommon for
dynamic models to require features specific to certain datasets or action classes [5, 6], or non-trivial
forms of pre-processing, such as tracking [8], manual annotation [7], etc. The second dimension,
again inspired by recent developments in image classification [9, 10], is to represent actions in
terms of intermediate-level semantic concepts, or attributes [11, 12]. This introduces a layer of
abstraction that improves the generalization of the representation, enables modeling of contextual
relationships [13], and simplifies knowledge transfer across activity classes [11].
In this work, we propose a representation that combines the benefits of these two types of extensions.
This consists of modeling the dynamics of human activities in the attributes space. The idea is to
exploit the fact that an activity is usually defined as a sequence of semantic events. For example, the
activity ?storing an object in a box? is defined as the sequence of the action attributes ?remove (hand
from box)?, ?grab (object)?, ?insert (hand in box)?, and ?drop (object)?. The representation of
1
the action as a sequence of these attributes makes the characterization of the ?storing object in
box? activity more robust (to confounding factors such as diversity of grabbing styles, hand motion
speeds, or camera motions) than dynamic representations based on low-level features. It is also
more discriminant than semantic representations that ignore dynamics, i.e., that simply record the
occurrence (or frequency) of the action attributes ?remove?, ?grab?, ?insert?, and ?drop?. In the
absence of information about the sequence in which these attributes occur, the ?store object in box?
activity cannot be distinguished from the ?retrieve object from box? activity, defined as the sequence
?insert (hand in box)?, ?grab (object)?, ?remove (hand from box)?, and ?drop (object)?. In summary,
the modeling of attribute dynamics is 1) more robust and flexible than the modeling of visual (lowlevel) dynamics, and 2) more discriminant than the modeling of attribute frequencies.
In this work, we address the problem of modeling attribute dynamics for activities. As is usual in
semantics-based recognition [11], we start by representing video in a semantic feature space, where
each feature encodes the probability of occurrence of an action attribute in the video, at a given
time. We then propose a generative model, denoted the binary dynamic system (BDS), to learn both
the distribution and dynamics of different activities in this space. The BDS is a non-linear dynamic
system, which combines binary observation variables with a hidden Gauss-Markov state process.
It can be interpreted as either 1) a generalization of binary principal component analysis (binary
PCA) [14], which accounts for data dynamics, or 2) an extension of the classical linear dynamic
system (LDS), which operates on a binary observation space. For activity recognition, the BDS has
the appeal of accounting for the two distinguishing properties of the semantic activity representation:
1) that semantic vectors define probability distributions over a space of binary attributes; and 2) that
these distributions evolve according to smooth trajectories that reflect the dynamics of the underlying
activity. Its advantages over previous representations are illustrated by the introduction of BDSbased activity classifiers. For this, we start by proposing an efficient BDS learning algorithm, which
combines binary PCA and a least squares problem, inspired by the learning procedure in dynamic
textures [15]. We then derive a similarity measure between BDSs, which generalizes the BinetCauchy kernel from the LDS literature [16]. This is finally used to design activity classifiers, which
are shown to outperform similar classifiers derived from the kernel dynamic systems (KDS) [6], and
state-of-the-art approaches for dynamics-based [4] and attribute-based [11] action recognition.
2
Prior Work
One of the most popular representations for activity recognition is the BoF, which reduces video to
an collection of orderless spatiotemporal descriptors [2, 3]. While robust, the BoF ignores the temporal structure of activities, and has limited power for fine-grained activity discrimination. A number
of approaches have been proposed to characterize this structure. One possibility is to represent actions in terms of limb or torso motions, spatiotemporal shape models, or motion templates [17, 18].
Since they require detection, segmentation, tracking, or 3D structure recovery of body parts, these
representations can be fragile. A robust alternative is to model the temporal structure of the BoF.
This can be achieved with generalizations of popular still image recognition methods. For example,
Laptev et al. extend pyramid matching to video, using a 3D binning scheme that roughly characterizes the spatio-temporal structure of video [3]. Niebles et al. employ a latent SVM that augments
the BoF with temporal context, which they show to be critical for understanding realistic motion [4].
All these approaches have relatively coarse modeling of dynamics. More elaborate models are usually based on generative representations. For example, Laxton et al. model a combination of object
contexts and action sequences with a dynamic Bayesian network [5], while Gaidon et al. reduce
each activity to three atomic actions and model their temporal distributions [7]. These methods
rely on action-class specific features and require detailed manual supervision. Alternatively, several researchers have proposed to model BoF dynamics with LDSs. For example, Kellokumpu et al.
combine dynamic textures [15] and local binary patterns [19], Li et al. perform a discriminant canonical correlation analysis on the space of action dynamics [8], and Chaudhry et al. map frame-wise
motion histograms to a reproducing kernel Hilbert space (RKHS), where they learn a KDS [6].
Recent research in image recognition has shown that various limitations of the BoF can be overcome
with representations of higher semantic level [10]. The features that underly these representations
are confidence scores for the appearance of pre-defined visual concepts in images. These concepts
can be object attributes [9], object classes [20, 21], contextual classes [13], or generic visual concepts [22]. Lately, semantic attributes have also been used for action recognition [11], demonstrating
the benefits of a mid-level semantic characterization for the analysis of complex human activities.
2
0.5/0.8
. . .
. . .
. . .
. . .
. . .
. . .
. . .
run
0.5/0.2
1/0
. . .
. . .
. . .
. . .
. . .
. . .
. . .
land
jump
0.8/0.3
0.2/0.7
Figure 1: Left: key frames of activities ?hurdle race? (top) and ?long jump? (bottom); Right: attribute transition probabilities of the two activities (?hurdle race? / ?long jump?) for attributes ?run?, ?jump?, and ?land?.
The work also suggests that, for action categorization, supervised attribute learning is far more useful
than unsupervised learning, resembling a similar observation from image recognition [20]. However, all of these representations are BoF-like, in the sense that they represent actions as orderless
feature collections, reducing an entire video sequence to an attribute vector. For this reason, we
denote them holistic attribute representations.
The temporal evolution of semantic concepts, throughout a video sequence, has not yet been exploited as a cue for action understanding. There has, however, been some progress towards this
type of modeling in the text analysis literature, where temporal extensions of latent Dirichlet allocation (LDA) have been proposed. Two representatives are the dynamic topic model (DTM) [23]
and the topic over time (TOT) model [24]. Although modeling topic dynamics, these models are not
necessarily applicable to semantic action recognition. First, like the underlying LDA, they are unsupervised models, and thus likely to underperform in recognition tasks [11, 10]. Second, the joint
goal of topic discovery and modeling topic dynamics requires a complex graphical model. This is
at odds with tractability, which is usually achieved by sacrificing the expressiveness of the temporal
model component.
3
Modeling the Dynamics of Activity Attributes
In this section, we introduce a new model, the binary dynamic system, for joint representation of the
distribution and dynamics of activities in action attribute space.
3.1
Semantic Representation
Semantic representations characterize video as a collection of descriptors with explicit semantics [10, 11]. They are obtained by defining a set of semantic concepts (or attributes, scene classes,
etc), and learning a classifier to detect each of those concepts. Given a video v ? X to analyze, each
classifier produces a confidence score for the presence of the associated concept. The ensemble of
classifiers maps the video to a semantic space S, according to ? : X ? S = [0, 1]K , ?(v) =
(?1 (v), ? ? ? , ?K (v))T , where ?i (v) is the confidence score for the presence of the i-th concept.
In this work, the classification score is the posterior probability of a concept c given video v,
i.e., ?c (v) = p(c|v) under a certain video representation, e.g., the popular BoF histogram of spatiotemporal descriptors. As the video sequence v progresses with time t, the semantic encoding defines
a trajectory {? t (v)} ? S. The benefits of semantic representations for recognition, namely a higher
level of abstraction (which leads to better generalization than appearance-based representations),
substantial robustness to the performance of the visual classifiers ?i (v), and intrinsic ability to account for contextual relationships between concepts, have been previously documented in the literature [13]. No attention has, however, been devoted to modeling the dynamics of semantic encodings
of video. Figure 1 motivates the importance of such modeling for action recognition, by considering
two activity categories (?long jump? and ?hurdle race?), which involve the same attributes, with
roughy the same probabilities, but span very different trajectories in S. Modeling these dynamics
can substantially enhance the ability of a classifier to discriminate between complex activities.
3.2
Binary PCA
The proposed representation is a generalization of binary PCA [14], a dimensionality reduction
technique for binary data, belonging to the generalized exponential family PCA [25]. It fits a linear
model to binary observations, by embedding the natural parameters of Bernoulli distributions in a
low-dimensional subspace. Let Y denote a K ? ? binary matrix (Ykt ? {0, 1}, e.g., the indicator of
3
occurrence of attribute k at time t) where each column is a vector of K binary observations sampled
from a multivariate Bernoulli distribution
ykt
Ykt ? B(ykt ; ?kt ) = ?kt
(1 ? ?kt )1?ykt = ?(?kt )ykt ?(??kt )1?ykt , ykt ? {0, 1}.
(1)
?
log( 1??
)
The log-odds ? =
is the natural parameter of the Bernoulli distribution, and ?(?) =
(1 + e?? )?1 is the logistic function. Binary PCA finds a L-dimensional (L K) embedding of the
natural parameters, by maximizing the log-likelihood of the binary matrix Y
i
X h
L = log P (Y ; ?) =
Ykt log ?(?kt ) + (1 ? Ykt ) log ?(??kt )
(2)
k,t
under the constraint
K?L
? = CX + u1T ,
L??
K
(3)
?
where C ? R
,X ?R
, u ? R and 1 ? R is the vector of all ones. Each column of C
is a basis vector of a latent subspace and the t-th column of X contains the coordinates of the t-th
binary vector in this basis (up to a translation by u).
Binary PCA is not directly applicable to attribute-based recognition, where the goal is to fit the
vectors of confidence scores {? t } produced by a set of K attribute classifiers (and not a sample of
binary attribute vectors per se). To overcome this problem, we maximize the expected log-likelihood
of the data Y (which is the lower bound to the log expected likelihood of the data Y , by Jensen?s
inequality). Since E[y t ] = ? t , it follows from (2) that
i
X h
EY [L] =
?kt log ?(?kt ) + (1 ? ?kt ) log ?(??kt ) .
(4)
k,t
The proposed extension of binary PCA consists of maximizing this expected log-likelihood under
the constraint of (3). It can be shown that, in the absence of the constraint, the maximum occurs
when ?(?kt ) = ?kt , ?k, t. As in PCA, (3) forces ?(?kt ) to lie on a subspace of S, i.e.,
?(?kt ) = ?
?kt ? ?kt .
(5)
The difference between the expected log-likelihood of the true scores {? t } and the binary PCA
scores {?(? t ) = ?(Cxt + u)} (?(?) ? [?(?1 ), ? ? ? , ?(?K )]T ) is
E[?L({? t }; {?(? t )})]
=
=
=
EY log(P (Y ; {? t })) ? EY log(P (Y ; {?(? t )}))
X
?kt
1 ? ?kt
?kt log
+ (1 ? ?kt ) log
k,t
?(?kt )
?(??kt )
X
KL[B(y; ? t )||B(y; ?(? t ))],
t
(6)
(7)
(8)
where KL(B(y; ?)||B(y; ? 0 )) is the Kullback-Leibler (KL) divergence between two multivariate
Bernoulli distributions of parameters ? and ? 0 . By maximizing the expected log-likelihood (4), the
optimal projection {? ?t } of the attribute score vectors {? t } on the subspace of (3) also minimizes the
KL divergence of (8). Hence, for the optimal natural parameters {? ?t }, the approximation of (5) is the
best in the sense of KL divergence, the natural similarity measure between probability distributions.
3.3
Binary Dynamic Systems
A discrete time linear dynamic system (LDS) is defined by
xt+1 = Axt + v t
,
y t = Cxt + wt + u
L
K
(9)
where xt ? R and y t ? R (of mean u) are the hidden state and observation variable at
L?L
time t, respectively; A ? R
is the state transition matrix that encodes the underlying dynamK?L
ics; C ? R
the observation matrix that linearly maps the state to the observation space; and
x1 = ?0 + v 0 ? N (?0 , S0 ) an initial condition. Both state and observations are subject to additive Gaussian noise processes v t ? N (0, Q) and wt ? N (0, R). Since the noise is Gaussian and
L < K, the matrix C can be interpreted as a PCA basis for the observation space (L eigenvectors
of the observation covariance). The state vector xt then encodes the trajectory of the PCA coefficients (projection on this basis) of the observed data over time. This interpretation is, in fact, at the
core of the popular dynamic texture (DT) [15] representation for video. While the LDS parameters
4
Algorithm 1: Learning a binary dynamic system
Input : a sequence of attribute score vectors {? t }?t=1 , state space dimension n.
?
Binary PCA: {C, X, u} = B-PCA({?
t }t=1 , n) using the method of [14].
t2
Estimate state parameters (Xt1 ? xt1 , ? ? ? , xt2 ):
1
A = X2? (X1? ?1 )? ; V = (X)?2 ? A(X)?1 ?1 ; Q = ? ?1
V (V )T ;
P?
P?
1
1
T
?0 = ? t=1 xt ; S0 = ? ?1 t=1 (xt ? ?0 )(xt ? ?0 ) .
Output: {A, C, Q, u, ?0 , S0 }
can be learned by maximum likelihood, using an expectation-maximization (EM) algorithm [26],
the DT decouples the learning of observation and state variables. Observation parameters are first
learned by PCA, and state parameters are then learned with a least squares procedure. This simple
approximate learning algorithm tends to perform very well, and is widely used in computer vision.
The proposed binary dynamic system (BDS) is defined as
xt+1 = Axt + v t
,
y t ? B(y; ?(Cxt + u))
L
(10)
K
where xt ? R and u ? R are the hidden state variable and observation bias, respectively; A ?
RL?L is the state transition matrix; and C ? RK?L the observation matrix. The initial condition is
given by x1 = ?0 + v 0 ? N (?0 , S0 ); and the state noise process is v t ? N (0, Q). Like the LDS
of (9), the BDS can be interpreted as combining a (now binary) PCA observation component with
a Gauss-Markov process for the state sequence. As in binary PCA, for attribute-based recognition
the binary observations y t are replaced by the attribute scores ? t , their log-likelihood under (10)
by the expected log-likelihood, and the optimal solution minimizes the approximation of (5) for
the most natural definition of similarity (KL divergence) between probability distributions. This is
conceptually equivalent to the behavior of the canonical LDS of (9), which determines the subspace
that best approximates the observations in the Euclidean sense, the natural similarity measure for
Gaussian data. Note that other extensions of the LDS, e.g., kernel dynamic systems (KDS) that rely
on a non-linear kernel PCA (KPCA) [27] of the observation space but still assume an Euclidean
measure (Gaussian noise) [28, 6], do not share this property. We will see, in the experimental
section, that the BDS is a better model of attribute dynamics.
3.4
Learning
Since the Gaussian state distribution of an LDS is a conjugate prior for the (Gaussian) conditionaldistribution of its observations given the state, maximum-likelihood estimates of LDS parameters
are tractable. The LDS parameters ?LDS = {A, C, Q, R, ?0 , S0 , u} of (9) can thus be estimated
with an EM algorithm [26]. For the BDS, where the state is Gaussian but the observations are not,
the expectation step is intractable. Hence, approximate inference is required to learn the parameters
?BDS = {A, C, Q, ?0 , S0 , u} of (10). In this work, we resort to the approximate DT learning
procedure, where observation and state components are learned separately [15]. The binary PCA
basis is learned first, by maximizing the expected log-likelihood of (4) subject to the constraint
of (3). Since the Bernoulli distribution is a member of exponential family, (4) is concave in ?, but
not in C, X and u jointly. We rely on a procedure introduced by [14], which iterates between the
optimization with respect to one of the variables C, X and u, with the remaining two held constant.
Each iteration is a convex sub-problem that can be solved efficiently with a fixed-point auxiliary
function (see [14] for details). Once the latent embedding C ? , X ? and u? of the attribute sequence
in the optimal subspace is recovered, the remaining parameters are estimated by solving a leastsquares problem for A and Q, and using standard maximum likelihood estimates for the Gaussian
parameters of the initial condition (?0 and S0 ) [15]. The procedure is summarized in Algorithm 1.
4
Measuring Distances between BDSs
The design of classifiers that account for attribute dynamics requires the ability to quantify similarity
between BDSs. In this section, we derive the BDS counterpart to the popular Binet-Cauchy kernel (BCK) for the LDS, which evaluates the similarity of the output sequences of two LDSs. Given
5
LDSs ?a and ?b driven by identical noise processes v t and wt with observation sequences y (a)
and y (b) , [16] propose a family of BCKs
hX?
i
(a)
(b)
KBC (?a , ?b ) = Ev,w
e??t (y t )T W y t ,
(11)
t=0
where W is a semi-definite positive weight matrix and ? > 0 a temporal discounting factor. To
(a)
(b)
extend (11) to BDSs ?a and ?b , we note that (y t )T W y t is the inner product of an Euclidean
(a)
(b)
(a)
(b)
(a)
(b)
output space of metric d2 (y t , y t ) = (y t ? y t )T W (y t ? y t ). For BDSs, whose obser(a)
(b)
vations y t are Bernouli distributed with parameters {?(? t )}, for ?a , and {?(? t )}, for ?b , this
distance measure is naturally replaced by the KL divergence between Bernoulli distributions
"
DBC (?a , ?b ) = Ev
?
X
??t
e
(a)
(b)
KL(B(?(? t ))||B(?(? t )))
+
#
(b)
(a)
KL(B(?(? t ))||B(?(? t )))
t=0
= Ev
(12)
X
?
t=0
??t
e
T
(a)
(b)
(a)
(b)
?(? t ) ? ?(? t )
?t ? ?t
,
where ? t = Cxt + u. The distance term at time t can be rewritten as
(a)
(b)
(a)
(?(? t ) ? ?(? t ))T (? t
(a)
(b)
? ? t ) = (? t
(b)
(b)
? t (? (a)
? ? t ),
? ? t )T W
t
(13)
? t a diagonal matrix whose k-th diagonal element is W
? t,k = (?(?(a) ) ? ?(?(b) ))/(?(a) ?
with W
t,k
t,k
t,k
(b)
? (a,b) ) (where, by the mean value theorem, ?
? (a,b) is some real value between ?
? (a) and
? ) = ? 0 (?
t,k
t,k
t,k
t,k
? (b) ). This reduces (13) to a form similar to (11), although with a time varying weight matrix Wt .
?
t,k
It is unclear whether (12) can be computed in closed-form. We currently rely on the approximation
P?
? (a) ) ? ?(?
? (b) ))T (?
? (a) ? ?
? (b) ), where ?
? is the mean of ?.
DBC (?a , ?b ) ? t=0 e??t (?(?
t
t
t
t
5
Experiments
Several experiments were conducted to evaluate the BDS as a model of activity attribute dynamics. In all cases, the BoF was used as low-level video representation, interest points were detected
with [2], and HoG/HoF descriptors [3] computed at their locations. A codebook of 3000 visual
words was learned via k-means, from the entire training set, and a binary SVM with histogram
intersection kernel (HIK) and probability outputs [29] trained to detect each attribute using the attribute definition same as [11]. The probability for attribute k at time t was used as attribute score
?tk , which was computed over a window of 20 frames, sliding across a video.
5.1
Weizmann Activities
To obtain some intuition on the performance of different algorithms considered, we first used complex activity sequences synthesized from the Weizmann dataset [17]. This contains 10 atomic action
classes (e.g., skipping, walking) annotated with respect to 30 lower-level attributes (e.g., ?one-armmotion?), and performed by 9 people. We created activity sequences by concatenating Weizmann
actions. A sequence of degree n (n = 4, 5, 6) is composed of n atomic actions, performed by the
same person. The row of images at the top of Figure 2 presents an example of an activity sequence of
degree 5. The images shown at the top of the figure are keyframes from the atomic actions (?walk?,
?pjump?, ?wave1?, ?wave2?, ?wave2?) that compose this activity sequence. The black curve (labeled ?Sem. Seq?) in the plot at the bottom of the figure shows the score of the ?two-arms-motion?
attribute, as a function of time. 40 activity categories were defined per degree n (total of 120 activity
categories) and a dataset was assembled per category, containing one activity sequence per person (9
people, 1080 sequences in total). Overall, the activity sequences differ in the number, category, and
temporal order of atomic actions. Since the attribute ground truth is available for all atomic actions
in this dataset, it is possible to train clean attribute models. Hence, all performance variations can
be attributed to the quality of the attribute-based inference of different approaches.
We started by comparing the binary PCA representation that underlies the BDS to the PCA and
KPCA decompositions of the LDS and KDS. In all cases we projected a set of attribute score vectors
? t }, and
{? t } into the low-dimensional PCA subspace, computed the reconstructed score vectors {?
? t ), as reported in Figure 3. The kernel used for KPCA was
the KL divergence KL(B(y, ? t )||B(y, ?
6
6
#
0
!%#
PCA
kernel?PCA
binary?PCA
4
>?1
!%!'
7?1
78 >?1
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78 7?1
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2
0
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#!!
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$"!
!
Figure 2: Top: key frames from the activity sequence class ?walkpjump-wave1-wave2-wave2?. Bottom: score of ?two-arms-motion?
attribute for video of this activity. True scores in black, and scores
sampled from the BDS (red) and KDS (blue). Also shown is the KLdivergence between sampled and original scores, for both models.
?4
0
1
2
3
4
5
6
7
n
Figure 3:
Log KL-divergence between original and reconstructed attribute scores, v.s. number of PCA components n, on Weizmann activities for
PCA, kernel PCA, and binary PCA.
Table 1: Classification Accuracy on Weizmann Activities and Olympic Sports Datasets
Dataset
BoF
Holistic Attri.
DTM
TOT
KDS
BDS
Weizmann Activities
Olympic Sports
57.8%
56.8%
72.6%
63.5%
84.6%
47.1%
88.2%
53.3%
90.2%
62.3%
94.8%
65.7%
the logit kernel K(? 1 , ? 2 ) = ? ?1 (? 1 )T ? ?1 (? 2 ), where ? ?1 (?) is the element-wise logit function.
Figure 3 shows the average log-KL divergence, over the entire dataset, as a function of the number of
PCA components used in the reconstruction. Binary PCA outperformed both PCA and KPCA. The
improvements over KPCA are particularly interesting since the latter uses the logistic transformation
that distinguishes binary PCA from PCA. This is explained by the Euclidean similarity measure that
underlies the assumption of Gaussian noise in KPCA, as discussed in Section 3.3. To gain some
more insight on the different models, a KDS and a BDS were learned from the 30 dimensional
attribute score vectors of the activity sequence in Figure 2. A new set of attribute score vectors were
then sampled from each model. The evolution of the scores sampled for the ?two-arms-motion?
attribute are shown in the figure (in red/blue for BDS/KDS). Note how the scores sampled from the
BDS approximate the original attribute scores better than those sampled from the KDS, which is
confirmed by the KL-divergences between the original attribute scores and those sampled from the
two models (also shown in the figure).
We next evaluated the benefits of different dynamics representations for activity recognition. Recognition rates were obtained with a 9-fold leave-one-out-cross-validation (LOOCV), where, per trial,
the activities of one subject were used as test set and those of the remaining 8 as training set. We
compared the performance of classifiers based on the KDS and BDS with a BoF classifier, a holistic
attribute classifier that ignores attribute dynamics (using a single attribute score vector computed
from the entire video sequence) and the dynamic topic models DTM [23] and TOT [24] from the
text literature. For the latter, the topics were equated to the activity attributes and learned with supervision (using the SVMs discussed above). Unsupervised versions of the topic models had worse
performance and are omitted. Classification was performed with Bayes rule for topic models, and a
nearest-neighbor classifier for the remaining methods. For BDS, distances were measured with (12),
while for the KDS we tried the Binet-Cauchy, X 2 , intersection and logit kernels, and reported the
best results. X 2 distance was used for the BoF and holistic attribute classifiers. The classification
accuracy of all classifiers is shown in Table 1. BDS and KDS had the best performance, followed by
the dynamic topic models, and the dynamics insensitive methods (BoF and holistic). Note that the
difference between the holistic classifier and the best dynamic model is of approximately 22%. This
shows that while attributes are important (14.8% improvement over BoF) they are not the whole
story. Problems involving fine-grained activity classification, i.e., discrimination between activities
composed of similar actions executed in different sequence, requires modeling of attribute dynamics.
Among dynamic models, the BDS outperformed the KDS, and topic models DTM and TOT.
5.2 Olympic Sports
The second set of experiments was performed on the Olympic Sports dataset [4]. This contains
YouTuBe videos of 16 sport activities, with a total of 783 sequences. Some activities are sequences
7
Table 2: Fine-grained Classification Accuracy on Olympic Sports by BDS
clean&jerk
(snatch)
long-jump
(triple-jump)
snatch
(clean&jerk)
triple-jump
(long-jump)
BDS
Holistic
85% (9%)
73% (21%)
80% (2%)
72% (20%)
78% (10%)
65% (27%)
62% (14%)
38% (43%)
Table 3: Mean Average Precision on Olympic Sports Dataset
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Laptev et al. [3]
( BoF )
Niebles et al. [4]
( BDS )
Liu et al. [11]
( Attr. / B+A )
B+A+D
62.0%
( 67.8% )
72.1%
(73.2%)
74.4%
(72.9% / 73.3%)
76.5%
@6B6>C ;5/823.= )!=
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Figure 4: Scatter plot of accuracy
gain on Olympic Sports by BDS.
of atomic actions, whose temporal structure is critical for discrimination from other classes (e.g.,
?clean and jerk? v.s.?snatch?, and ?long-jump? v.s.?triple-jump?). Since attribute labels are only
available for whole sequences, the training sets of the attribute classifiers are much noisier than
in the previous experiment. This degrades the quality of attribute models. The dataset was split
into 5 subsets, of roughly the same size, and results reported by 5-fold cross-validation. The
DTM and TOT classifiers were as above, and all others were implemented with an SVM of kernel K? (i, j) = exp(? ?1 d2 (i, j)), based on the distance measures d(i, j) of the previous section.
Table 1 shows that dynamic modeling again has the best performance. However, the gains over the
holistic attribute classifier are smaller than in Weizmann. This is due to two factors. First, the noisy
attributes make the dynamics harder to model. Note that the robustness of the dynamic models to
this noise varies substantially. As before, topic models have the weakest performance and the BDS
outperforms the KDS. Second, since fine grained discrimination is not needed for all categories,
attribute dynamics are not always necessary. This is confirmed by Figure 4, which presents a scatter
plot of the gain (difference in accuracy) of the BDS classifier over the holistic classifier, as a function of the accuracy of the latter. Each point corresponds to an activity. Note the strongly negative
correlation between the two factors: the largest gains occur for the most difficult classes for the
holistic classifier. Table 2 details these results for the two pairs of classes with most confusable attributes. Numbers outside brackets correspond to ground-truth category, numbers in brackets to the
confusing class (percentage of ground-truth examples assigned to it). BDS has dramatically better
performance for these classes. Overall, despite the attribute noise and the fact that dynamics are not
always required for discrimination, the BDS achieves the best performance on this dataset.
Finally, we compare the BDS classifier to classifiers from the literature. Three approaches, representative of the state-of-the art in classification with the BoF [3], dynamic representations [4],
and attributes [11], were selected as benchmarks. These were compared to our implementation
of BoF (kernel using only word histograms), attributes (the holistic classifier of Table 1), dynamics (the BDS classifier), and multiple kernel classifiers combining 1) BoF and attributes (B+A), and
2) BoF, attributes, and dynamics (B+A+D). All multiple kernels combinations were achieved by
cross-validation. The mean average precisions of all 1-vs-all classifiers are reported in Table 3. The
numbers in each column report to directly comparable classifiers, e.g., B+A is directly comparable
to [11], which jointly classifies BoF histograms and hollistic attribute vectors with a latent SVM.
Note that the BDS classifier outperforms the state-of-the-art in dynamic classifiers (Niebles et al.
[4]), which accounts for the dynamics of the BoF but not action attributes. This holds despite the fact
that our attribute categories (only 40 specified attributes) and classifiers (simple SVMs) are much
simpler than the best in the literature [11] , which uses both the data-driven and the 40 specified
attributes as ours, plus a latent SVM as the classifier. The use of a stronger attribute detection architecture could potentially further improve these results. Note also that the addition of the BDS kernel
to the simple attribute representation (B+A+D) far outperforms the use of the more sophisticated attribute classifier of [11], which does not account for attribute dynamics. This illustrates the benefits
of modeling the dynamics of attributes. The combination of BoF, attributes, and attribute dynamics
achieves the overall best performance on this dataset.
Acknowledgements
This work was partially supported by the NSF award under Grant CCF-0830535. We also thank
Jingen Liu for providing the attribute annotations.
8
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+,,-./,0123.14356786,1+88.69-8:
References
[1] J. K. Aggarwal and M. S. Ryoo, ?Human activity analysis: A review,? ACM Computing Surveys, vol. 43,
no. 16, pp. 1?16, 2011.
[2] P. Doll?ar, V. Rabaud, G. Cottrell, and S. Belongie, ?Behavior recognition via sparse spatio-temporal
features,? ICCV VS-PETS, 2005.
[3] I. Laptev, M. Marsza?ek, C. Schmid, and B. Rozenfeld, ?Learning realistic human actions from movies,?
CVPR, 2008.
[4] J. C. Niebles, C.-W. Chen, and L. Fei-Fei, ?Modeling temporal structure of decomposable motion segments for activity classification,? ECCV, 2010.
[5] B. Laxton, J. Lim, and D. Kriegman, ?Leveraging temporal, contextual and ordering constraints for recognizing complex activities in video,? CVPR, 2007.
[6] R. Chaudhry, A. Ravichandran, G. Hager, and R. Vidal, ?Histograms of oriented optical flow and binetcauchy kernels on nonlinear dynamical systems for the recognition of human actions,? CVPR, 2009.
[7] A. Gaidon, Z. Harchaoui, and C. Schmid, ?Actom sequence models for efficient action detection,? CVPR,
2011.
[8] B. Li, M. Ayazoglu, T. Mao, O. Camps, and M. Sznaier, ?Activity recognition using dynamic subspace
angles,? CVPR, 2011.
[9] C. H. Lampert, H. Nickisch, and S. Harmeling, ?Learning to detect unseen object classes by between-class
attribute transfer,? CVPR, 2009.
[10] N. Rasiwasia and N. Vasconcelos, ?Holistic context models for visual recognition,? IEEE Trans. Pattern
Analysis and Machine Intelligence, vol. 34, no. 5, pp. 902?917, 2012.
[11] J. Liu, B. Kuipers, and S. Savarese, ?Recognizing human actions by attributes,? CVPR, 2011.
[12] A. Fathi and G. Mori, ?Action recognition by learning mid-level motion features,? CVPR, 2008.
[13] N. Rasiwasia and N. Vasconcelos, ?Holistic context modeling using semantic co-occurrences,? CVPR,
2009.
[14] A. I. Schein, L. K. Saul, and L. H. Ungar, ?A generalized linear model for principal component analysis
of binary data,? AISTATS, 2003.
[15] G. Doretto, A. Chiuso, Y. N. Wu, and S. Soatto, ?Dynamic textures,? Int?l J. Computer Vision, vol. 51,
no. 2, pp. 91?109, 2003.
[16] S. V. N. Vishwanathan, A. J. Smola, and R. Vidal, ?Binet-cauchy kernels on dynamical systems and its
application to the analysis of dynamic scenes,? Int?l J. Computer Vision, vol. 73, no. 1, pp. 95?119, 2006.
[17] L. Gorelick, M. Blank, E. Shechtman, M. Irani, and R. Basri, ?Actions as space-time shapes,? IEEE Trans.
Pattern Analysis and Machine Intelligence, vol. 29, no. 12, pp. 2247?2253, 2007.
[18] N. ?Ikizler and D. A. Forsyth, ?Searching for complex human activities with no visual examples,? Int?l J.
Computer Vision, vol. 80, no. 3, pp. 337?357, 2008.
[19] V. Kellokumpu, G. Zhao, and M. Pietik?ainen, ?Human activity recognition using a dynamic texture based
method,? BMVC, 2008.
[20] N. Rasiwasia and N. Vasconcelos, ?Scene classification with low-dimensional semantic spaces and weak
supervision,? CVPR, 2008.
[21] A. Quattoni, M. Collins, and T. Darrell, ?Learning visual representations using images with captions,?
CVPR, 2007.
[22] N. Rasiwasi, P. J. Moreno, and N. Vasconcelos, ?Bridging the gap: Query by semantic example,? IEEE
Trans. Multimedia, vol. 9, no. 5, pp. 923?938, 2007.
[23] D. M. Blei and J. D. Lafferty, ?Dynamic topic models,? ICML, 2006.
[24] X. Wang and A. McCallum, ?Topics over time: a non-markov continuous-time model of topical trends,?
ACM SIGKDD, 2006.
[25] M. Collins, S. Dasgupta, and R. E. Schapire, ?A generalization of principal component analysis to the
exponential family,? NIPS, 2002.
[26] R. H. Shumway and D. S. Stoffer, ?An approach to time series smoothing and forecasting using the em
algorithm,? Journal of Time Series Analysis, vol. 3, no. 4, pp. 253?264, 1982.
[27] B. Sch?olkopf, A. Smola, and K.-R. M?uller, ?Nonlinear component analysis as a kernel eigenvalue problem,? Neural Computation, vol. 10, pp. 1299?1319, 1998.
[28] A. B. Chan and N. Vasconcelos, ?Classifying video with kernel dynamic textures,? CVPR, 2007.
[29] C.-C. Chang and C.-J. Lin, ?LIBSVM: A library for support vector machines,? ACM Trans. on Intelligent
Systems and Technology, vol. 2, no. 3, pp. 27:1?27:27, 2011.
9
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4,040 | 4,657 | Active Learning of Model Evidence
Using Bayesian Quadrature
Michael A. Osborne
University of Oxford
[email protected]
Carl E. Rasmussen
University of Cambridge
[email protected]
David Duvenaud
University of Cambridge
[email protected]
Roman Garnett
Carnegie Mellon University
[email protected]
Stephen J. Roberts
University of Oxford
[email protected]
Zoubin Ghahramani
University of Cambridge
[email protected]
Abstract
Numerical integration is a key component of many problems in scientific computing, statistical modelling, and machine learning. Bayesian Quadrature is a modelbased method for numerical integration which, relative to standard Monte Carlo
methods, offers increased sample efficiency and a more robust estimate of the
uncertainty in the estimated integral. We propose a novel Bayesian Quadrature
approach for numerical integration when the integrand is non-negative, such as
the case of computing the marginal likelihood, predictive distribution, or normalising constant of a probabilistic model. Our approach approximately marginalises
the quadrature model?s hyperparameters in closed form, and introduces an active learning scheme to optimally select function evaluations, as opposed to using
Monte Carlo samples. We demonstrate our method on both a number of synthetic
benchmarks and a real scientific problem from astronomy.
1
Introduction
The fitting of complex models to big data often requires computationally intractable integrals to be
approximated. In particular, machine learning applications often require integrals over probabilities
Z
Z = h?i = ?(x)p(x)dx,
(1)
where ?(x) is non-negative. Examples include computing marginal likelihoods, partition functions,
predictive distributions at test points, and integrating over (latent) variables or parameters in a model.
While the methods we will describe are applicable to all such problems, we will explicitly consider computing model evidences, where ?(x) is the unnormalised likelihood of some parameters
x1 , . . . , xD . This is a particular challenge in modelling big data, where evaluating the likelihood
over the entire dataset is extremely computationally demanding.
There exist several standard randomised methods for computing model evidence, such as annealed
importance sampling (AIS) [1], nested sampling [2] and bridge sampling. For a review, see [3].
These methods estimate Z given the value of the integrand on a set of sample points, whose size is
limited by the expense of evaluating ?(x). It is well known that convergence diagnostics are often
unreliable for Monte Carlo estimates of partition functions [4, 5, 6]. Most such algorithms also have
parameters which must be set by hand, such as proposal distributions or annealing schedules.
An alternative, model-based, approach is Bayesian Quadrature (BQ) [7, 8, 9, 10], which specifies
a distribution over likelihood functions, using observations of the likelihood to infer a distribution
1
Z
?(x)
samples
GP mean
GP mean ? SD
expected Z
p(Z|samples)
draw from GP
draw from GP
draw from GP
x
R
Figure 1: Model-based integration computes a posterior for the integral Z = ?(x)p(x)dx, conditioned on sampled values of the function ?(x). For this plot, we assume a Gaussian process model
for ?(x) and a broad Gaussian prior p(x). The variously probable integrands permitted under the
model will give different possible values for Z, with associated differing probabilities.
for Z (see Figure 1). This approach offers improved sample efficiency [10], crucial for expensive
samples computed on big data. We improve upon this existing work in three ways:
Log-GP: [10] used a GP prior on the likelihood function; this is a poor model in this case, unable to
express the non-negativity and high dynamic range of most likelihood functions. [11] introduced an
approximate means of exploiting a GP on the logarithm of a function (henceforth, a log-GP), which
better captures these properties of likelihood functions. We apply this method to estimate Z, and
extend it to compute Z?s posterior variance and expected variance after adding a sample.
Active Sampling: Previous work on BQ has used randomised or a priori fixed sampling schedules.
We use active sampling, selecting locations which minimise the expected uncertainty in Z.
Hyperparameter Marginalisation: Uncertainty in the hyperparameters of the model used for
quadrature has previously been ignored, leading to overconfidence in the estimate of Z. We introduce a tractable approximate marginalisation of input scale hyperparameters.
From a Bayesian perspective, numerical integration is fundamentally an inference and sequential
decision making problem: Given a set of function evaluations, what can we infer about the integral,
and how do we decide where to next evaluate the function. Monte Carlo methods, including MCMC,
provide simple but generally suboptimal and non-adaptive answers: compute a sample mean, and
evaluate randomly. Our approach attempts to learn about the integrand as it evaluates the function
at different points, and decide based on information gain where to evaluate next. We compare
our approach against standard Monte Carlo techniques and previous Bayesian approaches on both
simulated and real problems.
2
Bayesian Quadrature
Bayesian quadrature [8, 10] is a meansR of performing Bayesian inference about the value of a
potentially nonanalytic integral, hf i := f (x)p(x)dx. For clarity, we henceforth assume the domain of integration X = R, although all results generalise to Rn . We assume a Gaussian density
p(x) := N (x; ?x , ?x ), although other convenient forms, or, if necessary, the use of an importance
re-weighting trick (q(x) = q(x)/p(x)p(x) for any q(x)), allow any other integral to be approximated.
Quadrature involves evaluating f (x) at a vector of sample points xs , giving f s := f (xs ). Often this
evaluation is computationally expensive; the consequent sparsity of samples introduces uncertainty
about the function f between them, and hence uncertainty about the integral hf i.
Previous work on BQ chooses a Gaussian process (GP) [12] prior for f , with mean ?f and Gaussian
covariance function
K(x1 , x2 ) := h2 N (x1 ; x2 , w) .
(2)
Here hyperparameter h species the output scale, while hyperparameter w defines a (squared) input
scale over x. These scales are typically fitted using type two maximum likelihood (MLII); we will
later introduce an approximate means of marginalising them in Section 4. We?ll use the following
dense notation for the standard GP expressions for the posterior mean m, covariance C, and variance
2
log ?(x)
?(x)
input scale
input scale
x
x
Figure 2: A GP fitted to a peaked log-likelihood function is typically a better model than GP fit to
the likelihood function (which is non-negative and has high dynamic range). The former GP also
usually has the longer input scale, allowing it to generalise better to distant parts of the function.
V , respectively: mf |s (x? ) := m(f? |f s ), Cf |s (x? , x?? ) := C(f? , f?? |f s ) and Vf |s (x? ) := V (f? |f s ).
Note that this notation assumes implicit conditioning on hyperparameters. Where required for disambiguation, we?ll make this explicit, as per mf |s,w (x? ) := m(f? |f s , w) and so forth.
Variables possessing a multivariate Gaussian distribution are jointly Gaussian distributed with any
affine transformations of those variables. Because integration is affine, we can hence use computed
samples f s to perform analytic Gaussian process inference about the value of integrals over f (x),
such as hf i. The mean estimate for hf i given f s is
ZZ
m(hf i|f s ) =
hf i p(hf i|f ) p(f |f s ) dhf i df
ZZ
Z
=
hf i ? hf i ? f (x) p(x) dx N f ; mf |s , Cf |s dhf i df
Z
= mf |s (x) p(x) dx ,
(3)
which is expressible in closed-form due to standard Gaussian identities [10]. The corresponding
closed-form expression for the posterior variance of hf i lends itself as a natural convergence diagnostic. Similarly, we can compute the posteriors for integrals over the product of multiple, independent
functions. For example, we can calculate the posterior mean m(hf gi|f s , g s ) for an integral
R
f (x)g(x)p(x)dx. In the following three sections, we will expand upon the improvements this
paper introduces in the use of Bayesian Quadrature for computing model evidences.
3
Modelling Likelihood Functions
We wish to evaluate the evidence (1), an integral over non-negative likelihoods, ?(x). Assigning
a standard GP prior to ?(x) ignores prior information about the range and non-negativity of ?(x),
leading to pathologies such as potentially negative evidences (as observed in [10]). A much better
prior would be a GP prior on log ?(x) (see Figure 2). However, the resulting integral is intractable,
Z Z
m(Z|log ?s ) =
exp log ?(x) p(x) dx N log ?; mlog ?|s , Clog ?|s dlog ? ,
(4)
as (4) does not possess the affine property exploited in (3). To progress, we adopt an approximate
inference method inspired by [11] to tractably integrate under a log-GP prior.1 Specifically, we
linearise the problematic exponential term around some point log ?0 (x), as
exp log ?(x) ? exp log ?0 (x) + exp log ?0 (x) log ?(x) ? log ?0 (x)
(5)
The integral (4) consists of the product of Z and a GP for log ?. The former is ? exp log ?, the
latter is ? exp ?(log ? ? m)2 , effectively permitting only a small range of log ? functions. Over
this narrow region, it is reasonable to assume that Z does not vary too dramatically, and can be
approximated as linear in log ?, as is assumed by (5). Using this approximation, and making the
definition ?log ?|s := mlog ?|s ? log ?0 , we arrive at
Z
Z
m(Z|log ?s ) ? m(Z|log ?0 , log ?s ) := ?0 (x)p(x) dx + ?0 (x)?log ?|s (x)p(x) dx . (6)
1
In practice, we use the transform log (?(x) + 1), allowing us to assume the transformed quantity has zero
mean. For the sake of simplicity, we omit this detail in the following derivations.
3
log ?(xs )
log ?(x)
log m ?(x)|?(xs )
m log ?(x)|log ?(xs )
?c
m ?log ?|s (x)|?c
4
?(xs )
?(x)
m ?(x)|?(xs )
final approx
20
0
log ?(x)
?(x)
40
2
0
x
x
Figure 3: Our approximate use of a GP for log ?(x) improves upon the use of a
Here the ?final approx? is m?|s (1 + ?log ?|s ), from (5) and (6).
GP
for ?(x) alone.
We now choose ?0 to allow us to resolve the first integral in (6). First, we introduce a secondary GP
model for ?, the non-log space, and choose ?0 := m?|s , where m?|s is the standard GP conditional
mean for ? given observations ?(xs ). For both GPs2 (over both log and non-log spaces), we take
zero prior means and Gaussian covariances of the form (2). It is reasonable to use zero prior means:
?(x) is expected to be negligible except at a small number of peaks. If a quantity is dependent upon
the GP prior for ?, it will be represented as conditional on ?s ; if dependent upon the former GP prior
over log ?, it will be conditional upon log ?s . We expect ?log ?|s (x) to be small everywhere relative
to the magnitude of log ?(x) (see Figure 3). Hence log ?0 is close to the peaks of the Gaussian over
log ?, rendering our linearisation appropriate. For ?0 , the first integral in (6) becomes tractable.
Unfortunately, the second integral in (6) is non-analytic due to the log ?0 term within ?log ?|s . As
such, we perform another stage of Bayesian quadrature by treating ?log ?|s as an unknown function
of x. For tractability, we assume this prior is independent of the prior for log ?. We use another GP
for ?log ?|s , with zero prior mean and Gaussian covariance (2). A zero prior mean here is reasonable:
?log ?|s is exactly zero at xs , and tends to zero far away from xs , where both mlog ?|s and log ?0 are
given by the compatible prior means for log ? and ?. We must now choose candidate points xc at
which to evaluate the ?log ?|s function (note we do not need to evaluate ?(xc ) in order to compute
?c := ?log ?|s (xc )). xc should firstly include xs , where we know that ?log ?|s is equal to zero.
We select the remainder of xc at random on the hyper-ellipses (whose axes are defined by the input
scales for ?) surrounding existing observations; we expect ?log ?|s to be extremised at such xc . We
limit ourselves to a number of candidates that scales linearly with the dimensionality of the integral
for all experiments.
Given these candidates, we can now marginalise (6) over ?log ?|s to give
m(Z|log ?s ) ? m(Z|log ?0 , log ?s , ?c ) = m(Z|?s ) + m h??log ?|s i ?s , ?c ,
(7)
where both terms are analytic as per Section 2; m(Z|?s ) is of the form (3). The correction factor,
the second term in (7), is expected to be small, since ?log ?|s is small. We extend the work of [11]
to additionally calculate the variance in the evidence,
2
V (Z|log ?0 , log ?s , ?c ) = S(Z | log ?0 , log ?s ) ? m(Z|log ?0 , log ?s , ?c ) ,
(8)
where the second moment is
and hence
2
S(Z | log ?0 , log ?s ) := m h? Clog ?|s ?i log ?s + m(Z|log ?0 , log ?s , ?c ) ,
V (Z|log ?0 , log ?s , ?c ) = m h? Clog ?|s ?i log ?s
ZZ
:=
m?|s (x)m?|s (x? )Clog ?|s (x, x? )p(x)p(x? )dxdx? ,
(9)
(10)
which is expressible in closed form, although space precludes us from doing so. This variance can
be employed as a convergence diagnostic; it describes our uncertainty in the model evidence Z.
2
Note that separately modelling ? and log ? is not inconsistent: we use the posterior mean of the GP for ?
only as a convenient parameterisation for ?0 ; we do not treat this GP as a full probabilistic model. While this
modelling choice may seem excessive, this approach provides significant advantages in the sampling efficiency
of the overall algorithm by approximately capturing the non-negativity of our integrand and allowing active
sampling.
4
data
mean
variance
approx. marginalised length scale
true marginalised length scale
expected variance
f (x)
10
8
6
x
sample
x
(a)
(b)
Figure 4: a) Integrating hyperparameters increases the marginal posterior variance (in regions whose
mean varies as the input scales change) to more closely match the true posterior marginal variance.
b) An example showing the expected uncertainty in the evidence after observing the likelihood
function at that location. p(x) and l(x) are plotted at the top in green and black respectively, the
next sample location in red. Note the model discovering a new mode on the right hand side, sampling
around it, then moving on to other regions of high uncertainty on the left hand side.
In summary, we have described a linearisation approach to exploiting a GP prior over log-likelihoods;
this permitted the calculation of the analytic posterior mean (7) and variance (10) of Z. Note that our
approximation will improve with increasing numbers of samples: ?log ?|s will eventually be small
everywhere, since it is clamped to zero at each observation. The quality of the linearisation can also
be improved by increasing the number of candidate locations, at the cost of slower computation.
4
Marginalising hyperparameters
We now present a novel means of approximately marginalising the hyperparameters of the GP used
to model the log-integrand, log ?. In previous approaches to Bayesian Quadrature, hyperparameters
were estimated using MLII, which approximates the likelihood as a delta function. However, ignoring the uncertainty in the hyperparameters can lead to pathologies. In particular, the reliability of
the variance for Z depends crucially upon marginalising over all unknown quantities.
The hyperparameters of most interest are the input scales w for the GP over the log-likelihood;
these hyperparameters can have a powerful influence on the fit to a function. We use MLII to fit all
hyperparameters other than w. Marginalisation of w is confounded by the complex dependence of
our predictions upon these input scales. We make the following essential approximations:
Flat prior: We assume that the prior for w is broad, so that our posterior is the normalised likelihood.
Laplace approximation: p(log ?s |w) is taken as Gaussian with mean equal to the MLII value w
? and
with diagonal covariance Cw , diagonal elements fitted using the second derivatives of the likelihood.
We represent the posterior mean for log ? conditioned on w
? as m
? := mlog ?|s,w? .
GP mean affine in w: Given the narrow width of the likelihood for w, p(log ?|log ?s , w) is approximated as having a GP mean which is affine in w around the MLII values, and a constant covariance;
m
?
mlog ?|s,w ? m
? + ??w
(w ? w)
? and Clog ?|s,w ? Clog ?|s,w? .
The implication of these approximations is that the marginal posterior mean over log ? is simply
m
?
m
?
Cw ??w
.
m
? log ?|s := mlog ?|s,w? . The marginal posterior variance is C?log ?|s := Clog ?|s,w? + ??w
An example of our approximate posterior is depicted in Figure 4a. Our approximations give the
marginal posterior mean for Z:
m(Z|log
?
?0 , log ?s , ?c ) := m(Z|log ?0 , log ?s , ?c , w)
? ,
(11)
of the form (7). The marginal posterior variance
ZZ
? m(x)
?
? m(x
? ?)
?
?
?
?
V (Z|log ?0 , log ?s , ?c ) =
dx dx m?|s (x) m?|s (x ) Clog ?|s (x, x ) +
Cw
?w
?w
(12)
is possible, although laborious, to express analytically, as with (10).
5
5
Active Sampling
One major benefit of model-based integration is that samples can be chosen by any method, in
contrast to Monte Carlo methods, which typically must sample from a specific distribution. In
this section, we describe a scheme to select samples xs sequentially, by minimising the expected
uncertainty in the evidence that remains after taking each additional sample.3 We take the variance
in the evidence as our loss function, and proceed according to Bayesian decision theory.
Surprisingly, the posterior variance of a GP model with fixed hyperparameters does not depend
on the function values at sampled locations at all; only the location of those samples matters. In
traditional Bayesian quadrature, the evidence is an affine transformation of the sampled likelihood
values, hence its estimate for the variance in the evidence is also independent of likelihood values.
As such, active learning with fixed hyperparameters is pointless, and the optimal sampling design
can be found in advance [13].
In Section 3, we took Z as an affine transform of the log-likelihood, which we model with a GP. As
the affine transformation (5) itself depends on the function values (via the dependence of log ?0 ), the
conclusions of the previous paragraph do not apply, and active learning is desirable. The uncertainty
over the hyperparameters of the GP further motivates active learning: without assuming a priori
knowledge of the hyperparameters, we can?t evaluate the GP to precompute a sampling schedule.
The approximate marginalisation of hyperparameters permits an approach to active sampling that
acknowledges the influence new samples may have on the posterior over hyperparameters.
Active sampling selects a new sample xa so as to minimise the expected variance in the evidence
after adding the sample to the model
of ?. The objective is therefore to choose
the xa that minimises
the expected loss; xa = argminxa V (Z|log ?0 , log ?s,a ) | log ?0 , log ?s (note xa is implicitly
conditioned on, as usual for function inputs) where the expected loss is
Z
2
V (Z|log ?0 , log ?s,a ) | log ?0 , log ?s = S(Z | log ?0 , log ?s ) ? m(Z|log ?0 , log ?a,s , ?c )
?m
?a
?m
?T
a
?
? N log ?a ; m
? a , Ca +
dlog ?a , (13)
Cw
?w
?w
and we define m
? a := m(log ?a |log ?s , w)
? and C?a := V (log ?a |log ?s , w).
? The first term in (13),
the second moment, is independent of the selection of xa and can hence be safely ignored for active
sampling (true regardless of the model chosen for the likelihood). The second term, the negative
expected squared mean, can be resolved analytically4 for any trial xa (we omit the laborious details).
Importantly, we do not have to make a linearisation approximation for this new sample. That is, the
GP posterior over log ?a can be fully exploited when performing active sampling.
In order to minimise the expected variance, the objective in (13) encourages the maximisation of the
expected squared mean of Z. Due to our log-GP model, one means the method can use to do this
is to seek points where the log-likelihood is predicted to be large: which we call exploitation. The
objective in (13) naturally balances exploitation against exploration: the choice of points where our
current variance in the log-likelihood is significant (see Figure 4b). Note that the variance for log ?a
is increased by approximate integration over hyperparameters, encouraging exploration.
6
Experiments
We now present empirical evaluation of our algorithm in a variety of different experiments.
Metrics: We judged our methods according to three metrics, all averages over N similar experiments indexed by i. Define Zi as the ground truth evidence for the ith experiment, m(Zi ) as its
estimated mean and V (Zi ) as its predicted variance. Firstly, we computed the average log error,
3
We also expect such samples to be useful not just for estimating the evidence, but also for any other related
expectations, such as would beRrequired to perform prediction
using the model.
4
Here we use the fact that exp(c y) N y; m, ? 2 dy = exp(c m + 1/2 c2 ? 2 ). We assume that ?log ?|s
does not depend on log ?a , only its location xa : we know ?(xa ) = 0 and assume ?log ?|s elsewhere remains
unchanged.
6
PN
:= N1 i=1 |log m(Zi ) ? log Zi | . Next we computed the negative log-density of the truth,
PN
assuming experiments are independent, ? log p(Z) = ? i=1 log N (Zi ; m(Zi ), V (Zi )), which
quantifies the accuracy of our variance estimates. We also computed the calibration C, defined
as the fraction of ?
experiments in which the?ground truth lay within our 50% confidence interval
m(Zi ) ? 0.6745 V (Zi ), m(Zi ) + 0.6745 V (Zi ) . Ideally, C would be 50%: any higher, and a
method is under-confident, any lower and it is over-confident.
ALE
Methods: We first compared against simple Monte Carlo (SMC). SMC generates samples
PN
x1 , . . . , xN from p(x), and estimates Z by Z? = 1/N n=1 ?(xn ). An estimate of the variance
of Z? is given by the standard error of ?(x). As an alternative Monte Carlo technique, we implemented Annealed Importance Sampling (AIS) using a Metropolis-Hastings sampler. The inverse
temperature schedule was linear as in [10], and the proposal width was adjusted to attain approximately a 50% acceptance rate. Note that a single AIS chain provides no ready means of determining
the posterior variance for its estimate of Z. Our first model-based method was Bayesian Monte
Carlo (BMC) ? the algorithm used in [10]. Here samples were drawn from the AIS chain above, and
a GP was fit to the likelihood samples. For this and other methods, where not otherwise mentioned,
GP hyperparameters were selected using MLII .
We then tested four novel methods. Firstly, Bayesian Quadrature (BQ), which employed the linearisation approach of Section 3 to modeling the log-transformed likelihood values. The samples
supplied to it were drawn from the same AIS chain as used above, and 400 candidate points were permitted. BQ* is the same algorithm as BQ but with hyperparameters approximately marginalised, as
per Section 4. Note that this influences only the variance of the estimate; the means for BQ and BQ *
are identical. The performance of these methods allow us to quantify to what extent our innovations
improve estimation given a fixed set of samples.
Next, we tested a novel algorithm, Doubly Bayesian Quadrature (BBQ). The method is so named
for the fact that we use not only Bayesian inference (with a GP over the log-transformed likelihood)
to compute the posterior for the evidence, but also Bayesian decision theory to select our samples
actively, as described in Section 5. BBQ* is identical, but with hyperparameters approximately
marginalised. Both algorithms demonstrate the influence of active sampling on our performance.
Problems: We used these methods to evaluate evidences given Gaussian priors and a variety of
likelihood functions. As in [10] and [11], we focus on low numbers of samples; we permitted tested
methods 150 samples on synthetic integrands, and 300 when using real data. We are motivated by
real-world, big-data, problems where evaluating likelihood samples is expensive, making it desirable
to determine the techniques for evidence estimation that can operate best when permitted only a
small number of samples. Ground truth Z is available for some integrals; for the non-analytic
integrals, Z was estimated by a run of SMC with 105 samples.
We considered seven synthetic examples. We firstly tested using single Gaussians, in one, four,
ten and twenty dimensions. We also tested on mixtures of two Gaussians in one dimension (two
examples, alternately widely separated and overlapping) and four dimensions (a single example).
We additionally tested methods on a real scientific problem: detecting a damped Lyman-? absorber
(DLA) between the Earth and an observed quasar from spectrographic readings of the quasar. DLAs
are large objects consisting primarily of neutral hydrogen gas. The statistical properties of DLAs
inform us about the distribution of neutral hydrogen in the universe, which is of fundamental cosmological importance. We model the quasar spectra using a GP; the presence of a DLA is represented as an observation fault with known dynamics [14]. This model has five hyperparameters to
be marginalised, to which we assign priors drawn from the large corpus of data obtained from the
Sloan Digital Sky Survey (SDSS) [15]. We tested over four datasets; the expense of evaluating a GP
likelihood sample on the large datasets available from the SDSS (140TB of data have been released
in total) motivates the small sample sizes considered.
Evaluation Table 1 shows combined performance on the synthetic integrands listed above. The
calibration scores C show that all methods5 are systematically overconfident, although our approaches are at least as well calibrated as alternatives. On average, BBQ * provides an estimate
5
Because a single AIS chain gives no estimate of uncertainty, it has no likelihood or calibration scores.
7
?10?3
4
Z3
2
1
10
20
30
40
Number of samples
? log p(Z)
2
SMC
BMC
BBQ*
True value
0
?2
?4
?6
50
50
100
Number of samples
(a)
150
(b)
Figure 5: a) The posterior distribution over Z for several methods on a one-dimensional example
as the number of samples increases. Shaded regions denote ?2 SD?s from the mean. The shaded
regions for SMC and BMC are off the vertical scale of this figure. b) The log density of the true
evidence for different methods (colours identical to those in a), compared to the true Z (in black).
The integrand is the same as that in Figure 4b.
Table 2: Combined Real Results
Table 1: Combined Synthetic Results
Method
SMC
AIS
BMC
BQ
BQ *
BBQ
BBQ *
? log p(Z)
> 1000
N/A
> 1000
> 1000
> 1000
13.597
?11.909
ALE
1.101
1.695
2.695
6.760
6.760
0.919
0.271
C
Method
0.286
N/A
0.143
0.429
0.429
0.286
0.286
SMC
AIS
BMC
BQ
BQ *
BBQ
BBQ *
? log p(Z)
5.001
N/A
9.536
37.017
33.040
3.734
74.242
ALE
0.632
2.146
1.455
0.635
0.635
0.400
1.732
C
0.250
N/A
0.500
0.000
0.000
0.000
0.250
of Z which is closer to the truth than the other methods given the same number of samples, and assigns much higher likelihood to the true value of Z. BBQ * also achieved the lowest error on five, and
best likelihood on six, of the seven problems, including the twenty dimensional problem for both
metrics. Figure 5a shows a case where both SMC and BBQ * are relatively close to the true value,
however BBQ *?s posterior variance is much smaller. Figure 5b demonstrates the typical behaviour
of the active sampling of BBQ *, which quickly concentrates the posterior distribution at the true Z.
The negative likelihoods of BQ * are for every problem slightly lower than for BQ (? log p(Z) is
on average 0.2 lower), indicating that the approximate marginalisation of hyperparameters grants a
small improvement in variance estimate.
Table 2 shows results for the various methods on the real integration problems. Here BBQ is clearly
the best performer; the additional exploration induced by the hyperparameter marginalisation of
BBQ * may have led to local peaks being incompletely exploited. Exploration in a relatively high
dimensional, multi-modal space is inherently risky; nonetheless, BBQ * achieved lower error than
BBQ on two of the problems.
7
Conclusions
In this paper, we have made several advances to the BQ method for evidence estimation. These are:
approximately imposing a positivity constraint6 , approximately marginalising hyperparameters, and
using active sampling to select the location of function evaluations. Of these contributions, the active
learning approach yielded the most significant gains for integral estimation.
Acknowledgements
M.A.O.
was funded by the ORCHID project (http://www.orchid.ac.uk/).
6
Our approximations mean that we cannot guarantee non-negativity, but our approach improves upon alternatives that make no attempt to enforce the non-negativity constraint.
8
References
[1] R.M. Neal. Annealed importance sampling. Statistics and Computing, 11(2):125?139, 2001.
[2] J. Skilling. Nested sampling. Bayesian inference and maximum entropy methods in science and engineering, 735:395?405, 2004.
[3] M.H. Chen, Q.M. Shao, and J.G. Ibrahim. Monte Carlo methods in Bayesian computation. Springer,
2000.
[4] R. M. Neal. Probabilistic inference using Markov chain Monte Carlo methods. Technical Report CRGTR-93-1, University of Toronto, 1993.
[5] S.P. Brooks and G.O. Roberts. Convergence assessment techniques for Markov chain Monte Carlo. Statistics and Computing, 8(4):319?335, 1998.
[6] M.K. Cowles, G.O. Roberts, and J.S. Rosenthal. Possible biases induced by MCMC convergence diagnostics. Journal of Statistical Computation and Simulation, 64(1):87, 1999.
[7] P. Diaconis. Bayesian numerical analysis. In S. Gupta J. Berger, editor, Statistical Decision Theory and
Related Topics IV, volume 1, pages 163?175. Springer-Verlag, New York, 1988.
[8] A. O?Hagan. Bayes-Hermite quadrature. Journal of Statistical Planning and Inference, 29:245?260,
1991.
[9] M. Kennedy. Bayesian quadrature with non-normal approximating functions. Statistics and Computing,
8(4):365?375, 1998.
[10] C. E. Rasmussen and Z. Ghahramani. Bayesian Monte Carlo. In S. Becker and K. Obermayer, editors,
Advances in Neural Information Processing Systems, volume 15. MIT Press, Cambridge, MA, 2003.
[11] M.A. Osborne, R. Garnett, S.J. Roberts, C. Hart, S. Aigrain, N.P. Gibson, and S. Aigrain. Bayesian
quadrature for ratios. In Proceedings of the Fifteenth International Conference on Artificial Intelligence
and Statistics (AISTATS 2012), 2012.
[12] C. E. Rasmussen and C. K. I. Williams. Gaussian Processes for Machine Learning. MIT Press, 2006.
[13] T. P. Minka. Deriving quadrature rules from Gaussian processes. Technical report, Statistics Department,
Carnegie Mellon University, 2000.
[14] R. Garnett, M.A. Osborne, S. Reece, A. Rogers, and S.J. Roberts. Sequential bayesian prediction in the
presence of changepoints and faults. The Computer Journal, 53(9):1430, 2010.
[15] Sloan Digital Sky Survey, 2011. http://www.sdss.org/.
9
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4,041 | 4,658 | Why MCA? Nonlinear sparse coding with spike-andslab prior for neurally plausible image encoding
Jacquelyn A. Shelton,
Philip Sterne, J?org Bornschein, Abdul-Saboor Sheikh,
Frankfurt Institute for Advanced Studies
Goethe-University Frankfurt, Germany
{shelton, sterne, bornschein, sheikh}@fias.uni-frankfurt.de
?
J?org Lucke
Frankfurt Institute for Advanced Studies
Physics Dept., Goethe-University Frankfurt, Germany
[email protected]
Abstract
Modelling natural images with sparse coding (SC) has faced two main challenges:
?exibly representing varying pixel intensities and realistically representing lowlevel image components. This paper proposes a novel multiple-cause generative
model of low-level image statistics that generalizes the standard SC model in two
crucial points: (1) it uses a spike-and-slab prior distribution for a more realistic
representation of component absence/intensity, and (2) the model uses the highly
nonlinear combination rule of maximal causes analysis (MCA) instead of a linear combination. The major challenge is parameter optimization because a model
with either (1) or (2) results in strongly multimodal posteriors. We show for the
?rst time that a model combining both improvements can be trained ef?ciently
while retaining the rich structure of the posteriors. We design an exact piecewise Gibbs sampling method and combine this with a variational method based
on preselection of latent dimensions. This combined training scheme tackles both
analytical and computational intractability and enables application of the model
to a large number of observed and hidden dimensions. Applying the model to
image patches we study the optimal encoding of images by simple cells in V1
and compare the model?s predictions with in vivo neural recordings. In contrast
to standard SC, we ?nd that the optimal prior favors asymmetric and bimodal activity of simple cells. Testing our model for consistency we ?nd that the average
posterior is approximately equal to the prior. Furthermore, we ?nd that the model
predicts a high percentage of globular receptive ?elds alongside Gabor-like ?elds.
Similarly high percentages are observed in vivo. Our results thus argue in favor
of improvements of the standard sparse coding model for simple cells by using
?exible priors and nonlinear combinations.
1
Introduction
Sparse Coding (SC) is one of the most popular algorithms for feature learning and has become a
standard approach in Machine Learning, Computational Neuroscience, Computer Vision, and other
related ?elds. It was ?rst introduced as a model for the encoding of visual data in the primary visual
cortex of mammals [1] and became the standard model to describe coding in simple cells. Following
early recording studies [2] on simple cells, they were de?ned to be cells responding to localized,
oriented and bandpass visual stimuli ? sparse coding offered an optimal encoding explanation of
such responses by assuming that visual components are (a) independent, (b) linearly superimposed,
1
and (c) mostly inactive, with only a small subset of active components for a given image patch.
More formally, sparse coding assumes that each observation y = (y1 , . . . , yD ) is associated with a
(continuous or discrete) sparse latent variable s = (s1 , . . . , sH ), where sparsity implies that most of
the components sh in s are zero or close-to zero. Each data point is generated according to the data
?
model
p(y | ?) = p(y | s, ?) p(s | ?) ds
(1)
s
?
with s integrating (or summing) over all hidden states and ? denoting the model
? parameters. Typically, p(y | s, ?) is modelled as a Gaussian with a mean ? de?ned as ? = h sh Wh , i.e. as a
linear superposition of basis vectors Wh ? RD . The most typical choice of prior over p(s | ?) is a
Laplace distribution (which corresponds to L1 regularization).
The sparse coding generative model has remained essentially the same since its introduction, with
most work focusing on ef?cient inference of optimal model parameters ? (e.g., [3, 4]), usually exploiting unimodality of the resulting posterior probabilities. The standard form of the model offers
many mathematically convenient advantages, but the inherent assumptions may not be appropriate
if the goal is to accurately model realistic images. First, it has been pointed out that visual components ? such as edges ? are either present or absent and this is poorly modelled with a Laplace prior
because it lacks exact zeros. Recently, spike-and-slab distributions have been a favored alternative
(e.g. [5, 6, 7, 8]) as they enable the modelling of visual component absence/presence (the spike) as
well as the component?s intensity distribution (the slab). Second, it has been pointed out that image
components do not linearly superimpose to generate images, contrary to the standard sparse coding
assumption. Alternatively, various nonlinear combinations of visual components have been investigated [9, 10, 11, 12]. Either modi?cation (spike-and-slab prior or nonlinearities) leads to multimodal
posteriors, making parameter optimization dif?cult. As a result these modi?cations have so far only
been investigated separately. For linear sparse coding with a spike-and-slab prior the challenge for
learning has been overcome by applying factored variational EM approaches [13, 5] or sampling
[6]. Similarly, models with nonlinear superposition of components could be ef?ciently trained by
applying a truncated variational EM approach [14, 12], but avoiding the analytical intractability
introduced by using a continuous prior distribution.
In this work we propose a sparse coding model that for the ?rst time combines both of these improvements ? a spike-and-slab distribution and nonlinear combination of components ? in order to form
a more realistic model of images. We address the optimization of our model by using a combined
approximate inference approach with preselection of latents (for truncated variational EM [14]) in
combination with Gibbs sampling [15]. First, we show on arti?cial data that the method ef?ciently
and accurately infers all model parameters, including data noise and sparsity. Second, using natural image patches we show the model yields results consistent with in vivo recordings and that the
model passes a consistency check which standard SC does not. Third, we show our model performs
on par with other models on the functional benchmark tasks of denoising and inpainting.
2
The Generative Model: Nonlinear Spike-and-Slab Sparse Coding
We formulate the multi-causal data generation process as the probabilistic generative model:
p(yd | s, ?) = N (yd ; max{sh Wdh }, ? 2 ),
h
(2)
where maxh considers all H latent components and takes the h yielding the maximum value for
sh Wdh , and where sh has a spike-and-slab distribution given by sh = bh zh and parameterized by:
p(bh | ?) = B(bh ; ?) = ? bh (1 ? ?)1?bh
p(zh | ?) =
2
N (zh ; ?pr , ?pr
),
(3)
(4)
in this notation the spike is de?ned in Eq. 3 (parameterized by ?) and the slab is de?ned in Eq. 4
(parameterized by ?pr and ?pr ). The observation noise has a single parameter; ?. The columns of
the matrix W = (Wdh ) are the generative ?elds, Wh , one associated with each latent variable sh .
We denote the set of all parameters with ?. We will be interested in working with the posterior over
the latents given by
p(y|s, ?) p(s|?)
p(s|y, ?) = ?
.
(5)
? , ?) p(s? |?) ds?
p(y|s
?
s
2
A
C
generative ?elds
B
standard SC
D
spike-and-slab SC
(linear)
sum
spike-and-slab SC
(non-linear)
max
Figure 1: Generation according to different sparse coding generative models using the same generative ?elds. A 20 generative ?elds in the form of random straight lines. B Examples of patches
generated according to three generative models all using the ?elds in A. Top row: standard linear
sparse coding with Laplace prior. Middle row: linear sparse coding with spike-and-slab prior. Bottom row: spike-and-slab sparse coding with max superposition. The latter two use the same prior
parameterization (with positive slab). Generated patches are individually scaled to ?ll the color
space (with zero ?xed to green). C A natural image with two patches highlighted (magni?cations
show their preprocessed from). D Linear and nonlinear superposition of two single components for
comparison with the actual superposition in C.
As in standard sparse coding, the model assumes independent latents and given the latent variables,
the observations are distributed according to a Gaussian distribution. Unlike standard sparse coding,
the latent variables are not distributed according to a Laplace prior and the generative ?elds (or basis functions) are not combined linearly. Fig. 1 illustrates the model differences between a Laplace
prior and a spike-and-slab prior and the differences between linear and nonlinear superposition. As
can be observed, standard sparse coding results in strong interference when basis functions overlap.
For spike-and-slab sparse coding most components are exactly zero but interference between them
remains strong because of their linear superposition. Combining a spike-and-slab prior with nonlinear composition allows minimal interference between the bases and ensures that latents can be
exactly zero, which creates very multimodal posteriors since data must be explained by either one
cause or another. For comparison, the combination of two real image components is highlighted in
Fig. 1C (lower patch). Linear and nonlinear superposition of two basis functions resembling single
components is shown in Fig. 1D. This suggests that superposition de?ned by max represents a better
model of occluding components (compare [11, 12]).
In this paper we use expectation maximization (EM) to estimate the model parameters ?, and we use
sampling after latent preselection [15] to represent the posterior distribution over the latent space.
Optimization in the EM framework entails setting the free-energy to zero and solving for the model
parameters (M-step equations) (e.g., [16]). As an example we obtain the following formula for the
estimate of image noise:
?
?2
?
?
1 ???
(n)
,
(6)
max Whd skh ? yd
?
?2 =
h
N DK n
d
k
where we average over all N observed data points, D observed dimensions, and K Gibbs samples.
However, this notation is rather unwieldy for a simple underlying idea. As such we will use the
following notation:
??
?
(n)
?
? 2 = Wdh sh ? yd
,
(7)
where we maximize for h and average over n and d. That is, we denote the expectation values ? . ??
to mean the following:
?
?
p(s|y(n) , ?) f (s) ?(h is max) ds
?
s?
,
(8)
?f (s)? =
p(s|y(n) , ?) ?(h is max) ds
s
n
3
linear combination
spike & slab prior
p(s)
data likelihood
p(yd | s)
s2
?
point-wise maximum
posterior
p(s | yd )
?
data likelihood
p(yd | s)
?
posterior
p(s | yd )
?
s1
Figure 2: Illustration of a H=2-dimensional spike-and-slab prior over latents and the multimodal
posterior distribution induced by this prior for both linear and nonlinear data likelihoods.
where ? is the indicator function denoting the domain to integrate over, namely where h is the
maximum. This allows a condensed expression of the rest of the update equations:
?
? hd = ?sh yd ? ,
W
2
?
?sd ?
(9)
?
? = ??(s)?,
?
?pr = ?sh ?? ,
2
?
?pr
= ?(sh ? ?
?pr )2 ??
(10)
where we observe that in order to optimize the parameters we need to calculate several expectation
values with respect to the posterior distribution. As discussed however, the posterior distribution of
a model with a spike-and-slab prior in both the linear and nonlinear cases is strongly multimodal
and such posteriors are dif?cult to infer and represent. This is illustrated in Fig. 2. Calculating
expectations of this posterior is analytically intractable, thus we use Gibbs sampling to approximate
the expectations.
3
Inference: Exact Gibbs Sampling with Preselection of Latents
In order to ef?ciently handle the intractabilities posed by our model and the complex posterior
(multimodal, high dimensional) illustrated in Fig. 2, we take a combined approximate inference
approach. Speci?cally we do exact Gibbs sampling from the posterior after we have preselected the
most relevant set of latent states using a truncated variational form of EM. Preselection is not strictly
necessary, but signi?cantly helps with the computational intractability faced in high dimensions. As
such, we will ?rst descibe the sampling step and preselection only later.
Gibbs Sampling. Our main technical contribution towards ef?cient inference in this model is an
exact Gibbs sampler for the multimodal posterior. Previous work has used Gibbs sampling in
combination with spike-and-slab models [17], and for increased ef?ciency in sparse Bayesian inference [18]. Our aim is to construct a Markov chain with the target density given by the conditional
posterior distribution:
D
?
p(yd |sh , sH\h , ?).
(11)
p(sh |sH\h , y, ?) ? p(sh |?)
d=1
We see from Eq. 11 that the distribution factorizes into D + 1 factors: a single factor for the prior
and D factors for each likelihood. For the point-wise maximum nonlinear case we are considering,
the likelihood of a single yd is a piecewise function de?ned as follows:
p(yd |sh , sH\h , ?) = N (yd ; max
{Wdh? sh? }, ? 2 )
h?
?
{Wdh? sh? }, ? 2 )
?
? N (yd ; max
h? \h
??
?
?
=
?
constant
?
2
N (yd ; Wdh sh , ? )
(12)
= exp(ld (sh ))
if sh < Pd
(13)
= exp(rd (sh ))
if sh ? Pd ,
where the transition point is de?ned as the point where sh Wdh becomes the maximal cause:
Pd = max
{Wdh? sh? } / Wdh .
?
h \h
4
(14)
Log Likeilhoods
log(p(y2 | sh , s\h ))
B
exp
? log(p(sh | y, s\h ))
E
? p(sh | y, s\h )
P2
??
F
CDF
? s
h
log(p(sh ))
C
Posterior
Log Prior
+
P1
D
Log Posterior
log(p(y1 | sh , s\h ))
A
sh
sh
Figure 3: Illustration of the Gibbs sampler for an MCA-induced posterior. Left column: three
contributing factors for the posterior ? p(sh | s\h , y, ?) in logspace. A and B: Log likelihood
functions each de?ned by a transition point Pd and left and right pieces rd (sh ) and ld (sh ). C Log
prior, which consists of an overall gaussian and the Dirac-peak at sh = 0. D Log posterior, the sum
of functions A, B, and C consists of D + 1 pieces plus the Dirac-peak at sh = 0. E Exponentiation
of the D log posterior. F CDF for sh from which we do inverse transform sampling.
We refer to the two pieces of yd in Eq. 13 as the left piece of the function when sh < Pd and right
piece when sh ? Pd . The left is constant w.r.t. sh because the data is explained by another cause
when sh < Pd , and the right is a truncated Gaussian when considered a PDF of sh (see Fig. 3A).
We take the logarithm of p(yd |sh , sH\h , ?), which transforms the equation into a left-piece constant
and right-piece quadratic function that we can easily sum together. The sum of these D functions
results in one function with D + 1 segments mi (sh ), with transition points Pd given by the individual functions. We ?rst sort the functions according to their transition points, denoted here by
? = argsortd (Pd ), such that we can ef?ciently calculate the summation over these functions:
?i?1
?D
mi (sh ) = j=1 r?(j) (sh ) + u=i l?(u) (sh )
for 1 ? i ? D + 1,
(15)
where the left and right pieces are referred to as li (sh ) and ri (sh ) (as in Eq. 13), respectively. Since
all pieces li (sh ) and ri (sh ) are polynomials of 2nd degree, the result is still a 2nd degree polynomial.
We incoorporate the prior in two steps. The Gaussian slab of the prior is taken into account by
adding its 2nd degree polynomial to all the pieces mi (sh ), which also ensures that every piece is a
Gaussian.
To construct the piecewise cumulative distribution function (CDF), we relate each segment mi (sh )
to the Gaussian ? exp(mi (sh )) it de?nes. Next, the Bernoulli component of the prior is accounted
for by introducing the appropriate step into the CDF at sh = 0 (see Fig. 3F). Once the CDF is constructed, we simulate each sh from the exact conditional distribution (sh ? p(sh |s\h = s\h , y, ?))
by inverse transform sampling. Fig. 3 illustrates the entire process.
Preselection. To reduce the computational load of inference in our model, we can optionally preselect the most relevant latent variables before doing Gibbs sampling. This can be formulated as a
variational approximation to exact inference [14] where the posterior distribution p(s | y (n) , ?) is
approximated by the distribution qn (s; ?) which only has support on a subset Kn of the latent state
space:
p(s | y (n) , ?) ? qn (s; ?) =?
p(s | y (n) , ?)
p(s ? | y (n) , ?)
?(s ? Kn )
(16)
s ? ?Kn
where ?(s ? Kn ) = 1 if s ? Kn and zero otherwise. The subsets Kn are chosen in a data-driven
way using a deterministic selection function, they vary per data point y (n) , and should contain most
of the probability mass p(s | y) while also being signi?cantly smaller than the entire latent space.
Using such subsets Kn , Eqn. 16 results in good approximations to the posteriors. We de?ne Kn as
Kn = {s | for all h ?? I : sh = 0} where I contains the indices of the latents estimated to be most
relevant for y (n) . To obtain these latent indices we use a selection function of the form:
?
?2 ? ?
?
(17)
Sh (y (n) ) = ?Wh ? y (n) ?2 ?Wh ?2
to select the H ? < H highest scoring latents for I. This boils down to selecting the H ? dictionary
elements that are most similar to each datapoint, hence being most likely to have generated the
datapoint. We then sample from this reduced latent set.
5
A
B
D
EMsteps
30
?pr
2
00
E
H?
?
?gt = 2
4.5
4.0
3.5
3.0
2.5
2.0
1.5
0
H?gt = 2
EMsteps
30
5
4
3
2
1
0
F3.5
?pr
C5
2.0
?pr gt = 2
?pr gt = 0.5
0
EMsteps
30
0.0
0
EMsteps
30
Figure 4: Results of 10 experimental runs with 30 EM iterations on the same arti?cial ground-truth
data generated according to the model. We accurately recover ground-truth parameters which are
plotted with dotted lines. A Random selection of data y(n) , B The set of learned generative ?elds
Wh , C Data noise ?, D Sparsity H ? ?, E Prior standard dev. ?pr , F Prior mean ?pr .
4
Experiments
We ?rst investigate the performance of our algorithm on ground-truth arti?cial data. Second, we
apply our model to natural image patches and compare with in vivo recording from various sources.
Third, we investigate the applicability of our algorithm on functional benchmark tasks. All experiments were performed using a parallel implementation of the EM algorithm [19]. Small scale
experiments were run on a single multicore machine, while larger scale experiments were typically
run on a cluster with 320 CPU cores in parallel.
For all experiments described, 1/3 of the samples drawn are used for burn-in, and 2/3 are the samples
used for computations. We initialize our parameters by setting the ?pr and ? equal to the standard
deviation observed in the data, the prior mean ?pr is initialized to the observed data mean. W is
initialized at the observed data mean with additive Gaussian ?
noise of the ? observed in the data, but
D
we enforce the constraint that ?Wdh ? = 1 such that ?h?H d=1 Wdh = D, or that each Wdh is
approximately equal to one.
Arti?cial Data. The goal of the ?rst set of experiments is to verify that our model and inference
method produce an algorithm that can (1) recover ground-truth parameters from data that is generated according to the model and (2) that it reliably converges to locally optimal solutions. We
generate ground-truth data with N = 2, 000 consisting of D = 5 ? 5 = 25 observed and H = 10
hidden dimensions according to our model: N images with overlapping ?bars? of varying intensities
and with Gaussian observation noise of variance ?gt = 2 (Fig. 4A). On average, each data point
contains two bars, ? = H2 . Results (Fig. 4B-F) show that our algorithm converges to globally optimal solutions and recovers the generating ground-truth parameters. Here we drew 30 samples from
the posterior and set H ? = H, but investigation of a range of sample number and H ? values yields
the same results, suggesting that our approximation parameters do not have an effect on our results
(see Supp. Material for more experiments on this dataset).
Natural Image Patches. We applied our model to N = 50, 000 image patches of 16 ? 16 pixels.
The patches were extracted from the Van Hateren natural image database [20] and subsequently
preprocessed using pseudo-whitening [1]. We split the image patches into a positive and negative
? = 16 ? 16 is converted into a datapoint
channel to ensure yd ? 0: each image patch y? of size D
? by assigning yd = [?
yd ]+ and yD+d
=
[??
yd ]+ , where [x]+ = x for x > 0 and
of size D = 2 D
?
[x]+ = 0 otherwise. This can be motivated by the transfer of visual information by center-on and
center-off cells of the mammalian lateral geniculate nucleus (LGN). In a ?nal step, as a form of local
contrast normalization, we scaled each image patch so that 0 ? yd ? 10.
After 50 EM iterations with 100 samples per datapoint the model parameters had converged and
the learned dictionary elements Wh represent a variety of Gabor-Wavelet and Difference of Gaussians (DoG) like shapes (see Fig. 5A). We observe a mean activation of ?pr = 0.47, with standard
deviation ?pr = 0.13, i.e., we infer a strongly bimodal prior (Fig. 5D). The ?nal sparseness was
?H = 6.2, which means that an average of roughly six latent variables were active in every image
patch. The inferred observation noise was ? = 1.4. To quantitatively interpret the learned ?elds, we
6
A
B
C
30
D
0.8
ny
Usrey at. al.
Niell Styker
?pr = 0.47
0.6
0.4
?pr = 0.13
p(sh )
?H = 6.2
E
0.0
0.2
0.2
0.4
sh
0.6
0.0
0.0
0.8
0.6
0.4
0.2
0.6
0.4
0.2
0.0
0.2
0.4
s3
0.6
0.8
0.0
0.2
0.4
nx
0.6
0.8
0.8
p(s3 )
0.8
p(s2 )
0.8
D. Ringach
10
0
p(s1 )
1.0
15
5
0.0
1.2
20
SS-MCA
% globular RFs
25
0.6
0.4
0.2
0.0
0.2
0.4
s2
0.6
0.8
0.0
0.0
0.2
0.4
s3
0.6
0.8
Figure 5: Results after training our model on N = 50, 000 image patches of size 16 ? 16 using
H=500 latent units. A Selection of inferred dictionary elements Wh . The full set of elements
is shown in the supplementary material. B After ?tting with Gabor wavelets and DoG?s, 135 ?elds
(27%) are classi?ed as being globular. The fraction of globular ?elds measured in vivo are shown for
comparison. C nx /ny Gabor statistics plot of estimated receptive ?elds (blue circles, see Supp. D
Fig.D) overlayed with the distribution reported by Ringach (in vivo Macaque, red triangles). We
intentionally exclude ?elds best ?t by DoG?s, removing the typical cluster observed at 0-0 (see
Supp. D). D Visualization of the prior inferred by our model: On average ?H = 6.2 dictionary
elements are active per datapoint. E Histogram of the actual activation of three selected dictionary
elements: A Gabor wavelet, a DoG and a ?eld encoding low-frequency input. The bimodal pattern
closely resembles the prior activation inferred in D.
perform reverse correlation on the learned generative ?elds and ?t the resulting estimated receptive
?elds with Gabor wavelets and DoGs (see Supp. D for details). Next, we classify the ?elds as either
orientation-sensitive Gabor wavelets or ?globular? ?elds best matched by DoGs. In Fig. 5C) we then
plot only the ?elds classi?ed as Gabors, leaving out all DoG ?elds.
Notably, the proportion of globular ?elds predicted by the model (Fig. 5B) is similarly high as those
found in different species [21, 22, 23] (see next section for a discussion). Fig. 5D-E compares the
optimal prior distribution with the average posterior distribution for several latent variables (with
their associated generative ?elds shown in insets). It is a necessary condition of the correct model of
the data that the posterior averaged over the datapoints y(n) matches the prior, since the following
holds (compare, e.g., [24]):
?
lim N1 n p(s | y(n) , ?) = p(s | ?).
(18)
N ??
Our model satis?es this condition; the average posterior over these ?elds closely resembles the
optimal prior, which is a test standard sparse coding fails (see [17] for a discussion).
Functional Tasks. We also apply our model to the task of image inpainting and image denoising.
Given that we propose our model to be able to realistically model low-level image statistics, we
expect it to perform well on these tasks. Results show that our algorithm performs on par with the
latest benchmarks obtained by other algorithms. See Supp. Material for details and examples.
5
Discussion
In this work, we de?ned and studied a sparse coding model that, for the ?rst time, combines a spikeand-slab prior with a nonlinear combination of dictionary elements. To address the optimization of
our model, we designed an exact piecewise Gibbs sampling method combined with a variational
method based on preselection of latent dimensions. This combined training scheme tackles both
7
analytical and computational intractability and enables application of the model to a large number of
observed and hidden dimensions. The learning algorithm derived for the model enables the ef?cient
inference of all model parameters including sparsity and prior parameters.
The spike-and-slab prior used in this study can parameterize prior distributions which are symmetric
and unimodal (spike on top of the Guassian) as well as strongly bimodal distributions with the Gaussian mean being signi?cantly different from zero. However, inferring the correct prior distribution
requires sophisticated inference and learning schemes. Standard sparse coding with MAP-based approximation only optimizes the basis functions [25, 4]. Namely, the prior shape remains ?xed except
for its weighting factor (the regularization parameter) which is typically only inferred indirectly (if at
all) using cross-validation. Very few sparse coding approaches infer prior parameters directly. One
example is an approach using a mixture-of-Gaussians (MoG) prior [17] which applies Gibbs sampling for inference. The MoG prior can model multimodality but in numerical experiments on image
patches the mixture components were observed to converge to a monomodal prior ? which may be
caused by the assumed linear superposition or by the Gibbs sampler not mixing suf?ciently. When
the MoG prior was ?xed to be trimodal, no instructive generative ?elds were observed [17]. Another
example of sparse coding with prior inference is a more recent approach which uses a parameterized
student-t distribution as prior and applies sampling to infer the sparsity [26]. A student-t distribution
cannot model multimodality, however. The work in [27] uses a trimodal prior for image patches but
shape and sparsity remain ?xed, i.e. the study does not answer how optimal such a prior may be. In
contrast, we have shown in this study that the prior shape and sparsity level can be inferred from image data. The resulting prior is strongly bimodal and control experiments con?rm a high consistency
of the prior with the average posterior (Fig. 5D-E). Standard sparse coding approaches typically fail
in such controls which may be taken as early evidence for bimodal or multimodal priors being more
optimal (see [17]).
Together with a bimodal prior, our model infers Gabor and difference-of-Gaussian (DoG) functions
as the optimal basis functions for the used image patches. While Gabors are the standard outcome
of sparse coding, DoGs have not been predicted by sparse coding until very recently. Indeed, DoG
or ?globular? ?elds were identi?ed as the main qualitative difference between experimental measurements of V1 simple cells and theoretical predictions [21]. A number of studies have since
shown that globular ?elds can emerge in applications of computational models to image patches
[28, 27, 29, 30, 31, 12, 32]. One study [29] has shown that globular ?elds can be obtained with standard sparse coding by choosing speci?c values for overcompleteness and sparsity (i.e. prior shape
and sparsity are not inferred from data). The studies [27, 31, 32] assume a restricted set of values for
latent variables and yield relatively high proportion of globular ?elds suggesting that the emergence
of globular ?elds is due to hard constraints on the latents. On the other hand, the studies [28, 30, 12]
suggest that globular ?elds are a consequence of occlusion nonlinearities. Our study argues in favor
of the occlusion interpretation for the emergence of globular ?elds because the model studied here
shows that high percentages of globular ?elds emerge with a prior that is (a) inferred from data and
(b) allows for a continuous distribution of latent values.
In summary, the main results obtained by applying the novel model to preprocessed images are:
(1) the observation that a bimodal prior is preferred over a unimodal one for optimal image coding,
and (2) that high percentages of globular ?elds are predicted. The sparse bimodal prior is consistent
with sparse and positive neural activtiy for the encoding of image components in V1, and the high
percentage of globular ?elds is consistent with recent in vivo recordings of simple cells. Our model
therefore links improvements on optimal image encoding to a high consistency with neural data.
Acknowledgements. We acknowledge support by the German Research Foundation (DFG) in the project LU
1196/4-2, by the German Federal Ministry of Education and Research (BMBF), project 01GQ0840, and by the
LOEWE Neuronale Koordination Forschungsschwerpunkt Frankfurt (NeFF). Furthermore, we acknowledge
support by the Frankfurt Center for Scienti?c Computing (CSC).
References
[1] B. Olshausen and D. Field. Emergence of simple-cell receptive ?eld properties by learning a sparse code
for natural images. Nature, 381:607?9, 1996.
[2] D. H. Hubel and T. N. Wiesel. Receptive ?elds of single neurones in the cat?s striate cortex. The Journal
of Physiology, 1959.
8
[3] M. Seeger. Bayesian inference and optimal design for the sparse linear model. Journal of Machine
Learning Research, 9:759?813, June 2008.
[4] H. Lee, A. Battle, R. Raina, and A. Ng. Ef?cient sparse coding algorithms. In Advances in Neural
Information Processing Systems, volume 20, pages 801?08, 2007.
[5] M. Titsias and M. L?azaro-Gredilla. Spike and slab variational inference for multi-task and multiple kernel
learning. In Advances in Neural Information Processing Systems, 2011.
[6] S. Mohamed, K. Heller, and Z. Ghahramani. Evaluating Bayesian and L1 approaches for sparse unsupervised learning. In ICML, 2012.
[7] I. Goodfellow, A. Courville, and Y. Bengio. Large-scale feature learning with spike-and-slab sparse
coding. In ICML, 2012.
[8] J?org L?ucke and Abdul-Saboor Sheikh. Closed-form EM for sparse coding and its application to source
separation. In LVA/ICA, LNCS, pages 213?221. Springer, 2012.
[9] E. Saund. A multiple cause mixture model for unsupervised learning. Neural Computation, 1995.
[10] P. Dayan and R. S. Zemel. Competition and multiple cause models. Neural Computation, 1995.
[11] J. L?ucke and M. Sahani. Maximal causes for non-linear component extraction. Journal of Machine
Learning Research, 9:1227?67, 2008.
[12] G. Puertas, J. Bornschein, and J. L?ucke. The maximal causes of natural scenes are edge ?lters. In
Advances in Neural Information Processing Systems, volume 23, pages 1939?47. 2010.
[13] I. Goodfellow, A. Courville, and Y. Bengio. Spike-and-slab sparse coding for unsupervised feature discovery. In NIPS Workshop on Challenges in Learning Hierarchical Models. 2011.
[14] J?org L?ucke and Julian Eggert. Expectation truncation and the bene?ts of preselection in training generative
models. Journal of Machine Learning Research, 11:2855?900, 2010.
[15] J. Shelton, J. Bornschein, A.-S. Sheikh, P. Berkes, and J. L?ucke. Select and sample - a model of ef?cient
neural inference and learning. Advances in Neural Information Processing Systems, 24, 2011.
[16] R. Neal and G. Hinton. A view of the EM algorithm that justi?es incremental, sparse, and other variants.
In M. I. Jordan, editor, Learning in Graphical Models. Kluwer, 1998.
[17] B. Olshausen and K. Millman. Learning sparse codes with a mixture-of-Gaussians prior. Advances in
Neural Information Processing Systems, 12:841?847, 2000.
[18] X. Tan, J. Li, and P. Stoica. Ef?cient sparse Bayesian learning via Gibbs sampling. In ICASSP, pages
3634?3637, 2010.
[19] J. Bornschein, Z. Dai, and J. L?ucke. Approximate EM learning on large computer clusters. In NIPS
Workshop: LCCC. 2010.
[20] J. H. van Hateren and A. van der Schaaf. Independent component ?lters of natural images compared with
simple cells in primary visual cortex. Proceedings of the Royal Society of London B, 265:359?66, 1998.
[21] D. Ringach. Spatial structure and symmetry of simple-cell receptive ?elds in macaque primary visual
cortex. Journal of Neurophysiology, 88:455?63, 2002.
[22] W. M. Usrey, M. P. Sceniak, and B. Chapman. Receptive Fields and Response Properties of Neurons in
Layer 4 of Ferret Visual Cortex. Journal of Neurophysiology, 89:1003?1015, 2003.
[23] C. Niell and M. Stryker. Highly Selective Receptive Fields in Mouse Visual Cortex. The Journal of
Neuroscience, 28(30):7520?7536, 2008.
[24] P. Berkes, G. Orban, M. Lengyel, and J. Fiser. Spontaneous Cortical Activity Reveals Hallmarks of an
Optimal Internal Model of the Environment. Science, 331(6013):83?87, January 2011.
[25] B. Olshausen and D. Field. Sparse coding with an overcomplete basis set: A strategy employed by V1?
Vision Research, 37(23):3311?3325, December 1997.
[26] P. Berkes, R. Turner, and M. Sahani. On sparsity and overcompleteness in image models. Advances in
Neural Information Processing Systems, 21, 2008.
[27] M. Rehn and F. Sommer. A network that uses few active neurones to code visual input predicts the diverse
shapes of cortical receptive ?elds. Journal of Computational Neuroscience, 22(2):135?46, 2007.
[28] J. L?ucke. A dynamical model for receptive ?eld self-organization in V1 cortical columns. In Proc.
International Conference on Arti?cial Neural Networks, LNCS 4669, pages 389 ? 398. Springer, 2007.
[29] B. A. Olshausen, C. Cadieu, and D.K. Warland. Learning real and complex overcomplete representations
from the statistics of natural images. Proc. SPIE, (7446), 2009.
[30] J. L?ucke. Receptive ?eld self-organization in a model of the ?ne-structure in V1 cortical columns. Neural
Computation, 21(10):2805?45, 2009.
[31] M. Henniges, G. Puertas, J. Bornschein, J. Eggert, and J. L?ucke. Binary Sparse Coding. In Proceedings
LVA/ICA, LNCS 6365, pages 450?57. Springer, 2010.
[32] J. Zylberberg, J. Murphy, and M. Deweese. A Sparse Coding Model with Synaptically Local Plasticity
and Spiking Neurons Can Account for the Diverse Shapes of V1 Simple Cell Receptive Fields. PLoS
Computational Biology, 7(10):e1002250, 2011.
[33] M. Zhou, H. Chen, J. Paisley, L. Ren, G. Sapiro, and L. Carin. Non-parametric Bayesian dictionary
learning for sparse image representations 1. In NIPS Workshop. 2009.
9
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4,042 | 4,659 | Topic-Partitioned Multinetwork Embeddings
Peter Krafft?
CSAIL
MIT
[email protected]
?
Juston Moore? , Bruce Desmarais? , Hanna Wallach?
Department of Computer Science, ? Department of Political Science
University of Massachusetts Amherst
?
{jmoore, wallach}@cs.umass.edu
?
[email protected]
Abstract
We introduce a new Bayesian admixture model intended for exploratory analysis of communication networks?specifically, the discovery and visualization of
topic-specific subnetworks in email data sets. Our model produces principled visualizations of email networks, i.e., visualizations that have precise mathematical
interpretations in terms of our model and its relationship to the observed data.
We validate our modeling assumptions by demonstrating that our model achieves
better link prediction performance than three state-of-the-art network models and
exhibits topic coherence comparable to that of latent Dirichlet allocation. We
showcase our model?s ability to discover and visualize topic-specific communication patterns using a new email data set: the New Hanover County email network.
We provide an extensive analysis of these communication patterns, leading us to
recommend our model for any exploratory analysis of email networks or other
similarly-structured communication data. Finally, we advocate for principled visualization as a primary objective in the development of new network models.
1
Introduction
The structures of organizational communication networks are critical to collaborative problem solving [1]. Although it is seldom possible for researchers to directly observe complete organizational
communication networks, email data sets provide one means by which they can at least partially observe and reason about them. As a result?and especially in light of their rich textual detail, existing
infrastructure, and widespread usage?email data sets hold the potential to answer many important scientific and practical questions within the organizational and social sciences. While some
questions may be answered by studying the structure of an email network as a whole, other, more
nuanced, questions can only be answered at finer levels of granularity?specifically, by studying
topic-specific subnetworks. For example, breaks in communication (or duplicated communication)
about particular topics may indicate a need for some form of organizational restructuring. In order to
facilitate the study of these kinds of questions, we present a new Bayesian admixture model intended
for discovering and summarizing topic-specific communication subnetworks in email data sets.
There are a number of probabilistic models that incorporate both network and text data. Although
some of these models are specifically for email networks (e.g., McCallum et al.?s author?recipient?
topic model [2]), most are intended for networks of documents, such as web pages and the links
between them [3] or academic papers and their citations [4]. In contrast, an email network is more
naturally viewed as a network of actors exchanging documents, i.e., actors are associated with nodes
while documents are associated with edges. In other words, an email network defines a multinetwork
in which there may be multiple edges (one per email) between any pair of actors. Perhaps more
importantly, much of the recent work on modeling networks and text has focused on tasks such as
?
Work done at the University of Massachusetts Amherst
1
=
+
+
Figure 1: Our model partitions an observed email network (left) into topic-specific subnetworks
(right) by associating each author?recipient edge in the observed network with a single topic.
predicting links or detecting communities. Instead, we take a complementary approach and focus on
exploratory analysis. Specifically, our goal is to discover and visualize topic-specific subnetworks.
Rather than taking a two-stage approach in which subnetworks are discovered using one model and
visualized using another, we present a single probabilistic model that partitions an observed email
network into topic-specific subnetworks while simultaneously producing a visual representation of
each subnetwork. If network modeling and visualization are undertaken separately, the resultant visualizations may not directly reflect the model and its relationship to the observed data. Rather, these
visualizations provide a view of the model and the data seen through the lens of the visualization
algorithm and its associated assumptions, so any conclusions drawn from such visualizations can be
biased by artifacts of the visualization algorithm. Producing principled visualizations of networks,
i.e., visualizations that have precise interpretations in terms of an associated network model and its
relationship to the observed data, remains an open challenge in statistical network modeling [5].
Addressing this open challenge was a primary objective in the development of our new model.
In order to discover and visualize topic-specific subnetworks, our model must associate each author?
recipient edge in the observed email network with a topic, as shown in Figure 1. Our model draws
upon ideas from latent Dirichlet allocation (LDA) [6] to identify a set of corpus-wide topics of
communication, as well as the subset of topics that best describe each observed email. We model
network structure using an approach similar to that of Hoff et al.?s latent space model (LSM) [7] so
as to facilitate visualization. Given an observed network, LSM associates each actor in the network
with a point in K-dimensional Euclidean space. For any pair of actors, the smaller the distance
between their points, the more likely they are to interact. If K = 2 or K = 3, these interaction
probabilities, collectively known as a ?communication pattern?, can be directly visualized in 2- or
3-dimensional space via the locations of the actor-specific points. Our model extends this idea by
associating a K-dimensional Euclidean space with each topic. Observed author?recipient edges are
explicitly associated with topics via the K-dimensional topic-specific communication patterns.
In the next section, we present the mathematical details of our new model and outline a corresponding inference algorithm. We then introduce a new email data set: the New Hanover County (NHC)
email network. Although our model is intended for exploratory analysis, we test our modeling assumptions via three validation tasks. In Section 4.1, we show that our model achieves better link
prediction performance than three state-of-the-art network models. We also demonstrate that our
model is capable of inferring topics that are as coherent as those inferred using LDA. Together,
these experiments indicate that our model is an appropriate model of network structure and that
modeling this structure does not compromise topic quality. As a final validation experiment, we
show that synthetic data generated using our model possesses similar network statistics to those of
the NHC email network. In Section 4.4, we showcase our model?s ability to discover and visualize
topic-specific communication patterns using the NHC network. We give an extensive analysis of
these communication patterns and demonstrate that they provide accessible visualizations of emailbased collaboration while possessing precise, meaningful interpretations within the mathematical
framework of our model. These findings lead us to recommend our model for any exploratory analysis of email networks or other similarly-structured communication data. Finally, we advocate for
principled visualization as a primary objective in the development of new network models.
2
Topic-Partitioned Multinetwork Embeddings
In this section, we present our new probabilistic generative model (and associated inference algorithm) for communication networks. For concreteness, we frame our discussion of this model in
2
terms of email data, although it is generally applicable to any similarly-structured communication
data. The generative process and graphical model are provided in the supplementary materials.
(d)
(d)
A single email, indexed by d, is represented by a set of tokens w(d) = {wn }N
n=1 that comprise the
text of that email, an integer a(d) ? {1, ..., A} indicating the identity of that email?s author, and a
(d)
set of binary variables y (d) = {yr }A
r=1 indicating whether each of the A actors in the network is
a recipient of that email. For simplicity, we assume that authors do not send emails to themselves
(d)
(i.e., yr = 0 if r = a(d) ). Given a real-world email data set D = {{w(d) , a(d) , y (d) }}D
d=1 , our
model permits inference of the topics expressed in the text of the emails, a set of topic-specific
K-dimensional embeddings (i.e., points in K-dimensional Euclidean space) of the A actors in the
network, and a partition of the full communication network into a set of topic-specific subnetworks.
As in LDA [6], a ?topic? t is characterized by a discrete distribution over V word types with probability vector ?(t) . A symmetric Dirichlet prior with concentration parameter ? is placed over
? = {?(1) , ..., ?(T ) }. To capture the relationship between the topics expressed in an email and
that email?s recipients, each topic t is also associated with a ?communication pattern?: an A ? A
(t)
matrix of probabilities P (t) . Given an email about topic t, authored by actor a, element par is the
probability of actor a including actor r as a recipient of that email. Inspired by LSM [7], each communication pattern P (t) is represented implicitly via a set of A points in K-dimensional Euclidean
(t)
(t)
(t)
(t)
(t)
(t)
such that par = pra = ?(b(t) ? ksa ? sr k)
space S (t) = {sa }A
a=1 and a scalar bias term b
(t)
with sa ? N (0, ?12 I) and b(t) ? N (?, ?22 ).1 If K = 2 or K = 3, this representation enables
each topic-specific communication pattern to be visualized in 2- or 3-dimensional space via the locations of the points associated with the A actors. It is worth noting that the dimensions of each
(t)
K-dimensional space have no inherent meaning. In isolation, each point sa conveys no information; however, the distance between any two points has a precise and meaningful interpretation in the
generative process. Specifically, the recipients of any email associated with topic t are more likely
to be those actors near to the email?s author in the Euclidean space corresponding to that topic.
Each email, indexed by d, has a discrete distribution over topics ? (d) . A symmetric Dirichlet prior
(d)
with concentration parameter ? is placed over ? = {? (1) , ..., ? (D) }. Each token wn is associated
(d)
(d)
(d)
(d)
with a topic assignment zn , such that zn ? ? (d) and wn ? ?(t) for zn = t. Our model does
not include a distribution over authors; the generative process is conditioned upon their identities.
(d)
The email-specific binary variables y (d) = {yr }A
r=1 indicate the recipients of email d and thus the
presence (or absence) of email-specific edges from author a(d) to each of the A ? 1 other actors.
Consequently, there may be multiple edges (one per email) between any pair of actors, and D defines
a multinetwork over the entire set of actors. We assume that the complete multinetwork comprises T
(d)
topic-specific subnetworks. In other words, each yr is associated with some topic t and therefore
(t)
(d)
with topic-specific communication pattern P (t) such that yr ? Bern(par ) for a(d) = a. A natural
(d)
way to associate each yr with a topic would be to draw a topic assignment from ? (d) in a manner
(d)
analogous to the generation of zn ; however, as outlined by Blei and Jordan [8], this approach can
result in the undesirable scenario in which one subset of topics is associated with tokens, while another (disjoint) subset is associated with edges. Additionally, models of annotated data that possess
this exchangeable structure tend to exhibit poor generalization [3, 8]. A better approach, advocated
(d)
by Blei and Jordan, is to draw a topic assignment for each yr from the empirical distribution over
(d)
topics defined by z . By definition, the set of topics associated with edges will therefore be a subset of the topics associated with tokens. One way of simulating this generative process is to associate
(d)
(d)
each yr with a position n = 1, . . . , max (1, N (d) ) and therefore with the topic assignment zn at
(d)
(d)
that position2 by drawing a position assignment xr ? U(1, . . . , max (1, N (d) )) for each yr . This
(d)
(t)
(d)
(d)
indirect procedure ensures that yr ? Bern(par ) for a(d) = a, xr = n, and zn = t, as desired.
The function ?(?) is the logistic function, while the function k ? k is the l2 -norm.
Emails that do not contain any text (i.e., N (d) = 0) convey information about the frequencies of communication between their authors and recipients. As a result, we do not omit such emails from D; instead, we
(d)
(d)
augment each one with a single, ?dummy? topic assignment z1 for which there is no associated token w1 .
1
2
3
2.1
Inference
(d) D
For real-world data D = {w(d) , a(d) , y (d) }D
}d=1 , authors A =
d=1 , the tokens W = {w
(d) D
(d) D
{a }d=1 , and recipients Y = {y }d=1 are observed, while ?, ?, S = {S (t) }Tt=1 , B = {b(t) }Tt=1 ,
(d) D
Z = {z (d) }D
}d=1 are unobserved. Dirichlet?multinomial conjugacy allows ?
d=1 , and X = {x
and ? to be marginalized out [9], while typical values for the remaining unobserved variables can
be sampled from their joint posterior distribution using Markov chain Monte Carlo methods. In
this section, we outline a Metropolis-within-Gibbs sampling algorithm that operates by sequentially
(t)
(d)
(d)
resampling the value of each latent variable (i.e., sa , bt , zn , or xr ) from its conditional posterior.
(d)
Since zn is a discrete random variable, new values may be sampled directly using
P (zn(d) = t | wn(d) = v, W\d,n , A, Y, S, B, Z\d,n , X , ?, ?)
?
(v|t)
yr(d)
1?yr(d)
+ V? Q
(t)
(t)
?
?(N (t|d) + ? ) N\d,n
(p
)
(1
?
p
)
(d)
(t)
\d,n
T
a(d) r
a(d) r
r:xr =n
N\d,n +?
?
(d)
(d)
?
y
1?y
Q
r
r
?
(t)
(t)
(1 ? pa(d) r )
r:r6=a(d) (pa(d) r )
for N (d) > 0
otherwise,
where subscript ?\d, n? denotes a quantity excluding data from position n in email d. Count N (t) is
the total number of tokens in W assigned to topic t by Z, of which N (v|t) are of type v and N (t|d)
(d)
belong to email d. New values for discrete random variable xr may be sampled directly using
(t)
(d)
P (x(d)
r = n | A, Y, S, B, zn = t, Z\d,n ) ? (pa(d) r )
yr(d)
(t)
(1 ? pa(d) r )
1?yr(d)
.
(t)
New values for continuous random variables sa and b(t) cannot be sampled directly from their
conditional posteriors, but may instead be obtained using the Metropolis?Hastings algorithm. With
(t)
(t)
(t)
a non-informative prior over sa (i.e., sa ? N (0, ?)), the conditional posterior over sa is
Y
N (1|a,r,t) +N (1|r,a,t)
N (0|a,r,t) +N (0|r,a,t)
(t) (t)
P (s(t)
(p(t)
(1 ? p(t)
,
a | A, Y, S \a , b , Z, X ) ?
ar )
ar )
r:r6=a
where count N (1|a,r,t) =
(d)
PD
(d)
= a) 1(yr = 1)
d=1 1(a
N (d)
n=1
P
(d)
(d)
1(xr = n) 1(zn = t) .3 Counts
N (1|r,a,t) , N (0|a,r,t) , and N (0|r,a,t) are defined similarly. Likewise, with an improper, noninformative prior over b(t) (i.e., b(t) ? N (0, ?)), the conditional posterior over b(t) is
P (b(t) | A, Y, S (t) , Z, X ) ?
A Y
Y
(p(t)
ar )
N (1|a,r,t) +N (1|r,a,t)
N (0|a,r,t) +N (0|r,a,t)
(1 ? p(t)
ar )
.
a=1 r:r<a
3
Data
Due to a variety of factors involving personal privacy concerns and the ownership of content by
email service providers, academic researchers rarely have access to organizational email data. For
example, the Enron data set [10]?arguably the most widely studied email data set?was only released because of a court order. The public record is an alternative source of organizational email
data. Public record data sets are widely available and can be continually updated, yet remain relatively untapped by the academic community. We therefore introduce and analyze a new public
record email data set relevant to researchers in the organizational and social sciences as well as machine learning researchers. This data set consists of emails between the managers of the departments
that constitute the executive arm of government at the county level for New Hanover County, North
Carolina. In this semi-autonomous local government, county managers act as executives, and the individual departments are synonymous with the individual departments and agencies in, for instance,
the U.S. federal government. Therefore, not only does this email data set offer a view into the communication patterns of the managers of New Hanover County, but analyses of it also serve as case
studies in modeling inter-agency communications in the U.S. federal government administration.
3
The function 1(?) evaluates to one if its argument evaluates to true and evaluates to zero otherwise.
4
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200
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?
Average Topic Coherence
?90 ?80 ?70 ?60 ?50
?
?
?
Average F1 Score
0.2
0.3
?
?
?
0.0
Average F1 Score
0.2
0.3
?
?
?
0.0
0.1
?
?
?
?110
0.4
0.4
?
?
50
100
150
Number of Topics
(c)
200
?
?
?
?
?
?
?
?
0
?
?
?
50
100
150
Number of Topics
?
?
?
200
(d)
Figure 2: Average link prediction performance for (a) the NHC email network and (b) the Enron data
set. For MMSB and LSM, we only report results obtained using the best-performing hyperparameter
values. Average topic coherence scores for (c) the NHC email network and (d) the Enron data set.
The New Hanover County (NHC) email network comprises the complete inboxes and outboxes of
30 department managers from the month of February, 2011. In total, there are 30,909 emails, of
which 8,097 were authored by managers. Of these 8,097 emails, 1,739 were sent to other managers
(via the ?To? or ?Cc? fields), excluding any emails sent from a manager to him- or herself only.
For our experiments, we used these 1,739 emails between 30 actors. To verify that our model is
applicable beyond the NHC email network, we also performed two validation experiments using
the Enron email data set [10]. For this data set, we treated each unique @enron email address
as an actor and used only those emails between the 50 most active actors (determined by the total
numbers of emails sent and received). Emails that were not sent to at least one other active actor (via
the ?To? or ?Cc? fields) were discarded. To avoid duplicate emails, we retained only those emails
from ? sent mail?, ?sent?, or ?sent items? folders. These steps resulted in a total of 8,322 emails
involving 50 actors. Both data sets were preprocessed to concatenate the text of subject lines and
message bodies and to remove any stop words, URLs, quoted text, and (where possible) signatures.
4
Experiments
Our model is primarily intended as an exploratory analysis tool for organizational communication
networks. In this section, we use the NHC email network to showcase our model?s ability to discover
and visualize topic-specific communication subnetworks. First, however, we test our underlying
modeling assumptions via three quantitative validation tasks, as recommended by Schrodt [11].
4.1
Link Prediction
In order to gauge our model?s predictive performance, we evaluated its ability to predict the recipients of ?test? emails, from either the NHC email network or the Enron data set, conditioned on the
text of those emails and the identities of their authors. For each test email d, the binary variables
(d)
indicating the recipients of that email, i.e., {yr }A
r=1 , were treated as unobserved. Typical values
for these variables were sampled from their joint posterior distribution and compared to the true
values to yield an F1 score. We formed a test split of each data set by randomly selecting emails
with probability 0.1. For each data set, we averaged the F1 scores over five random test splits.
We compared our model?s performance with that of two baselines and three existing network models, thereby situating it within the existing literature. Given a test email authored by actor a, our
simplest baseline na??vely predicts that actor a will include actor r as a recipient of that email with
probability equal to the number of non-test emails sent from actor a to actor r divided by the total number of non-test emails sent by actor a. Our second baseline is a variant of our model in
which each topic-specific communication pattern P (t) is represented explicitly via A(A + 1) / 2
probabilities drawn from a symmetric Beta prior with concentration parameter ?. Comparing our
model to this variant enables us to validate our assumption that topic-specific communication patterns can indeed be accurately represented by a set of A points (one per actor) in K-dimensional
Euclidean space. We also compared our model?s performance to that of three existing network models: a variant of Erosheva et al.?s model for analyzing scientific publications [4], LSM [7], and the
5
mixed-membership stochastic blockmodel (MMSB) [12]. Erosheva et al.?s model can be viewed as
(d)
a variant of our model in which the topic assignment for each yr is drawn from ? (d) instead of
(d)
the empirical distribution over topics defined by z . Like our second baseline, each topic-specific
communication pattern is represented explicitly via probabilities drawn from a symmetric Beta prior
with concentration parameter ?; however, unlike this baseline, each one is represented using A prob(t)
(t)
abilities such that par = pr . LSM can be viewed as a network-only variant of our model in which
text is not modeled. As a result, there are no topics and a single communication pattern P . This pattern is represented implicitly via a set of A actor-specific points in K-dimensional Euclidean space.
Finally, MMSB is a widely-used model for mixed-membership community discovery in networks.
(d)
For our model and all its variants, typical values for {yr }A
r=1 can be sampled from their joint
posterior distribution using an appropriately-modified version of the Metropolis-within-Gibbs algorithm in Section 2.1. In all our experiments, we ran this algorithm for 40,000?50,000 iterations. On
iteration i, we defined each proposal distribution to be a Gaussian distribution centered on the value
from iteration i ? 1 with covariance matrix max (1, 100 / i) I, thereby resulting in larger covariances
for earlier iterations. Beta?binomial conjugacy allows the elements of P (t) to be marginalized out
in both our second baseline and Erosheva et al.?s model. For MMSB, typical values can be sampled using a modified version of Chang?s Gibbs sampling algorithm [13]. We ran this algorithm
for 5,000 iterations. For all models involving topics, we set concentration parameter ? to 1 for the
NHC network and 2 for the Enron data set. For both data sets, we set concentration parameter ? to
0.01V .4 We varied the number of topics from 1 to 200. In order to facilitate visualization, we used
2-dimensional Euclidean spaces for our model. For LSM, however, we varied the dimensionality of
the Euclidean space from 1 to 200. We report only those results obtained using the best-performing
dimensionality. For our second baseline and Erosheva et al.?s model, we set concentration parameter
? to 0.02. For MMSB, we performed a grid search over all hyperparameter values and the number
of blocks and, as with LSM, report only those results obtained using the best-performing values.5
F1 scores, averaged over five random test splits of each data set, are shown in Figure 2. Although our
model is intended for exploratory analysis, it achieves better link prediction performance than the
other models. Furthermore, the fact that our model outperforms our second baseline and Erosheva
et al.?s model validates our assumption that topic-specific communication patterns can indeed be
accurately represented by a set of A actor-specific points in 2-dimensional Euclidean space.
4.2
Topic Coherence
When evaluating unsupervised topic models, topic coherence metrics [14, 15] are often used as a
proxy for subjective evaluation of semantic coherence. In order to demonstrate that incorporating
network data does not impair our model?s ability to model text, we compared the coherence of topics
inferred using our model with the coherence of topics inferred using LDA, our second baseline, and
Erosheva et al.?s model. For each model, we varied the number of topics from 1 to 200 and drew five
samples from the joint posterior distribution over the unobserved random variables in that model. We
evaluated the topics resulting from each sample using Mimno et al.?s coherence metric [14]. Topic
coherence, averaged over the five samples, is shown in Figure 2. Our model achieves coherence
comparable to that of LDA. This result, when combined with the results in Section 4.1, demonstrates
that our model can achieve state-of-the-art predictive performance while producing coherent topics.
4.3
Posterior Predictive Checks
We used posterior predictive checking to assess the extent to which our model is a ?good fit? for the
NHC email network [16, 17]. Specifically, we defined four network statistics (i.e., four discrepancy
functions) that summarize meaningful aspects of the NHC network: generalized graph transitivity,
the dyad intensity distribution, the vertex degree distribution, and the geodesic distance distribution.6
We then generated 1,000 synthetic networks from the posterior predictive distribution implied by our
4
These values were obtained by slice sampling typical values for the concentration parameters in LDA.
They are consistent with the concentration parameter values used in previous work [9].
5
These values correspond to a Dir(0.1, . . . , 0.1) prior over block memberships, a Beta(0.1, 0.1) prior over
diagonal entries of the blockmodel, a Beta(0.01, 0.01) prior over off-diagonal entries, and 30 blocks.
6
These statistics are defined in the supplementary materials.
6
50
60
?
?
40
0.655
0.665
Transitivity
(a)
400
200
0
0.645
600
?
?
?
?
?
?
?
20
0
Simulated Quantile
100
1.2
800
Degree
Simulated Quantile
Frequency
150
1.4
?
?
80
200
??
??
??
???
?
?
?
??
?
0
0
20
40
?
?
??
?????
???
??
????
60
1.0
0.8
0.6
0.4
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Observed Quantile
Actor (Sorted by Observed Degree)
(b)
(c)
Observed Quantile
(d)
Figure 3: Four posterior predictive checks of our model using the NHC email network and 100
topics: (a) a histogram of the graph transitivity of the synthetic networks, with the graph transitivity
of the NHC email network indicated by a vertical line; (b) a quantile?quantile plot comparing the
distribution of dyadic intensities in the synthetic networks to that of the observed network; (c) a box
plot indicating the sampled degree of each manager in the synthetic networks, with managers sorted
from highest to lowest observed degree and their observed degrees indicated by a line; and (d) a
quantile?quantile plot comparing the observed and synthetic geodesic distance distributions.
model and the NHC network. We applied each discrepancy function to each synthetic network to
yield four distributions over the values of the four network statistics. If our model is a ?good fit?
for the NHC network, these distributions should be centered around the values of the corresponding
discrepancy functions when computed using the observed NHC network. As shown in Figure 3,
our model generates synthetic networks with dyad intensity, vertex degree, and geodesic distance
distributions that are very similar to those of the NHC network. The distribution over synthetic graph
transitivity values is not centered around the observed graph transitivity, but the observed transitivity
is not sufficiently far into the tail of the distribution to warrant reparameterization of our model.
4.4
Exploratory Analysis
In order to demonstrate our model?s novel ability to discover and visualize topic-specific communication patterns, we performed an exploratory analysis of four such patterns inferred from the NHC
email network using our model. These patterns are visualized in Figure 4. Each pattern is represented implicitly via a single set of A points in 2-dimensional Euclidean space drawn from their joint
posterior distribution. The recipients of any email associated with topic t are more likely to be those
actors near to the email?s author in the Euclidean space corresponding to that topic. We selected the
patterns in Figure 4 so as to highlight the types of insights that can be obtained using our model.
Although many structural properties may be of interest, we focus on modularity and assortativity.
For each topic-specific communication pattern, we examined whether there are active, disconnected
components in that topic?s Euclidean space (i.e., high modularity). The presence of such components indicates that there are groups of actors who engage in within- but not between-group communication about that topic. We also used a combination of node proximity and node coloration
to determine whether there is more communication between departments that belong to the same
?division? in the New Hanover County government organizational chart than between departments
within different divisions (i.e., assortativity). In Figure 4, we show one topic that exhibits strong
modularity and little assortativity (the ?Public Signage? topic), one topic that exhibits strong assortativity and little modularity (the ?Broadcast Messages? topic), and one topic that exhibits both
strong assortativity and strong modularity (the ?Meeting Scheduling? topic). The ?Public Relations?
topic, which includes communication with news agencies, is mostly dominated by a cluster involving many departments. Finally, the ?Meeting Scheduling? topic displays hierarchical structure, with
two assistant county managers located at the centers of groups that correspond to their divisions.
Exploratory analysis of communication patterns is a powerful tool for understanding organizational
communication networks. For example, examining assortativity can reveal whether actual communication patterns resemble official organizational structures. Similarly, if a communication pattern
exhibits modularity, each disconnected component may benefit from organizational efforts to facilitate inter-component communication. Finally, structural properties other than assortativity and
modularity may also yield scientific or practical insights, depending on organizational needs.
7
Public Signage
Broadcast Messages
15
change signs sign process
ordinance
17
fw fyi bulletin summary
week legislative
?
PM
CC
?
400
300
HL
?
?
AM
SF
?
?
PS
?
DS
200
EG
?
?
EG
LB
?
AM
?
?
DS
?
FN
PS
?
100
200
?
PI
TX
?
?
BG
LB
?
FS
?
0
?
HR
MS
?
?
CM
RD
?
AM
AM
EM PG
?
?
IT
CE
?
EM
?
RD
?
PG
?
CA
?
CC
?
SF
?
?
TX
?
FS
?
?300
?
SS
?
?
?200
EV
?
?
IT
MS
?
?100
?
PM
?200
?
RM
0
?
BG
EL
?
?400
?
EV
?
CM
CA
?
?
FN
?
YS
?
PI
?
SS
RM
?
YS
?
?
VS
?
HL
HR
?
VS
?
EL
?
CE
?
?400
?200
0
200
400
?300
?200
?100
0
100
200
300
Public Relations
Meeting Scheduling
31
city breakdown information
give
63
meeting march board
agenda week
EL
?
FS
?
400
400
SF
?
CA
?
PS
?
RM
?
200
200
RD
?
EG
?
RM
?
?
HR
?
MS
PI
?
?
DS
0
FS
?CE
?
EV
?
?
CM
? EL
PG
?
RD
?
?
TX
CC
?
?
LB
?
CE
?
EM
?
AM
?
HR
BG
?
?
FN
?
HL
0
?
PM
?
? SS
LB
?
HL
?
EM
?
AM
?
?200
IT
?
PG
?
BG
CA
?
MS
?
?200
?
PS
?
CM
?
EG
?
SS
IT
?
TX
?
0
?
?600
?400
CC
?
?400
AM
YS
?
?
AM
VS
?
DS
?
PI
?200
?
FN
YS
??
?400
?
PM
VS
?
SF
?
Assistant County Manager
Budget
Cooperative Extension
County Attorney
County Commissioners
County Manager
Development Services
Elections
Emergency Management
Engineering
Environmental Management
Finance
Fire Services
Health
Human Resources
Information Technology
Library
Museum
Parks and Gardens
Planning and Inspections
Pretrial Release Screening
Property Management
Register of Deeds
Risk Management
Sheriff
Social Services
Tax
?
Veteran Services
Youth Empowerment Services
AM
BG
CE
CA
CC
CM
DS
EL
EM
EG
EV
FN
FS
HL
HR
IT
LB
MS
PG
PI
PS
PM
RD
RM
SF
SS
TX
VS
YS
EV
?
200
400
600
?200
0
200
400
Figure 4: Four topic-specific communication patterns inferred from the NHC email network. Each
pattern is labeled with a human-selected name for the corresponding topic, along with that topic?s
most probable words in order of decreasing
probability. The size of each manager?s acronym in
q
(t)
(t)
(t)
topic t?s pattern (given by 0.45 + 1.25 da / maxa da , where da is the degree of actor a in that
subnetwork) indicates how often that manager communicates about that topic. Managers? acronyms
are colored according to their respective division in the New Hanover County organizational chart.
The acronym ?AM? appears twice in all plots because there are two assistant county managers.
5
Conclusions
We introduced a new Bayesian admixture model for the discovery and visualization of topic-specific
communication subnetworks. Although our model is intended for exploratory analysis, the validation experiments described in Sections 4.1 and 4.2 demonstrate that our model can achieve stateof-the-art predictive performance while exhibiting topic coherence comparable to that of LDA. To
showcase our model?s ability to discover and visualize topic-specific communication patterns, we
introduced a new data set (the NHC email network) and analyzed four such patterns inferred from
this data set using our model. Via this analysis, were are able to examine the extent to which actual
communication patterns resemble official organizational structures and identify groups of managers
who engage in within- but not between-group communication about certain topics. Together, these
predictive and exploratory analyses lead us to recommend our model for any exploratory analysis of
email networks or other similarly-structured communication data. Finally, our model is capable of
producing principled visualizations of email networks, i.e., visualizations that have precise mathematical interpretations in terms of this model and its relationship to the observed data. We advocate
for principled visualization as a primary objective in the development of new network models.
Acknowledgments
This work was supported in part by the Center for Intelligent Information Retrieval and in part by
the NSF GRFP under grant #1122374. Any opinions, findings, and conclusions or recommendations
expressed in this material are those of the authors and do not necessarily reflect those of the sponsors.
8
References
[1] W. Mason and D.J. Watts. Collaborative learning in networks. Proceedings of the National
Academy of Sciences, 109(3):764?769, 2012.
[2] A. McCallum, A. Corrada-Emmanuel, and X. Wang. Topic and role discovery in social networks. In Proceedings of the International Joint Conference on Artificial Intelligence, 2005.
[3] J. Chang and D.M. Blei. Relational topic models for document networks. In Proceedings of
the Twelfth International Conference on Artificial Intelligence and Statistics, 2009.
[4] E. Erosheva, S. Fienberg, and J. Lafferty. Mixed-membership models of scientific publications.
Proceedings of the National Academy of Sciences, 101(Suppl. 1), 2004.
[5] S.E Fienberg. A brief history of statistical models for network analysis and open challenges.
Journal of Computational and Graphical Statistics, 22, 2012.
[6] D.M. Blei, A.Y. Ng, and M.I. Jordan. Latent Dirichlet allocation. Journal of Machine Learning
Research, 3:993?1022, 2003.
[7] P.D. Hoff, A.E. Raftery, and M.S. Handcock. Latent space approaches to social network analysis. Journal of the American Statistical Association, 97(460):1090?1098, 2002.
[8] D.M. Blei and M.I. Jordan. Modeling annotated data. In Proceedings of the Twenty-Sixth
Annual International ACM SIGIR Conference on Research and Development in Information
Retrieval, pages 127?134, 2003.
[9] T.L. Griffiths and M. Steyvers. Finding scientific topics. Proceedings of the National Academy
of Sciences, 101(Suppl. 1), 2004.
[10] B. Klimt and Y. Yang. Introducing the Enron corpus. In Proceedings of the First Conference
on Email and Anti-Spam, 2004.
[11] P.A Schrodt. Seven deadly sins of contemporary quantitative political analysis. In Proceedings
of the Annual American Political Science Association Meeting and Exhibition, 2010.
[12] E.M. Airoldi, D.M. Blei, S.E. Fienberg, and E.P. Xing. Mixed membership stochastic blockmodels. Journal of Machine Learning Research, 9:1981?2014, 2008.
[13] J. Chang. Uncovering, Understanding, and Predicting Links. PhD thesis, Princeton Unversity,
2011.
[14] D. Mimno, H.M. Wallach, E.T.M. Leenders, and A. McCallum. Optimizing semantic coherence in topic models. In Proceedings of the Conference on Empirical Methods in Natural
Language Processing, 2011.
[15] D. Newman, J.H. Lau, K. Grieser, and T. Baldwin. Automatic evaluation of topic coherence. In
Proceedings of Human Language Technologies: The Annual Conference of the North American
Chapter of the Association for Computational Linguistics, pages 100?108, 2010.
[16] D.R. Hunter, M.S. Handcock, C.T. Butts, S.M. Goodreau, and M. Morris. ergm: A package
to fit, simulate and diagnose exponential-family models for networks. Journal of Statistical
Software, 24(3):1?29, 2008.
[17] D. Mimno and D.M. Blei. Bayesian checking for topic models. In Proceedings of the Conference on Empirical Methods in Natural Language Processing, pages 227?237, 2011.
9
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4,043 | 466 | A Contrast Sensitive Silicon Retina with
Reciprocal Synapses
Kwabena A. Boahen
Computation and Neural Systems
California Institute of Technology
Pasadena, CA 91125
Andreas G. Andreou
Electrical and Computer Engineering
Johns Hopkins University
Baltimore, MD 21218
Abstract
The goal of perception is to extract invariant properties of the underlying world. By computing contrast at edges, the retina reduces incident
light intensities spanning twelve decades to a twentyfold variation. In one
stroke, it solves the dynamic range problem and extracts relative reflectivity, bringing us a step closer to the goal. We have built a contrastsensitive silicon retina that models all major synaptic interactions in the
outer-plexiform layer of the vertebrate retina using current-mode CMOS
circuits: namely, reciprocal synapses between cones and horizontal cells,
which produce the antagonistic center/surround receptive field, and cone
and horizontal cell gap junctions, which determine its size. The chip has
90 x 92 pixels on a 6.8 x 6.9mm die in 2/lm n-well technology and is fully
functional.
1
INTRODUCTION
Retinal cones use both intracellular and extracellular mechanisms to adapt their
gain to the input intensity level and hence remain sensitive over a large dynamic
range. For example, photochemical processes within the cone modulate the photo currents while shunting inhibitory feedback from the network adjusts its membrane conductance. Adaptation makes the light sensitivity inversely proportional
to the recent input level and the membrane conductance proportional to the background intensity. As a result, the cone's membrane potential is proportional to the
ratio between the input and its spatial or temporal average, i.e. contrast. We have
764
A Contrast Sensitive Silicon Retina with Reciprocal Synapses
developed a contrast- sensitive silicon retina using shunting inhibition.
This silicon retina is the first to include variable inter-receptor coupling, allowing
one to trade-off resolution for enhanced signal-to-noise ratio, thereby revealing
low-contrast stimuli in the presence of large transistor mismatch. In the vertebrate
retina, gap junctions between photoreceptors perform this function [5]. At these
specialized synapses, pores in the cell membranes are juxtaposed, allowing ions
to diffuse directly from one cell to another [6]. Thus, each receptor's response is a
weighted average over a local region. The signal-to-noise ratio increases for features
larger than this region-in direct proportion to the space constant [5].
Our chip achieves a four-fold improvement in density over previous designs [2].
We use innovative current-mode circuits [7] that provide very high functionality
while faithfully modeling the neurocircuitry. A bipolar phototransistor models the
photo currents supplied by the outer-segment of the cone. We use a novel singletransistor implementation of gap junctions that exploits the physics of MaS transistors. Chemical synapses are also modeled very efficiently with a single device.
Mahowald and Mead's pioneering silicon retina [2] coded the logarithm of contrast.
However, a logarithmic encoding degrades the signal-to-noise ratio because large
signals are compressed more than smaller ones. Mead et. al. have subsequently
improved this design by including network-level adaptation [4] and adaptive photoreceptors [3, 4] but do not implement shunting inhibition. Our silicon retina was
designed to encode contrast directly using shunting inhibition.
The remainder of this paper is organized as follows. The neurocircuitry of the
distal retina is described in Section 2. Diffusors and the contrast-sensitive silicon retina circuit are featured in Section 3. We show that a linearized version of
this circuit computes the regularized solution for edge detection. Responses from a
one-dimensional retina showing receptive field organization and contrast sensitivity, and images from the two-dimensional chip showing spatial averaging and edge
enhancement are presented in Section 4. Section 5 concludes the paper.
Cones
I
Synapses
_____X )Hog:n:tal
~ Gap
Junctions
Figure 1: Neurocircuitry of the outer-plexiform layer. The white and black
triangles are excitatory and inhibitory chemical synapses, respectively. The
grey regions between adjacent cells are electrical gap junctions.
765
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Boahen and Andreou
2
THE RETINA
The outer plexiform layer of the retina produces the well-known antagonistic center/surround receptive field organization first described in detail by Kuffler in the
cat [11). The functional neurocircuitry, based on the red cone system in the turtle [10, 8, 6], is shown in Figure 1. Cones and horizontal cells are coupled by gap
junctions, forming two syncytia within which signals diffuse freely. The gap junctions between horizontal cells are larger in area (larger number of elementary pores),
so signals diffuse relatively far in the horizontal cell syncytium. On the other hand,
signals diffuse poorly in the cone syncytium and therefore remain relatively strong
locally. When light falls on a cone, its activity increases and it excites adjacent horizontal cells which reciprocate with inhibition. Due to the way signals spread, the
excitation received by nearby cones is stronger than the inhibition from horizontal
cells, producing net excitation in the center. Beyond a certain distance, however,
the reverse is true and so there is net inhibition in the surround.
The inhibition from horizontal cells is of the shunting kind and this gives rise to
to contrast sensitivity. Horizontal cells depolarize the cones by closing chloride
channels while light hyperpolarizes them by closing sodium channels [9, I). The
cone's membrane potential is given by
v
=
+ gD Vnet
gNa + gCI + gD
gNaENa
(1)
where the conductances are proportional to the number of channels that are open
and voltages are referred to the reversal potential for chloride. gD and Vnet describe
the effect of gap junctions to neighboring cones. Since the horizontal cells pool
signals over a relatively large area, gCI will depend on the background intensity.
Therefore, the membrane voltage will be proportional to the ratio between the
input, which determines gNa, and the background.
(a)
(b)
Figure 2: (a) Diffusor circuit. (b) Resistor circuit. The diffusor circuit simulates the currents in this linear resistive network.
A Contrast Sensitive Silicon Retina with Reciprocal Synapses
3
SILICON MODELS
In the subthreshold region of operation, a MOS transistor mimics the behavior of a
gap junction. Current flows by diffusion: the current through the channel is linearly
proportional to the difference in carrier concentrations across it [2]. Therefore, the
channel is directly analogous to a porous membrane and carrier concentration is the
analog of ionic species concentration. In conformity with the underlying physics, we
call transistors in this novel mode of operation diffusors. The gate modulates the
carrier concentrations at the drain and the source multiplicatively and therefore sets
the diffusivity. In addition to offering a compact gap junction with electronically
adjustable 'area,' the diffusor has a large dynamic range-at least five decades.
A current-mode diffusor circuit is shown in Figure 2a. The currents through the
diode-connected well devices Ml and M2 are proportional to the carrier concentrations at either end of the diffusor M 3 ? Consequently, the diffusor current is proportional to the current difference between Ml and M 2 ? Starting with the equation
describing subthreshold conduction [2, p. 36], we obtain an expression for the current IpQ in terms of the currents Ip and IQ, the reference voltage Vre / and the bias
voltage VL :
(2)
=
For simplicity, voltages and currents are in units of VT
kT/q, and 10 , the zero
bias current, respectively; all devices are assumed to have the same Ii and 10 ? The
ineffectiveness of the gate in controlling the channel potential, measured by Ii ~ 0.75,
introd uces a small nonideality. There is a direct analogy between this circuit and
the resistive circuit shown in Figure 2b for which I pQ == (Cz/Cl){IQ - Ip). The
currents in these circuits are identical if Cz/C l == exp(IiVL - Vre /) and Ii == l.
Increasing VL or reducing Vre / has the same effect as increasing C 2 or reducing C l .
Chemical synapses are also modeled using a single MOS transistor. Synaptic inputs
to the turtle cone have a much higher resistance, typically O.6GO or more [1],
than the input conductance of a cone in the network which is 50MO or less [8].
Thus the synaptic inputs are essentially current sources. This also holds true for
horizontal cells which are even more tightly coupled. Accordingly, chemical synapses
are modeled by a MOS transistor in saturation. In this regime, it behaves like a
current source driving the postsynapse controlled by a voltage in the presynapse.
The same applies to the light-sensitive input supplied by the cone outer-segment;
its peak conductance is about OAGO in the tiger salamander [9]. Therefore, the
cone outer-segment is modeled by a bipolar phototransistor, also in saturation,
which produces a current proportional to incident light intensity.
Shunting inhibition is not readily realized in silicon because the 'synapses' are current sources. However, to first order, we achieve the same effect by modulating the
gap junction diffusitivity gD (see Equation 1). In the silicon retina circuit, we set
VL globally for a given diffusitivity and control Vre / locally to implement shunting
inhibition.
A one-dimensional version of the current-mode silicon retina circuit is shown in
Figure 3. This is a direct mapping of the neurocircuitry of the outer-plexiform
layer (shown in Figure 1) onto silicon using one transistor per chemical synapse/gap
junction. Devices Ml and M2 model the reciprocal synapses. M4 and Ms model
767
768
Boahen and Andreou
VDD
I
Figure 3: Current-mode Outer-Plexiform Circuit.
the gap junctions; their diffusitivities are set globally by the bias voltages VG and
VF. The phototransistor M6 models the light-sensitive input from the cone outer
segment. The transistor M 3 , with a fixed gate bias Vu, is analogous to a leak in
the horizontal cell membrane that counterbalances synaptic input from the cone.
The circuit operation is as follows. The currents Ic and IH represent the responses
of the cone and the horizontal cell, respectively. These signals are actually in
the post-synaptic circuit-the nodes with voltage Vc and VH correspond to the
presynaptic signals but they encode the logarithm of the response. Increasing the
photocurrent will cause Vc to drop, turning on M2 and increasing its current Ic;
this is excitation. Ic pulls VH down, turning on Ml and increasing its current IH;
another excitatory effect. I H , in turn, pulls Vc up, turning off M2 and reducing its
current Ic; this is inhibition.
The diffusors in this circuit behave just like those in Figure 2 although the well
devices are not diode- connected. The relationship between the currents given by
Equation 2 still holds because the voltages across the diffusor are determined by the
currents through the well devices. However, the reference voltage for the diffusors
between 'cones' (M4) is not fixed but depends on the 'horizontal cell' response. Since
IH = exp(VDD - KVH ), the diffusitivity in the cone network will be proportional to
the horizontal cell response. This produces shunting inhibition.
3.1
RELATION TO LINEAR MODELS
Assuming the horizontal cell activities are locally very similar due to strong coupling, we can replace the cone network diffusitivity by g = (IH)g, where (IH) is the
local average. Now we treat the diffusors between the 'cones' as if they had a fixed
A Contrast Sensitive Silicon Retina with Reciprocal Synapses
diffusitivity fJ; the diffusitivity in the 'horizontal cell' network is denoted by h. Then
the equations describing the full two-dimensional circuit on a square grid are:
I(xm,Yn)
+ fJ
L
{Ic(xi,Yj) - Ic(xm,Yn)}
(3)
i = m? 1
j = n ? 1
Iu
L
+h
{IH(xm,Yn) - IH(xi,Yj)}
(4)
?1
?1
i= m
j
n
=
This system is a special case of the dual layer outer plexiform model proposed by
Vagi [12]-we have the membrane admittances set to zero and the synaptic strengths
set to unity. Using the second-difference approximation for the laplacian, we obtain
the continuous versions of these equations
I(x, y) + fJV 2 Ic(x, y)
Iu - hV 2 IH(x, y)
IH(x, y)
Ic(x,y)
(5)
(6)
with the internode distance normalized to unity. Solving for IH(x, Y), we find
)..V 2 V 2 I H(x,y) +IH(x,y) = I(x,y)
(7)
This is the biharmonic equation used in computer vision to find an optimally smooth
interpolating function 'IH(x,y)' for the noisy, discrete data 'I(x,y)' [13]. The coefficient).. = fJh is called the regularizing parameter; it determines the trade-off
between smoothing and fitting the data. In this context, the function of the horizontal cells is to compute a smoothed version of the image while the cones perform
edge detection by taking the laplacian of the smoothed image as given by Equation 6.
The space constant of the solutions is )..1/4 [13]. This predicts that the receptive
field size of our retina circuit will be weakly dependent on the input intensity since
fJ is proportional to the horizontal cell activity.
4
CHIP PERFORMANCE
Data from the one-dimensional chip showing receptive field organization is in Figure 4. As the 'cone' coupling increases, the gain decreases and the excitatory and
inhibitory subregions of the receptive field become larger. Increasing the 'horizontal
cell' coupling also enlarges the receptive field but in this case the gain increases. This
is because stronger diffusion results in weaker signals locally and so the inhibition
decreases. Figure 5(a) shows the variation of receptive field size with intensityroughly doubling in size for each decade. This indicates a one-third power dependence which is close to the theoretical prediction of one-fourth for the linear model.
The discrepancy is due to the body effect on transistor M2 (see Figure 3) which
makes the diffusor strength increase with a power of 1/ K,2.
Contrast sensitivity measurements are shown in Figure 5(b). The S-shaped curves
are plots of the Michaelis-Menten equation used by physiologists to fit responses of
cones [6]:
v - V.
In
(8)
-
maz
In
+ un
769
770
Boahen and Andreoli
---ii
55
50
50
45
45
-.40
40
535
-<
...,
dv 35
~ 30
~ 30
'"'
'"'
825
u 25
.fr
20-+--~~
...,
...,E.. 20
o
15
015
~
:=
:=
~~"In
10
10
50
5
10
15
Node Position
(a)
20
50
25
5
(b)
10
15
Node Position
20
25
Figure 4: Receptive fields measured for 25 x 1 pixel chip; arrows indicate
increasing diffu80r gate voltages. The inputs were 50DA at the center and
lOnA elsewhere, and the output current Iu was set to 20nA. (a) Increasing
inter-receptor diffusor voltages in l5mV steps. (b) Increasing inter-horizontal
cell diffusor voltages in 50m V steps.
---<=
-...,
:=
...,~
:=
60
50
50
40
40
-.30
<
--=
30
:; 20
...,~
20
:=
0
010
10
0
0
-10
0
(a)
5
10
15
Node Position
20
25
-10
10- 11
(b)
10- 10
10- 9
Input (A)
10- 8
10- 7
Figure 5: (a) Dependence of receptive field on intensity; arrows indicate increasing intensity. Center inputs were 500pA, 5nA, 15nA, 50nA, and 500nA.
The background input was always one-fifth of the center input. (b) Contrast
sensitivity measurements at two background intensity levels. Lines are fits of
the Michaelis-Menten equation.
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4,044 | 4,660 | Priors for Diversity in Generative
Latent Variable Models
Ryan P. Adams
School of Engineering and Applied Sciences
Harvard University
Cambridge, MA 02138
[email protected]
James Y. Zou
School of Engineering and Applied Sciences
Harvard University
Cambridge, MA 02138
[email protected]
Abstract
Probabilistic latent variable models are one of the cornerstones of machine learning. They offer a convenient and coherent way to specify prior distributions over
unobserved structure in data, so that these unknown properties can be inferred via
posterior inference. Such models are useful for exploratory analysis and visualization, for building density models of data, and for providing features that can
be used for later discriminative tasks. A significant limitation of these models,
however, is that draws from the prior are often highly redundant due to i.i.d. assumptions on internal parameters. For example, there is no preference in the prior
of a mixture model to make components non-overlapping, or in topic model to
ensure that co-occurring words only appear in a small number of topics. In this
work, we revisit these independence assumptions for probabilistic latent variable
models, replacing the underlying i.i.d. prior with a determinantal point process
(DPP). The DPP allows us to specify a preference for diversity in our latent variables using a positive definite kernel function. Using a kernel between probability
distributions, we are able to define a DPP on probability measures. We show how
to perform MAP inference with DPP priors in latent Dirichlet allocation and in
mixture models, leading to better intuition for the latent variable representation
and quantitatively improved unsupervised feature extraction, without compromising the generative aspects of the model.
1
Introduction
The probabilistic generative model is an important tool for statistical learning because it enables rich
data to be explained in terms of simpler latent structure. The discovered structure can be useful in
its own right, for explanatory purposes and visualization, or it may be useful for improving generalization to unseen data. In the latter case, we might think of the inferred latent structure as providing
a feature representation that summarizes complex high-dimensional interaction into a simpler form.
The core assumption behind the use of latent variables as features, however, is that the salient statistical properties discovered by unsupervised learning will be useful for discriminative tasks. This
requires that the features span the space of possible data and represent diverse characteristics that
may be important for discrimination. Diversity, however, is difficult to express within the generative
framework. Most often, one builds a model where the feature representations are independent a
priori, with the hope that a good fit to the data will require employing a variety of latent variables.
There is reason to think that this does not always happen in practice, and that during unsupervised
learning, model capacity is often spent improving the density around the common cases, not allocating new features. For example, in a generative clustering model based on a mixture distribution,
multiple mixture components will often be used for a single ?intuitive group? in the data, simply because the shape of the component?s density is not a close fit to the group?s distribution. A generative
1
mixture model will happily use many of its components to closely fit the density of a single group of
data, leading to a highly redundant feature representation. Similarly, when applied to a text corpus,
a topic model such as latent Dirichlet allocation [1] will place large probability mass on the same
stop words across many topics, in order to fine-tune the probability assigned to the common case. In
both of these situations, we would like the latent groupings to uniquely correspond to characteristics
of the data: that a group of data should be explained by one mixture component, and that common
stop words should be one category of words among many. This intuition expresses a need for diversity in the latent parameters of the model that goes beyond what is highly likely under the posterior
distribution implied by an independent prior.
In this paper, we propose a modular approach to diversity in generative probabilistic models by
replacing the independent prior on latent parameters with a determinantal point process (DPP). The
determinantal point process enables a modeler to specify a notion of similarity on the space of
interest, which in this case is a space of possible latent distributions, via a positive definite kernel.
The DPP then assigns probabilities to particular configurations of these distributions according to
the determinant of the Gram matrix. This construction naturally leads to a generative latent variable
model in which diverse sets of latent parameters are preferred over redundant sets.
The determinantal point process is a convenient statistical tool for constructing a tractable point
process with repulsive interaction. The DPP is more general than the Poisson process (see, e.g.,
[2]), which has no interaction, but more tractable than Strauss [3] and Gibbs/Markov [4] processes
(at the cost of only being able to capture anticorrelation). Hough et al. [5] provides a useful survey
of probabilistic properties of the determinantal point process, and for statistical properties, see, e.g.,
Scardicchio et al. [6] and Lavancier et al. [7]. There has also been recent interest in using the DPP
within machine learning for modeling sets of structures [8], and for conditionally producing diverse
collections of objects [9]. The approach we propose here is different from this previous work in that
we are suggesting the use of a determinantal point process within a hierarchical model, and using it
to enforce diversity among latent variables, rather than as a mechanism for diversity across directly
observed discrete structures.
2
Diversity in Generative Latent Variable Models
In this paper we consider generic directed probabilistic latent variable models that produce distributions over a set of N data, denoted {xn }N
n=1 , which live in a sample space X . Each of these data
has a latent discrete label zn , which takes a value in {1, 2, ? ? ? , J}. The latent label indexes into a
set of parameters {?j }Jj=1 . The parameters determined by zn then produce the data according to a
distribution f (xn | ?zn ). Typically we use independent priors for the ?j , here denoted by ?(?), but
the distribution over the latent indices zn may be more structured. Taken together this leads to the
generic joint distribution:
"
J
N
p({xn , zn }N
n=1 , {?j }j=1 ) = p({zn }n=1 )
N
Y
n=1
#
f (xn | ?zn )
J
Y
?(?j ).
(1)
j=1
The details of each distribution are problem-specific, but this general framework appears in many
contexts. For example, in a typical mixture model, the zn are drawn independently from a multinomial distribution and the ?j are the component-specific parameters. In an admixture model such as
latent Dirichlet allocation (LDA) [1], the ?j may be ?topics?, or distributions over words. In an admixture, the zn may share structure based on, e.g., being words within a common set of documents.
These models are often thought of as providing a principled approach for feature extraction. At
training time, one either finds the maximum of the posterior distribution p({?j }Jj=1 | {xn }N
n=1 ) or
collects samples from it, while integrating out the data-specific latent variables zn . Then when
presented with a test case x? , one can construct a conditional distribution over the corresponding
unknown variable z ? , which is now a ?feature? that might usefully summarize many related aspects
of x? . However, this interpretation of the model is suspect; we have not asked the model to make
the zn variables explanatory, except as a byproduct of improving the training likelihood. Different ?j
may assign essentially identical probabilities to the same datum, resulting in ambiguous features.
2
(a) Independent Points
(c) Independent Gaussians
(e) Independent Multinomials
(b) DPP Points
(d) DPP Gaussians
(f) DPP Multinomials
Figure 1: Illustrations of the determinantal point process prior. (a) 25 independent uniform draw in the unit
square; (b) a draw from a DPP with 25 points; (c) ten Gaussian distributions with means uniformly drawn on
the unit interval; (d) ten Gaussian distributions with means distributed according to a DPP using the probability
product kernel; (e) five random discrete distributions; (f) five random discrete distributions drawn from a DPP
on the simplex with the probability product kernel [10].
2.1
Measure-Valued Determinantal Point Process
In this work we propose an alternative to the independence
assumption of the standard latent variable
Q
model. Rather than specifying p({?j }Jj=1 ) = j ?(?j ), we will construct a determinantal point
process on sets of component-specific distributions {f (x | ?j )}Jj=1 . Via the DPP, it will be possible
for us to specify a preference for sets of distributions that have minimal overlap, as determined via a
positive-definite kernel function between distributions. In the case of the simple parametric families
for f (?) that we consider here, it is appropriate to think of the DPP as providing a ?diverse? set of
parameters ? = {?j }Jj=1 , where the notion of diversity is expressed entirely in terms of the resulting
probability measure on the sample space X . After MAP inference with this additional structure, the
hope is that the ?j will explain substantially different regions of X ? appropriately modulated by
the likelihood ? and lead to improved, non-redundant feature extraction at test time.
We will use ? to denote the space of possible ?. A realization from a point process on ? produces
a random finite subset of ?. To construct a determinantal point process, we first define a positive
definite kernel on ?, which we denote K : ? ? ? ? R. The probability density associated with a
particular finite ? ? ? is given by
p(? ? ?) ? |K? |,
(2)
where K? is the |?| ? |?| positive definite Gram matrix that results from applying K(?, ?0 ) to the
elements of ?. The eigenspectrum of the kernel on ? must be bounded to [0, 1]. The kernels we will
0
focus on in this paper are composed of two parts:
p 1) a positive definite correlation function R(?, ? ),
where R(?, ?) = 1, and 2) the ?prior kernel? ?(?)?(?0 ), which expresses our marginal preferences
for some parameters over others. These are combined to form the kernel of interest:
p
K(?, ?0 ) = R(?, ?0 ) ?(?)?(?0 ),
(3)
p
p
which leads to the matrix form K? = ? R? ?, where ? = diag([ ?(?1 ), ?(?2 ), ? ? ? ]).
Note that if R(?, ?0 ) = 0 when ? 6= ?0 , this construction recovers the Poisson process with intensity
measure ?(?). Note also in this case that if the cardinality of ? is predetermined, then this recovers
the traditional independent prior. More interesting, however, are R(?, ?0 ) with off-diagonal structure
that induces interaction within the set. Such kernels will always induce repulsion of the points so that
diverse subsets of ? will tend to have higher probability under the prior. See Fig. 1 for illustrations
of the difference between independent samples and the DPP for several different settings.
3
2.2
Kernels for Probability Distributions
The determinantal point process framework allows us to construct a generative model for repulsion,
but as with other kernel-based priors, we must define what ?repulsion? means. A variety of positive
definite functions on probability measures have been defined, but in this work we will use the probability product kernel [10]. This kernel is a natural generalization of the inner product for probability
distributions. The basic kernel has the form
Z
0
K(?, ? ; ?) =
f (x | ?)? f (x | ?0 )? dx
(4)
X
for ? > 0. As we require a correlation kernel, we use the normalized variant given by
p
R(?, ?0 ; ?) = K(?, ?0 ; ?)/ K(?, ? ; ?)K(?0 , ?0 ; ?).
(5)
This kernel has convenient closed forms for several distributions of interest, which makes it an ideal
building block for the present model.
2.3
Replicated Determinantal Point Process
A property that we often desire from our prior distributions is that they have the ability to become arbitrarily strong. That is, under the interpretation of a Bayesian prior as ?inferences from
previously-seen data?, we would like to be able to imagine an arbitrary amount of such data and
construct a highly-informative prior when appropriate. Unfortunately, the standard determinantal
point process does not provide a knob to turn to increase its strength arbitrarily.
For example, take a DPP on a Euclidean space and consider a point t, an arbitrary unit vector w and
a small scalar . Construct two pairs of points using a ? > 1: a ?near? pair {t, t + w)}, and a ?far?
pair {t, t + ?w}. We wish to find some small such that the ?far? configuration is arbitrarily more
likely than the ?near? configuration under the DPP. That is, we would like the ratio of determinants
r() =
p({t, t + ?w})
1 ? R(t, t + ?w)2
=
,
p({t, t + w)})
1 ? R(t, t + w))2
(6)
to be unbounded as approaches zero. The objective is to have a scaling parameter that can cause the
determinantal prior to be arbitrarily strong relative to the likelihood terms. If we perform a Taylor
expansion of the numerator and denominator around = 0, we get
1 ? (R(t, t) + 2?w ddt?R(t, t?) t?=t )
r() ?
= ?.
(7)
)
1 ? (R(t, t) + 2w d?R(t, t?)
dt
t?=t
We can see that, when near zero, this ratio captures the difference in distances, but not in a way
that can be rescaled to greater effect. This means that there exist finite data sets that we cannot
overwhelm with any DPP prior. To address this issue, we augment the determinantal point process
with an additional parameter ? > 0, so that the probability of a finite subset ? ? ? becomes
p(? ? ?) ? |K? |? .
(8)
For integer ?, it can be viewed as a set of ? identical ?replicated realizations? from determinantal
point processes, leaving our generative view intact. The replicate of ? is just ? ? = {? copies of ?}
and the corresponding K?? is a ?|?| ? ?|?| block diagonal matrix where each block is a replicate
of K? . This maps well onto the view of a prior as pseudo-data; our replicated DPP asserts that
we have seen ? previous such data sets. As in other pseudo-count priors, we do not require in
practice that ? be an integer, and under a penalized log likelihood view of MAP inference, it can be
interpreted as a parameter for increasing the effect of the determinantal penalty.
2.4
Determinantal Point Process as Regularization.
In addition to acting as a prior over distributions in the generative setting, we can also view the DPP
as a new type of ?diversity? regularizer on learning. The goal is to solve
? ? = argmin L(?; {xn }N
n=1 ) ? ? ln |K? |,
???
4
(9)
?k
?
K
?
?m
zmn
wmn
Nm
M
Figure 2: Schematic of DPP-LDA. We replace the standard plate notation for i.i.d topics in LDA with a ?doublestruck plate? to indicate a determinantal point process.
choosing the best set of parameters ? from ?. Here L(?) is a loss function that depends on the data
and the discrimination function, with parameters ?. From Eqn. (3),
X
ln |K? | = ln |R? | +
ln ?(?j ).
(10)
?j ??
If L(?) = ? ln p({xn }N
n=1 |?), then the resulting optimization is simply MAP estimation. In this
framework, we can combine the DPP penalty with any other regularizer on ?, for example the
sparsity-inducing `1 regularizer. In the following sections, we give empirical evidence that this
diversity improves generalization performance.
3
MAP Inference
In what follows, we fix the cardinality |?|. Viewing the kernel K? as a function of ?, the gradient
?1 ?K?
?
?? log |K? | = trace(K?
?? ). This allows application of general gradient-based optimization
algorithms for inference. In particular, we can optimize ? as a modular component within an offthe-shelf expectation maximization (EM) algorithm. Here we examine two illustrative examples of
generative latent variable models into which we can directly plug our DPP-based prior.
Diversified Latent Dirichlet Allocation Latent Dirichlet allocation (LDA) [1] is an immensely
popular admixture model for text and, increasingly, for other kinds of data that can be treated as a
?bag of words?. LDA constructs a set of topics ? distributions over the vocabulary ? and asserts
that each word in the corpus is explained by one of these topics. The topic-word assignments are
unobserved, but LDA attempts to find structure by requiring that only a small number of topics be
represented in any given document.
In the standard LDA formulation, the topics are K discrete distributions ?k over a vocabulary of
size V , where ?kv is the probability of topic k generating word v. There are M documents and
the mth document has Nm words. Document m has a latent multinomial distribution over topics,
denoted ?m and each word in the document wmn has a topic index zmn drawn from ?m . While
classical LDA uses independent Dirichlet priors for the ?k , here we ?diversify? latent Dirichlet
allocation by replacing this prior with a DPP. That is, we introduce a correlation kernel
PV
(?kv ?k0 v )?
,
(11)
R(?k , ?k0 ) = qP v=1 qP
V
V
2?
2?
?
?
0
v=1 kv
v=1 k v
which approaches one as ?k becomes more similar to ?k0 . In the application below of DPP-LDA, we
use ? = 0.5. We use ?(?k ) = Dirichlet(?), and write the resulting prior as p(?) ? |K? |. We call
this model ?DPP-LDA?, and illustrate it with a graphical model in Figure 2. We use a ?double-struck
plate? in the graphical model to represent the DPP, and highlight how it can be used as a drop-in
replacement for the i.i.d. assumption.
To perform MAP learning of this model, we construct a modified version of the standard variational
EM algorithm. As in variational EM for LDA, we define a factored approximation
q(?m , zm |?m , ?m ) = q(?m |?m )
N
Y
n=1
5
q(zmn |?mn ).
(12)
LDA
typical
the
to
and
in
of
is
it
for
that
can
the
to
and
it
of
is
in
that
for
you
?stop words?
DPP-LDA
?Christianity? ?space?
?OS?
?politics?
the
of
that
you
by
one
all
but
do
my
jesus
matthew
prophecy
christians
church
messiah
psalm
isaiah
prophet
lord
file
pub
usr
available
export
font
lib
directory
format
server
ms
myers
god
president
but
package
options
dee
believe
groups
and
in
at
from
some
their
with
your
who
which
space
nasa
astronaut
mm
mission
pilot
shuttle
military
candidates
ww
Table 1: Top ten words from representative topics learned in LDA and DPP-LDA.
In this approximation, each document m has a Dirichlet approximation to its posterior over topics,
given by ?m . ?m is an N ? K matrix in which the nth row, denoted ?mn , is a multinomial distribution over topics for word wmn . For the current estimate of ?kv , ?m and ?m are iteratively optimized.
See Blei et al. [1] for more details. Our extension of variational EM to include the DPP does not
require alteration of these steps.
The inclusion of the determinantal point process prior does, however, effect the maximization step.
The diversity prior introduces an additional penalty on ?, so that the M-step requires solving
( M N K V
)
m XX
XX
?
(v)
? = argmax
?mnk wmn ln ?kv + ? ln |K? | ,
(13)
?
m=1 n=1 k=1 v=1
subject to the constraints that each row of ? sum to 1. For ? = 0, this optimization procedure yields
PM PNm
(v)
?
the standard update for vanilla LDA, ?kv
? m=1 n=1
?mnk wmn . For ? > 0 we use gradient
descent to find a local optimal ?.
Diversified Gaussian Mixture Model The mixture model is a popular model for generative clustering and density estimation. Given J components, the probability of the data is given by
p(xn | ?) =
J
X
?j f (xn | ?j ).
(14)
j=1
Typically, the ?k are taken to be independent in the prior. Here we examine determinantal point
process priors for the ?k in the case where the components are Gaussian.
For Gaussian mixture models, the DPP prior is particularly tractable. As in DPP-LDA, we use the
probability product kernel, which in this case also has a convenient closed form [10]. Let f1 =
N (?1 , ?1 ) and f2 = N (?2 , ?2 ) be two Gaussians, the product kernel is:
D
D
? ?))
? 12 (|?1 ||?2 |)? ?2 exp(? ? (?T ??1 ?1 + ?T ??1 ?2 ? ?
K(f1 , f2 ) = (2?)(1?2?) 2 ?? 2 |?|
?T ??
2 2
2 1 1
?1
? = (?1 + ?2 )?1 and ?
where ?
? = ??1
1 ?1 + ?2 ?2 . In the special case of a fixed, isotropic
covariance ? 2 I and ? = 1, the kernel is
0 2
2
1
K(f (? | ?), f (? | ?0 )) =
e?||??? || /(4? )
(15)
2
D/2
(4?? )
where D is the data dimensionality.
In the standard EM algorithm for Gaussian mixtures, one typically introduces latent binary variables
znj , which indicate that datum n belongs to component j. The E-step computes the responsibility
vector ?(znj ) = E[znj ] ? ?j N (xn |?j , ?j ). This step is identical for DPP-GMM. The update for
PN
the component weights is also the same: ?j = N1 n=1 ?(znj ). The difference between this procedure and the standard EM approach is that the M-step for the DPP-GMM optimizes the objective
function (summarizing {?j , ?j }Jj=1 by ? for clarity):
?
?
N X
J
?X
?
? ? = argmax
?(znj ) [ln ?j + ln N (xn |?j , ?j )] + ? ln |K? | .
(16)
?
??? ?
n=1 j=1
6
% change in test accuracy
0.24
k=20
k=25
test accuracy
0.23
0.22
0.21
0.2
0.19
0
0.1
1
?
10
1.5
?=0.01
1
?=0.001
?=0.1
0.5
0
?0.5
0
100
Figure 3: Effect of ? on classification error.
5
10
15
20
% increase in inter?centroid distance
Figure 4: Effect of centroid distance on test error.
Closely related to DPP-GMM is DPP-K-means. The kernel acts on the set of centroids as in
Eqn. (15), with ? 2 now just a constant scaling term. Let ? = {?j } and znj be the hard assignment indicator, the maximization step is:
?
?
N X
J
?X
?
? ? = argmax
znj ||xn ? ?j ||2 + ? ln |K? | .
(17)
?
??? ?
n=1 j=1
With the product kernel, the similarity between two Gaussians decays exponentially as the distance
between their means increases. In practice, we find that when the number of mixture components |?|
is large, K? is well approximated by a sparse matrix.
4
Experiment I: diversified topic modeling.
We tested LDA and DPP-LDA on the unfiltered 20 Newsgroup corpus, without removing any stopwords. A common frustration with vanilla LDA is that applying LDA to unfiltered data returns
topics that are dominated by stop-words. This frustrating phenomenon occurs even as the number of
topics is varied from K = 5 to K = 50. The first two columns of Table 1 show the ten most frequent
words from two representative topics learned by LDA using K = 25 . Stop-words occur frequently
across all documents and thus are unhelpfully correlated with topic-specific informative keywords.
We repeated the experiments after removing a list of 669 most common stop-words. However, the
topics inferred by regular LDA are still dominated by secondary stop-words that are not informative.
DPP-LDA automatically groups common stop words into a few topics. By finding stop-word-specific
topics, the majority of the remaining topics are available for more informative words. Table 1 shows
a sample of topics learned by DPP-LDA on the unfiltered 20 Newsgroup corpus (K = 25, ? = 104 ).
As we vary K or increase ? we observe robust grouping of stop-words into a few topics. High
frequency words that are common across many topics significantly increase the similarity between
the topics, as measured by the product kernel on the ? distributions. This similarity incurs a large
penalty in DPP and so the objective actively pushes the parameters of LDA away from regions where
stop words occupy large probability mass across many topics.
Features learned from DPP-LDA leads to better document classification. It is common to use the ?m ,
the document specific posterior distribution over topics, as feature vectors in document classification.
We inferred {?m,train } on training documents from DPP-LDA variational EM, and then trained a
support vector machine (SVM) classifier on {?m,train } with the true topic labels from 20 Newsgroups. On test documents, we fixed the parameters ? and ? to the values inferred from the training
set, and used variational EM to find MAP estimates of {?m,test }. The mean test classification accuracy for a range of ? values is plotted in Figure 3. The setting ? = 0 corresponds to vanilla LDA. In
each trial, we use the same training set for DPP-LDA on a range of ? values. DPP-LDA with ? = 1
consistently outperforms LDA in test classification (p < 0.001 binomial test). Large values of ?
decrease classification performance.
5
Experiment II: diverse clustering.
Mixture models are often a useful way to learn features for classification. The recent work of Coates
et al. [11], for example, shows that even simple K-means works well as a method of extracting
7
training set size
K
K-means
DPP K-means
gain (%)
?
500
1000
2000
5000
10000
30
30
60
150
300
34.81
43.32
52.05
61.03
66.36
36.21
44.27
52.55
61.23
66.65
1.4
0.95
0.50
0.20
0.29
0.01
0.01
0.01
0.001
0.001
Table 2: Test classification accuracy on CIFAR-10 dataset.
features for image labeling. In that work, K-means gave state of art results on the CIFAR-10 object
recognition task. Coates et al. achieved these results using a patch-wise procedure in which random
patches are sampled from images for training. Each patch is a 6-by-6 square, represented as a point
in a 36 dimensional vector space. Patches from the training images are combined and clustered using
K-means. Each patch is then represented by a binary K-dimensional feature vector: the k th entry
is one if the patch is closer to the centroid k than its average distance to centroids. Roughly half of
the feature entries are zero. Patches from the same image are then pooled to construct one feature
vector for the whole image. An SVM is trained on these image features to perform classification.
We reason that DPP-K-means may produce more informative features since the cluster centroids
will repel each other into more distinct positions in pixel space. We replicated the experiments from
Coates et al., using their publicly-available code for identical pre- and post-processing. With this
setup, ? = 0 recovers regular K-means, and reproduces the results from Coates et al. [11]. We
applied DPP-K-means to the CIFAR-10 dataset, while varying the size of the training set. For each
training set size, we ran regular K-means for a range of values of K and select the K that gives the
best test accuracy for K-means. Then we compare the performance with DPP-K-means using the
same K. For up to 10000 images in the training set, DPP-K-means leads to better test classification
accuracy compared to the simple K-means. The comparisons are performed on matched settings:
for a given randomly sampled training set and a centroid initialization, we generate the centroids
from both K-means and DPP-K-means. The two sets of centroids were used to extract features
and train classifiers, which are then tested on the same test set of images. DPP-K-means consistently outperforms K-means in generalization accuracy (p < 0.001 binomial test). For example,
for training set of size 1000, with k = 30, we ran 100 trials, each with an random training set and
initialization, DPP-K-means outperformed K-means in 94 trials. As expected given its role as a regularizer, improvement from DPP-K-means is more significant for smaller training sets. For the full
CIFAR-10 with 50000 training images, DPP-K-means does not consistently outperform K-means.
Next we ask if there is a pattern between how far the DPP pushes apart the centroids and classification accuracy on the test set. Focusing on 1000 training images and k = 30, for each randomly
sampled training set and centroid initialization, we compute the mean inter-centroid distance for Kmeans and DPP-K-means. We compute the test accuracy for each set of centroids. Fig. 4 bins
the relative increase in inter-centroid distance into 10 bins. For each bin, we show the 25th , 50th ,
and 75th percentile of changes in test accuracy. Test accuracy is maximized when the inter-centroid
distances increase by about 14% from K-means centroids, corresponding to ? = 0.01.
6
Discussion.
We have introduced a general approach to including a preference for diversity into generative probabilistic models. We showed how a determinantal point process can be integrated as a modular component into existing learning algorithms, and discussed its general role as a diversity regularizer. We
investigated two settings where diversity can be useful: learning topics from documents, and clustering image patches. Plugging a DPP into latent Dirichlet allocation allows LDA to automatically
group stop-words into a few categories, enabling more informative topics in other categories. In
both document and image classification tasks, there exists an intermediate regime of diversity (as
controlled by the hyperparameter ?) that leads to consistent improvement in accuracy when compared to standard i.i.d. models. A computational bottleneck can come from inverting the M ? M
kernel matrix K, where M is the number of latent distributions. However in many settings such as
LDA, M is much smaller than the data size. We expect that there are many other settings where
DPP-based diversity can be usefully introduced into a generative probabilistic model: in the emission parameters of HMM and more general time series, and as a mechanism for transfer learning.
8
References
[1] David M. Blei, Andrew Y. Ng, and Michael I. Jordan. Latent Dirichlet allocation. Journal of
Machine Learning Research, 3:993?1022, 2003.
[2] J. F. C. Kingman. Poisson Processes. Oxford University Press, Oxford, United Kingdom,
1993.
[3] David J. Strauss. A model for clustering. Biometrika, 62(2):467?475, August 1975.
[4] Jesper M?ller and Rasmus Plenge Waagepetersen. Statistical Inference and Simulation for
Spatial Point Processes. Monographs on Statistics and Applied Probability. Chapman and
Hall/CRC, Boca Raton, FL, 2004.
[5] J. Ben Hough, Manjunath Krishnapur, Yuval Peres, and Blint Vir?ag. Determinantal processes
and independence. Probability Surveys, 3:206?229, 2006.
[6] Antonello Scardicchio, Chase E. Zachary, and Salvatore Torquato. Statistical properties of
determinantal point processes in high-dimensional Euclidean spaces. Physical Review E, 79(4),
2009.
[7] Fr?ed?eric Lavancier, Jesper M?ller, and Ege Rubak. Statistical aspects of determinantal point
processes. http://arxiv.org/abs/1205.4818, 2012.
[8] Alex Kulesza and Ben Taskar. Structured determinantal point processes. In Advanced in Neural
Information Processing Systems 23, 2011.
[9] Alex Kulesza and Ben Taskar. Learning determinantal point processes. In Proceedings of the
27th Conference on Uncertainty in Artificial Intelligence, 2011.
[10] Tony Jebara, Risi Kondor, and Andrew Howard. Probability product kernels. Journal of
Machine Learning Research, 5:819?844, 2004.
[11] Adam Coates Honglak Lee and Andrew Ng. An analysis of single-layer networks in unsupervised feature learning. In Proceedings of the 14th International Conference on Artificial
Intelligence and Statistics, 2011.
9
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4,045 | 4,661 | Unsupervised Structure Discovery for Semantic
Analysis of Audio
Bhiksha Raj
Language Technologies Institute
Carnegie Mellon University
Pittsburgh, PA 15213
[email protected]
Sourish Chaudhuri
Language Technologies Institute
Carnegie Mellon University
Pittsburgh, PA 15213
[email protected]
Abstract
Approaches to audio classification and retrieval tasks largely rely on detectionbased discriminative models. We submit that such models make a simplistic assumption in mapping acoustics directly to semantics, whereas the actual process
is likely more complex. We present a generative model that maps acoustics in
a hierarchical manner to increasingly higher-level semantics. Our model has two
layers with the first layer modeling generalized sound units with no clear semantic
associations, while the second layer models local patterns over these sound units.
We evaluate our model on a large-scale retrieval task from TRECVID 2011, and
report significant improvements over standard baselines.
1
Introduction
Automatic semantic analysis of multimedia content has been an active area of research due to potential implications for indexing and retrieval [1?7]. In this paper, we limit ourselves to the analysis
of the audio component of multimedia data only. Early approaches for semantic indexing of audio
relied on automatic speech recognition techniques to generate semantically relevant keywords [2].
Subsequently, supervised approaches were developed for detecting specific (potentially semantically
relevant) sounds in audio streams [6, 8?10], e.g. gunshots, laughter, music, crowd sounds etc., and
using the detected sounds to characterize the audio files. While this approach has been shown to be
effective on certain datasets, it requires data for each of the various sounds expected in the dataset.
Further, such detectors will not generalize across datasets with varying characteristics; e.g. audio
libraries are studio-quality, while user-generated Youtube-style content are noisy.
In order to avoid the issues that arise with using supervised, detection-based systems, unsupervised
approaches were developed to learn sound dictionaries from the data [7, 11, 12],. Typically, these
methods use clustering techniques on fixed length audio segments to learn a dictionary, and then
characterize new data using this dictionary. However, characterizing audio data with elements from
an audio dictionary (supervised or unsupervised) for semantic analysis involves an implicit assumption that the acoustics map directly to semantics. In reality, we expect the mapping to be more
complex, because acoustically similar sounds can be produced by very different sources. Thus, to
accurately identify the underlying semantics, we would need to effectively use more (and perhaps,
deeper) structure, such as the sound context, while making inferences.
In this paper, we present a novel hierarchical, generative framework that can be used for deeper
analysis of audio, and which attempts to model the underlying process that humans use in analyzing audio. Further, since most audio datasets do not contain detailed hierarchical labels that our
framework would require, we present unsupervised formulations for two layers in this hierarchical
framework, building on previous work for the first layer, and developing a model for the second.
1
However, since detailed annotations are not available, we cannot directly evaluate the induced structure on test data. Instead, we use features derived from this structure to characterize audio, and
evaluate these characterizations in a large-scale audio retrieval task with semantic categories, where
our model significantly improves over state-of-the-art baselines. A further benefit of the induced
structure is that the generated segments may be used for annotation by humans, thus removing the
need for the annotator to scan the audio to identify and mark segment boundaries, making the annotation process much faster [13].
In Section 2, we introduce a novel framework for mapping acoustics to semantics for deeper analysis
of audio. Section 3 describes the process of learning the lower-level acoustic units in the framework,
while Section 4 describes a generative model that automatically identifies patterns over and segments
these acoustic units. Section 5 describes our experiments and results, and we conclude in Section 6.
2
A Hierarchical Model for (Audio) Perception
The world around us is structured in space and time, and the evolution over time of naturally occurring phenomena is related to the previous states. Thus, changes in real-world scenes are sequential
by nature and the human brain can perceive this sequentiality and use it to learn semantic relationships between the various events to analyze scenes; e.g. the movement of traffic and people at an
intersection are governed by the traffic laws. In this section, we present a hierarchical model that
maps observed scene characteristics to semantics in a hierarchical fashion. We present this framework (and our experiments) in the context of audio, but it should apply to other modalities (e.g.
video), that require semantic analysis of information sequences.
Traditional detection-based approaches, that assign each frame or a sequence of frames of prespecified length to sound categories/clusters, are severely limited in their ability to account for context. In addition to context, we need to consider the possibility of polysemy in sounds? semantically
different sounds may be acoustically similar; e.g. a dull metallic sound may be produced by a hammer striking an object, a baseball bat hitting a ball, or a car collision. The sound alone doesn?t
provide us with sufficient information to infer the semantic context. However, if the sound is followed by applause, we guess the context to be baseball, screams or sirens suggest an accident, while
monotonic repetitions of the metallic sound suggest someone using a hammer. In order to automatically analyze scenes better, we need more powerful models that can handle temporal context.
In Figure 1a, we present a conceptual representation of a hierarchical framework that envisions a
system to perform increasingly complex analysis of audio. The grey circles closest to the observed
audio represent short-duration lower-level acoustic units which produce sounds that human ears can
perceive, such as the clink of glass, thump produced by footsteps, etc. These units have acoustic
characteristics, but no clear associated semantics since the semantics may be context dependent.
Sequences of these units, however, will have interpretable semantics? we refer to these as events
marked by grey rectangles in Figure 1a. The annotations in blue correspond to (usually unavailable)
human labels for these events. Further, these events themselves likely influence future events, shown
by the arrows, e.g. the loud cheering in the audio clip is because a hitter hit a home run.
Figure 1b shows the kind of structured information that we envision parsing from the audio. The
lowest level, indexed by a, correspond to the lower-level units. The event layer in Figure 1b has
been further divided into 2, where the lower level (indexed by v) correspond to observable events
(e.g. hit-ball, cheering), whereas the higher level (e) corresponds to a semantic event (e.g. battingin-run), and the root node represents the semantic category (baseball, in this case). The cost of
obtaining such hierarchical annotations would be very high due to the complexity of the annotation
task. Typically, audio datasets contain only a category or genre label for each audio file. As a result,
models for learning such structure must be able to operate in an unsupervised framework.
This framework for semantic analysis of audio is the first effort to extract deeper semantic structure,
to the best of our knowledge. In this paper, we deal only with the 2 lowest levels in Figure 1b.
We build on previous work to automatically learn lower level units unsupervised from audio data
[14]. We then develop a generative model to learn event patterns over the lower-level units, which
correspond to the second layer in Figure 1b. We represent the audio as a sequence of 39-dimensional
feature vectors, each comprising 13 Mel-Frequency Cepstral Coefficients and 13-dimensional? and
?? features.
2
Figure 1: Conceptual representation of the proposed hierarchical framework (a) Left figure: Conceptualizing increasingly complex semantic analysis; (b) Right figure: An example semantic parse
for baseball .
3
Unsupervised Learning of the Acoustic Unit Lexicon
At the the lowest level of the hierarchical structure specified by the model of Figure 1a is a sequence
of atomic acoustic units, as described earlier. In reality, the number of such acoustic units is very
large, possibly even infinite. Moreover, annotated training data from which they may be learned are
largely unavailable.
For the task of learning a lexicon of lower-level acoustic units, we leverage an unsupervised learning
framework proposed in [14], which employs a the generative model shown in Figure 2 to describe
audio recordings. We define a finite set of audio symbols A, and corresponding to each symbol a ?
A, we define an acoustic model ?a , and we refer to the set of all acoustic models as ?. According
to the model, in order to generate a recording, a transcription T comprising a sequence of symbols
from A is first generated, according to a language model (LM) distribution with parameters H.
Thereafter, for each symbol at in T , a variable-length audio segment Dat is generated in accordance
with ?at . The final audio D comprises the concatenation of the audio segments corresponding to all
the symbols in T . Similar to [14], we represent each acoustic unit as a 5-state HMM with gaussian
mixture output densities.
The parameters of the model may be learnt using an iterative EM-algorithm shown in Algorithm 1.
The learnt parameters ?a for each symbol a ? A allow us to decode any new audio file in terms of
the set of symbols. While these symbols are not guaranteed to have any semantic interpretations, we
expect them to capture acoustically consistent phenomena, and we see later that they do so in Figure
7. The symbols may hence be interpreted as representing generalized acoustic units (representing
clusters of basic sound units). As in [14], we refer to these units as ?Acoustic Unit Descriptors? or
AUDs.
Algorithm 1 Algorithm for Learning Acoustic Unit Lexicons ? (r + 1)-th iteration. Di : the i-th
audio file; Ti : Di ?s s transcript in terms of AUDs; ?: set of AUD parameters and H: the LM
Tir+1
r+1
?
argmaxT P (T |Di ; H r ; ?r )
Y
= argmax?
P (Di |Tir+1 ; ?)
=
(1)
(2)
Di
H r+1
=
argmaxH
Y
Di
3
P (Tir+1 ; H)
(3)
Figure 2: The generative model for generating audio from the acoustic units. H and ? are the language model and acoustic model parameters, T is
the latent transcript and D is the observed data.
4
Figure 3: The unigram based generative model for
segmentation. Only cn1 is observed.
A Generative Model for Inducing Patterns over AUDs
As discussed in Section 2, we expect that audio data are composed of a sequence of semantically
meaningful events which manifest themselves in various acoustic forms, depending on the context.
The acoustic unit (AUD) lexicon described in Section 3 automatically learns the various acoustic
manifestations from a dataset but do not have interpretable semantic meaning. Instead, we expect to
find semantics in the local patterns over the AUDs. In this section, we introduce a generative model
for the second layer in Fig 1a where the semantically interpretable acoustic events generate lower
level AUDs (and thus, the observed audio).
The distribution of AUDs for a specific event will be stochastic in nature (e.g. segments for a
cheering event may contain any or all of claps, shouts, speech, music), and the distribution of the
events themselves are stochastic and category-dependent. Again, while the number of such events
can be expected to be very large, we assume that for a given dataset, a limited number of events can
describe the event space fairly well. Further, we expect that the distribution of naturally occurring
events in audio will follow the power law properties typically found in natural distributions [15, 16].
We encode these intuitions in a generative model where we impose a power-law prior on the distribution of events. Events, drawn from this distribution, then generate lower level acoustic units
(AUDs) corresponding to the sounds that are to be produced. Because this process is stochastic, different occurrences of the same event may produce different sequences of AUDs, which are variants
of a common underlying pattern.
The generative model is shown in Figure 3. We assume K audio events in the vocabulary, and M
distinct AUD tokens, and we can generate a corpus of D documents as follows: for each document
d, we first draw a unigram distribution U for the events based on a power-law prior ?. We then draw
Nd event tokens from the distribution for the events. Each event token can generate a sequence of
AUDs of variable length n, where n is drawn from an event specific distribution ?. n AUDs (cn1 )
are now drawn from the multinomial AUD-emission distribution for the event ?event . Thus, in this
model, each audio document is a bag of events and each occurrence of an event is a bag of AUDs;
the events themselves are distributions over AUDs.
At training time, only the AUD token sequences are observed. We referring to the observed AUD
tokens as as X , the latent variables as Z and the parameters for our process (?, ? and ?) as ?,
we can write the joint probability of all the variables in this model as shown in Equation 4. In the
following subsections, we will outline a framework for training the parameter set for this model. We
can then use these parameters to estimate the latent events present in audio based on an observed
AUD stream (the AUD stream is obtained by decoding audio as described in Section 3).
P (X , Z, ?) =
Y
d
P(Ud ; ?)
Y
d
n
d
d
P(wid |Ud )P(nd
i |wi ; ?)P(c1 |wi , ni ; ?)
(4)
i
This formulation of unsupervised event induction from AUD streams bears some similarities to
approaches in text processing for discovering word boundaries and morphological segmentation
[17?20] where a token stream is input to the system and we wish to learn models that can appro4
priately segment new sequences. Unlike those, however, we model each event as a bag of AUDs
as opposed to an AUD sequence for two reasons. First, AUD sequences (and indeed, the observed
audio) for different instances of the same event will have innate variations. Second, in the case of the
audio, presence of multiple sounds may result in noisy AUD streams so that text character streams
which are usually clean are not directly analogous; instead, noisy, badly spelt text might be a better
analogy.
We chose the 2-parameter (r, p) Negative Binomial (Equation 5) distribution for ?, which approaches the Poisson distribution as r tends to infinity, and the r controls deviation from the Poisson.
The power law prior is imposed by a 1-parameter (s) distribution shown in Equation 6 (w(k) represents the k-th most frequent word), where the parameter s is drawn from N (?, ? 2 ). For English
text, the value of s has been observed to be very close to 1.
n ? N B(r, p), s.t. P (n = k) =
k+r?1 k
p (1 ? p)r
k
(5)
1
s
k
P (w(k) ; s, n) = Pi=n
(6)
1
i=1 is
Various methods can be used for parameter learning. In Section 4.1, we present an HMM-like
model that is used to estimate the parameters in an Expectation-Maximization (EM) framework [21].
Section 4.2 describes how the learning framework is used to update parameter estimates iteratively.
4.1
Latent Variable Estimation in the Learning Framework
Figure 4: An example automaton for a word of
maximum length 3. a, b, c and d represent the
probabilities of lengths 0 to 3 given the parameters
r and p for the negative binomial distribution.
Figure 5: An automaton with the K word automatons in parallel for decoding a token stream
We construct an automaton for each of the K events? an example is shown in Figure 4. This example
allows a maximum length of 3, and has 4 states for lengths 0 to 3 and a fifth dummy terminal state.
The state for length 01 behaves as the start state, while F is the terminal state. An AUD is emitted
whenever the automaton enters any non-final state. The transition probabilities in the automaton are
governed by the negative binomial parameters for that event. Based on these, states can skip to the
final state, thus accounting for variable lengths of events in terms of number of AUDs. We define
S as the set of all start states for events, so that Si =start state of event i. Since we model event
occurrences as bags of AUDs, AUD emission probabilities are shared by all states for a given event.
The automatons for the events are now put together as shown in Figure 5? the black circle represents
a dummy start state, and terminal states for each event can transition to this start state. Pd (wi )
represents the probability of the event wi given the unigram distribution for the document d. Now,
given a sequence of observed tokens, we can use the automaton in Figure 5 to compute a forward
table and backward table, in exactly the same manner as in HMMs. At training time, we combine
the forward and backward tables to obtain our expected counts, while at test time, we can use the
Viterbi algorithm to simply obtain the most likely decode for the observation sequence in terms of
the latent events.
1
We do not permit length 0 in our experiments, instead forcing a minimum length
5
Let us refer to the forward table as ? where ?(i, t) = P (state = i, t|ct1 ), and let ? refer to the
backward table where ?(i, t) = P (state = i, t|cnt+1 ). We can compute the likelihood of being in
state i (and extend that to being in word i) at time-step t given the entire observation sequence:
P (state = i, t)
=
P (wi , t)
=
?(i, t) ? ?(i, t)
P
j ?(j, t) ? ?(j, t)
P
k?wi ?(k, t) ? ?(k, t)
P
j ?(j, t) ? ?(j, t)
(7)
(8)
The forward-backward tables are constructed with our current estimates and the sufficient expected
counts are obtained using these estimates, which are then used to update the parameters.
4.2
Parameter Estimation
We obtain the EM-update equations by maximizing the (log-)likelihood from Equation 4. The
forward-backward tables for each AUD stream are used to obtain the sufficient counts, as described
in Section 4.1. To update the AUD emission probabilities, ?ij (AUD j emitted by event i), we use:
E ? Step :
X
r
P (Z|X ; ? )?ij (Z) =
T
X
P (wi , t)I(ct = j)
(9)
t=1
Z
P hP
M ? step
:
i
P (Z|X ; ?r )?ij (Z)
hP
i
?ij = P
M P
r )? (Z)
P
(Z|X
;
?
ij
j=1
d
Z
d
Z
(10)
Here, ?ij (Z) refers to the count of character j emitted by word i in the latent sequence Z. I(ct = j)
represents an indicator function that is 1 when the the token at time-step t is j, and 0 otherwise.
To update the NB parameters for each event, we compute the top-N paths through each training
sequence in the E-step (we used N = 50, but ideally, N should be as large as possible). Thus, if for
word i, we have a set of m occurrences in these paths of lengths n1 , n2 , ..., nm , we can estimate r
and p using Equation 11. p has a closed form solution (Eqn 12) but Eqn 13 for r needs an iterative
numerical solution. [?() is the digamma function]
m
Y
L=
N B(x = ni ; r, p)
(11)
i=1
p=
m
X
Pm ni
i=1 m
P
m
r + i=1 nmi
(12)
?(ni + r) ? m ? ?(r) + m ? ln(
i=1
r+
r
Pm
ni
i=1 m
)=0
(13)
To estimate the N (?, ? 2 ) for the power-law parameter s, we compute expected event frequencies
Efi for all events for each AUD stream. This can be done using the forward-backward table as
shown in Equation 14 and 15. The Zipf parameter is estimated as the slope of the best-fit line
between the log-expected-frequencies (Y ) and log-rank (X = [log rank 1]T ). The set of s values
in the corpus are used to estimate the ? and ? 2 .
E ? Step : count(wi ) =
T
X
P (state = Si , t)
(14)
t=1
count(wi )
Efi = P
j count(wj )
(15)
M ? step : sd = (Y X + )0 , ?d ? D
?? , ? 2? = arg max
2
?,?
6
i=D
Y
i=1
(16)
P (si |N (?, ? 2 ))
(17)
Figure 6: Oracle Experiment 1 emission distribution (L) True distribution; (R) Learnt distribution
Figure 7: Instances of log-spectra for 2 AUDs, with all occurrences across files concatenated. (L)
Predominantly music; (R) Predominantly speech. The y-axis correspond to frequency bins
5
Experiments
In this section, we present a pair of oracle experiments to verify that the segmentation model performs as expected. Then, we present results using the 2-level hierarchical model on the event kit of
the 2011 TRECVID Multimedia Event Detection (MED) task [22].
Oracle Experiment 1: We picked five words {the, cat, ate, blue, man} and used our model to
generate 100 documents (s ? N (1.5, 0.2)), and use this as the data to learn words unsupervised.
The results of learning are sensitive to initialization as is expected with EM-based algorithms, but
end up fairly close to the true parameters. The learned words are distributions over characters, and
we would expect the algorithm to recover a distribution close to the true distribution. Figure 6
compares the true and learned distributions over characters for the five words. We notice that the
learned one is not as sparse, but the higher-probability characters are very similar. To quantify the
error, the average error per parameter is 0.003 averaged over 5 random initializations.
Oracle Experiment 2: We performed a similar experiment with web URLs concatenated together,
since web URLs have a clear structure beginning with ?http://? or ?www? and containing ?.com?.
Again, the learner automatically identified the most frequent word to be one which had highest
emission probabilities for {?.?, ?c?, ?o?, ?m?} and the second most frequent word with {?h?, ?t?, ?p?,
?/?, ?:?, ?w?} characters having high probabilities. The respective segments identified for those words
while decoding conforms to our expectations and usually corresponded to ?.com? and ?http://www?.
TRECVID 2011 MED data: The MED data was used for our experiments, and contains 15 semantically defined categories (e.g. board trick, feeding an animal? full list at [22]), and was to be used to
build detectors for each semantic category, so that given a new file, it can predict whether it belongs
to any of those categories or not. All our reported results use 8-fold cross validation on the entire
event kit. Performance is evaluated using the Area Under the Missed Detection Rate against False
Alarm Rate curve (AUC, henceforth) for each category (hence, the lower the better).
The entire data was used to learn a lexicon of AUDs, and each file was decoded using the models.
Figure 7 shows two examples of log-spectra of all frames spanned by an AUD concatenated together.
In both cases, one can see structural consistency, showing that the AUDs find acoustically similar
segments as desired. While the AUDs are not required to have clear semantic interpretations, listening to the concatenated instances shows that the AUD on the left primarily spans music segments
while the right consists primarily of speech? speech formant structures are visible in the image.
We then use the decoded AUD sequences as character streams to learn parameters for the second
layer of observable acoustic events spanning local AUD sequences. Based on the learnt models, we
can decode the data in terms of these events. Since there are no annotations available, these events
7
Table 1: Performance summary across MED11 dataset (lower is better)
System
Average AUC
Best Performance in #categories
#SSI-over-VQ
#SSI-over-FOLEY
#SSI-over-AUD-FREQ
#SSI-over-EVENT-FREQ
VQ
0.2971
0
0
0
0
0
FOLEY
0.2624
0
5
0
0
0
AUD-FREQ
0.2174
2
15
15
0
11
EVENT-FREQ
0.2297
1
12
11
2
0
COMB
0.1842
12
15
15
12
15
are not assigned semantics, but listening to multiple instances concatenated together shows similar
phenomena being captured. One such event consists of sequences of sounds that relate to crowds
with loud cheering and a babble of voices in a party being subsumed within the same event.
We use the decoded AUDs and event sequences for each file to characterize the MED11 data,
and evaluate the effect of using the AUDs layer and the event layer individually (AUD-FREQ and
EVENT-FREQ, respectively) and together (COMB). For the AUD-FREQ characterization, each file
is represented by a k-dimensional feature vector (one for each of the k AUDs in the vocabulary, with
the frequency of occurrence of the AUD in the file being the feature value). The EVENT-FREQ
characterization is similarly set up while COMB is a concatenation of the two.
We compare our models with two baselines that are commonly used in such tasks. The first is a VQ
baseline (VQ) where a set of audio words is learned unsupervised by applying K-Means on the data
at the frame level. The second uses an audio library to create a supervised sound library from the
480 sound types in the Foley Sound Library [23], and we characterize each file using occurrence
information of these sounds in the file (FOLEY).
These feature representations were used to train a random forest classifier [24] with 500 trees for
each class. Table 1 summarizes the performance of the various feature settings on the MED11 data
(lower AUC is better). We used the best performing lexicon size for the various systems? 4096 clusters for the VQ, 480 Foley audio events, 1024 AUDs, and 128 acoustic events. The AUC numbers
reported are averaged across all 15 categories in the dataset. The last 4 rows of the column indicate
how feature sets compare pairwise by noting the number of categories in which one improves over
the other with statistical significance (p < 0.05, in a paired t-test). For some events, the performance
of different feature settings is not statistically significantly different. We observe that, overall, the
AUD-FREQ feature set seems to perform the best by itself, but combining it with the second layer
of events in our hierarchical framework results in the best overall system on average AUC, and the
combination outperforms AUDs alone on 12 of the 15 categories with statistical significance.
6
Conclusion and Future Work
Although the results above only show gains obtained in objective evaluations on a standard largescale retrieval tasks, the ?events? discovered by the learning algorithm have deeper significance?
they represent automatically learned characterizations of longer-scale acoustic phenomena with semantic import. This work presents an initial approach to extracting such deeper semantic features
from audio based on local patterns of low-level acoustic units. There are a few directions we hope
to explore in the future. Since the discovered latent events and acoustic units do not have true labels,
we would like to explore ways to leverage tags, knowledge bases and human annotators to induce
labels. In such settings, we would like to explore non-parametric techniques that can grow the event
set based on data. The distribution characteristics here are simple unigrams without additional structure; we would like to explore other models with appropriate priors for each layer in the hierarchical
model. Finally, we would like to use such event structure to study co-occurrences and dependencies
of acoustic event types that might allow us to predict sounds in the future based on the context.
Acknowledgments
This work was supported by funding from Charles Stark Draper Labs, Cambridge, USA.
8
References
[1] E. Wold, T. Blum, D. Keislar, and J.W. Wheaton. Content-based classication, search, and retrieval of
audio. IEEE Multimedia, 3:27?36, 1996.
[2] A.G. Hauptmann and M.J. Witbrock. Informedia: News-on-demand multimedia information acquisition
and retrieval. In Proceedings of Intelligent Multimedia Information Retrieval, pages 213?239. AAAI
Press, 1997.
[3] G. Guo and S.Z. Li. Content-based audio classication and retrieval by support vector machines. IEEE
Transactions on Neural Nets, 14, 2003.
[4] M. Slaney. Mixture of probability experts for audio retrieval and indexing. In Proceedings of the International Conference of Multimedia and Expo, 2002.
[5] M. Slaney. Semantic audio retrieval. In Proceedings of the International Conference on Acoustic Speech
and Signal Processing, 2002.
[6] S.F. Chang, D. Ellis, W. Jiang, K. Lee, A. Yanagawa, A. Loui, and J. Luo. Large-scale multimodal
semantic concept detection for consumer video. In Proceedings of the MIR workshop, ACM-Multimedia,
2007.
[7] S. Sundaram and S. Narayanan. Classication of sound clips by two schemes: using onomatopoeia and
semantic labels. In Proceedings of the IEEE International Conference of Multimedia and Expo, 2008.
[8] Z. Liu, J. Huang, and Y. Wang. Classification of tv programs based on audio information using hidden
markov model. In Proceedings of the 2nd IEEE Workshop on Multimedia Signal Processing, 1998.
[9] S. Berrani, G. Manson, and P. Lechat. A non-supervised approach for repeated sequence detection in tv
broadcast streams. In Signal Processing: Image Communication, volume 23, pages 525?537, 2008.
[10] G. Friedland, L. Gottlieb, and A. Janin. Using artistic markers and speaker identification for narrativetheme navigation of seinfeld episodes. In Workshop on Content-Based Audio/Video Analysis for Novel
TV Services, 11th IEEE International Symposium on Multimedia, 2009.
[11] S. Kim, S. Sundaram, P. Georgiou, and S. Narayanan. Audio scene understanding using topic models. In
Proceedings of the NIPS Workshop on Applications for Topic Models: Text and Beyond, 2009.
[12] S. Kim, S. Sundaram, P. Georgiou, and S. Narayanan. Acoustic stopwords for unstructured audio information retrieval. In Proceedings of the 18th European Signal Processing Conference, 2010.
[13] X. Zhu. Semi-supervised learning with graphs. PhD Thesis, 2005.
[14] S. Chaudhuri, M. Harvilla, and B. Raj. Unsupervised learning of acoustic unit descriptors for audio
content representation and classification. In Proceedings of Interspeech, 2011.
[15] W. Li. Random texts exhibit zipf?s-law-like word frequency distribution. IEEE Transactions on Information Theory, 38:1842?1845, 1992.
[16] J. Eeckhout. Gibrat?s law for (all) cities. American Economic Review, 94:1429?1451, 2004.
[17] D. Mochihashi, T. Yamada, and N. Ueda. Bayesian unsupervised word segmentation with nested pitmanyor language modeling. In Proceedings of the 47th Meeting of the Association for Computational Linguistics, 2009.
[18] H. Poon, C. Cherry, and K. Toutanova. Unsupervised morphological segmentation with log-linear models.
In Proceedings of the 47th Meeting of the Association for Computational Linguistics, 2009.
[19] S. Goldwater, T.L. Griffiths, and M. Johnson. A bayesian framework for word segmentation: Exploring
the effects of context. Cognition, 112:21?54, 2009.
[20] M. Johnson and S. Goldwater. Improving nonparametric bayesian inference: Experiments on unsupervised word segmentation with adaptor grammars. In Proceedings of Human Language Technologies:
North American Chapter of the Association for Computational Linguistics, 2009.
[21] A. P. Dempster, N.M. Laird, and D.B. Rubin. Maximum likelihood from incomplete data via the EM
algorithm. Journal of the Royal Statistical Society, Series B, 39:1?38, 1977.
[22] TRECVID Multimedia Event Detection Task. http://www.nist.gov/itl/iad/mig/med11.cfm. 2011.
[23] The Art of Foley. http://www.sound-ideas.com/artfoley.html. 2005.
[24] L. Breiman. Random forests. Machine Learning, 45:5?32, 2001.
9
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4,046 | 4,662 | Learning Partially Observable Models Using
Temporally Abstract Decision Trees
Erik Talvitie
Department of Mathematics and Computer Science
Franklin & Marshall College
Lancaster, PA 17604
[email protected]
Abstract
This paper introduces timeline trees, which are partial models of partially observable environments. Timeline trees are given some specific predictions to make and
learn a decision tree over history. The main idea of timeline trees is to use temporally abstract features to identify and split on features of key events, spread arbitrarily far apart in the past (whereas previous decision-tree-based methods have
been limited to a finite suffix of history). Experiments demonstrate that timeline
trees can learn to make high quality predictions in complex, partially observable
environments with high-dimensional observations (e.g. an arcade game).
1
Introduction
Learning a model of a high-dimensional environment can pose a significant challenge, but the ability
to make predictions about future events is key to good decision making. One common approach is
to avoid learning a complete, monolithic model of the environment, and to instead focus on learning
partial models that are only capable of making a restricted set of predictions (for instance, predictions about some particular aspect of the environment, or predictions about future rewards). Partial
models can often be simpler to learn than a complete model. In some cases they can be combined
to form complete, structured models, which can then be used for planning purposes (e.g. factored
MDPs [1], collections of partial models [2]). In other cases, partial models can be directly useful for
control (e.g. U-Tree [3], prediction profile models [4]).
This paper introduces timeline trees which are partial models for partially observable environments.
Timeline trees are focused on capturing a particular kind of partial observability; they assume that
their predictions can be made by recalling a (finite) sequence of events in the past that may have
occurred far apart from each other in time. While not all partially observable phenomena take this
form, a good deal of everyday partial observability has this flavor. For instance, you may know that
your keys are in the next room because you remember putting them there. Most of the experiences
since that event are probably irrelevant to making predictions about the location of your keys.
The main idea of timeline trees is to build a decision tree over history. As with similar approaches,
the decision tree can split on features of observations in recent history. However, a timeline tree may
also establish new timestamps in the past and is able split on features of observations surrounding
those events as well. For instance, there could be a timestamp representing the last time the agent
saw its keys, and then features of the neighboring observations could identify the keys? location. In
this way, timeline trees can make use of information arbitrarily spread out in history.
1
2
Partial Models
This paper will focus on discrete dynamical systems. Specifically, time procedes in discrete steps
t = 1, 2, 3, . . .. At every step t, the agent selects an action at from a finite set A and the environment (stochastically) emits an observation ot , taken from a finite set O. The history at time
t is the sequence of actions and observations from the beginning of time, up through time t:
def
ht = a1 o1 a2 o2 . . . at ot . In the general partially observable case, the observation emitted at each
step may depend upon the entire history (and the agent?s action). So, an agent wishing to predict the
next observation must model the conditional probability distribution Pr(Ot+1 | Ht , At+1 ).
If one is able to predict the next observation at any history and for any action (that is, if one has
access to this conditional distribution), one can compute the probability of any future sequence of
observations given any future sequence of actions and the history [5]. Such a model is called a
complete model because in any situtation, it is capable of making any prediction about the future.
Examples in the partially observable setting include POMDPs [6, 7] and PSRs [5, 8]. A partial
model is any model that does not represent this full conditional distribution.
This paper will focus on partial models that make conditional predictions about abstract features of
the next observation, though many of the ideas can be straightforwardly adapted to work with predictions of other forms. Formally, let ? and ? be many-to-one mappings over the set of observations
O. The task of the partial model at time t will be to predict the value of ?(ot+1 ), conditioned on the
value of ?(ot+1 ). So it represents the distribution Pr(?(Ot+1 ) | Ht , At+1 , ?(Ot+1 )). For example,
in the experiment in Section 5.3, observations are images and multiple partial models are learned,
each predicting the color of a single pixel, conditioned on the colors of pixels above and to the left.
2.1
Related Work: Partial Models in Partially Observable Environments
McCallum?s U-Tree [3] learns a decision tree over history, where the leaves of the tree map to the
expected discounted sum of rewards at the associated history (though the method could be adapted
to make other predictions, as in [9]). McCallum used binary features of the form ?Feature X takes
value Y at time-step t ? k,? where t is the current time-step. Thus the decision tree learns an
abstraction both over observations (which could be high-dimensional in their own right) and over
the sequence of observations (by using features from multiple time-steps). However, because it can
only consider a finite number of such features, U-Tree has a finite memory horizon; events that occur
before some arbitrary cutoff in the past cannot be taken into account when making predictions.
Timeline trees are an extension of UTree that allow it to use features of observations arbitrarily far
back in the past, though they are not the first attempt to address this issue. Looping predictive suffix
trees (LPSTs) [10] are prediction suffix trees [11] that allow nodes to loop back to their ancestors.
Local agent state representations (LASR) [12] map histories to a real number, and then learn a
direct mapping from that number to the target predictions. McCallum [13] and Mahmud [14] both
developed incremental hill-climbing algorithms to learn finite state machines (FSMs), where each
state is associated with predictions about future rewards, and the transitions depend on both the
action taken and the observation received. Prediction profile models [4] are similar FSMs, but rather
than hill-climbing, they are learned by pre-processing the data and then applying standard complete
model learning methods (they were demonstrated using POMDPs and LPSTs).
All of these approaches can, in principle, represent arbitrarily long-range dependencies in time.
However, unlike U-Tree, they all treat observations as atomic, which limits their applicability to
truly high-dimensional systems. Furthermore, despite their theoretical representational capacity,
their learning algorithms have difficulty discovering long-range dependencies in practice.
The learning algorithm for LPSTs first learns a full suffix tree, and then adds loops as appropriate. Thus, to capture very long-range dependencies, one must first build a very deep suffix tree.
McCallum reported that his FSM-learning method was often unable to detect long-range temporal
dependencies (since this typically involves multiple elaborations of the FSM, none of which would
individually seem valuable to the hill-climbing algorithm). Mahmud?s similar approach would likely
suffer a similar limitation. The learning algorithms for LASR and prediction profile models both rely
on estimating predictions at particular histories. Because estimates will only be accurate for histories that appear many times, these algorithms can only be effectively applied to data consisting of
many short trajectories, which limits their ability to discover long-range dependencies in practice.
2
It should be noted that prediction profile models have been combined with an additional preprocessing step that learns an abstraction before the prediction profile model is learned [15]. Because
of this, and because their formulation most closely fits the setting of this paper, the experiments in
Section 5 will directly compare against the performance of prediction profile models.
3
Timeline Trees
The goal of timeline trees is to combine the strengths of U-Tree with the ability to attend to events
arbitrarily far apart in history (rather than limited to a finite suffix). Unlike several of the above
approaches, timeline trees are not arbitrarily recurrent (they do not contain loops except in a limited,
implicit sense), which does restrict their representational capacity. However, in exchange they retain
the straightfoward decision tree training of U-Tree, which allows them to simultaneously learn an
abstraction over both the history sequence and high-dimensional observations and which furthermore allows them to discover long-range temporal depencies in practice (and not just in principle).
3.1
Timestamps
The decision tree built by U-Tree splits on features of observations at some temporal offset from
the current timestep. The key idea of timeline trees is to allow multiple timestamps in history and
to allow splits on features of observations at temporal offsets relative to any of these timestamps.
Timeline trees take a set F of binary features, where each feature f (ht , k) takes the history at time t,
ht , and a timestep 0 < k ? t + 11 and returns 1 or 0. For example, if the observations are images, f
could return 1 if a black pixel existed anywhere at step k ? 1 but did not exist at step k. It is assumed
that f (ht , k) makes use of no information after step k (though it may access timestep k or before).
For a fixed vector ? of timestamps, the model is a standard decision tree, and only a small extension
of U-Tree (which fixed the number of timestamps to 1: the current timestep). Each internal node in
the tree is associated with a feature f , a timestamp index i, and a temporal offset ? (which may be
negative) and has two children representing histories where the value of f (ht , ? [i] + ?) is 0 or 1, respectively. The leaves of the tree are associated with estimates of Pr(?(Ot+1 ) | ht , at+1 , ?(Ot+1 )).
To use the timeline tree to make a prediction, one simply follows a path from the root to a leaf in the
tree, choosing the appropriate child at each node according to the feature value f (ht , ? [i] + ?).
Timeline trees? real strength lies in their ability to add new timestamps. They do this via a special
type of feature. For every feature f ? F, there is an additional timestamp feature ? f . The feature
? f (ht , j, k) = 1 if there is some timestep m such that j < m < k where f (ht , m) = 1. More
importantly, the greatest such m (that is, the time of the most recent occurence of f ), call it mf , is
added as a timestamp to all nodes in the subtree where ? f = 1.
When making a prediction for ht , one maintains a growing vector ? of timestamps (in order of least
to most recent). Beginning at the root there is only one timestamp: ?root = ht + 1i, where t is the
current timestep. As one travels from the root to a leaf, one may encounter a node associated with
timestamp feature ? f . Such a node is also associated with an index i into the current timestamp
vector. If ? f (ht , ? [i ? 1], ? [i]) = 1, the path moves to the corresponding child and adds mf to ?
def
(let ? [0] = ?1). Nodes further down in the tree may refer to this new timestamp. As such, the tree
is able to establish timestamps based on the occurence of key events (the presence of some feature).
Timestamp features are a form of temporal abstraction; they refer to an event in the past, but abstract
away how long ago it was. They are limited, however. There are systems that would require an infinite timeline tree that approaches in Section 2.1 can capture easily (see Section 5.2). Nevertheless,
they do capture a natural and intuitive form of partial observability, as can be seen by example.
3.2
An Example
As a simple illustrative example, consider an agent that must keep track of its key. The agent?s key
can be in room A or B, or in the agent?s pocket (where it starts). The agent has three actions: move
1
For simplicity?s sake, the notation f (ht , k) hides the fact that the features may also depend on at+1 and
?(ot+1 ). For this discussion, assume that k may equal t + 1 if the feature makes use of only these aspects of
time t + 1. If k = t + 1 and the feature refers to other information, assume f (ht , k) = 0.
3
Figure 1: The Key World example.
(which switches its location), stay (which does nothing), and pocket. The last action transfers the
key between the agent?s pocket and the current room (in either direction) unless the key is in neither
(in which case it does nothing). The agent can observe its location and whether the key is in the
room. A diagram is shown in the left of Figure 1 (missing arrows are self-loops).
On the right of Figure 1 an example timeline tree is shown that can predict whether the agent will
see the key in the next timestep. At the root, there is only one timestamp: t+1, where t is the current
step. The root checks if the agent can currently see the key. If so, the agent will only see the key
in the next step if it stays. Otherwise, the agent must remember where the key was last seen. The
square-shaped node is meant to indicate a timestamp feature, which checks if the agent has ever seen
the key before the only timestamp. If not, the key is in the agent?s pocket. If so, a new timestamp m
is added that marks the last time the key was seen. If the agent put the key in its pocket after m, it
must take the key out to see it. Otherwise, it must be in the other room.
4
Learning Timeline Trees
Timeline trees can be learned using standard decision tree induction algorithms (e.g. ID3 [16] and
C4.5 [17]). The leaves of the tree contain the estimated predictions (counts of the occurrences of the
various values of ?(o) associated with histories mapping to that leaf). The tree starts as just the root
(not associated with a feature). Each phase of training expands a single leaf by associating it with a
feature and adding the appropriate children under it. At each phase every candidate expansion (every
leaf and every feature) is tried and the one that results in the highest information gain between the
predictions of the original tree and the expanded tree is greedily selected.
The main difference in timeline trees is that different features may be available in different leaf
nodes (because different timestamps will be available). Specifically for each leaf n, all features of
the form f (?, ?n [i] + k) are considered for all timestamp indices i ? {1, . . . , |?n |} and all integer
offsets k in some finite range. Similarly, all timestamp features of the form ? f (?, ?n [i ? 1], ?n [i]) are
considered for all timestamp indices i. In the experiments below, candidate expansions also include
all combinations of timestamp features and regular features (essentially two expansions at once).
These compound features take the form of first splitting on a timestamp feature, and then splitting
the resulting ?1 child? with a regular feature. This allows the tree to notice that a timestamp is useful
for the subsequent splits it allows, even if it is not inherently informative itself. For instance, in
the Key World, knowing whether the agent has ever seen the key may not be very informative, but
knowing that the pocket action was taken immediately after seeing the key is very informative.
Note that compound features will tend to result in higher information gain than simple features. As a
result, there will be a bias toward selecting compound features, which is not necessarily desireable.
To combat this, the information gain of compound features was penalized by a factor of ?. In
the experiments below, ? = 0.5. Also note that because the information gain measurement used to
choose expansions is estimated from a finite number of samples, expanding the tree until information
gain is zero for all candidates will typically result in overfitting. Thus, some form of early stopping
is common. In this implementation expansions are only considered if they make a statistically
significant change to the predictions (as measured by a likelihood ratio test). The statistical test
requires a significance level, ?, which controls the probability of detecting a spurious difference.
Applying the test several times to the same data set compounds the danger of such an error, so ?
should be set quite low. In the experiments below, ? = 10?10 .
4
(a) Shooting Gallery
(b) Three Card Monte
(c) Snake
Figure 2: Experiment Domains
5
Experiments
In this section, timeline trees will be evaluated in three problems to which prediction profile models
have been previously applied. In each problem a set of features and a set of training trajectories are
provided. For various amounts of training trajectories, timeline trees are learned and their prediction
accuracy is evaluated (as well as their usefulness for control). Results are averaged over 20 trials.
Note that for prediction profile models a completely new model is learned for each batch of training
trajectories. For timeline trees, the new data is simply added to the existing tree and new splits are
made until the algorithm stops. This strategy is effective for timeline trees since the initial splits can
often be made with relatively little data (this not possible for prediction profile models).
In addition to evaluating timeline trees, two variants will also be evaluated. One (labeled ?Finite
Suffix?) does not use any timestamp features at all. Thus, it is similar to U-Tree (splitting on features
of a finite suffix of history). The other (labeled ?No Timestamps?) includes timestamp features,
but does not use them to create new timestamps. This variant is meant to evaluate whether any
performance benefit is due to the form of the features or due to the addition of new timestamps.
5.1
Shooting Gallery
In this example, from Talvitie and Singh [4], the agent is in a shooting gallery (see Figure 2(a)). Its
gun is aimed at a fixed position (marked by the ?X?) and it must shoot a target that moves around
the grid, bouncing off the edges and obstacles (an example trajectory is pictured). If the target is in
the crosshairs in the step after the agent shoots, the agent gets a reward of 10. Otherwise it gets a
reward of -5. Whenever the agent hits the target, the gallery resets (obstacles are placed randomly)
and an special observation is emitted. The gallery may also reset on its own with a 0.01 chance.
Clearly the agent must predict whether the target will be in the crosshairs in the next timestep, but the
target?s movement is stochastic and partially observable. At every step it either moves in its current
direction with probability 0.7 or stays in place with probability 0.3. The agent must remember the
direction of the ball the last time it moved. This problem is also fairly high-dimensional. There are
roughly 4,000,000 possible observations, and even more hidden states. Because of the large number
of observations Talvitie and Singh [4] hand-crafted an observation abstraction and applied it to the
training data before learning the prediction profile models. Their abstraction pays attention only to
the position of the target and the configuration of the obstacles in its immediate neighborhood. By
constrast, timeline trees learn an abstraction over both observations and the history sequence.
Experimental Setup: The prediction profile models were trained on trajectories of length 4, generated by the uniform random policy. Though short trajectories are necessary for training prediction
profile models, the timeline trees tended to overfit to the short trajectories. In short trajectories, a
feature like ?Has the target ever been in the crosshairs?? might seem spuriously meaningful. During testing, which takes place on one long trajectory, this feature would be much less informative.
Therefore, the tree models were trained on fewer, longer trajectories (of length 40).
To train the tree models, a binary feature was provided for each color (target, obstacle, background,
or reset) for each pixel in the image. There was also a feature for each action. The maximum
temporal offset from a timestamp was set to 2.
The learned models are evaluated by using their predictions as features for a policy gradient algorithm, OLGARB [18]. Good predictions about the color under the cross-hairs should lead to a good
policy. For the details of how the predictions are encoded for OLGARB, see [4]. To evaluate the
learned models, OLGARB is run for 1,000,000 steps. The average reward obtained and the root
5
mean squared error (RMSE) of the probabilities provided by the model are reported (at each step,
the model?s probability that the target will be in the crosshairs is compared to the true probability).
Both the timeline trees and the prediction profile models are
able to learn to make good predictions, but timeline trees do so
with less data. Remember that timeline trees are learning from
raw images whereas the prediction profile models have been
provided a hand-crafted abstraction. The tree models without
timestamps are only able to make good predictions in histories
where the target has recently moved, which limits their performance. The ?No Timestamp? variant is outperformed by the
?Finite Suffix? model, which indicates that, despite the longer
training trajectories, it may still be overfitting.
5.2
Avg. Reward (20 Trials)
Control Performance
0.025
Optimal
True
0.02
0.015
TimelineTree
Prediction Profile
Finite Suffix
No Timestamps
0.01
0.005
0
0
1e+06
2e+06
3e+06
4e+06
# Training Steps
0.09
Avg. RMSE (20 Trials)
Results: Figure 3 shows the results. The line marked ?Prediction Profile? shows the best results reported by Talvitie and
Singh [4]; the other curves show the performance of timeline
trees and the comparison variants. In the control performance
graph, the dashed line marked ?Optimal? shows the average
performance of the optimal policy. The dashed line marked
?True? shows the average performance of OLGARB when
given the true predictions as features. This is the best performance a learned model could hope for.
Prediction Performance
0.08
TimelineTree
Prediction Profile
Finite Suffix
No Timestamps
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
1e+06
2e+06
3e+06
4e+06
# Training Steps
Figure 3: Shooting gallery results.
Three Card Monte
The next example, also from Talvitie and Singh [4], is one for which the decision tree approach
would not be appropriate. While illustrating the limitations of timeline trees in comparison to more
expressive methods, it also demonstrates that they can represent useful knowledge that the simpler
tree-based methods cannot. The problem is based on the simple game ?Three Card Monte?. There
are three face down cards on the table, one of which is the ace. A dealer repeatedly chooses two
cards and swaps their positions. Eventually the dealer asks the agent to flip over the ace. If the agent
succeeds, it gets a reward of 1; if it fails it gets a reward of -1. For a detailed specification, see [4].
Note that to do well in this game, the agent only needs to make the prediction, ?If I flip card 1, will
it be the ace?? (and the corresponding predictions for the other 2 cards) at any history. It does not,
for instance, need to predict which cards will be swapped in the next time step. A complete model
would attempt to make this prediction, which would mean not only modeling the movement of the
cards, but also the decision making process of the dealer! The dealer in these experiments choses
the swap it has chosen least frequently so far with probability 0.5. With probability 0.4 it choses
uniformly randomly between the other two swaps. With probability 0.1, it asks for a guess. Since
modeling the dealer?s behavior requires counting the number of times each swap has been selected,
a complete POMDP model of this system would require infinitely many states.
Further note that the entire sequence of swap observations since the last time the ace?s position was
observed is important for predicting the ace?s location. Since timeline trees? primary strength is
ignoring sections of history to focus on a few key events, they would not be expected to model this
problem well. Prediction profile models, on the other hand, are able to track the ace?s location with
a 3-state machine (pictured in Figure 2(b)).
Experimental Setup: Training and evaluation were the same as in the Shooting Gallery (above)
except the prediction profile models were given length 10 trajectories and the tree-based models
were given length 100 trajectories. The features provided to the trees were encodings of the atomic
actions and observations. There was a binary feature indicating each observation, action, and each
action-observation pair. The maximum time offset from a timestamp was 10 steps (both positive
and negative). The specification of the prediction profile models implicitly encodes the fact that the
agent?s action at+1 is important to the predictions (i.e. which card it flips). For fairness, the tree
models were also seeded with these features (they were split on the agent?s action before training).
Results: Figure 4(a) presents the control performance results and Figure 4(b) shows the prediction
error results. The prediction profile models are able to perfectly track the ace?s location after 100,000
6
0.7
Optimal
True
0.6
0.4
Prediction Profile
TimelineTree
No Timestamps
Finite Suffix
0.2
0
-0.2
-0.4
Prediction Performance
0.6
0.5
0.4
Prediction Profile
TimelineTree
No Timestamps
Finite Suffix
0.3
0.2
0.1
0
-0.1
-0.2
0
250000
500000
# Training Steps
(a)
750000
0
250000
500000
Control (Expert Training)
Avg. Reward (20 Trials)
Control Performance
Avg. RMSE (20 Trials)
Avg. Reward (20 Trials)
1
0.8
750000
# Training Steps
(b)
TimelineTree (Expert)
No Timestamps (Expert)
Finite Suffix (Expert)
TimelineTree
No Timestamps
Finite Suffix
0.15
0.1
0.05
0
-0.05
-0.1
0
2.5e+07
5e+07
7.5e+07
1e+08
# Training Steps
(c)
Figure 4: Results in Three Card Monte.
training steps. As expected, the tree methods perform poorly (negative average reward indicates
more wrong guesses than right), though timeline trees have marginally better control performance.
Part of the difficulty is that randomly generated training data is quite different than what the agent
will encounter during testing (the random agent flips cards over frequently, while the learning agent
eventually flips a card only when prompted to). Figure 4(c) shows the control performance of the
tree models trained with expert-generated data instead (generated by the optimal policy). The dashed
lines show the results for random training data for comparison. Expert-trained timeline trees are
eventually good enough to allow the agent to achieve positive average reward, though they do require
a great deal of data to do so (note the changes to the axes). Though the expert training improves the
performance of the limited variants as well, neither achieves positive average reward. So, though
their representational limitations do prevent all three tree-based methods from performing well in
this problem, timeline trees? ability to create new timestamps seems to allow them to make some
meaningful (and useful) predictions that the others cannot.
5.3
Snake
The final example is an arcade game called ?Snake? from Talvitie [15] (see Figure 2(c)). In this
problem, multiple partial models will be learned and combined to form a complete, structured model
which can be used for planning. The agent controls the head of a snake. The snake?s body trails
behind, and the tail does exactly what the head did, at a delay. There are 10 food pellets on the screen
and the goal is to eat them in a particular order (indicated by the shades of grey in Figure 2(c)). If
the snake ever runs into the wrong pellet, its own body, or the edge of the screen, the game is over
and the agent receives -0.01 reward. Whenever the snake eats a pellet, the agent gets 1 reward and
the tail stays still for 5 timesteps, making the snake?s body longer. In addition, there is a 0.2 chance
each step that the tail will not move, so the snake is always growing, imposing some time-pressure.
This version of Snake has two sources of partial observability. The tail shadows the head and, in
addition, the pellet the snake must eat next is invisible. Initially, all 10 pellets are shown, but when
the first pellet is eaten, the next one disappears, and only reappears if the snake?s head is adjacent
to it. When that pellet is eaten, the next pellet disappears, and so on. To do well, the agent must
remember the location of the next pellet. The observations are 20 ? 20 images. There are over 1030
distinct possible observations and even more hidden states.
Experimental Setup: For every location (x, y) and every color c, a timeline tree model was used
to predict whether the pixel at (x, y) would next be color c. These models jointly predict the entire
next observation. The models were ordered, and each model could condition on outcomes predicted
by the previous models (so ? gave the portion of the image predicted by models earlier in the order).
In this case, the models were ordered by color, then by location (column-major order). The color
order was head, tail, body, the 10 food pellet colors (in increasing order), and finally background.
For each color, training data was pooled data across all locations (rather than learn a separate model
for each coordinate). However, features were provided that gave the position of the model (there
was a binary feature for each column and each row), so the tree could choose to attend to position
information if necessary. There was also a binary feature for each action and several pixel-based
features. There was a feature for each color of each pixel in a 5 ? 5 square around the pixel being
predicted. There were also features for the same square of pixels indicating whether each pixel had
just changed from one color to another color (for all pairs of colors). Finally, there were features
indicating whether a particular color or particular color change existed anywhere in the image.
7
Prediction profile models were applied to this problem by Talvitie [15]. Similarly, multiple prediction profile models were learned, each responsible for a particular prediction in particular situations
(called ?histories of interest?). For instance, one model type predicted whether a particular pixel
would contain the head in the next timestep, but only when the head was in the immediate neighborhood of that pixel. Before training the prediction profile models, an abstraction learning process was
applied to the data that mapped each action-observation sequence between histories of interest to a
single abstract observation. Thus, even though the raw data might contain very long trajectories, the
abstract data consisted of only short trajectories. The reader is referred to Talvitie [15] for a detailed
description of this approach. The main thing to note is that the hand-crafted structure indicates to
each model which key events it should attend to, and which stretches of history can be ignored. By
contrast, timeline trees learn this information.
As in [15], the training data was generated by running UCT [19] (a sample-based planning algorithm) on the true model (with a 0.25 chance of taking a random action). Each training trajectory
is a full game (typically a few hundred steps long). The learned set of partial models was evaluated
collectively. The joint model was given to UCT and its average planning performance over 100 test
games was compared to that of UCT with a perfect model. The probability given by the model for
the observation at each step was compared to the true probability and the RMSE has been reported.
In terms of control performance, the ?No Timestamps? variant performs nearly identically to the full timeline tree. This
is because they are equally good at tracking the invisible food
pellet. The model checks if each pixel has ever contained a
food pellet and if it has ever contained the head. If ?yes? and
?no,? respectively then it must contain food. This can be expressed without creating new timestamps. However, the ?No
Timestamps? model cannot fully represent a model of the tail?s
movement (which requires remembering what the head did
when it was in the tail?s position). The timeline tree incurs
substantially less prediction error, indicating that it is able to
model the tail more accurately.
6
Control Performance
Avg. Score (20 Trials)
12
TimelineTree
No Timestamps
Prediction Profile
Finite Suffix
10
True
8
6
4
2
0
0
2000
4000
6000
8000
10000
# Training Trajectories
0.5
Avg. RMSE (20 Trials)
Results: The results are shown in Figure 5. Despite the
hand-crafted structure provided to the prediction profile models, timeline trees learn higher quality models with less data.
In fact, UCT appears to perform better using the timeline tree
model than the true model (marked ?True?). This is due to
a coincidental interaction between the model and UCT. The
learned model mistakenly predicts that the snake may not die
when it moves off the edge of the screen (a rare event in the
training data). This emboldens UCT to consider staying near
the edges, which can be necessary to escape a tight spot.
Prediction Performance
TimelineTree
No Timestamps
Prediction Profile
Finite Suffix
0.4
0.3
0.2
0.1
0
0
2000
4000
6000
8000
10000
# Training Trajectories
Figure 5: Results in Snake.
Conclusions
In these experiments, timeline trees learned to capture long-range dependencies in complex, partially
observable systems with high-dimensional observations. The assumption that the predictions of
interest depend on only a few key events in the past is limiting in the sense that there are simple
partial models that timeline trees cannot easily capture (e.g. Three Card Monte), but it does reflect
a broad, natural class of partially observable phenomena (the examples here, for instance, were not
designed with timeline trees in mind). In problems that do match timeline trees? inductive biases,
they have been shown to outperform the more expressive prediction profile models.
There are many possible directions in which to consider extending timeline trees. More sophisticated
decision tree induction methods could help with sample complexity and overfitting. Regression tree
methods could extend timeline trees into environments with continuous dimensions. The timestamp
features used here are only one of many possible types of temporally abstract features that could be
devised. Of particular interest is whether the ideas here can be combined with approaches described
in Section 2.1 in order to increase expressive power, while retaining the benefits of timeline trees.
8
References
[1] Craig Boutilier, Thomas Dean, and Steve Hanks. Decision-theoretic planning: Structural assumptions
and computational leverage. Journal of Artificial Intelligence Research, 11:1?94, 1999.
[2] Erik Talvitie and Satinder Singh. Simple local models for complex dynamical systems. In Advances in
Neural Information Processing Systems 21 (NIPS), pages 1617?1624, 2009.
[3] Andrew K. McCallum. Reinforcement Learning with Selective Perception and Hidden State. PhD thesis,
Rutgers University, 1995.
[4] Erik Talvitie and Satinder Singh. Learning to make predictions in partially observable environments
without a generative model. Journal of Artificial Intelligence Research (JAIR), 42:353?392, 2011.
[5] Michael Littman, Richard Sutton, and Satinder Singh. Predictive representations of state. In Advances in
Neural Information Processing Systems 14 (NIPS), pages 1555?1561, 2002.
[6] George E. Monahan. A survey of partially observable markov decisions processes: Theory, models, and
algorithms. Management Science, 28(1):1?16, 1982.
[7] Anthony R. Cassandra, Leslie Pack Kaelbling, and Michael L. Littman. Acting optimally in partially observable stochastic domains. In Proceedings of the Twelfth National Conference on Artificial Intelligence
(AAAI), volume 2, pages 1023?1028, 1994.
[8] Satinder Singh, Michael R. James, and Matthew R. Rudary. Predictive state representations: A new theory
for modeling dynamical systems. In Uncertainty in Artificial Intelligence: Proceedings of the Twentieth
Conference (UAI), pages 512?519, 2004.
[9] Alicia Peregrin Wolfe and Andrew G. Barto. Decision tree methods for finding reusable MDP homomorphisms. In Proceedings of the Twenty-First National Conference on Artificial Intelligence (AAAI),
2006.
[10] Michael Holmes and Charles Isbell. Looping suffix tree-based inference of partially observable hidden
state. In Proceedings of the Twenty-Third International Conference on Machine Learning (ICML), pages
409?416, 2006.
[11] Dana Ron, Yoram Singer, and Naftali Tishby. The power of amnesia. In Advances in Neural Information
Processing Systems 6, pages 176?183, 1994.
[12] Monica Dinculescu and Doina Precup. Approximate predictive representations of partially observable
systems. In Proceedings of the Twenty-Seventh International Conference on Machine Learning (ICML),
pages 895?902, 2010.
[13] R. Andrew McCallum. Overcoming incomplete perception with utile distinction memory. In Proceedings
of the Tenth International Conference on Machine Learning (ICML), pages 190?196, 1993.
[14] M. M. Hassan Mahmud. Constructing states for reinforcement learning. In Proceedings of the TwentySeventh International Conference on Machine Learning (ICML), pages 727?734, 2010.
[15] Erik Talvitie. Simple Partial Models for Complex Dynamical Systems. PhD thesis, University of Michigan,
Ann Arbor, MI, 2010.
[16] J. Ross Quinlan. Induction of decision trees. Machine Learning, 1:81?106, 1986.
[17] J. Ross Quinlan. C4.5: Programs for Machine Learning. Morgan Kaufman Publishers Inc., San Francisco,
CA, 1993.
[18] Lex Weaver and Nigel Tao. The optimal reward baseline for gradient-based reinforcement learning. In
Uncertainty in Artificial Intelligence: Proceedings of the Seventeenth Conference (UAI), pages 538?545,
2001.
[19] Levente Kocsis and Csaba Szepesv?ari. Bandit based monte-carlo planning. In Proceedings of the Seventeenth European Conference on Machine Learning (ECML), pages 282?293, 2006.
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4,047 | 4,663 | Accelerated Training for Matrix-norm
Regularization: A Boosting Approach
Xinhua Zhang?, Yaoliang Yu and Dale Schuurmans
Department of Computing Science, University of Alberta, Edmonton AB T6G 2E8, Canada
{xinhua2,yaoliang,dale}@cs.ualberta.ca
Abstract
Sparse learning models typically combine a smooth loss with a nonsmooth
penalty, such as trace norm. Although recent developments in sparse approximation have offered promising solution methods, current approaches either apply
only to matrix-norm constrained problems or provide suboptimal convergence
rates. In this paper, we propose a boosting method for regularized learning that
guarantees accuracy within O(1/) iterations. Performance is further accelerated
by interlacing boosting with fixed-rank local optimization?exploiting a simpler
local objective than previous work. The proposed method yields state-of-the-art
performance on large-scale problems. We also demonstrate an application to latent multiview learning for which we provide the first efficient weak-oracle.
1
Introduction
Our focus in this paper is on unsupervised learning problems such as matrix factorization or latent
subspace identification. Automatically uncovering latent factors that reveal important structure in
data is a longstanding goal of machine learning research. Such an analysis not only provides understanding, it can also facilitate subsequent data storage, retrieval and processing. We focus in
particular on coding or dictionary learning problems, where one seeks to decompose a data matrix
? = U V that minimizes reconstruction error while satisfying
X into an approximate factorization X
other properties like low rank or sparsity in the factors. Since imposing a bound on the rank or
number of non-zero elements generally makes the problem intractable, such constraints are usually
replaced by carefully designed regularizers that promote low rank or sparse solutions [1?3].
Interestingly, for a variety of dictionary constraints and regularizers, the problem is equivalent to
? [1, 4]. One intensively studied
a matrix-norm regularized problem on the reconstruction matrix X
example is the trace norm, which corresponds to bounding the Euclidean norm of the code vectors in
U while penalizing V via its `21 norm. To solve trace norm regularized problems, variational methods that optimize over U and V only guarantee local optimality, while proximal gradient algorithms
? [5, 6] can achieve an accurate (global) solutions in O(1/?) iterations, but these
that operate on X
require singular value thresholding [7] at each iteration, preventing application to large problems.
Recently, remarkable promise has been demonstrated for sparse approximation methods. [8] converts the trace norm problem into an optimization over positive semidefinite (PSD) matrices, then
solves the problem via greedy sparse approximation [9, 10]. [11] further generalizes the algorithm
from trace norm to gauge functions [12], dispensing with the PSD conversion. However, these
schemes turn the regularization into a constraint. Despite their theoretical equivalence, many practical applications require the solution to the regularized problem, e.g. when nested in another problem.
In this paper, we optimize the regularized objective directly by reformulating the problem in the
framework of `1 penalized boosting [13, 14], allowing it to be solved with a general procedure developed in Section 2. Each iteration of this procedure calls an oracle to find a weak hypothesis
?
Xinhua Zhang is now at the National ICT Australia (NICTA), Machine Learning Group.
1
(typically a rank-one matrix) yielding the steepest local reduction of the (unregularized) loss. The
associated weight is then determined by accounting for the `1 regularization. Our first key contribution is to establish that, when the loss is convex and smooth, the procedure finds an accurate
solution within O(1/) iterations. To the best of our knowledge, this is the first O(1/) objective
value rate that has been rigorously established for `1 regularized boosting. [15] considered a similar
boosting approach, but required totally corrective updates. In addition, their rate characterizes the
diminishment of the gradient, and is O(1/2 ) as opposed to O(1/) established here. [9?11, 16?18]
establish similar rates, but only for the constrained version of the problem.
We also show in Section 3 how the empirical performance of `1 penalized boosting can be greatly improved by introducing an auxiliary rank-constrained local-optimization within each iteration. Interlacing rank constrained optimization with sparse updates has been shown effective in semi-definite
programming [19?21]. [22] applied the idea to trace norm optimization by factoring the reconstruction matrix into two orthonormal matrices and a positive semi-definite matrix. Unfortunately, this
strategy creates a very difficult constrained optimization problem, compelling [22] to resort to manifold techniques. Instead, we use a simpler variational representation of matrix norms that leads to
a new local objective that is both unconstrained and smooth. This allows the application of much
simpler and much more efficient solvers to greatly accelerate the overall optimization.
Underlying standard sparse approximation methods is an oracle that efficiently selects a weak hypothesis (using boosting terminology). Unfortunately these oracle problems are extremely challenging except in limited cases [3, 11]. Our next major contribution, in Section 4, is to formulate an
efficient oracle for latent multiview factorization models [2, 4], based on a positive semi-definite
relaxation that we prove incurs no gap.
Finally, we point out that our focus in this paper is on the optimization of convex problems that relax
the ?hard? rank constraint. We do not explicitly minimize the rank, which is different from [23].
?
Notation We use ?K to denote the gauge induced by set K; k?k to denote the dual norm of k?k;
and k?kF , k?ktr and k?ksp to denote the Frobenius norm, trace norm and spectral norm respectively.
P
kXkR,1 denotes the row-wise norm i kXi: kR , while hX, Y i := tr(X 0 Y ) denotes the inner product. The notation X < 0 will denote positive semi-definite; X:i and Xi: stands for the i-th column
and i-th row of matrix X; and diag {ci } denotes a diagonal matrix with the (i, i)-th entry ci .
2
The Boosting Framework with `1 Regularization
Consider a coding problem where one is presented an n?m matrix Z, whose columns correspond to
m training examples. Our goal is to learn an n?k dictionary matrix U , consisting of k basis vectors,
and a k ? m coefficient matrix V , such that U V approximates Z under some loss L(U V ). We suppress the dependence on the data Z throughout the paper. To remove the scaling invariance between
U and V , it is customary to restrict the bases, i.e. columns of U , to the unit ball of some norm k?kC .
Unfortunately, for a fixed k, this coding problem is known to be computationally tractable only for
the squared loss. To retain tractability for a variety of convex losses, a popular and successful recent
approach has been to avoid any ?hard? constraint on the number of bases, i.e. k, and instead impose
regularizers on the matrix V that encourage a low rank or sparse solution.
To be more specific, the following optimization problem lies at the heart of many sparse learning
models [e.g. 1, 3, 4, 24, 25]:
min
min L(U V? ) + ?kV? kR,1 ,
(1)
U :kU:i kC ?1 V?
where ? ? 0 specifies the tradeoff between loss and regularization. The k?kR norm in the block R-1
norm provides the flexibility of promoting useful structures in the solution, e.g. `1 norm for sparse
solutions, `2 norm for low rank solutions, and block structured norms for group sparsity. To solve
(1), we first reparameterize the rows of V? by V?i: = ?i Vi: , where ?i ? 0 and kVi: kR ? 1. Now (1)
can be reformulated by introducing the reconstruction matrix X := U V? :
X
(1) = min L(X) + ?
min
kV? kR,1 = min L(X) + ?
min
?i , (2)
X
U,V? :kU:i kC ?1,U V? =X
X
?,U,V :??0,U ?V =X
i
where ? = diag{?i }, and U and V in the last minimization also carry norm constraints. (2) is
illuminating in two respects. First it reveals that the regularizer essentially seeks a rank-one decomposition of the reconstruction matrix X, and penalizes the `1 norm of the combination coefficients
as a proxy of the ?rank?. Second, the regularizer in (2) is now expressed precisely in the form of the
2
Algorithm 1: The vanilla boosting algorithm.
Require: The weak hypothesis set A in (3).
1: Set X0 = 0, s0 = 0.
2: for k = 1, 2, . . . do
3:
Hk ? argmin h?L(Xk?1 ), Hi.
Algorithm 2: Boosting with local search.
Require: A set of weak hypotheses A.
1: Set X0 = 0, U0 = V0 = ?0 = [ ], s0 = 0.
2: for k = 1, 2, . . . do
3:
(uk , vk ) ? argmin h?L(Xk?1 ), uv0 i.
uv0 ?A
H?A
4:
(ak , bk ) ?
argmin L(aXk?1 +bHk ) + ?(ask +b).
4:
a?0,b?0
5:
(k)
(k?1)
(k)
(k?1)
?i ? ak ?i
, Ai ? Ai
,?i<k
(k)
(k)
?k ? bk , Ak ? Hk .
Pk
(k) (k)
Xk ? i=1 ?i Ai = ak Xk?1+bk Hk ,
Pk
(k)
sk ? i=1 ?i = ak sk?1 + bk .
5:
6:
6:
7:
7: end for
8:
(ak , bk ) ?
argmin L(aXk?1+b uk vk0 )+?(ask+b).
a?0,b?0
p
?
?k?1 ak ?k?1 , bk uk ),
Uinit ? (U
p
?
Vinit ? ( ak ?k?1 V?k?1 , bk vk )0 .
Locally optimize g(U, V ) with initial
value (Uinit , Vinit ). Get a solution (Uk ,Vk ).
Xk ? Uk Vk , ?k ? diag{kU:i kC kVi: kR },
Pk
2
2
sk ? 21 i=1 (kU:i kC + kVi: kR ).
end for
gauge function ?K induced by the convex hull K of the set1
A = {uv0 : kukC ? 1, kvkR ? 1}.
(3)
Since K is convex and symmetric (?K = K), the gauge function ?K is in fact a norm, hence the
support function of A defines the dual norm ||| ? ||| (see e.g. [26, Proposition V.3.2.1]):
|||?||| := max tr(X 0 ?) =
X?A
max
u,v:kukC ?1,kvkR ?1
?
u0 ?v = max
k?0 ukR = max
u:kukC ?1
v:kvkR ?1
?
k?vkC ,
(4)
and the gauge function ?K is simply its dual norm ||| ? |||? . For example, when k ? kR = k ? kC = k ? k2 ,
we have ||| ? ||| = k ? ksp , so the regularizer (as the dual norm) becomes k ? ktr . Another special
case of this result was found in [4, Theorem 1], where again k ? kR = k ? k2 but k ? kC is more
complicated than k ? k2 . Note that the original proofs in [1, 4] are somewhat involved. Moreover,
this gauge function framework is flexible enough to subsume a number of structurally regularized
problems [11, 12], and it is certainly possible to devise other k ? kR and k ? kC norms that would
induce interesting matrix norms.
The gauge function framework also allows us to develop an efficient boosting algorithm for (2), by
resorting to the following equivalent problem:
X
X
{?i? , A?i } := argmin f ({?i , Ai }), where f ({?i , Ai }) := L
? i Ai + ?
?i . (5)
?i ?0,Ai ?A
i
The optimal solution X ? of (2) can be easily recovered as
terminology, A corresponds to the set of weak hypotheses.
2.1
i
? ?
i ?i Ai . Note that in the boosting
P
The boosting algorithm
To solve (5) we propose the boosting strategy presented in Algorithm 1. At each iteration, a weak
hypothesis Hk that yields the most rapid local decrease of the loss L is selected. Then Hk is combined with the previous ensemble by tuning its weights to optimize the regularized objective. Note
that in Step 5 all the weak hypotheses selected in the previous steps are scaled by the same value.
As the `1 regularizer requires the sum of all the weights, we introduce a variable sk that recursively
updates this sum in Step 6. In addition, Xk is used only in Step 3 and 4, which do not require its
explicit expansion in terms of the elements of A. Therefore this expansion of Xk does not need to
be explicitly maintained and Step 5 is included only for conceptual clarity.
2.2
Rate of convergence
We prove the convergence rate of Algorithm 1, under the standard assumption:
Assumption 1 L is bounded from below and has bounded sub-level sets. The problem (5) admits
at least one minimizer X ? . L is differentiable and satisfies the following inequality for all ? ?
1
Recall that the gauge function ?K is defined as ?K (X) := inf{
3
P
i
?i :
P
i
?i Ai = X, Ai ? K, ?i ? 0}.
[0, 1] and A, B in the (smallest) convex set that contains both X ? and the sub-level set of f (0):
2
L((1 ? ?)A + ?B) ? L(A) + ? hB ? A, ?L(A)i + CL2? . Here CL > 0 is a finite constant that
depends only on L and X ? .
Theorem 1 (Rate of convergence) Under Assumption 1, Algorithm 1 finds an accurate solution
to (5) in O(1/) steps. More precisely, denoting f ? as the minimum of (5), then
(k)
(k)
f ({?i , Ai }) ? f ? ?
4CL
.
k+2
(6)
The proof is given in Appendix A. Note that the rate is independent of the regularization constant ?.
2
; it should be clear that
In the proof we fix the variable a in Step 4 of Algorithm 1 to be simply k+2
setting a by line search will only accelerate the convergence. An even more aggressive scheme is
(k)
the totally corrective update [15], which in Step 4 finds the weights for all Ai ?s selected so far:
!
k
k
X
X
(k)
min L
? i Ai
+?
?i .
(7)
?i ?0
i=1
i=1
(k)
But in this case we will have to explicitly maintain the expansion of Xt in terms of the Ai ?s.
For boosting without regularization, the 1/ rate of convergence is known to be optimal [27]. We
conjecture that 1/ is also a lower bound for regularized boosting.
Extensions Our proof technique allows the regularizer to be generalized toP
the form h(?P
K (X)),
where h is a convex non-decreasing function over [0, ?). In (5), this replaces i ?i with h( i ?i ).
By taking h(x) as the indicator h(x) = 0 if x ? 1; ? otherwise, our rate can be straightforwardly
translated into the constrained setting.
3
Local Optimization with Fixed Rank
In Algorithm 1, Xk is determined by searching in the conic hull of Xk?1 and Hk .2 Suppose there
exists some auxiliary procedure that allows Xk to be further improved somehow to Yk (e.g. by local
greedy search), then the overall optimization can benefit from it. The only challenge, nevertheless,
is how to restore the ?context? from Yk , especially the bases Ai and their weights ?i .
In particular, suppose we have an auxiliary function g and the following procedure is feasible:
1. Initialization: given an ensemble {?i , Ai }, there exists a S such that g(S) ? f ({?i , Ai }).
2. Local optimization: some (local) optimizer can find a T such that g(T ) ? g(S).
3. Recovery: one can recover an ensemble {?i , Bi : ?i ? 0, Bi ? A} such that f ({?i , Bi }) ? g(T ).
Then obviously the new ensemble {?i , Bi } improves upon {?i , Ai }. This local search scheme can
(k)
(k)
be easily embedded into Algorithm 1 as follows. After Step 5, initialize SPby {?i , Ai }. Perform
P
local optimization and recover {?i , Bi }. Then replace Step 6 by Xk = i ?i Bi and sk = i ?i .
The rate of convergence will directly carry over. However, the major challenge here is the potentially
expensive step of recovery because little assumption or constraint is made on T .
Fortunately, a careful examination of Algorithm 1 reveals that a complete recovery of {?i , Bi } is not
required. Indeed, only two ?sufficient statistics? are needed: Xk and sk , and therefore it suffices to
recover them only. Next we will show how this can be accomplished efficiently in (2) . Two simple
propositions will play a key role. Both proofs can be found in Appendix C.
Proposition 1 For the gauge ?K induced by K, the convex hull of A in (3), we have
1X
2
2
kU:i kC + kVi: kR .
?K (X) =
min
U,V :U V =X 2
i
(8)
2
This does not mean Xk is a minimizer of L(X) + ??K (X) in that cone, because the bases are not
optimized simultaneously. Incidentally, this also shows why working with (5) turns out to be more convenient.
4
P
2
If k?kR = k?kC = k ? k2 , then ?K becomes the trace norm (as we saw before), and i (kU:i kC +
2
2
2
kVi: kR ) is simply kU kF + kV kF . Then Proposition 1 is a well-known variational form of the trace
norm [28]. This motivates us to choose the auxiliary
function as
?X
2
2
g(U, V ) = L(U V ) +
kU:i kC + kVi: kR .
(9)
2 i
Proposition 2 For any U ? Rm?k and V ? Rk?n, there exist ?i ? 0, ui ? Rm , and vi ? Rn such that
k
k
k
X
X
1X
2
2
0
kU:i kC + kVi: kR . (10)
UV =
?i ui vi , kui kC ? 1, kvi kR ? 1,
?i =
2 i=1
i=1
i=1
Now we can specify concrete details for local optimization in the context of matrix norms:
1. Initialize: given {?i ? 0, ui vi0 ? A}ki=1 , set (Uinit , Vinit ) to satisfy g(Uinit , Vinit ) = f ({?i , ui vi0 }):
?
?
?
?
and
Vinit = ( ?1 v1 , . . . , ?k vk )0 .
(11)
Uinit = ( ?1 u1 , . . . , ?k uk ),
2. Locally optimize g(U, V ) with initialization (Uinit , Vinit ), to obtain a solution (U ? , V ? ).
? i, v
? i } from (U ? , V ? ).
3. Recovery: use Proposition 2 to (conceptually) recover {?i , u
The key advantage of this procedure is that Proposition 2 allows Xk and sk to be computed directly
from (U ? , V ? ), keeping the recovery completely implicit:
k
k
k
X
X
1X
2
2
? iv
? i0 = U ? V ? , and sk =
Xk =
?i u
?i =
kU:i? kC + kVi:? kR .
(12)
2 i=1
i=1
i=1
In addition, Proposition 2 ensures that locally improving the solution does not incur an increment
in the number of weak hypotheses. Using the same trick, the (Uinit , Vinit ) in (11) for the (k + 1)-th
iteration can also be formulated in terms of (U ? , V ? ). Different from the local optimization for
trace norm in [21] which naturally works on the original objective, our scheme requires a nontrivial
(variational) reformulation of the objective based on Propositions 1 and 2.
? and V? in Step 5 denote the column-wise
The final algorithm is summarized in Algorithm 2, where U
and row-wise normalized versions of U and V , respectively. Compared to the local optimization in
[22], which is hampered by orthogonal and PSD constraints, our (local) objective in (9) is unconstrained and smooth for many instances of k?kC and k?kR . This is plausible because no other constraints (besides the norm constraint), such as orthogonality, are imposed on U and V in Proposition
2. Thus the local optimization we face, albeit non-convex in general, is more amenable to efficient
solvers such as L-BFGS.
Remark Consider if one performs totally corrective update as in (7). Then all of the coefficients
and weak hypotheses from (U ? , V ? ) have to be considered, which can be computationally expensive. For example, in the case of trace norm, this leads to a full SVD on U ? V ? . Although U ? and V ?
usually have low rank, which can be exploited to ameliorate the complexity, it is clearly preferable
to completely eliminate the recovery step, as in Algorithm 2.
4
Latent Generative Model with Multiple Views
Underlying most boosting algorithms is an oracle that identifies the steepest descent weak hypothesis
(Step 3 of Algorithm 1). Approximate solutions often suffice [8, 9]. When k?kR and k?kC are both
Euclidean norms, this oracle can be efficiently computed via the leading left and right singular vector
pair. However, for most other interesting cases like low rank tensors, such an oracle is intractable
[29]. In this section we discover that for an important problem of multiview learning, the oracle can
be surprisingly solved in polynomial time, yielding an efficient computational strategy.
Multiview learning analyzes multi-modal data, such as heterogeneous descriptions of text, image and
video, by exploiting the implicit conditional independence structure. In this case, beyond a single
dictionary U and coefficient matrix V that model a single view Z (1) , multiple dictionaries U (k) are
needed to reconstruct multiple views Z (k) , while keeping the latent representation V shared across
all views. Formally the problem in multiview factorization is to optimize [2, 4]:
min
(1)
U (1) :kU:i kC ?1
...
min
min
(k)
U (k) :kU:i kC ?1 V
5
k
X
t=1
Lt (U (t) V ) + ? kV kR,1 .
(13)
We can easily re-express the problem as an equivalent ?single? view formulation (1) by stacking all
(t)
{U (t) } into the rows of a big matrix U , with a new column norm kU:i kC := max kU:i kC . Then
t=1...k
the constraints on U (t) in (13) can be equivalently written as kU:i kC ? 1, and Algorithm 2 can be
directly applied with two specializations. First the auxiliary function g(U, V ) in (9) becomes
2
?X
?X
(t)
(t) 2
2
2
g(U, V ) = L(U V )+
max kU:i kC +kVi: kR = L(U V )+
max kU k +kVi: kR
t=1...k
2 i
2 i t=1...k :i C
which can be locally optimized. The only challenge left is the oracle problem in (4), which takes the
following form when all norms are Euclidean:
2
X
2
0
0
0
(14)
?t ut
max
u ?v = max k? uk =
max
.
kuk ?1,kvk?1
kuk ?1
u:?t,kut k?1
C
C
t
[4, 24] considered the case where k = 2 and showed that exact solutions to (14) can be found efficiently. But their derivation does not seem to extend to k > 2. Fortunately there is still an interesting
and tractable scenario. Consider multilabel classification with a small number of classes, and U (1)
and U (2) are two views of features (e.g. image and text). Then each class label corresponds to a
view and the corresponding ut is univariate. Since there must be an optimal solution on the extreme
points of the feasible region, we can enumerate {?1, 1} for ut (t ? 3) and for each assignment solve
a subproblem of the following form that instantiates (14) (c is a constant vector)
(QP )
2
max k?01 u1 + ?02 u2 + ck , s.t. ku1 k ? 1, ku2 k ? 1.
(15)
u1 ,u2
Due to inhomogeneity, the technique in [4] is not applicable. Rewrite (15) in matrix form
(QP ) min hM0 , zz0 i s.t. hM1 , zz0 i ? 0 hM2 , zz0 i ? 0 hI00 , zz0 i = 1,
(16)
z
!
!
!
!
r
0
c0 ?01 c0 ?02
?1
?1
I
0
where z= u1 , M0 = ? ?1 c ?1 ?01 ?1 ?02 , M1 =
, M2 =
,
u2
?2 c ?2 ?01 ?2 ?02
0
I
and I00 is a zero matrix with only the (1, 1)-th entry being 1. Let X = zz0 , a semi-definite programming relaxation for (QP ) can be obtained by dropping the rank-one constraint:
(SP ) min hM0 , Xi , s.t. hM1 , Xi ? 0, hM2 , Xi ? 0, hI00 , Xi = 1, X 0. (17)
X
Its dual problem, which is also the Lagrange dual of (QP ), can be written as
(SD)
max y0 , s.t. Z := M0 ? y0 I00 + y1 M1 + y2 M2 0, y1 ? 0,
y0 ,y1 ,y2
y2 ? 0. (18)
(SD) is a convex problem that can be solved efficiently by, e.g., cutting plane methods. (SP ) is
also a convex semidefinite program (SDP) amenable for standard SDP solvers. However further
recovering the solution to (QP ) is not straightforward, because there may be a gap between the
optimal values of (SP ) and (QP ). The gap is zero (i.e. strong duality between (QP ) and (SD))
only if the rank-one constraint that (SP ) dropped from (QP ) is automatically satisfied, i.e. if (SP )
has a rank-one optimal solution.
Fortunately, as one of our main results, we prove that strong duality always holds for the particular
problem originating from (15). Our proof utilizes some recent development in optimization [30],
and is relegated to Appendix D.
5
Experimental Results
We compared our Algorithm 2 with three state-of-the-art solvers for trace norm regularized objectives: MMBS3 [22], DHM [15], and JS [8]. JS was proposed for solving the constrained problem:
minX L(X) s.t. kXktr ? ?, which makes it hard to compare with solvers for the penalized problem: minX L(X) + ? kXktr . As a workaround, we first chose a ?, and found the optimal solution
X ? for the penalized problem. Then we set ? = kX ? ktr and finally solved the constrained problem
by JS. In this case, it is only fair to compare how fast L(X) (loss) is decreased by various solvers,
rather than L(X) + ? kXk? (objective). DHM is sensitive to the estimate of the Lipschitz constant
of the gradient of L, which we manually tuned for a small value such that DHM still converges.
Since the code for MMBS is specialized to matrix completion, it was used only in this comparison.
Traditional solvers such as proximal methods [6] were not included because they are much slower.
3
http://www.montefiore.ulg.ac.be/?mishra/softwares/traceNorm.html
6
MovieLens?1m, ? = 50
10
5
10
?2
10
0
9
MovieLens?10m, ? = 50
10
Objective and loss (training)
Ours?obj
Ours?loss
MMBS?obj
MMBS?loss
DHM?obj
DHM?loss
JS?loss
Objective and loss (training)
Objective and loss (training)
MovieLens?100k, ? = 20
6
7
10
6
10
Ours?obj
Ours?loss
MMBS?obj
MMBS?loss
DHM?obj
DHM?loss
JS?loss
10
7
10
6
0
2
10
10
Running time (seconds)
8
10
2
10
Running time (seconds)
10 0
10
4
10
Ours?obj
Ours?loss
MMBS?obj
MMBS?loss
DHM?obj
DHM?loss
JS?loss
2
4
10
10
Running time (seconds)
6
10
(a) Objective & loss vs time (loglog) (a) Objective & loss vs time (loglog) (a) Objective & loss vs time (loglog)
MovieLens?1m, ? = 50
MovieLens?100k, ? = 20
0.8
Test error
Test error
0.7
0.6
0.5
0.9
0.8
0.8
0.7
0.7
0.6
0.5
0.4
0.4
0.3
0.2
?2
10
Ours
MMBS
DHM
JS
0.3
0.2
0
2
10
10
Running time (seconds)
MovieLens?10m, ? = 50
0.9
Test error
0.9
Ours
MMBS
DHM
JS
0.5
0.4
0.3
0.2
0
10
0.6
2
10
Running time (seconds)
4
10
0.1 0
10
Ours
MMBS
DHM
JS
2
4
10
10
Running time (seconds)
6
10
(b) Test NMAE vs time (semilogx) (b) Test NMAE vs time (semilogx) (b) Test NMAE vs time (semilogx)
Figure 1: MovieLens100k.
Figure 2: MovieLens1M.
Figure 3: MovieLens10M.
Comparison 1: Matrix completion We first compared all methods on a matrix completion problem, using the standard datasets MovieLens100k, MovieLens1M, and MovieLens10M [6, 8, 21],
which are sized 943 ? 1682, 6040 ? 3706, and 69878 ? 10677 respectively (#user ? #movie). They
contain 105 , 106 and 107 movie ratings valued from 1 to 5, and the task is to predict the rating for
a user on a movie. The training set was constructed by randomly selecting 50% ratings for each
user, and the prediction is made on the rest 50% ratings. In Figure 1 to 3, we show how fast various
algorithms drive down the training objective, training loss L (squared Euclidean distance), and the
normalized mean absolute error (NMAE) on the test data [see, e.g., 6, 8]. We tuned the ? to optimize
the test NMAE.
From Figure 1(a), 2(a), 3(a), it is clear that it takes much less amount of CPU time for our method to
reduce the objective value (solid line) and the loss L (dashed line). This implies that local search and
partially corrective updates in our method are very effective. Not surprisingly MMBS is the closest
to ours in terms of performance because it also adopts local optimization. However it is still slower
because their local search is conducted on a constrained manifold. In contrast, our local search
objective is entirely unconstrained and smooth, which we manage to solve efficiently by L-BFGS.4
JS, though applied indirectly, is faster than DHM in reducing the loss. We observed that DHM kept
running coordinate descent with a constant step size, while the totally corrective update was rarely
taken. We tried accelerating it by using a smaller value of the estimate of the Lipschitz constant of
the gradient of L, but it leads to divergence after a rapid decrease of the objective for the first few
iterations. A hybrid approach might be useful.
We also studied the evolution of the NMAE performance on the test data. For this we compared the
matrix reconstruction after each iteration against the ground truth. As plotted in Figure 1(b), 2(b),
3(b), our approach achieves comparable (or better) NMAE in much less time than all other methods.
Comparison 2: multitask and multiclass learning Secondly, we tested on a multiclass classification problem with synthetic dataset. Following [15], we generated a dataset of D = 250 features
and C = 100 classes. Each class c has 10 training examples and 10 test examples drawn independently and identically from a class-specific multivariate Gaussian N (?c , ?c ). ?c ? R250 has
the last 200 coordinates being 0, and the top 50 coordinates were chosen uniformly random from
{?1, 1}. The (i, j)-th element of ?c is 22 (0.5)|i?j| . The task is to predict the class membership of
a given example. We used the logistic loss for a model matrix W ? RD?C . In particular, for each
4
http://www.cs.ubc.ca/?pcarbo/lbfgsb-for-matlab.html
7
where for any class c,
p(c|xi ;W ) = Zi?1 exp(W:c0 xi ),
X
Zi =
exp(W:c0 xi ).
c
Synthetic multiclass, ? = 0.01
4
10
Ours?obj
Ours?loss
DHM?obj
DHM?loss
JS?loss
1
10
0
10
?1
10
0
Objective and loss (training)
Li (W ) = ? log p(yi |xi ; W ),
2
Objective and loss (training)
training example xi with label
yi ? {1, .., C}, we defined an
individual loss Li (W ) as
2
10
10
Running time (seconds)
School multitask, ? = 0.1
10
Ours?obj
Ours?loss
DHM?obj
DHM?loss
JS?loss
3
10
2
10
1
10
0
2
10
10
Running time (seconds)
(a) Objective & loss vs time (loglog) (a) Objective & loss vs time (loglog)
Objective and loss (training)
Test error
Test regression error
Then L(W ) is defined as the
Multiclass, ? = 0.01
School Multitask, ? = 0.1
average of Li (W ) over the
1000
Ours
Ours
whole training set. We found
0.95
DHM
DHM
800
that ? = 0.01 yielded the
JS
0.9
JS
lowest test classification er600
0.85
ror; the corresponding results
0.8
400
are given in Figure 4. Clearly,
0.75
200
the intermediate models out0.7
put by our scheme achieve
0.65
0
0
2
0
2
10
10
comparable (or better) train10
10
Running time (seconds)
Running time (seconds)
ing objective and test error in
(b) Test error vs time (semilogx)
(b) Test error vs time (semilogx)
orders of magnitude less time
than those generated by DHM Figure 4: Multiclass classifica- Figure 5: Multitask learning for
tion with synthetic datset.
school dataset.
and JS.
Multiview Flickr, ?=0.001
2
We also applied the solvers to a multitask learning problem with
10
Ours?obj
the school dataset [25]. The task is to predict the score of
Ours?loss
15362 students from 139 secondary schools based on a number
Alt?obj
Alt?loss
of school-specific and student-specific attributes. Each school is
1
10
considered as a task for which a predictor is learned. We used the
first random split of training and testing data provided by [25] 5 ,
and set ? so as to achieve the lowest test squared error. Again,
as shown in Figure 5 our approach is much faster than DHM and
0
10
2
3
JS in finding the optimal solution for training objective and test
10
10
Running time (seconds)
error. As the problem requires a large ?, the trace norm penalty
Figure 6: Multiview training.
is small, making the loss close to the objective.
Comparison 3: Multiview learning Finally we perform an initial test on our global optimization
technique for learning latent models with multiple views. We used the Flickr dataset from NUSWIDE [31]. Its first view is a 634 dimensional low-level feature, and the second view consists of
1000 dimensional tags. The class labels correspond to the type of animals and we randomly chose 5
types with 20 examples in each type. The task is to train the model in (13) with ? = 10?3 . We used
squared loss for the first view, and logistic loss for the other views.
We compared our method with a local optimization approach to solving (13). The local method first
fixes all U (t) and minimizes V , which is a convex problem that can be solved by FISTA [32]. Then
it fixes V and optimizes U (t) , which is again convex. We let Alt refer to the scheme that alternates
these updates to convergence. From Figure 6 it is clear that Alt is trapped by a locally optimal
solution, which is inferior to a globally optimal solution that our method is guaranteed to find. Our
method also reduces both the objective and the loss slightly faster than Alt.
6
Conclusion and Outlook
We have proposed a new boosting algorithm for a wide range of matrix norm regularized problems.
It is closely related to generalized conditional gradient method [33]. We established the O(1/)
convergence rate, and showed its empirical advantage over state-of-the-art solvers on large scale
problems. We also applied the method to a novel problem, latent multiview learning, for which we
designed a new efficient oracle. We plan to study randomized boosting with `1 regularization [34] ,
and to extend the framework to more general nonlinear regularization [3].
5
http://ttic.uchicago.edu/?argyriou/code/mtl_feat/school_splits.tar
8
References
[1] F. Bach, J. Mairal, and J. Ponce. Convex sparse matrix factorizations. arXiv:0812.1869v1, 2008.
[2] H. Lee, R. Raina, A. Teichman, and A. Ng. Exponential family sparse coding with application to selftaught learning. In IJCAI, 2009.
[3] D. Bradley and J. Bagnell. Convex coding. In UAI, 2009.
[4] X. Zhang, Y-L Yu, M. White, R. Huang, and D. Schuurmans. Convex sparse coding, subspace learning,
and semi-supervised extensions. In AAAI, 2011.
[5] T. K. Pong, P. Tseng, S. Ji, and J. Ye. Trace norm regularization: Reformulations, algorithms, and multitask learning. SIAM Journal on Optimization, 20(6):3465?3489, 2010.
[6] K-C Toh and S. Yun. An accelerated proximal gradient algorithm for nuclear norm regularized least
squares problems. Pacific Journal of Optimization, 6:615?640, 2010.
[7] J-F Cai, E. J. Cand?es, and Z. Shen. A singular value thresholding algorithm for matrix completion. SIAM
Journal on Optimization, 20(4):1956?1982, 2010.
[8] M. Jaggi and M. Sulovsky. A simple algorithm for nuclear norm regularized problems. In ICML, 2010.
[9] E. Hazan. Sparse approximate solutions to semidefinite programs. In LATIN, 2008.
[10] K. L. Clarkson. Coresets, sparse greedy approximation, and the Frank-Wolfe algorithm. In SODA, 2008.
[11] A. Tewari, P. Ravikumar, and I. S. Dhillon. Greedy algorithms for structurally constrained high dimensional problems. In NIPS, 2011.
[12] V. Chandrasekaran, B. Recht, P. A. Parrilo, and A. S. Willsky. The convex geometry of linear inverse
problems. Foundations of Computational Mathematics, 12(6):805?849, 2012.
[13] Y. Bengio, N.L. Roux, P. Vincent, O. Delalleau, and P. Marcotte. Convex neural networks. In NIPS, 2005.
[14] L. Mason, J. Baxter, P. L. Bartlett, and M. Frean. Functional gradient techniques for combining hypotheses. In Advances in Large Margin Classifiers, pages 221?246, Cambridge, MA, 2000. MIT Press.
[15] M. Dudik, Z. Harchaoui, and J. Malick. Lifted coordinate descent for learning with trace-norm regularizations. In AISTATS, 2012.
[16] S. Shalev-Shwartz, N. Srebro, and T. Zhang. Trading accuracy for sparsity in optimization problems with
sparsity constraints. SIAM Journal on Optimization, 20:2807?2832, 2010.
[17] X. Yuan and S. Yan. Forward basis selection for sparse approximation over dictionary. In AISTATS, 2012.
[18] T. Zhang. Sequential greedy approximation for certain convex optimization problems. IEEE Trans.
Information Theory, 49(3):682?691, 2003.
[19] S. Burer and R. Monteiro. Local minima and convergence in low-rank semidefinite programming. Mathematical Programming, 103(3):427?444, 2005.
[20] M. Journee, F. Bach, P.-A. Absil, and R. Sepulchre. Low-rank optimization on the cone of positive
semidefinite matrices. SIAM Journal on Optimization, 20:2327C?2351, 2010.
[21] S. Laue. A hybrid algorithm for convex semidefinite optimization. In ICML, 2012.
[22] B. Mishra, G. Meyer, F. Bach, and R. Sepulchre. Low-rank optimization with trace norm penalty. Technical report, 2011. http://arxiv.org/abs/1112.2318.
[23] S. Shalev-Shwartz, A. Gonen, and O. Shamir. Large-scale convex minimization with a low-rank constraint. In ICML, 2011.
[24] M. White, Y. Yu, X. Zhang, and D. Schuurmans. Convex multi-view subspace learning. In NIPS, 2012.
[25] A. Argyriou, T. Evgeniou, and M. Pontil. Convex multi-task feature learning. Machine Learning, 73(3):
243?272, 2008.
[26] J-B Hiriart-Urruty and C. Lemar?echal. Convex Analysis and Minimization Algorithms, I and II, volume
305 and 306. Springer-Verlag, 1993.
[27] I. Mukherjee, C. Rudin, and R. Schapire. The rate of convergence of Adaboost. In COLT, 2011.
[28] N. Srebro, J. Rennie, and T. Jaakkola. Maximum-margin matrix factorization. In NIPS, 2005.
[29] C. Hillar and L-H Lim. Most tensor problems are NP-hard. arXiv:0911.1393v3, 2012.
[30] W. Ai and S. Zhang. Strong duality for the CDT subproblem: A necessary and sufficient condition. SIAM
Journal on Optimization, 19:1735?1756, 2009.
[31] T.S. Chua, J. Tang, R. Hong, H. Li, Z. Luo, and Y.T. Zhang. A real-world web image database from
national university of singapore. In International Conference on Image and Video Retrieval, 2009.
[32] A. Beck and M. Teboulle. A fast iterative shrinkage-thresholding algorithm for linear inverse problems.
SIAM Journal on Imaging Sciences, 2(1):183?202, 2009.
[33] K. Bredies, D. Lorenz, and P. Maass. A generalized conditional gradient method and its connection to an
iterative shrinkage method. Computational Optimization and Applications, 42:173?193, 2009.
[34] Y. Nesterov. Efficiency of coordinate descent methods on huge-scale optimization problems. SIAM
Journal on Optimization, 22(2):341?362, 2012.
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4,048 | 4,664 | Mirror Descent Meets Fixed Share
(and feels no regret)
Nicol? Cesa-Bianchi
Universit? degli Studi di Milano
[email protected]
G?bor Lugosi
ICREA & Universitat Pompeu Fabra, Barcelona
[email protected]
Pierre Gaillard
Ecole Normale Sup?rieure?, Paris
[email protected]
Gilles Stoltz
Ecole Normale Sup?rieure?, Paris &
HEC Paris, Jouy-en-Josas, France
[email protected]
Abstract
Mirror descent with an entropic regularizer is known to achieve shifting regret
bounds that are logarithmic in the dimension. This is done using either a carefully
designed projection or by a weight sharing technique. Via a novel unified analysis,
we show that these two approaches deliver essentially equivalent bounds on a notion of regret generalizing shifting, adaptive, discounted, and other related regrets.
Our analysis also captures and extends the generalized weight sharing technique
of Bousquet and Warmuth, and can be refined in several ways, including improvements for small losses and adaptive tuning of parameters.
1
Introduction
Online convex optimization is a sequential prediction paradigm in which, at each time step, the
learner chooses an element from a fixed convex set S and then is given access to a convex loss
function defined on the same set. The value of the function on the chosen element is the learner?s
loss. Many problems such as prediction with expert advice, sequential investment, and online regression/classification can be viewed as special cases of this general framework. Online learning
algorithms are designed to minimize the regret. The standard notion of regret is the difference
between the learner?s cumulative loss and the cumulative loss of the single best element in S. A
much harder criterion to minimize is shifting regret, which is defined as the difference between the
learner?s cumulative loss and the cumulative loss of an arbitrary sequence of elements in S. Shifting
regret bounds are typically expressed in terms of the shift, a notion of regularity measuring the length
of the trajectory in S described by the comparison sequence (i.e., the sequence of elements against
which the regret is evaluated). In online convex optimization, shifting regret bounds for convex subsets S ? Rd are obtained for the projected online mirror descent (or follow-the-regularized-leader)
algorithm. In this case the shift is typically computed in terms of the p-norm of the difference of
consecutive elements in the comparison sequence ?see [1, 2] and [3].
We focus on the important special case when S is the simplex. In [1] shifting bounds are shown for
projected mirror descent with entropic regularizers using a 1-norm to measure the shift.1 When the
comparison sequence is restricted to the corners of the simplex (which is the setting of prediction
with expert advice), then the shift is naturally defined to be the number of times the trajectory moves
?
Ecole Normale Sup?rieure, Paris ? CNRS ? INRIA, within the project-team CLASSIC
Similar 1-norm shifting bounds can also be proven using the analysis of [2]. However, without using
entropic regularizers it is not clear how to achieve a logarithmic dependence on the dimension, which is one of
the advantages of working in the simplex.
1
1
to a different corner. This problem is often called ?tracking the best expert? ?see, e.g., [4, 5, 1, 6, 7],
and it is well known that exponential weights with weight sharing, which corresponds to the fixedshare algorithm of [4], achieves a good shifting bound in this setting. In [6] the authors introduce a
generalization of the fixed-share algorithm, and prove various shifting bounds for any trajectory in
the simplex. However, their bounds are expressed using a quantity that corresponds to a proper shift
only for trajectories on the simplex corners.
In this paper we offer a unified analysis of mirror descent, fixed share, and the generalized fixed
share of [6] for the setting of online convex optimization in the simplex. Our bounds are expressed
in terms of a notion of shift based on the total variation distance. Our analysis relies on a generalized
notion of shifting regret which includes, as special cases, related notions of regret such as adaptive
regret, discounted regret, and regret with time-selection functions. Perhaps surprisingly, we show
that projected mirror descent and fixed share achieve essentially the same generalized regret bound.
Finally, we show that widespread techniques in online learning, such as improvements for small
losses and adaptive tuning of parameters, are all easily captured by our analysis.
2
Preliminaries
For simplicity, we derive our results in the setting of online linear optimization. As we show in the
supplementary material, these results can be easily extended to the more general setting of online
convex optimization through a standard linearization step.
Online linear optimization may be cast as a repeated
and the environ game between the forecaster
ment as follows. We use ?d to denote the simplex q ? [0, 1]d : kqk1 = 1 .
Online linear optimization in the simplex. For each round t = 1, . . . , T ,
bt = (b
1. Forecaster chooses p
p1,t , . . . , pbd,t ) ? ?d
2. Environment chooses a loss vector `t = (`1,t , . . . , `d,t ) ? [0, 1]d
b>
3. Forecaster suffers loss p
t `t .
>
b T = PT p
The goal of the forecaster is to minimize the accumulated loss, e.g., L
t=1 b t `t . In the now
classical problem of prediction with expert advice, the goal of the forecaster is to compete with the
PT
best fixed component (often called ?expert?) chosen in hindsight, that is, with mini=1,...,T t=1 `i,t ;
or even to compete with a richer class of sequences of components. In Section 3 we state more
specifically the goals considered in this paper.
We start by introducing our main algorithmic tool, described in Figure 1, a share algorithm whose
formulation generalizes the seemingly unrelated formulations of the algorithms studied in [4, 1, 6]. It
is parameterized by the ?mixing functions? ?t : [0, 1]td ? ?d for t > 2 that assign probabilities to
past ?pre-weights? as defined below. In all examples discussed in this paper, these mixing functions
are quite simple, but working with such a general model makes the main ideas more transparent. We
then provide a simple lemma that serves as the starting point2 for analyzing different instances of
this generalized share algorithm.
Lemma 1. For all t > 1 and for all q t ? ?d , Algorithm 1 satisfies
bt ? q t
p
>
`t 6
d
vi,t+1
?
1X
qi,t ln
+ .
? i=1
pbi,t
8
Proof. By Hoeffding?s inequality (see, e.g., [3, Section A.1.1]),
?
?
d
d
X
?
1 ?X
pbj,t e?? `j,t ? + .
pbj,t `j,t 6 ? ln
?
8
j=1
j=1
(1)
Pd
By definition of vi,t+1 , for all i = 1, . . . , d we then have j=1 pbj,t e?? `j,t = pbi,t e?? `i,t /vi,t+1 ,
b>
which implies p
pi,t ) + ?/8. The proof is concluded by taking a
t `t 6 `i,t + (1/?) ln(vi,t+1 /b
convex aggregation with respect to q t .
2
We only deal with linear losses in this paper. However, it is straightforward that for sequences of ??expconcave loss functions, the additional term ?/8 in the bound is no longer needed.
2
Parameters: learning rate ? > 0 and mixing functions ?t for t > 2
b1 = v 1 = (1/d, . . . , 1/d)
Initialization: p
For each round t = 1, . . . , T ,
bt ;
1. Predict p
2. Observe loss `t ? [0, 1]d ;
3. [loss update] For each j = 1, . . . , d define
pbj,t e?? `j,t
vj,t+1 = Pd
the current pre-weights, and v t+1 = (v1,t+1 , . . . , vd,t+1 );
bi,t e?? `i,t
i=1 p
Vt+1 = vi,s 16i6d, 16s6t+1 the d ? (t + 1) matrix of all past and current pre-weights;
bt+1 = ?t+1 Vt+1 .
4. [shared update] Define p
Algorithm 1: The generalized share algorithm.
3
A generalized shifting regret for the simplex
We now introduce a generalized notion of shifting regret which unifies and generalizes the notions of
discounted regret (see [3, Section 2.11]), adaptive regret (see [8]), and shifting regret (see [2]). For
a fixed horizon T , a sequence of discount factors ?t,T > 0 for t = 1, . . . , T assigns varying weights
to the instantaneous losses suffered at each round. We compare the total loss of the forecaster with
the loss of an arbitrary sequence of vectors q 1 , . . . , q T in the simplex ?d . Our goal is to bound the
regret
T
T
X
X
b>
?t,T p
?t,T q >
`
?
t `t
t t
t=1
t=1
in terms of the ?regularity? of the comparison sequence q 1 , . . . , q T and of the variations of the
d
discounting weights ?t,T . By setting ut = ?t,T q >
t ? R+ , we can rephrase the above regret as
T
X
b>
kut k1 p
t `t ?
t=1
T
X
u>
t `t .
(2)
t=1
In the literature on tracking the best expert [4, 5, 1, 6], the regularity of the sequence u1 , . . . , uT is
measured as the number of times ut 6= ut+1 . We introduce the following regularity measure
m(uT1 ) =
T
X
DTV (ut , ut?1 )
(3)
t=2
P
where for x = (x1 , . . . , xd ), y = (y1 , . . . , yd ) ? Rd+ , we define DTV (x, y) = xi >yi (xi ? yi ).
Note that when x, y ? ?d , we recover the total variation distance DTV (x, y) = 21 kx ? yk1 , while
for general x, y ? Rd+ , the quantity DTV (x, y) is not necessarily symmetric and is always bounded
by kx ? yk1 . The traditional shifting regret of [4, 5, 1, 6] is obtained from (2) when all ut are such
that kut k1 = 1.
4
Projected update
The shifting variant of the EG algorithm analyzed in [1] is a special case of the generalized share
algorithm in which the function ?t+1 performs a projection of the pre-weights on the convex set
d
??
d = [?/d, 1] ? ?d . Here ? ? (0, 1) is a fixed parameter. We can prove (using techniques similar
to the ones shown in the next section?see the supplementary material) the following bound which
generalizes [1, Theorem 16].
3
Theorem 1. For all T > 1, for all sequences `1 , . . . , `t ? [0, 1]d of loss vectors, and for all
u1 , . . . , uT ? Rd+ , if Algorithm 1 is run with the above update, then
T
X
b>
kut k1 p
t `t ?
t=1
T
X
u>
t `t 6
t=1
T
X
ku1 k1 ln d m(uT1 ) d ?
+
ln +
+?
kut k1 .
?
?
?
8
t=1
(4)
This bound can be optimized by a proper tuning of ? and ? parameters. We show a similarly tuned
(and slightly better) bound in Corollary 1.
5
Fixed-share update
Next, we consider a different instance of the generalized share algorithm corresponding to the update
pbj,t+1 =
d
X
?
i=1
?
+ (1 ? ?)1i=j vi,t+1 = + (1 ? ?)vj,t+1 ,
d
d
06?61
(5)
Despite seemingly different statements, this update in Algorithm 1 can be seen to lead exactly to the
fixed-share algorithm of [4] for prediction with expert advice. We now show that this update delivers
a bound on the regret almost equivalent to (though slightly better than) that achieved by projection
on the subset ??
d of the simplex.
Theorem 2. With the above update, for all T > 1, for all sequences `1 , . . . , `T of loss vectors
`t ? [0, 1]d , and for all u1 , . . . , uT ? Rd+ ,
T
X
t=1
b>
kut k1 p
t `t
?
T
X
t=1
u>
t `t
T
ku1 k1 ln d ? X
+
kut k1
6
?
8 t=1
m(uT1 ) d
ln +
+
?
?
PT
t=2
kut k1 ? m(uT1 )
1
ln
.
?
1??
Note that if we only consider vectors of the form ut = q t = (0, . . . , 0, 1, 0, . . . , 0) then m(q T1 )
corresponds to the number of times q t+1 6= q t in the sequence q T1 . We thus recover [4, Theorem 1]
and [6, Lemma 6] from the much more general Theorem 2.
The fixed-share forecaster does not need to ?know? anything in advance about the sequence of
the norms kut k for the bound above to be valid. Of course, in order to minimize the obtained
upper bound, the tuning parameters ?, ? need to be optimized and their values will depend on the
PT
maximal values of m(uT1 ) and t=1 kut k1 for the sequences one wishes to compete against. This
is illustrated in the following corollary, whose proof is omitted. Therein, h(x) = ?x ln x ? (1 ?
x) ln(1 ? x) denotes the binary entropy function for x ? [0, 1]. We recall3 that h(x) 6 x ln(e/x)
for x ? [0, 1].
Corollary 1. Suppose Algorithm 1 is run with the update (5). Let m0 > 0 and U0 > 0. For all T >
1, for all sequences `1 , . . . , `T of loss vectors `t ? [0, 1]d , and for all sequences u1 , . . . , uT ? Rd+
PT
with ku1 k1 + m(uT1 ) 6 m0 and t=1 kut k1 6 U0 ,
v
v
u
! u
!
T
T
X
X
u
u U0 m0
U
m
e
U
0
0
0
>
t
bt `t ?
kut k1 p
u>
m0 ln d + U0 h
6t
ln d + ln
t `t 6
2
U
2
m0
0
t=1
t=1
whenever ? and ? are optimally chosen in terms of m0 and U0 .
Proof of Theorem 2. Applying Lemma 1 with q t = ut / kut k1 , and multiplying by kut k1 , we get
for all t > 1 and ut ? Rd+
>
b>
kut k1 p
t `t ? ut `t 6
3
d
1X
vi,t+1
?
ui,t ln
+ kut k1 .
? i=1
pbi,t
8
As can be seen by noting that ln 1/(1 ? x) < x/(1 ? x)
4
(6)
We now examine
X
d
d
d
X
X
vi,t+1
1
1
1
1
ui,t ln
=
? ui,t?1 ln
+
? ui,t ln
.
ui,t ln
ui,t?1 ln
pbi,t
pbi,t
vi,t
vi,t
vi,t+1
i=1
i=1
i=1
(7)
For the first term on the right-hand side, we have
d
X
X
1
vi,t
1
1
ui,t ln
? ui,t?1 ln
=
(ui,t ? ui,t?1 ) ln
+ ui,t?1 ln
pbi,t
vi,t
pbi,t
pbi,t
i=1
i : ui,t >ui,t?1
X
vi,t
1
+ui,t ln
+
(ui,t ? ui,t?1 ) ln
. (8)
vi,t
pbi,t
i : ui,t <ui,t?1 |
{z
}
60
In view of the update (5), we have 1/b
pi,t 6 d/? and vi,t /b
pi,t 6 1/(1 ? ?). Substituting in (8), we
get
d
X
1
1
? ui,t?1 ln
ui,t ln
pbi,t
vi,t
i=1
?
?
X
X
X
d ?
1
6
(ui,t ? ui,t?1 ) ln +
ui,t?1 +
ui,t ? ln
?
1
?
?
i: ui,t <ui,t?1
i : ui,t >ui,t?1
i: ui,t >ui,t?1
?
?
d
X
d ?X
1
= DTV (ut , ut?1 ) ln +
.
ui,t ?
(ui,t ? ui,t?1 )? ln
?
1
?
?
i=1
i : ui,t >ui,t?1
{z
}
|
=kut k1 ?DTV (ut ,ut?1 )
The sum of the second term in (7) telescopes. Substituting the obtained bounds in the first sum of
the right-hand side in (7), and summing over t = 2, . . . , T , leads to
!
d
T
T X
X
X
vi,t+1
d
1
T
T
ui,t ln
6 m(u1 ) ln +
kut k1 ? m(u1 ) ln
pbi,t
?
1??
t=2
t=2 i=1
+
d
X
ui,1 ln
i=1
We hence get from (6), which we use in particular for t = 1,
T
X
>
b>
kut k1 p
t `t ? ut `t 6
t=1
6
1
1
? ui,T ln
.
vi,2
vi,T +1
|
{z
}
60
d
T
1
?X
1X
ui,1 ln
+
kut k1
? i=1
pbi,1
8 t=1
PT
kut k1 m(uT1 )
m(uT1 ) d
1
+
ln + t=2
ln
.
?
?
?
1??
Applications
We now show how our regret bounds can be specialized to obtain bounds on adaptive and discounted
regret, and on regret with time-selection functions. We show regret bounds only for the specific
instance of the generalized share algorithm using update (5); but the discussion below also holds up
to minor modifications for the forecaster studied in Theorem 1.
Adaptive regret was introduced by [8] and can be viewed as a variant of discounted regret where
the monotonicity assumption is dropped. For ?0 ? {1, . . . , T }, the ?0 -adaptive regret of a forecaster
is defined by
( s
)
s
X >
X
?0 ?adapt
bt `t ? min
RT
=
max
p
q > `t .
(9)
[r, s] ? [1, T ]
s + 1 ? r 6 ?0
t=r
5
q??d
t=r
The fact that this is a special case of (2) clearly emerges from the proof of Corollary 2 below here.
Adaptive regret is an alternative way to measure the performance of a forecaster against a changing
environment. It is a straightforward observation that adaptive regret bounds also lead to shifting
regret bounds (in terms of hard shifts). In this paper we note that these two notions of regret share
an even tighter connection, as they can be both viewed as instances of the same alma mater notion
of regret, i.e., the generalized shifting regret introduced in Section 3. The work [8] essentially
considered the case of online convex optimization with exp-concave loss function; in case of general
convex functions, they also mentioned that the greedy projection forecaster of [2] enjoys adaptive
regret guarantees. This is obtained in much the same way as we obtain an adaptive regret bound for
the fixed-share forecaster in the next result.
Corollary 2. Suppose that Algorithm 1 is run with the shared update (5). Then for all T > 1, for
all sequences `1 , . . . , `T of loss vectors `t ? [0, 1]d , and for all ?0 ? {1, . . . , T },
s
r
?0
?0
1
?0 ?adapt
RT
6
?0 h
+ ln d 6
ln(ed?0 )
2
?0
2
whenever ? and ? are chosen optimally (depending on ?0 and T ).
As mentioned in [8], standard lower bounds on the regret show that the obtained bound is optimal
up to the logarithmic factors.
Proof. For 1 6 r 6 s 6 T and q ? ?d , the regret in the right-hand side of (9) equals the
regret considered in Theorem 2 against the sequence uT1 defined as ut = q for t = r, . . . , s and
0 = (0, . . . , 0) for the remaining t. When r > 2, this sequence is such that DTV (ur , ur?1 ) =
DTV (q, 0) = 1 and DTV (us+1 , us ) = DTV (0, q) = 0 so that m(uT1 ) = 1, while ku1 k1 = 0.
When r = 1, we have ku1 k1 = 1 and m(uT1 ) = 0. In all cases, m(uT1 ) + ku1 k1 = 1, that
is, m0 = 1. Specializing the bound of Theorem 2 with the additional choice U0 = ?0 gives the
result.
Discounted regret was introduced in [3, Section 2.11] and is defined by
max
q??d
T
X
>
b>
?t,T p
t `t ? q `t .
(10)
t=1
The discount factors ?t,T measure the relative importance of more recent losses to older losses. For
instance, for a given horizon T , the discounts ?t,T may be larger as t is closer to T . On the contrary,
in a game-theoretic setting, the earlier losses may matter more then the more recent ones (because of
interest rates), in which case ?t,T would be smaller as t gets closer to T . We mostly consider below
monotonic sequences of discounts (both non-decreasing and non-increasing). Up to a normalization,
we assume that all discounts ?t,T are in [0, 1]. As shown in [3],
P a minimal requirement to get nontrivial bounds is that the sum of the discounts satisfies UT = t6T ?t,T ? ? as T ? ?.
A natural objective is to show
? that the quantity in (10) is o(UT ), for instance, by bounding it by
something of the order of UT . We claim that Corollary 1 does so, at least whenever the sequences
(?t,T ) are monotonic for all T . To support this claim, we only need to show that m0 = 1 is a suitable
value to deal with (10). Indeed, for all T > 1 and for all q ? ?d , the measure of regularity involved
in the corollary satisfies
T
X
k?1,T qk1 + m (?t,T q)t6T = ?1,T +
?t,T ? ?t?1,T + = max ?1,T , ?T,T 6 1 ,
t=2
where the second equality follows from the monotonicity assumption on the discounts.
The values of the discounts for all t and T are usually known in advance. However, the horizon T
is not. Hence, a calibration issue may arise. The online tuning of the parameters ? and
? ? shown
in Section 7.3 entails a forecaster that can get discounted regret bounds of the order UT for all
T . The fundamental reason for this is that the discounts only come in the definition of the fixedshare forecaster via their sums. In contrast, the forecaster discussed in [3, Section 2.11] weighs each
instance t directly with ?t,T (i.e., in the very definition of the forecaster) and enjoys therefore no
regret guarantees for horizons other than T (neither before T nor after T ). Therein, the knowledge
6
of the horizon T is so crucial that it cannot be dealt with easily, not even with online calibration of
the parameters or with a doubling trick. We insist that for the fixed-share forecaster, much flexibility
is gained as some of the discounts ?t,T can change in a drastic manner for a round T to values
?t,T +1 for the next round. However we must admitqthat the bound of [3, Section 2.11]
qPis smaller
P
2
than the one obtained above, as it of the order of
t6T ?t,T , in contrast to our
t6T ?t,T
bound. Again, this improvement was made possible because of the knowledge of the time horizon.
As for the comparison to the setting of discounted losses of [9], we note that the latter can be cast as
a special case of our setting (since the discounting weights take the special form ?t,T = ?t . . . ?T ?1
therein, for some sequence ?s of positive numbers). In particular, the fixed-share forecaster can
satisfy the bound stated in [9, Theorem 2], for instance, by using the online tuning techniques of
Section 7.3. A final reference to mention is the setting of time-selection functions of [10, Section 6],
which basically corresponds to knowing in advance the weights kut k1 of the comparison sequence
u1 , . . . , uT the forecaster will be evaluated against. We thus generalize their results as well.
7
Refinements and extensions
We now show that techniques for refining the standard online analysis can be easily applied to our
framework. We focus on the following: improvement for small losses, sparse target sequences, and
dynamic tuning of parameters. Not all of them where within reach of previous analyses.
7.1
Improvement for small losses
The regret bounds of the fixed-share forecaster can be significantly improved when the cumulative
loss of the best sequence of experts is small. The next result improves on Corollary 1 whenever
L0 U0 . For concreteness, we focus on the fixed-share update (5).
Corollary 3. Suppose Algorithm 1 is run with the update (5). Let m0 > 0, U0 > 0, and L0 > 0.
For all T > 1, for all sequences `1 , . . . , `T of loss vectors `t ? [0, 1]d , and for all sequences
PT
PT
u1 , . . . , uT ? Rd+ with ku1 k1 + m(uT1 ) 6 m0 , t=1 kut k1 6 U0 , and t=1 u>
t `t 6 L0 ,
v
u
!
T
T
X
X
u
e U0
e U0
>
>
t
bt `t ?
ut `t 6 L0 m0 ln d + ln
kut k1 p
+ ln d + ln
m0
m0
t=1
t=1
whenever ? and ? are optimally chosen in terms of m0 , U0 , and L0 .
Here again, the parameters ? and ? may be tuned online using the techniques shown in Section 7.3.
The above refinement is obtained by mimicking the analysis of Hedge forecasters for small losses
(see, e.g., [3, Section 2.4]). In particular, one should substitute Lemma 1 with the following lemma
in the analysis carried out in Section 5; its proof follows from the mere replacement of Hoeffding?s
inequality by [3, Lemma A.3], which states that for all ? ? R and for all random variable X taking
values in [0, 1], one has ln E[e??X ] 6 (e?? ? 1)EX.
d
1X
vi,t
1 ? e?? >
bt `t ? q >
p
`
6
q
ln
for all q t ? ?d .
Lemma 2. Algorithm 1 satisfies
i,t
t t
?
? i=1
pbi,t+1
7.2
Sparse target sequences
The work [6] introduced forecasters that are able to efficiently compete with the best sequence of
experts among all those sequences that only switch a bounded number of times and also take a
small number of different values. Such ?sparse? sequences of experts appear naturally in many
applications. In this section we show that their algorithms in fact work very well in comparison with
a much larger class of sequences u1 , . . . , uT that are ?regular??that is, m(uT1 ), defined in (3) is
Pd
small?and ?sparse? in the sense that the quantity n(uT1 ) = i=1 maxt=1,...,T ui,t is small. Note
that when q t ? ?d for all t, then two interesting upper bounds can be provided. First, denoting
T
the union of the supports of these convex
combinations by S ? {1, . . . , d}, we have n(q 1 ) 6 |S|,
T
the cardinality of S. Also, n(q 1 ) 6 {q t , t = 1, . . . , T } , the cardinality of the pool of convex
combinations. Thus, n(uT1 ) generalizes the notion of sparsity of [6].
7
Here we consider a family of shared updates of the form
wj,t
pbj,t = (1 ? ?)vj,t + ?
,
0 6 ? 6 1,
(11)
Zt
where the wj,t are nonnegative weights that may depend on past and current pre-weights and Zt =
Pd
i=1 wi,t is a normalization constant. Shared updates of this form were proposed by [6, Sections 3
and 5.2]. Apart from generalizing the regret bounds of [6], we believe that the analysis given below
is significantly simpler and more transparent. We are also able to slightly improve their original
bounds.
We focus on choices of the weights wj,t that satisfy the following conditions: there exists a constant
C > 1 such that for all j = 1, . . . , d and t = 1, . . . , T ,
vj,t 6 wj,t 6 1
and
C wj,t+1 > wj,t .
(12)
T
T
The next result improves on Theorem 2 when T d and n(u1 ) m(u1 ), that is, when the
dimension (or number of experts) d is large but the sequence uT1 is sparse. Its proof can be found in
the supplementary material; it is a variation on the proof of Theorem 2.
Theorem 3. Suppose Algorithm 1 is run with the shared update (11) with weights satisfying the
conditions (12). Then for all T > 1, for all sequences `1 , . . . , `T of loss vectors `t ? [0, 1]d , and
for all sequences u1 , . . . , uT ? Rd+ ,
T
X
b>
kut k1 p
t `t
?
T
X
t=1
t=1
u>
t `t
T
n(uT1 ) ln d n(uT1 ) T ln C
?X
6
+
+
kut k1
?
?
8 t=1
PT
kut k1 ? m(uT1 )
m(uT1 ) maxt6T Zt
1
+
ln
+ t=2
ln
.
?
?
?
1??
Corollaries 8 and 9 of [6] can now be generalized (and even improved); we do so?in the supplementary material?by showing two specific instances of the generic update (11) that satisfy (12).
7.3
Online tuning of the parameters
The forecasters studied above need their parameters ? and ? to be tuned according to various quantities, including the time horizon T . We show here how the trick of [11] of having these parameters
vary over time can be extended to our setting. For the sake of concreteness we focus on the fixedshare update, i.e., Algorithm 1 run with the update (5). We respectively replace steps 3 and 4 of its
description by the loss and shared updates
, d
?t
X ? ?t
?t
?t?1 ?? `
t j,t
vj,t+1 = pbj,t e
pbi,tt?1 e??t `i,t
and
pj,t+1 =
+ (1 ? ?t ) vj,t+1 , (13)
d
i=1
for all t > 1 and all j ? {1, . . . , d}, where (?? ) and (?? ) are two sequences of positive numbers,
indexed by ? > 1. We also conventionally define ?0 = ?1 . Theorem 2 is then adapted in the
following way (when ?t ? ? and ?t ? ?, Theorem 2 is exactly recovered).
Theorem 4. The forecaster based on the updates (13) is such that whenever ?t 6 ?t?1 and ?t 6
?t?1 for all t > 1, the following performance bound is achieved. For all T > 1, for all sequences
`1 , . . . , `T of loss vectors `t ? [0, 1]d , and for all u1 , . . . , uT ? Rd+ ,
!
T
T
T
X
X
kut k1 X
1
1
>
>
bt `t ?
kut k1 p
+
kut k1
?
ln d
ut `t 6
?1
?t
?t?1
t=1
t=1
t=2
T
+
T
X ?t?1
m(uT1 ) d(1 ? ?T ) X kut k1
1
ln
+
ln
+
kut k1 .
?T
?T
?t?1
1 ? ?t t=1 8
t=2
Due to space constraints, we provide an illustration of this bound only in the supplementary material.
Acknowledgments
The authors acknowledge support from the French National Research Agency (ANR) under grant
EXPLO/RA (?Exploration?exploitation for efficient resource allocation?) and by the PASCAL2
Network of Excellence under EC grant no. 506778.
8
References
[1] M. Herbster and M. Warmuth. Tracking the best linear predictor. Journal of Machine Learning
Research, 1:281?309, 2001.
[2] M. Zinkevich. Online convex programming and generalized infinitesimal gradient ascent. In
Proceedings of the 20th International Conference on Machine Learning, ICML 2003, 2003.
[3] N. Cesa-Bianchi and G. Lugosi. Prediction, learning, and games. Cambridge University Press,
2006.
[4] M. Herbster and M. Warmuth. Tracking the best expert. Machine Learning, 32:151?178, 1998.
[5] V. Vovk. Derandomizing stochastic prediction strategies. Machine Learning, 35(3):247?282,
Jun. 1999.
[6] O. Bousquet and M.K. Warmuth. Tracking a small set of experts by mixing past posteriors.
Journal of Machine Learning Research, 3:363?396, 2002.
[7] A. Gy?rgy, T. Linder, and G. Lugosi. Tracking the best of many experts. In Proceedings of the
18th Annual Conference on Learning Theory (COLT), pages 204?216, Bertinoro, Italy, Jun.
2005. Springer.
[8] E. Hazan and C. Seshadhri. Efficient learning algorithms for changing environments. Proceedings of the 26th International Conference of Machine Learning (ICML), 2009.
[9] A. Chernov and F. Zhdanov. Prediction with expert advice under discounted loss. In Proceedings of the 21st International Conference on Algorithmic Learning Theory, ALT 2010, pages
255?269. Springer, 2008.
[10] A. Blum and Y. Mansour. From extermal to internal regret. Journal of Machine Learning
Research, 8:1307?1324, 2007.
[11] P. Auer, N. Cesa-Bianchi, and C. Gentile. Adaptive and self-confident on-line learning algorithms. Journal of Computer and System Sciences, 64:48?75, 2002.
9
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4,049 | 4,665 | Statistical Consistency of Ranking Methods in A
Rank-Differentiable Probability Space
Yanyan Lan
Institute of Computing Technology
Chinese Academy of Sciences
[email protected]
Jiafeng Guo
Institute of Computing Technology
Chinese Academy of Sciences
[email protected]
Xueqi Cheng
Institute of Computing Technology
Chinese Academy of Sciences
[email protected]
Tie-Yan Liu
Microsoft Research Asia
[email protected]
Abstract
This paper is concerned with the statistical consistency of ranking methods. Recently, it was proven that many commonly used pairwise ranking methods are
inconsistent with the weighted pairwise disagreement loss (WPDL), which can
be viewed as the true loss of ranking, even in a low-noise setting. This result
is interesting but also surprising, given that the pairwise ranking methods have
been shown very effective in practice. In this paper, we argue that the aforementioned result might not be conclusive, depending on what kind of assumptions
are used. We give a new assumption that the labels of objects to rank lie in a
rank-differentiable probability space (RDPS), and prove that the pairwise ranking
methods become consistent with WPDL under this assumption. What is especially inspiring is that RDPS is actually not stronger than but similar to the low-noise
setting. Our studies provide theoretical justifications of some empirical findings
on pairwise ranking methods that are unexplained before, which bridge the gap
between theory and applications.
1 Introduction
Ranking is a central problem in many applications, such as document retrieval, meta search, and
collaborative filtering. In recent years, machine learning technologies called ?learning to rank? have
been successfully applied. A learning-to-rank process can be described as follows. In training, a
number of sets (queries) of objects (documents) are given and within each set the objects are labeled
by assessors, mainly based on multi-level ratings. The target of learning is to create a model that
provides a ranking over the objects that best respects the observed labels. In testing, given a new set
of objects, the trained model is applied to generate a ranked list of the objects.
Ideally, the learning process should be guided by minimizing a true loss such as the weighted pairwise disagreement loss (WPDL) [11], which encodes people?s knowledge on ranking evaluation.
However, the minimization can be very difficult due to the nonconvexity of the true loss. Alternatively, many learning-to-rank methods minimize surrogate loss functions. For example, RankSVM
[14], RankBoost [12], and RankNet [3] minimize the hinge loss, the exponential loss, and the crossentropy loss, respectively.
In machine learning, statistical consistency is regarded as a desired property of a learning method
[1, 21, 20], which reveals the statistical connection between a surrogate loss function and the true
loss. Statistical consistency in the context of ranking have been actively studied in recent years
1
[8, 9, 19, 11, 2, 18]. According to the studies in [11], many existing pairwise ranking methods
are, surprisingly, inconsistent with WPDL, even in a low-noise setting. However, as we know, the
pairwise ranking methods have been shown to work very well in practice, and have been regarded
as state-of-the-art even today [15, 16, 17]. For example, the experimental results in [2] show that
a weighted preorder loss in RankSVM [4] can outperform a consistent surrogate loss in terms of
NDCG (See Table 2 in [2]).
The contradiction between theory and application inspires us to revisit the statistical consistency of
pairwise ranking methods. In particular, we will study whether there exists a new assumption on the
probability space that can make statistical consistency naturally hold, and how this new assumption
compares with the low-noise setting used in [11].
To perform our study, we first derive a sufficient condition for statistical consistency of ranking
methods called rank-consistency, which is in nature very similar to edge-consistency in [11] and
order-preserving in [2]. Then we give an assumption on the probability space where ratings (labels)
of objects come from, which we call a rank-differentiable probability space (RDPS). Intuitively,
RDPS reveals the reason why an object (denoted as object A) should be ranked higher than another
object (denoted as object B). That is, the probability of any ratings consistent with the preference1 is
larger than that of its dual ratings (obtained by exchanging the labels of object A and object B while
keeping others unchanged). We then prove that with the RDPS assumption, the weighted pairwise
surrogate loss, which is a generalization of many surrogate loss functions used in existing pairwise
ranking methods (e.g., the preorder loss in RankSVM [2], the exponential loss in RankBoost [12],
and the logistic loss in RankNet [3]), is statistically consistent with WPDL.
Please note that our theoretical result contradicts the result obtained in [11], mainly due to the
different assumptions used. What is interesting, and to some extent inspiring, is that our RDPS
assumption is not stronger than the low-noise setting used in [11], and in some sense they are very
similar to each other (although they focus on different aspects of the probability space). We then
conducted detailed comparisons between them to gain more insights on what affects the consistency
of ranking.
According to our theoretical analysis, we argue that it is not yet appropriate to draw any conclusion
about the inconsistency of pairwise ranking methods, especially because it is hard to know what
the probability space really is. In this sense, we think the pairwise ranking methods are still good
choices for real ranking applications, due to their good empirical performances.
The rest of this paper is organized as follows. Sections 2 defines the consistency problem formally
and provides a sufficient condition under which consistency with WPDL is achieved for ranking
methods. Section 3 gives the main theoretical results, including formal definition of RDPS and
conditions of statistical consistency of pairwise ranking methods. Further discussions on whether
RDPS is a strong assumption and why our results contradict with that in [11] are presented in Section
4. Conclusions are presented in Section 5.
2
Preliminaries of Statistical Consistency
Let x = {x1 , ? ? ? , xm } be a set of objects to be ranked. Suppose the labels of the objects are given
as multi-level ratings r = (r1 , ? ? ? , rm ) from space R, where ri denotes the label of xi . Without
loss of generality, we adopt K-level ratings used in [7], that is, ri ? {0, 1, ? ? ? , K ?1}. If ri > rj , xi
should be ranked higher than xj . Assume that (x, r) is a random variable of space X ? R according
to a probability measure P . Following existing literature, let f be a ranking function that gives a
score to each object to produce a ranked list and denote F as the space of all ranking functions.
In this paper, we adopt the weighted pairwise disagreement loss (WPDL) defined in [11, 10] as the
true loss to evaluate f :
?
?
l0 (?, G) =
aG
aG
(1)
ij 1{?i ??j } +
ij 1{?i <?j } ,
i<j
i>j
where ? = (?1 , ? ? ? , ?m ) = (f (x1 ), ? ? ? , f (xm )), G is a directed acyclic graph (DAG for short)
with edge i ? j to represent the preference that xi should be ranked higher than xj , and aG
ij is a
non-negative penalty indexed by i ? j on graph G.
1
Here, consistency with the preference means that the rating of object A is larger than that of object B.
2
Specifically, in the setting of multi-level ratings, i ? j is constructed between pair (i, j) with
2
ri > rj , and aG
ij is thus just relevant to the labels of the two objects. For ease of representation , we
replace aG
ij with D(ri , rj ), and WPDL becomes the following form:
?
l0 (f ; x, r) =
D(ri , rj )1{f (xi )?f (xj )?0} ,
(2)
i,j:ri >rj
where 1{?} is an indicator function3 and D(ri , rj ) is a weight function s.t. (1) ?ri ?= rj , D(ri , rj ) >
0; (2) ?ri , rj , D(ri , rj ) = D(rj , ri ); (3) ?ri < rj < rk , D(ri , rj ) ? D(ri , rk ), D(rj , rk ) ? D(ri , rk ).
The conditional expected true risk and the expected true risk of f are then defined as:
?
R0 (f |x) = Er|x l0 (f ; x, r) =
l0 (f ; x, r)P (r|x), R0 (f ) = Ex [Er|x l0 (f ; x, r)].
(3)
r?R
Due to the nonconvexity of the true loss, it is infeasible to minimize the true risk in Eq.(3). As is
done in the literature of machine learning, we adopt a surrogate loss l? to minimize in place of l0 .
The conditional expected surrogate risk and the expected surrogate risk of f are then defined as:
?
(4)
R? (f |x) = Er|x l? (f ; x, r) =
l? (f ; x, r)P (r|x), R? (f ) = Ex [Er|x l? (f ; x, r)].
r?R
Statistical consistency is a desired property for a good surrogate loss, which measures whether the
expected true risk of the ranking function obtained by minimizing a surrogate loss converges to the
expected true risk of the optimal ranking in the large sample limit.
Definition 1. We say a ranking method that minimizes a surrogate loss l? is statistically consistent
with respect to the true loss l0 , if ??1 > 0, ??2 > 0, such that for any ranking function f ? F,
R? (f ) ? inf h?F R? (h) + ?2 implies R0 (f ) ? inf h?F R0 (h) + ?1 .
We then introduce a property of the surrogate loss, called rank-consistency, which is a sufficient
condition for the statistical consistency of the surrogate loss, as indicated by Theorem 1.
Definition 2. We say a surrogate loss l? is rank-consistent with the true loss l0 , if ?x, for any
ranking function f ? F such that R0 (f |x) > inf h?F R0 (h|x), the following inequality holds:
inf R? (h|x) < inf{R? (g|x) : g ? F , g(xi ) ? g(xj ), for (i, j) where f (xi ) ? f (xj ).}.
h?F
(5)
Rank-consistency can be viewed as a generalization of infinite sample consistency for classification
proposed in [20] (also referred to as ?classification-calibrated? in [1]) to ranking on a set of objects.
It is also similar to edge-consistent in [11] and order-preserving in [2].
Theorem 1. If a surrogate loss l? is rank-consistent with the true loss l0 on the function space F,
then it is statistically consistent with the true loss l0 on F.
We omit the proof since it is a straightforward extension of the proof for Theorem 3 in [20]. The
proof is also similar to Lemma 3, 4, 5 and Theorem 6 in [11].
3
Main Results
In this section, we present our main theoretical results: with a new assumption on the probability
space, many commonly used pairwise ranking algorithms can be proved consistent with WPDL.
3.1
A Rank-Differentiable Probability Space
First, we give a new assumption named a rank-differentiable probability space (RDPS for short),
with which many pairwise ranking methods will be rank-consistent with WPDL. Hereafter, we will
also refer to data from RDPS as having a rank-differentiable property.
2
Here we do not distinguish i > j and i < j, because they are just introduced to avoid minor technical
issues as stated in [11]. Furthermore, it will not influence the consistency results.
3
1A = 1,if A is true and 1A = 0,if A is false.
3
Before introducing the definition of RDPS, we give two definitions, an equivalence class of ratings
and dual ratings. Intuitively, we say two ratings are equivalent if they induce the same ranking
or preference relationships. And we say two ratings are the dual ratings with respect to a pair of
objects, if the two ratings just exchange the ratings of the two objects while keeping the ratings of
other objects unchanged. The formal definitions are given as follows.
Definition 3. A ratings r is called equivalent to ?r, denoted as r ? ?r, if P(r) = P(?r). Where
P(r) = {(i, j) : ri > rj .} and P(?r) = {(i, j) : r?i > r?j .} stand for the preference relationships
induced by r and ?r, respectively. Therefore, an equivalence class of the ratings r, denoted as [r], is
defined as the set of ratings which are equivalent to r. That is, [r] = {?r ? R : ?r ? r.}.
Definition 4. Let R(i, j) = {r ? R : ri > rj .}, r? is called the dual ratings of r ? R(i, j) with
respect to (i, j) if rj? = ri , ri? = rj , rk? = rk , ?k ?= i, j.
Now we give the definition of RDPS. An intuitive explanation on this definition is that there exists
a unique equivalence class of ratings that for each induced pairwise preference relationship, the
probability will be able to separate the two dual ratings with respect to that pair.
Definition 5. Let R(i, j) = {r ? R : ri > rj .}, a probability space is called rank-differentiable
with (i, j), if for any r ? R(i, j), P (r|x) ? P (r? |x), and there exists at least one ratings r ?
R(i, j), s.t. P (r|x) > P (r? |x), where r? is the dual ratings of r.
Definition 6. A probability space is called rank-differentiable, if there exists an equivalence class
[r?], s.t. P(r? ) = {(i, j) : the probability space is rank-differentiable with(i, j).}, where P(r? ) =
{(i, j) : ri? > rj? .}. We will also call this probability space a RDPS or rank-differentiable with [r? ].
Please note that [r? ] in Definition 6 is unique, which can be directly proved by Definition 3.
Definition 5 implies that if a probability space is rank-differentiable with (i, j), the optimal ranking
function will rank xi higher than xj , as shown in the following theorem. The proof is similar to
that of Theorem 4, thus we omit it here for space limitation. Hereafter, we will call this property
?separability on pairs?.
Theorem 2. ?x ? X , let f ? ? F be an optimal ranking function that R0 (f ? |x) = inf f ?F R0 (f |x).
If the probability space is rank-differentiable with (i, j), we have f ? (xi ) > f ? (xj ).
Further considering the ?transitivity4 over pairs? of a ranking function, Definition 6 implies that if a
probability space is rank-differentiable with [r? ], the optimal ranking function will induce the same
preference relationships, as shown in the following theorem.
Theorem 3. ?x ? X , let f ? ? F be an optimal ranking function that R0 (f ? |x) = inf f ?F R0 (f |x).
If the probability space is rank-differentiable with [r? ], for any (i, j) ? P(r? ), we have f ? (xi ) >
f ? (xj ), where P(r? ) = {(i, j) : ri? > rj? .}.
3.2
Conditions of Statistical Consistency
With RDPS as the new assumption, we study the statistical consistency of pairwise ranking methods.
First, we define the weighted pairwise surrogate loss as
?
l? (f ; x, r) =
D(ri , rj )?(f (xi ) ? f (xj )),
(6)
i,j:ri >rj
where ? is a convex function. The surrogate losses used in many existing pairwise ranking methods
can be regarded as special cases of this weighted pairwise surrogate loss, such as the hinge loss in
RankSVM [14], the exponential loss in RankBoost [12], the cross-entropy loss in RankNet [3] and
the preorder loss in [2]. For the weighted pairwise surrogate loss, we get its sufficient condition
of statistical consistency as shown in Theorem 5. In order to prove this theorem, we first prove
Theorem 4.
Theorem 4. We assume the probability space is rank-differentiable with an equivalence class [r? ].
Suppose that ?(?) : R ? R in the weighted pairwise surrogate loss is a non-increasing function
such that ?(z) < ?(?z), ?z > 0. ?x ? X , let f ? F be a ranking function such that R? (f |x) =
4
Transitivity means that if xi is ranked higher than xj and xj is ranked higher than xk , xi must be ranked
higher than xk .
4
inf h?F R? (h|x), then for any object pair (xi , xj ), ri? > rj? , we have f (xi ) ? f (xj ). Moreover, if
?(?) is differentiable and ?? (0) < 0, we have f (xi ) > f (xj ).
Proof. (1) We assume that f (xi ) < f (xj ), and define f ? as the function such that f ? (xi ) =
f (xj ), f ? (xj ) = f (xi ), f ? (xk ) = f (xk ), ?k ?= i, j. We can then get the following equation,
R? (f ? |x) ? R? (f |x)
? ?
=
[D(rk , rj )?D(rk , ri )][?(f (xk )?f (xi ))??(f (xk )?f (xj ))][P (r|x)?P (r? |x)]
r,r? , k:rj <ri <rk
r?R(i,j)
+
?
?
D(ri , rk )[?(f (xj )?f (xk ))??(f (xi )?f (xk ))][P (r|x)?P (r? |x)]
r,r? , k:rj <rk <ri
r?R(i,j)
+
?
?
D(rk , rj )[?(f (xk )?f (xi ))??(f (xk )?f (xj ))][P (r|x)?P (r? |x)]
?
r,r , k:rj <rk <ri
r?R(i,j)
+
?
?
[D(ri , rk )?D(rj , rk )][?(f (xj )?f (xk ))??(f (xi )?f (xk ))][P (r|x)?P (r? |x)]
r,r? , k:rk <rj <ri
r?R(i,j)
+[?(f (xj )?f (xi ))??(f (xi )?f (xj ))]
?
D(ri , rj )[P (r|x)?P (r? |x)]
?
r,r ,
r?R(i,j)
According to the conditions of RDPS, the requirements of the weight function D in Section 2 and
the assumption that ? is a non-increasing function such that ?(z) < ?(?z), ?z > 0, we can obtain
?
R? (f ? |x) ? R? (f |x) ? [?(f (xj )?f (xi ))??(f (xi )?f (xj ))]
D(ri , rj )[P (r|x)?P (r? |x)] < 0.
r,r? ,
r?R(i,j)
This is a contradiction with R? (f )=inf h?F R? (h|x). Therefore, we have proven that f (xi )?f (xj ).
(2) Now we assume
that f (xi ) = f (xj ) = f0 . From the assumption R? (f |x) = inf h?F R? (h|x), we
?R? (f |x)
? (f |x)
can get ?f (xi ) = 0, ?R
?f (xj ) = 0. Accordingly, we can obtain the two following equations:
f0
f0
?
?
(7)
A1 P (r|x) + A2 P (r? |x) = 0,
B1 P (r|x) + B2 P (r? |x) = 0,
r,r? ,
r?R(i,j)
r,r? ,
r?R(i,j)
where,
?
A1 = B2 =
D(rk , ri )[??? (f (xk )?f0 )] +
k:rj <ri <rk
+
?
A 2 = B1 =
k:rj <rk <ri
D(ri , rk )? (f0 ?f (xk )) + D(ri , rj )?? (0).
D(rk , rj )[??? (f (xk )?f0 )] +
k:rj <ri <rk
+
?
D(ri , rk )?? (f0 ?f (xk ))
?
k:rk <rj <ri
?
?
?
D(rk , rj )[??? (f (xk )?f0 )]
k:rj <rk <ri
?
D(rj , rk )? (f0 ?f (xk )) + D(ri , rj )[??? (0)].
k:rk <rj <ri
?
If ? (0) < 0, based on the requirements of RDPS and the weight function D, we can get
?
(A1 ? B1 )P (r|x) + (A2 ? B2 )P (r? |x)
r,r? ,
r?R(i,j)
=
?
?
(A1 ? A2 )[P (r|x) ? P (r? |x)] ? 2?? (0)
D(ri , rj )[P (r|x) ? P (r? |x)] < 0.
r,r? ,
r?R(i,j)
r,r? ,
r?R(i,j)
This is a contradiction with Eq.(7). Therefore, we actually have proven that f (xi ) > f (xj ).
5
Figure 1: Relationships among order-preserving, rank-differentiable and low-noise.
Theorem 5. Let ?(?) be a non-negative, non-increasing and differentiable function such that
?? (0) < 0. Then the weighted pairwise surrogate loss is consistent with WPDL under the assumption of RDPS.
Proof. We assume that the probability space is rank-differentiable with an equivalence class [r? ].
Then for any object pair (xi , xj ), ri? > rj? , we are going to prove that
?
R?|x
= inf R? (h|x) < inf{R? (f |x) : f ? F, f (xi ) ? f (xj ).}
h?F
(8)
because from Theorem 3 this implies the rank-consistency condition in Eq.(5) holds.
Suppose Eq.(8) is not true, then we can find a sequence of functions {fm } such that 0 = fm (xi ) ?
?
fm (xj ), and limm R? (fm |x) = R?|x
. We can further select a subsequence such that for each pair
(i, j), fm (xi ) ? fm (xj ) converges (may also converge to ??). This leads to a limit function f ,
with properly defined f (xi ) ? f (xj ), even when either f (xi ) or f (xj ) is ??. This implies that
?
R? (f |x) = R?|x
and 0 = f (xi ) ? f (xj ). However, this violates Theorem 4. Thus, Eq.(8) is true.
Therefore, we have proven that the weighted pairwise surrogate loss is consistent with WPDL.
Many commonly used pairwise surrogate losses, such as the preorder loss in RankSVM [2], the
exponential loss in RankBoost [12], and the logistic loss in RankNet[3], satisfy the conditions in
Theorem 5, thus they are consistent with WPDL. In other words, we have shown that statistical
consistency of pairwise ranking methods is achieved under the assumption of RDPS.
4
Discussions
In Section 3, we have shown that statistical consistency of pairwise ranking methods is achieved
with the assumption of RDPS. Considering the contradicting conclusion drawn in [11], a natural
question is whether the RDPS is stronger than the low-noise setting used in [11]. In this section we
will make some discussions on this issue.
4.1
Relationships of RDPS with Previous Work
Here, we discuss the relationships between the rank-differentiable property and the assumptions used
in some previous works (including the order-preserving property in [19] and the low-noise setting
in [11]). According to our analysis, we find that the rank-differentiable property is not a strong
assumption on the probability space. Actually, it is a weaker assumption than the order-preserving
property and is very similar to the low-noise setting. A sketch map of the relationships between the
three assumptions is presented in Figure 1, where the low-noise probability spaces stands for a set
of spaces satisfying the low-noise setting. Detailed discussions are given as follows.
6
1. Rank-Differentiable vs. Order-Preserving
The rank-differentiable property is defined on the space of multi-level ratings while the orderpreserving property is defined on the permutation space. To understand their relationship, we need
to put them onto the same space. Actually, we can restrict the space of multi-level ratings to the
permutation space by setting K = m ? 1 and requiring the ratings of each two objects to be different. After doing so, it is not difficult to see that the rank-differentiable property is weaker than the
order-preserving property, as shown in the following theorem.
Theorem 6. Let K = m ? 1. For each permutation y ? Y, where y(i) stands for the position of
y
xi , define the corresponding ratings ry = (r1y , ? ? ? , rm
) as riy = m ? y(i), i = 1, ? ? ? , n. Assume
y
that P (r ) = P (y), and P (r) = 0 if there does not exist a permutation y s.t. r = ry . If the
probability space is order-preserving with respect to m?1 pairs (j1 , j2 ), (j2 , j3 ),? ? ?, (jm?1 , jm ),
it is rank-differentiable with the equivalence class [r? ], where rj?i > rj?i+1 , i = 1, ? ? ? , m, but the
converse is not always true.
2. Rank-Differentiable vs. Low-Noise
The rank-differentiable property is defined on the space of multi-level ratings while the low-noise
setting is defined on the space of DAGs. According to the correspondence between ratings and
DAGs (as stated in Section 2), we can restrict the space of DAGs to the space of multi-level ratings.
Consequently, we obtain the relationship between the rank-differentiable property and the low-noise
setting as follows:
(1) Mathematically, the inequalities in the low-noise setting can be viewed as the combinations of
the corresponding inequalities in the rank-differentiable property. They are similar to each other in
their forms and the rank-differentiable property is not stronger than the low-noise setting.
(2) Intuitively, the rank-differentiable property induces ?separability on pairs? and ?transitivity over
pairs? as described in Theorem 2 and 3, while the low-noise setting aims to explicitly express the
transitivity over pairs, but fails in achieving it.
Let us use an example to illustrate the above points. Suppose there are three objects to be ranked in
the setting of three-level ratings (K = 3). Furthermore, suppose that the ratings of every two objects
are different and all the graphs are fully connected DAGs in the setting of [11]. We order the ratings
and DAGs as:
r1 = (2, 1, 0), r2 = (1, 2, 0), r3 = (2, 0, 1), r4 = (0, 2, 1), r5 = (1, 0, 2), r6 = (0, 1, 2).
G1 = {(1 ? 2),(2 ? 3),(1 ? 3)}, G2 = {(2 ? 1),(1 ? 3),(2 ? 3)}, G3 = {(1 ? 3),(3 ? 2),(1 ? 2)},
G4 = {(2 ? 3),(3 ? 1),(2 ? 1)}, G5 = {(3 ? 1),(1 ? 2),(3 ? 2)}, G6 = {(3 ? 2),(2 ? 1),(3 ? 1)},
Therefore ri , Gi have one-to-one correspondence, we can set the probability as P (ri |x) =
i
P (Gi |x) = Pi and define aG
kl = D(rik , ril ), i = 1, ? ? ? , 6; k, l = 1, 2, 3.
Considering conditions in the definition of RDPS, rank-differentiable with [r1 ] requires the following inequalities to hold and at least one inequalities in (9) and (10) to hold strictly.
P1 ? P2 ? 0, P3 ? P4 ? 0, P5 ? P6 ? 0,
(9)
P4 ? P6 ? 0, P2 ? P5 ? 0, P1 ? P3 ? 0,
(10)
We assume there are edges 1 ? 2 and 2 ? 3 in the difference graph. Then the low-noise setting in
Definition 8 of [11] requires that a13 ? a31 ? a12 ? a21 + a23 ? a32 , where,
a12 ? a21 = D(2, 1)(P1 ? P2 ) + D(2, 0)(P3 ? P4 ) + D(1, 0)(P5 ? P6 ),
a23 ? a32 = D(2, 1)(P4 ? P6 ) + D(2, 0)(P2 ? P5 ) + D(1, 0)(P1 ? P3 ),
a13 ? a31 = D(2, 1)(P3 ? P5 ) + D(2, 0)(P1 ? P6 ) + D(1, 0)(P2 ? P4 ).
According to the above example,
(1) a12 ? a21 and a23 ? a32 are exactly the combinations of the terms in (9) and (10), respectively.
Thus, if the probability space is rank-differentiable with [r1 ], we can only get a12 ? a21 > 0, a23 ?
a32 > 0, but not the inequalities in the low-noise setting. This indicates that our rank-differentiable
property is not stronger than the low-noise setting.
7
(2) With the assumption that aij ? aji > 0 can guarantee the optimal ranking with which xi is
ranked higher than xj , it seems that the low-noise setting intends to make the preferences of 1 ? 2
and 2 ? 3 transitive to 1 ? 3. However, the assumption is not always true. Instead, the rankdifferentiable property can naturally induce the ?transitivity over pairs? (See Theorem 2 and 3).
In this sense, the rank-differentiable property is much more powerful than the low-noise setting,
although not stronger.
4.2 Explanation on Theoretical Contradiction
On one hand, different conclusions on the consistency of pairwise ranking methods have been obtained in our work and in [11]. On the other hand, we have shown that there exists an connection
between the rank-differentiable property and the low-noise setting (see Figure 1). Therefore, one
may get confused by the contradicting results and may wonder what will happen if a probability
space satisfies both the rank-differentiable property and the low-noise setting. In this subsection, we
will make discussions on this issue.
Please note that we adopt the multi-level ratings as the labeling strategy (as stated clearly in Section
2) in our analysis. With this setting, the graph space G in [11] will not contain all the DAGs. For
example, considering a three-graph case, the graph G2 = {(1, 2, 3) : (2 ? 3), (3 ? 1)} in the proof
of Theorem 11 of [11] (the main negative result on the consistency of pairwise surrogate losses)
actually does not exist. That is because if 2 ? 3 and 3 ? 1 exist in a graph G, we can get that
r2 > r3 , r3 > r1 according to the correspondence between graphs and ratings as stated in Section 2.
Therefore, we can immediately get r2 > r1 . Once again according to the correspondence between
graphs and ratings, we will get that 2 ? 1 should be contained in graph G, which contradicts with
G2 . Thus, G2 will not exist in the setting of multi-level ratings. However, in the proof of [11], they
do not take the constraint of multi-level ratings into consideration, thus deduce contradict results.
From the above discussions, we can see that our theoretical results contradict with that in [11] mainly
because the two works consider different settings and assumptions. If a probability space satisfies
both the rank-differentiable property and the low-noise setting, the pairwise ranking methods will be
consistent with WPDL in the setting of multi-level ratings but inconsistent in the setting of DAGs.
One may argue that the setting of multi-level ratings is not as general as the DAG setting, however,
please note that multi-level ratings are the dominant setting in the literature of ?learning to rank?
[13, 16, 15, 6] and have been widely used in many applications such as web search and document
retrieval [17, 5]. Therefore, we think the setting of multi-level ratings is general enough and our
result has its value to the mainstream research of learning to rank.
To sum up, based on all the discussions in this paper, we argue that it is not yet appropriate to draw
any conclusion about the inconsistency of pairwise ranking methods, especially because it is hard to
know what the probability space really is. In this sense, we think the pairwise ranking methods are
still good choices for real ranking applications, due to their good empirical performances.
5
Conclusions
In this paper, we have discussed the statistical consistency of ranking methods. Specifically, we
argue that the previous results on the inconsistency of commonly-used pairwise ranking methods
are not conclusive, depending on the assumptions about the probability space. We then propose a
new assumption, which we call a rank-differentiable probability space (RDPS), and prove that the
pairwise ranking methods are consistent with the same true loss as in previous studies under this
assumption. We show that RDPS is not a stronger assumption than the assumptions used in previous
work, indicating that our finding is similarly reliable to previous ones.
Acknowledgments
This research work was funded by the National Natural Science Foundation of China under Grant
No. 60933005, No. 61173008, No. 61003166 , No. 61203298 and 973 Program of China under
Grants No. 2012CB316303.
8
References
[1] P. L. Bartlett, M. I. Jordan, and J. D. McAuliffe. Convexity, classification, and risk bounds.
Journal of the American Statistical Association, 101(473):138?156, 2006.
[2] D. Buffoni, C. Calauzenes, P. Gallinari, and N. Usunier. Learning scoring functions with
order-preserving losses and standardized supervision. In Proceedings of the 28th International
Conference on Machine Learning (ICML 2011), pages 825?832, 2011.
[3] C. Burges, T. Shaked, E. Renshaw, A. Lazier, M. Deeds, N. Hamilton, and G. Hullender.
Learning to rank using gradient descent. In Proceedings of the 22th International Conference
on Machine Learning (ICML 2005), pages 89?96, 2005.
[4] O. Chapelle. Training a support vector machine in the primal. Neural Computation, 19:1155?
1178, 2007.
[5] O. Chapelle and Y. Chang. Yahoo! learning to rank challenge overview. Journal of Machine
Learning Research - Proceedings Track, 14:1?24, 2011.
[6] O. Chapelle, Y. Chang, and T.-Y. Liu. Future directions in learning to rank. Journal of Machine
Learning Research - Proceedings Track, 14:91?100, 2011.
[7] W. Chen, T.-Y. Liu, Y. Lan, Z. Ma, and H. Li. Ranking measures and loss functions in learning
to rank. In 24th Annual Conference on Neural Information Processing Systems (NIPS 2009),
pages 315?323, 2009.
[8] S. Cl?emenc?on, G. Lugosi, and N. Vayatis. Ranking and scoring using empirical risk minimization. In Proceedings of the 18th Annual Conference on Learning Theory (COLT 2005), volume
3559, pages 1?15, 2005.
[9] D. Cossock and T. Zhang. Subset ranking using regression. In Proceedings of the 19th Annual
Conference on Learning Theory (COLT 2006), pages 605?619, 2006.
[10] O. Dekel, C. D. Manning, and Y. Singer. Log-linear models for label ranking. In 18th Annual
Conference on Neural Information Processing Systems (NIPS 2003), 2003.
[11] J. C. Duchi, L. W. Mackey, and M. I. Jordan. On the consistency of ranking algorithms. In
Proceedings of the 27th International Conference on Machine Learning (ICML 2010), pages
327?334, 2010.
[12] Y. Freund, R. Iyer, R. E. Schapire, and Y. Singer. An efficient boosting algorithm for combining
preferences. Journal of Machine Learning Research, 4:933?969, 2003.
[13] R. Herbrich, K. Obermayer, and T. Graepel. Large margin rank boundaries for ordinal regression. In Advances in Large Margin Classifiers., pages 115?132, 1999.
[14] T. Joachims. Optimizing search engines using clickthrough data. In Proceedings of the 8th
ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD
2002), pages 133?142, 2002.
[15] H. Li, T.-Y. Liu, and C. Zhai. Learning to rank for information retrieval (lr4ir 2008). SIGIR
Forum, 42:76?79, 2008.
[16] T.-Y. Liu. Learning to rank for information retrieval. Foundation and Trends on Information
Retrieval, 3:225?331, 2009.
[17] T.-Y. Liu, J. Xu, T. Qin, W.-Y. Xiong, and H. Li. Letor: Benchmark dataset for research
on learning to rank for information retrieval. In SIGIR ?07 Workshop, San Francisco, 2007.
Morgan Kaufmann.
[18] P. D. Ravikumar, A. Tewari, and E. Yang. On ndcg consistency of listwise ranking methods.
Journal of Machine Learning Research - Proceedings Track, 15:618?626, 2011.
[19] F. Xia, T.-Y. Liu, J. Wang, W. S. Zhang, and H. Li. Listwise approach to learning to rank
- theory and algorithm. In Proceedings of the 25th International Conference on Machine
Learning (ICML 2008), 2008.
[20] T. Zhang. Statistical analysis of some multi-category large margin classification methods.
Journal of Machine Learning Research, 5:1225?1251, 2004.
[21] T. Zhang. Statistical behavior and consistency of classification methods based on convex risk
minimization. Annuals of Statistics, 32:56?85, 2004.
9
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4,050 | 4,666 | Online Regret Bounds for Undiscounted Continuous
Reinforcement Learning
Ronald Ortner??
Montanuniversitaet Leoben
8700 Leoben, Austria
[email protected]
?
?
Daniil Ryabko?
INRIA Lille-Nord Europe, e? quipe SequeL
59650 Villeneuve d?Ascq, France
[email protected]
Abstract
We derive sublinear regret bounds for undiscounted reinforcement learning in continuous state space. The proposed algorithm combines state aggregation with the
use of upper confidence bounds for implementing optimism in the face of uncertainty. Beside the existence of an optimal policy which satisfies the Poisson equation, the only assumptions made are H?older continuity of rewards and transition
probabilities.
1
Introduction
Real world problems usually demand continuous state or action spaces, and one of the challenges for
reinforcement learning is to deal with such continuous domains. In many problems there is a natural
metric on the state space such that close states exhibit similar behavior. Often such similarities can
be formalized as Lipschitz or more generally H?older continuity of reward and transition functions.
The simplest continuous reinforcement learning problem is the 1-dimensional continuum-armed
bandit, where the learner has to choose arms from a bounded interval. Bounds on the regret with
respect to an optimal policy under the assumption that the reward function is H?older continuous have
been given in [15, 4]. The proposed algorithms apply the UCB algorithm [2] to a discretization of the
problem. That way, the regret suffered by the algorithm consists of the loss by aggregation (which
can be bounded using H?older continuity) plus the regret the algorithm incurs in the discretized setting. More recently, algorithms that adapt the used discretization (making it finer in more promising
regions) have been proposed and analyzed [16, 8].
While the continuous bandit case has been investigated in detail, in the general case of continuous
state Markov decision processes (MDPs) a lot of work is confined to rather particular settings, primarily with respect to the considered transition model. In the simplest case, the transition function
is considered to be deterministic as in [19], and mistake bounds for the respective discounted setting have been derived in [6]. Another common assumption is that transition functions are linear
functions of state and action plus
? some noise. For such settings sample complexity bounds have
? T ) bounds for the regret after T steps are shown in [1]. However,
been given in [23, 7], while O(
there is also some research considering more general transition dynamics under the assumption that
close states behave similarly, as will be considered here. While most of this work is purely experimental [12, 24], there are also some contributions with theoretical guarantees. Thus, [13] considers
PAC-learning for continuous reinforcement learning in metric state spaces, when generative sampling is possible. The proposed algorithm is a generalization of the E3 algorithm [14] to continuous
domains. A respective adaptive discretization approach is suggested in [20]. The PAC-like bounds
derived there however depend on the (random) behavior of the proposed algorithm.
Here we suggest a learning algorithm for undiscounted reinforcement learning in continuous state
space. The proposed algorithm is in the tradition of algorithms like UCRL2 [11] in that it implements
1
the ?optimism in the face of uncertainty? maxim, here combined with state aggregation. Thus, the
algorithm does not need a generative model or access to ?resets:? learning is done online, that is, in
a single continual session of interactions between the environment and the learning policy.
? (2+?)/(2+2?) ) for MDPs with 1-dimensional
For our algorithm we derive regret bounds of O(T
state space and H?older-continuous rewards and transition probabilities with parameter ?. These
bounds also straightforwardly generalize to dimension d where the regret is bounded by
? (2d+?)/(2d+2?) ). Thus, in particular, if rewards and transition probabilities are Lipschitz, the
O(T
? (2d+1)/(2d+2)) ) in dimension d and O(T
? 3/4 ) in dimension 1. We also
regret is bounded by O(T
?
present an accompanying lower bound of ?( T ). As far as we know, these are the first regret
bounds for a general undiscounted continuous reinforcement learning setting.
2
Preliminaries
We consider the following setting. Given is a Markov decision process (MDP) M with state space
S = [0, 1]d and finite action space A. For the sake of simplicity, in the following we assume d = 1.
However, proofs and results generalize straightforwardly to arbitrary dimension, cf. Remark 5 below.
The random rewards in state s under action a are assumed to be bounded in [0, 1] with mean r(s, a).
The transition probability distribution in state s under action a is denoted by p(?|s, a).
We will make the natural assumption that rewards and transition probabilities are similar in close
states. More precisely, we assume that rewards and transition probabilities are H?older continuous.
Assumption 1. There are L, ? > 0 such that for any two states s, s0 and all actions a,
|r(s, a) ? r(s0 , a)| ? L|s ? s0 |? .
Assumption 2. There are L, ? > 0 such that for any two states s, s0 and all actions a,
p(?|s, a) ? p(?|s0 , a)
? L|s ? s0 |? .
1
For the sake of simplicity we will assume that ? and L in Assumptions 1 and 2 are the same.
We also assume existence of an optimal policy ? ? : S ? A which gives optimal average reward
?? = ?? (M ) on M independent of the initial state. A sufficient condition for state-independent
optimal reward is geometric convergence of ? ? to an invariant probability measure. This is a natural
condition which e.g. holds for any communicating finite state MDP. It also ensures (cf. Chapter 10
of [10]) that the Poisson equation holds for the optimal policy. In general, under suitable technical
conditions (like geometric convergence to an invariant probability measure ?? ) the Poisson equation
Z
?? + ?? (s) = r(s, ?(s)) +
p(ds0 |s, ?(s)) ? ?? (s0 )
(1)
S
relates the rewards and transition probabilities under any measurable policy ? to its average reward ?? and the bias function ?? : S ? R of ?. Intuitively, the bias is the difference in accumulated
rewards when starting in a different
state. Formally, the bias is defined by the Poisson equation (1)
R
and the normalizing equation S ?? d?? = 0 (cf. e.g. [9]). The following result follows from the
bias definition and Assumptions 1 and 2 (together with results from Chapter 10 of [10]).
Proposition 3. Under Assumptions 1 and 2, the bias of the optimal policy is bounded.
Consequently, it makes sense to define the bias span H(M ) of a continuous state MDP M satisfying
Assumptions 1 and 2 to be H(M ) := sups ??? (s) ? inf s ??? (s). Note that since inf s ??? (s) ? 0
by definition of the bias, the bias function ??? is upper bounded by H(M ).
We are interested in algorithms which can compete with the optimal policy ? ? and measure their
PT
performance by the regret (after T steps) defined as T ?? (M ) ? t=1 rt , where rt is the random
reward obtained by the algorithm at step t. Indeed, within T steps no canonical or even bias optimal
optimal policy (cf. Chapter 10 of [10]) can obtain higher accumulated reward than T ?? + H(M ).
3
Algorithm
Our algorithm UCCR L, shown in detail in Figure 1, implements the ?optimism in the face of uncertainty maxim? just like UCRL2 [11] or REGAL [5]. It maintains a set of plausible MDPs M and
2
Algorithm 1 The UCCR L algorithm
Input: State space S = [0, 1], action space A, confidence parameter ? > 0, aggregation parameter n ? N, upper bound H on the bias span, Lipschitz parameters L, ?.
Initialization:
j
B Let I1 := 0, n1 , Ij := j?1
n , n for j = 2, 3, . . . , n.
B Set t := 1, and observe the initial state s1 and interval I(s1 ).
for episodes k = 1, 2, . . . do
B Let Nk (Ij , a) be the number of times action a has been chosen in a state ? Ij prior to
episode k, and vk (Ij , a) the respective counts in episode k.
Initialize episode k:
B Set the start time of episode k, tk := t.
B Compute estimates r?k (s, a) and p?agg
k (Ii |s, a) for rewards and transition probabilities, using
all samples from states in the same interval I(s), respectively.
Compute policy ?
?k :
? with H(M
? ) ? H and rewards r?(s, a) and
B Let Mk be the set of plausible MDPs M
transition probabilities p?(?|s, a) satisfying
q
7 log(2nAtk /?)
r?(s, a) ? r?k (s, a)
? Ln?? + 2 max{1,N
,
(2)
k (I(s),a)}
q
agg
56n log(2Atk /?)
.
(3)
p (?|s, a) ? p?agg
? Ln?? + max{1,N
?
k (?|s, a)
k (I(s),a)}
1
B
? k ? Mk such that
Choose policy ?
?k and M
? k ) = arg max{?? (M ) | M ? Mk }.
???k (M
(4)
Execute policy ?
?k :
while vk (I(st ), ?
?k (st )) < max{1, Nk (I(st ), ?
?k (st ))} do
B
Choose action at = ?
?k (st ), obtain reward rt , and observe next state st+1 .
B
Set t := t + 1.
end while
end for
? ? M and a policy ?
? ) is maxchooses optimistically an MDP M
? such that the average reward ??? (M
imized, cf. (4). Whereas for UCRL2 and REGAL the set of plausible MDPs is defined by confidence
intervals for rewards and transition probabilities for each individual state-action pair, for UCCR L
we assume an MDP to be plausible if its aggregated rewards and transition probabilities are within
a certain range. This range is defined by the aggregation error (determined by the assumed H?older
continuity) and respective confidence intervals, cf. (2), (3). Correspondingly, the estimates for rewards and transition probabilities for some state action-pair (s, a) are calculated from all sampled
values of action a in states close to s.
More precisely, for the aggregation UCCR L partitions the state space into intervals I1 := 0, n1 ,
k
Ik := k?1
n , n for k = 2, 3, . . . , n. The corresponding aggregated transition probabilities are
defined by
Z
pagg (Ij |s, a) :=
p(ds0 |s, a).
(5)
Ij
Generally, for a (transition) probability distribution p(?) over S we write pagg (?) for the aggregated probability distribution with respect to {I1 , I2 . . . , In }. Now, given the aggregated state space
{I1 , I2 . . . , In }, estimates r?(s, a) and p?agg (?|s, a) are calculated from all samples of action a in
states in I(s), the interval Ij containing s. (Consequently, the estimates are the same for states in
the same interval.)
As UCRL2 and REGAL, UCCR L proceeds in episodes in which the chosen policy remains fixed.
Episodes are terminated when the number of times an action has been sampled from some interval Ij
has been doubled. Only then estimates are updated and a new policy is calculated.
3
Since all states in the same interval Ij have the same confidence intervals, finding the optimal
? k, ?
? agg and
pair M
?k in (4) is equivalent to finding the respective optimistic discretized MDP M
k
agg
agg
agg
?
an optimal policy ?
?k on Mk . Then ?
?k can be set to be the extension of ?
?k to S, that is,
?
?k (s) := ?
?kagg (I(s)) for all s. However, due to the constraint on the bias even in this finite case
? agg and ?
efficient computation of M
?kagg is still an open problem. We note that the REGAL.C algok
rithm [5] selects optimistic MDP and optimal policy in the same way as UCCR L.
While the algorithm presented here is the first modification of UCRL2 to continuous reinforcement
learning problems, there are similar adaptations to online aggregation [21] and learning in finite state
MDPs with some additional similarity structure known to the learner [22].
4
Regret Bounds
For UCCR L we can derive the following bounds on the regret.
Theorem 4. Let M be an MDP with continuous state space [0, 1], A actions, rewards and transition probabilities satisfying Assumptions 1 and 2, and bias span upper bounded by H. Then with
probability 1 ? ?, the regret of UCCR L (run with input parameters n and H) after T steps is upper
bounded by
q
const ? nH AT log T? + const0 ? HLn?? T.
(6)
Therefore, setting n = T 1/(2+2?) gives regret upper bounded by
q
const ? HL A log T? ? T (2+?)/(2+2?) .
With no known upper bound on the bias span, guessing H by log T one still obtains an upper bound
? (2+?)/(2+2?) ).
on the regret of O(T
Intuitively, the second term in the regret bound of (6) is the discretization error, while the first term
corresponds to the regret on the discretized MDP. A detailed proof of Theorem 4 can be found in
Section 5 below.
Remark 5 (d-dimensional case). The general d-dimensional case can be handled as described for
dimension 1, with the only difference being that the discretization now has nd states, so that one
has nd instead of n in the first term of (6). Then choosing n = T 1/(2d+2?) bounds the regret by
? (2d+?)/(2d+2?) ).
O(T
Remark 6 (unknown horizon). If the horizon T is unknown then the doubling trick (executing the
algorithm in rounds i = 1, 2, . . . guessing T = 2i and setting the confidence parameter to ?/2i )
gives the same bounds.
Remark 7 (unknown H?older parameters). The UCCR L algorithm receives (bounds on) the
H?older parameters L as ? as inputs. If these parameters are not known, then one can still obtain
sublinear regret bounds albeit with worse dependence on T . Specifically, we can use the modelselection technique introduced in [17]. To do this, fix a certain number J of values for the constants
L and ?; each of these values will be considered as a model. The model selection consists in running
UCCR L with each of these parameter values for a certain period of ?0 time steps (exploration). Then
one selects the model with the highest reward and uses it for a period of ?00 time steps (exploitation),
while checking that its average reward stays within (6) of what was obtained in the exploitation
phase. If the average reward does not pass this test, then the model with the second-best average
reward is selected, and so on. Then one switches to exploration with longer periods ?1 , etc. Since
there are no guarantees on the behavior of UCCR L when the H?older parameters are wrong, none
of the models can be discarded at any stage. Optimizing over the parameters ?i and ?i0 as done
in [17], and increasing the number J of considered parameter values, one can obtain regret bounds
? (2+2?)/(2+3?) ), or O(T
? 4/5 ) in the Lipschitz case. For details see [17]. Since in this modelof O(T
selection process UCCR L is used in a ?black-box? fashion, the exploration is rather wasteful, and
thus we think that this bound is suboptimal. Recently, the results of [17] have been improved [18],
and it seems that similar analysis gives improved regret bounds for the case of unknown H?older
parameters as well.
The following is a complementing lower bound on the regret for continuous state reinforcement
learning.
4
Theorem 8. For any A, H > 1 and any reinforcement learning algorithm there is a continuous
state reinforcement learning problem with
? A actions and bias span H satisfying Assumption 1 such
that the algorithm suffers regret of ?( HAT ).
Proof. Consider the following reinforcement learning problem with state space [0, 1]. The state
space is partitioned into n intervals Ij of equal size. The transition probabilities for each action a
are on each of the intervals Ij concentrated and equally distributed on the same interval Ij . The
rewards on each interval Ij are also constant for each a and are chosen as in the lower bounds for a
multi-armed bandit problem [3] ?
with nA arms. That is, giving only one arm slightly higher reward,
it is known [3] that regret of ?( nAT ) can be forced upon any algorithm on the respective bandit
problem. Adding another action giving no reward and ?
equally distributing over the whole state
space, the bias span of the problem is n and the regret ?( HAT ).
Remark 9. Note that Assumption 2 does not hold in the example used in the proof of Theorem 8.
However, the transition probabilities are piecewise constant (and hence Lipschitz) and known to
the learner. Actually, it is straightforward to deal with piecewise H?older continuous rewards and
transition probabilities where the finitely many points of discontinuity are known to the learner. If
one makes sure that the intervals of the discretized state space do not contain any discontinuities, it
is easy to adapt UCCR L and Theorem 4 accordingly.
Remark 10 (comparison to bandits). The bounds of Theorems 4 and 8 cannot be directly compared to bounds for the continuous-armed bandit problem [15, 4, 16, 8], because the latter is no
special case of learning MDPs with continuous state space (and rather corresponds to a continuous
action space). Thus, in particular one cannot freely sample an arbitrary state of the state space as
assumed in continuous-armed bandits.
5
Proof of Theorem 4
For the proof of the main theorem we adapt the proof of the regret bounds for finite MDPs in [11]
and [5]. Although the state space is now continuous, due to the finite horizon T , we can reuse
some arguments, so that we keep the structure of the original proof of Theorem 2 in [11]. Some of
the necessary adaptations made are similar to techniques used for showing regret bounds for other
modifications of the original UCRL2 algorithm [21, 22], which however only considered finite-state
MDPs.
5.1
Splitting into Episodes
Let vk (s, a) be the number of times action a has been chosen in episode
P k when being in state s, and
denote the total number of episodes by m. Then setting ?k := s,a vk (s, a)(?? ? r(s, a)), with
probability at least 1 ? 12T?5/4 the regret of UCCR L after T steps is upper bounded by (cf. Section
4.1 of [11]),
q
Pm
5
8T
+ k=1 ?k .
(7)
8 T log
?
5.2
Failing Confidence Intervals
Next, we consider the regret incurred when the true MDP M is not contained in the set of plausible MDPs Mk . Thus, fix a state-action pair (s, a), and recall that r?(s, a) and p?agg (?|s, a) are the
estimates for rewards and transition probabilities calculated from all samples of state-action pairs
contained in the same interval I(s). Now assume that at step t there have been N > 0 samples of
action a in states in I(s) and that in the i-th sample a transition from state si ? I(s) to state s0i has
been observed (i = 1, . . . , N ).
First, concerning the rewards one obtains as in the proof of Lemma 17 in Appendix C.1 of [11] ? but
now using Hoeffding for independent and not necessarily identically distributed random variables
? that
n
q 7
o
?
Pr r?(s, a) ? E[?
r(s, a)] ? 2N
log 2nAt
?
.
(8)
?
60nAt7
5
Concerning the transition probabilities, we have for a suitable x ? {?1, 1}n
n
X
agg
agg
p (?|s, a) ? E[?
p (Ij |s, a) ? E[?
pagg (?|s, a)]
=
pagg (Ij |s, a)]
?
?
1
=
n
X
j=1
p?agg (Ij |s, a) ? E[?
pagg (Ij |s, a)] x(Ij )
j=1
=
1
N
N
X
x(I(s0i )) ?
Z
p(ds0 |si , a) ? x(I(s0 )) .
(9)
S
i=1
R
For any x ? {?1, 1}n , Xi := x(I(s0i ))? S p(ds0 |si , a)?x(I(s0 )) is a martingale difference sequence
PN
with |Xi | ? 2, so that by Azuma-Hoeffding inequality (e.g., Lemma 10 in [11]), Pr{ i=1 Xi ?
?} ? exp(??2 /8N ) and in particular
q
nP
o ? 7n
?
N
2At
? 2At
.
56nN
log
? n
Pr
X
?
i=1 i
?
2 20nAt7
A union bound over all sequences x ? {?1, 1}n then yields from (9) that
q
n
o
?
agg
2At
p (?|s, a) ? E[?
pagg (?|s, a)]
? 56n
Pr
?
.
(10)
log
?
N
?
20nAt7
1
Another union bound over all t possible values for N , all n intervals and all actions shows that the
?
confidence intervals in (8) and (10) hold at time t with probability at least 1 ? 15t
6 for the actual
counts N (I(s), a) and all state-action pairs (s, a). (Note that the equations (8) and (10) are the same
for state-action pairs with states in the same interval.)
PN
Now, by linearity of expectation E[?
r(s, a)] can be written as N1 i=1 r(si , a). Since the si are assumed to be in the same interval
it follows that |E[?
r(s, a)] ? r(s, a)| < Ln?? . Similarly,
I(s), ??
agg
agg
E[?
p (?|s, a)] ? p (?|s, a) 1 < Ln . Together with (8) and (10) this shows that with proba?
bility at least 1 ? 15t
6 for all state-action pairs (s, a)
q
7 log(2nAt/?)
r?(s, a) ? r(s, a) < Ln?? +
(11)
2 max{1,N (I(s),a)} ,
q
agg
56n log(2At/?)
p (?|s, a) ? pagg (?|s, a)
< Ln?? + max{1,N
(12)
?
(I(s),a)} .
1
This shows that the true MDP is contained in the set of plausible MDPs M(t) at step t with proba?
bility at least 1 ? 15t
6 , just as in Lemma 17 of [11]. The argument that
m
X
?k 1M 6?Mk ?
?
T
(13)
k=1
with probability at least 1 ? 12T?5/4 then can be taken without any changes from Section 4.2 of [11].
5.3
Regret in Episodes with M ? Mk
? k and ?
Now for episodes with M ? Mk , by the optimistic choice of M
?k in (4) we can bound
X
?
?k =
vk (s, ?
?k (s)) ? ? r(s, ?
?k (s))
s
?
X
=
X
vk (s, ?
?k (s)) ???k ? r(s, ?
?k (s))
s
X
vk (s, ?
?k (s)) ???k ? r?k (s, ?
?k (s)) +
vk (s, ?
?k (s)) r?k (s, ?
?k (s)) ? r(s, ?
?k (s)) .
s
s
Any term r?k (s, a) ? r(s, a) ? |?
rk (s, a) ? r?k (s, a)| + |?
rk (s, a) ? r(s, a)| is bounded according to
? k , M ? Mk , so that summarizing states in the same interval Ij
(2) and (11), as we assume that M
n X
q
X
X
7 log(2nAtk /?)
?
??
?k ?
vk (s, ?
?k (s)) ??k ? r?k (s, ?
?k (s)) + 2
vk (Ij , a) Ln + 2 max{1,Nk (Ij ,a)} .
s
j=1 a?A
6
Since max{1, Nk (Ij , a)} ? tk ? T , setting ?k := tk+1 ? tk to be the length of episode k we have
X
?k ?
vk (s, ?
?k (s)) ???k ? r?k (s, ?
?k (s))
s
q
+ 2Ln?? ?k +
14 log
2nAT
?
n X
X
p
j=1 a?A
vk (Ij , a)
.
max{1, Nk (Ij , a)}
(14)
We continue analyzing the first term on the right hand side of (14). By the Poisson equation (1) for
? k := ?
? ?? we can write
? k , denoting the respective bias by ?
?
?k on M
k
X
?
vk (s, ?
?k (s)) ??k ? r?k (s, ?
?k (s))
s
Z
X
? k (s0 ) ? ?
? k (s)
=
vk (s, ?
?k (s))
p?k (ds0 |s, ?
?k (s)) ? ?
S
s
X
=
vk (s, ?
?k (s))
Z
(15)
S
s
+
? k (s0 ) ? ?
? k (s)
p(ds0 |s, ?
?k (s)) ? ?
X
vk (s, ?
?k (s))
s
n Z
X
j=1
? k (s0 ).
p?k (ds0 |s, ?
?k (s)) ? p(ds0 |s, ?
?k (s)) ? ?
(16)
Ij
5.4
The True Transition Functions
agg
agg
? (?|s, a) ? pagg (?|s, a)
Now
p?k (?|s, a) ? pagg (?|s, a)
1 ?
p?agg
?agg
k (?|s, a) ? p
k (?|s, a) 1 + p
k
1
? k , M ? Mk . Hence, since by definition of
can be bounded by (3) and (12), because we assume M
? k , the term in (16) is bounded by
the algorithm H bounds the bias function ?
n Z
X
X
? k (s0 ) p?k (ds0 |s, ?
vk (s, ?
?k (s))
?
?k (s)) ? p(ds0 |s, ?
?k (s))
s
j=1
?
X
?
X
Ij
vk (s, ?
?k (s)) ? H ?
s
n
X
p?agg
?k (s)) ? pagg (Ij |s, ?
?k (s))
k (Ij |s, ?
j=1
q
56n log(2AT /?)
vk (s, ?
?k (s)) ? H ? 2 Ln?? + max{1,N
k (I(s),at )}
s
= 2HLn?? ?k + 4H
q
14n log
2AT
?
n X
X
p
j=1 a?A
vk (Ij , a)
,
max{1, Nk (Ij , a)}
(17)
while for the term in (15)
Z
X
? k (s0 ) ? ?
? k (s)
vk (s, ?
?k (s))
p(ds0 |s, ?
?k (s)) ? ?
S
s
=
tk+1 ?1 Z
X
S
t=tk
=
? k (s0 ) ? ?
? k (st )
p(ds0 |st , at ) ? ?
tk+1 ?1 Z
X
? k (s0 ) ? ?
? k (st+1 ) + ?
? k (st ) ? ?
? k (st ).
p(ds0 |st , at ) ? ?
k+1
k
S
t=tk
Let
R k(t)0 be the index 0of the episode time step t belongs to. Then the sequence Xt :=
? k(t) (s ) ? ?
? k(t) (st+1 ) is a sequence of martingale differences so that Azumap(ds |st , at ) ? ?
S
Hoeffding inequality shows (cf. Section 4.3.2 and in particular eq. (18) in [11]) that after summing
over all episodes we have
m tk+1
X
X?1 Z
? k (s0 ) ? ?
? k (st+1 ) + ?
? k (st ) ? ?
? k (st )
p(ds0 |st , at ) ? ?
k+1
k
k=1
t=tk
S
?H
7
q
5
2T
log
8T
?
+ HnA log2
8T
nA
,
(18)
where the second term comes from an upper bound on the number of episodes, which can be derived
analogously to Appendix C.2 of [11].
5.5
Summing over Episodes with M ? Mk
To conclude, we sum (14) over all the episodes with M ? Mk , using (15), (17), and (18). This
yields that with probability at least 1 ? 12T?5/4
m
m X
n X
q
X
X
vk (Ij , a)
p
?
?k 1M ?Mk ? 2HLn?? T + 4H 14n log 2AT
?
max{1, Nk (Ij , a)}
k=1
k=1 j=1 a?A
q
8T
+ H 52 T log 8T
+ HnA log2 nA
?
m X
n X
q
X
vk (Ij , a)
p
+ 2Ln?? T + 14 log 2nAT
.
(19)
?
max{1,
Nk (Ij , a)}
k=1 j=1 a?A
Analogously to Section 4.3.3 and Appendix C.3 of [11], one can show that
n XX
X
?
?
vk (Ij , a)
p
?
2 + 1 nAT ,
max{1, Nk (Ij , a)}
j=1 a?A k
and we get from (19) after some simplifications that with probability ? 1 ?
m
q
X
8T
?k 1M ?Mk ? H 52 T log 8T
+ HnA log2 nA
?
?
12T 5/4
k=1
q
+ (4H + 1) 14n log
2AT
?
?
2+1
?
nAT + 2(H + 1)Ln?? T .
(20)
Finally, evaluating (7) by summing ?k over all episodes, by (13) and (20) we have with probability
? 1 ? 4T?5/4 an upper bound on the regret of
m
m
q
X
X
5
8T
T
log
+
?
1
+
?k 1M ?Mk
k M ?M
/ k
8
?
k=1
?
q
5
8T
log
8T
?
k=1
?
+
T +H
q
+ (4H + 1) 14n log
2AT
?
q
5
2T
?
log
8T
?
2+1
?
+ HnA log2
8T
nA
nAT + 2(H + 1)Ln?? T.
A union bound over all possible values of T and further simplifications as in Appendix C.4 of [11]
finish the proof.
6
Outlook
We think that a generalization of our results to continuous action space should not pose any major
problems. In order to improve over the given bounds, it may be promising to investigate more
sophisticated discretization patterns.
The assumption of H?older continuity is an obvious, yet not the only possible assumption one can
make about the transition probabilities and reward functions. A more general problem is to assume
a set F of functions, find a way to measure the ?size? of F, and derive regret bounds depending on
this size of F.
Acknowledgments
The authors would like to thank the three anonymous reviewers for their helpful suggestions and
R?emi Munos for useful discussion which helped to improve the bounds. This research was funded by
the Ministry of Higher Education and Research, Nord-Pas-de-Calais Regional Council and FEDER
(Contrat de Projets Etat Region CPER 2007-2013), ANR projects EXPLO-RA (ANR-08-COSI004), Lampada (ANR-09-EMER-007) and CoAdapt, and by the European Community?s FP7 Program under grant agreements n? 216886 (PASCAL2) and n? 270327 (CompLACS). The first author
is currently funded by the Austrian Science Fund (FWF): J 3259-N13.
8
References
[1] Yasin Abbasi-Yadkori and Csaba Szepesv?ari. Regret bounds for the adaptive control of linear quadratic
systems. COLT 2011, JMLR Proceedings Track, 19:1?26, 2011.
[2] Peter Auer, Nicol`o Cesa-Bianchi, and Paul Fischer. Finite-time analysis of the multi-armed bandit problem. Mach. Learn., 47:235?256, 2002.
[3] Peter Auer, Nicol`o Cesa-Bianchi, Yoav Freund, and Robert E. Schapire. The nonstochastic multiarmed
bandit problem. SIAM J. Comput., 32:48?77, 2002.
[4] Peter Auer, Ronald Ortner, and Csaba Szepesv?ari. Improved rates for the stochastic continuum-armed
bandit problem. In Learning Theory, 20th Annual Conference on Learning Theory, COLT 2007, pages
454?468, 2007.
[5] Peter L. Bartlett and Ambuj Tewari. REGAL: A regularization based algorithm for reinforcement learning
in weakly communicating MDPs. In Proc. 25th Conference on Uncertainty in Artificial Intelligence, UAI
2009, pages 25?42, 2009.
[6] Andrey Bernstein and Nahum Shimkin. Adaptive-resolution reinforcement learning with polynomial
exploration in deterministic domains. Mach. Learn., 81(3):359?397, 2010.
[7] Emma Brunskill, Bethany R. Leffler, Lihong Li, Michael L. Littman, and Nicholas Roy. Provably efficient
learning with typed parametric models. J. Mach. Learn. Res., 10:1955?1988, 2009.
[8] S?ebastien Bubeck, R?emi Munos, Gilles Stoltz, and Csaba Szepesv?ari. Online optimization of ?-armed
bandits. In Advances in Neural Information Processing Systems 22, NIPS 2009, pages 201?208, 2010.
[9] On?esimo Hern?andez-Lerma and Jean Bernard Lasserre. Discrete-time Markov control processes, volume 30 of Applications of mathematics. Springer, 1996.
[10] On?esimo Hern?andez-Lerma and Jean Bernard Lasserre. Further topics on discrete-time Markov control
processes, volume 42 of Applications of mathematics. Springer, 1999.
[11] Thomas Jaksch, Ronald Ortner, and Peter Auer. Near-optimal regret bounds for reinforcement learning.
J. Mach. Learn. Res., 11:1563?1600, 2010.
[12] Nicholas K. Jong and Peter Stone. Model-based exploration in continuous state spaces. In Abstraction,
Reformulation, and Approximation, 7th International Symposium, SARA 2007, pages 258?272. Springer,
2007.
[13] Sham Kakade, Michael J. Kearns, and John Langford. Exploration in metric state spaces. In Machine
Learning, Proc. 20th International Conference, ICML 2003, pages 306?312, 2003.
[14] Michael J. Kearns and Satinder P. Singh. Near-optimal reinforcement learning in polynomial time. Mach.
Learn., 49:209?232, 2002.
[15] Robert Kleinberg. Nearly tight bounds for the continuum-armed bandit problem. In Advances Neural
Information Processing Systems 17, NIPS 2004, pages 697?704, 2005.
[16] Robert Kleinberg, Aleksandrs Slivkins, and Eli Upfal. Multi-armed bandits in metric spaces. In Proc.
40th Annual ACM Symposium on Theory of Computing, STOC 2008, pages 681?690, 2008.
[17] Odalric-Ambrym Maillard, R?emi Munos, and Daniil Ryabko. Selecting the state-representation in reinforcement learning. In Advances Neural Processing Systems 24, NIPS 2011, pages 2627?2635, 2012.
[18] Odalric-Ambrym Maillard, Phuong Nguyen, Ronald Ortner, and Daniil Ryabko. Optimal regret bounds
for selecting the state representation in reinforcement learning. accepted for ICML 2013.
[19] Gerhard Neumann, Michael Pfeiffer, and Wolfgang Maass. Efficient continuous-time reinforcement learning with adaptive state graphs. In Machine Learning: ECML 2007, 18th European Conference on Machine
Learning, pages 250?261, 2007.
[20] Ali Nouri and Michael L. Littman. Multi-resolution exploration in continuous spaces. In Advances in
Neural Information Processing Systems 21, NIPS 2008, pages 1209?1216, 2009.
[21] Ronald Ortner. Adaptive aggregation for reinforcement learning in average reward Markov decision
processes. Ann. Oper. Res., 2012. doi:10.1007/s10479-12-1064-y, to appear.
[22] Ronald Ortner, Daniil Ryabko, Peter Auer, and R?emi Munos. Regret bounds for restless Markov bandits.
In Proc. 23rd Conference on Algorithmic Learning Theory, ALT 2012, pages 214?228, 2012.
[23] Alexander L. Strehl and Michael L. Littman. Online linear regression and its application to model-based
reinforcement learning. In Advances Neural Information Processing Systems 20, NIPS 2007, pages 1417?
1424, 2008.
[24] William T. B. Uther and Manuela M. Veloso. Tree based discretization for continuous state space reinforcement learning. In Proc. 15th National Conference on Artificial Intelligence and 10th Innovative
Applications of Artificial Intelligence Conference, AAAI 98, IAAI 98, pages 769?774, 1998.
9
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4,051 | 4,667 | Entropy Estimations Using Correlated Symmetric
Stable Random Projections
Ping Li
Department of Statistical Science
Cornell University
Ithaca, NY 14853
[email protected]
Cun-Hui Zhang
Department of Statistics and Biostatistics
Rutgers University
New Brunswick, NJ 08901
[email protected]
Abstract
Methods for efficiently estimating Shannon entropy of data streams have important applications in learning, data mining, and network anomaly detections (e.g.,
the DDoS attacks). For nonnegative data streams, the method of Compressed
Counting (CC) [11, 13] based on maximally-skewed stable random projections
can provide accurate estimates of the Shannon entropy using small storage. However, CC is no longer applicable when entries of data streams can be below zero,
which is a common scenario when comparing two streams. In this paper, we
propose an algorithm for entropy estimation in general data streams which allow
negative entries. In our method, the Shannon entropy is approximated by the finite difference of two correlated frequency moments estimated from correlated
samples of symmetric stable random variables. Interestingly, the estimator for the
moment we recommend for entropy estimation barely has bounded variance itself,
whereas the common geometric mean estimator (which has bounded higher-order
moments) is not sufficient for entropy estimation. Our experiments confirm that
this method is able to well approximate the Shannon entropy using small storage.
1 Introduction
Computing the Shannon entropy in massive data have important applications in neural computation [17], graph estimation [5], query logs analysis in Web search [14], network anomaly detection [21], etc. (See NIPS2003 workshop on entropy estimation www.menem.com/?ilya/
pages/NIPS03). In modern applications, as massive datasets are often generated in a streaming
fashion, entropy estimation in data streams has become a challenging and interesting problem.
1.1 Data Streams
Massive data generated in a streaming fashion are difficult to transmit and store [15], as the processing is often done on the fly in one-pass of the data. The problem of ?scaling up for high
dimensional data and high speed data streams? is among the ?ten challenging problems in data mining research? [20]. Mining data streams at petabyte scale has become an important research area [1],
as network data can easily reach that scale [20].
In the standard turnstile model [15], a data stream is a vector At of length D, where D = 264 or
even D = 2128 is possible in network applications, e.g., (a pair of) IP addresses + port numbers. At
time t, there is an input stream at = (it , It ), it ? [1, D] which updates At by a linear rule:
At [it ] = At?1 [it ] + It .
(1)
where It is the increment/decrement of package size at t. For network traffic, normally At [i] ? 0,
which is called the strict turnstile model and suffices for describing certain natural phenomena. On
the other hand, the general turnstile model (which allows At [i] < 0) is often used for comparing
two streams, e.g., in network OD (origin-destination) flow analysis [21].
An important task is to compute the ?-th frequency moment F(?) and the Shannon entropy H:
F(?) =
D
X
?
|At [i]| ,
H=?
i=1
D
X
|At [i]|
i=1
F1
log
|At [i]|
,
F1
(2)
The exact computation of these summary statistics is not feasible because to do so one has to store
the entire vector At of length D, as the entries are time-varying. Also, many applications (such as
anomaly detections of network traffic) require computing the summary statistics in real-time.
1
1.2 Network Measurement, Monitoring, and Anomaly Detection
Network traffic is a typical example of high-rate data streams. Industries are now prepared to move
to 100 Gbits/second or Terabit/second Ethernet. An effective and reliable measurement of network
traffic in real-time is crucial for anomaly detection and network diagnosis; and one such measurement metric is the Shannon entropy [4, 8, 19, 2, 9, 21]. The exact entropy measurement in real-time
on high-speed links is however computationally prohibitive.
The Distributed Denial of Service (DDoS) attack is a representative example of network anomalies. A DDoS attack attempts to make computers unavailable to intended
users, either by forcing users to reset the computers or
by exhausting the resources of service-hosting sites. For
example, hackers may maliciously saturate the victim
machines by sending many external communication requests. DDoS attacks typically target sites such as banks,
credit card payment gateways, or military sites. A DDoS
attack normally changes the statistical distribution of network traffic, which could be reliably captured by the abnormal variations in the measurements of Shannon entropy [4]. See Figure 1 for an illustration.
source IP address: entropy value
10
9
8
7
6
5
4
3
2
1
0
200
400
600
800 1000
packet counts (thousands)
1200
Figure 1: This plot is reproduced from
a DARPA conference [4]. One can view
x-axis as the surrogate for time. Y-axis
is the measured Shannon entropy, which
exhibited a sudden sharp change at the
time when an attack occurred.
Apparently, the entropy measurements do not have to be
?perfect? for detecting attacks. It is however crucial that
the algorithms should be computationally efficient (i.e., real-time and one-pass) at low memory cost,
because the traffic data generating by large high-speed networks are enormous and transient.
1.3 Symmetric Stable Random Projections and Entropy Estimation Using Moments
It turns out that, for 0 < ? ? 2, one can use stable random projections to compute F(?) efficiently
because the Turnstile model (1) is a linear model and the random projection operation is also linear
(i.e., vector-matrix multiplication) [7]. Conceptually, we multiply the data stream vector At ? RD
by a random matrix R ? RD?k , resulting in a vector X = At ? R ? Rk with entries
xj = [At ? R]j =
D
X
rij At [i], j = 1, 2, ..., k
i=1
where rij ? S(?, 1) is a symmetric ?-stable random variable with unit scale [3, 22]: E(erij t ) =
?
e?|t| . The standard normal (or Cauchy) distribution is a special case with ? = 2 (or ? = 1).
In data stream computations, the matrix R is not materialized. The standard procedure is to
(re)generate entries of R on-demand [7] using pseudo-random numbers [16]. Thus, we only need
to store X ? Rk . When a stream element at = (it , It ) arrives, one updates the entries of X:
xj ? xj + It rit j , j = 1, 2, ..., k.
By property of stable distributions, the samples xj , j = 1 to k, are also i.i.d. stable
!
?
D
D
X
X
?
xj =
rij At [i] ? S ?, F(?) =
|At [i]|
i=1
(3)
(4)
i=1
Therefore, the task boils down to estimating the scale parameter from k i.i.d. stable samples.
Because the Shannon entropy is essentially the derivative of the frequency moment at ? = 1, the
popular approach is to approximate the Shannon entropy by the Tsallis entropy [18]:
?
!
F(?)
1
1? ?
T? =
.
(5)
??1
F(1)
which approaches the Shannon entropy H as ? ? 1. [21] used a slight variant of (5) but the
difference is not essential.1 In their approach, F(?) and F(1) are first estimated separately from
1
F
?F
[21] used (1+?)2? (1??) and estimated the two frequency moments independently. The subtle difference
between the finite difference approximations is not essential. It is the correlation that plays the crucial role.
2
two independent sets of samples. The estimated moments are then plugged in (5) to estimate the
Shannon entropy H. Immediately, we can see the problem here: the variance of the estimated T(?)
1
1
2
might be proportional to (??1)
2 = ?2 . (Recall var(cX) = c var(X)).
One question is how to choose ? (i.e., ?). [6] proposed a conservative criterion by choosing ?
?7
according to the worst case bias |H ? T? |. One can verify
? 14that
? ? = |1 ? ?| < 10 is likely in
[6]. In other words, the required sample size could be O 10 . In practice, [21] exploited the biasvariance tradeoff but they still had to use an excessive number of samples, e.g., 106 . In comparison,
using our proposed approach, it appears that 100 ? 1000 samples might be sufficient.
1.4 Our Proposal
We have made two key contributions. Firstly, instead of estimating F(?) and F(1) separately using
two independent sets of samples, we make them highly positively correlated. Intuitively, if the two
consistent estimators, denoted by F?(?) and F?(1) respectively, are highly positively
correlated, then
?
?
possibly their ratio
F?(?)
F? ?
can be close to 1 with small variance. Ideally, if V ar
?
(1)
variance of the estimated Tsallis entropy T?? =
1
??1
?
It turns out that finding an estimator with V ar
?
F
(?)
??
F
1?
?
F
(?)
??
F
?
?
F
(?)
??
F
? ?
= O ?2 , the
(1)
will be essentially independent of ?.
(1)
?
? ?
= O ?2 was not straightforward. It is known
(1)
that around ? = 1, the geometric mean estimator [10] is nearly statistically optimal. Interestingly,
our analysis and
show that using the geometric mean estimator, we can essentially only
? simulation
?
?
F
achieve V ar F?(?)
= O (?), which, albeit a large improvement, is not small sufficient to cancel
?
(1)
? 1 ?
the O ?2 term. Therefore, our second key component is a new estimator of T? using a moment
estimator which does not have (or barely has) finite variance. Even though such an estimator is not
good for estimating the single moment compared to the geometric mean, due to the high correlation,
? ?
F?
the ratio F?(?)
is still very well-behaved and its variance is essentially O ?2 , as shown in our
?
(1)
theoretical analysis and experiments.
1.5 Compressed Counting (CC) for Nonnegative Data Streams
The recent work [13] on Compressed Counting (CC) [11] provides an ideal solution to the problem
of entropy estimation in nonnegative data streams. Basically, for nonnegative data streams, i.e.,
At [i] ? 0 at all times and all locations, we can compute the first moment easily, because
F(1) =
D
X
|At [i]| =
i=1
D
X
i=1
At [i] =
t
X
Is
(6)
s=0
where Is is the increment/decrement at time s. In other words, we just need a single counter to
accumulate all the increments Is . This observation lead to the conjecture that estimating F(?) should
be also easy if ? ? 1, which consequently lead to the development of Compressed Counting which
used maximally-skewed stable random projections instead of symmetric stable projections.? The
?
most recent work of CC [13] provided a new moment estimator to achieve the variance ? O ?2 .
Unfortunately, for general data streams where entries can be negative, we have to resort to symmetric stable random projections. Fundamentally, the reason that skewed projections work well on
nonnegative data streams is because the data themselves are skewed. However, when we compare
two streams, the data become more or less symmetric and hence we must use symmetric projections.
1.6 Why Comparing the Difference of Two Streams?
In machine learning research and practice, people routinely use the difference between feature vectors. [21] used the difference between data streams from a slightly different motivation.
The goal of [21] is to measure the entropies of all OD pairs (origin-destination) in a network, because
entropy measurements are crucial for detecting anomaly events such as DDoS attacks and network
failures. They argued that the change of entropy of the traffic distribution may be invisible (i.e., too
small to be detected) in the traditional volume matrix even during the time when an attack occurs.
Instead, they proposed to measure the entropy from a number of locations across the network, i.e.,
3
by examining the entropy of every OD flow in the network. In a similar argument, a DDoS attack
may be invisible in terms of the traffic volume change, if the attack is launched outside the network.
While [21] successfully demonstrated that measuring the Shannon entropy of OD flows is effective
for detecting anomaly events, at that time they did not have the tools for efficiently estimating the
entropy. Using symmetric stable random projections and independent samples,? they
? needed a large
number of samples (e.g., 106 ) because their variance blows up at the rate of O ?12 .
For anomaly detection, reducing the sample size (k) is crucial because k determines the storage
and estimation speed; and it is often required to detect the events at real time. In addition, the
pseudo-random numbers have to be (re)-generated on the fly, at a cost proportional to k.
2 Our Proposed Algorithm
Recall that a data stream is a long vector At [i], i = 1 to D. At time t, an incoming element
at = (it , It ) updates one entry: At [it ] ? At?1 [it ] + It . Conceptually, we generate a random
matrix R ? RD?k whose entries are sampled from a stable distribution and multiply it with At :
X = At ? R. The matrix multiplication is linear and can be conducted incrementally as the new
stream elements arrive. R is not materialized; its entries are re-generated on demand using pseudorandom numbers, as the standard practice in data stream computations [7]. Our method does not
require At [i] ? 0 and hence it can handle the difference between two streams (e.g., the OD flows).
2.1 The Symmetric Stable Law
Our work utilizes the symmetric stable distribution. We adopt the standard approach [3] to sample
from the stable law S(?, 1) with index ? and unit scale. We generate two independent random
variables: w ? exp(1) and u ? unif om(??/2, ?/2) and feed them to a nonlinear transformation:
Z(?) = g(w, u, ?) =
sin(?u) h cos(u ? ?u) i(1??)/?
? S(?, 1),
w
(cos u)1/?
(7)
to obtain a sample from S(?, 1). An important property is that, for ?1 < ? < ?, the moment exists:
E|Z|? = (2/?)?(1 ? ?/?)?(?) sin(??/2). For convenience, we define
G(?, ?) = E|g(w, u, ?)|? =
2
?(1 ? ?/?)?(?) sin (??/2)
?
(8)
2.2 Our Recommended Estimator
Conceptually, we have two matrices of i.i.d. random numbers:
wij ? exp(1),
uij ? unif orm(??/2, ?/2),
i = 1, 2, ..., D, j = 1, 2, ..., k,
(9)
As new stream elements arrive, we incrementally maintain two sets of samples, i.e., for i = 1 to k,
xj =
D
X
At [i]g(wij , uij , 1),
yj =
i=1
D
X
At [i]g(wij , uij , ?)
(10)
i=1
Note that xj and yj are highly correlated because they are generated using the same random numbers
(with different ?). However, xi and yj are independent if i 6= j.
Our recommended estimator of the Tsallis entropy T? is
?
?
!2? ?
Pk p
?
|yj |
1
?
j=1
?1 ?
?
?
? Pk p
T??,0.5 =
1
??1
? 1 ? 2?
|x
|
j
j=1
(11)
where ? = 1 + ? > 1 and the meaning of 0.5 will soon be clear. When ? is sufficiently small, the
estimated Tsallis entropy will be sufficiently close to the Shannon entropy. A nice property is that its
1
variance is free of ?
or ?12 terms. While it is intuitively clear that it is beneficial to make xj and yj
highly correlated for the sake of reducing the variance, it might not be as intuitive why T??,0.5 (11)
is a good estimator for the entropy. We will explain why the obvious geometric mean estimator [10]
is not sufficient for entropy estimation.
4
3
The Geometric Mean Estimator
For estimating F(?) , the geometric mean estimator [10] is close to be statistically optimal (efficiency
? 80%) at ? ? 1. Thus, it was our first attempt to test the following estimator of the Tsallis entropy:
?
!
Qk
Qk
1/k
?(?),gm
|yj |?/k
F
1
j=1
j=1 |xj |
T??,gm =
1? ?
, where F?(?),gm = k
, F?(1),gm =
,
??1
G (?, ?/k)
Gk (1, 1/k)
F?(1),gm
where G() is defined in (8). After simplification, we obtain:
?
#?
"? ?
k
Y
? yj ??/k G(1, 1/k)
1
? ?
?1 ?
?.
T??,gm =
? xj ?
??1
G(?,
?/k)
j=1
3.1
(12)
Theoretical Analysis
The theoretical analysis of T?(?),gm , however, turns out to be difficult, as it requires computing
??
?
?
"? ?
#
? PD A [i]g(w , u , ?) ?s?/k
? yj ?s?/k
?
?
t
ij
ij
?,
E ?? ??
= E ?? Pi=1
s = 1, 2,
(13)
?
? D At [i]g(wij , uij , 1) ?
xj
i=1
where g() is defined in (7). We first provide the following Lemma:
Lemma 1 Let w ? exp(1) and u ? unif orm(??/2, ?/2) be two independent variables. Let
? = 1 + ? > 1, for small ? > 0. Then, for ? > ?1,
?
?
? g(w, u, ?) ??
?
?
E?
g(w, u, 1) ?
?
?
?
?
=1 ? 0.5772?? + 0.5772??2 ? 1.6386??3 + 1.6822? 2 ?2 + O ??4 + O ? 2 ?3
? (14)
Note that we need to keep higher order terms in order to prove Lemma 2, to show the properties of
the geometric mean estimator, when D = 1 (i.e., a stream with only one element).
Lemma 2 If D = 1, then
? 2?
? ?
?
? 1 ?2
? ?
?
?
1
E T??,gm =
? 2.0935 + 1.0614?2 + O ?3 + O
+O
k 2
k
k
k2
? ?
? ?
?
? 3.3645
?
1
V ar T??,gm =
+O
+O
k
k
k2
(15)
?
(16)
When D = 1, we know T? = H = 0. In this case, the geometric mean estimator T??,gm is
1
, which is very encouraging.
asymptotically unbiased with variance essentially free of ?
Will this result in Lemma 2 extend to general D? The answer is no, even for D = 2, i.e.,
yj
At [1]g(w1j , u1j , ?) + At [2]g(w2j , u2j , ?)
=
xj
At [1]g(w1j , u1j , 1) + At [2]g(w2j , u2j , 1)
Because g() is symmetric, it is possible that the denominator At [1]g(w1j , u1j , 1) +
At [2]g(w2j , u2j , 1) might be very small while the numerator At [1]g(w1j , u1j , ?) +
At [2]g(w2j , u2j , ?) is not too small. In other words, there will be more variations when
D > 1. In fact, our experiments in Sec. 3.2 and the theoretical analysis
? 1 ?of a more general estimator
in Sec. 4 both reveal that the variance of T??,gm
is
essentially
O
? , which is of course still a
? ?
substantial improvement over the previous O ?12 solution.
3.2
Experiments on the Geometric Mean Estimator (Correlated vs. Independent samples)
We present some experimental results for evaluating T??,gm , to demonstrate that (i) using correlation
does substantially reduce variance and hence reduces the? required
sample size, and (ii) the variance
?
1
(or MSE, the mean square error) of T??,gm is roughly O ?
.
5
We follow [13] by using static data to evaluate the accuracies of the estimators. The projected vector
X = At ? R is the same at the end of the stream, regardless of whether it is computed at once (i.e.,
static) or incrementally (i.e., dynamic). Following [13], we selected 4 word vectors from a chunk of
Web crawl data. For example, the entries for vector ?REVIEW? are the numbers of occurrences of
the word ?REVIEW? in each document. We group these 4 vectors into 2 pairs: ?THIS-HAVE? and
?DO-REVIEW? and we estimate the Shannon entropies of the two resultant difference vectors.
Figure 2 presents the mean square errors (MSE) of the estimated Shannon entropy, i.e., E(T?? ?H)2 ,
normalized by the truth (H 2 ). The left panels contain the results using independently sampling
(i.e., the prior work [21]) and the geometric mean estimator. The middle panels contain the results
using correlated sampling (i.e., this paper) and the geometric mean estimator (12). The right panels
multiply the results of the middle panels by ?? to
? illustrate that the variance of the geometric mean
1
estimator for entropy T??,gm is essentially O ?
. See more experiments in Figure 3.
6
6
100
2
10
0
10
k = 1000
?2
4
10
k = 10
2
10
k = 100
0
10
k = 1000
?2
10
10
THIS?HAVE : GM + Indep.
?5
10
?4
10
?3
?2
10
10
?=??1
0
10
?5
10
10
?4
?3
10
?2
10
10
?=??1
?1
10
10
6
Normalized MSE
2
100
10
0
10
k = 1000
?2
10
2
10
k = 10
0
10
k = 100
?2
?5
?4
10
?3
?2
10
10
?=??1
k = 100
?4
10
10
DO?REVIEW : GM + Corr.
4
k = 1000
10
DO?REVIEW : GM + Indep.
10
10
k = 1000
MSE ? ?
?4
10
?3
?2
10
10
?=??1
?1
0
10
0
10
DO?REVIEW : GM + Corr.
?2
k = 10
10
?3
k = 100
10
?4
k = 1000
10
MSE ? ?
?5
10
?1
10
?1
10
k = 10
10
k = 10
?3
10 ?5
10
0
Normalized MSE ? ?
6
4
THIS?HAVE : GM + Corr.
?2
10
?5
?1
10
Normalized MSE
10
THIS?HAVE : GM + Corr.
Normalized MSE ? ?
10
10
?1
10
k = 10
4
Normalized MSE
Normalized MSE
10
?5
10
?4
?3
10
?2
10
10
?=??1
?1
0
10
10
10 ?5
10
?4
10
?3
?2
10
10
?=??1
?1
10
0
10
Figure 2: Two pairs of word vectors were selected. We conducted symmetric random projections
using both independent sampling (left panels, as in [21]) and correlated sampling (middle panels, as
our proposal). The Tsallis entropy (of the difference vector) is estimated using the geometric mean
estimator (12) with three sample sizes k = 10, 100, and 1000. The normalized mean square errors
(MSE: E|T??,gm ? H|2 /H 2 ) verify that correlated sampling reduces the errors substantially.
4
The General Estimator
Since the geometric mean estimator could not satisfactorily solve the entropy estimation problem,
we resort to estimators which behave dramatically different from the geometric mean. Our recommended estimator T??,0.5 as in (11) is a special case (for ? = 0.5) of a more general family of
estimators [12], parameterized by ? ? (0, 1):
?
!
? Pk
!?/?
? Pk
!1/?
?(?),?
|yj |?
|xj |?
F
1
j=1
j=1
T??,? =
1? ?
, F?(?),? =
, F?(1),? =
??1
kG(?, ?)
kG(1, ?)
F?(1),?
which, after simplification, becomes
T??,? =
?
1 ?
1?
??1
Recall G(?, ?) is defined in (8), and
G(1,0.5)
G(?,0.5)
? Pk
?
j=1 |yj | G(1, ?)
Pk
? G(?, ?)
j=1 |xj |
=
!?/? ?
?
(17)
?
?
.
1
?(1? 2?
)
?
To better understand
F?(?),?
? , recall if Z ? S(?, 1), then E|Z| = G(?, ?) < ? if ?1 < ? < ?.
?
Therefore,
Pk
|yj |?
kG(?,?)
j=1
?/?
is an unbiased estimate of F(?) . To recover F(?) , we need to apply the
?
power ?/? operation.
? Thus, it is clear that, as long as 0 < ? < 1, F(?),? is a consistent estimator of
?
F(?) and E F?(?),? is finite. In particular, the variance of F?(?),? is bounded if 0 < ? < 0.5:
? ?
?
1
?
E F(?),? = F(?) + O
,
k
?
? ?
? F 2 ?2 G(?, 2?) ? G2 (?, ?)
1
(?)
?
V ar F(?),? =
+O
k ?2
G2 (?, ?)
k2
?
6
The variance is unbounded if ? = 0.5 and ? = 1, because G(1, 1) = ? (?(0) = ?). Interestingly,
when ? ? 0 and ? = 1, the asymptotic variance reaches the minimum. In fact, when ? ? 0, F?(?),?
converges to the geometric mean estimator F?(?),gm . A variant of F?(?),? was discussed in [12].
4.1
Theoretical Analysis
Based on Lemma 3 and Lemma 4 (which
? is?a fairly?technical
? proof), we know that the variance of the
?2??1
?
general estimator is essentially V ar T?,? = O
, for fixed ? ? (0, 1/2). In other words,
k
when ? is close to 0, the variance of the entropy estimator is essentially on the order of O (1/(k?)),
and while ? is close to 1/2, the variance is essentially O(1/k) as desired.
Lemma 3 For any fixed ? ? (0, 1),
?
?
? ?
?
?
?
? 2
O
E(|x
|
?
|y
|
)
1
1
1
1
+O
V ar T??,? = 2
?
k
k2
?
Lemma 4 Let 0 < ? ? 1/2 and ? = 1 + ?. Let ? ? (0, 1/2) and m be a positive integer no
smaller than 1/?. Then, there exists a universal constant M such that
?
?2
n
o.
e2
E |x1 |? ? |y1 |? ? M F 2? ?1+2??1/m m2 m + H
+ (1 ? 2?)?2
(1 ? 2?),
(2m)
(1)
?
e 2m = PD
where H
i=1
|At [i]| ?
[i]| 2m ?1/(2m)
log |AFt(1)
)
.
F(1)
?
We should clarify that our theoretical analysis is only applicable for fixed ? ? (0, 1/2). When
? = 0.5, the estimator T?(?),0.5 is still well-behaved, except we are unable to precisely analyze this
case. Also, since we do not compute the exact constant, it is possible that for some carefully chosen
? (data-dependent), T?(?),? with ? < 0.5 may exhibit smaller variance than T?(?),0.5 . We recommend
T?(?),0.5 for convenience because it essentially frees practitioners from carefully choosing ?.
4.2
Experimental Results
Figure 3 presents some empirical results, for testing the general estimator T??,? (17), using more
word vector pairs (including the same 2 pairs in Figure 2). We can see that when ? = 0.5, the
(normalized) MSEs become flat (as desired) as ? = ? ? 1 ? 0. When ? > 1/2, the MSEs increase
although the curves remain flat. When ? < 1/2, the MSEs blow up with increasing ?. Note that,
when ? < 1/2, it is possible to achieve smaller MSEs if we carefully choose ?.
How many samples (k) are needed? If the goal is to estimate the Shannon entropy?within a few
percentages of the the true value, then k = 100 ? 1000 should be sufficient, because M SE/H <
0.1 when k ? 100 as shown in Figure 3.
5 Conclusion
Entropy estimation is an important task in machine learning, data mining, network measurement,
anomaly detection, neural computations, etc. In modern applications, the data are often generated
in a streaming fashion and many operations on the streams can only be conducted in one-pass of the
data. It has been a challenging problem to estimate the Shannon entropy of data streams.
The prior work [21] achieved some success in entropy estimation using symmetric stable random
projections. However, even after aggressively exploiting the bias-variance tradeoff, they still need
to a large number of samples, e.g., 106 , which is prohibitive in both time and space, especially
considering that in streaming applications the pseudo-random numbers have to be re-generated on
the fly, the cost of which is directly proportional to the sample size.
In our approach, we approximate the Shannon entropy using two high correlated estimates of the
frequent comments. The positive correlation can substantially reduce the variance of the Shannon
entropy estimate. However, finding the appropriate estimator of the frequency moment is another
challenging task. We successfully find such an estimator and show that its variance (of the Shannon
entropy estimate) is very small. Experimental results demonstrate that about 100 ? 1000 samples
should be sufficient for achieving high accuracies.
7
Acknowledgement
The research of Ping Li is partially supported by NSF-IIS-1249316, NSF-DMS-0808864, NSF-SES1131848, and ONR-YIP-N000140910911. The research of Cun-Hui Zhang is partially supported
by NSF-DMS-0906420, NSF-DMS-1106753, NSF-DMS-1209014, and NSA-H98230-11-1-0205.
Normalized MSE
2
100
10
0
10
k = 1000
?2
10
?4
10
?3
?2
10
10
?=??1
?1
10
0
2
k = 10
0
k = 100
10
10
k = 1000
?5
10
10
Normalized MSE
100
2
10
0
10
k = 1000
?5
?4
10
?3
?2
10
10
?=??1
?1
10
0
10
10
k = 100
0
k = 1000
?4
?3
10
?2
10
10
?=??1
?1
10
0
10
6
0
10
k = 1000
2
10
k = 10
0
10
k = 100
10
FOOD?LOVE : GM + Indep.
?5
10
?4
10
?3
?2
10
10
?=??1
?1
10
0
10
?4
10
?3
?2
10
10
?=??1
?1
10
0
10
6
Normalized MSE
k = 10
10
100
2
10
0
10
k = 1000
?2
10
?5
?4
10
?3
?2
10
10
?=??1
?1
10
DATA?PAPER : GM + Corr.
k = 10
2
10
0
k = 100
?2
k = 1000
10
0
?5
10
10
?4
10
?3
?2
10
10
?=??1
?1
10
6
10
Normalized MSE
100
2
10
0
10
k = 1000
?2
10
?4
10
?3
?2
10
10
?=??1
2
k = 10
10
0
k = 100
10
?2
10
NEWS?WASHINGTON : GM + Indep.
?5
10
?1
10
?4
10
?3
?2
10
10
?=??1
?1
10
6
10
0
10
k = 1000
?2
k = 10
2
10
k = 100
0
10
?2
10
?4
10
?3
?2
10
10
?=??1
?1
10
k = 1000
10
MACHINE?LEARN : GM + Indep.
?5
k = 100
?2
10
k = 1000
?3
10
0
10
DATA?PAPER : Corr.? = 0.3
?5
10
?4
10
?3
?2
10
10
?=??1
?1
10
0
10
Normalized MSE
Normalized MSE
10
?4
10
?3
?2
10
10
?=??1
?1
10
10
k = 10
?2
10
k = 100
k = 1000
?3
10
k = 100
?2
10
k = 1000
?3
10
NEWS?WASHINGTON : Corr. ? = 0.3
?4
10
?3
?2
10
10
?=??1
?1
10
?3
?2
10
10
?=??1
?1
10
0
10
k = 10
k = 100
?2
10
k = 1000
?3
10
k = 100
10
k = 1000
10
?4 MACHINE?LEARN : Corr. ? = 0.3
10 ?5
?4
?3
?2
?1
0
10
10
10
10
10
10
?=??1
?3
?2
10
10
?=??1
?1
10
0
10
A?THE : Corr. ? = 0.7
?1
k = 10
k = 100
10
?2
k = 1000
10
?3
10
?4
10
?3
?2
10
10
?=??1
?1
10
0
10
FOOD?LOVE : Corr. ? = 0.7
?1
k = 10
k = 100
?2
k = 1000
10
10
?3
10
?3
?2
10
10
?=??1
?1
10
10 ?5
10
0
10
?4
10
?3
?2
10
10
?=??1
?1
10
0
10
0
10
DATA?PAPER : Corr.? = 0.5
?1
k = 10
?2
k = 100
10
10
k = 1000
?3
10
?4
10
?3
?2
10
10
?=??1
?1
10
?1
k = 10
k = 100
?2
k = 1000
10
10
?3
10
DATA?PAPER : Corr.? = 0.7
?4
10 ?5
?4
?3
?2
?1
10
10
10
10
10
?=??1
0
10
0
10
0
10
NEWS?WASHINGTON : Corr. ? = 0.5
?1
10
k = 10
k = 100
?2
10
k = 1000
?3
10
10
?4
10
?4
?4
10
?1
?3
?2
10
10
?=??1
?1
10
k = 10
k = 100
10
?2
10
k = 1000
?3
10
?4
?4
10
NEWS?WASHINGTON : Corr. ? = 0.7
10 ?5
10
0
10
0
?1
10
10
?4
10
?3
?4
?4
10
FOOD?LOVE : Corr. ? = 0.5
?1
10 ?5
10
0
10
0
10
0
10
10
k = 10
?1
10
k = 1000
?2
10 ?5
10
0
?1
?2
UNITED?STATES : Corr.? = 0.7
k = 10
k = 100
10
10
A?THE : Corr. ? = 0.5
10 ?5
10
0
10
?1
10
10 ?5
10
0
?4
10
?3
10
10
?=??1
?1
0
?1
?2
?2
?1
k = 10
10
10
?3
?4
10
?3
10
10
?=??1
?4
?4
10
10
?4
10
0
10 ?5
10
0
10
10
4
Normalized MSE
Normalized MSE
100
2
k = 1000
?3
10
MACHINE?LEARN : GM + Corr.
10
10
k = 100
?2
10
0
10
k = 10
4
10
10
0
10 ?5
10
0
10
k = 1000
?3
10
?1
k = 1000
?5
10
6
0
10
10
?4
0
10
?1
k = 1000
?3
10
10
?4
10
k = 10
k = 100
?2
10
10 ?5
10
0
10
k = 100
?2
10 ?5
10
10
NEWS?WASHINGTON : GM + Corr.
4
10
?1
10
?4
0
10
k = 10
4
k = 1000
?3
10
10 ?5
10
0
10
Normalized MSE
6
10
k = 100
?2
10
?4
?2
k = 10
10
10
?1
DO?REVIEW : Corr. ? = 0.7
?1
10
0
10
k = 10
4
?3
10
10
?=??1
0
10
0
10
?4
?4
10
UNITED?STATES : Corr.? = 0.5
?1
10 ?5
10
10
10
10
DATA?PAPER : GM + Indep.
10
10
0
10
Normalized MSE
6
10
4
k = 1000
?3
?4 FOOD?LOVE : Corr. ? = 0.3
10 ?5
?4
?3
?2
?1
10
10
10
10
10
?=??1
k = 1000
?5
10
?3
10
k = 10
4
10
?2
?2
10
k = 100
?2
10
0
Normalized MSE
Normalized MSE
2
10
?2
0
10
k = 1000
?3
10
10
k = 10
k = 100
k = 1000
10
10
k = 10
k = 100
?2
10
THIS?HAVE : Corr. ? = 0.7
?4
10 ?5
?4
?3
?2
?1
10
10
10
10
10
?=??1
0
10
DO?REVIEW : Corr. ? = 0.6
?1
10 ?5
10
0
10
k = 10
?1
FOOD?LOVE : GM + Corr.
100
?1
10
10
k = 10
4
?2
?4 A?THE : Corr. ? = 0.3
10 ?5
?4
?3
?2
10
10
10
10
?=??1
10
10
?3
10
10
?=??1
?1
10
0
10
k = 10
10
?1
10
?4
?4
10
10
k = 10
2
10
10
10
k = 1000
?3
10
?4 UNITED?STATES : Corr.? = 0.3
10 ?5
?4
?3
?2
?1
0
10
10
10
10
10
10
?=??1
0
A?THE : GM + Corr.
?5
10
6
Normalized MSE
10
10
?=??1
?1
4
10
A?THE : GM + Indep.
10
Normalized MSE
10
?2
10
?2
10
k = 100
?2
10
0
10
k = 10
4
10
?2
Normalized MSE
?3
6
10
Normalized MSE
?4
Normalized MSE
6
k = 10
0
10
UNITED?STATES : GM + Indep.
?5
DO?REVIEW : Corr. ? = 0.5
?1
10
10
UNITED?STATES : GM + Corr.
4
10
?2
10
10
?4
6
10
k = 10
4
10
?2
0
10 ?5
10
0
10
?3
10
10
?=??1
Normalized MSE
k = 1000
?4
10
Normalized MSE
?3
10
Normalized MSE
6
10
10 ?5
10
0
10
Normalized MSE
k = 100
k = 1000
?3
10
Normalized MSE
?2
10
DO?REVIEW : Corr. ? = 0.4
?4
10 ?5
?4
?3
?2
?1
10
10
10
10
10
?=??1
0
10
?1
10
Normalized MSE
k = 10
k = 10
k = 100
?2
10
Normalized MSE
DO?REVIEW : Corr. ? = 0.3
?4
10 ?5
?4
?3
?2
?1
10
10
10
10
10
?=??1
?2
Normalized MSE
k = 1000
?3
10
?3
10
10
?=??1
Normalized MSE
k = 100
?2
10
?1
10
10
?4
?4
10
0
10
Normalized MSE
Normalized MSE
k = 10
k = 1000
?3
10
?4
0
?1
k = 100
10 ?5
10
0
10
10
10
k = 10
?2
10
0
10
THIS?HAVE : Corr. ? = 0.6
?1
?4
10
?3
?2
10
10
?=??1
?1
10
0
10
0
10
MACHINE?LEARN : Corr. ? = 0.5
?1
10
?2
Normalized MSE
k = 1000
10
Normalized MSE
0
Normalized MSE
?3
10
THIS?HAVE : Corr. ? = 0.4
?4
10 ?5
?4
?3
?2
?1
10
10
10
10
10
?=??1
0
10
10
Normalized MSE
k = 100
10
Normalized MSE
k = 1000
?3
10
?2
10
0
10
THIS?HAVE : Corr. ? = 0.5
?1
Normalized MSE
k = 100
?2
10
k = 10
Normalized MSE
Normalized MSE
Normalized MSE
10
THIS?HAVE : Corr. ? = 0.3
?4
10 ?5
?4
?3
?2
?1
10
10
10
10
10
?=??1
Normalized MSE
0
10
?1
Normalized MSE
0
10
k = 10
?1
Normalized MSE
0
10
k = 10
k = 100
10
k = 1000
?3
10
?4
10 ?5
10
?4
10
?3
?2
10
10
?=??1
?1
10
0
10
?1
k = 10
k = 100
?2
k = 1000
10
10
?3
10
MACHINE?LEARN : Corr. ? = 0.7
?4
10 ?5
?4
?3
?2
?1
0
10
10
10
10
10
10
?=??1
Figure 3: The first two rows are the normalized MSEs for same two vectors used in Figure 2, for
estimating Shannon entropy using the general estimator T??,? with ? = 0.3, 0.4, 0.5, 0.6, 0.7. For
the rest of the rows, the leftmost panels are the results of using independent samples (i.e., the prior
work [21]) and the geometric mean estimator. The second column of panels are the results of using
correlated samples and the geometric mean estimator. The right three columns of panels are for the
proposed general estimator T??,? with ? = 0.3, 0.5, 0.7. We recommend ? = 0.5.
8
References
[1] Brian Babcock, Shivnath Babu, Mayur Datar, Rajeev Motwani, and Jennifer Widom. Models and issues
in data stream systems. In PODS, pages 1?16, Madison, WI, 2002.
[2] Daniela Brauckhoff, Bernhard Tellenbach, Arno Wagner, Martin May, and Anukool Lakhina. Impact of
packet sampling on anomaly detection metrics. In IMC, pages 159?164, Rio de Janeriro, Brazil, 2006.
[3] John M. Chambers, C. L. Mallows, and B. W. Stuck. A method for simulating stable random variables.
Journal of the American Statistical Association, 71(354):340?344, 1976.
[4] Laura Feinstein, Dan Schnackenberg, Ravindra Balupari, and Darrell Kindred. Statistical approaches to
DDoS attack detection and response. In DARPA Information Survivability Conference and Exposition,
pages 303?314, 2003.
[5] Anupam Gupta, John D. Lafferty, Han Liu, Larry A. Wasserman, and Min Xu. Forest density estimation.
In COLT, pages 394?406, Haifa, Israel, 2010.
[6] Nicholas J. A. Harvey, Jelani Nelson, and Krzysztof Onak. Streaming algorithms for estimating entropy.
In ITW, 2008.
[7] Piotr Indyk. Stable distributions, pseudorandom generators, embeddings, and data stream computation.
Journal of ACM, 53(3):307?323, 2006.
[8] Anukool Lakhina, Mark Crovella, and Christophe Diot. Mining anomalies using traffic feature distributions. In SIGCOMM, pages 217?228, Philadelphia, PA, 2005.
[9] Ashwin Lall, Vyas Sekar, Mitsunori Ogihara, Jun Xu, and Hui Zhang. Data streaming algorithms for
estimating entropy of network traffic. In SIGMETRICS, pages 145?156, Saint Malo, France, 2006.
[10] Ping Li. Estimators and tail bounds for dimension reduction in l? (0 < ? ? 2) using stable random
projections. In SODA, pages 10 ? 19, San Francisco, CA, 2008.
[11] Ping Li. Compressed counting. In SODA, New York, NY, 2009.
[12] Ping Li and Trevor J. Hastie. A unified near-optimal estimator for dimension reduction in l? (0 < ? ? 2)
using stable random projections. In NIPS, Vancouver, BC, Canada, 2007.
[13] Ping Li and Cun-Hui Zhang. A new algorithm for compressed counting with applications in shannon
entropy estimation in dynamic data. In COLT, 2011.
[14] Qiaozhu Mei and Kenneth Church. Entropy of search logs: How hard is search? with personalization?
with backoff? In WSDM, pages 45 ? 54, Palo Alto, CA, 2008.
[15] S. Muthukrishnan. Data streams: Algorithms and applications. Foundations and Trends in Theoretical
Computer Science, 1:117?236, 2 2005.
[16] Noam Nisan. Pseudorandom generators for space-bounded computations. In Proceedings of the twentysecond annual ACM symposium on Theory of computing, STOC, pages 204?212, 1990.
[17] Liam Paninski. Estimation of entropy and mutual information. Neural Comput., 15(6):1191?1253, 2003.
[18] Constantino Tsallis. Possible generalization of boltzmann-gibbs statistics. Journal of Statistical Physics,
52:479?487, 1988.
[19] Kuai Xu, Zhi-Li Zhang, and Supratik Bhattacharyya. Profiling internet backbone traffic: behavior models
and applications. In SIGCOMM ?05: Proceedings of the 2005 conference on Applications, technologies,
architectures, and protocols for computer communications, pages 169?180, Philadelphia, Pennsylvania,
USA, 2005.
[20] Qiang Yang and Xindong Wu. 10 challeng problems in data mining research. International Journal of
Information Technology and Decision Making, 5(4):597?604, 2006.
[21] Haiquan Zhao, Ashwin Lall, Mitsunori Ogihara, Oliver Spatscheck, Jia Wang, and Jun Xu. A data streaming algorithm for estimating entropies of od flows. In IMC, San Diego, CA, 2007.
[22] Vladimir M. Zolotarev. One-dimensional Stable Distributions. American Mathematical Society, Providence, RI, 1986.
9
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4,052 | 4,668 | Cardinality Restricted Boltzmann Machines
Kevin Swersky
Daniel Tarlow
Ilya Sutskever
Dept. of Computer Science
University of Toronto
[kswersky,dtarlow,ilya]@cs.toronto.edu
Ruslan Salakhutdinov?,?
Richard S. Zemel?
Dept. of Computer Science? and Statistics?
University of Toronto
[rsalakhu,zemel]@cs.toronto.edu
Ryan P. Adams
School of Eng. and Appl. Sciences
Harvard University
[email protected]
Abstract
The Restricted Boltzmann Machine (RBM) is a popular density model that is also
good for extracting features. A main source of tractability in RBM models is
that, given an input, the posterior distribution over hidden variables is factorizable
and can be easily computed and sampled from. Sparsity and competition in the
hidden representation is beneficial, and while an RBM with competition among
its hidden units would acquire some of the attractive properties of sparse coding,
such constraints are typically not added, as the resulting posterior over the hidden units seemingly becomes intractable. In this paper we show that a dynamic
programming algorithm can be used to implement exact sparsity in the RBM?s
hidden units. We also show how to pass derivatives through the resulting posterior
marginals, which makes it possible to fine-tune a pre-trained neural network with
sparse hidden layers.
1
Introduction
The Restricted Boltzmann Machine (RBM) [1, 2] is an important class of probabilistic graphical
models. Although it is a capable density estimator, it is most often used as a building block for
deep belief networks (DBNs). The benefit of using RBMs as building blocks for a DBN is that they
often provide a good initialization for feed-forward neural networks, and they can effectively utilize
large amounts of unlabeled data, which has led to success in a variety of application domains [3].
Despite the benefits of this approach, there is a disconnect between the unsupervised nature of
RBMs and the final discriminative task (e.g., classification) for which the learned features are used.
This disconnect has motivated the search for ways to improve task-specific performance, while still
retaining the unsupervised nature of the original model [4, 5]. One effective method for improving
performance has been the incorporation of sparsity into the learned representation. Approaches that
learn and use sparse representations have achieved good results on a number of tasks [6], and in the
context of computer vision, sparsity has been linked with learning features that are invariant to local
transformations [7]. Sparse features are also often more interpretable than dense representations
after unsupervised learning.
For directed models, such as sparse coding [8], sparsity can be enforced using a Laplace or spike
and slab prior [9]. For undirected models, introducing hard sparsity constraints directly into the
energy function often results in non-trivial dependencies between hidden units that makes inference
intractable. The most common way around this is to encourage sparsity during training by way of a
penalty function on the expected conditional hidden unit activations given data [10]. However, this
training-time procedure is a heuristic and does not guarantee sparsity at test time.
1
Recently, methods for efficiently dealing with highly structured global interactions within the graphical modeling framework have received considerable interest. One class of these interactions is based
on assigning preferences to counts over subsets of binary variables [11, 12]. These are known as
cardinality potentials. For example, the softmax distribution can be seen as arising from a cardinality potential that forces exactly one binary variable to be active. For general potentials over counts,
it would seem that the cost of inference would grow exponentially with the number of binary variables. However, efficient algorithms have been proposed that compute exact marginals for many
higher-order potentials of interest [12]. For achieving sparsity in RBMs, it turns out that a relatively
simple dynamic programming algorithm by Gail et al. [13] contains the key ingredients necessary
to make inference and learning efficient. The main idea behind these algorithms is the introduction
of auxiliary variables that store cumulative sums in the form of a chain or a tree.
In this paper, we show how to combine these higher-order potentials with RBMs by placing a cardinality potential directly over the hidden units to form a Cardinality-RBM (CaRBM) model. This will
allow us to obtain genuinely sparse representations, where only a small number of units are allowed
to be active. We further show how gradients can be backpropagated through inference using a recently proposed finite-difference method [14]. On a benchmark suite of classification experiments,
the CaRBM is competitive with current approaches that do not enforce sparsity at test-time.
2
Background
2.1
Restricted Boltzmann Machines
A Restricted Boltzmann Machine is a particular type of Markov random field that has a two-layer
architecture, in which the visible, stochastic units v ? {0, 1}Nv are connected to hidden stochastic
units h ? {0, 1}Nh . The probability of the joint configuration {v, h} is given by:
P (v, h) =
1
exp (v> W h + v> bv + h> bh ),
Z
(1)
where Z is the normalizing constant, and {W ? RNv ?Nh , bv ? RNv , bh ? RNh } are the model
parameters, with W representing visible-to-hidden symmetric interaction terms, and bv , bh representing visible and hidden biases respectively. The derivative of the log-likelihood with respect to
the model parameters1 W can be obtained from Eq. 1:
? log P (v; ?)
= EPdata [vh> ] ? EPmodel [vh> ],
?W
where EPdata [?] denotes an expectation with respect to the data distribution
Pdata (h, v; ?) = P (h | v; ?) Pdata (v),
1
N
(2)
(3)
P
where Pdata (v) =
n ?(v?vn ) represents the empirical distribution, and EPmodel [?] is an expectation with respect to the distribution defined by the model, as in Eq. 1. Exact maximum likelihood
learning in this model is intractable because exact computation of the expectation EPmodel [?] takes
time that is exponential in the number of visible or hidden units. Instead, learning can be performed
by following an approximation to the gradient, the ?Contrastive Divergence? (CD) objective [15].
After learning, the hidden units of the RBM can be thought of as features extracted from the input
data. Quite often, they are used to initialize a deep belief network (DBN), or they can be used
directly as inputs to some other learning system.
2.2
The Sparse RBM (SpRBM)
For many challenging tasks, such as object or speech recognition, a desirable property for the hidden
variables is to encode the data using sparse representations. That is, given an input vector v, we
would like the corresponding distribution P (h|v) to favour sparse configurations. The resulting
features are often more interpretable and tend to improve performance of the learning systems that
use these features as inputs. On its own, it is highly unlikely that the RBM will produce sparse
features. However, suppose we have some desired target expected sparsity ?. If qj represents a
1
The derivatives with respect to the bias terms take a similar form.
2
P
running average of the hidden unit marginals qj = 1/N n P (hj = 1|vn ), then we can add the
following penalty term to the log-likelihood objective [16]:
? (? log qj + (1 ? ?) log(1 ? qj )) ,
(4)
where ? represents the strength of the penalty. This penalty is proportional to the negative of the
KL divergence between the hidden unit marginals and the target sparsity probability. The derivative
with respect to the activity on any case n is proportional to ?(? ? qj ). Note that this is applied to
each hidden unit independently and has the intuitive property of encouraging each hidden unit to
activate with proportion ? across the dataset.
If the hidden unit activations are stored in a matrix where each row corresponds to a training example, and each column corresponds to a hidden unit, then this is enforcing sparsity in the columns of
the matrix. This is also referred to as lifetime sparsity. When using the SpRBM model, the hope is
that each individual example will be encoded by a sparse vector, corresponding to sparsity across
the rows, or population sparsity.
3
The Cardinality Potential
Consider a distribution of the form
?
?
N
N
X
Y
1
q(x) = ? ?
xj ?
?j (xj ),
Z
j=1
j=1
(5)
where x is a binary vector and Z is the normalizing constant. This distribution consists of noninteracting terms, with the exception of the ?(?) potential, which couples all of the variables together.
This is a cardinality potential (or ?counts potential?), because it depends only on the number of 1?s
in the vector x, but not on their identity. This distribution is useful for imposing sparsity because it
allows us to represent the constraint that the vector x can have at most k elements set to one.
There is an efficient exact inference algorithm for computing the normalizing constant and marginals
of this distribution. This can be interpreted as a dynamic programming algorithm [13, 17], or as an
instance of the sum-product algorithm [18]. We prefer the sum-product interpretation because it
makes clear how to compute marginal distributions over binary variables, how to compute marginal
distributions over total counts, and how to draw an exact joint sample from the model (pass messages
forwards, then sample backwards) and also lends itself towards extensions. In this view, we create N
auxiliary variables zj ? {1, . . . , N }. The auxiliary variables are then deterministically related to
Pj
the x variables by setting zj = k=1 xk , where zj represents the cumulative sum of the first j
binary variables.
More formally, consider the following joint distribution q?(x, z):
q?(x, z) =
N
Y
?j (xj ) ?
j=1
N
Y
?(xj , zj , zj?1 ) ? ?(zN ).
(6)
j=2
We let ?(xj , zj , zj?1 ) be a deterministic ?addition potential?, which assigns the value one to any
triplet (x, z, z0 ) satisfying z = x + z0 and zero otherwise. Note that the second product ranges
from j = 2, and that z1 is replaced with x1 . This notation represents the observation that zj can
Pj
be computed either as zj = k=1 xk , or more simply as zj = zj?1 + xj . The latter is preferable, because it induces a chain-structured dependency graph amongst the z and x variables. Thus,
the distribution q?(x, z) has two important properties. First, it is chain-structured, and therefore
we can perform exact inference using the sum-product algorithm. By leveraging the fact that at
most k are allowed to be on, the runtime can be made to be O(N k) by reducing the range of
each zi from {1, . . . , N } to {1, . . . , k + 1}. Second, the posterior q?(z|x) assigns a probability of
Pj
1 to the configuration z? that is given by zj? = k=1 xj for all j. This is a direct consequence of
PN
?
?
the sum-potentials ?(?) enforcing the constraint zj? = xj + zj?1
. Since zN
= j=1 xj , it follows
that q(x) = q?(x, z? ), and since q(z|x) concentrates all of its mass on z? , we obtain:
X
X
q?(x) =
q?(x, z) =
q?(z|x)?
q (x) = q?(x, z? ) = q(x).
(7)
z
z
3
This shows that q(x) is the marginal distribution of the chain-structured distribution q?(x, z). By
running the sum-product algorithm on q? we can recover the singleton marginals ?j (xj ), which
are also the marginals of q(?). We can likewise sample from q by computing all of the pairwise marginals ?j+1,j (zj+1 , zj ), computing the pairwise conditionals ?j+1,j (zj+1 |zj ), and sampling each zj sequentially, given zj?1 , to obtain a sample z. The vector x can be recovered
via xj = zj ? zj?1 . The basic idea behind this algorithm is given in [13] and the sum-product
interpretation is elaborated upon in [18].
There are many algorithmic extensions, such as performing summations in tree-structured distributions, which allow for more efficient inference with very large N (e.g. N > 1000) using fast Fourier
transforms [19, 18]. But in this work we only use the chain-structured distribution q? described above
with the restriction that there are only k states.
4
The Cardinality RBM (CaRBM)
The Cardinality Restricted Boltzmann Machine is defined as follows:
?
?
Nh
X
1
P (v, h) = exp v> W h + v> bv + h> bh ? ?k ?
hj ? ,
Z
j=1
(8)
where ?k is a potential given by ?k (c) = 1 if c ? k and 0 otherwise. Observe that the conditional
distribution P (h|v) assigns a non-zero probability mass to a vector h only if |h| ? k. The cardinality potential implements competition in the hidden layer because now, a data vector v can be
explained by at most k hidden units. This form of competition is similar to sparse coding in that there
may be many non-sparse configurations that assign high probability to the data, however only sparse
configurations are allowed to be used. Unlike sparse coding, however, the CaRBM learning problem involves maximizing the likelihood of the training data, rather than minimizing a reconstruction
cost. Using the techniques from the previous section, computing the conditional distribution P (h|v)
is tractable, allowing us to use learning algorithms like CD or stochastic maximum likelihood [20].
The conditional distribution P (v|h) is still factorial and easy to sample from.
Perhaps the best way to view the effect of the cardinality potential is to consider the case of k = 1
with the further restriction that configurations with 0 active hidden units are disallowed. In this case,
the CaRBM reduces to an ordinary RBM with a single multinomial hidden unit. A similar model to
the CaRBM is the Boltzmann Perceptron [21], which also introduces a term in the energy function
to promote competition between units; however, they do not provide a way to efficiently compute
marginals or draw joint samples from P (h|v). Another similar line of work is the Restricted Boltzmann Forest [22], which uses k groups of multinomial hidden units.
We should note that the actual marginal probabilities of the hidden units given the visible units are
not guaranteed to be sparse, but rather the distribution assigns zero mass to any hidden configuration
that is not sparse. In practice though, we find that after learning, the marginal probabilities do tend
to have low entropy. Understanding this as a form of regularization is a topic left for future work.
4.1
The Cardinality Marginal Nonlinearity
One of the most common ways to use an RBM is to consider it as a pre-training method for a
deep belief network [2]. After one or several RBMs are trained in a greedy layer-wise fashion,
the network is converted into a deterministic feed-forward neural network that is fine-tuned with
the backpropagation algorithm. The fine-tuning step is important for getting the best results with a
DBN model [23]. While it is easy to convert a stack of standard RBMs into a feed-forward neural
network, turning a stack of CaRBMs into a feed-forward neural network is less obvious, because it
is not clear what nonlinearity should be used.
Observe that in the case of a standard, binary-binary RBM, the selected nonlinearity is the sigmoid ?(x) ? 1/(1+exp(?x)). We can justify this choice by noticing that it is the expectation of the
conditional distribution P (h|v), namely
?(W > v + bh ) = EP (h|v) [h],
4
(9)
where the sigmoid is applied to the vector in an element-wise fashion. In particular, using the
conditional expectation as the nonlinearity is a fundamental ingredient in the variational lower bound
that justifies the greedy layer-wise procedure [2]. It also appears naturally when the score matching
estimator is applied to RBMs over Gaussian-distributed visible units [24, 25]. This justification
suggests that for the CaRBM, we should choose a nonlinearity ?(?) which will satisfy the following
equality:
?(W > v + bh ) = EP (h|v) [h],
(10)
where the conditional P (h|v) can be derived from Eq. 8. First note that such a nonlinear function
exists, because the distribution P (h|v) is completely determined by the total input W > v + bh .
Therefore, the feed-forward neural network that is obtained from a stack of CaRBMs uses a messagepassing algorithm to compute the nonlinearity ?(?). We should note that ? depends on k, the number
of units that can take on the value 1, but this is a constant that is independent of the input. In practice,
we keep k fixed to the k that was used in unsupervised training.
To compute gradients for learning the network, it is necessary to ?backpropagate? through ?, which
is equivalent to multiplying by the Jacobian of ?. Analytic computation of the Jacobian, however,
results in an overly expensive O(N 2 ) algorithm. We also note that it is possible to manually differentiate the computational graph of ? by passing the derivatives back through the sum-product
algorithm. While this approach is correct, it is difficult to implement and can be numerically unstable.
We propose an alternative approach to multiplying by the Jacobian of ?. Let x = W > v + bh be the
total input to the RBM?s hidden units, then the Jacobian J(x) is given by:
J(x)
= EP (h|v) [hh> ] ? EP (h|v) [h] EP (h|v) [h> ],
= EP (h|v) [hh> ] ? ?(x)?(x)> .
(11)
We need to multiply by the transpose of the Jacobian from the right, since by the chain rule,
?L
?? > ?L
?L
=
= J(x)>
,
?x
?x ??
??
(12)
where L is the corresponding loss function. One way to do this is to reuse the sample h ? P (h|v)
in order to obtain a rank-one unbiased estimate of EP (h|v) [hh> ], but we found this to be inaccurate. Luckily, Domke [14] makes two critical observations. First, the Jacobian J(x) is symmetric
(see Eq. 11). Second, it is easy to multiply by the Jacobian of any function using numerical differentiation, because multiplication by the Jacobian (without a transpose) is precisely a directional
derivative.
More formally, let f (x) be any differentiable function and J be its Jacobian. For any vector `, it can
be easily verified that:
lim
?0
f (x + `) ? f (x)
f (x) + J` + o() ? f (x)
o() J`
= lim
= lim
+
= J`.
?0
?0
(13)
Since ? is a differentiable function, we can compute J(x)` by a finite difference formula:
J(x)` ?
?(x + `) ? ?(x ? `)
.
2
(14)
Using the symmetry of the Jacobian of ?, we can backpropagate a vector of derivatives ?L/?? using
Eq. 14. Of the approaches we tried, we found this approach to provide the best combination of speed
and accuracy.
5
Experiments
The majority of our experiments were carried out on various binary datasets from Larochelle et
al [26], hence referred to as the Montreal datasets. Each model was trained using the CD-1 algorithm with stochastic gradient descent on mini-batches. For training the SpRBM, we followed the
guidelines from Hinton [27].
5
5.1
Training CaRBMs
One issue when training a model with lateral inhibition is that in the initial learning epochs, a small
group of hidden units can learn global features of the data and effectively suppress the other hidden
units, leading to ?dead units?. This effect has been noted before in energy-based models with competition [22]. One option is to augment the log-likelihood with the KL penalty given in Eq. 4. In the
SpRBM, this penalty term is used to encourage each hidden unit to be active a small number of times
across the training set, which indirectly provides sparsity per-example. In the CaRBM it is used to
ensure that each hidden unit is used roughly equally across the training set, while the per-example
sparsity is directly controlled.
We observed that dead units occurred only with a random initialization of the parameters and that
this was no longer an issue once the weights had been properly initialized. In our experiments, we
used the KL penalty during unsupervised learning, but not during supervised fine-tuning.
A related issue with SpRBMs is that if the KL penalty is set too high then it can create dead examples
(examples that activate no hidden units). Note that the KL penalty will not penalize this case as long
as the inter-example activations matches the target probability ?.
5.2
Comparing CaRBM with SpRBM
Both the CaRBM and SpRBM models attempt to achieve the same goal of sparsity in the hidden
unit activations. However, the way in which they accomplish this is fundamentally different.
For datasets such as MNIST, we found the two models to give qualitatively similar results. Indeed,
this seemed to be the case for several datasets. On the convex dataset, however, we noticed that the
models produced quite different results. The convex dataset consists of binary 28 ? 28-pixel images
of polygons (sometimes with multiple polygons per image). Figure 1 (a) shows several examples
from this dataset. Unlike the MNIST dataset, there is a large variation in the number of active pixels
in the inputs. Figure 1 (e) shows the distribution of the number of pixels taking the value 1. In some
examples, barely any pixels are active, while in others virtually every pixel is on.
For both models, we set the target sparsity to 10%. We next performed a grid search over the
strength of the KL penalty until we found a setting that achieved an average hidden unit population
sparsity that matched the target without creating dead examples (in the case of the SpRBM) or dead
units (in the case of the CaRBM). Figure 1 (d) and (h) show that both models achieve the desired
mean population sparsity. However, the SpRBM exhibits a heavy-tailed distribution over activations,
with some examples activating over half of the hidden units. By comparison, all inputs activate the
maximum number of allowable hidden units in the CaRBM, generating a spike at 10%. Indeed, in
the CaRBM, the hidden units suppress each other through competition, while in the SpRBM there
is no such direct competition. Figure 1 (b) and (f) display the learned weights. Both models appear
to give qualitatively similar results, although the CaRBM weights appear to model slightly more
localized features at this level of sparsity.
5.3
Classification Performance
To evaluate the classification performance of CaRBMs, we performed a set of experiments on the
Montreal datasets. We conducted a random search over hyperparameter settings as recommended
by Bergstra & Bengio [28], and set the target sparsity to be between 2.5% and 10%. Table 1 shows
that the CarBM and SpRBM achieve comparable performance. On this suite we found that the validation sets were quite small and prone to overfitting. For example, both the SpRBM and CaRBM
achieve 0.5% validation error on the rectangles dataset. Interestingly, for the convex dataset, the
SpRBM model, chosen by cross-validation, used a weak penalty strength and only achieved a population sparsity of 25%. As we increased the strength of the sparsity penalty, classification performance in the SpRBM degraded, but the desired sparsity level was still not achieved.
5.4
CIFAR-10 Patches
We extracted 16 ? 16 whitened image patches from the CIFAR-10 dataset [29] and trained both
models. Figure 2 (a) shows learned filters of the CaRBM model (both models behave similarly
6
0.0
(a)
0
400
(b)
0.05
800
(e)
0.2
0.5
0.0
(c)
(f)
0.10
0.3
0.6
(d)
0.15
0.0
(g)
0.5
1.0
(h)
Figure 1: (a),(e) Samples from the Convex dataset and the distribution of the number of pixels in
each image with the value 1. (b),(f) Visualization of the incoming weights to 25 randomly selected
hidden units in the SpRBM and CaRBM models respectively. (c),(g) The distribution of the mean
lifetime activations (across examples) of the hidden units in the SpRBM and CaRBM respectively.
(d),(h) The distribution of the mean population activations (within examples) of the hidden units in
the SpRBM and CaRBM respectively.
Dataset
rectangles
background im
background im rot
recangles im
RBM
4.05%
23.78%
58.21%
24.24%
SpRBM
2.66%
23.49%
56.48%
22.50%
CaRBM
5.60%
22.16%
56.39%
22.56%
Dataset
convex
mnist basic
mnist rot
background rand
RBM
20.66%
4.42%
14.83%
12.96%
SpRBM
18.52%
3.84%
13.11%
12.97%
CaRBM
21.13%
3.65%
12.40%
12.67%
Table 1: Test-set classification errors on the Montreal datasets.
and so we just display the CaRBM weights). Observe that the learned weights resemble Gabor-like
filters. These features are often considered to be beneficial for classification when modeling images.
5.5
Topic Modeling with the NIPS Dataset
One form of data with highly variable inputs is text, because some words are used much more
frequently than others. We applied the SpRBM and CaRBM to the NIPS dataset2 , which consists
of 13649 words and 1740 papers from NIPS conferences from 1987 to 1999. Each row corresponds
to a paper, each column corresponds to a word, and the entries are the number of times each word
appears in each paper. We binarized the dataset by truncating the word counts and train the SpRBM
and CaRBM models with 50 hidden units, searching over learning rates and KL penalty strengths
until 10% sparsity is achieved without dead units or examples. Once a model is learned, we define
a topic for a hidden unit by considering the 5 words with the highest connections to that unit. We
conjecture that sparse RBMs should be beneficial in learning interpretable topics because there will
be fewer ways for hidden units to collude in order to model a given input.
Table 2 shows the result of picking a general topic and finding the closest matching hidden unit from
each model. While all models discover meaningful topics, we found that the grouping of words
produced by the RBM tended to be less cohesive than those produced by the SpRBM or CaRBM.
For example, many of the hidden units contain the words ?abstract? and ?reference?, both of which
appear in nearly every paper.
Figure 2 (b)-(d) displays the effect that the KL penalty ? has on the population sparsity of the
SpRBM. For a fairly narrow range, if ? is too small then the desired sparsity level will not be met.
2
http://psiexp.ss.uci.edu/research/programs_data/toolbox.htm
7
Model
RBM
SpRBM
CaRBM
Computer Vision
Neuroscience
Bayesian Inference
images, pixel, computer,
quickly, stanford
visual, object, objects, images, vision
image, images, pixels, objects, recognition
inhibitory, organization, neurons, synaptic, explain
neurons, biology, spike,
synaptic, realistic
membrane,
resting,
inhibitory, physiol, excitatory
probability, bayesian, priors,
likelihood, covariance
conditional,
probability,
bayesian, hidden, mackay
likelihood, hyperparameters,
monte, variational, neal
Table 2: Topics learned by each model on the NIPS dataset. Each column corresponds to a chosen
topic, and each cell corresponds to a single hidden unit. The hidden unit is chosen as the best match
to the given topic from amongst all of the hidden units learned by the model in the row.
0.0
(a)
0.2
0.5
0.0
(b) ? = 0.1
0.2
(c) ? = 0.5
0.5
0.0
0.2
0.5
(d) ? = 1
Figure 2: (a) Weights of the CaRBM learned on 16?16 images patches sampled from the CIFAR-10
dataset. (b)-(c) Change in population sparsity with increasing KL penalty ? on the NIPS dataset.
The SpRBM is sensitive to ?, and can fail to model certain examples if ? is set too high.
As it is increased, the lifetime sparsity better matches the target but at the cost of an increasing
number of dead examples. This may hurt the generative performance of the SpRBM.
6
Conclusion
We have introduced cardinality potentials into the energy function of a Restricted Boltzmann Machine in order to enforce sparsity in the hidden representation. We showed how to use an auxiliary
variable representation in order to perform efficient posterior inference and sampling. Furthermore,
we showed how the marginal probabilities can be treated as nonlinearities, and how a simple finitedifference trick from Domke [14] can be used to backpropagate through the network. We found
that the CaRBM performs similarly to an RBM that has been trained with a sparsity-encouraging
regularizer, with the exception being datasets that exhibit a wide range of variability in the number
of active inputs (e.g. text), where the SpRBM seems to have difficulty matching the target sparsity.
It is possible that this effect may be significant in other kinds of data, such as images with high
amounts of lighting variation.
There are a number of possible extensions to the CaRBM. For example, the cardinality potentials
can be relaxed to encourage sparsity, but not enforce it, and they can be learned along with the other
model parameters. It would also be interesting to see if other high order potentials could be used
within the RBM framework. Finally, it would be worth exploring the use of the sparse marginal
nonlinearity in auto-encoder architectures and in the deeper layers of a deep belief network.
References
[1] P. Smolensky. Information processing in dynamical systems: foundations of harmony theory. In Parallel
Distributed Processing: Explorations in the Microstructure of Cognition, vol. 1, pages 194?281. MIT
Press, 1986.
[2] G.E. Hinton, S. Osindero, and Y.W. Teh. A fast learning algorithm for deep belief nets. Neural Computation, 18(7):1527?1554, 2006.
[3] H. Lee, R. Grosse, R. Ranganath, and A.Y. Ng. Convolutional deep belief networks for scalable unsupervised learning of hierarchical representations. In International Conference on Machine Learning,
2009.
8
[4] Y. Bengio, P. Lamblin, D. Popovici, and H. Larochelle. Greedy layer-wise training of deep networks.
Advances in Neural Information Processing Systems, 2007.
[5] J. Snoek, R. P. Adams, and H. Larochelle. Nonparametric guidance of autoencoder representations using
label information. Journal of Machine Learning Research, 13:2567?2588, 2012.
[6] J. Yang, K. Yu, Y. Gong, and T. Huang. Linear spatial pyramid matching using sparse coding for image
classification. In Computer Vision and Pattern Recognition, 2009.
[7] I. Goodfellow, Q. Le, A. Saxe, H. Lee, and A.Y. Ng. Measuring invariances in deep networks. Advances
in Neural Information Processing Systems, 2009.
[8] B.A. Olshausen and D.J. Field. Sparse coding with an overcomplete basis set: A strategy employed by
V1? Vision Research, 37(23):3311?3325, 1997.
[9] I. Goodfellow, A. Courville, and Y. Bengio. Large-scale feature learning with spike-and-slab sparse
coding. International Conference on Machine Learning, 2012.
[10] H. Lee, C. Ekanadham, and A. Ng. Sparse deep belief net model for visual area V2. Advances in Neural
Information Processing Systems, 2007.
[11] R. Gupta, A. Diwan, and S. Sarawagi. Efficient inference with cardinality-based clique potentials. In
International Conference on Machine Learning, 2007.
[12] D. Tarlow, I. Givoni, and R. Zemel. HOP-MAP: Efficient message passing for high order potentials. In
Artificial Intelligence and Statistics, 2010.
[13] M. H. Gail, J. H. Lubin, and L. V. Rubinstein. Likelihood calculations for matched case-control studies
and survival studies with tied death times. Biometrika, 68:703?707, 1981.
[14] J. Domke. Implicit differentiation by perturbation. Advances in Neural Information Processing Systems,
2010.
[15] G.E. Hinton. Training products of experts by minimizing contrastive divergence. Neural Computation,
14(8):1771?1800, 2002.
[16] V. Nair and G.E. Hinton. 3d object recognition with deep belief nets. Advances in Neural Information
Processing Systems, 2009.
[17] R. E. Barlow and K. D. Heidtmann. Computing k-out-of-n system reliability. IEEE Transactions on
Reliability, 33:322?323, 1984.
[18] D. Tarlow, K. Swersky, R. Zemel, R.P. Adams, and B. Frey. Fast exact inference for recursive cardinality
models. In Uncertainty in Artificial Intelligence, 2012.
[19] L. Belfore. An O(n) log2(n) algorithm for computing the reliability of k-out-of-n:G and k-to-l-out-of-n:G
systems. IEEE Transactions on Reliability, 44(1), 1995.
[20] T. Tieleman. Training restricted Boltzmann machines using approximations to the likelihood gradient. In
International Conference on Machine Learning, 2008.
[21] H.J. Kappen. Deterministic learning rules for Boltzmann machines. Neural Networks, 8(4):537?548,
1995.
[22] H. Larochelle, Y. Bengio, and J. Turian. Tractable multivariate binary density estimation and the restricted
Boltzmann forest. Neural Computation, 22(9):2285?2307, 2010.
[23] G.E. Hinton and R.R. Salakhutdinov. Reducing the dimensionality of data with neural networks. Science,
313(5786):504?507, 2006.
[24] K. Swersky, M. Ranzato, D. Buchman, B.M. Marlin, and N. de Freitas. On autoencoders and score
matching for energy based models. In International Conference on Machine Learning, 2011.
[25] P. Vincent. A connection between score matching and denoising autoencoders. Neural Computation,
23(7):1661?1674, 2011.
[26] H. Larochelle, D. Erhan, A. Courville, J. Bergstra, and Y. Bengio. An empirical evaluation of deep architectures on problems with many factors of variation. In International Conference on Machine Learning,
2007.
[27] G.E. Hinton. A practical guide to training restricted Boltzmann machines. Technical Report UTML-TR
2010003, Department of Computer Science, University of Toronto, 2010.
[28] J. Bergstra and Y. Bengio. Random search for hyper-parameter optimization. The Journal of Machine
Learning Research, 13:281?305, 2012.
[29] A. Krizhevsky. Learning multiple layers of features from tiny images. Master?s thesis, University of
Toronto, 2009.
9
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4,053 | 4,669 | Probabilistic Low-Rank Subspace Clustering
S. Derin Babacan
University of Illinois at Urbana-Champaign
Urbana, IL 61801, USA
[email protected]
Shinichi Nakajima
Nikon Corporation
Tokyo, 140-8601, Japan
[email protected]
Minh N. Do
University of Illinois at Urbana-Champaign
Urbana, IL 61801, USA
[email protected]
Abstract
In this paper, we consider the problem of clustering data points into lowdimensional subspaces in the presence of outliers. We pose the problem using a
density estimation formulation with an associated generative model. Based on this
probability model, we first develop an iterative expectation-maximization (EM) algorithm and then derive its global solution. In addition, we develop two Bayesian
methods based on variational Bayesian (VB) approximation, which are capable
of automatic dimensionality selection. While the first method is based on an alternating optimization scheme for all unknowns, the second method makes use of
recent results in VB matrix factorization leading to fast and effective estimation.
Both methods are extended to handle sparse outliers for robustness and can handle missing values. Experimental results suggest that proposed methods are very
effective in subspace clustering and identifying outliers.
1
Introduction
Modeling data using low-dimensional representations is a fundamental approach in data analysis,
motivated by the inherent redundancy in many datasets and to increase the interpretability of data
via dimensionality reduction. A classical approach is principal component analysis (PCA), which
implicitly models data to live in a single low-dimensional subspace within the high-dimensional
ambient space. However, a more suitable model in many applications is the union of multiple
low-dimensional subspaces. This modeling leads to the more challenging problem of subspace
clustering, which attempts to simultaneously cluster data points into multiple subspaces and find the
basis of the corresponding subspace for each cluster.
Mathematically, subspace clustering can be defined as follows: Let Y be the M ? N data matrix
consisting of N vectors {yi ? RM }N
i=1 , which are assumed be drawn from a union of K linear (or
affine) subspaces Sk of unknown dimensions dk = dim(Sk ) with 0 < dk < M . The subspace
clustering problem is to find the number of subspaces K, their dimensions {dk }K
k=1 , the subspace
bases, and the clustering of vectors yi into these subspaces.
Subspace clustering is widely investigated problem due to its application in a large number of fields,
including computer vision [6, 12, 23], machine learning [11, 22] and system identification [31]
(see [22, 28] for comprehensive reviews). Some of the common approaches include algebraicgeometric approaches such as generalized PCA (GPCA) [19, 29], spectral clustering [18], and mixture models [9, 26]. Recently, there has been a great interest in methods based on sparse and/or
low-rank representation of the data [5, 7, 8, 14?17, 25]. The general approach in these methods is to
first find a sparse/low-rank representation X of the data and then apply a spectral clustering method
on X. It has been shown that with appropriate modeling, X provides information about the seg1
mentation of the vectors into the subspaces. Two common models for X are summarized below.
? Sparse Subspace Clustering (SSC) [7, 25]: This approach is based on representing data
points yi as sparse linear combinations of other data points. A possible optimization formulation is
min ?kY ? Dk2F + kD ? DXk2F + ?kXk1 , subject to diag(X) = 0 ,
(1)
D,X
where k ? kF is the Frobenius norm and k ? k1 is the l1 -norm.
? Low-Rank Representation (LRR) [8, 14?17] : These methods are based on a principle
similar to SSC, but X is modeled as low-rank instead of sparse. A general formulation for
this model is
min ?kY ? Dk2F + kD ? DXk2F + ?kXk? ,
(2)
D,X
where k ? k? is the nuclear norm.
In these formulations, D is a clean dictionary and data Y is assumed to be the noisy version of D
possibly with outliers. When ? ? ?, Y = D, and thus the data itself is used as the dictionary
[7,15,25]. If the subspaces are disjoint or independent1 , the solution X in both formulations is shown
to be such that Xik 6= 0 only if data points yi and yk belong to the same subspace [7, 14, 15, 25].
That is, the sparsest/lowest rank solution is obtained when each point yi is represented as a linear
combination of points in its own subspace. The estimated X is used to define an affinity matrix [18]
such as |X| + |XT | and a spectral clustering algorithm, such as normalized cuts [24], is applied on
this affinity to cluster the data vectors. The subspace bases can then be obtained in a straightforward
manner using this clustering. These methods have also been extended to include sparse outliers.
In this paper, we develop probabilistic modeling and inference procedures based on a principle
similarly to LRR. Specifically, we formulate the problem using a latent variable model based on
the factorized form X = AB, and develop inference procedures for estimating A, B, D (and
possibly outliers), along with the associated hyperparameters. We first show a maximum-likelihood
formulation of the problem, which is solved using an expectation-maximization (EM) method. We
derive and analyze its global solution, and show that it is related to closed-form solution of the rankminimization formulation (2) in [8]. To incorporate automatic estimation of the latent dimensionality
of subspaces and the algorithmic parameters, we further present two Bayesian approaches: The first
one is based on same probability model as the EM method, but additional priors are placed on the
latent variables and variational Bayesian inference is employed for approximate marginalization to
avoid overfitting. The second one is based on a matrix-factorization formulation, and exploits the
recent results on Bayesian matrix factorization [20] to achieve fast estimation that is less prone
to errors due to alternating optimization. Finally, we extent both methods to handle large errors
(outliers) in the data, to achieve robust estimation.
Compared to deterministic methods, proposed Bayesian methods have the advantages of automatically estimating the dimensionality and the algorithmic parameters. This is crucial in unsupervised
clustering as the parameters can have a drastic effect on the solution, especially in the presence of
heavy noise and outliers. While our methods are closely related to Bayesian PCA [2, 3, 20] and
mixture models [9, 26], our formulation is based on a different model and leads to robust estimation
less dependent on the initialization, which is one of the main disadvantages of such methods.
2
Probabilistic Model for Low-Rank Subspace Clustering
In the following, without loss of generality we assume that M ? N and Y is full row-rank. We also
assume that each subspace is sufficiently sampled, that is, for each Si of dimension di , there exist at
least di data vectors yi in Y that span Si . As for notation, the expectations are denoted by h ? i, N
is the normal distribution, and diag() denotes the diagonal of a matrix. We do not differentiate the
variables from the parameters of the model to have a unified presentation throughout the paper.
We formulate the latent variable model as
yi = di + nY ,
di = DAbi + nD ,
i = 1, . . . , N
(3)
(4)
L
PK
L
1
The subspaces Sk are called independent if dim( K
the direct sum.
k=1 SK ) =
k=1 dim(Sk ) with
The subspaces are disjoint if they only intersect at the origin.
2
where D is M ? N , A is N ? N , and nY , nD are i.i.d. Gaussian noise independent of the data.
The associated probability model is given by2
p(yi |di ) = N yi | di , ?y2 IM ,
(5)
p(di |D, A, bi ) = N di | DAbi , ?d2 IM ,
(6)
p(bi ) = N (bi |0, IN ) .
(7)
QN
We model the components as independent such that p(Y|D) = i=1 p(yi |di ), p(D|A, B) =
QN
QN
i=1 p(di |D, A, bi ), and p(B) =
i=1 p(bi ). This model has the generative interpretation where
latent vectors bi are drawn from an isotropic Gaussian distribution, shaped by A to obtain Abi ,
which then chooses a sample of points from the dictionary D to generate the ith dictionary element
di . In this sense, matrix DA has a role similar to principal subspace matrix in probabilistic principal
component analysis (PPCA) [26]. However, notice that in contrast to this and related approaches
such as mixture of PPCAs [9, 26], the principal subspaces are defined using the data itself in (6).
In (5), the observations yi are modeled as corrupted versions of dictionary elements di with iid
Gaussian noise. Such separation of D and Y is not necessary if there are no outliers, as the presence
of noise nY and nD makes them unidentifiable. However, we use this general formulation to later
include outliers.
2.1
An Expectation-Maximization (EM) Algorithm
In (5) - (7), latent variables bi can be regarded as missing data and D, A as parameters, and an EM
algorithm can be devised for their joint estimation. The complete log-likelihood is given by
LC =
N
X
log p(yi , bi )
(8)
i=1
with p(yi , bi ) = p(yi |di ) p(di |D, A, bi ) p(bi ). The EM algorithm can be obtained by taking the
expectation of this log-likelihood with respect to (w.r.t.) B (E-step) and maximizing it w.r.t. D, A,
?d , and ?y (M-step). In the E-step, the distribution p(B|D, A, ?d2 ) is found as N (hBi, ?B ) with
hBi = ?B
1 T T
A D D,
?d2
??1
B =I+
1 T T
A D DA ,
?d2
(9)
and the expectation of the likelihood is taken w.r.t. this distribution. In the M-step, maximizing the
expected log-likelihood w.r.t. D and A in an alternating fashion yields the update equations
?1
?1
1
1
1
T
T
, A = hBi hBBT i
,
(10)
D = 2 Y 2 I + 2 h (I ? AB) (I ? AB) iB
?y
?y
?d
with hBBT i = BBT + N ?B . Finally, the estimates of ?d2 and ?y2 are found as
?d2 =
kD ? DAhBik2F + N tr(AT DT DA?B )
,
MN
?y2 =
kY ? Dk2F
.
MN
(11)
In summary, the maximum likelihood solution is obtained by an alternating iterative procedure
where first the statistics of B are calculated using (9), followed by the M-step updates for D, A, ?d ,
and ?y in (10) and (11), respectively.
2.2
Global Solution of the EM algorithm
Although the iterative EM algorithm above can be applied to estimate A, B, D, the global solutions
can in fact be found in closed form. Specifically, the optimal solution is found (see the supplementary) as either AhBi = 0 or
? ?2 VT ,
AhBi = Vq Iq ? N ?d2 ?
q
q
(12)
2
Here we assume that Abi 6= wi where wi is a zero vector with 1 as the ith coefficient, to have a proper
density. This is a reasonable assumption if each subspace is sufficiently sampled and the dictionary element di
belongs to one of them (i.e., it is not an outlier). Outliers are explicitly handled later.
3
?
? j = max(?j , N ?d ). Here, D = U?VT
? q is a q ? q diagonal matrix with coefficients ?
where ?
is the singular value decomposition (SVD) of D, and Vq contains
? its q right singular vectors that
correspond to singular values that are larger than or equal to N ?d . Hence, the solution (12) is
related to the rank-q shape interaction matrix (SIM) Vq VqT [6], while in addition it involves scaling
of the singular vectors via thresholded singular values of D.
Using AhBi in (10), the singular vectors of the optimal D and Y are found to be the same, and the
singular values ?j of D are related to the singular values ?j of Y as
?
(
?j + N ?y2 ??1
N ?d
j , if ?j >
?
?j =
(13)
?y2 +?d2
?j ? 2 ,
if ?j ? N ?d
d
This is a combination of two operations: down-scaling and the solutions a quadratic equation, where
the latter is a polynomial thresholding operation on the singular values ?j of Y (see supplementary).
Hence, the optimal D is obtained by applying the thresholding operation (13) on the singular values
of Y, where the shrinkage amount is small for large singular values so that they are preserved,
whereas small singular values are shrank by down-scaling. This is an interesting result, as there is
no explicit penalty on the rank of D in our modeling. As shown in [8], the nuclear norm formulation
(2) leads to a similar closed-form solution, but it requires the solution of a quartic equation.
Finally, at the stationary points, the noise variance ?d2 is found as
?d2 =
1
N ?q
N
X
?2q0 ,
(14)
q 0 =q+1
that is, the average of the squared discarded singular values of D when computing DAhBi. A
simple closed form expression of ?y2 cannot be found due to the polynomial thresholding in (13),
but it can simply be calculated using (11).
In summary, if ?y2 and ?d2 are given, the optimal D and AhBi are found by taking the SVD of Y
and applying shrinkage/thresholding operations on the singular values of Y. However, this method
requires setting ?y2 and ?d2 manually. When Y itself is used as the dictionary D (i.e., ?y2 = 0),
an alternative method is to choose q, the total number of independent dimensions to be retained in
DAhBi, calculate ?d2 from (14), and finally use (12) to obtain AhBi. However, when ?y2 6= 0,
q cannot directly be set and a trial-and-error procedure is required to find it. Although ?d2 and ?y2
can also be estimated automatically using the iterative EM procedure in Sec. 2.1, this method is
susceptible to local minima, as the trivial solution AhBi = 0 also maximizes the likelihood.
These issues can be overcome by employing a Bayesian estimation to automatically determine the
effective dimensionality of D and AB. We develop two methods towards this goal, which are
described next.
3
Variational Bayesian Low-Rank Subspace Clustering
Bayesian estimation of D, A and B can be achieved by treating them as latent variables to be
marginalized over to avoid overfitting and trivial solutions such as AB = 0. Here we develop
such a method based on the probability model in the previous section but with additional priors
introduced on A, B and the noise variances. Before presenting our complete probability model, we
first introduce the matrix-variate normal distribution as its use significantly simplifies the algorithm
derivation. For a M ? N matrix X, the matrix-variate normal distribution is given by [10]
NM
N
M
1
T
N (X|M, ?, ?) = (2?) 2 |?|? 2 |?|? 2 exp ? tr ??1 (X ? M) ??1 (X ? M)
(15)
2
where M is the mean, and ?, ? are M ? M row and N ? N column covariances, respectively.
To automatically determine the number of principal components in AB, we employ an automatic
relevance determination mechanism [21] on the columns of A and rows of B using priors p(A) =
N (A|0, I, CA ), p(B) = N (B|0, CB , I), where CA and CB are diagonal matrices with CA =
diag(cA,i ) and CB = diag(cB,i ), i = 1, . . . , N . Jeffrey?s priors are placed on cA,i and cB,i , and
they are assumed to be independent. To avoid scale ambiguity, the columns of A and rows of B can
also be coupled using the same set of hyperparameters CA = CB , as in [1].
4
For inference, we employ the variational Bayesian (VB) method [4] which leads to a fast algorithm.
Let q(D, A, B, CA , CB , ?d2 , ?y2 ) be the distribution that approximates the posterior. The variational
free energy is given by the following functional
F = h log q(D, A, B, CA , CB , ?d2 , ?y2 ) ? log p(Y, D, A, B, CA , CB , ?d2 , ?y2 )i .
(16)
Using the mean field approximation, the approximate posterior is factorized as
q(D, A, B, CA , CB , ?d2 , ?y2 ) = q(D) q(A) q(B) q(CA ) q(CB ) q(?d2 ) q(?y2 ). Using the priors defined above with the conditional distributions in (5) and (6), the approximating distributions
of D, A and B minimizing the free energy F are found as matrix-variate normal distributions3
q(D) = N (hDi, I, ?D ), q(A) = N (hAi, ?A , ?A ) and q(B) = N (hBi, ?B , I), with parameters
1
1
1
T
hDi = 2 Y ?D ,
??1
=
IN + 2 h (I ? AB) (I ? AB) i
(17)
D
2
h?y i
h?y i
h?d i
1
1
tr(?A hBBT i) hDT Di
tr(C?1
(18)
??1
A ?A ) I +
A =
N
N ?d2
1
1
??1
tr(?A hDT Di) hBBT i
tr(?A )C?1
(19)
A =
A +
N
N ?d2
1
1
T
T
T
T
(20)
hAiC?1
A + 2 hD DihAihBB i = 2 hD DihBi
?d
?d
1
1
?1
hBi = ?B 2 hAT DT Di,
??1
hAT DT DAi .
(21)
B = CB +
h?d i
h?d2 i
The estimate hAi in (20) is solved using fixed-point iterations. The hyperparameter updates are
given by
hc?1
A,i i =
N
,
hAT Aiii
hc?1
B,i i =
N
,
diag(hBBT iii )
(22)
hkD ? DABk2F i
hkY ? Dk2F i
,
h?y2 i =
.
(23)
MN
MN
Explicit forms of the required moments are given in the supplementary. In summary, the algorithm
alternates between calculating the sufficient statistics of the distributions of D, A and B, and the
updates of the hyperparameters cA,i , cB,i , ?d2 and ?y2 . The convergence can be monitored during
iterations using the variational free energy F. F is also useful in model comparison, which we use
for detecting outliers, as explained in Sec. 5.
h?d2 i =
Similarly to the matrix factorization approaches [2, 3, 13], automatic dimensionality selection is
invoked via hyperparameters cA,i and cB,i , which enforce sparsity in the columns and rows of A
and B, respectively. Specifically, when a particular set of variances cA,i , cB,i assume very small
values, the posteriors of the ith column of A and ith row of B will be concentrated around zero,
such that the effective number of principal directions in AB will be reduced. In practice, this is
performed via thresholding of variances cA,i , cB,i with a small threshold (e.g., 10?10 ).
4
A Factorization-Based Variational Bayesian Approach
Another Bayesian method can be developed by further investigating the probability model. Essentially, the estimates of A and B is based on the factorization of D and are independent of Y.
Thus, one can apply a matrix factorization method to D, and relate this factorization to DAB to
find AB. Based on this idea, we modify the probabilistic model to p(D) = N (D|DL DR , I, ?12 I),
d
p(DL ) = N (DL |0, I, CL ), p(DR ) = N (DR |0, CR , I), where diagonal covariances CL and CR
are used to induce sparsity in the columns of DL and rows of DR , respectively. It has been shown
in [20] that when variational Bayesian inference is applied to this model, the global solution is found
analytically and given by
DL DR = U?F VT ,
(24)
3
The optimal distribution q(A) does not have a matrix-variate normal form. However, we force it to have
this form for computational efficiency (see supplementary for details).
5
where U, V contain the singular vectors of D, and ?F is a diagonal matrix, obtained by applying a
specific shrinkage method to the singular values of D [20]. The number of retained singular values
are therefore automatically determined. Then, setting DL DR equal to DAB, we obtain the solution
T
AB = Vf ??1
f ?F Vf , where the subscript f denotes the retained singular value and vectors.
The only modification to the method in the previous section is to replace the estimation of A and
T
B in (18)-(21) with the global solution Vf ??1
f ?F Vf . Thus, this method allows us to avoid the
alternating optimization for finding A and B, which potentially can get stuck in undesired local
minima. Although the probability model is slightly different than the one described in the previous
section, we anticipate that its global solution to be related to the factorization-based solution.
5
Robustness to Outliers
Depending on the application, the outliers might be in various forms. For instance in motion tracking
applications, an entire data point might become an outlier if the tracker fails at that instance. In
other applications, only a subset of coordinates might be corrupted with large errors. Both types
(and possibly others) can be handled in our modeling. The only required change in the model is in
the conditional distribution of the observations as
p(Y|D) = N (Y|D + E, ?y2 ) ,
(25)
where E is the sparse outlier matrix for which we introduce the prior
R
C
R
p(E) = N (E|0, CC
E , CE ) = N (vec(E)|0, CE ? CE ) .
(26)
R
The shape of the column covariance matrix CC
E and row covariance matrix CE depends on the nature
C
of outliers. If only entire data points might be corrupted, we can use CE = I and independent terms
R
R
in CR
E such that CE = diag(cE,i ), i = 1, . . . , N . When entire coordinates can be corrupted,
C
C
row-sparsity in E can be imposed using CR
E = I and CE = diag(cE,i ). In the first case, the VB
estimation rule becomes q(ei ) = N (hei i, I, ?ei ) with
!?1
1
1
1
hei i = ?ei 2 (yi ? hdi i) ?ei = diag
+
,
(27)
h?y i
h?y2 i hcR
E,i i
T
with the hyperparameter update hcR
E,i i = hei i hei i+tr (?ei ). The estimation rules for other outlier
models can be derived in a similar manner.
In the presence of outlier data points, there is an inherent unidentifiability between AB and E
which can prevent the detection of outliers and hence reduce the performance of subspace clustering.
Specifically, an outlier yi can be included in the sparse component as ei = yi or included in the
dictionary D with its own subspace, which leads to (AB)ii ? 1. To avoid the latter case, we
introduce a heuristic inspired by the birth and death method in [9]. During iterations, data points
yi with (AB)ii larger than a threshold (e.g., 0.95) are assigned to the sparse component ei . As
this might initially increase the variational energy F, we monitor its progress over a few iterations
and reject this ?birth? of the sparse component if F does not decrease below its original state.
This method is observed to be very effective in identifying outliers and alleviating the effect of the
initialization.
Finally, missing valuesQ
in Y can also be handled
by modifying the distribution of the observations
in (5) to p(yi |di ) = k?Zi N yik | dik , ?y2 , where Zi is the set containing the indices of the
observed entries in vector yi . The inference procedures can be modified with relative ease to accommodate this change.
6
Experiments
In this section, we evaluate the performance of the three algorithms introduced above, namely, the
EM method in Sec. 2.2, the variational Bayesian method in Sec. 3 (VBLR) and the factorizationbased method in Sec. 4 (VBLR-Fac). We also include comparisons with deterministic subspace
clustering and mixture of PPCA (MPPCA) methods. In all experiments, the estimated AB matrix is
used to find the affinity matrix and the normalized cuts algorithm [24] is applied to find the clustering
and hence the subspaces.
6
110
LRR (? = 0.01)
LRR (? = 0.16)
VBLR
VBLR?Fac
MPPCA
Clustering accuracy (%)
100
90
80
70
60
50
40
(a)
(b)
30
0
Figure 1: Clustering 1D subspaces (points in the same
cluster are in the same color) (a) MPPCA [3] result, (b)
the result of the EM algorithm (global solution). The
Bayesian methods give results almost identical to (b).
20
40
60
80
Percentage of Outliers (%)
100
Figure 2: Accuracy of clustering 5 independent subspaces of dimension 5 for different percentage of outliers.
Synthetic Data. We generated 27 line segments intersecting at the origin, as shown in Fig. 1, where
each contains 800 points slightly corrupted by iid Gaussian noise of variance 0.1. Each line can
be considered as a separate 1D subspace, and the subspaces are disjoint but not independent. We
first applied the mixture of PPCA [3] to which we provided the dimensions and the number of the
subspaces. This method is sensitive to the proximity of the subspaces, and in all of our trials gave
results similar to Fig. 1(a), where close lines are clustered together. On the other hand, the EM
method accurately clusters the lines into different subspaces (Fig. 1(b)), and it is extremely efficient
involving only one SVD. Both Bayesian methods VBLR and VBLR-Fac gave similar results and
accurately estimated the subspace dimensions, while the VB-variant of MPPCA [9] gave results
similar to Fig. 1(a).
Next, similarly to the setup in [15], we construct 5 independent subspaces {Si } ? R50 of dimension
5 with bases Ui generated as follows: We first generate a random 50 ? 5 orthogonal matrix U1 , and
then rotate it with random orthonormal matrices Ri to obtain Ui = Ri U1 , i = 2, 3, 4. Dictionary
D is obtained by sampling 25 points from each subspace using Di = Ui Vi where Vi are 5 ? 25
matrices with elements drawn from N (0, 1). Finally, Y is obtained by corrupting D with outliers
sampled from N (0, 1) and normalized to lie on the unit sphere. We applied our methods VBLR
and VBLR-Fac to cluster the data into 5 groups, and compare their performance with MPPCA
and LRR. Average clustering errors (over 20 trials) in Fig. 2 show that LRR and the proposed
methods provide much better performance than MPPCA. VBLR and VBLR-Fac gave similar results,
while VBLR-Fac converges much faster (generally about 10 vs 100 iterations). Although LRR also
gives very good results, its performance varies with its parameters. As an example, we included its
results obtained by the optimal and a slightly different parameter value, where in the latter case the
degradation in accuracy is evident.
Table 1: Clustering errors (%) on the Hopkins155 motion database
Method
Mean
Max
Std
GPCA [19]
30.51
55.67
11.79
LSA [30]
8.77
38.37
9.80
SSC [7]
3.66
37.44
7.21
LRR [15]
1.71
32.50
4.85
VBLR
1.75
35.13
4.92
VBLR-Fac
1.85
37.32
5.10
Real Data with Small Corruptions. The Hopkins155 motion database [27] is frequently used to
test subspace clustering methods. It consists of 156 sequences where each contains 39 to 550 data
vectors corresponding to either 2 or 3 motions. Each motion corresponds to a subspace and each
sequence is regarded as a separate clustering task. While most existing methods use a pre-processing
stage that generally involves dimensionality reduction using PCA, we do not employ pre-processing
and apply our Bayesian methods directly (the EM method cannot handle outliers and thus is not
included in the experiments). The mean and maximum clustering errors and the standard deviation
in the whole set are shown in Table 1. The proposed methods provide close to state-of-the-art
performance, while competing methods require manual tuning of their parameters, which can affect
their performance. For instance, the results of LRR is obtained by setting its parameter ? = 4,
while changing it to ? = 2.4 gives 3.13% error [15]. The method in [8], which is similar to our EM7
method except that it also handles outliers, achieves an error rate of 1.44%. Finally, the deterministic
method [17] achieves an error rate of 0.85% and to our knowledge, is the best performing method
in this dataset.
Real Data with Large Corruptions. To test our methods in real data with large corruptions, we
use the Extended Yale Database B [12] where we chose the first 10 classes that contain 640 frontal
face images. Each class contains 64 images and each image is resized to 48 ? 42 and stacked to
generate the data vectors. Figure 3 depicts some example images, where significant corruption due
to shadows and heavy noise is evident. The task is to cluster the 640 images into 10 classes. The
segmentation accuracies achieved by the proposed methods and some existing methods are listed in
Table 2, where it is evident that the proposed methods achieve state-of-art-performance. Example
recovered clean dictionary and sparse outlier components are shown in Fig. 3.
Table 2: Clustering accuracy (%) on the Extended Yale Database B
Method
Average
Y
LSA [30]
31.72
SSC [7]
37.66
VBLR
DAB
LRR [15]
62.53
E
VBLR
69.72
VBLR-Fac
67.62
VBLR-Fac
DAB
E
Figure 3: Examples of recovered clean data and large corruptions. Original images are shown in the
left column (denoted by Y), the clean dictionary elements obtained by VBLR and VBLR-Fac are
shown in columns denoted by DAB, and columns denoted by E show corruption captured by the
sparse element.
7
Conclusion
In this work we presented a probabilistic treatment of low dimensional subspace clustering. Using
a latent variable formulation, we developed an expectation-maximization method and derived its
global solution. We further proposed two effective Bayesian methods both based on the automatic
relevance determination principle and variational Bayesian approximation for inference. While the
first one, VBLR, relies completely on alternating optimization, the second one, VBLR-Fac, makes
use of the global solution of VB matrix factorization to eliminate one alternating step and leads
to faster convergence. Both methods have been extended to handle sparse large corruptions in the
data for robustness. These methods are advantageous over deterministic methods as they are able
to automatically determine the total number of principal dimensions and all required algorithmic
parameters. This property is particularly important in unsupervised settings. Finally, our formulation
can potentially be extended for modeling multiple nonlinear manifolds, by the use of kernel methods.
Acknowledgments. The authors thank anonymous reviewers for helpful comments. SDB acknowledges the Beckman Institute Postdoctoral Fellowship. SN thanks the support from MEXT Kakenhi
23120004. MND was partially supported by NSF CHE 09-57849.
8
References
[1] S. D. Babacan, M. Luessi, R. Molina, and A. K. Katsaggelos. Sparse Bayesian methods for low-rank
matrix estimation. IEEE Trans. Signal Proc., 60(8), Aug 2012.
[2] C. M. Bishop. Bayesian principal components. In NIPS, volume 11, pages 382?388, 1999.
[3] C. M. Bishop. Variational principal components. In Proc. of ICANN, volume 1, pages 514?509, 1999.
[4] C.M. Bishop. Pattern Recognition and Machine Learning. Springer, 2006.
[5] E. J. Cand`es, X. Li, Y. Ma, and J. Wright. Robust principal component analysis? CoRR, abs/0912.3599,
2009.
[6] J. P. Costeira and T. Kanade. A multibody factorization method for independently moving objects. Int. J.
Comput. Vision, 29(3):159?179, September 1998.
[7] E. Elhamifar and R. Vidal. Sparse subspace clustering. In CVPR, pages 2790?2797, 2009.
[8] P. Favaro, R. Vidal, and A. Ravichandran. A closed form solution to robust subspace estimation and
clustering. In CVPR, pages 1801?1807, 2011.
[9] Z. Ghahramani and M. J. Beal. Variational inference for Bayesian mixtures of factor analysers. In NIPS,
volume 12, pages 449?455, 2000.
[10] A. K. Gupta and D. K. Nagar. Matrix Variate Distributions. Chapman & Hall/CRC, New York, 2000.
[11] K. Huang and S. Aviyente. Sparse representation for signal classification. In NIPS, 2006.
[12] K.-C. Lee, J. Ho, and D. Kriegman. Acquiring linear subspaces for face recognition under variable
lighting. IEEE Trans. Pattern Anal. Machine Intell., 27:684?698, 2005.
[13] Y. J. Lim and T. W. Teh. Variational Bayesian approach to movie rating prediction. In Proc. of KDD Cup
and Workshop, 2007.
[14] G. Liu, Z. Lin, S. Yan, J. Sun, Y. Yu, and Y. Ma. Robust recovery of subspace structures by low-rank
representation. CoRR, abs/1010.2955, 2012.
[15] G. Liu, Z. Lin, and Y. Yu. Robust subspace segmentation by low-rank representation. In ICML, pages
663?670, 2010.
[16] G. Liu, H. Xu, and S. Yan. Exact subspace segmentation and outlier detection by low-rank representation.
In AISTATS, 2012.
[17] G. Liu and S. Yan. Latent low-rank representation for subspace segmentation and feature extraction. In
ICCV, 2011.
[18] U. Luxburg. A tutorial on spectral clustering. Statistics and Computing, 17(4):395?416, December 2007.
[19] Y. Ma, A. Yang, H. Derksen, and R. Fossum. Estimation of subspace arrangements with applications in
modeling and segmenting mixed data,. SIAM Review, 50(3):413?458, 2008.
[20] S. Nakajima and M. Sugiyama. Theoretical analysis of Bayesian matrix factorization. Journal of Machine
Learning Research, 12:2583?2648, 2011.
[21] R. M. Neal. Bayesian Learning for Neural Networks. Springer, 1996.
[22] H. Peterkriegel, P. Kroger, and A. Zimek. Clustering high-dimensional data: a survey on subspace slustering, pattern-based clustering, and correlation clustering. In Proc. KDD, 2008.
[23] S. Rao, R. Tron, R. Vidal, and Y. Ma. Motion segmentation in the presence of outlying, incomplete, or
corrupted trajectories. IEEE Trans. Pattern Anal. Machine Intell., 32(10):1832?1845, 2010.
[24] J. Shi and J. Malik. Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Machine Intell.,
22(8):888 ?905, aug 2000.
[25] M. Soltanolkotabi and E. J. Cand`es. A geometric analysis of subspace clustering with outliers. CoRR,
2011.
[26] M. E. Tipping and C. M. Bishop. Mixtures of probabilistic principal component analyzers. Neural
Comput., 11(2):443?482, February 1999.
[27] R. Tron and R. Vidal. A benchmark for the comparison of 3-d motion segmentation algorithms. In CVPR,
June 2007.
[28] R. Vidal. Subspace clustering. IEEE Signal Process. Mag., 28(2):52?68, 2011.
[29] R. Vidal, Y. Ma, and S. Sastry. Generalized principal component analysis (gpca). IEEE Trans. on PAMI,
27(12):1945?1959, 2005.
[30] J. Yan and M. Pollefeys. A general framework for motion segmentation: Independent, articulated, rigid,
non-rigid, degenerate and non-degenerate. In ECCV, volume 4, pages 94?106, 2006.
[31] C. Zhang and R. R. Bitmead. Subspace system identification for training-based MIMO channel estimation. Automatica, 41:1623?1632, 2005.
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4,054 | 467 | Network generalization for production:
Learning and producing styled letterforms
Igor Grebert
David G. Stork
541 Cutwater Ln. Ricoh Calif. Research Cen.
Foster City, CA
2882 Sand Hill Rd.# 115
94404
Menlo Park, CA 94025
Ron Keesing
Steve Mims
Dept. Physiology Electrical Engin.
Stanford U.
U. C. S. F.
San Francisco, CA Stanford, CA
94305
94143
Abstract
We designed and trained a connectionist network to generate
letterfonns in a new font given just a few exemplars from
that font. During learning. our network constructed a
distributed internal representation of fonts as well as letters.
despite the fact that each training instance exemplified both a
font and a letter. It was necessary to have separate but
interconnected hidden units for " letter" and "font"
representations - several alternative architectures were not
successful.
l. INTRODUCTION
Generalization from examples is central to the notion of cognition and
intelligent behavior (Margolis, 1987). Much research centers on
generalization in recognition, as in optical character recognition, speech
recognition. and so fonh. In all such cases, during the recognition event the
information content of the representation is reduced; sometimes
categorization is binary, representing just one bit of infonnation. Thus the
infonnation reduction in answering "Is this symphony by Mozan?" is very
large.
A different class of problems requires generalization for production, e.g.,
paint a portrait of Madonna in the style of Matisse. Here during the
production event a very low infonnational input ("Madonna," and
"Matisse") is used to create a very high informational output, including
color, fonn, etc. on the canvas. Such problems are a type of analogy. and
typically require the generalization system to abstract out invariants in both
the instance being presented (e.g., Madonna) and the style (e.g., Matisse),
and to integrate these representations in a meaningful way. This must be
1118
Network generalization for production: Learning and producing styled letterforms
done despite the fact that the system is never taught explicitly the features
that correspond to Matisse's style alone, nor to Madonna's face alone, and
is never presented an example of both simultaneously_
To explore this class of analogy and production issues, we addressed the
following problem, derived from Hofstadter (1985):
Given just a few letters in a new font, draw the remaining letters.
Connectionist networks have recently been applied to production problems
such as music composition (Todd, 1989), but our task is somewhat
different. Whereas in music composition, memory and context (in the form
of recurrent connections in a network) are used for pattern generation
(melody or harmony), we have no such temporal or other explicit context
information during the production of letterforms.
2. DATA, NETWORK AND TRAINING
Figure 1 illustrates schematically our class of problems, and shows a subset
of the data used to train our network. The general problem is to draw all the
remaining letterforms in a given font, such that those forms are recognizable
as letters in the style of that font.
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House and Benzene right) in Hofstadter's GridFont
system. There are 56 fundamental horizontal, vertical and
diagonal strokes, or "pixels," in the grid.
1119
1120
Grebert, Stork, Keesing, and Mims
Each letterfonn in Figure 1 has a recognizable letter identity and "style" (or
font). Each letter (columns) shares some invariant features as does each
font (rows), though it would be quite difficult to describe what is the
"same" in each of the a's for instance, or for all letters in Benzene right
font.
We trained our network with 26 letters in each of five fonts (Standard,
House, Slant, Benzene right and Benzene left), and just 14 letters in
the "test" font (Hunt four font). The task of the network was to
reconstruct the missing 12 letters in Hunt four font. We used a structured
three-level network (Figure 2) in which letter identity was represented in a
l-of-26 code (e.g., 010000 ... ~ b), and the font identity was represented in
a similar l-of-6 code. The letterfonns were represented as 56-element
binary vectors, with l' s for each stroke comprising the character, and were
provided to the output units by a teacher. (Note that this network is
"upside-down" from the typical use of connectionist networks for
categorization.) The two sections of the input layer were each fully
connected to the hidden layer, but the hidden layer-to-output layer
connections were restricted (Figures 3 and 4). Such restricted hidden-tooutput projections helped to prevent the learning of spurious and
meaningless correlations between strokes in widely separate grid regions.
There are unidirectional one-to-many intra-hidden layer connections from
the letter section to the font section within the hidden layer (Figure 3).
56 strokes
restricted
connections
I
/
44 letter hidden
fully interconnected \
26 letters
I
44 font hidden
tUllY
I
interconnected
6 fonts
Figure 2: Network used for generalization in production.
Note that the high-dimensional representation of strokes is at
the output of the network, while the low-dimensional
representation (a one-of-26 coding for letters and a one-ofsix for fonts) is the input. The net has one-to-many
connections from letter hidden units to font hidden units (cf.
Figure 3)
~
Network generalization for produc tion: Learning and producing styled letterforms
18 strokes
in ascender
region
~~I!?1'M?.; . ...?.~
,'< ','
1121
output units
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??? ? ?
? ????
font hidden units
letter hidden units
Figure 3: Expanded view of the hidden and output layers
of the network of Figure 2. Four letter hidden units and four
font hidden units (dark) project fully to the eighteen stroke
(output) units representing the ascender region of the
GridFont grid; these hidden units project to no other output
units. Each of the four letter hidden units also projects to all
four of the corresponding font hidden units. This basic
structure is repeated across the network (see text).
All connection weights, including intra-hidden layer weights, were adjusted
using backpropagation (Rumelhart, Hinton and Williams, 1986), with a
learning rate of TJ = 0.005 and momentum IX = 0.9. The training error
stopped decreasing after roughly 10,000 training epochs, where each epoch
consisted of one presentation of each of the 144 patterns (26 letters x
5 fonts + 14 letters) in random order.
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Figure 4: The number of hidden units projecting to each
region of the output. Four font hidden units and four letter
hidden units project to the 18 top strokes (ascender region)
of the output layer, as indicated. Ten font hidden units and
ten letter hidden units project to the next lower square region
(20 strokes), etc. This restriction prevents the learning of
meaningless correlations between particular strokes in the
ascender and descender regions (for instance). Such
spurious correlations disrupt learning and generalization only
with a small training set such as ours.
Grebert, Stork, Keesing, and Mims
1122
3. RESULTS AND CONCLUSIONS
In order to produce any letterfonn, we presented as input to the trained
network a (very sparse) l-of-26 and l-of-6 signal representing the target
letter and font; the letterfonns emerged at the output layer. Our network
reproduced nearly perfectly all the patterns in the training set.
Figure 5 shows untrained letterfonns generated by the network. Note that
despite irregularities, all the letters except z can be easily recognized by
humans. Moreover, the letterfonns typically share the common style of
Hunt four font - b, c, g, and p have the diamond-shaped "loop" of 0, q,
and other letters in the font; the g and y generated have the same right
descender, similar to that in several letters of the original font, and so on;
the I exactly matches the fonn designed by Hofstadter. Incidentally, we
found that some of the letterfonns produced by the network could be
considered superior to those designed by Hofstadter. For instance, the
generated w had the characteristic Hunt four diamond shape while the w
designed by Hostadter did not. We must stress, though, that there is no
"right" answer here; the letterforms provided by Hofstadter are merely one
possible solution. Just as there is no single "correct" portrait of Madonna in
the style of Matisse, so our system must be judged successful if the
letterforms produced are both legible and have the style implied by the other
letterforms in the test font
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Figure 5: Hofstadter's letterfonns from Hunt four font
(above), and the output of our network (below) for the
twelve letterforms that had never been presented during
training. Hofstadter's letterfonns serve merely as a guide; it
is not necessary that the network reproduce these exactly to
be judged successful.
Analysis of learned connection strengths (Grebert et al., 1992) reveals that
different internal representations were formed for letter and for font
characteristics, and that these are appropriate to the task at hand. The
particular letter hidden unit shown in Figure 6 effectively "shuts down" any
activity in the ascender region. Such a hidden unit would be useful when
Network generalization for production: Learning and producing styled letterforms
generating a, c, e, etc. Indeed this hidden unit receives strong input from
all letters that have no ascenders. The particular font hidden unit shown in
Figure 6 leads to excitation of the "loop" in Slant font, and is used in the
generation of 0, b, d, g, etc. in that font. We note further that our network
integrated style information (e.g., the diamond shape of the "loop" for the
b, g, the "dot" for the I, etc.) with the form information appropriate to the
particular letter being generated.
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hidden unit (left) and font hidden unit (right).
In general, the network does quite well. The only letterform quite poorly
represented is z. Evidently, the z letterform cannot be inferred from other
information, presumably because z does not consist of any of the simplest
fundamental features that make up a wide variety of other letters (left or
right ascenders, loops, crosses for t and f, dots, right or left descenders).
The average adult has seen perhaps as many as 10 6 distinct examples of
each letter in perhaps 10 10 presentations; in contrast, our network
experienced just five or six distinct examples of each letter in 104
presentations. Out of this tremendous number of letterforms, the human
virtually never experiences a g that has a disconnected descender (to take
one example), and would not have made the errors our network does. We
suspect that the errorS our network makes are similar to those a typical
westerner would exhibit in generating novel characters in a completely
foreign alphabet, such as Thai. Although our network similarly has
experienced only g's with connected descenders, it has a very small
database over which to generalize; it is to be expected, then, that the
network has not yet "deduced" the connectivity constraint for g. Indeed, it
is somewhat surprising that our network performs as well as it does, and
this gives us confidence that the architecture of Figure 2 is appropriate for
the production task.
:
~
".
'
0"
",
:
1123
1124
Grebert, Stork, Keesing, and Mims
This conclusion is supported by the fact that alternative architectures gave
very poor results. For instance a standard three-level backpropagation
network produced illegible letterfonns. Likewise, if the direct connections
between letter hidden units and the output units in Figure 2 were removed,
generalization perfonnance was severely compromised.
Our network parameters could have been "fine tuned" for improved
perfonnance but such fine tuning would be appropriate for our problem
alone, and not the general class of production problems. Even without such
fine tuning, though, it is clear that the architecture of Figure 2 can
successfully learn invariant features of both letter and font infonnation, and
integrate them for meaningful production of unseen letterfonns. We believe
this architecture can be applied to related problems, such as speech
production, graphic image generation, etc.
ACKNOWLEDGEMENTS
Thanks to David Rumelhart and Douglas Hofstadter for useful discussions.
Reprint requests should be addressed to Dr. Stork at the above address, or
[email protected].
REFERENCES
Grebert, Igor, David G. Stork, Ron Keesing and Steve Mims, "Connectionist generalization for production: An example from GridFont,"
Neural Networks (1992, in press).
Hofstadter, Douglas, "Analogies and Roles in Human and Machine
Thinking," Chapter 24, 547-603 in Metamagical Themas:
Questing for the Essence of Mind and Pattern Basic Books
(1985).
Margolis, Howard, Patterns, Thinking, and Cognition: A Theory
of Judgment U. Chicago Press (1987).
Rumelhart, David E., Geoffrey E. Hinton and Ron 1. Williams, "Learning
Internal Representations by Error Propagation," Chapter 8, pp. 318362 in Parallel Distributed Processing: Explorations in the
Microstructure of Cognition. Vol 1: Foundations D. E.
Rumelhart, and 1. L. McClelland (eds.) MIT Press (1986).
Todd, Peter M., "A Connectionist approach to algorithmic composition,"
Computer Music Journal, 13(4), 27-43, Winter 1989.
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4,055 | 4,670 | Rational inference of relative preferences
Paul R Schrater
Dept of Psychology
University of Minnesota
Nisheeth Srivastava
Dept of Computer Science
University of Minnesota
Abstract
Statistical decision theory axiomatically assumes that the relative desirability of
different options that humans perceive is well described by assigning them optionspecific scalar utility functions. However, this assumption is refuted by observed human behavior, including studies wherein preferences have been shown
to change systematically simply through variation in the set of choice options
presented. In this paper, we show that interpreting desirability as a relative comparison between available options at any particular decision instance results in a
rational theory of value-inference that explains heretofore intractable violations of
rational choice behavior in human subjects. Complementarily, we also characterize the conditions under which a rational agent selecting optimal options indicated
by dynamic value inference in our framework will behave identically to one whose
preferences are encoded using a static ordinal utility function.
1
Introduction
Normative theories of human choice behavior have long been based on how economic theory has
postulated they should be made. The standard version of the theory states that consumers seek to
maximize innate, stable preferences over the options they consume. Preferences are represented by
numerical encoding of value in terms of utilities, and subjects are presumed to select the option with
the maximum expected utility. The most difficult part of this theory is that preferences must exist
before decisions can be made. The standard response, in both economics and decision theory, to the
basic question ?Where do preferences come from?? is ?We?ll leave that one to the philosophers,
utilities are simply abstractions we assume for the work we do.?, which, while true, is not an answer.
While this question has been studied before in the form of learning utility values from behavior [5,
14, 10], human preferences exhibit patterns of behavior that are impossible to reconcile with the idea
that stable numerical representations of value can be ascribed to each item they choose between.
Behavioral experiments in the last half century have conclusively demonstrated (see [18] for a
comprehensive review) that human choice strongly violates the key axioms that the existence of
stable utility values depends on. A particular subset of these violations, called context effects, wound
the utility maximization program the most deeply, since such violations cannot be explained away
as systematic distortions of underlying utility and/or probability representations [22]. Consider for
instance, the ?frog legs? thought problem, pictured in Figure 1, introduced by Luce and Raiffa in
their seminal work [15]. No possible algebraic reformulation of option-specific utility functions
can possibly explain preference reversals of the type exhibited in the frog legs example. Preference
reversals elicited through choice set variation have been observed in multiple empirical studies,
using a variety of experimental tasks, and comprise one of the most powerful criticisms of the use
of expected utility as a normative standard in various economic programs, e.g. in public goods
theory. However, for all its problems, the mathematical simplicity of the utility framework and lack
of principled alternatives has allowed it to retain its central role in microeconomics [12], machine
learning [1], computational cognitive science [7] and neuroscience [11].
1
(a) When asked to select between just (b) When presented with an additional third
salmon and steak, the diner picks salmon, in- menu item, the diner picks steak, indicating
dicating salmon steak by his choice
steak salmon
Figure 1: Illustration of Luce?s ?frog legs? thought experiment. No possible absolute utility assignation to individual items can account for the choice behavior exhibited by the diner in this experiment.
The frog legs example is illustrative of reversals in preference occuring solely through variation in
the set of options a subject has to choose from.
Our contribution in this paper is the development of a rational model that infers preferences from
limited information about the relative value of options. We postulate that there is a value inference
process that predicts the relative goodness of items in enabling the agent to achieve its homeostatic
and other longer-range needs (e.g. survival and reproductive needs). While this process should be
fully explicated, we simply don?t know enough to make detailed mathematical models. However,
we show that we only have to postulate that feedback from decisions provides limited information
about the relative worth of options within the choice set for a decision to retrieve an inductive representation of value that is equivalent to traditional preference relations. Thus, instead of assuming
utilities as being present in the environment, we learn an equivalent sense of option desirability from
information in a limited format that depends on the set of options in the decision set. This inductive methodology naturally makes choice sets informative about the value of options, and hence
affords simple explanations for context effects. We show how to formalize the idea of relative value
inference, and that it provides a new rational foundation for understanding the origins of human
preferences.
2
Human Preferences via Value Inference
We begin by reviewing and formalizing probabilistic decision-making under uncertainty. An agent
selects between possibilities x in the world represented by the set X . The decision-making problem
can be formulated as one wherein the agent forms a belief b(x), x ? X about the relative desirability
of different possibilities in X and uses this belief to choose an element or subset X ? ? X . When
these beliefs satisfy the axioms of utility, the belief function simply the expected utility associated
with individual possibilities u(x), u : X ? R.
We assume these desirabilities must be learned from experience, suggesting a reinforcement learning approach. The agent?s belief about the relative desirability of the world is constantly updated by
information that it receives about the desirability of options in terms of value signals r(x). Belief
updating produces transition dynamics on bt (x). Given a sequence of choices, the normative expectation is for agents to select possibilities in a way that maximizes their infinite-horizon cumulative
discounted desirability,
?
X
arg max
? t bt (x).
(1)
x(t)
t
The sequence of choices selected describes the agent?s expected desirability maximizing behavior
in a belief MDP-world.
2
From a Bayesian standpoint, it is critical to describe the belief updating about the desirability of different states. Let p(x|r(1:t) ) represent the belief a value x is the best option given a sequence of value
signals. Since the agent learns this distribution from observing r(x) signals from the environment,
an update of the form,
p(x|r(t) ) = p(r(t) |x) ? p(x|{r(1) , r(2) ? ? ? , r(t?1) }),
(2)
reflects the basic process of belief formation via value signals. When value signals are available for
every option, independent of other options, the likelihood term p(r|x) in Equation (2) is a probabilistic representation of observed utility, which remains unaffected in the update by the agent?s history
of sampling past possibilities and hence is invariant to transition probabilities. Such separation between utilities and probabilities in statistical decision theory is called probabilistic sophistication,
an axiom that underlies almost all existing computational decision theory models [11].
The crux of our new approach is that we assume that value signals p(r|x) are not available for
every option . Instead, we assume we get partial information about the value of one or more options
within the set of options c available in the decision instance t. In this case value signals are hidden
for most options x. However, the set of options c ? C ? P(X )1 observed can now potentially be
used as auxiliary information to impute values for options whose value has not been observed. In
such a scenario, the agent requires a more sophisticated inference process,
Z
1
p(x|r(1:t) ) =
p(x, c, r(1:t) ),
p(r(1:t) ) c
Z
1
=
[p(rt |x, c)p(c|x)] ? p(x|{r(1) , r(2) ? ? ? , r(t?1) }).
p(r(1:t) ) c
Importantly, we concentrate on understanding the meaning of utility in this framework. As in the
case of value observability for all options, a probabilistic representation of utility under indirect
observability must be equivalent to,
R
R
p(r, x, c)
p(r|x, c)p(x|c)p(c)
p(r, x)
p(r|x) =
=Rc
= c R
.
(3)
p(x)
p(x|c)p(c)
p(x|c)p(c)
c
c
The resulting prediction of value of an option couples value signals received across decision instances with different option sets, or contexts. The intuition behind this approach is contained in the
frog leg?s example - the set of options become informative about the hidden state of the world, like
whether the restaurant has a good chef.
Naively, one could assume that altering existing theory to include this additional source of information would be an incremental exercise. However, a formidable epistemological difficulty arises as
soon as we attempt to incorporate context into utility-based accounts of decision-making. To see
this, let us assume that we have defined a measure of utility u(x, c) that is sensitive to the context c
of observing possibility x. Now, for such a utility measure, if it is true that for any two possibilities
{xi , xj } and any two contexts {ck , cl },
u(xi , ck ) > u(xj , ck ) ? u(xi , cl ) > u(xj , cl ),
then the choice behavior of an agent maximizing u(x, c) would be equivalent to one maximizing u(x). Thus, for the inclusion of context to have any effect, there must exist at least some
{xi , xj , ck , cl } for which the propositions u(xi , ck ) > u(xj , ck ) and u(xi , cl ) < u(xj , cl ) can hold
simultaneously.
Note however, that the context in this operationalization is simply a collection of other possibilities,
i.e. c ? X which ultimately implies u(x, c) = u(X ? ) = u(X ), X ? = {x, c} ? X . Such a
measure could assign absolute numbers to each of the possibilities, but any such static assignment
would make it impossible for the propositions u(x1 , X ) > u(x2 , X ) and u(x1 , X ) < u(x2 , X )
to hold simultaneously, as is desired of a context-sensitive utility measure. Thus, we see that it is
impossible to design a utility function u such that u : X ? C ? R. If we wish to incorporate the
effects of context variation on the desirability of a particular world possibility, we must abandon a
foundational premise of existing statistical decision theory - the representational validity of absolute
utility.
1
P(?) references the power set operation throughout this paper.
3
3
Rational decisions without utilities
In place of the traditional utility framework, we define an alternative conceptual partitioning of the
world X as a discrete choice problem. In this new formulation, at any decision instant t, agents
observe the feasibility of a subset o(t) ? X of all the possibilities in the world. In the following
exposition, we use y t to denote an indicator function on X encoding the possibilities observed as
o(t) ,
X
y t (x) =
?(x ? i),
i?o(t)
. An intelligent agent will encode its understanding of partial observability as a belief over which
possibilities of the world likely co-occur. We call an agent?s belief about the co-occurrence of
possibilities in the world its understanding of the context of its observation. We instantiate contexts
c as subsets of X that the agent believes will co-occur based on its history of partial observations of
the world and index them with an indicator function z on X , so that for context c(t) ,
X
z t (x) =
?(x ? i).
i?c(t)
Instead of computing absolute utilities on all x ? X , a context-aware agent evaluates the comparable
desirability of only those possibilities considered feasible in a particular context c. Hence, instead
of using scalar values to indicate which possibility is more preferable, we introduce preference
information into our system via a desirability function d that simply ?points? to the best option in a
given context, i.e. d(c) = B, where B is a binary relation (c, c, m) and mi = 1 iff ci ci0 ?ci0 ?
c \ {ci } and zero otherwise. The desirability indicated by d(c) can be remapped on to the larger
set of options by defining a relative desirability across all possibilities r(x) = m, x ? c and zero
otherwise.
Recall now that we have already defined what we mean by utility in our system in Equation 3.
Instantiated in the discrete choice setting, this can be restated as a probabilistic definition of relative
desirability at decision instant t as,
PC
p(r(t) |x, c)p(x|c)p(c)
R(t) (x) = p(r(t) |x) = c PC
,
(4)
c p(x|c)p(c)
where it is understood that p(c) = p(c|{o1 , o2 , ? ? ? , ot?1 }) is a distribution on the set of all possible
contexts inferred from the agent?s observation history. From the definition of desirability, we can
also obtain a simple definition of p(r|x, c) as p(ri |xi , c) = 1 iff ri xi = 1 and zero otherwise.
To instantiate Eqn (4) concretely, it is finally necessary to define a specific form for the likelihood
term p(x|c). While multiple mathematical forms can be proposed for this expression, depending on
quantitative assumptions about the amount of uncertainty intrinsic to the observation, the underlying
intuition must remain one that obtains the highest possible value for c = o and penalizes mismatches
in set membership. Such definitions can be introduced in the mathematical definition of the elementwise mismatch probability, p(?yit |zit ). Since p(xi |c(t) ) = 1 ? p(?yit |zit ), we can use these elementwise probabilities to compute the likelihood of any particular observation o(t) as,
? (t)
?
|o |
|c(t) |
|X |
[
[
X
(t) (t)
t
t ?
?
P (o |c ) = 1 ? p
{?yi }|
{zi } , = 1 ? ?
p(?yit |zit ),
i
i
i
where ? is a parameter controlling the magnitude of the penalty imposed for each mismatch observed.
This likelihood function can then be used to update the agent?s posterior belief about the contexts it
considers viable at decision instance t, given its observation history as,
p(o(t) |c)p(c|{o(1) , o(2) , ? ? ? , o(t?1) })
,
p(c(t) |{o(1) , o(2) , ? ? ? , o(t) }) = PC
(t)
(1) , o(2) , ? ? ? , o(t?1) })
c p(o |c)p(c|{o
(5)
To outline a decision theory within this framework, observe that, at decision instant t, a Bayesian
agent could represent its prior preference for different world possibilities in the form of a probability
4
distribution over the possible outcomes in X , conditioned on desirability information obtained in
earlier decisions, p(x|c(t) , {r(1) , r(2) , ? ? ? r(t?1) }). New evidence for the desirability of outcomes
observed in context c(t) is incorporated using p(r(t) |x, c(t) ), a distribution encoding the relative
desirability information obtained from the environment at the current time step, conditioned on the
context in which the information is obtained. This formulation immediately yields the belief update,
p(x|c(t) , r(t) ) ? p(r(t) |c(t) , x) ? p(x|c(t) , {r(1) , r(2) , ? ? ? r(t?1) }),
(6)
to obtain a posterior probability encoding the desirability of different possibilities x, while also
accounting tractably for the context in which desirability information is obtained at every decision
instance. Defining a choice function to select the mode of the posterior belief completes a rational
context-sensitive decision theory.
4
Demonstrations
To demonstrate the value of the relative desirability-based encoding of preferences, in Section 4.1,
we describe situations in which the influence of context shifting significantly affects human preference behavior in ways that utility-based decision theories have historically been hard-pressed to
explain. Complementarily, in Section 4.2 we characterize conditions under which the relative desirability framework yields predictions of choice behavior equivalent to that predicted by ordinal utility
theories, and hence, is an equivalent representation for encoding preferences.
4.1
Where context matters ...
In this section, we show how our inductive theory of context-sensitive value inference leads, not
surprisingly, to a simple explanation for the major varieties of context effects seen in behavioral
experiments. These are generally enumerated as attraction, similarity, comparison and reference
point effects [2]. Interestingly, we find that each of these effects can be described as a special case
of the frog legs example, with the specialization arising out of additional assumptions made about
the relationship of the new option added to the choice set. Table 1, with some abuse of notation,
describes this relationship between the effects in set-theoretic terms. Space constraints necessitate
Effect name
Frog legs
Similarity
Attraction
Compromise
Reference point
Description
c1 ? {X, Y } ? X Y , c2 ? {X, Y, Z} ? Y X
c1 ? {X, Y } ? X Y , c2 ? {X, Y, Z} ? Y X
c1 ? {X, Y } ? X ? Y , c2 ? {X, Y, Z} ? X Y
c1 ? {X, Y } ? X Y , c2 ? {X, Y, Z} ? Y X
c1 ? {X, Y } ? X ? Y , c2 ? {X, Y, Z} ? X (?) Y
Assumptions
Z?X
X Z
Y (c) X, Z
Z X
Table 1: A unified description of context effects. indicates stochastic preference for one item
over another. (c) indicates that the preference in question holds only in some observation contexts.
(?) indicates that the preference in question is stochastically weaker than before.
an abbreviate description of our results. Detailed descriptions of these effects, supplemented with
an explanation of how they may be elicited in our framework, is provided in SI. We use available
space to completely describe how the most general version of preference reversal, as seen in the frog
legs example, emerges from our framework and provide a brief overview of the other results. To
instantiate our likelihood definition in (5), we define a specific mismatch probability,
1
p(?yit |zit ) =
(1 ? zit )yit + (1 ? yit )zit ,
(7)
|X |
with ? = 1 for all our demonstrations.
In the frog legs example, the reversal in preferences is anecdotally explained by the diner originally
forming a low opinion of the restaurant?s chef, given the paucity of choices on the menu, deciding to
pick the safe salmon over a possibly a burnt steak. However, the waiter?s presenting frog legs as the
daily special suddenly raises the diner?s opinion of the chef?s abilities, causing him to favor steak.
This intuition maps very easily into our framework of choice selection, wherein the diner?s partial
5
menu observations o1 = {steak, salmon} and o2 = {steak, salmon, frog legs} are associated with two
separate contexts c1 and c2 of observing the menu X . Bad experiences related to ordering steak in
menus typically observed under context c1 (interpretable as ?cheap restaurants?) may be encoded by
defining the vector m = {1, 0, 0, 0} for c1 and good experiences ordering steak off menues observed
in context c2 (interpretable as ?upscale restaurants?) as m = {0, 1, 0, 0} for c2 . Then, by definition,
p(r|salmon, c1 ) > p(r|steak, c1 ), while p(r|salmon, c2 ) < p(r|steak, c2 ). For the purposes of this
demonstration, let us assume these probability pairs, obtained through the diner?s past experiences in
restaurants to be {0.7, 0.3} and {0.3, 0.7} respectively. Now, when the waiter first offers the diner a
choice between steak or salmon, the diner computes relative desirabilities using (4), where the only
context for the observation is {salmon, steak}. Hence, the relative desirabilities of steak and salmon
are computed over a single context, and are simply R(salmon) = 0.7, R(steak) = 0.3. When the
diner is next presented with the possibility of ordering frog legs, he now has two possible contexts to
evaluate the desirability of his menu options: {salmon, steak} and {salmon, steak, frog legs}. Based
on the sequence of his history of experience with both contexts, the diner will have some posterior
belief p(c) = {p, 1 ? p} on the two contexts. Then, the relative desirability of salmon, after having
observed frog legs on the menu can be calculated using (4) as,
p(r|salmon, c1 )p(salmon|c1 )p(c1 ) + p(r|salmon, c2 )p(salmon|c2 )p(c2 )
,
p(salmon|c1 )p(c1 ) + p(salmon|c2 )p(c2 )
0.7 ? 1 ? p + 0.3 ? 1 ? (1 ? p)
=
= 0.7p + 0.3(1 ? p).
1 ? p + 1 ? (1 ? p)
R(salmon) =
Similarly, we obtain R(steak) = 0.3p + 0.7(1 ? p). Clearly, for 1 ? p > p, R(steak) > R(salmon),
and the diner would be rational in switching his preference. Thus, through our inferential machinery,
we retrieve the anecdotal explanation for the diner?s behavior: if he believes that he is more likely
to be in a good restaurant (with probability (1 ? p)) than not, he will prefer steak.
Along identical lines, making reasonable assumptions about the contexts of past observations, our
decision framework accomodates parsimonious explanations for each of the other effects detailed in
Table 1. Attraction effects are traditionally studied in market research settings where a consumer is
unsure about which of two items to prefer. The introduction of a third item that is clearly inferior to
one of the two earlier options leads the consumer towards preferring that particular earlier option.
Our framework elicits this behavior through the introduction of additional evidence of the desirability of one of the options from a new context, causing the relative desirability of this particular
option to rise. Similarity effects arise when, given that a consumer prefers one item to another, giving him further options that resemble his preferred item causes him to subsequently prefer the item
he earlier considered inferior. This effect is elicited simply as a property of division of probability
among multiple similar options, resulting in reduced desirabiliy of the previously superior option.
Compromise effects arise when the introduction of a third option to a choice set where the consumer
already prefers one item to another causes the consumer to consider the previously inferior option
as a compromise between the formerly superior option and the new option, and hence prefer it. We
find that the compromise effect arises through a combination of reduction in the desirability of the
superior option through negative comparions with the new item and increase in the desirability of
the formerly inferior item through positive comparisons with the new item, and that this inference
occurs automatically in our framework assuming equal history of comparisons between the existing choice set items and the new item. Reference point effects have typically not been associated
with explicit studies of context variation, and may in fact be used to reference a number of behavior
patterns that do not satisfy the definition we provide in Table 1. Our definition of the reference
point effect is particularized to explain data on pain perception collected by [23], demonstrating
relativity in evaluation of objectively identical pain conditions depending on the magnitude of alternatively experienced pain conditions. In concord with empirical observation, we show that the
relative (un)desirability of an intermediate pain option reduces upon the experience of greater pain,
a simple demonstration of prospect relativity that utility-based accounts of value cannot match.
Competing hypotheses that seek to explain these behaviors are either normative and static, (e.g. extended discrete choice models ( [13] provides a recent review), componential context theory [21],
quantum cognition [8]) or descriptive and dynamic, (specifically, decision field theory [3]). In contrast, our approach not only takes a dynamic inductive view of value elicitation, it retains a normativity criterion (Bayes rationality) for falsifying observed predictions, a standard that is expected of
any rational model of decision-making [6].
6
4.2
... and where it doesn?t
It could be conjectured that the relative desirability indicator d will be an inadequate representation
of preference information compared with scalar utility signals assigned to each world possibility,
which would leave open the possibility that we may have retrieved a context-sensitive decision
theory at the expense of theoretical assurance of rational choice selection, as has been the case in
many previous attempts cited above. Were this conjecture to be true, it would severely limit the
scope and applicability of our proposal. To anticipate this objection, we theoretically prove that
our framework reduces to the standard utility-based representation of preferences under equivalent
epistemic conditions, showing that our theory retains equivalent rational representational ability
as utility theory in simple, and simply extends this representational ability to explain preference
behaviors that utility theory can?t.
What does it mean for a measure to represent preference information? To show that a utility function
u completely represents a preference relation on X it is sufficient [12] to show that, ?x1 , x2 ?
X , x1 x2 ? u(x1 ) > u(x2 ). Hence, equivalently, to show that our measure of relative desirability
R also completely represents preference information, it should be sufficient to show that, for any two
possibilities xi , xj ? X , and for any observation context c
xi xj ? R(xi ) > R(xj ).
(8)
In SI, we prove that (8) holds at decision instant t under three conditions,
(I) Context consistency: ?c ? C, s.t. xi xj ? xi xj ?c ? Cij , {xi , xj } ? Cij ? C.
(II) Transitivity between contexts: if xi xj in c1 and xj xk in c2 , ?c ? C, xi xk .
(t)
(t)
(III) Symmetry in context observability: ?xi , xj ? X , limt?? |Ci\j | = |Cj\i |.2 .
Of the three assumptions we need to prove this equivalence result, (I) and (II) simply define a stable preference relation across observation contexts and find exact counterparts in the completeness
and transitivity assumptions necessary for representing preferences using ordinal utility functions.
(III), the only additional assumption we require, ensures that the agent?s history of partial observations of the environment does not contain any useful information. The restriction of infinite data
observability, while stringent and putatively implausible, actually uncovers an underlying epistemological assumption of utility theory, viz. that utility/desirability values can somehow be obtained
directly from the environment. Any inference based preference elicitation procedure will therefore
necessarily need infinite data to attain formal equivalence with the utility representation. Finally,
we point out that our equivalence result does not require us to assume continuity or the equivalent Archimedean property to encode preferences, as required in ordinal utility definitions. This is
because the continuity assumption is required as a technical condition in mapping a discrete mathematical object (a preference relation) to a continuous utility function. Since relative desirability is
defined constructively on Q ? Q, |Q| < ?, a continuity assumption is not needed.
5
Discussion
Throughout this exegesis, we have encountered three different representations of choice preferences:
relative (ordinal) utilities, absolute (cardinal) utilities and our own proposal, viz. relative desirability.
Each representation leads to a slightly different definition of rationality, so that, assuming a rational
set selection function ? in each case we have,
? Economic rationality: x ? ?(X ) ? @y ? X , s.t. y x, predominantly used in human
preference modeling in neoclassical economics [12]], e.g. discrete choice modeling [9].
? VNM-rationality: x ? ?(X ) ? @y ? X , s.t. u(y) > u(x), predominantly used in
studying decision-making under risk [19], e.g. reinforcement learning [1].
? Bayes rationality: x ? ?(X ) ? @y ? X , s.t. R(y, {H}) > R(x, {H}), which we have
proposed. The term {H} here is shorthand for {o1 , o2 , ? ? ? , ot?1 }, {r1 , r2 , ? ? ? rt?1 }, the
entire history of choice set and relative desirability observations made by an agent leading
up to the current decision instance.
2
The notation Ci\j references the subset of all observed contexts that contain xi but not xj .
7
Bayes rationality simply claims that value inference with the same history of partial observations
will lead to a consistent preference for a particular option in discrete choice settings. In Section
4.2, we have shown conditions on choice set observations under which Bayes-rationality will be
equivalent to economic rationality. VNM-rationality is a further specialization of economic rationality, valid for preference relations that, in addition to being complete, transitive and continuous (as
required for economic preferences representable via ordinal utilities) also satisfy an independence
of irrelevant attributes (IIA) assumption [16]. Bayes-rationality specializes to economic rationality
once we instantiate the underlying intuitions behing the completeness and transitivity assumptions
in a context-sensitive preference inference theory. Therefore, rational value inference in the form
we propose can formally replace static assumptions about preference orderings in microeconomic
models that currently exclusively use ordinal utilities [12]. As such, context-sensitive preference
elicitation is immediately useful for the nascent agent-based economic modeling paradigm as well
as in dynamic stochastic general equilibrium models of economic behavior. Further work is necessary to develop a context-sensitive equivalent of the IIA assumption, which is necessary for our
system to be directly useful in modeling decision-making behaviors under uncertainty. However,
even in its current form, our inference model can be used in conjunction with existing ?inverse planning? models of utility elicitation from choice data [17] that infer absolute utilities from choice data
using extraneous constraints on the form of the utility function from the environment. In such a
synthesis, our model could generate a preference relation sensitive to action set observability, which
inverse planning models could use along with additional information from the environment to generate absolute utilities that account for observational biases in the agent?s history.
A philosophically astute reader will point out a subtle flaw in our inferential definition of rationality.
Namely, while we assume an intuitive notion of partial observability of the world, in practice, our
agents compile desirability statistics on the set of all possibilities, irrespective of whether they have
ever been observed, a problem that is rooted in an inherent limitation of Bayesian epistemology
of being restricted to computing probabilities over a fixed set of hypotheses. How can a desirability representation that assumes that observers maintain probabilistic preferences over all possible
states of the world be more epistemologically realistic than one that assumes that observers maintain scalar utility values over the same state space3 ? As a partial response to this criticism, we point
out that we do not require an ontic commitment to the computation of joint probability distributions
on all x ? X . In practice, it is likely that Bayesian computations are implemented in the brain via
sampling schemes that, in hierarchical formulations, allow approximating information of the joint
distribution as a set of the most likely marginals (in our case, relative desirability in typical observation contexts). Neural implementations of such sampling schemes have been proposed in the recent
cognitive science literature [20]. Devising a sampling scheme that matches the intuition of context
retrieval from memory to supplement our value-inference scheme presents a promising direction for
future research.
Another straightforward extension of our framework would imbue observable world possibilities
with attributes, resulting in the possibility of deriving a more general definition of contexts as clusters in the space of attributes. Such an extension would result in the possibility of transferring preferences to entirely new possibilities, allowing the set X to be modified dynamically, which would
further address the epistemological criticism above. Even further, such an extension maps directly
to the intuition of value inference resulting from organisms? monitoring of internal need states, here
modeled as attributes. Canini?s recent modeling of transfer learning using hierarchical Dirichlet
processes [4] provides most of the mathematical apparatus required to perform such an extension,
making this a promising direction for future work in our project.
In conclusion, it has long been recognized that state-specific utility representations of the desirability
of options are insufficient to capture the rich variety of systematic behavior patterns that humans exhibit. In this paper, we show that reformulating the atomic unit of desirability as a context-sensitive
?pointer? to the best option in the observed set recovers a rational way of representing desirability in
a manner sufficiently powerful to describe a broad range of context effects in decisions. Since it is
likely that preferences for options do not exist a priori and are induced via experience, our present
proposal is expected to approximate the true mechanisms for the emergence of context-sensitive
preference variation better than alternative static theories, while retaining normativity criteria missing in alternative dynamic accounts.
3
One could argue that we are essentially observing the state space (to be able to index using its membership),
but pretending to not observe it.
8
References
[1] A.G. Barto and R.S. Sutton. Reinforcement Learning: an introduction. Univesity of Cambridge
Press, 1998.
[2] J. R. Busemeyer, R. Barkan, S. Mehta, and A. Chaturvedi. Context effects and models of
preferential choice: implications for consumer behavior. Marketing Theory, 7(1):39?58, 2007.
[3] J.R. Busemeyer and J.T. Townsend. Decision field theory: A dynamic cognition approach to
decision making. Psychological Review, 100:432?459, 1993.
[4] K. Canini, M. Shashkov, and T. Griffiths. Modeling transfer learning in human categorization
with the hierarchical dirichlet process. In ICML, pages 151?158, 2010.
[5] U. Chajewska, D. Koller, and D. Ormoneit. Learning an agent?s utility function by observing
behavior. In ICML, pages 35?42, 2001.
[6] N. Chater. Rational and mechanistic perspectives on reinforcement learning. Cognition,
113(3):350 ? 364, 2009. Reinforcement learning and higher cognition.
[7] N. Daw and M. Frank. Reinforcement learning and higher level cognition: Introduction to special issue. Cognition, 113(3):259 ? 261, 2009. Reinforcement learning and higher cognition.
[8] L. Gabora and D. Aerts. Contextualizing concepts using a mathematical generalization of the
quantum formalism. Joural of Experimental and Theoretical Artificial Intelligence, 14(4):327?
358, 2002.
[9] D. Hensher, J. Rose, and W. Greene. Applied Choice Analysis: A Primer. Cambridge University Press, 2005.
[10] A. Jern, C. Lucas, and C. Kemp. Evaluating the inverse decision-making approach to preference learning. In NIPS, pages 2276?2284, 2011.
[11] D. Kahneman. Perception, action and utility: the tangled skein. In M. Rabinovich, K. Friston,
and P. Varona, editors, Principles of Brain Dynamics: Global State Interactions. MIT Pres,
2012.
[12] D. Kreps. A Course in Microeconomic Theory, pages 17?69. Princeton University Press, 1990.
[13] W. Leong and D. Hensher. Embedding decision heuristics in discrete choice models: A review.
Transport Reviews, 32(3):313?331, 2012.
[14] C.G. Lucas, T. Griffiths, F. Xu, and C. Fawcett. A rational model of preference learning and
choice prediction by children. In NIPS, pages 985?992, 2008.
[15] R. D. Luce and H. Raiffa. Games and Decisions: Introduction and Critical Survey. Wiley,
New York, 1957.
[16] J.v. Neumann and O. Morgenstern. Theory of Games and Economic Behavior. Princeton
University Press, 1953.
[17] A. Y. Ng and S. J. Russell. Algorithms for inverse reinforcement learning. In Proceedings of
the Seventeenth International Conference on Machine Learning, ICML ?00, pages 663?670,
2000.
[18] M. Rabin. Psychology and economics. Journal of Economic Literature, 36(1):pp. 11?46, 1998.
[19] S.J. Russell and P. Norvig. Artificial Intelligence: A Modern Approach. MIT Press, 1998.
[20] L. Shi and T. Griffiths. Neural Implementation of Hierarchical Bayesian Inference by Importance Sampling. In Y. Bengio, D. Schuurmans, J. Lafferty, C. K. I. Williams, and A. Culotta,
editors, Advances in Neural Information Processing Systems 22, pages 1669?1677. 2009.
[21] A. Tversky and I. Simonson. Context-dependent preferences. Management Science, 39(10):pp.
1179?1189, 1993.
[22] I. Vlaev, N. Chater, N. Stewart, and G. Brown. Does the brain calculate value? Trends in
Cognitive Sciences, 15(11):546 ? 554, 2011.
[23] I. Vlaev, B. Seymour, R.J. Dolan, and N. Chater. The price of pain and the value of suffering.
Psychological Science, 20(3):309?317, 2009.
9
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4,056 | 4,671 | Bayesian Probabilistic Co-Subspace Addition
Lei Shi
Baidu.com, Inc
[email protected]
Abstract
For modeling data matrices, this paper introduces Probabilistic Co-Subspace Addition (PCSA) model by simultaneously capturing the dependent structures among
both rows and columns. Briefly, PCSA assumes that each entry of a matrix is generated by the additive combination of the linear mappings of two low-dimensional
features, which distribute in the row-wise and column-wise latent subspaces respectively. In consequence, PCSA captures the dependencies among entries intricately, and is able to handle non-Gaussian and heteroscedastic densities. By
formulating the posterior updating into the task of solving Sylvester equations,
we propose an efficient variational inference algorithm. Furthermore, PCSA is
extended to tackling and filling missing values, to adapting model sparseness, and
to modelling tensor data. In comparison with several state-of-art methods, experiments demonstrate the effectiveness and efficiency of Bayesian (sparse) PCSA on
modeling matrix (tensor) data and filling missing values.
1 Introduction
This paper focuses on modeling data matrices by simultaneously capturing the dependent structures
among both rows and columns, which is especially useful for filling missing values. Using Gaussian
Process (GP), Xu et al [25] modified the kernel to incorporate relational information and drew
outputs from GPs. Widely used in geostatistics, Linear Models of Corregionalization (LMC) [5]
learns the covariance structures over the vectorized data matrix. In [12, 16], Bayesian probabilistic
matrix factorization (PMF) is investigated via modeling the row-wise and column-wise specific
variances and inferred based on suitable priors. Probabilistic Matrix Addition (PMA) [1] describes
the covariance structures among rows and columns, showing promising results compared with GP
regression, PMF and LMC. However, both LMC and PMA are inefficient on large scale matrices.
On high dimensional data, subspace structures are usually designed in statistical models with reduced numbers of free parameters, leading to improvement on both learning efficiency and accuracy
[3, 11, 24]. Equipping PMA with the subspace structures, this paper proposes a simple yet novel
generative Probabilistic Co-Subspace Addition (PCSA) model, which, as its name, assumes that all
entries in a matrix come from the sums of linear mappings of latent features in row-wise and columnwise hidden subspaces. Including many existing models as its special cases (see Section 2.1), PCSA
is able to capture the dependencies among entries intricately, fit the non-Gaussian and heteroscedastic density, and extract the hidden features in the co-subspaces.
We propose a variational Bayesian algorithm for inferring both the parameters and the latent dimensionalities of PCSA. For quick and stable convergence, we formulate the posterior updating
procedure into solving Sylvester equations [10]. Furthermore, Bayesian PCSA is implemented in
three extensions. First, missing values in data matrices are easily tackled and filled by iterating
with the variational inference. Second, with a Jeffreys prior, Bayesian sparse PCSA is implemented
with an adaptive model sparseness [4]. Finally, we extend the PCSA on matrix data (i.e., 2nd-order
tensor) to PCSA-k for modelling tensor data with an arbitrary order k.
1
On the task of filling missing values in matrix data, we compare (sparse) PCSA with several stateof-art models/approaches, including PMA, Robust Bayesian PMF and Bayesian GPLVM [21]. The
datasets under consideration range from multi-label classification data, user-item rating data for
collaborative filtering, and face images. Further on tensor structured face image data, PCSA is compared with the M2 SA method [6] that uses consecutive SVDs on all modes of the tensor. Although
simple and not designed for any particular application, through experiments PCSA shows results
promisingly comparable to or better than the competing approaches.
2
PCSA Model and Variational Bayesian Inference
2.1
Probabilistic Co-Subspace Addition (PCSA)
The PCSA model defines distributions over real valued matrices. Letting X ? RD1 ?D2 be an
observed matrix with D1 ? D2 without loss of generality1 , we start by outlining a generative model
for X. Consider two hidden variables y ? N (y|0d1 , Id1 ) and z ? N (z|0d2 , Id2 ) with d1 < D1
and d2 < D2 , where 0d denotes a d-dim vector with all entries being zeros and Id denotes a
d ? d identity matrix. Using the concatenation nomenclature of Matlab, two matrices of hidden
factors Y = [y?1 , . . . , y?D2 ] ? Rd1 ?D2 and Z = [z?1 , . . . , z?D1 ] ? Rd2 ?D1 are column-wise
independently generated, respectively. Through two linear mapping matrices A ? RD1 ?d1 and
B ? RD2 ?d2 , each entry xij ? X is independent given Y and Z by xij = ai? y?j + bj? z?i + eij ,
where ai? is the i-th row of A. Each eij ? N (eij |0, 1/? ) is independently Gaussian distributed and
independent from Y, Z. The generative process of X thus is:
? Get Y by independently drawing each vector y?j ? N (y?j |0d1 , Id1 ) for j = 1, . . . , D2 ;
? Get Z by independently drawing each vector z?i ? N (z?i |0d2 , Id2 ) for i = 1, . . . , D1 ;
? Get E ? RD1 ?D2 by independently drawing each element eij ? N (eij |0, 1/? ) for ?i, j;
? Get X = AY + (BZ)? + E given Y and Z, i.e., additively combines the co-subspaces.
Given parameters ? = {A, B, ? }, the joint distribution of X, Y and Z is p(X, Y, Z|?) =
[
D2
?
j=1
N (y?j |0d1 , Id1 )] ? [
D1
?
N (z?i |0d2 , Id2 )] ? [
i=1
D1 ?
D2
?
N (xij |ai? y?j + bj? z?i , 1/? )].
(1)
i=1 j=1
Properties and relations to existing work. Albeit its simple generative process, PCSA owns
meaningful properties and can be viewed as an extension of several existing models.
? Intricate dependencies between entries in X. Although each entry xij ? X is independent given Y and Z, the PCSA model captures the dependencies along rows as well as
columns in the joint X. Particularly, assuming D1 is the data dimensionality while D2 is
the sample size, the samples (column vectors) in X is dependent from each other by PCSA. When B is constrained as 0, PCSA will degenerate to Probabilistic PCA (PPCA) [3],
which insists the sample i.i.d. assumption.
? Non-Gaussianity and heteroscedasticity. If we still consider D1 as the data dimensionality and D2 as the sample size, the PCSA model handles the non-Gaussianity in samples of
X. As an extreme example, if all columns of Z? B? are discretized to take values from a
set of n vectors, PCSA degenerates to Mixture of PPCA [7, 20] with n components, whose
subspace loadings are the same. That is, learning such a PCSA model actually implements
the group PPCA [24] throughout different components. Also, if marginalizing Z, we are
describing the column samples of X with a dependent heteroscedasticity.
? Co-subspace feature extraction. Although able to describe the row-wise and columnwise covariances, PMA [1] requires estimating and inverting two (large) kernel matrices
with sizes D1 ? D1 and D2 ? D2 respectively, which is intractable for many real world
applications. In contrast, PCSA has (D1 d1 + D2 d2 + 1) free parameters and inverts smaller
matrices, and recovers PMA when d1 = D1 and d2 = D2 . Moreover, PCSA is able to
extract the hidden features Y and Z simultaneously.
1
Otherwise, we can transpose X. This assumption is for efficient Sylvester equation solving in the sequel.
2
2.2
Variational Bayesian Inference
Given X and the hidden dimensionalities (d1 , d2 ), we can estimate PCSA?s parameters ? =
{A, B, ? } by maximizing the likelihood p(X|?). However, the capacity control is essential to generalization ability, for which we proceed to deliver a variational Bayesian inference on PCSA.
By introducing hyper-parameters ? = [?1 , . . . , ?d1 ]? and ? = [?1 , . . . , ?d2 ]? for a hierarchical
Normal-Gamma prior on (A, ?) and (B, ?) respectively [3, 7, 20], we have the prior p(?) as
p(? ) = ?(? |u? , v ? ),
d1
d1
?
?
p(A|?) =
N (a?i |0D1 , ID1 /?i ), p(?) =
?(?i |u?i , vi? ),
p(?, ?, ?) = p(? )p(A, ?)p(B, ?),
p(A, ?) = p(A|?)p(?),
p(B, ?) = p(B|?)p(?), p(B|?) =
i=1
d2
?
i=1
d2
?
N (b?i |0D2 , ID2 /?i ), p(?) =
i=1
?
?(?i |u?
i , vi ), (2)
i=1
where ?(?|u, v) denotes a Gamma distribution with a shape parameter u and an inverse scale parameter v. Each column a?i of the mapping matrix A priori independently follows a spherical Gaussian
with a precision scalar ?i , i.e., an automatic relevance determination (ARD) type prior [14]. Each
precision ?i further follows a Gamma prior for completing the specification of the Bayesian model.
?
It is computationally intractable to evaluate the marginal likelihood p(X) = p(X|?)p(?)d?,
where ? = {Z, Y, ?, ?, ?} represents the set of all parameters and latent variables. Since MCMC
samplers are inefficient for high dimensional data, this paper chooses variational inference instead
[11], which introduces a distribution Q(?) and approximates maximizing the log marginal like?
lihood log p(X) by maximizing a lower bound L(Q) = Q(?) log p(X,?)
Q(?) d?. For tractability,
Q(?) is factorized into the following mean-field form:
Q(?) = Q(Y)Q(Z)Q(A)Q(B)Q(? )Q(?)Q(?), Q(Y) =
D2
?
Q(y?i ), Q(Z) =
i=1
Q(A) =
d1
?
i=1
Q(a?i ),
Q(B) =
d2
?
Q(b?i ),
Q(?) =
i=1
d1
?
Q(?i ),
D1
?
Q(z?i ),
i=1
Q(?) =
i=1
d2
?
Q(?i ).
(3)
i=1
Maximizing L(Q) w.r.t. the above Q(?) for ?? ? ? leads to the following explicit conjugate forms
? Y ), Q(z?i ) = N (z?i |?
? Z ),
Q(y?t ) = N (y?t |?
y?t , ?
z?i , ?
?
Q(?i ) = ?(?i |?
u?i , v?i? ), Q(?i ) = ?(?i |?
u?
,
v
?
),
i
i
? ?i , ? B ID ), Q(? ) = ?(? |?
Q(a?i ) = N (a?i |?
a?i , ? A ID1 ), Q(b?i ) = N (b?i |b
u? , v?? ). (4)
2
? = [?
? Y
? and Z.
? During
??d1 ] and similarly for B,
For expression simplicity, we denote A
a?1 , . . . , a
? and Z
? are bundled and conditional on each other:
maximizing L(Q), the solutions of Y
[
]
? = ?? ?SA A
? ? (X ? Z
? ?B
? ? ), ?
? Y = SA , SA = (1 + ?? ?D1 ? A )Id + ?? ?A
? ?A
? ?1 ,
Y
1
[
]
? = ?? ?SB B
? ? (X ? A
? Y)
? ?,
? Z = SB , SB = (1 + ?? ?D2 ? B )Id + ?? ?B
? ?B
? ?1 , (5)
Z
?
2
where ??? is expectation and ?? ? = u
?? /?
v ? . Directly updating by the above converges neither quickly
nor stably. Instead after putting one equation into the other, we attain a Sylvester equation [10] and
? is obtained by solving LZ ZL
? Z ?Z
? + LZ = 0,
can efficiently solve it by many tools. Then Z
1
2
3
)
(
?
?
A
?
Z
B
Z
A
Z
2 B ?? ?
? ? , (6)
? X ID ? ?? ?AS
? A
? , L = ?? ?S B
? A
with L1 = ?? ? S B B, L2 = AS
3
1
? 2 . Given Y
? and Z,
? updating (A,
? B)
? is similar.
whose solution is further put into Eq.(5) to update Y
The remainders of Q(?) in Eq.(4) are updated as
( ?1 )
[
]?1
,
? A = d1 /tr SY
,
SY = ?? ?KY + diag(???)
( ?1 )
[
]?1
B
Z
Z
Z
,
? = d2 /tr S
,
S = ?? ?K + diag(???)
2
?Y
? ? + D2 ?
?Y ,
KY = Y
?Z
? ? + D1 ?
? Z,
KZ = Z
? first is based on the assumption D1 ? D2 for learning efficiency.
The choice of computing Z
3
D1
D2
D1 D2
? ? = u? +
? ? = u? +
,
u
1d ,
u
1d ,
2
2 1
2 2
D1
?Y A
? ?A
? + ? A KY ) + D2 tr(?
? ZB
? ?B
? + ? B KZ ) + 1 ||(X ? A
?Y
? ?Z
? ?B
? ? )||2 ,
v?? = v ? +
tr(?
F
2
2
2
A
B
1
1
? ? A)
? + D1 ? 1d ,
? ? B)
? + D2 ? 1d ,
? ? = v? + diag(B
? ? = v? + diag(A
v
v
(7)
1
2
2
2
2
2
u
?? = u? +
v1? , . . . , u
??
vd?2 ]? , tr(?) stands for trace, diag(?)
where ??? = [?
u?1 /?
v1? , . . . , u
??d1 /?
vd? 1 ]? , ??? = [?
u?
1 /?
d2 /?
inter-converts between a vector and a diagonal matrix, and || ? ||F is the Frobenius norm.
In implementation, all Gamma priors in Eq.(2) are set to be vague as ?(?|10?3 , 10?3 ). During
? and B
? will be pushed to approach zeros, which actually makes
learning, redundant columns of A
Bayesian model selection on hidden dimensionalities d1 and d2 .
3
3.1
Extensions
Filling Missing Values
In many real applications, X is usually partially observed with some missing entries. The goal here
is to infer not only the PCSA model but also the missing values in X based on the model structure.
e where the missing values
Similar to the settings of PMA in [1], let us begin with a full matrix X,
are randomly filled. We denote M = {(i, j) : x
?ij is missing} as the index set of the missing
e is the observed matrix and update Q(?) by
values therein. In each iteration, we ?pretend? that X
Eqs.(6?7). Then given Q(?), the missing entries {?
xij : (i, j) ? M} are updated by maximizing
? = arg maxX L(Q) = A
?Y
? +Z
? ?B
? ? . This updating manner plays a
L(Q), i.e., x
?ij = x
?ij with X
role of adaptive regularization [2], and performs well in experiments as to be shown in Section 4.
Moreover, filling missing values in PMA [1] needs to infer the column and row factors by either
? ?Z
? that were estimated already in
Gibbs sampling or MAP. In contrast, PCSA directly employs Y
the variational inference, and thus saves the computing cost.
3.2
Bayesian Sparse PCSA
As discussed above, PCSA describes observations by mapping hidden features (Y and Z) in the
co-subspaces through A and B respectively, i.e., A and B serve similarly to the transformation
matrix in Factor Analysis and PPCA. For high dimensional data, the parameters A and B probably
suffer from inaccurate estimations and are difficult to interpret. Sparsification is one popularlyused method to improve model interpretability in the literature. In this part, we extend to provide a
Bayesian treatment on the sparse PCSA model.
LASSO [19] encourages model sparseness by adding an ?1 regularizer, which is equivalent to a
Laplacian prior. In [4], the sparseness is adaptively controlled by assigning a hierarchical NormalJeffreys (NJ) prior. Paper [9] showed that the NJ prior performs better than the Laplacian on sparse
PPCA. In this paper, we choose to adopt the NJ prior for learning a sparse PCSA model.
Different from Eq.(2), each column of A and B follows a hierarchical Normal-Jeffreys prior:
p(A|?A ) =
d1
?
N (a?i |0, ?iA ID1 ),
p(?A ) =
d1
?
1
,
?A
i=1 i
with ?A = [?1A , . . . , ?dA1 ]? ,
N (b?i |0, ?iB ID2 ),
p(?B ) =
d2
?
1
,
?B
i=1 i
with ?B = [?1B , . . . , ?dB2 ]? , (8)
i=1
p(B|?B ) =
d2
?
i=1
which also encourages the variances in ?A and ?B of redundant dimensions to approach zeros.
The prior on ? remains the same as in Eq.(2). Still under the variational inference framework,
we now let ? = {Z, Y, ?} and Q(?) = Q(Y)Q(Z)Q(?) takes the conjugate form same as in
?
A
,?B )
Eq.(4). In consequence, we optimize L(Q; ?A , ?B ) = Q(?) log p(X,?|?
d? w.r.t. Q(?),
Q(?)
?A and ?B , where L(Q; ?A , ?B ) ? log p(X|?A , ?B ). Posterior inference remains the same
as above, except that all appearances of ??? and ??? are replaced with [1/?1A , . . . , 1/?dA1 ]T and
4
[1/?1B , . . . , 1/?dB2 ]T respectively. Then given Q(?), the variances ?A and ?B are updated via
? ? A)
? + ? A ] and ?B = 1 [diag(B
? ? B)
? + ? B ].
?A = D11+2 [diag(A
D2 +2
3.3
Modeling High-Order Tensor Data
Up till now, we have been talking about modeling X when it is a matrix, and this part extends the
PCSA model and its Bayesian inference to cover the cases when X is structured as a tensor. Tensors
are higher-order generalizations of vectors (1st-order tensors) and matrices (2nd-order tensors) [6].
Each dimension of a tensor is called as a mode, and the order of a tensor is determined as the number
of its modes. Let us denote tensors with open-face uppercase letters (e.g., X, Y, Z), in comparison
with the bold uppercase letters (e.g., X, Y, Z) for matrices. A kth-order tensor X can be denoted
by X ? RD1 ?D2 ?...?Dk , where its dimensionalities in each mode are D1 , D2 , . . . , Dk respectively.
An element and a (1st-mode) vector of X are denoted by xj1 j2 ...jk and x?j2 ...jk respectively, where
1 ? ji ? Di for each i = 1, . . . , k. Moreover, the 1st-mode flattening transform of X, denoted by
F(X) ? RD1 ?(D2 D3 ...Dk ) , is obtained by concatenating all the (1st-mode) vectors of X. Vice versa,
a ([D1 , . . . , Dk ])-tensorization of a matrix X ? RD1 ?(D2 ...Dk ) is defined as T(X, [D1 , . . . , Dk ]) ?
RD1 ?D2 ?...?Dk , so that T(F(X), [D1 , . . . , Dk ]) = X. An ith mode-shift transform is defined as
M(X, i) ? RDi ?Di+1 ?...?Dk ?D1 ?...?Di?1 , which shifts the modes sequentially in a cycle and until
the ith-mode in X becomes the 1st-mode in M(X, i).
Based on the above definitions, the PCSA model describes a kth-order tensor data XD1 ?...?Dk
through the following generative process: (i) for each mode i, all elements of the hidden tensor
(i)
Y(i) ? Rdi ?Di+1 ?...?Dk ?D1 ?...?Di?1 are assumed i.i.d. drawn from N (yji ji+1 ...jk j1 ji?1 |0, 1);
?k
(i) (i)
(ii) draw each element xj1 ...jk ? N (xj1 ...jk | i=1 aji ? y?ji+1 ...jk j1 ...ji?1 , 1/? ), i.e., X is actually
generated by a mode-shift co-subspace addition:
k
( (
)
)
?
? (i) , [Di , . . . , Dk , D1 , . . . , Di?1 ] , k + 2 ? i ,
X=E+
M T X
(9)
i=1
where each X = A(i) F(Y(i) ) and the matrix A(i) ? RDi ?di maps Y(i) to X. Shortly named as
PCSA-k, this model has latent tensors {Y(i) }ki=1 and parameters ? = {? } ? {A(i) }ki=1 with latent
scales {di }ki=1 . When k = 2, PCSA-2 is exactly the PCSA in Section 2.1 on matrix data. Also, it
can be imagined as a kind of group Factor Analysis [24].
? (i)
Same as Eq.(2), each column of A(i) takes a hierarchical Normal-Gamma prior, and the Bayesian
inference in Section 2.2 can be trivially extended for covering PCSA-k model. Please see the details
in the supplementary materials. Except the involvement of the tensor structure and its operators,
there is another difference compared with the variational posterior updating based on a matrix X.
Remembering (Q(Y), Q(Z)) and (Q(A), Q(B)) pairwise were decoupled and updated by solving
Sylvester equations, we can decouple neither {Q(Y(i) )}i nor {Q(A(i) )}i into Sylvester equations
for the general k > 2. Instead, sequentially for each i = 1, . . . , k, we update only Q(Y(i) ) (or
Q(A(i) )) and keep the remaining {Q(Y(u) )}u?=i (or {Q(A(u) )}u?=i ) fixed.
4
4.1
Experimental Results
Predicting Missing Entries in Weight Matrices
On Emotions and CAL500 Data. The proposed PCSA model can be viewed as a rather direct
extension of the PMA model, which showed advantages over GPR, LMC and PMF in [1]. Following
[1], the first experiment compares PCSA with PMA in filling the missing entries of a truncated logodds matrix in multi-label classification. For n samples and m classes, the class memberships can
be represented as an n ? m binary matrix G. A truncated log-odds matrix X is constructed with
xij = c if gij = 1 and xij = ?c if gij = 0, where c is nonzero constant. In experiments, certain
entries xij are assumed to be missing and filled as x
?ij by an algorithm, and the performance is
evaluated by the class membership prediction accuracy based on sign(?
xij ).
Two multi-label classification datasets are under consideration, namely Emotions [22] and
CAL500 [23]. Already used in [1], the Emotions contains 593 samples with 72 numeric at5
tributes in 6 classes, and the number of classes that each sample belongs to ranges from 1 to 3. The
constructed X for Emotions is thus 593 ? 6. The CAL500 contains 502 samples with 68 numeric
attributes in 174 classes, and the min and max numbers of classes that each sample belongs to are
13 and 48 respectively. The constructed X for CAL500 is thus 502 ? 174, i.e., its size is larger and
more balanced than the one for Emotions.
To test the capability in dealing with missing values, the proportion of the missing labels is increased
from 10% to 50%, with 5% as a step size. Instead of Gibbs sampling, the MAP inference is used in
PMA implementation for a fair comparison. After 10 independent runs on each dataset, Fig.1 reports the error rates for recovering the missing labels in the truncated log-odds matrices, by Bayesian
PCSA, Bayesian sparse PCSA and PMA. On the relatively unbalanced Emotions data, PCSA outperforms sparse PCSA when the missing proportion is no larger than 40%, while sparse PCSA takes
over the advantage when too many entries are missing due to the increasing importance of model
sparsity. On the more balanced CAL500 data, the sparse PCSA keeps a slight outperformance over
PCSA, again due to the sparsity. Moreover, PCSA and sparse PCSA always perform considerably
better than PMA on both datasets. Table 1 reports the average time cost, where sparse PCSA shows
a little quicker convergence than PCSA. Both are much quicker than PMA, since they do not need
to either invert large covariances or infer the factor during filling missing values (see Section 3.1).
dataset:
PCSA
sparse PCSA
PMA
Figure 1: Error rates of 10 independent runs for
recovering the missing labels in Emotions (left)
and CAL500 (right) data.
Emotions
4.0
3.5
22.9
CAL500
17.3
11.6
198.3
Table 1: Average time cost (in seconds)
on each dataset throughout 10 independent runs and all missing proportions.
On MovieLens and JesterJoke Data. In many real applications, e.g. collaborative filtering,
the size of the matrix X is much larger than the above. We proceed to consider on two larger
weight datasets: the MovieLens100K data3 and the JesterJoke3 data [8]. Particularly, the
MovieLens100K dataset contains 100K ratings of 943 users on 1682 movies, which are ordinal
values on the scale [1, 5]. The JesterJoke3 data contains ratings of 24983 users who have rated
between 15 and 35 pieces of the total 100 jokes, where the ratings are continuous in [?10.0, 10.0].
Recently in [12], Robust Bayesian Matrix Factorization (RBMF) was proposed by adopting a
Student-t prior in probabilistic matrix factorization, and showed promising results on predicting
entries on both MovieLens100K and JesterJoke3 data. Following [12], in each run we randomly choose 70% of the ratings for training, and use the remaining ratings as the missing values
, the performance is evalrt }Tt=1?
for testing. Given the true test ratings {rt }Tt=1 and the predictions {?
?T
uated based on the rooted mean squared error (RMSE), i.e., RMSE = T1 t=1 (rt ? r?t )2 , and
?T
the mean absolute error (MAE), i.e., MAE = T1 t=1 |rt ? r?t |.
After 10 independent runs, the average RMSE and MAE values obtained by (sparse) PCSA are
reported in Table 2, in comparison with the best results by RBMF (i.e., RBMF-RR) collected from
[12]. Since PMA runs inefficiently on high dimensional data as in Table 1, it is not considered to fill
the ratings in this experiment. It is observed that the performance by PCSA on predicting ratings is
comparable with RBMF. On both RMSE and MAE scores, the sparse PCSA further improves the
correctness and performs similarly to or better than RBMF.
Table 2: Average RMSE and MAE on MovieLens100K (left) and JesterJoke3 (right).
model
RMSE MAE
model
RMSE MAE
PCSA
0.903 0.708
PCSA
4.446 3.447
sparse PCSA
0.898 0.706
sparse PCSA
4.413 3.434
RBMF-RR [12] 0.900 0.705
RBMF-RR [12] 4.454 3.439
3
Downloaded from www.grouplens.org/node/73.
6
4.2
Completing Partially Observed Images
We consider two greyscale face image datasets, namely Frey [15] and ORL [17]. Specifically, Frey
has 1965 images of size 28 ? 20 taken from one person, and the data X is thus a 560 ? 1965 matrix;
ORL has 400 images of size 64 ? 64 taken from 40 persons (10 images per person), and the data X
is thus a 4096 ? 400 matrix. Applied on these matrices, the PCSA model is expected to extract the
latent correlations among pixels and images. In [13], Neil Lawrence proposed a Gaussian process
latent variable model (GPLVM) for modeling and visualizing high dimensional data. Recently a
Bayesian GPLVM [21] was developed and showed much improved performance on filling pixels in
partially observed Frey faces. This experiment compares PCSA with Bayesian GPLVM4 .
While PCSA can utilize the partial observed samples, the Bayesian GPLVM cannot. Thus in each
run, we randomly pick nf images as fully observed, and a half pixels of the remaining images are
further randomly chosen as missing values. Same as [21], Bayesian GPLVM uses the nf images for
training and then infers the missing pixels. In contrast, (sparse) PCSA uses all images as a whole
matrix. In order to test the robustness, the nf for Frey is decreased gradually from 1000 to 200,
and for ORL is decreased gradually from 300 to 50. Performance is evaluated by the correlation
coefficient (CORR) and the MAE between the filled pixels and the ground truth.
On Frey and ORL data respectively, Fig.2 and Fig.3 report the CORR and MAE values of 10
independent runs by PCSA, sparse PCSA and Bayesian GPLVM. Both PCSA and sparse PCSA
perform more accurately than Bayesian GPLVM in completing the missing pixels, and PCSA gives
the best matching. Also, (sparse) PCSA shows promising stability against the decreasing fully
observed sample size nf , and this tendency is kept even when we assign all images are partially
observed (i.e., nf = 0), as exemplified by Fig.4. The results by Bayesian GPLVM deteriorate more
obviously, because the partially observed images have no contribution during learning. Furthermore,
the advantage of PCSA becomes more significant, as we shift from the Frey data for a single
person, to the ORL data for multiple persons. It indirectly reflects the importance of extracting the
correlations among different images, rather than keeping them independent. Sparse PCSA performs
worse than PCSA in this task, mainly because it leads to a little too many sparse dimensions.
Figure 2: Results of 10 runs on Frey faces.
Figure 3: Results of 10 runs on ORL faces.
Figure 4: Reconstruction examples by PCSA when all images are partially observed: Frey (left)
and ORL (right). Three rows from top are true, observed, and reconstructed images, respectively.
4.3
Completing Partially Observed Image Tensor
We proceed to consider modeling the face image data arranged in a tensor. The dataset under consideration is a subset of the CMU PIE database [18], and totally has 5100 face images from 30 individuals. Each person?s face exhibits 170 images corresponding to 170 different pose-and-illumination
combinations. Each normalized image has 32 ? 32 greyscale pixels, and the dataset is thus a tensor
X ? R1024?30?170 , whose three modes correspond to pixel, identity, and pose/illumination, respectively. Figure 5 shows some image examples of two persons. The PCSA-k model (with k = 3 on the
4
We use the code in http://staffwww.dcs.shef.ac.uk/people/N.Lawrence/vargplvm/.
7
3rd-order tensor X) in Section 3.3 is expected to extract the co-subspace structures (i.e., correlations
among pixels, identities, and poses/illuminations respectively) and fill the missing values in X. In
[6], an M2 SA method was proposed to conduct multilinear subspace analysis with missing values
on the tensor data, via consecutive SVD dimension reductions on each mode.
Figure 5: Typical normalized face images from the CMU PIE database.
true:
filled:
true:
filled:
Figure 6: Typical missing images filled by PCSA-3. Original images (in the odd rows) are
randomly picked and removed, and PCSA-3 fills the images in the even rows.
Table 3: Average CORR (left) and MAE (right) of 10 runs
by PCSA-3 and M2 SA on the CMU PIE data.
missing proportion: 10%
20%
30%
PCSA-3
0.937 0.926 0.908
M2 SA
0.928 0.914 0.893
missing proportion: 10% 20% 30%
PCSA-3
14.6 18.3 21.5
M2 SA
17.8 21.9 24.8
Here, the randomly drawn missing values are not pixels as in Section 4.2 but images. Compared with
the true missing images, the goodness of the filled missing images is evaluated again by CORR and
MAE. Still to test the capability in dealing with missing values, the proportion of the missing images
is considered as 10%, 20% and 30%, respectively. After 10 independent runs for each proportion, the
averages CORR and average MAE of filing the missing images by PCSA-3 and M2 SA are compared
in Table 3. During implementing M2 SA, the ratio of the subspace rank over the original rank is set
as 0.6 according to Fig.9 in [6]. As shown in Table 3, PCSA-3 achieves the better performance in
all cases. For demonstration, Fig.6 shows some filled missing images when the missing proportion
is 20%, which match the original images steadily well.
5 Concluding Remarks
We have introduced the Probabilistic Co-Subspace Addition (PCSA) model, which simultaneously
captures the dependent structures among both rows and columns in data matrices (tensors). Variational inference is proposed on PCSA for an approximate Bayesian learning, and the posteriors
can be efficiently and stably updated by solving Sylvester equations. Capable to fill missing values,
PCSA is extended to not only sparse PCSA with the help of a Jeffreys prior, but also PCSA-k that
models arbitrary kth-order tensor data. Although somewhat simple and not designed for any particular application, the experiments demonstrate the effectiveness and efficiency of PCSA on modeling
matrix (tensor) data and filling missing values. The performance by PCSA may be further improved
by considering nonlinear mappings with the kernel trick, which however is not that direct due to the
coupling inner products between the co-subspaces.
Acknowledgments
The author would like to thank the anonymous reviewers for their useful comments on this paper.
8
References
[1] A. Agovic, A. Banerjee, and S. Chatterjee. Probabilistic matrix addition. In Proc. ICML, pages 1025?
1032, 2011.
[2] C. M. Bishop. Training with noise is equivalent to Tikhonov regularization. Neural Computation,
7(1):108?116, 1995.
[3] C. M. Bishop. Variational principal components. In Proc. ICANN?1999, volume 1, pages 509?514, 1999.
[4] M. A. T. Figueiredo. Adaptive sparseness using Jeffreys prior. In Advances in NIPS, volume 14, pages
679?704. MIT Press, Cambridge, MA, 2002.
[5] A. E. Gelfand and S. Banerjee. Multivariate spatial process models. In A. E. Gelfand, P. Diggle, P.Guttorp,
and M. Fuentes, editors, Handbook of Spatial Statistics. CRC Press, 2010.
[6] X. Geng, K. Smith-Miles, Z.-H. Zhou, and L. Wang. Face image modeling by multilinear subspace
analysis with missing values. IEEE Trans. Syst., Man, Cybern. B, Cybern., 41(3):881?892, 2011.
[7] Z. Ghahramani and G. Hinton. The EM algorithm for mixtures of factor analyzers. Technical Report
CRG-TR-96-1, Department of Computer Science, University of Toronto, Toronto, Canada, 1997.
[8] K. Goldberg, T. Roeder, D. Gupta, and C. Perkins. Eigentaset: A constant time collaborative filtering
algorithm. Information Retrieval, 4(2):133?151, 2001.
[9] Y. Guan and J. Dy. Sparse probabilistic principal component analysis. In Proc. AISTATS?2009, JMLR
W&CP, volume 5, pages 185?192. 2009.
[10] D. Y. Hu and L. Reichel. Krylov-subspace methods for the Sylvester equation. Linear Algebra and Its
Applications, 172:283?313, 1992.
[11] M. I. Jordan, editor. Learning in graphical models. MIT Press, Cambridge MA, 1999.
[12] B. Lakshimanarayan, G. Bouchard, and C. Archambeau. Robust Bayesian matrix factorisation. In Proc.
AISTATS?2011, JMLR W&CP, volume 15, pages 425?433. 2011.
[13] N. D. Lawrence. Gaussian process latent variable models for visualisation of high dimensional data. In
Advances in NIPS, volume 16, pages 329?336. MIT Press, Cambridge, MA, 2003.
[14] R. M. Neal. Bayesian Learning for Neural Networks. Springer-Verlag, New York, 1996.
[15] S. Roweis, L. K. Saul, and G. Hinton. Global coordination of local linear models. In Advances in NIPS,
volume 14, pages 889?896. MIT Press, Cambridge, MA, 2002.
[16] R. Salakhutdinov and A. Mnih. Probabilistic matrix factorization. In Advances in NIPS, volume 20, pages
1257?1264. MIT Press, Cambridge, MA, 2008.
[17] F. Samaria and A. Harter. Parameterisation of a stochastic model for human face identification. In Proc.
2nd IEEE Workshop on Applications of Computer Vision, pages 138?142, 1994.
[18] T. Sim, S. Baker, and M. Bsat. The CMU pose, illumination, and expression database. IEEE Trans. Patten
Anal. Mach. Intell., 25(12):1615?1618, 2003.
[19] R. Tibshirani. Regression shrinkage and selection via the lasso. J. R. Stat. Soc. B, 58(1):267?288, 1996.
[20] M. E. Tipping and C. M. Bishop. Mixtures of probabilistic principal component analyzers. Neural
Computation, 11(2):443?482, 1999.
[21] M. Titsias and N. Lawrence. Bayesian Gaussian process latent variable model. In Proc. AISTATS?2009,
JMLR W&CP, volume 9, pages 844?851. 2010.
[22] K. Trohidis, G. Tsoumakas, G. Kalliris, and I. Vlahavas. Multilabel classification of music into emotions.
In Proc. Intl. Conf. on Music Information Retrieval (ISMIR), pages 325?330, 2008.
[23] D. Turnbull, L. Barrington, D. Torres, and G. Lanckriet. Semantic annotation and retrieval of music and
sound effects. IEEE Trans. Audio, Speech and Lang. Process., 16(2):467?476, 2008.
[24] S. Virtanen, A. Klami, S. A. Khan, and S. Kaski. Bayesian group factor analysis. In Proc. AISTATS?2012,
JMLR W&CP, volume 22, pages 1269?1277. 2012.
[25] Z. Xu, K. Kersting, and V. Tresp. Multi-relational learning with Gaussian processes. In Proc. IJCAI?2009,
pages 1309?1314, 2009.
9
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4,057 | 4,672 | Context-Sensitive Decision Forests
for Object Detection
Peter Kontschieder1 Samuel Rota Bul`o2 Antonio Criminisi3
Pushmeet Kohli3 Marcello Pelillo2 Horst Bischof1
1
ICG, Graz University of Technology, Austria
2
DAIS, Universit`a Ca? Foscari Venezia, Italy
3
Microsoft Research Cambridge, UK
Abstract
In this paper we introduce Context-Sensitive Decision Forests - A new perspective
to exploit contextual information in the popular decision forest framework for the
object detection problem. They are tree-structured classifiers with the ability to
access intermediate prediction (here: classification and regression) information
during training and inference time. This intermediate prediction is available for
each sample and allows us to develop context-based decision criteria, used for
refining the prediction process. In addition, we introduce a novel split criterion
which in combination with a priority based way of constructing the trees, allows
more accurate regression mode selection and hence improves the current context
information. In our experiments, we demonstrate improved results for the task of
pedestrian detection on the challenging TUD data set when compared to state-ofthe-art methods.
1
Introduction and Related Work
In the last years, the random forest framework [1, 6] has become a very popular and powerful tool
for classification and regression problems by exhibiting many appealing properties like inherent
multi-class capability, robustness to label noise and reduced tendencies to overfitting [7]. They
are considered to be close to an ideal learner [13], making them attractive in many areas of computer vision like image classification [5, 17], clustering [19], regression [8] or semantic segmentation [24, 15, 18]. In this work we show how the decision forest algorithm can be extended to include
contextual information during learning and inference for classification and regression problems.
We focus on applying random forests to object detection, i.e. the problem of localizing multiple
instances of a given object class in a test image. This task has been previously addressed in random
forests [9], where the trees were modified to learn a mapping between the appearance of an image
patch and its relative position to the object category centroid (i.e. center voting information). During
inference, the resulting Hough Forest not only performs classification on test samples but also casts
probabilistic votes in a generalized Hough-voting space [3] that is subsequently used to obtain object
center hypotheses. Ever since, a series of applications such as tracking and action recognition [10],
body-joint position estimation [12] and multi-class object detection [22] have been presented. However, Hough Forests typically produce non-distinctive object hypotheses in the Hough space and
hence there is the need to perform non-maximum suppression (NMS) for obtaining the final results.
While this has been addressed in [4, 26], another shortcoming is that standard (Hough) forests treat
samples in a completely independent way, i.e. there is no mechanism that encourages the classifier
to perform consistent predictions.
Within this work we are proposing that context information can be used to overcome the aforementioned problems. For example, training data for visual learning is often represented by images in
form of a (regular) pixel grid topology, i.e. objects appearing in natural images can often be found in
a specific context. The importance of contextual information was already highlighted in the 80?s with
1
Figure 1: Top row: Training image, label image, visualization of priority-based growing of tree (the lower,
the earlier the consideration during training.). Bottom row: Inverted Hough image using [9] and breadth-first
training after 6 levels (26 = 64 nodes), Inverted Hough image after growing 64 nodes using our priority queue,
Inverted Hough image using priority queue shows distinctive peaks at the end of training.
a pioneering work on relaxation labelling [14] and a later work with focus on inference tasks [20]
that addressed the issue of learning within the same framework. More recently, contextual information has been used in the field of object class segmentation [21], however, mostly for high-level
reasoning in random field models or to resolve contradicting segmentation results. The introduction of contextual information as additional features in low-level classifiers was initially proposed
in the Auto-context [25] and Semantic Texton Forest [24] models. Auto-context shows a general
approach for classifier boosting by iteratively learning from appearance and context information. In
this line of research [18] augmented the feature space for an Entanglement Random Forest with a
classification feature, that is consequently refined by the class posterior distributions according to
the progress of the trained subtree. The training procedure is allowed to perform tests for specific,
contextual label configurations which was demonstrated to significantly improve the segmentation
results. However, the
In this paper we are presenting Context-Sensitve Decision Forests - A novel and unified interpretation of Hough Forests in light of contextual sensitivity. Our work is inspired by Auto-Context and
Entanglement Forests, but instead of providing only posterior classification results from an earlier
level of the classifier construction during learning and testing, we additionally provide regression
(voting) information as it is used in Hough Forests. The second core contribution of our work is
related to how we grow the trees: Instead of training them in a depth- or breadth-first way, we propose a priority-based construction (which could actually consider depth- or breadth-first as particular
cases). The priority is determined by the current training error, i.e. we first grow the parts of the tree
where we experience higher error. To this end, we introduce a unified splitting criterion that estimates the joint error of classification and regression. The consequence of using our priority-based
training are illustrated in Figure 1: Given the training image with corresponding label image (top
row, images 1 and 2), the tree first tries to learn the foreground samples as shown in the color-coded
plot (top row, image 3, colors correspond to index number of nodes in the tree). The effects on the
intermediate prediction quality are shown in the bottom row for the regression case: The first image
shows the regression quality after training a tree with 6 levels (26 = 64 nodes) in a breadth-first way
while the second image shows the progress after growing 64 nodes according to the priority based
training. Clearly, the modes for the center hypotheses are more distinctive which in turn yields to
more accurate intermediate regression information that can be used for further tree construction.
Our third contribution is a new family of split functions that allows to learn from training images
containing multiple training instances as shown for the pedestrians in the example. We introduce a
test that checks the centroid compatibility for pairs of training samples taken from the context, based
on the intermediate classification and regression derived as described before. To assess our contributions, we performed several experiments on the challenging TUD pedestrian data set [2], yielding a
significant improvement of 9% in the recall at 90% precision rate in comparison to standard Hough
Forests, when learning from crowded pedestrian images.
2
2
Context-Sensitive Decision Trees
This section introduces the general idea behind the context-sensitive decision forest without references to specific applications. Only in Section 3 we show a particular application to the problem
of object detection. After showing some basic notational conventions that are used in the paper, we
provide a section that revisits the random forest framework for classification and regression tasks
from a joint perspective, i.e. a theory allowing to consider e.g. [1, 11] and [9] in a unified way.
Starting from this general view we finally introduce the context-sensitive forests in 2.2.
Notations. In the paper we denote vectors using boldface lowercase (e.g. d, u, v) and sets by
using uppercase calligraphic (e.g. X , Y) symbols. The sets of real, natural and integer numbers are
denoted with R, N and Z as usually. We denote by 2X the power set of X and by 1 [P ] the indicator
function returning 1 or 0 according to whether the proposition P is true or false. Moreover, with
P(Y) we denote the set of probability distributions having Y as sample space and we implicitly
assume that some ?-algebra is defined on Y. We denote by ?(x) the Dirac delta function. Finally,
Ex?Q [f (x)] denotes the expectation of f (x) with respect to x sampled according to distribution Q.
2.1 Random Decision Forests for joint classification and regression
A (binary) decision tree is a tree-structured predictor1 where, starting from the root, a sample is
routed until it reaches a leaf where the prediction takes place. At each internal node of the tree the
decision is taken whether the sample should be forwarded to the left or right child, according to a
binary-valued function. In formal terms, let X denote the input space, let Y denote the output space
and let T dt be the set of decision trees. In its simplest form a decision tree consists of a single node
(a leaf ) and is parametrized by a probability distribution Q ? P(Y) which represents the posterior
probability of elements in Y given any data sample reaching the leaf. We denote this (admittedly
rudimentary) tree as L F (Q) ? T td . Otherwise, a decision tree consists of a node with a left and
a right sub-tree. This node is parametrized by a split function ? : X ? {0, 1}, which determines
whether to route a data sample x ? X reaching it to the left decision sub-tree tl ? T dt (if ?(x) = 0)
or to the right one tr ? T dt (if ?(x) = 1). We denote such a tree as N D (?, tl , tr ) ? T td . Finally,
a decision forest is an ensemble F ? T td of decision trees which makes a prediction about a data
sample by averaging over the single predictions gathered from all trees.
Inference. Given a decision tree t ? T dt , the associated posterior probability of each element in
Y given a sample x ? X is determined by finding the probability distribution Q parametrizing the
leaf that is reached by x when routed along the tree. This is compactly presented with the following
definition of P (y|x, t), which is inductive in the structure of t:
?
if t = L F (Q)
?Q(y)
P (y | x, t ) = P (y | x, tl ) if t = N D (?, tl , tr ) and ?(x) = 0
(1)
?
P (y | x, tr ) if t = N D (?, tl , tr ) and ?(x) = 1 .
Finally, the combination of the posterior probabilities derived from the trees in a forest F ? T dt can
be done by an averaging operation [6], yielding a single posterior probability for the whole forest:
1 X
P (y|x, F) =
P (y|x, t) .
(2)
|F|
t?F
Randomized training.
A random forest is created by training a set of random decision trees independently on random
subsets of the training data D ? X ?Y. The training procedure for a single decision tree heuristically
optimizes a set of parameters like the tree structure, the split functions at the internal nodes and the
density estimates at the leaves in order to reduce the prediction error on the training data. In order
to prevent overfitting problems, the search space of possible split functions is limited to a random
set and a minimum number of training samples is required to grow a leaf node. During the training
procedure, each new node is fed with a set of training samples Z ? D. If some stopping condition
holds, depending on Z, the node becomes a leaf and a density on Y is estimated based on Z.
Otherwise, an internal node is grown and a split function is selected from a pool of random ones in
a way to minimize some sort of training error on Z. The selected split function induces a partition
1
we use the term predictor because we will jointly consider classification and regression.
3
of Z into two sets, which are in turn becoming the left and right childs of the current node where
the training procedure is continued, respectively.
We will now write this training procedure in more formal terms. To this end we introduce a function
?(Z) ? P(Y) providing a density on Y estimated from the training data Z ? D and a loss function
L(Z | Q) ? R penalizing wrong predictions on the training samples in Z, when predictions are given
according to a distribution Q ? P(Y). The loss function L can be further decomposed in terms of a
loss function `(?|Q) : Y ? R acting on each sample of the training set:
X
L(Z | Q) =
`(y | Q) .
(3)
(x,y)?Z
Also, let ?(Z) be a set of split functions randomly generated for a training set Z and given a split
function ? ? ?(Z), we denote by Zl? and Zr? the sets identified by splitting Z according to ?, i.e.
Zl? = {(x, y) ? Z : ?(x) = 0}
and
Zr? = {(x, y) ? Z : ?(x) = 1} .
We can now summarize the training procedure in terms of a recursive function g : 2X ?Y ? T ,
which generates a random decision tree from a training set given as argument:
(
L F (?(Z))
if some stopping condition holds
g(Z) =
(4)
?
?
N D ?, g(Zl ), g(Zr )
otherwise .
Here, we determine the optimal split function ? in the pool ?(Z) as the one minimizing the loss we
incur as a result of the node split:
n
o
0
0
? ? arg min L(Zl? ) + L(Zr? ) : ?0 ? ?(Z)
(5)
where we compactly write L(Z) for L(Z|?(Z)), i.e. the loss on Z obtained with predictions driven
by ?(Z). A typical split function selection criterion commonly adopted for classification and regression is information gain. The equivalent counterpart in terms of loss can be obtained by using
a log-loss, i.e. `(y|Q) = ? log(Q(y)). A further widely used criterion is based on Gini impurity,
which can be expressed in this setting by using `(y|Q) = 1 ? Q(y).
Finally, the stopping condition that is used in (4) to determine whether to create a leaf or to continue
branching the tree typically consists in checking |Z|, i.e. the number of training samples at the node,
or the loss L(Z) are below some given thresholds, or if a maximum depth is reached.
2.2 Context-sensitive decision forests
A context-sensitive (CS) decision tree is a decision tree in which split functions are enriched with the
ability of testing contextual information of a sample, before taking a decision about where to route
it. We generate contextual information at each node of a decision tree by exploiting a truncated
version of the same tree as a predictor. This idea is shared with [18], however, we introduce some
novelties by tackling both, classification and regression problems in a joint manner and by leaving
a wider flexibility in the tree truncation procedure. We denote the set of CS decision trees as T .
The main differences characterizing a CS decision tree t ? T compared with a standard decision
tree are the following: a) every node (leaves and internal nodes) of t has an associated probability
distribution Q ? P(Y) representing the posterior probability of an element in Y given any data
sample reaching it; b) internal nodes are indexed with distinct natural numbers n ? N in a way to
preserve the property that children nodes have a larger index compared to their parent node; c) the
split function at each internal node, denoted by ?(?|t0 ) : X ? {0, 1}, is bound to a CS decision
tree t0 ? T , which is a truncated version of t and can be used to compute intermediate, contextual
information.
Similar to Section 2.1 we denote by L F (Q) ? T the simplest CS decision tree consisting of a single
leaf node parametrized by the distribution Q, while we denote by N D (n, Q, ?, tl , tr ) ? T , the rest of
the trees consisting of a node having a left and a right sub-tree, denoted by tl , tr ? T respectively,
and being parametrized by the index n, a probability distribution Q and the split function ? as
described above.
As shown in Figure 2, the truncation of a CS decision tree at each node is obtained by exploiting
the indexing imposed on the internal nodes of the tree. Given a CS decision tree t ? T and m ? N,
4
1
1
4
2
3
6
2
5
4
3
(b) The truncated version t(<5)
(a) A CS decision tree t
Figure 2: On the left, we find a CS decision tree t, where only the internal nodes are indexed. On the right, we
see the truncated version t(<5) of t, which is obtained by converting to leaves all nodes having index ? 5 (we
marked with colors the corresponding node transformations).
we denote by t(<m) a CS decision tree derived from t in which only the internal nodes having index
< m are kept and the internal nodes with index ? m having a parent with index < m, or being the
root node, are converted into leaves. Finally, all nodes left-over are pruned away.
Inference. The inference process, given a CS decision tree t ? T , is equivalent to the one introduced for standard decision trees, with the only difference that a split function in a node indexed
by n can use the truncated version of the same decision tree t(<n) to additionally exploit contextual information while taking decisions about where to route samples. In the specific, the posterior
probability of y ? Y given a sample x ? X is inductively defined as:
?
if t = L F (Q)
?Q(y)
(6)
P (y | x, t) = P (y | x, tl ) if t = N D (n, ?, ?, tl , tr ) and ?(x | t(<n) ) = 0
?
P (y | x, tr ) if t = N D (n, ?, ?, tl , tr ) and ?(x | t(<n) ) = 1 .
The same posterior probabilities with respect to a forest F ? T can be obtained as in (2).
Prioritized node training. The training process for CS decision forests consists in training an
ensemble of CS decision trees independently on random subsets of the training set D ? X ? Y.
Each CS decision tree is trained in an iterative way and, similar to the case of standard decision
trees, a decision about whether to branch new nodes or produce a leaf is taken based on a subset
of the training samples Z ? D. However, in contrast to the standard setting, the learning process
depends on the order in which nodes of the tree are grown because split functions depend on t(<m)
which in turn is affected by the node ordering. In other words, we impose an explicit ordering on
the recursive calls of function g described in Equation (4). This ordering is determined by means
of a priority queue, where the priority associated to each function call is determined according to
a cost value. This cost can for instance be the depth at which a new node will be grown by the
recursive call, in which case we enforce a breadth-first ordering, or the negative loss ?L(Z) defined
as in (3), Z being the subset of the training data argument of the function call. This second option
is particularly interesting because it forces the tree to split first the nodes where the training error
measured in terms of the loss function is the highest. This indeed allows to reduce the uncertainty
uniformly during the tree growth and in turn results in more reliable contextual information.
Whenever a new node is grown, it takes the time at which it was extracted from the priority queue
as index. It is easy to see that the indexing deriving from this procedure never violates the property
that children of a node have an index larger than the parent node. The split function selection is
performed according to (5), the only difference being the type of split functions that are generated,
which can exploit t(<m) to test contextual information.
3
Application to Object Detection
In this section we employ the CS decision trees for the problem of object detection, following a
solution setting similar to [9]. Specifically, we adopt a patch-based abstraction of an image and the
aim of the tree-based predictor is to jointly predict, for each patch, the foreground/background class
it belongs to and a displacement vector pointing to the object?s center. By collecting all the object
position hypotheses from all foreground patches, we can setup a Hough space in which objects can
be detected from the vote modes.
An image I : Z2 ? F is a function mapping pixels to elements of a feature space F. The feature space here may include a variety of image cues, like color information, gradients, filter bank
5
responses, etc. . We denote by I(u) ? F the feature vector associated to pixel u and by I the set of
images, and by I(u)k the kth element of the feature vector associated to u. The input space X for
our learning problem is a set of patches, each represented as a pair (u, I) ? Z2 ? I, pixel u being
the center of the patch in image I.
The output space Y is a set of pairs (c, d), where c ? {0, 1} is a binary class label indicating the
presence of an object and d ? Z2 is the displacement of the object?s center. Hence, if a training
sample (u, I) ? X has (c, d) ? Y as the ground-truth prediction then we have in image I at location
u either a background pixel (c = 0) or a foreground pixel (c = 1), i.e. belonging to an object and,
if the second case holds, u + d is the center of the object to which the pixel belongs. Note that Y
encodes both the classification and regression part of the object detection task.
The loss function `(c, d|Q) that we employ for the computation of L(Z|Q) in (3) is given by
`(c, d | Q) = E(c0 ,d0 )?Q [1 [c 6= c0 ] + 1 [(c, c0 ) = (1, 1)] (1 ? K? (d ? d0 ))]
(7)
where K? (x) = exp(?kxk2 /? 2 ). This quantity measures the expected loss that we incur by predicting (c0 , d0 ) in place of (c, d), where (c0 , d0 ) is sampled according to Q. The term under expectation behaves as a 0/1 loss for all combinations of class labels, excepting the case c = c0 = 1
where also the correct prediction of the displacement vector is taken into account. Indeed, even if a
pixel belonging to an object is correctly labelled, we incur a high loss if the object?s center position
estimation is completely wrong. This is taken into account with the second term.
The density estimation function ?(Z), which generates the posterior distributions stored in the tree
leaves, is different depending if we are at an internal node or at a leaf of the tree. In both cases
it provides distributions that factorize in two marginal distributions, for the class labels and the
displacement vector, respectively. The marginal over the class labels is always a discrete distribution
providing the probability of drawing a sample of a given class from the set Z. The difference is
with respect to the marginal over the displacement vector. We have a point-wise and uni-modal
distribution at the internal nodes, while we keep track of multiple modes at the leaves.
Let q ? P({0, 1}) be the marginal distribution over the class labels defined as q(c) = |Zc |/|Z|,
where Z0 and Z1 are the sets of background and foreground samples in Z, respectively. At the
internal node level ?(Z) returns a probability distribution Qn ? P(Y) defined as
Qn (c, d) = q(c)?(d ? d? ) .
Here, d? represents the single point-wise mode of the marginal distribution with respect to d (i.e. the
second term), which is determined in a way to minimize the loss L(Z|Qn ) over the training samples.
A local solution of the minimization problem can be found by iterating the following procedure2
.
X
X
K? (d ? d? ) .
d? ?
d K? (d ? d? )
(x,(c,d))?Z1
(x,(c,d))?Z1
At the leaf level, instead, ?(Z) returns a probability distribution Ql ? P(Y) defined as:
X
Ql (c, d) = q(c)
?(d ? d0 )/|Z1 | .
(x,(c0 ,d0 ))?Z1
Here, the second term, i.e. the marginal over d, is uniform over the set of displacement vectors
belonging to foreground samples reaching the leaf.
We define finally a novel type of split function, which performs a test by exploiting the contextual
information. This test is particularly interesting because it allows to check whether two pixels are
expected to belong to the same object instance. The new split function ?(cs) (u, I|t, h1 , h2 , ? ) takes
as input a sample (u, I) ? X and it is parametrized by a CS decision tree t ? T that is used
for generating the contextual information, by two relative displacement vectors h1 , h2 ? R2 that
identify the position of two pixels relative to u and by a threshold ? . The definition of our contextsensitive split functions ?(cs) is as follows:
?(cs) (u, I|t, h1 , h2 , ? ) = 1 E(c,d,c0 ,d0 )?P1 ?P2 [1 [(c, c0 ) = (1, 1)]K? (d ? d0 )] < ?
(8)
2
In the experiments conducted, we never exceeded 10 iterations for finding a mode.
6
where Pj = P (?|(u + hj , I), t), with j = 1, 2, are the posterior probabilities obtained from tree t
given samples at position u+h1 and u+h2 of image I, respectively. Please note that this test should
not be confused with the regression split criterion in [9], which tries to partition the training set in a
way to group examples with similar voting direction and length. Besides the novel context-sensitive
split function we employ also standard split functions performing tests on X as defined in [24].
4
Experiments
To assess our proposed approach, we have conducted several experiments on the task of pedestrian
detection. Detecting pedestrians is very challenging for Hough-voting based methods as they typically exhibit strong articulations of feet and arms, yielding to non-distinctive hypotheses in the
Hough space. We evaluated our method on the TUD pedestrian data base [2] in two different ways:
First, we show our detection results with training according to the standard protocol using 400
training images (where each image contains a single annotation of a pedestrian) and evaluation on
the Campus and Crossing scenes, respectively (Section 4.1). With this experiment we show the
improvement over state-of-the-art approaches when learning can be performed with simultaneous
knowledge about context information. In a second variation (Section 4.2), we use the images of
the Crossing scene (201 images) as a training set. Most images of this scene contain more than
four persons with strong overlap and mutual occlusions. However, instead of using the original annotation which covers only pedestrians with at least 50% overlap (1008 bounding boxes), we use
the more accurate, pixel-wise ground truth annotations of [23] for the entire scene that includes all
persons and consists of 1215 bounding boxes. Please note that this annotation is even more detailed
than the one presented in [4] with 1018 bounding boxes. The purpose of the second experiment is
to show that our context-sensitive forest can exploit the availability of multiple training instances
significantly better than state-of-the-art.
The most related work and therefore also the baseline in our experiments is the Hough Forest [9]. To
guarantee a fair comparison, we use the same training parameters for [9] and our context sensitive
forest: We trained 20 trees and the training data (including horizontally flipped images) was sampled
homogeneously per category per image. The patch size was fixed to 30 ? 30 and we performed
1600 node tests for finding the best split function parameters per node. The trees were stopped
growing when < 7 samples were available. As image features, we used the the first 16 feature
channels provided in the publicly available Hough Forest code of [9]. In order to obtain the object
detection hypotheses from the Hough space, we use the same Non-maximum suppression (NMS)
technique in all our experiments as suggested in [9]. To evaluate the obtained hypotheses, we use
the standard PASAL-VOC criterion which requires the mutual overlap between ground truth and
detected bounding boxes to be ? 50%. The additional parameter of (7) was fixed to ? = 7.
4.1 Evaluation using standard protocol training set
The standard training set contains 400 images where each image comes with a single pedestrian
annotation. For our experiments, we rescaled the images by a factor of 0.5 and doubled the training image set by including also the horizontally flipped images. We randomly chose 125 training
samples per image for foreground and background, resulting in 2 ? 400 ? 2 ? 125 = 200k training
samples per tree. For additional comparisons, we provide the results presented in the recent work
on joint object detection and segmentation of [23], from which we also provide evaluation results
of the Implicit Shape Model (ISM) [16]. However, please note that the results of [23] are based on
a different baseline implementation. Moreover, we show the results of [4] when using the provided
code and configuration files from the first authors homepage. Unfortunately, we could not reproduce
the results of the original paper.
First, we discuss the results obtained on the Campus scene. This data set consists of 71 images
showing walking pedestrians at severe scale differences and partial occlusions. The ground truth
we use has been released with [4] and contains a total number of 314 pedestrians. Figure 3, first
row, plot 1 shows the precision-recall curves when using 3 scales (factors 0.3, 0.4, 0.55) for our
baseline [9] (blue), results from re-evaluating [4] (cyan, 5 scales), [23] (green) and our ContextSensitive Forest without and with using the priority queue based tree construction (red/magenta). In
case of not using the priority queue, we trained the trees according to a breadth-first way. We obtain
a performance boost of ? 6% in recall at a precision of 90% when using both, context information
and the priority based construction of our forest. The second plot in the first row of Figure 3 shows
the results when the same forests are tested on the Crossing scene, using the more detailed ground
7
TUD?Crossing (3 scales)
1
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Barinova et al. CVPR?10, 5 scales
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Riemenschneider et al. ECCV?12
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Leibe et al. IJCV?08 (1 scale)
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Figure 3: Precision-Recall Curves for detections, Top row: Standard training (400 images), evaluation on
Campus and Crossing (3 scales). Bottom row: Training on Crossing annotations of [23], evaluation on Campus,
3 and 5 scales. Right images: Qualitative examples for Campus (top 2) and Crossing (bottom 2) scenes. (green)
correctly found by our method (blue) ground truth (red) wrong association (cyan) missed detection.
truth annotations. The data set shows walking pedestrians (Figure 3, right side, last 2 images) with
a smaller variation in scale compared to the Campus scene but with strong mutual occlusions and
overlaps. The improvement with respect to the baseline is lower (? 2% gain at a precision of 90%)
and we find similar developments of the curves. However, this comes somewhat expectedly as the
training data does not properly reflect the occlusions we actually want to model.
4.2 Evaluation on Campus scene using Crossing scene as training set
In our next experiment we trained the forests (same parameters) on the novel annotations of [23]
for the Crossing scene. Please note that this reduces the training set to only 201 images (we did not
include the flipped images). Qualitative detection results are shown in Figure 3, right side, images
1 and 2. From the first precison-recall curve in the second row of Figure 3 we can see, that the
margin between the baseline and our proposed method could be clearly improved (gain of ? 9%
recall at precision 90%) when evaluating on the same 3 scales. With evaluation on 5 scales (factors
0.34, 0.42, 0.51, 0.65, 0.76) we found a strong increase in the recall, however, at the cost of loosing
2 ? 3% of precision below a recall of 60%, as illustrated in the second plot of row 2 in Figure 3.
While our method is able to maintain a precision above 90% up to a recall of ? 83%, the baseline
implementation drops already at a recall of ? 20%.
5
Conclusions
In this work we have presented Context-Sensitive Decision Forests with application to the object detection problem. Our new forest has the ability to access intermediate prediction (classification and
regression) information about all samples of the training set and can therefore learn from contextual
information throughout the growing process. This is in contrast to existing random forest methods
used for object detection which typically treat training samples in an independent manner. Moreover, we have introduced a novel splitting criterion together with a mode isolation technique, which
allows us to (a) perform a priority-driven way of tree growing and (b) install novel context-based
test functions to check for mutual object centroid agreements. In our experimental results on pedestrian detection we demonstrated superior performance with respect to state-of-the-art methods and
additionally found that our new algorithm can significantly better exploit training data containing
multiple training objects.
Acknowledgements. Peter Kontschieder acknowledges financial support of the Austrian Science
Fund (FWF) from project ?Fibermorph? with number P22261-N22.
8
References
[1] Y. Amit and D. Geman. Shape quantization and recognition with randomized trees. Neural Computation,
1997.
[2] M. Andriluka, S. Roth, and B. Schiele. People-tracking-by-detection and people-detection-by-tracking.
In (CVPR), 2008.
[3] D. H. Ballard. Generalizing the hough transform to detect arbitrary shapes. Pattern Recognition, 13(2),
1981.
[4] O. Barinova, V. Lempitsky, and P. Kohli. On detection of multiple object instances using hough transforms. In (CVPR), 2010.
[5] A. Bosch, A. Zisserman, and X. Mu?
noz. Image classification using random forests and ferns. In (ICCV),
2007.
[6] L. Breiman. Random forests. In Machine Learning, 2001.
[7] A. Criminisi, J. Shotton, and E. Konukoglu. Decision forests: A unified framework for classification, regression, density estimation, manifold learning and semi-supervised learning. In Foundations and Trends
in Computer Graphics and Vision, volume 7, pages 81?227, 2012.
[8] A. Criminisi, J. Shotton, D. Robertson, and E. Konukoglu. Regression forests for efficient anatomy
detection and localization in CT scans. In MICCAI-MCV Workshop, 2010.
[9] J. Gall and V. Lempitsky. Class-specific hough forests for object detection. In (CVPR), 2009.
[10] J. Gall, A. Yao, N. Razavi, L. Van Gool, and V. Lempitsky. Hough forests for object detection, tracking,
and action recognition. (PAMI), 2011.
[11] P. Geurts, D. Ernst, and L. Wehenkel. Extremely randomized trees. Machine Learning, 2006.
[12] R. Girshick, J. Shotton, P. Kohli, A. Criminisi, and A. Fitzgibbon. Efficient regression of general-activity
human poses from depth images. In (ICCV), 2011.
[13] T. Hastie, R. Tibshirani, and J. H. Friedman. The Elements of Statistical Learning. Springer, 2009.
[14] R. A. Hummel and S. W. Zucker. On the foundations of relaxation labeling. (PAMI), 5(3):267?287, 1983.
[15] P. Kontschieder, S. Rota Bul`o, H. Bischof, and M. Pelillo. Structured class-labels in random forests for
semantic image labelling. In (ICCV), 2011.
[16] B. Leibe, A. Leonardis, and B. Schiele. Robust object detection with interleaved categorization and
segmentation. (IJCV), 2008.
[17] R. Mar?ee, P. Geurts, J. Piater, and L. Wehenkel. Random subwindows for robust image classification. In
(CVPR), 2005.
[18] A. Montillo, J. Shotton, J. Winn, J. E. Iglesias, D. Metaxas, and A. Criminisi. Entangled decision forests
and their application for semantic segmentation of CT images. In (IPMI), 2011.
[19] F. Moosmann, B. Triggs, and F. Jurie. Fast discriminative visual codebooks using randomized clustering
forests. In (NIPS), 2006.
[20] M. Pelillo and M. Refice. Learning compatibility coefficients for relaxation labeling processes. (PAMI),
16(9):933?945, 1994.
[21] A. Rabinovich, A. Vedaldi, C. Galleguillos, E. Wiewiora, and S. Belongie. Objects in context. In (ICCV),
2007.
[22] N. Razavi, J. Gall, and L. Van Gool. Scalable multi-class object detection. In (CVPR), 2011.
[23] H. Riemenschneider, S. Sternig, M. Donoser, P. M. Roth, and H. Bischof. Hough regions for joining
instance localization and segmentation. In (ECCV), 2012.
[24] J. Shotton, M. Johnson, and R. Cipolla. Semantic texton forests for image categorization and segmentation. In (CVPR), 2008.
[25] Z. Tu. Auto-context and its application to high-level vision tasks. In (CVPR), 2008.
[26] O. Woodford, M. Pham, A. Maki, F. Perbet, and B. Stenger. Demisting the hough transform for 3d shape
recognition and registration. In (BMVC), 2011.
9
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4,058 | 4,673 | Ancestor Sampling for Particle Gibbs
Fredrik Lindsten
Div. of Automatic Control
Link?oping University
[email protected]
Michael I. Jordan
Dept. of EECS and Statistics
University of California, Berkeley
[email protected]
Thomas B. Sch?on
Div. of Automatic Control
Link?oping University
[email protected]
Abstract
We present a novel method in the family of particle MCMC methods that we refer
to as particle Gibbs with ancestor sampling (PG-AS). Similarly to the existing
PG with backward simulation (PG-BS) procedure, we use backward sampling
to (considerably) improve the mixing of the PG kernel. Instead of using separate forward and backward sweeps as in PG-BS, however, we achieve the same
effect in a single forward sweep. We apply the PG-AS framework to the challenging class of non-Markovian state-space models. We develop a truncation strategy
of these models that is applicable in principle to any backward-simulation-based
method, but which is particularly well suited to the PG-AS framework. In particular, as we show in a simulation study, PG-AS can yield an order-of-magnitude
improved accuracy relative to PG-BS due to its robustness to the truncation error.
Several application examples are discussed, including Rao-Blackwellized particle
smoothing and inference in degenerate state-space models.
1
Introduction
State-space models (SSMs) are widely used to model time series and dynamical systems. The
strong assumptions of linearity and Gaussianity that were originally invoked in state-space inference
have been weakened by two decades of research on sequential Monte Carlo (SMC) and Markov
chain Monte Carlo (MCMC). These Monte Carlo methods have not, however, led to substantial
weakening of a further strong assumption, that of Markovianity. It remains a major challenge to
develop inference algorithms for non-Markovian SSMs:
xt+1 ? f (xt+1 | ?, x1:t ),
yt ? g(yt | ?, x1:t ),
(1)
where ? ? ? is a static parameter with prior density p(?), xt is the latent state and yt is the observation at time t, respectively. Models of this form arise in many different application scenarios,
either from direct modeling or via a transformation or marginalization of a larger model. We provide
several examples in Section 5.
To tackle the challenging problem of inference for non-Markovian SSMs, we work within the framework of particle MCMC (PMCMC), a family of inferential methods introduced in [1]. The basic idea
in PMCMC is to use SMC to construct a proposal kernel for an MCMC sampler. Assume that we
observe a sequence of measurements y1:T . We are interested in finding the density p(x1:T , ? | y1:T ),
i.e., the joint posterior density of the state sequence and the parameter. In an idealized Gibbs sampler we would target this density by sampling as follows: (i) Draw ?? | x1:T ? p(? | x1:T , y1:T );
(ii) Draw x?1:T | ?? ? p(x1:T | ?? , y1:T ). The first step of this procedure can be carried out
exactly if conjugate priors are used. For non-conjugate models, one option is to replace Step (i)
with a Metropolis-Hastings step. However, Step (ii)?sampling from the joint smoothing density
p(x1:T | ?, y1:T )?is in most cases very difficult. In PMCMC, this is addressed by instead sampling
a particle trajectory x?1:T based on an SMC approximation of the joint smoothing density. More
precisely, we run an SMC sampler targeting p(x1:T | ?? , y1:T ). We then sample one of the particles
1
at the final time T , according to their importance weights, and trace the ancestral lineage of this
particle to obtain the trajectory x?1:T . This overall procedure is referred to as particle Gibbs (PG).
The flexibility provided by the use of SMC as a proposal mechanism for MCMC seems promising for
tackling inference in non-Markovian models. To exploit this flexibility we must address a drawback
of PG in the high-dimensional setting, which is that the mixing of the PG kernel can be very poor
when there is path degeneracy in the SMC sampler [2, 3]. This problem has been addressed in the
generic setting of SSMs by adding a backward simulation step to the PG sampler, yielding a method
denoted PG with backward simulation (PG-BS). It has been found that this considerably improves
mixing, making the method much more robust to a small number of particles as well as larger data
records [2, 3].
Unfortunately, however, the application of backward simulation is problematic for non-Markovian
models. The reason is that we need to consider full state trajectories during the backward simulation
pass, leading to O(T 2 ) computational complexity (see Section 4 for details). To address this issue,
we develop a novel PMCMC method which we refer to as particle Gibbs with ancestor sampling
(PG-AS) that achieves the effect of backward sampling without an explicit backward pass. As part
of our development, we also develop a truncation method geared to non-Markovian models. This
method is a generic method that is also applicable to PG-BS, but, as we show in a simulation study in
Section 6, the effect of the truncation error is much less severe for PG-AS than for PG-BS. Indeed,
we obtain up to an order of magnitude increase in accuracy in using PG-AS when compared to
PG-BS in this study.
Since we assume that it is straightforward to sample the parameter ? of the idealized Gibbs sampler,
we will not explicitly include sampling of ? in the subsequent sections to simplify our presentation.
2
Sequential Monte Carlo
We first review the standard auxiliary SMC sampler, see e.g. [4,5]. Let ?t (x1:t ) for t = 1, . . . , T be
a sequence of unnormalized densities on Xt , which we assume can be evaluated pointwise in linear
time. Let ??t (x1:t ) be the corresponding normalized probability densities. For an SSM we would typm
N
ically have ??t (x1:t ) = p(x1:t | y1:t ) and ?t (x1:t ) = p(x1:t , y1:t ). Assume that {xm
1:t?1 , wt?1 }m=1
is a weighted particle system targeting ??t?1 (x1:t?1 ). This particle system is propagated to time t by
sampling independently from a proposal kernel,
at
wat ?t?1
t
Mt (at , xt ) = P t?1
Rt (xt | xa1:t?1
).
(2)
l
l
w
?
l t?1 t?1
In this formulation, the resampling step is implicit and corresponds to sampling the ancestor indices
m
m
at . Note that am
t is the index of the ancestor particle of xt . When we write x1:t we refer to the
m
m
m
ancestral path of xt . The factors ?t = ?t (x1:t ), known as adjustment multiplier weights, are
used in the auxiliary SMC sampler to increase the probability of sampling ancestors that better can
describe the current observation [5]. The particles are then weighted according to wtm = Wt (xm
1:t ),
where the weight function is given by
?t (x1:t )
Wt (x1:t ) =
,
(3)
?t?1 (x1:t?1 )?t?1 (x1:t?1 )Rt (xt | x1:t?1 )
for t ? 2. The procedure is initiated by sampling from a proposal density xm
1 ? R1 (x1 ) and
assigning importance weights w1m = W1 (xm
1 ) with W1 (x1 ) = ?1 (x1 )/R1 (x1 ). In PMCMC it is
instructive to view this sampling procedure as a way of generating a single sample from the density
?(x1:T , a2:T ) ,
N
Y
R1 (xm
1 )
m=1
N (T ?1)
T Y
N
Y
m
Mt (am
t , xt )
(4)
t=2 m=1
on the space XN T ? {1, . . . , N }
. Here we have introduced the boldface notation xt =
{x1t , . . . , xN
t } and similarly for the ancestor indices.
3
Particle Gibbs with ancestor sampling
PMCMC methods is a class of MCMC samplers in which SMC is used to construct proposal kernels [1]. The validity of these methods can be assessed by viewing them as MCMC samplers on an
2
extended state space in which all the random variables generated by the SMC sampler are seen as
auxiliary variables. The target density on this extended space is given by
??T (xk1:T )
?(x1:T , a2:T )
?(x1:T , a2:T , k) ,
.
(5)
b1 QT
NT
R1 (x1 ) t=2 Mt (abt t , xbt t )
By construction, this density admits ??T (xk1:T ) as a marginal, and can thus be used as a surrogate for
the original target density ??T [1]. Here k is a variable indexing one of the particles at the final time
1:T
point and b1:T corresponds to the ancestral path of this particle: xk1:T = xb1:T
= {xb11 , . . . , xbTT }.
bt+1
. The PG
These indices are given recursively from the ancestor indices by bT = k and bt = at+1
b2:T
sampler [1] is a Gibbs sampler targeting ? using the following sweep (note that b1:T = {a2:T , bT }),
?b2:T
1:T
2:T
1:T
1:T
1. Draw x?,?b
, a?,?b
? ?(x?b
| xb1:T
, b1:T ).
1:T
2:T
1:T , a2:T
?,?b1:T
2:T
1:T
2:T
2. Draw k ? ? ?(k | x1:T
, a?,?b
, xb1:T
, ab2:T
).
2:T
?b1:T
Here we have introduced the notation x?m
= {x1t , . . . , xm?1
, xm+1
, . . . , xN
=
t
t
t
t }, x1:T
?bT
?b1
{x1 , . . . , xT } and similarly for the ancestor indices. In [1], a sequential procedure for sampling
from the conditional density appearing in Step 1 is given. This method is known as conditional SMC
(CSMC). It takes the form of an SMC sampler in which we condition on the event that a prespecified
b1:T
= x01:T , with indices b1:T , is maintained throughout the sampler (see Algorithm 1 for a
path x1:T
related procedure). Furthermore, the conditional distribution appearing in Step 2 of the PG sampler
is shown to be proportional to wTk , and it can thus straightforwardly be sampled from.
1:T
, b1:T ?1 } in this sweep. Hence, the PG
Note that we never sample new values for the variables {xb1:T
sampler is an ?incomplete? Gibbs sampler, since it does not loop over all the variables of the model.
It still holds that the PG sampler is ergodic, which intuitively can be explained by the fact that the
collection of variables that is left out is chosen randomly at each iteration. However, it has been
observed that the PG sampler can have very poor mixing, especially when N is small and/or T is
large [2, 3]. The reason for this poor mixing is that the SMC path degeneracy causes the collections
of variables that are left out at any two consecutive iterations to be strongly dependent.
We now turn to our new procedure, PG-AS, which aims to address this fundamental issue. Our
idea is to sample new values for the ancestor indices b1:T ?1 as part of the CSMC procedure1 . By
adding these variables to the Gibbs sweep, we can considerably improve the mixing of the PG
?b2:T
?b1:T
1:T
, b1:T )
| xb1:T
, a2:T
kernel. The CSMC method is a sequential procedure to sample from ?(x1:T
?,?b1:t?1
?,?b2:t?1
?,?bt
?,?bt
b1:T
?bt
?bt
, at
} ? ?(xt , at | x1:t?1 , a2:t?1 , x1:T , b1:T ), for
by sampling according to {xt
t = 1, . . . , T . After having sampled these variables at time t, we add a step in which we generate a
new value for bt?1 (= abt t ), resulting in the following sweep:
10 . (CSMC with ancestor sampling) For t = 1, . . . , T , draw
?,?b
t
t
t
1:T
x?,?b
, a?,?b
? ?(x?b
, at?bt | x1:t?11:t?1 , a?2:t?1 , xb1:T
, bt?1:T ),
t
t
t
?,?b
t
1:T
(a?,b
=) b?t?1 ? ?(bt?1 | x1:t?11:t?1 , a?2:t?1 , xb1:T
, bt:T ).
t
?,?b1:T
1:T
20 . Draw (k ? =) b?T ? ?(bT | x1:T
, a?2:T , xb1:T
).
It can be verified that this corresponds to a partially collapsed Gibbs sampler [6] and will thus leave
? invariant. To determine the conditional densities from which the ancestor indices are drawn,
consider the following factorization, following directly from (3),
?t (x1:t ) = Wt (x1:t )?t?1 (x1:t?1 )Rt (xt | x1:t?1 )?t?1 (x1:t?1 )
P l
bt?1 bt?1
l
bt?1
bt?1
bt
bt
l wt?1 ?t?1 wt?1 ?t?1
P l
? ?t (x1:t ) = wt
Rt (xbt t | x1:t?1
)?t?1 (x1:t?1
)
bt?1
l
w
?
wt?1
l t?1 t?1
!
t?1
t
YX
Y
bt
l l
= . . . = wt
ws ?s R1 (xb11 )
Mt (abss , xbss ).
(6)
s=1
s=2
l
1:T
xb1:T
1
Ideally, we would like to include the variables
as well, but this is in general not possible since it
would be similar to sampling from the original target density (which we assume is infeasible).
3
Furthermore, we have
b
b
t+1:T
t+1:T
?(bt | x1:t , a2:t , xt+1:T
, bt+1:T ) ? ?(x1:t , a2:t , xt+1:T
, bt:T )
?
t
?t (xb1:t
)
?T (xk1:T )?(x1:t , a2:t )
?T (xk1:T )
?
.
bt
bs
bs
b1 Qt
b1 Qt
?t (x1:t ) R1 (x1 ) s=2 Ms (abss , xbss )
R1 (x1 ) s=2 Ms (as , xs )
(7)
By plugging (6) into the numerator we get,
b
t+1:T
?(bt | x1:t , a2:t , xt+1:T
, bt+1:T ) ? wtbt
?T (xk1:T )
.
t
?t (xb1:t
)
(8)
Hence, to sample a new ancestor index for the conditioned path at time t + 1, we proceed as follows.
bt+1:T
Given x0t+1:T (= xt+1:T
) we compute the backward sampling weights,
m
wt|T
= wtm
0
?T ({xm
1:t , xt+1:T })
,
?t (xm
1:t )
(9)
m
for m = 1, . . . , N . We then set bt = m with probability proportional to wt|T
.
It follows that the proposed CSMC with ancestor sampling (Step 10 ), conditioned on {x01:T , b1:T },
can be realized as in Algorithm 1. The difference between this algorithm and the CSMC sampler
derived in [1] lies in the ancestor sampling step 2(b) (where instead, they set abt t = bt?1 ). By
introducing the ancestor sampling, we break the strong dependence between the generated particle
trajectories and the path on which we condition. We call the resulting method, defined by Steps 10
and 20 above, PG with ancestor sampling (PG-AS).
Algorithm 1 CSMC with ancestor sampling, conditioned on {x01:T , b1:T }
1. Initialize (t = 1):
b1
0
(a) Draw xm
1 ? R1 (x1 ) for m 6= b1 and set x1 = x1 .
m
m
(b) Set w1 = W1 (x1 ) for m = 1, . . . , N .
2. for t = 2, . . . , T :
bt
0
m
(a) Draw {am
t , xt } ? Mt (at , xt ) for m 6= bt and set xt = xt .
m
.
(b) Draw abt t with P (abt t = m) ? wt?1|T
am
m
m
m
t
(c) Set xm
1:t = {x1:t?1 , xt } and wt = Wt (x1:t ) for m = 1, . . . , N .
The idea of including the variables b1:T ?1 in the PG sampler has previously been suggested by
Whiteley [7] and further explored in [2, 3]. This previous work, however, accomplishes this with
a explicit backward simulation pass, which, as we discuss in the following section, is problematic
for our applications to non-Markovian SSMs. In the PG-AS sampler, instead of requiring distinct
forward and backward sequences of Gibbs steps as in PG with backward simulation (PG-BS), we
obtain a similar effect via a single forward sweep.
4
Truncation for non-Markovian state-space models
We return to the problem of inference in non-Markovian SSMs of the form shown in (1). To employ
backward sampling, we need to evaluate the ratio
T
Y
p(x1:T , y1:T )
?T (x1:T )
=
=
g(ys | x1:s )f (xs | x1:s?1 ).
?t (x1:t )
p(x1:t , y1:t )
s=t+1
(10)
In general, the computational cost of computing the backward sampling weights will thus be O(T ).
This implies that the cost of generating a full backward trajectory is O(T 2 ). It is therefore computationally prohibitive to employ backward simulation type of particle smoothers, as well as the PG
samplers discussed above, for general non-Markovian models.
4
Probability
1
0.8
padpt. = 12
padpt. = 5
0.6
0.4
0.2
0
0
50
100
200 0
150
50
100
150
200
Figure 1: Probability under Pep as a function of the truncation level p for two different systems; one
5 dimensional (left) and one 20 dimensional (right). The N = 5 dotted lines correspond to Pep (m)
for m ? {1, . . . , N }, respectively (N.B. two of the lines overlap in the left figure). The dashed
vertical lines show the value of the truncation level padpt. , resulting from the adaption scheme with
? = 0.1 and ? = 10?2 . See Section 6.2 for details on the experiments.
To make progress, we consider non-Markovian models in which there is a decay in the influence
of the past on the present, akin to that in Markovian models but without the strong Markovian
assumption. Hence, it is possible to obtain a useful approximation when the product in (10) is
truncated to a smaller number of factors, say p. We then replace (9) with the approximation,
p,m
w
et|T
= wtm
0
?t+p ({xm
1:t , xt+1:t+p })
.
?t (xm
1:t )
(11)
The following proposition formalizes our assumption.
Proposition 1. Let P and Pep be the probability distributions on {1, . . . , N }, defined by
the backward sampling weight (9) and the truncated backward sampling weights (11), respectively. Let hs (k) = g(yt+s | xk1:t , x0t+1:t+s )f (x0t+s | xk1:t , x0t+1:t?s ) and assume that
maxk,l (hs (k)/hs (l) ? 1) ? A exp(?cs), for some constants A and c > 0. Then, DKLD (P kPep ) ?
C exp(?cp) for some constant C, where DKLD is the Kullback-Leibler divergence (KLD).
Proof. Provided in the supplemental material.
From (11), we see that we can compute the backward weights in constant time under the truncation
within the PG-AS framework. The resulting approximation can be quite useful; indeed, in our
experiments we have seen that even p = 1 can lead to very accurate inferential results. In general,
however, it will not be known a priori how to set the truncation level p for any given problem. To
address this problem, we propose to use an adaption of the truncation level. Since the approximative
weights (11) can be evaluated sequentially, the idea is to start with p = 1 and then increase p until
the weights have, in some sense, converged. In particular, in our experimental work, we have used
the following simple approach.
Let Pep be the discrete probability measure defined by (11). Let ?p = DTV (Pep , Pep?1 ) be the total
variation (TV) distance between the distributions for two consecutive truncation levels. We then
compute the exponentially decaying moving average of the sequence ?p , with forgetting factor ? ?
[0, 1], and stop when this falls below some threshold ? ? [0, 1]. This adaption scheme removes the
requirement to specify p directly, but instead introduces the design parameters ? and ? . However,
these parameters are much easier to reason about ? a small value for ? gives a rapid response to
changes in ?p whereas a large value gives a more conservative stopping rule, improving the accuracy
of the approximation at the cost of higher computational complexity. A similar trade off holds for
the threshold ? as well. Most importantly, we have found that the same values for ? and ? can be
used for a wide range of models, with very different mixing properties.
To illustrate the effect of the adaption rule, and how the distribution Pep typically evolves as we
increase p, we provide two examples in Figure 1. These examples are taken from the simulation
study provided in Section 6.2. Note that the untruncated distribution P is given for the maximal
value of p, i.e., furthest to the right in the figures. By using the adaptive truncation, we can stop the
evaluation of the weights at a much earlier stage, and still obtain an accurate approximation of P .
5
5
Application areas
In this section we present examples of problem classes involving non-Markovian SSMs for which
the proposed PG-AS sampler can be applied. Numerical illustrations are provided in Section 6.
5.1
Rao-Blackwellized particle smoothing
One popular approach to increase the efficiency of SMC samplers for SSMs is to marginalize over
one component of the state, and apply an SMC sampler in the lower-dimensional marginal space.
This leads to what is known as the Rao-Blackwellized particle filter (RBPF) [8?10]. The same
approach has also been applied to state smoothing [11,12], but it turns out that Rao-Blackwellization
is less straightforward in this case, since the marginal state-process will be non-Markovian. As an
example, a mixed linear/nonlinear Gaussian SSM (see, e.g., [10]) with ?nonlinear state? xt and
?linear state? zt , can be reduced to xt ? p(xt | x1:t?1 , y1:t?1 ) and yt ? p(yt | x1:t , y1:t?1 ). These
conditional densities are Gaussian and can be evaluated for any fixed marginal state trajectory x1:t?1
by running a conditional Kalman filter to marginalize the zt -process.
In order to apply a backward-simulation-based method (e.g., a particle smoother) for this model, we
need to evaluate the backward sampling weights (9). In a straightforward implementation2 , we thus
need to run N Kalman filters for T ? t time steps, for each t = 1, . . . , T ? 1. The computational
complexity of this calculation can be reduced by employing the truncation proposed in Section 4.
5.2
Particle smoothing for degenerate state-space models
Many dynamical systems are most naturally modelled as degenerate in the sense that the transition kernel of the state process does not admit any dominating measure. For instance, consider a
nonlinear system with additive noise of the form,
?t = f (?t?1 ) + G?t?1 ,
yt = g(?t ) + et ,
(12)
where G is a tall matrix, and consequently rank(G) < dim(?t ). That is, the process noise covariance
matrix is singular. SMC samplers can straightforwardly be applied to this type of models, but it is
more problematic to address the smoothing problem using particle methods. The reason is that the
backward kernel also will be degenerate and it cannot be approximated in a natural way by the
forward filter particles, as is normally done in backward-simulation-based particle smoothers.
A possible remedy for this issue is to recast the degenerate SSM as a non-Markovian model in
T
a lower-dimensional space. Let G = U [? 0] V T with unitary U and V be a singular value
decomposition of G and let,
xt
?V T ?t?1
T
T
T
, U ?t = U f (U U ?t?1 ) +
.
(13)
zt
0
For simplicity we assume that z1 is known. If this is not the case, it can be included in the system
state or seen as a static parameter of the model. Hence, the sequence z1:t is ?(x1:t?1 )-measurable
and we can write zt = zt (x1:t?1 ). With vt , ?V T ?t and by appropriate definitions of the functions
fx and h, the model (12) can thus be rewritten as, xt = fx (x1:t?1 ) + vt?1 and yt = h(x1:t ) + et ,
which is a non-degenerate, non-Markovian SSM. By exploiting the truncation proposed in Section 4
we can thus apply PG-AS to do inference in this model.
5.3
Additional problem classes
There are many more problem classes in which non-Markovian models arise and in which backwardsimulation-based methods can be of interest. For instance, the Dirichlet process mixture model
(DPMM, see, e.g., [13]) is a popular nonparametric Bayesian model for mixtures with an unknown
number of components. Using a Polya urn representation, the mixture labels are given by a nonMarkovian stochastic process, and the DPMM can thus be seen as a non-Markovian SSM. SMC has
2
For the specific problem of Rao-Blackwellized smoothing in conditionally Gaussian models, a backward
simulator which can be implemented in O(T ) computational complexity has recently been proposed in [11].
This is based on the idea of propagating information backward in time as the backward samples are generated.
6
?1
10
PG w. ancestral sampling
PG w. backward simulation
?2
10
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Figure 2: Rao-Blackwellized state smoothing using PG. Running RMSEs for five independent runs
of PG-AS (?) and PG-BS (?), respectively. The truncation level is set to p = 1. The solid line
corresponds to a run of an untruncated FF-BS.
previously been used for inference in DPMMs [14, 15]. An interesting venue for future work is to
use the PG-AS sampler for these models. A second example in Bayesian nonparametrics is Gaussian
process (GP) regression and classification (see, e.g., [16]). The sample path of the GP can be seen as
the state-process in a non-Markovian SSM. We can thus employ PMCMC, and in particular PG-AS,
to address these inference problems.
An application in genetics, for which SMC has been been successfully applied, is reconstruction
of phylogenetic trees [17]. A phylogenetic tree is a binary tree with observation at the leaf nodes.
SMC is used to construct the tree in a bottom up fashion. A similar approach has also been used
for Bayesian agglomerative clustering, in which SMC is used to construct a binary clustering tree
based on Kingman?s coalescent [18]. The generative models for the trees used in [17, 18] are in fact
Markovian, but the observations give rise to a conditional dependence which destroys the Markov
property. To employ backward simulation to these models, we are thus faced with problems of a
similar nature as those discussed in Section 4.
6
Numerical evaluation
This section contains a numerical evaluation of the proposed method. We consider linear Gaussian
systems, which is instructive since the exact smoothing density then is available, e.g., by running
a modified Bryson-Frazier (MBF) smoother [19]. For more details on the experiments, and for
additional (nonlinear) examples, see [20].
6.1
RBPS: Linear Gaussian state-space model
As a first example, we consider Rao-Blackwellized particle smoothing (RBPS) in a single-output
4th-order linear Gaussian SSM. We generate T = 100 samples from the system and run PG-AS and
PG-BS, marginalizing three out of the four states using an RBPF, i.e., dim(xt ) = 1. Both methods
are run for R = 10000 iterations using N = 5 particles. The truncation level is set to p = 1, leading
to a coarse approximation. We discard the first 1000 iterations and then compute running means
of the state trajectory x1:T . From these, we then compute the running root mean squared errors
(RMSEs) r relative to the true posterior means (computed with an MBF smoother). Hence, if no
approximation would have been made, we would expect r ? 0, so any static error can be seen as
the effect of the truncation. The results for five independent runs from both PG samplers are shown
in Figure 2. First, we note that both methods give accurate results. Still, the error for PG-AS is close
to an order of magnitude less than for PG-BS. Furthermore, it appears as if the error for PG-AS
would decrease further, given more iterations, suggesting that the bias caused by the truncation is
dominated by the Monte Carlo variance, even after R = 10000 iterations.
For further comparison, we also run an untruncated forward filter/backward simulator (FF-BS) particle smoother [21], using N = 5000 forward filter particles and M = 500 backward trajectories
(with a computational complexity of O(N M T 2 )). The resulting RMSE value is shown as a solid
line in Figure 2. These results suggest that PMCMC samplers, such as the PG-AS, indeed can be
serious competitors to more ?standard? particle smoothers. Even with p = 1, PG-AS outperforms
7
d=2
?1
?1
10
?2
10
?2
?2
10
?3
10
?3
p=1
p=2
p=3
Adapt. (3.8)
10
d = 20
?1
10
10
10
d=5
PG w. ancestral sampling
PG w. backward simulation
?3
p=1
p=5
p = 10
Adapt. (5.9)
10
p=1
p=5
p = 10
Adapt. (10.6)
Figure 3: Box plots of the RMSE errors for PG-AS (black) and PG-BS (gray), for 150 random
systems of different dimensions d (left, d = 2; middle, d = 5; right, d = 20). Different values for
the truncation level p are considered. The rightmost boxes correspond to an adaptive threshold and
the values in parentheses are the average over all systems and MCMC iterations (the same for both
methods). The dots within the boxes show the median errors.
FF-BS in terms of accuracy and, due to the fact that the ancestor sampling allows us to use as few
as N = 5 particles at each iteration, at a lower computational cost.
6.2
Random linear Gaussian systems with rank deficient process noise covariances
To see how the PG samplers are affected by the choice of truncation level p and by the mixing
properties of the system, we evaluate them on random linear Gaussian SSMs of different orders.
We generate 150 random systems, using the MATLAB function drss from the Control Systems
Toolbox, with model orders 2, 5 and 20 (50 systems for each model order). The number of outputs
are taken as 1, 2 and 4 for the different model orders, respectively. The systems are then simulated
for T = 200 time steps, driven by Gaussian process noise entering only on the first state component.
Hence, the rank of the process noise covariance is 1 for all systems.
We run the PG-AS and PG-BS samplers for 10000 iterations using N = 5 particles. We consider
different fixed truncation levels, as well as an adaptive level with ? = 0.1 and ? = 10?2 . Again,
we compute running posterior means (discarding 1000 samples) and RMSE values relative the true
posterior mean. Box plots are shown in Figure 3. Since the process noise only enters on one of the
state components, the mixing tends to deteriorate as we increase the model order. Figure 1 shows
how the probability distributions on {1, . . . , N } change as we increase the truncation level, in two
representative cases for a 5th and a 20th order system, respectively. By using an adapted level,
we can obtain accurate results for systems of different dimensions, without having to change any
settings between the runs.
7
Discussion
PG-AS is a novel approach to PMCMC that makes use of backward simulation ideas without needing an explicit backward pass. Compared to PG-BS, a conceptually similar method that does require
an explicit backward pass, PG-AS has advantages, most notably for inference in the non-Markovian
SSMs that have been our focus here. When using the proposed truncation of the backward weights,
we have found PG-AS to be more robust to the approximation error than PG-BS. Furthermore, for
non-Markovian models, PG-AS is easier to implement than PG-BS, since it requires less bookkeeping. It can also be more memory efficient, since it does not require us to store intermediate quantities
that are needed for a separate backward simulation pass, as is done in PG-BS. Finally, we note that
PG-AS can be used as an alternative to PG-BS for other inference problems to which PMCMC can
be applied, and we believe that it will prove attractive in problems beyond the non-Markovian SSMs
that we have discussed here.
Acknowledgments
This work was supported by: the project Calibrating Nonlinear Dynamical Models (Contract number: 621-2010-5876) funded by the Swedish Research Council and CADICS, a Linneaus Center
also funded by the Swedish Research Council.
8
References
[1] C. Andrieu, A. Doucet, and R. Holenstein, ?Particle Markov chain Monte Carlo methods,?
Journal of the Royal Statistical Society: Series B, vol. 72, no. 3, pp. 269?342, 2010.
[2] N. Whiteley, C. Andrieu, and A. Doucet, ?Efficient Bayesian inference for switching statespace models using discrete particle Markov chain Monte Carlo methods,? Bristol Statistics
Research Report 10:04, Tech. Rep., 2010.
[3] F. Lindsten and T. B. Sch?on, ?On the use of backward simulation in the particle Gibbs sampler,?
in Proceedings of the 2012 IEEE International Conference on Acoustics, Speech and Signal
Processing (ICASSP), Kyoto, Japan, Mar. 2012.
[4] A. Doucet and A. Johansen, ?A tutorial on particle filtering and smoothing: Fifteen years later,?
in The Oxford Handbook of Nonlinear Filtering, D. Crisan and B. Rozovsky, Eds. Oxford
University Press, 2011.
[5] M. K. Pitt and N. Shephard, ?Filtering via simulation: Auxiliary particle filters,? Journal of the
American Statistical Association, vol. 94, no. 446, pp. 590?599, 1999.
[6] D. A. V. Dyk and T. Park, ?Partially collapsed Gibbs samplers: Theory and methods,? Journal
of the American Statistical Association, vol. 103, no. 482, pp. 790?796, 2008.
[7] N. Whiteley, ?Discussion on Particle Markov chain Monte Carlo methods,? Journal of the
Royal Statistical Society: Series B, 72(3), p 306?307, 2010.
[8] R. Chen and J. S. Liu, ?Mixture Kalman filters,? Journal of the Royal Statistical Society: Series
B, vol. 62, no. 3, pp. 493?508, 2000.
[9] A. Doucet, S. J. Godsill, and C. Andrieu, ?On sequential Monte Carlo sampling methods for
Bayesian filtering,? Statistics and Computing, vol. 10, no. 3, pp. 197?208, 2000.
[10] T. Sch?on, F. Gustafsson, and P.-J. Nordlund, ?Marginalized particle filters for mixed linear/nonlinear state-space models,? IEEE Transactions on Signal Processing, vol. 53, no. 7,
pp. 2279?2289, Jul. 2005.
[11] S. S?arkk?a, P. Bunch, and S. Godsill, ?A backward-simulation based Rao-Blackwellized particle smoother for conditionally linear Gaussian models,? in Proceedings of the 16th IFAC
Symposium on System Identification, Brussels, Belgium, Jul. 2012.
[12] W. Fong, S. J. Godsill, A. Doucet, and M. West, ?Monte Carlo smoothing with application
to audio signal enhancement,? IEEE Transactions on Signal Processing, vol. 50, no. 2, pp.
438?449, Feb. 2002.
[13] N. L. Hjort, C. Holmes, P. Mller, and S. G. Walker, Eds., Bayesian Nonparametrics.
bridge University Press, 2010.
Cam-
[14] S. N. MacEachern, M. Clyde, and J. S. Liu, ?Sequential importance sampling for nonparametric Bayes models: The next generation,? The Canadian Journal of Statistics, vol. 27, no. 2, pp.
251?267, 1999.
[15] P. Fearnhead, ?Particle filters for mixture models with an unknown number of components,?
Statistics and Computing, vol. 14, pp. 11?21, 2004.
[16] C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning.
Press, 2006.
MIT
[17] A. Bouchard-C?ot?e, S. Sankararaman, and M. I. Jordan, ?Phylogenetic inference via sequential
Monte Carlo,? Systematic Biology, vol. 61, no. 4, pp. 579?593, 2012.
[18] Y. W. Teh, H. Daum?e III, and D. Roy, ?Bayesian agglomerative clustering with coalescents,?
Advances in Neural Information Processing, pp. 1473?1480, 2008.
[19] G. J. Bierman, ?Fixed interval smoothing with discrete measurements,? International Journal
of Control, vol. 18, no. 1, pp. 65?75, 1973.
[20] F. Lindsten, M. I. Jordan, and T. B. Sch?on, ?Ancestor sampling for particle Gibbs,? arXiv.org,
arXiv:1210.6911, Oct. 2012.
[21] S. J. Godsill, A. Doucet, and M. West, ?Monte Carlo smoothing for nonlinear time series,?
Journal of the American Statistical Association, vol. 99, no. 465, pp. 156?168, Mar. 2004.
9
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4,059 | 4,674 | Feature Clustering for Accelerating
Parallel Coordinate Descent
Chad Scherrer
Independent Consultant
Yakima, WA
[email protected]
Ambuj Tewari
Department of Statistics
University of Michigan
Ann Arbor, MI
[email protected]
Mahantesh Halappanavar
Pacific Northwest National Laboratory
Richland, WA
[email protected]
David J Haglin
Pacific Northwest National Laboratory
Richland, WA
[email protected]
Abstract
Large-scale `1 -regularized loss minimization problems arise in high-dimensional
applications such as compressed sensing and high-dimensional supervised learning, including classification and regression problems. High-performance algorithms and implementations are critical to efficiently solving these problems.
Building upon previous work on coordinate descent algorithms for `1 -regularized
problems, we introduce a novel family of algorithms called block-greedy coordinate descent that includes, as special cases, several existing algorithms such as
SCD, Greedy CD, Shotgun, and Thread-Greedy. We give a unified convergence
analysis for the family of block-greedy algorithms. The analysis suggests that
block-greedy coordinate descent can better exploit parallelism if features are clustered so that the maximum inner product between features in different blocks is
small. Our theoretical convergence analysis is supported with experimental results using data from diverse real-world applications. We hope that algorithmic
approaches and convergence analysis we provide will not only advance the field,
but will also encourage researchers to systematically explore the design space of
algorithms for solving large-scale `1 -regularization problems.
1
Introduction
Consider the `1 -regularized loss minimization problem
n
min
w
1X
`(yi , (Xw)i ) + kwk1 ,
n i=1
(1)
where X 2 IRn?p is the design matrix, w 2 IRp is a weight vector to be estimated, and the
loss function ` is such that `(y, ?) is a convex differentiable function for each y. This formulation
includes `1 -regularized least squares (Lasso) (when `(y, t) = 12 (y t)2 ) and `1 -regularized logistic
regression (when `(y, t) = log(1+exp( yt))). In recent years, coordinate descent (CD) algorithms
have been shown to be efficient for this class of problems [Friedman et al., 2007; Wu and Lange,
2008; Shalev-Shwartz and Tewari, 2011; Bradley et al., 2011].
Motivated by the need to solve large scale `1 regularized problems, researchers have begun to explore parallel algorithms. For instance, Bradley et al. [2011] developed the Shotgun algorithm.
More recently, Scherrer et al. [2012] have developed ?GenCD?, a generic framework for expressing
1
parallel coordinate descent algorithms. Special cases of GenCD include Greedy CD [Li and Osher,
2009; Dhillon et al., 2011], the Shotgun algorithm of [Bradley et al., 2011], and Thread-Greedy CD
[Scherrer et al., 2012].
In fact, the connection between these three special cases of GenCD is much deeper, and more fundamental, than is obvious under the GenCD abstraction. As our first contribution, we describe a
general randomized block-greedy that includes all three as special cases. The block-greedy algorithm has two parameter: B, the total number of feature blocks and P , the size of the random subset
of the B blocks that is chosen at every time step. For each of these P blocks, we greedily choose, in
parallel, a single feature weight to be updated.
Second, we present a non-asymptotic convergence rate analysis for the randomized block-greedy
coordinate descent algorithms for general values of B 2 {1, . . . , p} (as the number of blocks cannot
exceed the number of features) and P 2 {1, . . . , B}. This result therefore applies to stochastic CD,
greedy CD, Shotgun, and thread-greedy. Indeed, we build on the analysis and insights in all of these
previous works. Our general convergence result, and in particular its instantiation to thread-greedy
CD, is novel.
Third, based on the convergence rate analysis for block-greedy, we optimize a certain ?block spectral
radius? associated with the design matrix. This parameter is a direct generalization of a similar
spectral parameter that appears in the analysis of Shotgun. We show that the block spectral radius
can be upper bounded by the maximum inner product (or correlation if features are mean zero)
between features in distinct blocks. This motivates the use of correlation-based feature clustering to
accelerate the convergence of the thread-greedy algorithm.
Finally, we conduct an experimental study using a simple clustering heuristic. We observe dramatic
acceleration due to clustering for smaller values of the regularization parameter, and show characteristics that must be paid particularly close attention for heavily regularized problems, and that can
be improved upon in future work.
2
Block-Greedy Coordinate Descent
Scherrer et al. [2012] describe ?GenCD?, a generic framework for parallel coordinate descent algorithms, in which a parallel coordinate descent algorithm can be determined by specifying a select
step and an accept step. At each iteration, features chosen by select are evaluated, and a proposed
increment is generated for each corresponding feature weight. Using this, the accept step then determines which proposals are to be updated.
In these terms, we consider the block-greedy algorithm that takes as
part of the input a partition of the features into B blocks. Given this,
each iteration selects features corresponding to a set of P randomly
selected blocks, and accepts a single feature from each block, based
on an estimate of the resulting reduction in the objective function.
The pseudocode for the randomized block-greedy coordinate descent
is given by Algorithm 1. The algorithm can be applied to any function
of the form F + R where F is smooth and convex, and R is convex
and separable across coordinates. Our objective function (1) satisfies
these conditions. The greedy step chooses a feature within a block for Figure 1: The design space
which the guaranteed descent in the objective function (if that feature of block-greedy coordinate
alone were updated) is maximized. This descent is quantified by |?j |, descent.
which is defined precisely in the next section. To arrive at an heuristic
understanding, it is best to think of |?j | as being proportional to the
absolute value |rj F (w)| of the jth entry in the gradient of the smooth part F . In fact, if R is zero
(no regularization) then this heuristic is exact.
The two parameters, B and P , of the block-greedy CD algorithm have the ranges B 2 {1, . . . , p}
and P 2 {1, . . . , B}. Setting these to specific values gives many existing algorithms. For instance
when B = p, each feature is a block on its own. Then, setting P = 1 amounts to randomly choosing
a single coordinate and updating it which gives us the stochastic CD algorithm of Shalev-Shwartz
and Tewari [2011]. Shotgun [Bradley et al., 2011] is obtained when B is still p but P 1. Another
2
Algorithm 1 Block-Greedy Coordinate Descent
Parameters: B (no. of blocks) and P ? B (degree of parallelism)
while not converged do
Select a random subset of size P from the B available blocks
Set J to be the features in the selected blocks
Propose increment ?j , j 2 J
Accept J 0 = {j : ?j has maximal absolute value in its block}
Update weight wj
wj ?j for all j 2 J 0
// parallel
// parallel
extreme is the case when all the features constitute a single block. That is, B = 1. Then blockgreedy CD is a deterministic algorithm and becomes the greedy CD algorithm of Li and Osher
[2009]; Dhillon et al. [2011]. Finally, we can choose non-trivial values of B that lie strictly between
1 and p. When this is the case, and we choose to update all blocks in parallel each time (P = B),
we arrive at the thread-greedy algorithm of Scherrer et al. [2012]. Figure 1 shows a schematic
representation of the parameterization of these special cases.
3
Convergence Analysis
Of course, there is no reason to expect block-greedy CD to converge for all values of B and P . In
this section, we give a sufficient condition for convergence and derive a convergence rate assuming
this condition.
Bradley et al. express the convergence criteria for Shotgun algorithm in terms of the spectral radius (maximal eigenvalue) ?(XT X). For block-greedy, the corresponding quantity is a bit more
complicated. We define
?block = max ?(M )
M 2M
where M is the set of all B ? B submatrices that we can obtain from XT X by selecting exactly one
index from each of the B blocks. The intuition is that if features from different blocks are almost
orthogonal then the matrices M in M will be close to identity and will therefore have small ?(M ).
Highly correlated features within a block do not increase ?block .
As we said above, we will assume that we are minimizing a ?smooth plus separable? convex function
F + R where the convex differentiable function F : Rp ! R satisfies a second order upper bound
F (w +
) ? F (w) + rF (w)T
+
T
2
XT X
In our case, this inequality will hold as soon as `00 (y, t) ? for any y,
Ptp(where differentiation is
w.r.t. t). The function R is separable across coordinates: R(w) =
j=1 r(wj ). The function
kwk1 is clearly separable.
The quantity ?j appearing in Algorithm 1 serves to quantify the guaranteed descent (based on second
order upper bound) if feature j alone is updated and is obtained as a solution of the one-dimensional
minimization problem
?j = argmin rj F (w)? +
?
2
? 2 + r(wj + ?)
r(wj ) .
Note that if there is no regularization, then ?j is simply rj F (w)/ = gj / (if we denote
rj F (w) by gj for brevity). In the general case, by first order optimality conditions for the above
one-dimensional convex optimization problem, we have gj + ?j +?j = 0 where ?j is a subgradient
of r at wj + ?j . That is, ?j 2 @r(wj + ?j ). This implies that r(wj + ?j ) r(w0 ) ? ?j (wj + ?j w0 )
for any w0 .
Theorem 1. Let P be chosen so that
?=
(P
1)(?block
(B 1)
3
1)
is less than 1. Suppose the randomized block-greedy coordinate algorithm is run on a smooth plus
separable convex function f = F + R to produce the iterates {wk }k 1 . Then the expected accuracy
after k steps is bounded as
f (w? )] ? C
E [f (wk )
B R12
1
? .
(1 ?)P k
Here the constant C only depends on (Lipschitz and smoothness constants of) the function F , R1 is
an upper bound on the norms {kwk w? k1 }k 1 , and w? is any minimizer of f .
Proof. We first calculate the expected change in objective function following the Shotgun analysis.
We will use wb to denote wjb (similarly for ?b , gb etc.)
?
0
E [f (w ) f (w)] = P Eb ?b gb + (?b )2 + r(wb + ?b ) r(wb )
2
?
?
+ P (P 1)Eb6=b0 ?b ? ?b0 ? ATjb Ajb0
2
Define the B ? B matrix M (that depends on the current iterate w) with entries Mb,b0 = ATjb Ajb .
Then, using r(wb + ?b ) r(wb ) ? ?b ?b , we continue
?
?
P T
P (P 1) ? >
?
? g + ?T ? + ? T ? +
? M ? ?T ?
B
2
2B(B 1)
Above (with some abuse of notation), ?, ? and g are B length vectors with components ?b , ?b and
gb respectively. By definition of ?block , we have ? > M ? ? ?block ? T ?. So, we continue
?
P T
P (P 1)
?
? g + ?T ? gT ?
?T ? +
(?block 1)? T ?
B
2
2B(B 1)
where we used ? =
g
?. Simplifying we get
E [f (w0 )
f (w)] ?
where
?=
(P
P
[ 1 + ?] k?k22
2B
1)(?block
(B 1)
1)
should be less than 1.
P
Now note that k?k22 = b |?jb |2 = k?k21,2 where the ?infinity-2? norm k ? k1,2 of a p-vector is, by
definition, as follows: take the `1 norm within a block and take the `2 of the resulting values. Note
that in the second step above, we moved from a B-length ? to a p length ?.
This gives us
E [f (w0 )
(1
f (w)] ?
?)P
k?k21,2 .
2B
For the rest of the proof, assume = 0. In that case ? = g/ . Thus, convexity and the fact that
the dual norm of the ?infinity-2? norm is the ?1-2? norm, give
f (w)
f (w? ) ? rf (w)T (w
w? ) ? krf (w)k1,2 ? kw
w? k1,2
Putting the last two inequalities together gives (for any upper bound R1 on kw
w? k1,2 )
(1 ?)P
E [f (w0 ) f (w)] ?
(f (w) f (w? ))2 .
2 BR12
Defining the accuracy ?k = f (wk )
w? k1
f (w? ), we translate the above into the recurrence
E [?k+1
(1 ?)P ? 2 ?
E ?k
2 BR12
?k ] ?
4
kw
? ?
and by Jensen?s we have (E [?k ])2 ? E ?k2 and therefore
E [?k+1 ]
E [?k ] ?
(1 ?)P
(E [?k ])2
2 BR12
which solves to (up to a universal constant factor)
E [?k ] ?
2 BR12 1
?
(1 ?)P k
Even when > 0, we can still relate k?k1,2 to f (w) f (w? ) but the argument is a little more
involved. We refer the reader to the supplementary for more details.
In particular, consider the case where all blocks are updated in parallel as in the thread-greedy
coordinate descent algorithm of Scherrer et al. [2012]. Then P = B and there is no randomness in
the algorithm, yielding the following corollary.
Corollary 2. Suppose the block-greedy coordinate algorithm with B = P (thready-greedy) is run
on a smooth plus separable convex function f = F + R to produce the iterates {wk }k 1 . If
?block < 2, then
?
?
1
f (wk ) f (w? ) = O
.
(2 ?block )k
4
Feature Clustering
The convergence analysis of section 3 shows that we need to minimize the block spectral radius.
Directly finding a clustering that minimizes ?block is a computationally daunting task. Even with
B
equal-sized blocks, the number of possible partitions is p!/ Bp . In the absence of an efficient
search strategy for this enormous space, we find it convenient to work instead in terms of the inner
product of features from distinct blocks. The following proposition makes the connection between
these approaches precise.
Proposition 3. Let S 2 RB?B be positive semidefinite, with Sii = 1, and |Sij | < " for i 6= j. Then
the spectral radius of S has the upper bound
?(S) ? 1 + (B
1) " .
Proof. Let x be the eigenvector corresponding to the largest eigenvalue of S, scaled so that kxk1 =
1. Then
0
1
X
X
X
X
@|xi | + "
? (S) = kSxk1 =
xi + Sij
xj ?
|xj |A = 1 + (B 1) "
i
i
j6=i
j6=i
Proposition 3 tells us that we can partition the features into clusters using a heuristic approach that
strives to minimize the maximum absolute inner product between the features (columns of the design
matrix) Xi and Xj where i and j are features in different blocks.
4.1
Clustering Heuristic
Given p features and B blocks, we wish to distribute the features evenly among the blocks, attempting to minimize the absolute inner product between features across blocks. Moreover, we require an
approach that is efficient, since any time spent clustering could instead have been used for iterations
of the main algorithm. We describe a simple heuristic that builds uniform-sized clusters of features.
To construct a given block, we select a feature as a ?seed?, and assign the nearest features to the seed
(in terms of absolute inner product) to be in the same block. Because inner products with very sparse
features result in a large number of zeros, we choose at each step the most dense unassigned feature
as the seed. Algorithm 2 provides a detailed description. This heuristic requires computation of
O(Bp) inner products. In practice it is very fast?less than three seconds for even the large KDDA
dataset.
5
Algorithm 2 A heuristic for clustering p features into B blocks, based on correlation
U
{1, ? ? ? , p}
for b = 1 to B 1 do
s
arg maxj2U NNZ(Xj )
for j 2 U do
cj
|hXs , Xj i|
Jb
{j yielding the dp/Be largest values of cj }
U
U \Jb
JB
U
return {Jb |b = 1, ? ? ? , B}
Name
N EWS 20
R EUTERS
R EAL S IM
KDDA
# Features
1, 355, 191
47, 237
20, 958
20, 216, 830
# Samples
19, 996
23, 865
72, 309
8, 407, 752
# Nonzeros
9, 097, 916
1, 757, 800
3, 709, 083
305, 613, 510
// parallel
Source
Keerthi and DeCoste [2005]
Lewis et al. [2004]
RealSim
Lo et al. [2011]
Table 1: Summary of input characteristics.
5
Experimental Setup
Platform All our experiments are conducted on a 48-core system comprising of 4 sockets and 8
banks of memory. Each socket is an AMD Opteron processor codenamed Magny-Cours, which is a
multichip processor with two 6-core chips on a single die. Each 6-core processor is equipped with a
three-level memory hierarchy as follows: (i) 64 KB of L1 cache for data and 512 KB of L2 cache
that are private to each core, and (ii) 12 MB of L3 cache that is shared among the 6 cores. Each
6-core processor is linked to a 32 GB memory bank with independent memory controllers leading
to a total system memory of 256 GB (32 ? 8) that can be globally addressed from each core. The
four sockets are interconnected using HyperTransport-3 technology1 .
Datasets A variety of datasets were chosen2 for experimentation; these are summarized in Table 1. We consider four datasets: (i) N EWS 20 contains about 20, 000 UseNet postings from 20
newsgroups. The data was gathered by Ken Lang at Carnegie Mellon University circa 1995. (ii)
R EUTERS is the RCV1-v2/LYRL2004 Reuters text data described by Lewis et al. [2004]. In this
term-document matrix, each example is a training document, and each feature is a term. Nonzero
values of the matrix correspond to term frequencies that are transformed using a standard tf-idf normalization. (iii) R EAL S IM consists of about 73, 000 UseNet articles from four discussion groups:
simulated auto racing, simulated aviation, real auto racing, and real aviation. The data was gathered
by Andrew McCallum while at Just Research circa 1997. We consider classifying real vs simulated
data, irrespective of auto/aviation. (iv) KDDA represents data from the KDD Cup 2010 challenge
on educational data mining. The data represents a processed version of the training set of the first
problem, algebra 2008 2009, provided by the winner from the National Taiwan University. These
four inputs cover a broad spectrum of sizes and structural properties.
Implementation For the current work, our empirical results focus on thread-greedy coordinate descent with 32 blocks. At each iteration, a given thread must step through the nonzeros of each of its
features to compute the proposed increment (the ?j of Section 3) and the estimated benefit of choosing that feature. Once this is complete, the thread (without waiting) enters the line search phase,
where it remains until all threads are being updated by less than the specified tolerance. Finally, all
updates are performed concurrently. We use OpenMP?s atomic directive to maintain consistency.
Testing framework
We compare the effect of clustering to randomization (i.e. features are randomly assigned to blocks),
for a variety of values for the regularization parameter . To test the effect of clustering for very
1
Further details on AMD Opteron can be found at http://www.amd.com/us/products/
embedded/processors/opteron/Pages/opteron-6100-series.aspx.
2
from http://www.csie.ntu.edu.tw/?cjlin/libsvmtools/datasets/
6
5
3
4
10 ? REL
10 ? REL
6
6
5
4
3
2
5
15
10
15
(a) N EWS 20,
4
0 = 10
5
0
5
10
15
10
15
NNZ
NNZ
103
Time (min)
Time (min)
(b) R EUTERS,
4
0 = 10
0
50
100
150
0
50
100
150
1
10
NNZ
102
10
Time (min)
0
104
0
10
103
15
5
102
10
0
10 102 103 104
10 ? REL
2
1
5
15
104
0
10
6.920 6.925 6.930
7
7
7
6
5
4
3
10 ? REL
2
1
0
5
10 102 103 104
NNZ
0
Time (min)
(c) R EAL S IM,
4
0 = 10
(d) KDDA,
6
0 = 10
Figure 2: Convergence results. For each dataset, we show the regularized expected loss (top) and
number of nonzeros (bottom), using powers of ten as regularization parameters. Results for randomized features are shown in black, and those for clustered features are shown in red. Note that the
allowed running time for KDDA was ten times that of other datasets.
Active blocks
Iterations per second
NNZ @ 1K sec
Objective @ 1K sec
NNZ @ 10K iter
Objective @ 10K iter
= 10 4
Randomized Clustered
32
6
153
12.9
184
215
0.472
0.591
74
203
0.570
0.593
= 10 5
Randomized Clustered
32
32
152
12.3
797
8592
0.264
0.321
82
8812
0.515
0.328
= 10 6
Randomized Clustered
32
32
136
12.3
1248
19473
0.206
0.136
110
19919
0.472
0.141
Table 2: The effect of feature clustering, for R EUTERS.
sparse weights, we first let 0 be the largest power of ten that leads to any nonzero weight estimates. This is followed by the next three consecutive powers of ten. For each run, we measure the
regularized expected loss and the number of nonzeros at one-second intervals. Times required for
clustering and randomization are negligible, and we do not report them here.
6
Results
Figure 2 shows the regularized expected loss (top) and number of nonzeros (bottom), for several
values of the regularization parameter . Black and red curves indicate randomly-permuted features
and clustered features, respectively. The starting value of was 10 4 for all data except KDDA,
which required = 10 6 in order to yield any nonzero weights.
In the upper plots, within a color, the order of the 4 curves, top to bottom, corresponds to successively
decreasing values of . Note that a larger value of results in a sparser solution with greater
regularized expected loss and a smaller number of nonzeros. Thus, for each subfigure of Figure 2,
the order of the curves in the lower plot is reversed from that of the upper plot.
Overall, results across datasets are very consistent. For large values of , the simple clustering
heuristic results in slower convergence, while for smaller values of we see considerable benefit.
Due to space limitations, we choose a single dataset for which to explore results in greater detail.
Of the datasets we tested, R EUTERS might reasonably lead to the greatest concern. Like the other
datasets, clustered features lead to slow convergence for large and fast convergence for small .
However, R EUTERS is particularly interesting because for = 10 5 , clustered features seem to
provide an initial benefit that does not last; after about 250 seconds it is overtaken by the run with
randomized features.
7
(a) Block density
NNZ
102
10
1
Block
103
104
7
5
4
3
10 ? REL
2
?
?? ??
??????
?????????????????
?
???????
?
?
?
?
??
??
??
?????
????????????
1
105
104
NNZ
6
106
?
10
102
103
104
105
Iterations
(b) Regularized expected loss
1
10
102
103
104
105
Iterations
(c) Number of nonzeros
Figure 3: A closer look at performance characteristics for R EUTERS.
Table 2 gives a more detailed summary of the results for R EUTERS, for the three largest values of .
The first row of this table gives the number of active blocks, by which we mean the number of blocks
containing any nonzeros. For an inactive block, the corresponding thread repeatedly confirms that
all weights remain zero without contributing to convergence.
In the most regularized case = 10 4 , clustered data results in only six active blocks, while for
other cases every block is active. Thus in this case features corresponding to nonzero weights are
colocated within these few blocks, severely limiting the advantage of parallel updates.
In the second row, we see that for randomized features, the algorithm is able to get through over ten
times as many iterations per second. To see why, note that the amount of work for a given thread is
a linear function of the number of nonzeros over all of the features in its block. Thus, the block with
the greatest number of nonzeros serves as a bottleneck.
The middle two rows of Figure 2 summarize the state of each run after 1000 seconds. Note that for
this test, randomized features result in faster convergence for the two largest values of .
For comparison, the final two rows of Figure 2 provide a similar summary based instead on the
number of iterations. In these terms, clustering is advantageous for all but the largest value of .
Figure 3 shows the source of this problem. First, Figure 3a shows the number of nonzeros in all
features for each of the 32 blocks. Clearly the simple heuristic results in poor load-balancing. For
comparison, Figures 3b and 3c show convergence rates as a function of the number of iterations.
7
Conclusion
We have presented convergence results for a family of randomized coordinate descent algorithms
that we call block-greedy coordinate descent. This family includes Greedy CD, Thread-Greedy
CD, Shotgun, and Stochastic CD. We have shown that convergence depends on ?block , the maximal
spectral radius over submatrices of XT X resulting from the choice of one feature for each block.
Even though a simple clustering heuristic helps for smaller values of the regularization parameter,
our results also show the importance of considering issues of load-balancing and the distribution of
weights for heavily-regularized problems.
A clear next goal in this work is the development of a clustering heuristic that is relatively well
load-balanced and distributes weights for heavily-regularized problems evenly across blocks, while
maintaining good computational efficiency.
Acknowledgments
The authors are grateful for the helpful suggestions of Ken Jarman, Joseph Manzano, and our anonymous reviewers.
Funding for this work was provided by the Center for Adaptive Super Computing Software - MultiThreaded Architectures (CASS-MT) at the U.S. Department of Energy?s Pacific Northwest National Laboratory. PNNL is operated by Battelle Memorial Institute under Contract DE-ACO676RL01830.
8
References
J Friedman, T Hastie, H H?ofling, and R Tibshirani. Pathwise coordinate optimization. Annals of
Applied Statistics, 1(2):302?332, 2007.
T Wu and K Lange. Coordinate descent algorithms for lasso penalized regression. Annals of Applied
Statistics, 2:224?244, 2008.
S Shalev-Shwartz and A Tewari. Stochastic methods for `1 -regularized loss minimization. Journal
of Machine Learning Research, 12:1865?1892, 2011.
J K Bradley, A Kyrola, D Bickson, and C Guestrin. Parallel Coordinate Descent for L1-Regularized
Loss Minimization. In Proceedings of the 28th International Conference on Machine Learning,
pages 321?328, 2011.
C Scherrer, A Tewari, M Halappanavar, and D Haglin. Scaling up Parallel Coordinate Descent
Algorithms. In International Conference on Machine Learning, 2012.
Y Li and S Osher. Coordinate Descent Optimization for `1 Minimization with Application to Compressed Sensing ; a Greedy Algorithm Solving the Unconstrained Problem. Inverse Problems and
Imaging, 3:487?503, 2009.
I S Dhillon, P Ravikumar, and A Tewari. Nearest neighbor based greedy coordinate descent. In
Advances in Neural Information Processing Systems 24, pages 2160?2168, 2011.
D Lewis, Y Yang, T Rose, and F Li. RCV1 : A New Benchmark Collection for Text Categorization
Research. Journal of Machine Learning Research, 5:361?397, 2004.
S S Keerthi and D DeCoste. A modified finite Newton method for fast solution of large scale linear
SVMs. Journal of Machine Learning Research, 6:341?361, 2005.
RealSim. Document classification data gathered by Andrew McCallum., circa 1997. URL:http:
//people.cs.umass.edu/?mccallum/data.html.
Hung-Yi Lo, Kai-Wei Chang, Shang-Tse Chen, Tsung-Hsien Chiang, Chun-Sung Ferng, Cho-Jui
Hsieh, Yi-Kuang Ko, Tsung-Ting Kuo, Hung-Che Lai, Ken-Yi Lin, Chia-Hsuan Wang, Hsiang-Fu
Yu, Chih-Jen Lin, Hsuan-Tien Lin, and Shou de Lin. Feature engineering and classifier ensemble
for KDD Cup 2010, 2011. To appear in JMLR Workshop and Conference Proceedings.
9
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4,060 | 4,675 | Perfect Dimensionality Recovery
by Variational Bayesian PCA
Shinichi Nakajima
Nikon Corporation
Tokyo, 140-8601, Japan
[email protected]
Masashi Sugiyama
Tokyo Institute of Technology
Tokyo 152-8552, Japan
[email protected]
Ryota Tomioka
The University of Tokyo
Tokyo 113-8685, Japan
[email protected]
S. Derin Babacan
University of Illinois at Urbana-Champaign
Urbana, IL 61801, USA
[email protected]
Abstract
The variational Bayesian (VB) approach is one of the best tractable approximations to the Bayesian estimation, and it was demonstrated to perform well in many
applications. However, its good performance was not fully understood theoretically. For example, VB sometimes produces a sparse solution, which is regarded
as a practical advantage of VB, but such sparsity is hardly observed in the rigorous
Bayesian estimation. In this paper, we focus on probabilistic PCA and give more
theoretical insight into the empirical success of VB. More specifically, for the situation where the noise variance is unknown, we derive a sufficient condition for
perfect recovery of the true PCA dimensionality in the large-scale limit when the
size of an observed matrix goes to infinity. In our analysis, we obtain bounds for
a noise variance estimator and simple closed-form solutions for other parameters,
which themselves are actually very useful for better implementation of VB-PCA.
1
Introduction
Variational Bayesian (VB) approximation [1] was proposed as a computationally efficient alternative
to rigorous Bayesian estimation. The key idea is to force the posterior to be factorized, so that the
integration?a typical intractable operation in Bayesian methods?can be analytically performed
over each parameter with the other parameters fixed. VB has been successfully applied to many
applications [4, 7, 20, 11].
Typically, VB solves a non-convex optimization problem with an iterative algorithm [3], which
makes theoretical analysis difficult. An important exceptional case is the matrix factorization (MF)
model [11, 6, 19] with no missing entry in the observed matrix. Recently, the global analytic solution
of VBMF has been derived and theoretical properties such as the mechanism of sparsity induction
have been revealed [15, 16]. These works also posed thought-provoking relations between VB
and rigorous Bayesian estimation: The VB posterior is actually quite different from the true Bayes
posterior (compare the left and the middle graphs in Fig. 1), and VB induces sparsity in its solution
but such sparsity is hardly observed in rigorous Bayesian estimation (see the right graph in Fig. 1).1
These facts might deprive the justification of VB based solely on the fact that it is one of the best
tractable approximations to the Bayesian estimation.
1
Also in mixture models, inappropriate model pruning by VB approximation was discussed [12].
1
Bayes p osterior (V = 1)
VB p osterior (V = 1)
3
3
0.1 0.2
0.2
0.3
MAP estimators:
2
0.3
2 (A, B ) ? (? 1, ? 1)
3
VB estimator : (A, B ) = (0, 0)
0.05
0.
05
!
U
0.1
1
0.
1
0.1
?1
0.0
5
?2
?1
0.3
0.1
?3
?3
0
A
0.05
?2
0.2
0.3 0.2
?2
5
0
2
0.15
0.0
0.2
0.3
0.3
0.2
0.1
?1
0.3
0.2
0.1
B
0.2
0.
05
0.1
0.
1
1
0.
0
0.3
0.15
0.1
0.2
0.3
1
0.1
0.2
0.3
0.05
B
1
FB
MAP
VB
EFB
EMAP
EVB
1
2
3
?3
?3
?2
?1
0
A
1
2
3
1
2
3
V
Figure 1: Dissimilarities between VB and the rigorous Bayesian estimation. (Left and Center) the
Bayes posterior and the VB posterior of a 1 ? 1 MF model, V = BA + E, when V = 1 is observed
(E is a Gaussian noise). VB approximates the Bayes posterior having two modes by an origincentered Gaussian, which induces sparsity. (Right) Behavior of estimators of U = BA! , given the
observation V . The VB estimator (the magenta solid curve) is zero when V ? 1, which means exact
sparsity. On the other hand, FB (fully-Bayesian or rigorous Bayes; blue crosses) shows no sign of
sparsity. Further discussion will be provided in Section 5.2. All graphs are quoted from [15].
Since the probabilistic PCA [21, 18, 3] is an instance of MF, the global analytic solution derived
in [16] for MF can be utilized for analyzing the probabilistic PCA. Indeed, automatic dimensionality selection of VB-PCA, which is an important practical advantage of VB-PCA, was theoretically
investigated in [17]. However, the noise variance, which is usually unknown in many realistic applications of PCA, was treated as a given constant in that analysis.2 In this paper, we consider a more
practical and challenging situation where the noise variance is unknown, and theoretically analyze
VB-PCA.
It was reported that noise variance estimation fails in some Bayesian approximation methods, if the
formal rank is set to be full [17]. With such methods, an additional model selection procedure is
required for dimensionality selection [14, 5]. On the other hand, we theoretically show in this paper
that VB-PCA can estimate the noise variance accurately, and therefore automatic dimensionality selection works well. More specifically, we establish a sufficient condition that VB-PCA can perfectly
recover the true dimensionality in the large-scale limit when the size of an observed matrix goes to
infinity. An interesting finding is that, although the objective function minimized for noise variance
estimation is multimodal in general, only a local search algorithm is required for perfect recovery.
Our results are based on the random matrix theory [2, 5, 13, 22], which elucidates the distribution
of singular values in the large-scale limit.
In the development of the above theoretical analysis, we obtain bounds for the noise variance estimator and simple closed-form solutions for other parameters. We also show that they can be nicely
utilized for better implementation of VB-PCA.
2
Formulation
In this section, we introduce the variational Bayesian matrix factorization (VBMF).
2.1
Bayesian Matrix Factorization
Assume that we have an observation matrix V ? RL?M , which is the sum of a target matrix U ?
RL?M and a noise matrix E ? RL?M :
V = U + E.
In the matrix factorization model, the target matrix is assumed to be low rank, which can be expressed as the following factorizability:
U = BA! ,
2
If the noise variance is known, we can actually show that dimensionality selection by VB-PCA is outperformed by a naive strategy (see Section 3.3). This means that VB-PCA is not very useful in this setting.
2
where A ? RM ?H and B ? RL?H . $ denotes the transpose of a matrix or vector. Thus, the rank
of U is upper-bounded by H ? min(L, M ).
In this paper, we consider the probabilistic matrix factorization (MF) model [19], where the observation noise E and the priors of A and B are assumed to be Gaussian:
!
"
p(V |A, B) ? exp ? 2?1 2 'V ? BA! '2Fro ,
(1)
! 1 !
""
! 1 !
""
?1 !
?1 !
p(A) ? exp ? 2 tr ACA A
,
p(B) ? exp ? 2 tr BCB B
.
(2)
Here, we denote by ' ? 'Fro the Frobenius norm, and by tr(?) the trace of a matrix. We assume that
L ? M . If L > M , we may simply re-define the transpose V ! as V so that L ? M holds.3 Thus,
this does not impose any restriction. We assume that the prior covariance matrices CA and CB are
diagonal and positive definite, i.e.,
CA = diag(c2a1 , . . . , c2aH ),
CB = diag(c2b1 , . . . , c2bH )
for cah , cbh > 0, h = 1, . . . , H. Without loss of generality, we assume that the diagonal entries of
the product CA CB are arranged in non-increasing order, i.e., cah cbh ? cah! cbh! for any pair h < h$ .
Throughout the paper, we denote a column vector of a matrix by a bold lowercase letter, and a row
vector by a bold lowercase letter with a tilde, namely,
$
%!
# M )! ? RM ?H , B = (b1 , . . . , bH ) = #
A = (a1 , . . . , aH ) = (#
a1 , . . . , a
b1 , . . . , #
bL
? RL?H .
2.2
Variational Bayesian Approximation
The Bayes posterior is given by
p(A, B|V ) =
p(V |A,B)p(A)p(B)
,
p(V )
(3)
where p(Y ) = )p(V |A, B)*p(A)p(B) . Here, )?*p denotes the expectation over the distribution p.
Since this expectation is intractable, we need to approximate the Bayes posterior.
Let r(A, B), or r for short, be a trial distribution. The following functional with respect to r is called
the free energy:
&
'
&
'
r(A,B)
r(A,B)
F (r) = log p(V |A,B)p(A)p(B)
= log p(A,B|V
? log p(V ).
(4)
)
r(A,B)
r(A,B)
In the last equation, the first term is the Kullback-Leibler (KL) divergence from the trial distribution
to the Bayes posterior, and the second term is a constant. Therefore, minimizing the free energy (4)
amounts to finding a distribution closest to the Bayes posterior in the sense of the KL divergence. A
general approach to Bayesian approximate inference is to find the minimizer of the free energy (4)
with respect to r in some restricted function space.
In the VB approximation, the independence between the entangled parameter matrices A and B is
assumed:
r = r(A)r(B).
(5)
Under this constraint, an iterative algorithm for minimizing the free energy (4) was derived [3, 11].
Let r( be such a minimizer, and we define the MF solution by the mean of the target matrix U :
)
*
( = BA!
U
.
(6)
r
!(A,B)
(.
In the context of PCA where V is a data matrix, the solution is given as the subspace spanned by U
The MF model has hyperparameters (CA , CB ) in the priors (2). By manually choosing them, we
can control regularization and sparsity of the solution (e.g., the PCA dimensions). A popular way
to set the hyperparameter in the Bayesian framework is again based on the minimization of the free
energy (4):
(A , C
(B ) = argminC ,C (minr F (r; CA , CB |V )) .
(C
A
B
We refer to this method as an empirical VB (EVB) method. When the noise variance ? 2 is unknown,
it can also be estimated as
!
"
?
(2 = argmin?2 minr,CA ,CB F (r; CA , CB , ? 2 |V ) .
3
When the number of samples is larger (smaller) than the data dimensionality in the PCA setting, the
observation matrix V should consist of the columns (rows), each of which corresponds to each sample.
3
3
Simple Closed-Form Solutions of VBMF
Recently, the global analytic solution of VBMF has been derived [16]. However, it is given as a
solution of a quartic equation (Corollary 1 in [16]), and it is not easy to use for further analysis due
to its complicated expression. In this section, we derive much simpler forms, which will be used for
analyzing VB-PCA in the next section.
3.1
VB Solution
Our new analytic-form solution only involves linear and quadratic equations, which is summarized
in the following theorem (the proof is omitted due to the space limitation):
Theorem 1 Let
V =
+H
h=1
(7)
?h ? bh ? !
ah
be the singular value decomposition (SVD) of V with its singular values {?h } arranged in nonincreasing order, and the associated right and left singular vectors {? ah , ? bh }. Then, the VB solution can be written as a truncated shrinkage SVD as follows:
H
,
??hVB if ?h ? ? VB
,
VB
VB
!
VB
(
h
U
=
?
(h ? bh ? ah ,
where
?
(h =
(8)
0
otherwise.
h=1
Here, the truncation threshold and the shrinkage estimator are, respectively, given by
.
12
/
32
/
2
(L+M )
(L+M )
0
VB
?2
?
=?
+
+
+ ?
? LM ,
2c2a c2b
2
h
2
??hVB = ?h 1 ?
h
?2
2
2?h
2
h
2c2a c2b
2
4
M + L + (M ? L)2 +
h
(9)
h
2
4?h
c2a c2b
h
h
33
.
(10)
We can also derive a simple closed-form expression of the VB posterior (omitted).
3.2 EVB Solution
Combining Theorem 1 with the global EVB solution (Corollary 2 in [16]), we have the following
theorem (the proof is omitted):
Theorem 2 Let
?=
(11)
L
M,
and let ? = ?(?) (> 1) be the zero-cross point of the following decreasing function:
$ %
?
? (?; ?) = ? ( ??) + ? ??? ,
where
?(x) = log(x+1)
? 12 .
x
Then, the EVB solution can be written as a truncated shrinkage SVD as follows:
5 EVB
H
,
??h
if ?h ? ? EVB ,
EVB
EVB
!
EVB
(
U
=
?
(h ? bh ? ah ,
where
?
(h
=
0
otherwise.
(12)
(13)
h=1
Here, the truncation threshold and the shrinkage estimator are, respectively, given by
4
$
%
?
EVB
?
= ? M + L + LM ? + ?1 ,
6
7
4$
%
??hEVB =
?h
2
1?
(M +L)? 2
2
?h
1?
+
(M +L)? 2
2
?h
2
?
4LM ? 4
4
?h
.
(14)
(15)
The EVB threshold (14) involves ?, which needs to be numerically computed. We can easily prepare
a table of the values for 0 < ? ? 1 beforehand, like the cumulative Gaussian probability used in
statistical tests. Fig. 2 shows the EVB threshold (14) by a red solid curve labeled as ?EVB?.
4
3
2.5
p (u)
?
?/ M ? 2
2
1.5
x
4
u(0.1)
u(1)
1
1
0.5
0
0
EVB
MPUL
VBFL
0.2
0.4
0.6
0.8
0
0
1
?
Figure 2: Thresholds.
3.3
2
6
?=1
? = 0.1
!u" p ( u )
2
1
2
3
4
u = ? 2 /(? 2 ?M )
5
Figure 3: Mar?cenko-Pastur law.
0
0
? 0 (x)
? (x)
2
4
x
6
8
Figure 4: ?0 (x) and ?(x).
Large-Scale Limiting Behavior of EVB When Noise Variance Is Known
Here, we first introduce a result from random matrix theory [13, 22], and then discuss the behavior
of EVB when the noise variance is known.
Assume that E ? RL?M is a random matrix such that each element is independently drawn from a
distribution with mean zero and variance ? 2? (not necessarily Gaussian). Let u1 , u2 , . . . , uL be the
eigenvalues of M ?1 2? EE ! , and define the empirical distribution of the eigenvalues by
p(u) =
1
L
(?(u1 ) + ?(u2 ) + ? ? ? + ?(uL )) ,
where ?(u) denotes the Dirac measure at u. Then the following proposition holds:
Proposition 1 (Mar?cenko-Pastur law) [13, 22] In the large-scale limit when L and M go to infinity with its ratio ? = L/M fixed, the probability measure of the empirical distribution of the
eigenvalue u of ?2?1M EE ! converges almost surely to
?
(u?u)(u?u)
p(u)du =
?(u < u < u)du,
(16)
2??u
? 2
? 2
where u = (1 ? ?) , u = (1 + ?) , and ?(?) denotes an indicator function such that
?(condition) = 1 if the condition is true and ?(condition) = 0 otherwise.
Fig. 3 shows the Mar?cenko-Pastur (MP) distributions for ? = 0.1, 1. The mean )u*p(u) = 1 (which
is constant for any 0 < ? ? 1) and the upper-limit u = u(?) for ? = 0.1, 1 are indicated by
arrows. Note that the MP distribution appears for a single sample matrix; different from standard
?large-sample? theories, we do not need many sample matrices (this property is sometimes called
self-averaging). This single-sample property of the MP distribution is highly useful in our analysis
because we are working with a single observation matrix in the MF scenario.
Proposition 1 states that all singular values of the random matrix E are almost surely upper-bounded
by
?
?
?
? MPUL = M ? 2? u = ( L + M )? ? ,
(17)
which we call the Mar?cenko-Pastur upper-limit (MPUL). This property can be used for designing
estimators robust against noise [10, 9]. Although EVB-PCA was proposed independently from the
random matrix theory [3], its good performance can be proven with a related property to Proposition 1, as shown in Section 4.
When the noise variance is known, we can set the parameter to ? = ? ? in Eq.(1). We depicted
MPUL (17) for this case in Fig. 2. We can see that MPUL lower-bounds the EVB threshold (14)
(actually this is true regardless of the value of ? > 0). This implies a nice behavior of EVB, i.e.,
EVB eliminates all noise components in the large-scale limit. However, a simple optimal strategy?
discarding the components with singular values smaller than ? MPUL ?outperforms EVB, because
signals lying between the gap [? MPUL , ? EVB ) are discarded by EVB. Therefore, EVB is not very
useful when ? 2? is known. In Section 4, we investigate the behavior of EVB in a more practical and
challenging situation where ? 2? is unknown and is also estimated from observation.
In Fig. 2, we also depicted the VB threshold (9) with almost flat prior cah , cbh ? ? (labeled
as ?VBFL?) for comparison. Actually, this coincides with the mean of the MP distribution, i.e.,
limcah ,cah ?? (? VB
)2 /(M ? 2 ) = )u*p(u) = 1. This implies that VBFL retains a lot of noise comh
ponents, and does not perform well even when ? 2? is known.
5
4
Analysis of EVB When Noise Variance Is Unknown
In this section, we derive bounds of the VB-based noise variance estimator, and obtain a sufficient
condition for perfect dimensionality recovery in the large-scale limit.
4.1
Bounds of Noise Variance Estimator
The simple closed-form solution obtained in Section 3 is the global minimizer of the free energy
(4), given ? 2 . Using the solution, we can explicitly describe the free energy as a function of ? 2 . We
obtain the following theorem (the proof is omitted):
Theorem 3 The noise variance estimator ?
(2 EVB is the global minimizer of
$
%
$ 2 %
2
+H
+L
?
?
?(? ?2 ) = h=1 ? M h?2 + h=H+1 ?0 M h?2 ,
"
? !
where ? (x) = ?0 (x) + ? (x > x) ?1 (x) ,
x = 1 + ? + ? ? + ??1 ,
$
% ?
?
? +1 ?
?0 (x) = x ? log x,
?1 (x) = log ( ??(x) + 1) + ? log ?(x)
??(x),
?
? is a constant defined in Theorem 2, and ?(x) is a function of x (> x) defined by
2
3
8
2
?(x) = 2?1 ? (x ? (1 + ?)) + (x ? (1 + ?)) ? 4? .
(18)
(19)
(20)
(21)
Note that x and ?(?h2 /(? 2 M )) are rescaled versions of the squared EVB threshold (14) and the
EVB shrinkage estimator (15), respectively, i.e., x = (? EVB )2 /(? 2 M ) and ?(?h2 /(? 2 M )) =
?
?h ??hEVB /(? 2 M L).
The functions ?0 (x) and ? (x) are depicted in Fig. 4. We can prove the convexity of ?0 (x) and
quasi-convexity of ? (x), which are useful properties in our theoretical analysis.
( EVB be the rank estimated by VB, which satisfies ?
( EVB and
Let H
(hEVB > 0 for h = 1, . . . , H
EVB
EVB
(
?
(h
= 0 for h = H
+ 1, . . . , H. Then we have the following theorem:
( EVB is upper-bounded as
Theorem 4 H
( EVB ? H = min
H
$9
L
1+?
:
%
? 1, H ,
(22)
and the noise variance estimator ?
(2 EVB is bounded as follows:
2
3
"L
+L
?2
h=H+1 h
1
2
2
max ? H+1 , M L?H(1+?)
<?
(2 EVB ? LM
h=1 ?h ,
(
)
?
for h = 0,
?
??2
?h
where ? 2h = M
for h = 1, . . . , L,
x
?
?
0
for h = L + 1.
(23)
(24)
(Sketch of proof) First, we show that a global minimizer w.r.t. ? 2 exists in (?L2 /M, ?12 /M ), and it
( the derivative of ? w.r.t. ? ?2 is written as
is a stationary point. Given a hypothetic H,
?(? ?2 ) ?
1 ??
L ?? ?2
= ?? 2 +
"H
#
h=1
"
EVB
2
?h (?h ??
?h
?h
)+ Lh=H+1
#
LM
.
Eq.(15) implies the following bounds:
?
?
!
"
(M + L)? 2 < ?h ?h ? ??hEVB < ( M + L)2 ? 2 for ?h > ? EVB ,
(25)
(26)
which allows us to bound ? by simple inequalities. Finding a condition prohibiting ? to be zero
proves the theorem.
!
Theorem 4 states that EVB discards the (L ? /L/(1 + ?)0 + 1) ? 1 smallest components, regardless
of the observed values {?h }. For example, the half components are always discarded when the
matrix is square (i.e., ? = L/M = 1). The smallest singular value ?L is always discarded, and
?
(2 EVB > ?L2 /(M (L ? (L ? 1)(1 + ?)) > ?L2 /M always holds.
6
?
?
?
?
?
?
0.5
0
0
1
2
3
=
=
=
=
=
=
0.0
0.1
0.2
0.4
0.6
0.8
4
5
1
?
?
?
?
?
?
0.5
0
0
1
2
y
(a) ? = 1
Succes s Rate
Succes s Rate
Succes s Rate
1
3
y
=
=
=
=
=
=
4
0.0
0.1
0.2
0.4
0.6
0.8
5
1
?
?
?
?
?
?
0.5
0
0
1
2
3
=
=
=
=
=
=
4
0.0
0.1
0.2
0.4
0.6
0.8
5
y
(b) ? = 0.5
(c) ? = 0.1
Figure 5: Success rate of dimensionality recovery in numerical simulation for M = 200. The
threshold for the guaranteed recovery (the second inequality in Eq.(28)) is depicted with a vertical
bar with the same color and line style.
4.2
Perfect Recovery Condition
Here, we derive a sufficient condition for perfect recovery of the true PCA dimensionality in the
large-scale limit.
Assume that the observation matrix V is generated as
V = U ? + E,
(27)
?
where U is a true signal matrix with rank H ? and the singular values {?h? }, and each element of
the noise matrix E is subject to a distribution with mean zero and variance ? 2? . We rely on a result
[2, 5] on the eigenvalue distribution of the spiked covariance model [8]. The following theorem
guarantees the accuracy of VB-PCA:
Theorem 5 Assume H ? H ? (i.e., we set the rank of the MF model sufficiently large), and denote
the relevant rank (dimensionality) ratio by
?
? = HL .
?
In the large-scale limit with finite ? and H , EVB implemented with a local search algorithm for
( EVB = H ? , if ? = 0 or
the noise variance ? 2 estimation almost surely recovers the true rank, i.e., H
$
%
x?1
?2
2?
? < x1 and ?H
(28)
? >
1?x? ? ? ? M ? ,
where x is defined in Eq.(19).
(Sketch of proof) We first show that, in the large-scale limit and when ? = 0, Eq.(25) is equal to
zero if and only if ? 2 = ? 2? . This means the perfect recovery in the no-signal case. ? 2h defined in
Eq.(24) is actually the thresholding point of estimated ?
(2 for the h-th component to be discarded.
( EVB = H ? if and only if ? 2 ?
Therefore, H
(2 < ? 2H ? . Using Eq.(26), we can obtain a
H +1 < ?
sufficient condition that a local minimum exists only in this range, which proves the theorem.
!
Note that ? ? 0 in the large scale limit. However, we treated ? as a positive value in Theorem 5,
hoping that the obtained result can approximately hold in a practical situation when L and M are
large but finite. The obtained result well explains the dependency on ? in the numerical simulation
below.
Theorem 5 guarantees that, if the true rank H ? is small enough compared with L and the smallest
?
2?
signal ?H
, VB-PCA works perfectly. It is important to note
? is large enough compared with ?
that, although the objective function (18) is non-convex and possibly multimodal in general, perfect
recovery does not require global search, but only a local search, of the objective function for noise
variance estimation.
Fig. 5 shows numerical results for M = 200 and ? = 1, 0.5, 0.1. E was drawn from the Gaussian
distribution with variance ? 2? = 1, and signal singular values were drawn from the uniform distribution on [yM ? 2? , 10M ] for different y (the horizontal axis of the graphs indicates y). The vertical
( EVB = H ? , over 100 trials. If the
axis indicates the success rate of dimensionality recovery, i.e., H
condition for ? (the first inequality in Eq.(28)) is violated, the corresponding line is depicted with
markers. Otherwise, the threshold of y for the guaranteed recovery (the second inequality in Eq.(28))
is indicated by a vertical bar with the same color and line style. We can see that the guarantee by
Theorem 5 approximately holds even in this small matrix size, although it is slightly conservative.
7
5
Discussion
Here, we discuss implementation of VB-PCA, and the origin of sparsity of VB.
5.1
Implementation
Implementation of VB-PCA (VB-MF) based on the result given in [16] involves a quartic equation.
This means that we need to use a highly complicated analytic-form solution, derived by, e.g., Ferrari?s method, of a quartic equation, or rely on a numerical quartic solver, which is computationally
less efficient. The theorems we gave in this paper can actually simplify the implementation greatly.
A table of ? defined in Theorem 2 should be prepared beforehand. Given an observed matrix V ,
we perform SVD and obtain the singular values. After that, in our new implementation, we first
directly estimate the noise variance based on Theorem 3, using any 1-D local search algorithm?
Theorem 4 helps restrict the search range. Then we obtain the noise variance estimator ?
(2 EVB . For
a dimensionality reduction purpose, simply discarding all the components such that ? 2h < ?
(2 EVB
( EVB is needed, Theorem 2
gives the result (here ? 2h is defined by Eq.(24)). When the estimator U
2
2 EVB
gives the result for ? = ?
(
. The VB posterior is also easily computed (its description is
omitted). In this way, we can perform VB-PCA, whose performance is guaranteed, very easily.
5.2
Origin of Exact Sparsity
Sparsity is regarded as a practical advantage of VB. Nevertheless, as discussed in Section 1, it is
not necessarily a property inherent in the rigorous Bayesian estimation. Actually, at least in MF,
the sparsity is induced by the independence assumption between A and B. Let us go back to Fig.1,
where the Bayes posterior for V = 1 is shown in the left graph. In this scalar factorization model, the
( = BA toward the positive
probability mass in the first and the third quadrants pulls the estimator U
direction, and the mass in the second and the fourth quadrants toward the negative direction. Since
the Bayes posterior is skewed and more mass is put in the first and the third quadrants, it is natural
that the Bayesian estimator ?
( = )BA*p(A,B|V ) is positive. This is true even if V > 0 is very small.
On the other hand, the VB posterior (the middle graph) is prohibited to be skewed because of the
independent assumption, and thus it has to wait until V grows so that one of the modes has a enough
probability mass. This is the cause of sparsity in VBMF. The Bayes posterior (left graph) implies
that, if we restrict the posterior to be Gaussian, but allow to have correlation between A and B, exact
sparsity will not be observed.
It is observed that the Bayesian estimation gives a sparse solution when the hyper parameters
(CA , CB ) are optimized. This estimator is also depicted as blue diamonds labeled as EFB (empirical fully-Bayesian) in the right graph of Fig.1. Note that, in this case, the independence between
?1/2
?1/2
A and CA
(as well as B and CB ), which are strongly correlated in the prior (2) and hence in
the Bayes posterior, is forced?the point estimation of CA (as well as CB ) breaks the correlation
because approximating by the delta function induces the independence from all other parameters.
Further investigation on the relation between the independence constraint and the sparsity induction
is our future work.
6
Conclusion
In this paper, we considered the variational Bayesian PCA (VB-PCA) when the noise variance is
unknown. Analyzing the behavior of the noise variance estimator, we derived a sufficient condition
for VB-PCA to perfectly recover the true dimensionality. Our result theoretically supports the usefulness of VB-PCA. In our theoretical analysis, we obtained bounds for a noise variance estimator
and simple closed-form solutions for other parameters, which were shown to be useful for better
implementation of VB-PCA.
Acknowledgments
SN, RT, and MS thank the support from MEXT Kakenhi 23120004, MEXT Kakenhi 22700138, and
the FIRST program, respectively. SDB was supported by a Beckman Postdoctoral Fellowship.
8
References
[1] H. Attias. Inferring parameters and structure of latent variable models by variational Bayes.
In Proceedings of the Fifteenth Conference Annual Conference on Uncertainty in Artificial
Intelligence (UAI-99), pages 21?30, San Francisco, CA, 1999. Morgan Kaufmann.
[2] J. Baik and J. W. Silverstein. Eigenvalues of large sample covariance matrices of spiked population models. Journal of Multivariate Analysis, 97(6):1382?1408, 2006.
[3] C. M. Bishop. Variational principal components. In Proc. of ICANN, volume 1, pages 514?509,
1999.
[4] Z. Ghahramani and M. J. Beal. Graphical models and variational methods. In Advanced Mean
Field Methods, pages 161?177. MIT Press, 2001.
[5] D. C. Hoyle. Automatic PCA dimension selection for high dimensional data and small sample
sizes. Journal of Machine Learning Research, 9:2733?2759, 2008.
[6] A. Ilin and T. Raiko. Practical approaches to principal component analysis in the presence of
missing values. JMLR, 11:1957?2000, 2010.
[7] T. S. Jaakkola and M. I. Jordan. Bayesian parameter estimation via variational methods. Statistics and Computing, 10:25?37, 2000.
[8] I. M. Johnstone. On the distribution of the largest eigenvalue in principal components analysis.
Annals of Statistics, 29:295?327, 2001.
[9] N. El Karoui. Spectrum estimation for large dimensional covariance matrices using random
matrix theory. Annals of Statistics, 36(6):2757?2790, 2008.
[10] O. Ledoit and M. Wolf. A well-conditioned estimator for large-dimensional covariance matrices. Journal of Multivariate Analysis, 88(2):365?411, 2004.
[11] Y. J. Lim and T. W. Teh. Variational Bayesian approach to movie rating prediction. In Proceedings of KDD Cup and Workshop, 2007.
[12] D. J. C. Mackay. Local minima, symmetry-breaking, and model pruning in variational
free energy minimization. Available from http://www.inference.phy.cam.ac.uk/
mackay/minima.pdf. 2001.
[13] V. A. Marcenko and L. A. Pastur. Distribution of eigenvalues for some sets of random matrices.
Mathematics of the USSR-Sbornik, 1(4):457?483, 1967.
[14] T. P. Minka. Automatic choice of dimensionality for PCA. In Advances in NIPS, volume 13,
pages 598?604. MIT Press, 2001.
[15] S. Nakajima and M. Sugiyama. Theoretical analysis of Bayesian matrix factorization. Journal
of Machine Learning Research, 12:2579?2644, 2011.
[16] S. Nakajima, M. Sugiyama, and S. D. Babacan. Global solution of fully-observed variational
Bayesian matrix factorization is column-wise independent. In Advances in Neural Information
Processing Systems 24, 2011.
[17] S. Nakajima, M. Sugiyama, and S. D. Babacan. On Bayesian PCA: Automatic dimensionality
selection and analytic solution. In Proceedings of 28th International Conference on Machine
Learning (ICML2011), Bellevue, WA, USA, Jun. 28?Jul.2 2011.
[18] S. Roweis and Z. Ghahramani. A unifying review of linear Gaussian models. Neural Computation, 11:305?345, 1999.
[19] R. Salakhutdinov and A. Mnih. Probabilistic matrix factorization. In J. C. Platt, D. Koller,
Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems 20,
pages 1257?1264, Cambridge, MA, 2008. MIT Press.
[20] M. Sato, T. Yoshioka, S. Kajihara, K. Toyama, N. Goda, K. Doya, and M. Kawato. Hierarchical
Bayesian estimation for MEG inverse problem. Neuro Image, 23:806?826, 2004.
[21] M. E. Tipping and C. M. Bishop. Probabilistic principal component analysis. Journal of the
Royal Statistical Society, 61:611?622, 1999.
[22] K. W. Wachter. The strong limits of random matrix spectra for sample matrices of independent
elements. Annals of Probability, 6:1?18, 1978.
9
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4,061 | 4,676 | Confusion-Based Online Learning and a
Passive-Aggressive Scheme
Liva Ralaivola
QARMA, Laboratoire d?Informatique Fondamentale de Marseille
Aix-Marseille University, France
[email protected]
Abstract
This paper provides the first ?to the best of our knowledge? analysis of online
learning algorithms for multiclass problems when the confusion matrix is taken
as a performance measure. The work builds upon recent and elegant results on
noncommutative concentration inequalities, i.e. concentration inequalities that
apply to matrices, and, more precisely, to matrix martingales. We do establish
generalization bounds for online learning algorithms and show how the theoretical
study motivates the proposition of a new confusion-friendly learning procedure.
This learning algorithm, called COPA (for COnfusion Passive-Aggressive) is a
passive-aggressive learning algorithm; it is shown that the update equations for
COPA can be computed analytically and, henceforth, there is no need to recourse
to any optimization package to implement it.
1
Introduction
This paper aims at promoting an infrequent way to tackle multiclass prediction problems: we advocate for the use of the confusion matrix ?the matrix which reports the probability of predicting
class q for an instance of class p for all potential label pair (p, q)? as the objective ?function? to
be optimized. This way, we step aside the more widespread viewpoint of relying on the misclassification rate ?the probability of misclassifying a point? as a performance measure for multiclass
predictors. There are obvious reasons for taking this perspective, among which we may name the following. First, the confusion information is a finer-grain information than the misclassification rate,
as it allows one to precisely identify the types of error made by a classifier. Second, the confusion
matrix is independent of the class distributions, since it reports conditional probability distributions:
a consequence is that a predictor learned to achieve a ?small? confusion matrix will probably be insensitive to class imbalance and it will also be robust to changes in class prior distributions between
train and test data. Finally, there are many application domains such as medicine, bioinformatics,
information retrieval, where the confusion matrix (or an estimate thereof) is precisely the object of
interest for an expert who wants to assess the relevance of an automatic prediction procedure.
Contributions. We essentially provide two contributions. On the one hand, we provide a statistical
analysis for the generalization ability of online learning algorithms producing predictors that aim at
optimizing the confusion. This requires us to introduce relevant statistical quantities that are taken
advantage of via a concentration inequality for matrix martingales proposed by [8]. Motivated by our
statistical analysis, we propose an online learning algorithm from the family of passive aggressive
learning algorithms [2]: this algorithm is inherently designed to optimize the confusion matrix and
numerical simulations are provided that show it reaches its goal.
Outline of the paper. Section 2 formalizes our pursued objective of targetting a small confusion error. Section 3 provides our results regarding the ability of online confusion-aware learning
1
procedures to achieve a small confusion together with the update equations for COPA, a new passiveaggressive learning procedure designed to control the confusion risk. Section 4 reports numerical
results that should be viewed as a proof of concept for the effectiveness of our algorithm.
2
2.1
Problem and Motivation
Notation
We focus on the problem of multiclass classification: the input space is denoted by X and the target
space is Y = {1, . . . , Q}. The training sequence Z = {Zt = (Xt , Yt )}Tt=1 is made of T identically
and independently random pairs Zt = (Xt , Yt ) distributed according to some unknown distribution
D over X ? Y. The sequence of input data will be referred to as X = {Xt }Tt=1 and the sequence
of corresponding labels Y = {Yt }Tt=1 . We may write that Z is distributed according to DT =
?Tt=1 D. Z 1:t denotes the subsequence Z 1:t = {(X? , Y? )}t? =1 . We use DX|y for the conditional
distribution on X given that Y = y; therefore, for a given sequence y = (y1 , . . . , yT ) ? Y T ,
DX|y = ?Tt=1 DX|yt is the distribution of the random sample X = {X1 , . . . , XT } over X T such
that Xt is distributed according to DX|yt . E[?] and EX|y respectively denote the expectation with
respect to D and DX|y .
For a sequence y of labels, T (y) = [T1 (y) ? ? ? TQ (y)] ? NQ is such that Tq (y) is the number of
times label q appears in y. Often, we will drop the dependence upon y for T (y). Throughout, we
make the simplifying assumption that Tq > 1 for all q ?otherwise, our analysis still holds but extra
care and notation must be taken for handling classes absent from Z.
The space of hypotheses we consider is H (e.g. H ? {f : f : X ? R}Q ), and A designates an
online learning algorithm that produces hypothesis ht ? H when it encounters a new example Zt .
Finally, ` = (`q|p )1?p,q?Q is a family of class-dependent loss functionals `q|p : H ? X ? R+ . For
a point x ? X of class y ? Y, `q|y (h, x) is the cost of h?s favoring class q over y for x.
Example 1 (Misclassification Loss). The family of (cost-sensitive) misclassification losses `misclass
is defined as
.
`misclass
(h, x) = ?[h(x)=q] Cyq ,
(1)
q|y
where Cpq ? R+ , ?p, q ? Y and ?[E] = 1 if E is true and 0 otherwise.
Example 2 (Hinge Loss). P
The family of multiclass hinge losses `hinge is such that, given W =
Q
{w1 , . . . , wQ } ? X with wq = 0 and hypothesis hW such that hW (x) = [hw1 , xi ? ? ? hwQ , xi]
1
.
hinge
`q|y (hW , x) = hwq , xi +
, where | ? |+ = max(0, ?).
(2)
Q ? 1 +
2.2
From the Confusion Matrix to the Confusion Risk, and their Minimization
Confusion matrix. In many situations, e.g. class-imbalanced datasets, it is important not to measure the quality of a predictor h based upon its classification rate
.
R(h) = PXY (h(X) 6= Y )
(3)
only; this may lead to erroneous conclusions regarding the quality of h. Indeed, if, for instance,
some class q is predominantly present in the data at hand, say P(Y = q) = 1 ? ?, for some small
? > 0, then the predictor hmaj that always outputs hmaj (x) = q regardless of x has a classification
error lower that is at most ?, whereas it never correctly predicts the class of data from classes p 6= q.
A more informative object is the confusion matrix C(h) ? RQ?Q of h, which is defined as:
.
C(h) = (P(h(X) = q)|Y = p)1?p,q?Q .
(4)
The nondiagonal entries of C(h) provides the information as to the types of confusion, and their
prevalence, h makes when predicting the class of some x. Let us now abuse the notation and denote
C(h) the confusion matrix where the diagonal entries are zeroed. It is straightforward to see, that, if
? = [P(Y = 1) ? ? ? P(Y = Q)] is the vector of class prior distributions then, the misclassification
rate R(h) (cf. (3)) can be retrieved as
R(h) = k?C(h)k1 ,
2
where k ? k1 denotes the 1-norm. This says that, with little additional information, the misclassification rate might be obtained from the confusion matrix, while the converse is not true.
Is is clear that having the confusion matrix C(h) be zero means that h is a perfect predictor. When
such situation is not possible (if the data are corrupted by noise, for instance), a valuable objective
might be to look for a classifier h having a confusion matrix as close to zero as possible. Here, we
choose to measure the closeness to zero of matrices through the operator norm k ? k, defined as:
kM vk2
.
kM k = max
,
v6=0 kvk2
(5)
where k ? k2 is the standard Euclidean norm ?kM k is merely the largest singular value of M . In
addition to being a valid norm, the operator norm has the following nice property, that will be of
help to us. Given two matrices A = (aij ) and B = (bij ) of the same dimensions
0 ? aij ? bij , ?i, j ? kAk ? kBk.
(6)
Given the equivalence between norms and the definition of the operator norm, it is easy to see that
p
R(h) ? QkC(h)k,
and targetting a small confusion matrix for h may have the consequence of implying a small misclassification risk R(h).
The discussion conducted so far brings us to a natural goal of multiclass learning, that of minimizing
the norm of the confusion matrix, i.e. that of solving the following optimization problem
min kC(h)k.
h?H
However, addressing this question directly poses a number of problems, both from the statistical and
algorithmic sides: a) it is not possible to compute C(h) as it depends on the unknown distribution D
and b) relying on empirical estimates of C(h) as is folk in statistical learning, requires to deal with
the indicator function ?[] that appears in (1), which is not optimization-friendly.
Confusion Risk. In order to deal with the latter problem and prepare the ground for tackling the
former one from a theoretical point of view, we now introduce and discuss the confusion risk, which
is parameterized by a family of loss functions that may be surrogate for the indicator loss ?[] .
Definition 1 (Confusion Risk). The confusion risk C` (h) of h is defined as
EX|p `q|p (h, X) if p 6= q,
.
C` (h) = c`pq 1?p,q?Q ? RQ?Q , with c`pq =
0
otherwise.
(7)
Observe that if the family `misclass of losses from Example 1 is retained, and Cpq = 1 for all p, q then
C`misclass (h) is precisely the confusion matrix discussed above (with the diagonal set to zero).
?
.
Similarly as before, the `-risk R` (h) is defined as R` (h) = k?C` (h)k1 and R` (h) ? QkC`(h) k.
The following lemma directly comes from Equation (6).
Lemma 1. Let h ? H. If 0 ? ?[h(x)6=p] ? `q|p (h, x) , ?x ? X , ?p, q ? Y, then kC(h)k ? kC` (h)k.
This says that if we recourse to surrogate ` that upper bounds the 0-1 indicator function then the
norm of the confusion risk is always larger than the norm of the confusion matrix.
Armed with the confusion risk and the last lemma, we may now turn towards the legitimate objective
min kC` (h)k,
h?H
a small value of kC` (h)k implying a small value of kC(h)k (which was our primary goal).
It is still impossible to solve this problem because of the unknown distribution D according to which
expectations are computed. However, as already evoked, it is possible to derive learning strategies
based on empirical estimates of the confusion risk, that ensures kC` (h)k will be small. The next
section is devoted to show how this could be done.
3
3
3.1
Bounds on the Confusion Risk via Online Learning and COPA
(Empirical) Online Confusion Risk
Assume an online learning algorithm A that outputs hypotheses from a family H: run on the traning
sequence Z ? DT , A outputs hypotheses h = {h0 , . . . , hT }, where h0 is an arbitrary hypothesis.
Definition 2 (Lb|p (?, ?) and L|p (?) matrices). Let ` = (`q|p ) be a family of losses and let p ? Y.
For h ? H and x ? X , we define
`v|u (h, x) if u = p and v 6= u,
.
Q?Q
b
b
b
L|p (h, x) = (luv )1?u,v?Q ? R
, with luv =
(8)
0
otherwise.
The matrix L|p (h) ? RQ?Q is naturally given by
L|p (h) = EX|p Lb|p (h, X).
(9)
Remark 1. Note that the only row that may be nonzero in Lb|p (h, x) and L|p (h) is the pth row.
Definition 3 ((Conditional) Online Confusion Risk). Let y ? Y T be a fixed sequence of labels and
Y a random sequence of T labels. Let h = {h0 , . . . , hT ?1 } be a sequence of T hypotheses.
The conditional and non-conditional confusion risks C`,y (h) and C`,Y (h) are defined as
T
. X 1
C`,y (h) =
L|yt (ht?1 ),
T
t=1 yt
T
. X 1
and, C`,Y (h) =
L|Yt (ht?1 ).
T
t=1 Yt
(10)
Remark 2. C`,Y (h) is a random variable. In order to provide our generalization bounds, it will be
more convenient for us to work with the conditional confusion C`,y (h). A simple argument will
then allow us to provide generalization bounds for C`,Y (h).
Definition 4. Let y ? Y T be a fixed sequence of labels. Let h = {h0 , . . . , hT ?1 } be the hypotheses
output by A when run on Z y = {(Xt , yt )}Tt=1 , such that Xt is distributed according to DX|yt .
The (non-)conditional empirical online confusion risks Cb`,y (h, X) and Cb`,Y (h, X) are
T
. X 1 b
Cb`,y (h, X) =
L|yt (ht?1 , Xt ),
T
t=1 yt
T
. X 1 b
and, Cb`,Y (h, X) =
L|Yt (ht?1 , Xt ).
T
t=1 Yt
(11)
We now are almost ready to provide our results. We just need to introduce a pivotal result that is a
corollary of Theorem 7.1 from [8], a proof of which can be found in the appendix.
Corollary 1 (Rectangular Matrix Azuma). Consider a sequence {Uk } of d1 ? d2 random matrices,
and a fixed sequence of scalars {Mk } such that
EUk |U1 ...Uk?1 Uk = 0, and kUk k ? Mk almost surely.
Then, for all t > 0,
n
X
o
t2
P
Uk
? t ? (d1 + d2) exp ? 2 ,
k
2?
3.2
. X 2
with ? 2 =
Mk .
k
New Results
This subsection reports our main results.
Theorem 1. Suppose that the losses in ` are such that 0 ? `q|p ? M for some M > 0. Fix y ? Y T .
For any ? ? (0; 1], it holds with probability 1 ? ? over the draw of X ? DX|y that
s
X 1
Q
b
log
(12)
C`,y (h) ? C`,y (h, X)
? M 2Q
p Tp
?
where h = {h0 , . . . , hT ?1 } is the set of hypotheses output by A when provided with {(Xt , yt )}Tt=1 .
4
Therefore, with probability at least 1 ? ?
kC`,y (h)k ?
Cb`,y (h, X)
+ M
s
2Q
X
p
1
Q
log .
Tp
?
(13)
Proof. The proof is straightforward using Corollary 1. Indeed, consider the random variable
1 b
. 1
L|yt (ht?1 ) ?
Ut =
L|yt (ht?1 , Xt ).
Tyt
Tyt
On the one hand, we observe that:
EXt |X 1:t?1 ,y Ut = EXt |X 1:t?1 ,y1:t Ut = 0,
since EXt |X 1:t?1 y1:t Lb|yt (ht?1 , Xt ) = L|yt (ht?1 ). On the other hand, introducing
.
?t,q = EXt |yt `q|yt (ht?1 , Xt ) ? `q|yt (ht?1 , Xt ),
we observe that
1
kUt k = sup kUt vk2 =
T
yt
v:kvk?1
sX
Mp
1
?2t,q ?
?t,q vq =
Q.
q6=yt
Tyt
Tyt
X
sup
v:kvk?1
q6=yt
where we used that the only row of Ut not to be zero is its yt th row (see Remark 1), the fact that
supv:kvk?1 v ? u = kuk2 and the assumption 0 ? `q|p ? M , which gives that |?t,q | ? M .
?
Using Corollary 1 on the matrix martingale {Ut }, where kUt k ? M Q/Tyt almost surely, gives
(
)
o
n
X
?2
P
Ut
? ? ? 2Q exp ?
P
t
2M 2 Q t T12
yt
Setting the right hand side to ? gives that, with probability at least 1 ? ?
s
X
X 1
2Q
Ut
? M 2Q
ln
.
t
t Ty2
?
t
P ?2 P ?1
gives (12). The triangle inequality |kAk ? kBk| ? kA ? Bk gives (13).
t Tyt =
p Tp
If one takes a little step back to fully understand Theorem 1, it may not be as rejoicing as expected.
Indeed, it provides a bound on the norm of the average confusion risks of hypotheses h0 , . . . , hT ?1 ,
which, from a practical point of view, does not say much about the norm of the confusion risk
of a specific hypothesis. In fact, as is usual in online learning [1], it provides a bound on the
confusion risk of the Gibbs classifier, which uniformly samples over h0 , . . . , hT ?1 before outputting
a prediction. Just as emphasized by [1], things may turn a little bit more enticing when the loss
functions ` are convex with respect to their first argument, i.e.
?h, h0 ? H, ?p, q ? Y, ?? ? [0, 1], `q|p (?h + (1 ? ?)h0 , x) ? ?`q|p (h) + (1 ? ?)`q|p (h0 , x). (14)
In that case, we may show the following theorem, that is much more compatible with the stated goal
of trying to find a hypothesis that has small (or at least, controlled) confusion risk.
Theorem 2. In addition to the assumptions of Theorem 1 we now assume that ` is made of convex
losses (as defined in (14)).
For any ? ? (0; 1], it holds with probability 1 ? ? over the draw of X ? DX|y that
s
T
X 1
Q
. 1X
b
log , with h =
ht?1 .
kC` (h)k ?
C`,y (h, X)
+ M 2Q
p Tp
?
T t=1
Proof. It is a direct consequence of the convexity of ` combined with Equation (6).
5
(15)
It is now time to give the argument allowing us to state results for the non-conditional online
confusion risks. If a random event E(A, B) defined with respect to random variables A and
B
1 ? ? for all possible values of b then PAB (E(A, B)) =
P is such that PA|B=b (E(A, b)) ? P
P
(E(A,
b))P
(B
=
b)
?
B
A|B=b
b
b (1 ? ?)PB (B = b) = 1 ? ?. The results of Theorem 1
and Theorem 2 may be therefore stated in terms of Y instead of y.
In light of the generic results of this subsection, we are now ready to motivate and derive a new
online learning algorithm that aims at a small confusion risk.
3.3
Online Learning with COPA
This subsection presents a new online algorithm, COPA (for COnfusion Passive-Aggressive learning). Before giving the full detail of our algorithm, we further discuss implications of Theorem 2.
A first message from Theorem 2 is that, provided the functions ` considered are convex, it is relevant
to use the average hypothesis h as a predictor. We indeed know by (15) that the confusion risk of h
is bounded by kCb`,y (h, X)k, plus some quantity directly related to the number of training data encountered. The second message from the theorem is that the focus naturally comes to kCb`,y (h, X)k
and the question as to how minimize this quantity.
According to Definition 4, Cb`,y (h, X) is the sum of instantaneous confusion matrices
Lb|yt (ht?1 , Xt )/Tyt . In light of (6), it does make sense to try to minimize each entry of
Lb|yt (h, Xt )/Tyt with respect to h to get ht , with the hope that the instantaneous risk of ht on
Xt+1 will be small: one may want to minimize the norm of Lb|yt (h, Xt )/Tyt and pose a problem
like the following:
1
b|y (h, Xt )
.
min
L
t
h
Tyt
However, as remarked before, Lb|yt has only one nonzero row, its yt th. Therefore, the operator norm
of Lb|yt (h, Xt )/Tyt is simply the Euclidean norm of its yt th row. Trying to minimize the square
Euclidean norm of this row might be written as
1 X 2
min 2
`q|yt (h, Xt ).
(16)
h Ty
t
q6=yt
This last equation is the starting point of COPA. To see how the connection is made, we make some
instantiations. The hypothesis
P space H is made of linear classifiers, so that a sequence of vectors
W = {w1 , . . . , wQ } with q wq = 0 defines a hypothesis hW . The family ` COPA builds upon is
1
, ?p, q ? Y.
`q|p (hW , x) = hwq , xi +
Q ? 1 +
In other words, COPA is an instance of Example 2. We focus on this setting because it is known that,
in the batch scenario, it provides Bayes consistent classifiers [3, 4]. Given a current set of vectors
t
Wt = {w1t , . . . , wQ
}, using (16), and implementing a passive-aggressive learning strategy [2], the
new set of vectors Wt+1 is searched as the solution of
2
Q
1 X
C X
1
t
2
min
wq ? wq 2 +
(17)
P
hwq , xi + Q ? 1 .
2Ty2
W, q wq =0 2
+
q=1
q6=y
It turns out that it is possible to find the solution of this optimization problem without having to
recourse to any involved optimization procedure. This is akin to a result obtained by [6], which
applies for another family of loss functions. We indeed prove the following theorem (proof in
supplementary material).
Theorem 3 (COPA update). Suppose we want to solve
2
Q
1 X
C X
1
t
2
min
wq ? wq 2 +
(18)
P
hwq , xi + Q ? 1 .
2
W, q wq =0 2
+
q=1
q6=y
6
Algorithm 1 COPA
Input: z = {(xt , yt )}Tt=1 training set (realization of Z), R number of epochs over z, C cost
Output: W = {w1 , . . . , wQ }, the classification vectors
? =0
0
w10 = . . . = wQ
for r=1 to R do
for t=1 to T do
receive (xt , yt )
compute ?? according to (20)
?q, perform the update: wq? +1 ? wq? ? ?q? ?
? ?? +1
end for
end for
P?
?q, wq ? ?1 k=1 wqk
Let `tq be defined as
1
Q
PI ?
q=1
?q? xt
.
`tq = hwq , xi + 1/(Q ? 1).
Let ? be a permutation defined over {1, . . . , Q ? 1} taking values in Y\{y} such that
`t?(1) ? . . . ? `t?(Q?1) .
Let I ? be the largest index I ? {1, . . . , Q ? 1} such that
`t?(I) +
I?1
X
kxk2
`t
> 0,
?Q ? (I ? 1)kxk2 q=1 ?(q)
. 1
with ? =
+ kxk2
C
.
?
If I ? is set to I ? = {?(1), . . . , ?(I ? )}, then we may define ?? = [?1? ? ? ? ?Q
] as
?
? 1 t kxk2
X
Q
.
`q +
s? (I ? )
if q ? I ?
? .
, where s? (I ? ) =
`tq .
?q =
?
Q
?
2
?
?Q ? I kxk
0
otherwise
q?I ?
(19)
(20)
and the vectors
.
wq? = wqt ?
I?
1 X ?
?
?q? ?
Q q=1 q
!
x, q = 1, . . . , Q
(21)
are the solution of optimization problem (18). These equations provide COPA?s update procedure.
The full COPA algorithm is depicted in Algorithm 1.
4
Numerical Simulations
We here report results of preliminary simulations of COPA on a toy dataset. We generate a set of
5000 samples according to three Gaussian distributions each of variance ? 2 I with ? = 0.5. One
of the Gaussian is centered at (0, 1), the other at (1, 0) and the last one at (?1, 0). The respective
weights of the Gaussian distributions are 0.9, 0.05 and 0.05. The first generated sample is used
to choose the parameter C of COPA with a half split of the data between train and test; 10 other
samples of size 5000 are generated and split as 2500 for learning and 2500 for testing and the results
are averaged on the 10 samples. We compare the results of COPA to that of a simple multiclass
perceptron procedure (the number of epochs for each algorithm is set to 5). As recommended by the
theory we average the classification vector to get our predictors (both for COPA and the perceptron).
The essential finding of these simulations is that COPA and the perceptron achieve similar rate of
classification accuracy, 0.85 and 0.86, respectively but the norm of the confusion matrix of COPA is
0.10 and that of the Perceptron is 0.18. This means COPA indeed does its job in trying to minimize
the confusion risk.
7
5
Conclusion
In this paper, we have provided new bounds for online learning algorithms aiming at controlling
their confusion risk. To the best of our knowledge, these results are the first of this kind. Motivated
by the theoretical results, we have proposed a passive-aggressive learning algorithm, COPA, that
has the seducing property that its updates can be computed easily, without having to resort to any
optimization package. Preliminary numerical simulations tend to support the effectiveness of COPA.
In addition to complementary numerical simulations, we want to pursue our work in several directions. First, we would like to know whether efficient update equation can be derived if a simple
hinge, instead of a square hinge is used. Second, we would like to see if a full regret-style analysis
can be made to study the properties of COPA. Also, we are interested in comparing the merits of our
theoretical results with those recently proposed in [5] and [7], which propose to address learning
with the confusion matrix from the algorithmic stability and the PAC-Bayesian viewpoints. Finally,
we would like to see how the proposed material can be of some use in the realm of structure prediction and by extension, in the case where the confusion matrix to consider is infinite-dimensional.
Acknowledgments. Work partially supported by Pascal 2 NoE ICT-216886-NOE, French ANR
Projects ASAP (ANR-09-DEFIS?001) and GRETA (ANR-12-BS02-0004).
Appendix
Theorem 4 (Matrix Azuma-Hoeffding, [8]). Consider a finite sequence {Xk } of self-adjoint matrices in dimension d, and a fixed sequence {Ak } of self-adjoint matrices that satisfy Ek?1 Xk = 0
and
Xk2 4 A2k , and Ak Xk = Xk Ak , almost surely.
Then, for all t ? 0,
n
X
o
2
2
P ?max
Xk ? t ? d ? e?t /2? ,
k
P
with ? =
k A2k
.
2
Proof of Corollary 1. To prove the result, it suffices to make use of the dilation technique and apply
Theorem 4. The self-adjoint dilation D(U ) of a matrix U ? Rd1 ?d2 is the self-adjoint matrix D(U )
of order d1 + d2 defined by
0 U
D(U ) =
U? 0
where U ? is the adjoint of U (as U has only real coefficient here, U ? is the transpose of U ).
As recalled in [8], kD(U )k = kU k and, therefore, the largest eigenvalue ?max of D2 (U ) is equal to
kU k2 and D2 (U ) 4 kU k2 I.
Considering Uk , we get that, almost surely:
D2 (Uk ) 4 Mk2 I,
and since dilation is a linear operator, we have that
EUk |U1 ???Uk?1 D(Uk ) = 0.
The sequence of matrices {D(Uk )} is therefore a matrix martingale that verifies the assumption of
Theorem 4, the application of which gives
n
X
o
t2
P ?max
D(Uk ) ? t ? (d1 + d2 ) exp ? 2 ,
k
2?
P
with ? 2 = k Mk2 . Thanks to the linearity of D,
X
X
X
?max
D(Uk ) = ?max D
Uk
=
Uk
,
k
k
which closes the proof.
8
k
References
[1] N. Cesa-Bianchi, A. Conconi, and C. Gentile. On the generalization ability of online learning
algorithms. IEEE Transactions on Information Theory, 50(9):2050?2057, 2004.
[2] Koby Crammer, Ofer Dekel, Joseph Keshet, Shai Shalev-Shwartz, and Yoram Singer. Online
passive-aggressive algorithms. Journal of Machine Learning Research, 7:551?585, 2006.
[3] Y. Lee. Multicategory support vector machines, theory, and application to the classification of
microarray data and satellite radiance data. Technical report, University of Wisconsin, 2002.
[4] Y. Lee, Y. Lin, and G. Wahba. Multicategory support vector machines. Journal of the American
Statistical Association, 99:67?81, march 2004.
[5] P. Machart and L. Ralaivola. Confusion matrix stability bounds for multiclass classification.
Technical report, Aix-Marseille University, 2012.
[6] S. Matsushima, N. Shimizu, K. Yoshida, T. Ninomiya, and H. Nakagawa. Exact passiveaggressive algorithm for multiclass classification using support class. In SDM 10, pages 303?
314, 2010.
[7] E. Morvant, S. Koc?o, and L. Ralaivola. PAC-Bayesian Generalization Bound on Confusion
Matrix for Multi-Class Classification. In John Langford and Joelle Pineau, editors, International
Conference on Machine Learning, pages 815?822, Edinburgh, United Kingdom, 2012.
[8] J. A. Tropp. User-friendly tail bounds for sums of random matrices. Foundations of Computational Mathematics, pages 1?46, 2011.
9
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4,062 | 4,677 | Augment-and-Conquer Negative Binomial Processes
Lawrence Carin
Dept. of Electrical and Computer Engineering
Duke University, Durham, NC 27708
[email protected]
Mingyuan Zhou
Dept. of Electrical and Computer Engineering
Duke University, Durham, NC 27708
[email protected]
Abstract
By developing data augmentation methods unique to the negative binomial (NB)
distribution, we unite seemingly disjoint count and mixture models under the NB
process framework. We develop fundamental properties of the models and derive
efficient Gibbs sampling inference. We show that the gamma-NB process can
be reduced to the hierarchical Dirichlet process with normalization, highlighting
its unique theoretical, structural and computational advantages. A variety of NB
processes with distinct sharing mechanisms are constructed and applied to topic
modeling, with connections to existing algorithms, showing the importance of
inferring both the NB dispersion and probability parameters.
1
Introduction
There has been increasing interest in count modeling using the Poisson process, geometric process
[1, 2, 3, 4] and recently the negative binomial (NB) process [5, 6]. Notably, it has been independently
shown in [5] and [6] that the NB process, originally constructed for count analysis, can be naturally
applied for mixture modeling of grouped data x1 , ? ? ? , xJ , where each group xj = {xji }i=1,Nj .
For a territory long occupied by the hierarchical Dirichlet process (HDP) [7] and related models,
the inference of which may require substantial bookkeeping and suffer from slow convergence [7],
the discovery of the NB process for mixture modeling can be significant. As the seemingly distinct
problems of count and mixture modeling are united under the NB process framework, new opportunities emerge for better data fitting, more efficient inference and more flexible model constructions.
However, neither [5] nor [6] explore the properties of the NB distribution deep enough to achieve
fully tractable closed-form inference. Of particular concern is the NB dispersion parameter, which
was simply fixed or empirically set [6], or inferred with a Metropolis-Hastings algorithm [5]. Under
these limitations, both papers fail to reveal the connections of the NB process to the HDP, and thus
may lead to false assessments on comparing their modeling abilities.
We perform joint count and mixture modeling under the NB process framework, using completely
random measures [1, 8, 9] that are simple to construct and amenable for posterior computation.
We propose to augment-and-conquer the NB process: by ?augmenting? a NB process into both
the gamma-Poisson and compound Poisson representations, we ?conquer? the unification of count
and mixture modeling, the analysis of fundamental model properties, and the derivation of efficient
Gibbs sampling inference. We make two additional contributions: 1) we construct a gamma-NB
process, analyze its properties and show how its normalization leads to the HDP, highlighting its
unique theoretical, structural and computational advantages relative to the HDP. 2) We show that
a variety of NB processes can be constructed with distinct model properties, for which the shared
random measure can be selected from completely random measures such as the gamma, beta, and
beta-Bernoulli processes; we compare their performance on topic modeling, a typical example for
mixture modeling of grouped data, and show the importance of inferring both the NB dispersion and
probability parameters, which respectively govern the overdispersion level and the variance-to-mean
ratio in count modeling.
1
1.1 Poisson process for count and mixture modeling
Before introducing the NB process, we first illustrate how the seemingly distinct problems of count
and mixture modeling can be united under the Poisson process. Denote ? as a measure space and
for each Borel set A ? ?, denote Xj (A) as a count random variable describing the number of observations in xj that reside within A. Given grouped data x1 , ? ? ? , xJ , for any measurable disjoint partition A1 , ? ? ? , AQ of ?, we aim to jointly model the count random variables {Xj (Aq )}. A natural
choice would be to define a Poisson process Xj ? PP(G), with a shared completely random mea
PQ
sure G on ?, such that Xj (A) ? Pois G(A) for each A ? ?. Denote G(?) = q=1 G(Aq ) and
e = G/G(?). Following Lemma 4.1 of [5], the joint distributions of Xj (?), Xj (A1 ), ? ? ? , Xj (AQ )
G
are equivalent under the following two expressions:
PQ
Xj (?) = q=1 Xj (Aq ),
Xj (Aq ) ? Pois G(Aq ) ;
(1)
e 1 ), ? ? ? , G(A
e Q ) . (2)
Xj (?) ? Poisson(G(?)), [Xj (A1 ), ? ? ? , Xj (Aq )] ? Mult Xj (?); G(A
Thus the Poisson process provides not only a way to generate independent counts from each Aq ,
but also a mechanism for mixture modeling, which allocates the observations into any measurable
e
disjoint partition {Aq }1,Q of ?, conditioning on Xj (?) and the normalized mean measure G.
To complete the model, we may place a gamma process [9] prior on the shared measure as
G ? GaP(c, G0 ), with concentration parameter c and base measure G0 , such that G(A) ?
Gamma(G0 (A), 1/c) for each A ? ?, where G0 can be continuous, discrete or a combination of
e = G/G(?) now becomes a Dirichlet process (DP) as G
e ? DP(?0 , G
e 0 ), where
both. Note that G
e
?0 = G0 (?) and G0 = G0 /?0 . The normalized gamma representation of the DP is discussed in
[10, 11, 9] and has been used to construct the group-level DPs for an HDP [12]. The Poisson process
has an equal-dispersion assumption for count modeling. As shown in (2), the construction of Poisson
processes with a shared gamma process mean measure implies the same mixture proportions across
groups, which is essentially the same as the DP when used for mixture modeling when the total
counts {Xj (?)}j are not treated as random variables. This motivates us to consider adding an additional layer or using a different distribution other than the Poisson to model the counts. As shown
below, the NB distribution is an ideal candidate, not only because it allows overdispersion, but also
because it can be augmented into both a gamma-Poisson and a compound Poisson representations.
2
Augment-and-Conquer the Negative Binomial Distribution
The NB distribution m ? NB(r, p) has the probability mass function (PMF) fM (m) = ?(r+m)
m!?(r) (1 ?
p)r pm . It has a mean ? = rp/(1?p) smaller than the variance ? 2 = rp/(1 ? p)2 = ?+r?1 ?2 , with
the variance-to-mean ratio (VMR) as (1?p)?1 and the overdispersion level (ODL, the coefficient of
the quadratic term in ? 2 ) as r?1 . It has been widely investigated and applied to numerous scientific
studies [13, 14, 15]. The NB distribution can be augmented into a gamma-Poisson construction as
m ? Pois(?), ? ? Gamma (r, p/(1 ? p)), where the gamma distribution is parameterized by its
shape r and scale p/(1 ? p). It can also be augmented under a compound Poisson representation
Pl
[16] as m = t=1 ut , ut ? Log(p), l ? Pois(?r ln(1 ? p)), where u ? Log(p) is the logarithmic
distribution [17] with probability-generating function (PGF) CU (z) = ln(1 ? pz)/ln(1 ? p), |z| <
p?1 . In a slight abuse of notation, but for added conciseness, in the following discussion we use
Pl
Pl
m ? t=1 Log(p) to denote m = t=1 ut , ut ? Log(p).
The inference of the NB dispersion parameter r has long been a challenge [13, 18, 19]. In this paper,
we first place a gamma prior on it as r ? Gamma(r1 , 1/c1 ). We then use Lemma 2.1 (below) to
infer a latent count l for each m ? NB(r, p) conditioning on m and r. Since l ? Pois(?r ln(1 ? p))
by construction, we can use the gamma Poisson conjugacy to update r. Using Lemma 2.2 (below),
we can further infer an augmented latent count l0 for each l, and then use these latent counts to
update r1 , assuming r1 ? Gamma(r2 , 1/c2 ). Using Lemmas 2.1 and 2.2, we can continue this
process repeatedly, suggesting that we may build a NB process to model data that have subgroups
within groups. The conditional posterior of the latent count l was first derived by us but was not
given an analytical form [20]. Below we explicitly derive the PMF of l, shown in (3), and find that
it exactly represents the distribution of the random number of tables occupied by m customers in a
Chinese restaurant process with concentration parameter r [21, 22, 7]. We denote l ? CRT(m, r)
as a Chinese restaurant table (CRT) count random
variable with such a PMF and as proved in the
Pm
supplementary material, we can sample it as l = n=1 bn , bn ? Bernoulli (r/(n ? 1 + r)).
2
Both the gamma-Poisson and compound-Poisson augmentations of the NB distribution and Lemmas
2.1 and 2.2 are key ingredients of this paper. We will show that these augment-and-concur methods
not only unite count and mixture modeling and provide efficient inference, but also, as shown in
Section 3, let us examine the posteriors to understand fundamental properties of the NB processes,
clearly revealing connections to previous nonparametric Bayesian mixture models.
Lemma 2.1. Denote s(m, j) as Stirling numbers of the first kind [17]. Augment m ? NB(r, p)
Pl
under the compound Poisson representation as m ? t=1 Log(p), l ? Pois(?r ln(1 ? p)), then
the conditional posterior of l has PMF
?(r)
Pr(l = j|m, r) = ?(m+r)
|s(m, j)|rj , j = 0, 1, ? ? ? , m.
(3)
Pj
Proof. Denote wj ?
t=1 Log(p), j = 1, ? ? ? , m. Since wj is the summation of j iid Log(p)
j
random variables, the PGF of wj becomes CWj (z) = CUj (z) = [ln(1 ? pz)/ln(1 ? p)] , |z| <
n
P
?
s(n,j)x
p?1 . Using the property that [ln(1 + x)]j = j! n=j
[17], we have Pr(wj = m) =
n!
(m)
CWj (0)/m! = (?1)m pj j!s(m, j)/(m![ln(1 ? p)]j ). Thus for 0 ? j ? m, we have Pr(L =
Pm
j|m, r) ? Pr(wj = m)Pois(j; ?r ln(1?p)) ? |s(m, j)|rj . Denote Sr (m) = j=0 |s(m, j)|rj , we
Qm?1
Qm?1
have Sr (m) = (m?1+r)Sr (m?1) = ? ? ? = n=1 (r +n)Sr (1) = n=0 (r +n) = ?(m+r)
?(r) .
ln(1?p)
, then m can also
Lemma 2.2. Let m ? NB(r, p), r ? Gamma(r1 , 1/c1 ), denote p0 = c?
1 ?ln(1?p)
be generated from a compound distribution as
Pl
Pl0
m ? t=1 Log(p), l ? t0 =1 Log(p0 ), l0 ? Pois(?r1 ln(1 ? p0 )).
(4)
Pl
Proof. Augmenting m leads to m ? t=1 Log(p), l ? Pois(?r ln(1 ? p)). Marginalizing out r
leads to l ? NB (r1 , p0 ). Augmenting l using its compound Poisson representation leads to (4).
3
Gamma-Negative Binomial Process
We explore sharing the NB dispersion across groups while the probability parameters are group
dependent. We define a NB process X ? NBP(G, p) as X(A) ? NB(G(A), p) for each A ? ? and
construct a gamma-NB process for joint count and mixture modeling as Xj ? NBP(G, pj ), G ?
GaP(c, G0 ), which can be augmented as a gamma-gamma-Poisson process as
Xj ? PP(?j ), ?j ? GaP((1 ? pj )/pj , G), G ? GaP(c, G0 ).
(5)
In the above PP(?) and GaP(?) represent the Poisson and gamma processes, respectively, as defined
in Section 1.1. Using Lemma 2.2, the gamma-NB process can also be augmented as
PLj
Xj ? t=1
Log(pj ), Lj ? PP(?G ln(1 ? pj )), G ? GaP(c, G0 );
(6)
P
P
PL0
?
ln(1?p
)
j
L = j Lj ? t=1 Log(p0 ), L0 ? PP(?G0 ln(1 ? p0 )), p0 = c?Pj ln(1?pj ) .
(7)
j
These three augmentations allow us to derive a sequence of closed-form update equations for inference with the gamma-NB process. Using the gamma Poisson conjugacy on (5), for each A ? ?, we
have ?j (A)|G, Xj , pj ? Gamma (G(A) + Xj (A), pj ), thus the conditional posterior of ?j is
?j |G, Xj , pj ? GaP 1/pj , G + Xj .
(8)
P
Define T ? CRTP(X, G) as a CRT process that T (A) = ??A T (?), T (?) ? CRT(X(?), G(?))
for each A ? ?. Applying Lemma 2.1 on (6) and (7), we have
Lj |Xj , G ? CRTP(Xj , G), L0 |L, G0 ? CRTP(L, G0 ).
(9)
If G0 is a continuous base measure and ?0 = G0 (?) is finite, we have G0 (?)? 0 ? ? ? ? and thus
P
P
P
L0 (?)|L, G0 = ??? ?(L(?) > 0) = ??? ?( j Xj (?) > 0)
(10)
which is equal to K + , the total number of used discrete atoms; if G0 is discrete as G0 =
P
P K ?0
?0
0
0
+
j Xj (?k ) > 0, thus L (?) ? K . In
k=1 K ??k , then L (?k ) = CRT(L(?k ), K ) ? 1 if
either case, let ?0 ? Gamma(e0 , 1/f0 ), with the gamma Poisson conjugacy on (6) and (7), we have
1
?0 |{L0 (?), p0 } ? Gamma e0 + L0 (?), f0 ?ln(1?p
(11)
0) ;
P
P
G|G0 , {Lj , pj } ? GaP c ? j ln(1 ? pj ), G0 + j Lj .
(12)
Since the data {xji }i are exchangeable within group j, the predictive distribution of a point Xji ,
conditioning on Xj?i = {Xjn }n:n6=i and G, with ?j marginalized out, can be expressed as
Xji |G, Xj?i ?
E[?j |G,Xj?i ]
E[?j (?)|G,Xj?i ]
=
G
G(?)+Xj (?)?1
3
+
Xj?i
G(?)+Xj (?)?1 .
(13)
3.1 Relationship with the hierarchical Dirichlet process
Using the equivalence between (1) and (2) and normalizing all the gamma processes in (5), denoting
e j = ?j /?j (?), ? = G(?), G
e = G/?, ?0 = G0 (?) and G
e 0 = G0 /?0 , we can re-express (5) as
?
ej , ?
e j ? DP(?, G),
e ? ? Gamma(?0 , 1/c), G
e ? DP(?0 , G
e0 )
Xji ? ?
(14)
which is an HDP [7]. Thus the normalized gamma-NB process leads to an HDP, yet we cannot return from the HDP to the gamma-NB process without modeling Xj (?) and ?j (?) as random variables. Theoretically, they are distinct in that the gamma-NB process is a completely
random measure, assigning independent random variables into any disjoint Borel sets {Aq }1,Q
of ?; whereas the HDP is not. Practically, the gamma-NB process can exploit conjugacy to
achieve analytical conditional posteriors for all latent parameters. The inference of the HDP is
a major challenge and it is usually solved through alternative constructions such as the Chinese
restaurant franchise (CRF) and stick-breaking representations [7, 23]. In particular, without analytical conditional posteriors, the inference of concentration parameters ? and ?0 is nontrivial
[7, 24] and they are often simply fixed [23]. Under the CRF metaphor ? governs the random
number of tables occupied by customers in each restaurant independently; further, if the base
e 0 is continuous, ?0 governs the random number of dishes selected by taprobability measure G
bles of all restaurants. One may apply the data augmentation method of [22] to sample ? and ?0 .
e 0 is discrete as G
e 0 = PK 1 ?? , which is of practical value and becomes a conHowever, if G
k
k=1 K
tinuous base measure as K ? ? [11, 7, 24], then using the method of [22] to sample ?0 is only
approximately correct, which may result in a biased estimate in practice, especially if K is not large
enough. By contrast, in the gamma-NB process, the shared gamma process G can be analytically
updated with (12) and G(?) plays the role of ? in the HDP, which is readily available as
P
1
G(?)|G0 , {Lj , pj }j=1,N ? Gamma ?0 + j Lj (?), c?P ln(1?p
(15)
)
j
j
and as in (11), regardless of whether the base measure is continuous, the total mass ?0 has an analytical gamma posterior whose shape parameter is governed by L0 (?), with L0 (?) = K + if G0 is conPK
tinuous and finite and L0 (?) ? K + if G0 = k=1 ?K0 ??k . Equation (15) also intuitively shows how
e j } in the gamma-NB process.
the NB probability parameters {pj } govern the variations among {?
In the HDP, pj is not explicitly modeled, and since its value becomes irrelevant when taking the normalized constructions in (14), it is usually treated as a nuisance parameter and perceived as pj = 0.5
when needed for interpretation purpose. Fixing pj = 0.5 is also considered in [12] to construct an
HDP, whose group-level DPs are normalized from gamma processes with the scale parameters as
pj
1?pj = 1; it is also shown in [12] that improved performance can be obtained for topic modeling by
learning the scale parameters with a log Gaussian process prior. However, no analytical conditional
posteriors are provided and Gibbs sampling is not considered as a viable option [12].
3.2 Augment-and-conquer inference for joint count and mixture modeling
For a finite continuous base measure, the gamma process G ? GaP(c, G0 ) can also be defined
?1 ?cr
with its L?evy measure on a product space R+ ? ?, expressed
R R as ?(drd?) = r e drG0 (d?) [9].
+
Since the Poisson intensity ? = ?(R+ ??) = ? and
r?(drd?) is finite, a draw from this
R+ ??
P?
process can be expressed as G = k=1 rk ??k , (rk , ?k ) ? ?(drd?), ?(drd?)? + ? ?(drd?) [9].
PK
Here we consider a discrete base measure as G0 = k=1 ?K0 ??k , ?k ? g0 (?k ), then we have G =
PK
k=1 rk ??k , rk ? Gamma(?0 /K, 1/c), ?k ? g0 (?k ), which becomes a draw from the gamma
process with a continuous base measure as K ? ?. Let xji ? F (?zji ) be observation i in group j,
PNj
linked to a mixture component ?zji ? ? through a distribution F . Denote njk = i=1
?(zji = k),
we can express the gamma-NB process with the discrete base measure as
PK
?k ? g0 (?k ), Nj = k=1 njk , njk ? Pois(?jk ), ?jk ? Gamma(rk , pj /(1 ? pj ))
rk ? Gamma(?0 /K, 1/c), pj ? Beta(a0 , b0 ), ?0 ? Gamma(e0 , 1/f0 )
(16)
where marginally we have njk ? NB(rk , pj ). Using the equivalence between (1) and (2), we
can equivalently express Nj and njk in the above model as Nj ? Pois (?j ) , [nj1 , ? ? ? , njK ] ?
PK
Mult (Nj ; ?j1 /?j , ? ? ? , ?jK /?j ), where ?j =
k=1 ?jk . Since the data {xji }i=1,Nj are fully
exchangeable, rather than drawing [nj1 , ? ? ? , njK ] once, we may equivalently draw the index
zji ? Discrete (?j1 /?j , ? ? ? , ?jK /?j )
(17)
4
PNj
for each xji and then let njk = i=1
?(zji = k). This provides further insights on how the seemingly disjoint problems of count and mixture modeling are united under the NB process framework.
Following (8)-(12), the block Gibbs sampling is straightforward to write as
Q
p(?k |?) ? zji =k F (xji ; ?k )g0 (?k ), Pr(zji = k|?) ? F (xji ; ?k )?jk
P
P
?
ln(1?pj )
(pj |?) ? Beta a0 + Nj , b0 + k rk , p0 = c?Pj ln(1?pj ) , (ljk |?) ? CRT(njk , rk )
j
P
P
1
(lk0 |?) ? CRT( j ljk , ?0 /K), (?0 |?) ? Gamma e0 + k lk0 , f0 ?ln(1?p
0)
P
1
(rk |?) ? Gamma ?0 /K + j ljk , c?P ln(1?p
, (?jk |?) ? Gamma(rk + njk , pj ). (18)
j)
j
which has similar computational complexity as that of the direct assignment block Gibbs sampling
of the CRF-HDP [7, 24]. If g0 (?) is conjugate to the likelihood F (x;
p(?|?)
P?), then the posterior
P
would be analytical. Note that when K ? ?, we have (lk0 |?) = ?( j ljk > 0) = ?( j njk > 0).
Using (1) and (2) and normalizing the gamma distributions, (16) can be re-expressed as
? j ), ?
? j ? Dir(??
zji ? Discrete(?
r ), ? ? Gamma(?0 , 1/c), r? ? Dir(?0 /K, ? ? ? , ?0 /K) (19)
which loses the count modeling ability and becomes a finite representation of the HDP, the inference
of which is not conjugate and has to be solved under alternative representations [7, 24]. This also
implies that by using the Dirichlet process as the foundation, traditional mixture modeling may
discard useful count information from the beginning.
4
The Negative Binomial Process Family and Related Algorithms
The gamma-NB process shares the NB dispersion across groups. Since the NB distribution has two
adjustable parameters, we may explore alternative ideas, with the NB probability measure shared
across groups as in [6], or with both the dispersion and probability measures shared as in [5]. These
constructions are distinct from both the gamma-NB process and HDP in that ?j has space dependent
e j = ?j /?j (?) no longer follows a Dirichlet process.
scales, and thus its normalization ?
It is natural to let the probability measure be drawn from a beta process [25, 26], which can be
defined by its L?evy measure on a product space [0, 1] ? ? as ?(dpd?) = cp?1 (1 ? p)c?1 dpB0 (d?).
A draw from the beta process
P? B ? BP(c, B0 ) with concentration parameter c and base measure B0
can be expressed as B = k=1 pk ??k . A beta-NB process
6] can be constructed by letting Xj ?
P[5,
?
NBP(rj , B), with a random draw expressed as Xj =
k=1 njk ??k , njk ? NB(rj , pk ). Under
this construction, the NB probability measure is shared and the NB dispersion parameters are group
dependent. As in [5], we may also consider a marked-beta-NB1 process that both the NB probability
and dispersion measures are shared, in which each point of the beta process is marked with an
independent gamma random variable. Thus a draw from the marked-beta process
P
P?becomes (R, B) =
?
(r
,
p
)?
,
and
the
NB
process
X
?
NBP(R,
B)
becomes
X
=
k
k
?
j
j
k
k=1 njk ??k , njk ?
k=1
NB(rk , pk ). Since the beta and NB processes are conjugate, the posterior of B is tractable, as shown
in [5, 6]. If it is believed that there are excessive number of zeros, governed by a process other
than the NB process, we may introduce a zero inflated NB process as XP
j ? NBP(RZj , pj ), where
?
Zj ? BeP(B) is drawn from the Bernoulli process [26] and (R, B) = k=1 (rk , ?k )??k is drawn
from a marked-beta process, thus njk ? NB(rk bjk , pj ), bjk = Bernoulli(?k ). This construction
can be linked to the model in [27] with appropriate normalization, with advantages that there is no
need to fix pj = 0.5 and the inference is fully tractable. The zero inflated construction can also be
linked to models for real valued data using the Indian buffet process (IBP) or beta-Bernoulli process
spike-and-slab prior [28, 29, 30, 31].
4.1
Related Algorithms
To show how the NB processes can be diversely constructed and to make connections to previous
parametric and nonparametric mixture models, we show in Table 1 a variety of NB processes, which
differ on how the dispersion and probability measures are shared. For a deeper understanding on
how the counts are modeled, we also show in Table 1 both the VMR and ODL implied by these
1
We may also consider a beta marked-gamma-NB process, whose performance is found to be very similar.
5
Table 1: A variety of negative binomial processes are constructed with distinct sharing mechanisms, reflected
with which parameters from rk , rj , pk , pj and ?k (bjk ) are inferred (indicated by a check-mark X), and the
implied VMR and ODL for counts {njk }j,k . They are applied for topic modeling of a document corpus, a
typical example of mixture modeling of grouped data. Related algorithms are shown in the last column.
Algorithms
rk rj pk
pj
?k
VMR
ODL
Related Algorithms
NB-LDA
X
X
(1 ? pj )?1
rj?1
LDA [32], Dir-PFA [5]
NB-HDP
X
0.5
2
rk?1
HDP[7], DILN-HDP [12]
NB-FTM
X
0.5 X
2
(rk )?1 bjk
FTM [27], S??-PFA [5]
Beta-NB
X X
(1 ? pk )?1
rj?1
BNBP [5], BNBP [6]
Gamma-NB
X
X
(1 ? pj )?1
rk?1
CRF-HDP [7, 24]
Marked-Beta-NB X
X
(1 ? pk )?1
rk?1
BNBP [5]
settings. We consider topic modeling of a document corpus, a typical example of mixture modeling of grouped data, where each a-bag-of-words document constitutes a group, each word is an
exchangeable group member, and F (xji ; ?k ) is simply the probability of word xji in topic ?k .
We consider six differently constructed NB processes in Table 1: (i) Related to latent Dirichlet
allocation (LDA) [32] and Dirichlet Poisson factor analysis (Dir-PFA) [5], the NB-LDA is also a
parametric topic model that requires tuning the number of topics. However, it uses a document dependent rj and pj to automatically learn the smoothing of the gamma distributed topic weights, and
it lets rj ? Gamma(?0 , 1/c), ?0 ? Gamma(e0 , 1/f0 ) to share statistical strength between documents, with closed-form Gibbs sampling inference. Thus even the most basic parametric LDA topic
model can be improved under the NB count modeling framework. (ii) The NB-HDP model is related to the HDP [7], and since pj is an irrelevant parameter in the HDP due to normalization, we set
it in the NB-HDP as 0.5, the usually perceived value before normalization. The NB-HDP model is
comparable to the DILN-HDP [12] that constructs the group-level DPs with normalized gamma processes, whose scale parameters are also set as one. (iii) The NB-FTM model introduces an additional
beta-Bernoulli process under the NB process framework to explicitly model zero counts. It is the
same as the sparse-gamma-gamma-PFA (S??-PFA) in [5] and is comparable to the focused topic
model (FTM) [27], which is constructed from the IBP compound DP. Nevertheless, they apply about
the same likelihoods and priors for inference. The Zero-Inflated-NB process improves over them by
allowing pj to be inferred, which generally yields better data fitting. (iv) The Gamma-NB process
explores the idea that the dispersion measure is shared across groups, and it improves over the NBHDP by allowing the learning of pj . It reduces to the HDP [7] by normalizing both the group-level
and the shared gamma processes. (v) The Beta-NB process explores sharing the probability measure
across groups, and it improves over the beta negative binomial process (BNBP) proposed in [6],
allowing inference of rj . (vi) The Marked-Beta-NB process is comparable to the BNBP proposed
in [5], with the distinction that it allows analytical update of rk . The constructions and inference
of various NB processes and related algorithms in Table 1 all follow the formulas in (16) and (18),
respectively, with additional details presented in the supplementary material.
Note that as shown in [5], NB process topic models can also be considered as factor analysis of
the term-document count matrix under the Poisson likelihood, with ?k as the kth factor loading
that sums to one and ?jk as the factor score, which can be further linked to nonnegative matrix
? j and r?, the
factorization [33] and a gamma Poisson factor model [34]. If except for proportions ?
absolute values, e.g., ?jk , rk and pk , are also of interest, then the NB process based joint count and
mixture models would apparently be more appropriate than the HDP based mixture models.
5
Example Results
Motivated by Table 1, we consider topic modeling using a variety of NB processes, which differ on
which parameters are learned and consequently how the VMR and ODL of the latent counts {njk }j,k
are modeled. We compare them with LDA [32, 35] and CRF-HDP [7, 24]. For fair comparison, they
are all implemented with block Gibbs sampling using a discrete base measure with K atoms, and
for the first fifty iterations, the Gamma-NB process with rk ? 50/K and pj ? 0.5 is used for
initialization. For LDA and NB-LDA, we search K for optimal performance and for the other
models, we set K = 400 as an upper-bound. We set the parameters as c = 1, ? = 0.05 and
a0 = b0 = e0 = f0 = 0.01. For LDA, we set the topic proportion Dirichlet smoothing parameter
as 50/K, following the topic model toolbox2 provided for [35]. We consider 2500 Gibbs sampling
iterations, with the last 1500 samples collected. Under the NB processes, each word xji would
6
(a)
(b)
1300
1500
1250
LDA
NB?LDA
NB?HDP
NB?FTM
Beta?NB
CRF?HDP
Gamma?NB
Marked?Beta?NB
1400
1200
1300
1200
1100
Perplexity
Perplexity
1150
1050
1000
K+=127
950
K =201
900
K+=107
900
K+=161 K+=177 K+=130
850
800
0
50
1100
1000
+
100
150
200
250
Number of topics
300
800
350
700
400
0.2
0.3
0.4
0.5
0.6
Training data percentage
0.7
0.8
Figure 1: Comparison of per-word perplexities on the held-out words between various algorithms. (a) With
60% of the words in each document used for training, the performance varies as a function of K in both LDA
and NB-LDA, which are parametric models, whereas the NB-HDP, NB-FTM, Beta-NB, CRF-HDP, GammaNB and Marked-Beta-NB all infer the number of active topics, which are 127, 201, 107, 161, 177 and 130,
respectively, according to the last Gibbs sampling iteration. (b) Per-word perplexities of various models as a
function of the percentage of words in each document used for training. The results of the LDA and NB-LDA
are shown with the best settings of K under each training/testing partition.
be assigned to a topic k based on both F (xji ; ?k ) and the topic weights {?jk }k=1,K ; each topic is
drawn from a Dirichlet base measure as ?k ? Dir(?, ? ? ? , ?) ? RV , where V is the number of unique
terms in the vocabulary and ? is a smoothing parameter.
word
P P Let vji denote the location ofP
Pxji in the
vocabulary, then we have (?k |?) ? Dir ? + j i ?(zji = k, vji = 1), ? ? ? , ? + j i ?(zji =
k, vji = V ) . We consider the Psychological Review2 corpus, restricting the vocabulary to terms
that occur in five or more documents. The corpus includes 1281 abstracts from 1967 to 2003, with
2,566 unique terms and 71,279 total word counts. We randomly select 20%, 40%, 60% or 80%
of the words from each document to learn a document dependent probability for each term v as
PS PK
PV PK
(s) (s) PS
(s) (s)
fjv = s=1 k=1 ?vk ?jk
s=1
v=1
k=1 ?vk ?jk , where ?vk is the probability of term v
in topic k and S is the total number of collected samples. We use {fjv }j,v to calculate the perword perplexity on the held-out words as in [5]. The final results are averaged from five random
training/testing partitions. Note that the perplexity per test word is the fair metric to compare topic
models. However, when the actual Poisson rates or distribution parameters for counts instead of the
mixture proportions are of interest, it is obvious that a NB process based joint count and mixture
model would be more appropriate than an HDP based mixture model.
Figure 1 compares the performance of various algorithms. The Marked-Beta-NB process has the
best performance, closely followed by the Gamma-NB process, CRF-HDP and Beta-NB process.
With an appropriate K, the parametric NB-LDA may outperform the nonparametric NB-HDP and
NB-FTM as the training data percentage increases, somewhat unexpected but very intuitive results,
showing that even by learning both the NB dispersion and probability parameters rj and pj in a
document dependent manner, we may get better data fitting than using nonparametric models that
share the NB dispersion parameters rk across documents, but fix the NB probability parameters.
Figure 2 shows the learned model parameters by various algorithms under the NB process framework, revealing distinct sharing mechanisms and model properties. When (rj , pj ) is used, as in the
NB-LDA, different documents are weakly coupled with rj ? Gamma(?0 , 1/c), and the modeling
results show that a typical document in this corpus usually has a small rj and a large pj , thus a large
ODL and a large VMR, indicating highly overdispersed counts on its topic usage. When (rj , pk ) is
used to model the latent counts {njk }j,k , as in the Beta-NB process, the transition between active
and non-active topics isP
very sharp that pk is either
P close to one or close to zero. That is because pk
controls the mean as E[ j njk ] = pk /(1 ? pk ) j rj and the VMR as (1 ? pk )?1 on topic k, thus
a popular topic must also have large pk and thus large overdispersion measured by the VMR; since
the counts {njk }j are usually overdispersed, particularly true in this corpus, a middle range pk indicating an appreciable mean and small overdispersion is not favored by the model and thus is rarely
observed. When (rk , pj ) is used, as in the Gamma-NB process, the transition
is much
P
Psmoother that
rk gradually decreases. The reason is that rk controls the mean as E[ j njk ] = rk j pj /(1 ? pj )
and the ODL rk?1 on topic k, thus popular topics must also have large rk and thus small overdispersion measured by the ODL, and unpopular topics are modeled with small rk and thus large overdispersion, allowing rarely and lightly used topics. Therefore, we can expect that (rk , pj ) would allow
2
http://psiexp.ss.uci.edu/research/programs data/toolbox.htm
7
NB?LDA
rj
2
NB?HDP
rk
NB?FTM
rk
10
Beta?NB
rj
2
30
Gamma?NB
rk
0
0
?4
10
10
10
10
?2
?4
0
500
1000
Document Index
10
0
pj
1
10
0
20
?2
10
?2
0
10
10
10
Marked?Beta?NB
rk
10
0
10
200
Topic Index
pj
1
0
400
?4
0
200
Topic Index
400
10
10
?4
0
10
500
1000
Document Index
?4
0
pk
?k
0
?2
10
?2
10
200
Topic Index
400
10
0
pj
200
Topic Index
400
pk
1
1
1
0.5
0.5
0.5
0.8
?1
10
0.6
0.5
0.4
?2
10
0.2
0
500
1000
Document Index
0
?3
0
500
1000
Document Index
10
0
200
Topic Index
400
0
0
200
Topic Index
400
0
0
500
1000
Document Index
0
0
200
Topic Index
400
Figure 2: Distinct sharing mechanisms and model properties are evident between various NB processes, by
comparing their inferred parameters. Note that the transition between active and non-active topics is very sharp
when pk is used and much smoother when rk is used. Both the documents and topics are ordered in a decreasing
order based on the number of words associated with each of them. These results are based on the last Gibbs
sampling iteration. The values are shown in either linear or log scales for convenient visualization.
more topics than (rj , pk ), as confirmed in Figure 1 (a) that the Gamma-NB process learns 177 active
topics, significantly more than the 107 ones of the Beta-NB process. With these analysis, we can
conclude that the mean and the amount of overdispersion (measure by the VMR or ODL) for the
usage of topic k is positively correlated under (rj , pk ) and negatively correlated under (rk , pj ).
When (rk , pk ) is used, as in the Marked-Beta-NB process, more diverse combinations ofPmean and
overdispersion would be allowed as both rk and pk are now responsible for the mean E[ j njk ] =
Jrk pk /(1?pk ). For example, there could be not only large mean and small overdispersion (large rk
and small pk ), but also large mean and large overdispersion (small rk and large pk ). Thus (rk , pk )
may combine the advantages of using only rk or pk to model topic k, as confirmed by the superior
performance of the Marked-Beta-NB over the Beta-NB and Gamma-NB processes. When (rk , ?k )
is used, as in the NB-FTM model, our results show that we usually have a small ?k and a large rk ,
indicating topic k is sparsely used across the documents but once it is used, the amount of variation
on usage is small. This modeling properties might be helpful when there are excessive number of
zeros which might not be well modeled by the NB process alone. In our experiments, we find the
more direct approaches of using pk or pj generally yield better results, but this might not be the
case when excessive number of zeros are better explained with the underlying beta-Bernoulli or IBP
processes, e.g., when the training words are scarce.
It is also interesting to compare the Gamma-NB and NB-HDP. From a mixture-modeling viewpoint,
fixing pj = 0.5 is natural as pj becomes irrelevant after normalization. However, from a count modeling viewpoint, this would make a restrictive assumption that each count vector {njk }k=1,K has
the same VMR of 2, and the experimental results in Figure 1 confirm the importance of learning pj
together with rk . It is also interesting to examine (15), which can be viewed as the concentration parameter ? in the HDP, allowing the adjustment of pj would allow a more flexible model assumption
on the amount of variations between the topic proportions, and thus potentially better data fitting.
6
Conclusions
We propose a variety of negative binomial (NB) processes to jointly model counts across groups,
which can be naturally applied for mixture modeling of grouped data. The proposed NB processes
are completely random measures that they assign independent random variables to disjoint Borel sets
of the measure space, as opposed to the hierarchical Dirichlet process (HDP) whose measures on
disjoint Borel sets are negatively correlated. We discover augment-and-conquer inference methods
that by ?augmenting? a NB process into both the gamma-Poisson and compound Poisson representations, we are able to ?conquer? the unification of count and mixture modeling, the analysis of
fundamental model properties and the derivation of efficient Gibbs sampling inference. We demonstrate that the gamma-NB process, which shares the NB dispersion measure across groups, can be
normalized to produce the HDP and we show in detail its theoretical, structural and computational
advantages over the HDP. We examine the distinct sharing mechanisms and model properties of
various NB processes, with connections to existing algorithms, with experimental results on topic
modeling showing the importance of modeling both the NB dispersion and probability parameters.
Acknowledgments
The research reported here was supported by ARO, DOE, NGA, and ONR, and by DARPA under
the MSEE and HIST programs.
8
References
[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]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
J. F. C. Kingman. Poisson Processes. Oxford University Press, 1993.
M. K. Titsias. The infinite gamma-Poisson feature model. In NIPS, 2008.
R. J. Thibaux. Nonparametric Bayesian Models for Machine Learning. PhD thesis, UC Berkeley, 2008.
K. T. Miller. Bayesian Nonparametric Latent Feature Models. PhD thesis, UC Berkeley, 2011.
M. Zhou, L. Hannah, D. Dunson, and L. Carin. Beta-negative binomial process and Poisson factor analysis. In AISTATS, 2012.
T. Broderick, L. Mackey, J. Paisley, and M. I. Jordan. Combinatorial clustering and the beta negative
binomial process. arXiv:1111.1802v3, 2012.
Y. W. Teh, M. I. Jordan, M. J. Beal, and D. M. Blei. Hierarchical Dirichlet processes. JASA, 2006.
M. I. Jordan. Hierarchical models, nested models and completely random measures. 2010.
R. L. Wolpert, M. A. Clyde, and C. Tu. Stochastic expansions using continuous dictionaries: L?evy
Adaptive Regression Kernels. Annals of Statistics, 2011.
T. S. Ferguson. A Bayesian analysis of some nonparametric problems. Ann. Statist., 1973.
H. Ishwaran and M. Zarepour. Exact and approximate sum-representations for the Dirichlet process. Can.
J. Statist., 2002.
J. Paisley, C. Wang, and D. M. Blei. The discrete infinite logistic normal distribution. Bayesian Analysis,
2012.
C. I. Bliss and R. A. Fisher. Fitting the negative binomial distribution to biological data. Biometrics,
1953.
A. C. Cameron and P. K. Trivedi. Regression Analysis of Count Data. Cambridge, UK, 1998.
R. Winkelmann. Econometric Analysis of Count Data. Springer, Berlin, 5th edition, 2008.
M. H. Quenouille. A relation between the logarithmic, Poisson, and negative binomial series. Biometrics,
1949.
N. L. Johnson, A. W. Kemp, and S. Kotz. Univariate Discrete Distributions. John Wiley & Sons, 2005.
S. J. Clark and J. N. Perry. Estimation of the negative binomial parameter ? by maximum quasi-likelihood.
Biometrics, 1989.
M. D. Robinson and G. K. Smyth. Small-sample estimation of negative binomial dispersion, with applications to SAGE data. Biostatistics, 2008.
M. Zhou, L. Li, D. Dunson, and L. Carin. Lognormal and gamma mixed negative binomial regression. In
ICML, 2012.
C. E. Antoniak. Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems.
Ann. Statist., 1974.
M. D. Escobar and M. West. Bayesian density estimation and inference using mixtures. JASA, 1995.
C. Wang, J. Paisley, and D. M. Blei. Online variational inference for the hierarchical Dirichlet process. In
AISTATS, 2011.
E. Fox, E. Sudderth, M. Jordan, and A. Willsky. Developing a tempered HDP-HMM for systems with
state persistence. MIT LIDS, TR #2777, 2007.
N. L. Hjort. Nonparametric Bayes estimators based on beta processes in models for life history data. Ann.
Statist., 1990.
R. Thibaux and M. I. Jordan. Hierarchical beta processes and the Indian buffet process. In AISTATS,
2007.
S. Williamson, C. Wang, K. A. Heller, and D. M. Blei. The IBP compound Dirichlet process and its
application to focused topic modeling. In ICML, 2010.
T. L. Griffiths and Z. Ghahramani. Infinite latent feature models and the Indian buffet process. In NIPS,
2005.
M. Zhou, H. Chen, J. Paisley, L. Ren, L. Li, Z. Xing, D. Dunson, G. Sapiro, and L. Carin. Nonparametric
Bayesian dictionary learning for analysis of noisy and incomplete images. IEEE TIP, 2012.
M. Zhou, H. Yang, G. Sapiro, D. Dunson, and L. Carin. Dependent hierarchical beta process for image
interpolation and denoising. In AISTATS, 2011.
L. Li, M. Zhou, G. Sapiro, and L. Carin. On the integration of topic modeling and dictionary learning. In
ICML, 2011.
D. Blei, A. Ng, and M. Jordan. Latent Dirichlet allocation. J. Mach. Learn. Res., 2003.
D. D. Lee and H. S. Seung. Algorithms for non-negative matrix factorization. In NIPS, 2000.
J. Canny. Gap: a factor model for discrete data. In SIGIR, 2004.
T. L. Griffiths and M. Steyvers. Finding scientific topics. PNAS, 2004.
9
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4,063 | 4,678 | Multiclass Learning Approaches:
A Theoretical Comparison with Implications
Amit Daniely
Department of Mathematics
The Hebrew University
Jerusalem, Israel
Sivan Sabato
Microsoft Research
1 Memorial Drive
Cambridge, MA 02142, USA
Shai Shalev-Shwartz
School of CS and Eng.
The Hebrew University
Jerusalem, Israel
Abstract
We theoretically analyze and compare the following five popular multiclass classification methods: One vs. All, All Pairs, Tree-based classifiers, Error Correcting
Output Codes (ECOC) with randomly generated code matrices, and Multiclass
SVM. In the first four methods, the classification is based on a reduction to binary
classification. We consider the case where the binary classifier comes from a class
of VC dimension d, and in particular from the class of halfspaces over Rd . We
analyze both the estimation error and the approximation error of these methods.
Our analysis reveals interesting conclusions of practical relevance, regarding the
success of the different approaches under various conditions. Our proof technique
employs tools from VC theory to analyze the approximation error of hypothesis
classes. This is in contrast to most previous uses of VC theory, which only deal
with estimation error.
1
Introduction
In this work we consider multiclass prediction: The problem of classifying objects into one of
several possible target classes. Applications include, for example, categorizing documents according
to topic, and determining which object appears in a given image. We assume that objects (a.k.a.
instances) are vectors in X = Rd and the class labels come from the set Y = [k] = {1, . . . , k}.
Following the standard PAC model, the learner receives a training set of m examples, drawn i.i.d.
from some unknown distribution, and should output a classifier which maps X to Y.
The centrality of the multiclass learning problem has spurred the development of various approaches
for tackling the task. Perhaps the most straightforward approach is a reduction from multiclass
classification to binary classification. For example, the One-vs-All (OvA) method is based on a
reduction of the multiclass problem into k binary problems, each of which discriminates between
one class to all the rest of the classes (e.g. Rumelhart et al. [1986]). A different reduction is the AllPairs (AP) approach in which all pairs of classes are compared to each other [Hastie and Tibshirani,
1998]. These two approaches have been unified under the framework of Error Correction Output
Codes (ECOC) [Dietterich and Bakiri, 1995, Allwein et al., 2000]. A tree-based classifier (TC) is
another reduction in which the prediction is obtained by traversing a binary tree, where at each node
of the tree a binary classifier is used to decide on the rest of the path (see for example Beygelzimer
et al. [2007]).
All of the above methods are based on reductions to binary classification. We pay special attention
to the case where the underlying binary classifiers are linear separators (halfspaces). Formally, each
w ? Rd+1 defines the linear separator hw (x) = sign(hw, x
?i), where x
? = (x, 1) ? Rd+1 is the
concatenation of the vector x and the scalar 1. While halfspaces are our primary focus, many of our
results hold for any underlying binary hypothesis class of VC dimension d + 1.
1
Other, more direct approaches to multiclass classification over Rd have also been proposed (e.g.
Vapnik [1998], Weston and Watkins [1999], Crammer and Singer [2001]). In this paper we analyze
the Multiclass SVM (MSVM) formulation of Crammer and Singer [2001], in which each hypothesis
is of the form hW (x) = argmaxi?[k] (W x
?)i , where W is a k ? (d + 1) matrix and (W x
?)i is the i?th
element of the vector W x
? ? Rk .
We theoretically analyze the prediction performance of the aforementioned methods, namely, OvA,
AP, ECOC, TC, and MSVM. The error of a multiclass predictor h : Rd ? [k] is defined to be the
probability that h(x) 6= y, where (x, y) is sampled from the underlying distribution D over Rd ? [k],
namely, Err(h) = P(x,y)?D [h(x) 6= y]. Our main goal is to understand which method is preferable
in terms of the error it will achieve, based on easy-to-verify properties of the problem at hand.
Our analysis pertains to the type of classifiers each method can potentially find, and does not depend
on the specific training algorithm. More precisely, each method corresponds to a hypothesis class,
H, which contains the multiclass predictors that may be returned by the method. For example, the
hypothesis class of MSVM is H = {x 7? argmaxi?[k] (W x
?)i : W ? Rk?(d+1) }.
A learning algorithm, A, receives a training set, S = {(xi , yi )}m
i=1 , sampled i.i.d. according to
D, and returns a multiclass predictor which we denote by A(S) ? H. A learning algorithm
is called an Empirical Risk Minimizer (ERM) if it returns a hypothesis in H that minimizes the
empirical error on the sample. We denote by h? a hypothesis in H with minimal error,1 that is,
h? ? argminh?H Err(h).
When analyzing the error of A(S), it is convenient to decompose this error as a sum of approximation error and estimation error:
Err(A(S)) =
Err(h? ) + Err(A(S)) ? Err(h? ) .
| {z }
|
{z
}
approximation
(1)
estimation
? The approximation error is the minimum error achievable by a predictor in the hypothesis
class, H. The approximation error does not depend on the sample size, and is determined
solely by the allowed hypothesis class2 .
? The estimation error of an algorithm is the difference between the approximation error,
and the error of the classifier the algorithm chose based on the sample. This error exists
both for statistical reasons, since the sample may not be large enough to determine the best
hypothesis, and for algorithmic reasons, since the learning algorithm may not output the
best possible hypothesis given the sample.
p For the ERM algorithm, the estimation error can
be bounded from above by order of C(H)/m where C(H) is a complexity measure of
H (analogous to the VC dimension) and m is the sample size. A similar term also bounds
the estimation error from below for any algorithm. Thus C(H) is an estimate of the best
achievable estimation error for the class.
When studying the estimation error of different methods, we follow the standard distribution-free
analysis. Namely, we will compare the algorithms based on the worst-case estimation error, where
worst-case is over all possible distributions D. Such an analysis can lead us to the following type
of conclusion: If two hypothesis classes have roughly the same complexity, C(H1 ) ? C(H2 ),
and the number of available training examples is significantly larger than this value of complexity,
then for both hypothesis classes we are going to have a small estimation error. Hence, in this
case the difference in prediction performance between the two methods will be dominated by the
approximation error and by the success of the learning algorithm in approaching the best possible
estimation error. In our discussion below we disregard possible differences in optimality which stem
from algorithmic aspects and implementation details. A rigorous comparison of training heuristics
would certainly be of interest and is left to future work.
For the approximation error we will provide even stronger results, by comparing the approximation
error of classes for any distribution. We rely on the following definition.
1
For simplicity, we assume that the minimum is attainable.
Note that, when comparing different hypothesis classes over the same distribution, the Bayes error is
constant. Thus, in the definition of approximation error, we do not subtract the Bayes error.
2
2
Definition 1.1. Given two hypothesis classes, H, H0 , we say that H essentially contains H0 if for
any distribution, the approximation error of H is at most the approximation error of H0 . H strictly
contains H0 if, in addition, there is a distribution for which the approximation error of H is strictly
smaller than that of H0 .
Our main findings are as follows (see a full comparison in Table 1). The formal statements are given
in Section 3.
? The estimation errors of OvA, MSVM, and TC are all roughly the same, in the sense that
?
C(H) = ?(dk)
for all of the corresponding hypothesis classes. The complexity of AP is
2
?
?(dk ). The complexity of ECOC with a code of length l and code-distance ? is at most
?
?
O(dl)
and at least d?/2. It follows that for randomly generated codes, C(H) = ?(dl).
Note that this analysis shows that a larger code-distance yields a larger estimation error and
might therefore hurt performance. This contrasts with previous ?reduction-based? analyses
of ECOC, which concluded that a larger code distance improves performance.
? We prove that the hypothesis class of MSVM essentially contains the hypothesis classes
of both OvA and TC. Moreover, these inclusions are strict. Since the estimation errors of
these three methods are roughly the same, it follows that the MSVM method dominates
both OvA and TC in terms of achievable prediction performance.
? In the TC method, one needs to associate each leaf of the tree to a label. If no prior
knowledge on how to break the symmetry is known, it is suggested in Beygelzimer et al.
[2007] to break symmetry by choosing a random permutation of the labels. We show that
whenever d k, for any distribution D, with high probability over the choice of a random
permutation, the approximation error of the resulting tree would be close to 1/2. It follows
that a random choice of a permutation is likely to yield a poor predictor.
? We show that if d k, for any distribution D, the approximation error of ECOC with a
randomly generated code matrix is likely to be close to 1/2.
? We show that the hypothesis class of AP essentially contains the hypothesis class of MSVM
(hence also that of OvA and TC), and that there can be a substantial gap in the containment.
Therefore, as expected, the relative performance of AP and MSVM depends on the wellknown trade-off between estimation error and approximation error.
Estimation error
Approximation
error
Testing run-time
TC
dk
? MSVM
? 1/2 when d k
d log(k)
OvA
dk
? MSVM
MSVM
dk
? AP
AP
dk 2
smallest
dk
dk
dk 2
random ECOC
dl
incomparable
? 1/2 when d k
dl
Table 1: Summary of comparison
The above findings suggest that in terms of performance, it may be wiser to choose MSVM over OvA
and TC, and especially so when d k. We note, however, that in some situations (e.g. d = k) the
prediction success of these methods can be similar, while TC has the advantage of having a testing
run-time of d log(k), compared to the testing run-time of dk for OvA and MSVM. In addition, TC
and ECOC may be a good choice when there is additional prior knowledge on the distribution or on
how to break symmetry between the different labels.
1.1
Related work
Allwein et al. [2000] analyzed the multiclass error of ECOC as a function of the binary error. The
problem with such a ?reduction-based? analysis is that such analysis becomes problematic if the
underlying binary problems are very hard. Indeed, our analysis reveals that the underlying binary
problems would be too hard if d k and the code is randomly generated. The experiments in Allwein et al. [2000] show that when using kernel-based SVM or AdaBoost as the underlying classifier,
OvA is inferior to random ECOC. However, in their experiments, the number of classes is small relative to the dimension of the feature space, especially if working with kernels or with combinations
of weak learners.
3
Crammer and Singer [2001] presented experiments demonstrating that MSVM outperforms OvA
on several data sets. Rifkin and Klautau [2004] criticized the experiments of Crammer and Singer
[2001], Allwein et al. [2000], and presented another set of experiments demonstrating that all methods perform roughly the same when the underlying binary classifier is very strong (SVM with a
Guassian kernel). As our analysis shows, it is not surprising that with enough data and powerful
binary classifiers, all methods should perform well. However, in many practical applications, we
will prefer not to employ kernels (either because of shortage of examples, which might lead to a
large estimation error, or due to computational constraint), and in such cases we expect to see a
large difference between the methods.
Beygelzimer et al. [2007] analyzed the regret of a specific training method for trees, called Filter
Tree, as a function of the regret of the binary classifier. The regret is defined to be the difference
between the learned classifier and the Bayes-optimal classifier for the problem. Here again we show
that the regret values of the underlying binary classifiers are likely to be very large whenever d k
and the leaves of the tree are associated to labels in a random way. Thus in this case the regret
analysis is problematic. Several authors presented ways to learn better splits, which corresponds to
learning the association of leaves to labels (see for example Bengio et al. [2011] and the references
therein). Some of our negative results do not hold for such methods, as these do not randomly attach
labels to tree leaves.
Daniely et al. [2011] analyzed the properties of multiclass learning with various ERM learners, and
have also provided some bounds on the estimation error of multiclass SVM and of trees. In this
paper we both improve these bounds, derive new bounds for other classes, and also analyze the
approximation error of the classes.
2
Definitions and Preliminaries
We first formally define the hypothesis classes that we analyze in this paper.
Multiclass SVM (MSVM): For W ? Rk?(d+1) define hW : Rd ? [k] by hW (x) =
argmaxi?[k] (W x
?)i and let L = {hW : W ? Rk?(d+1) }. Though NP-hard in general, solving
the ERM problem with respect to L can be done efficiently in the realizable case (namely, whenever
exists a hypothesis with zero empirical error on the sample).
Tree-based classifiers (TC): A tree-based multiclass classifier is a full binary tree whose leaves
are associated with class labels and whose internal nodes are associated with binary classifiers. To
classify an instance, we start with the root node and apply the binary classifier associated with
it. If the prediction is 1 we traverse to the right child. Otherwise, we traverse to the left child.
This process continues until we reach a leaf, and then we output the label associated with the leaf.
Formally, a tree for k classes is a full binary tree T together with a bijection ? : leaf(T ) ? [k],
which associates a label to each of the leaves. We usually identify T with the pair (T, ?). The set
of internal nodes of T is denoted by N (T ). Let H ? {?1}X be a binary hypothesis class. Given a
mapping C : N (T ) ? H, define a multiclass predictor, hC : X ? [k], by setting hC (x) = ?(v)
where v is the last node of the root-to-leaf path v1 , . . . vm = v such that vi+1 is the left (resp. right)
child of vi if C(vi )(x) = ?1 (resp. C(vi )(x) = 1). Let HT = {hC | C : N (T ) ? H}. Also, let
Htrees = ?T is a tree for k classes HT . If H is the class of linear separators over Rd , then for any tree T
the ERM problem with respect to HT can be solved efficiently in the realizable case. However, the
ERM problem is NP-hard in the non-realizable case.
Error Correcting Output Codes (ECOC): An ECOC is a code M ? Rk?l along with a bijection
? : [k] ? [k]. We sometimes identify ? with the identity function and M with (M, ?)3 . Given a
code M , and the result of l binary classifiers represented by avector u ? {?1, 1}l , the code
selects
Pl
l
?
?
a label via M : {?1, 1} ? [k], defined by M (u) = ? arg maxi?[k]
Mij uj . Given
j=1
binary classifiers h1 , . . . , hl for each column in the code matrix, the code assigns to the instance
? (h1 (x), . . . , hl (x)). Let H ? {?1}X be a binary hypothesis class. Denote by
x ? X the label M
3
The use of ? here allows us to later consider codes with random association of rows to labels.
4
HM ? [k]X the hypotheses class HM = {h : X ? [k] | ?(h1 , . . . , hl ) ? Hl s.t. ?x ? X , h(x) =
? (h1 (x), . . . , hl (x))}.
M
The distance of a binary code, denoted by ?(M ) for M ? {?1}k?l , is the minimal hamming
distance between any two pairs of rows in the code matrix. Formally, the hamming distance between
u, v ? {?1, +1}l is ?h (u, v) = |{r : u[r] 6= v[r]}|, and ?(M ) = min1?i<j?k ?h (M [i], M [j]).
The ECOC paradigm described in [Dietterich and Bakiri, 1995] proposes to choose a code with a
large distance.
One vs. All (OvA) and All Pairs (AP): Let H ? {?1}X and k ? 2. In the OvA method we train
k binary problems, each of which discriminates between one class and the rest of the classes. In the
AP approach all pairs of classes are compared to each other. This is formally defined as two ECOCs.
Define M OvA ? Rk?k to be the matrix whose (i, j) elements is 1 if i = j and ?1 if i 6= j. Then,
k
the hypothesis class of OvA is HOvA = HM OvA . For the AP method, let M AP ? Rk?(2) be such that
for all i ? [k] and 1 ? j < l ? k, the coordinate corresponding to row i and column (j, l) is defined
to be ?1 if i = j, 1 if i = l, and 0 otherwise. Then, the hypothesis class of AP is HAP = HM AP .
Our analysis of the estimation error is based on results that bound the sample complexity of multiclass learning. The sample complexity of an algorithm A is the function mA defined as follows: For
, ? > 0, mA (, ?) is the smallest integer such that for every m ? mA (, ?) and every distribution
D on X ? Y, with probability of > 1 ? ? over the choice of an i.i.d. sample S of size m,
Err(A(Sm )) ? min Err(h) + .
(2)
h?H
The first term on the right-hand side is the approximation error of H. Therefore, the sample complexity is the number of examples required to ensure that the estimation error of A is at most (with
high probability). We denote the sample complexity of a class H by mH (, ?) = inf A mA (, ?),
where the infimum is taken over all learning algorithms.
To bound the sample complexity of a hypothesis class we rely on upper and lower bounds on the
sample complexity in terms of two generalizations of the VC dimension for multiclass problems,
called the Graph dimension and the Natarajan dimension and denoted dG (H) and dN (H). For
completeness, these dimensions are formally defined in the appendix.
Theorem 2.1. Daniely et al. [2011] For every hypothesis class H, and for every ERM rule,
dN (H) + ln( 1? )
min{dN (H) ln(|Y|), dG (H)} + ln( 1? )
?
?
m
(,
?)
?
m
(,
?)
?
O
H
ERM
2
2
We note that the constants in the O, ? notations are universal.
3
Main Results
In Section 3.1 we analyze the sample complexity of the different hypothesis classes. We provide
lower bounds on the Natarajan dimensions of the various hypothesis classes, thus concluding, in
light of Theorem 2.1, a lower bound on the sample complexity of any algorithm. We also provide
upper bounds on the graph dimensions of these hypothesis classes, yielding, by the same theorem,
an upper bound on the estimation error of ERM. In Section 3.2 we analyze the approximation error
of the different hypothesis classes.
3.1
Sample Complexity
Together with Theorem 2.1, the following theorems estimate, up to logarithmic factors, the sample
complexity of the classes under consideration. We note that these theorems support the rule of thumb
that the Natarajan and Graph dimensions are of the same order of the number of parameters. The
?
first theorem shows that the sample complexity of MSVM depends on ?(dk).
Theorem 3.1. d(k ? 1) ? dN (L) ? dG (L) ? O(dk log(dk)).
Next, we analyze the sample complexities of TC and ECOC. These methods rely on an underlying
hypothesis class of binary classifiers. While our main focus is the case in which the binary hypothesis class is halfspaces over Rd , the upper bounds on the sample complexity we derive below holds
for any binary hypothesis class of VC dimension d + 1.
5
Theorem 3.2. For every binary hypothesis class of VC dimension d + 1, and for any tree T ,
dG (HT ) ? dG (Htrees ) ? O(dk log(dk)). If the underlying hypothesis class is halfspaces over
Rd , then also
d(k ? 1) ? dN (HT ) ? dG (HT ) ? dG (Htrees ) ? O(dk log(dk)).
Theorems 3.1 and 3.2 improve results from Daniely et al. [2011] where it was shown that b d2 cb k2 c ?
dN (L) ? O(dk log(dk)), and for every treedG (HT) ? O(dk log(dk)). Further it was shown that
if H is the set of halfspaces over Rd , then ?
dk
log(k)
? dN (HT ).
We next turn to results for ECOC, and its special cases OvA and AP.
Theorem 3.3. For every M ? Rk?l and every binary hypothesis class of VC dimension d,
dG (HM ) ? O(dl log(dl)). Moreover, if M ? {?1}k?l and the underlying hypothesis class is
halfspaces over Rd , then
d ? ?(M )/2 ? dN (HM ) ? dG (HM ) ? O(dl log(dl)) .
We note if the code has a large distance, which is the case, for instance, in random codes, then
?(M ) = ?(l). In this case, the bound is tight up to logarithmic factors.
Theorem 3.4. For any binary hypothesis class of VC dimension d, dG (HOvA ) ? O(dk log(dk)) and
dG (HAP ) ? O(dk 2 log(dk)). If the underlying hypothesis class is halfspaces over Rd we also have:
d(k ? 1) ? dN (HOvA ) ? dG (HOvA ) ? O(dk log(dk))
d k?1
? dN (HAP ) ? dG (HAP ) ? O(dk 2 log(dk)).
2
3.2
and
Approximation error
We first show that the class L essentially contains HOvA and HT for any tree T , assuming, of
course, that H is the class of halfspaces in Rd . We find this result quite surprising, since the sample
complexity of all of these classes is of the same order.
Theorem 3.5. L essentially contains Htrees and HOvA . These inclusions are strict for d ? 2 and
k ? 3.
One might suggest that a small increase in the dimension would perhaps allow us to embed L in HT
for some tree T or for OvA. The next result shows that this is not the case.
Theorem 3.6. Any embedding into a higher dimension that allows HOvA or HT (for some tree T
?
for k classes) to essentially contain L, necessarily embeds into a dimension of at least ?(dk).
The next theorem shows that the approximation error of AP is better than that of MSVM (and hence
also better than OvA and TC). This is expected as the sample complexity of AP is considerably
higher, and therefore we face the usual trade-off between approximation and estimation error.
Theorem 3.7. HAP essentially contains L. Moreover, there is a constant k ? > 0, independent of d,
such that the inclusion is strict for all k ? k ? .
For a random ECOC of length o(k), it is easy to see that it does not contain MSVM, as MSVM has
higher complexity. It is also not contained in MSVM, as it generates non-convex regions of labels.
We next derive absolute lower bounds on the approximation errors of ECOC and TC when d k.
Recall that both methods are built upon binary classifiers that should predict h(x) = 1 if the label
of x is in L, for some L ? [k], and should predict h(x) = ?1 if the label of x is not in L. As the
following lemma shows, when the partition of [k] into the two sets L and [k] \ L is arbitrary and
balanced, and k d, such binary classifiers will almost always perform very poorly.
Lemma 3.8. There exists a constant C > 0 for which the following holds. Let H ? {?1}X be any
hypothesis class of VC-dimension d, let ? ? (0, 1/2], and let D be any distribution over X ? [k]
such that ?i P(x,y)?D (y = i) ? 10
k . Let ? : [k] ? {?1} be a randomly chosen function which
is sampled according to one of the following rules: (1) For each i ? [k], each coordinate ?(i)
is chosen independently from the other coordinates and P(?(i) = ?1) = ?; or (2) ? is chosen
uniformly among all functions satisfying |{i ? [k] : ?(i) = ?1}| = ?k.
6
Let D? be the distribution over X ? {?1} obtained
by drawing
(x, y) according to D and replacing
d+ln( 1 )
?
it with (x, ?(y)). Then, for any ? > 0, if k ? C ?
, then with probability of at least 1 ? ?
?2
over the choice of ?, the approximation error of H with respect to D? will be at least ? ? ?.
As the corollaries below show, Lemma 3.8 entails that when k d, both random ECOCs with a
small code length, and balanced trees with a random labeling of the leaves, are expected to perform
very poorly.
Corollary 3.9. There is a constant C > 0 for which the following holds. Let (T, ?) be a tree
for k classes such that ? : leaf(T ) ? [k] is chosen uniformly at random. Denote by kL and kR
the number of leaves of the left and right sub-trees (respectively) that descend from root, and let
? = min{ kk1 , kk2 }. Let H ? {?1}X be a hypothesis class of VC-dimension d, let ? > 0, and let
D
d+ln( 1 )
?
be any distribution over X ? [k] such that ?i P(x,y)?D (y = i) ? 10
,
k . Then, for k ? C ?
?2
with probability of at least 1 ? ? over the choice of ?, the approximation error of HT with respect
to D is at least ? ? ?.
Corollary 3.10. There is a constant C > 0 for which the following holds. Let (M, ?) be an ECOC
where M ? Rk?l , and assume that the bijection ? : [k] ? [k] is chosen uniformly at random. Let
H ? {?1}X be a hypothesis class of VC-dimension d, let ? > 0,and let D be any
distribution over
dl log(dl)+ln( 1 )
?
X ? [k] such that ?i P(x,y)?D (y = i) ? 10
, with probability
k . Then, for k ? C ?
?2
of at least 1 ? ? over the choice of ?, the approximation error of HM with respect to D is at least
1/2 ? ?.
Note that the first corollary holds even if only the top level of the binary tree is balanced and splits
the labels randomly to the left and the right sub-trees. The second corollary holds even if the code
itself is not random (nor does it have to be binary), and only the association of rows with labels is
random. In particular, if the length of the code is O(log(k)), as suggested in Allwein et al. [2000],
?
and the number of classes is ?(d),
then the code is expected to perform poorly.
For an ECOC with a matrix of length ?(k) and d = o(k), we do not have such a negative result as
stated in Corollary 3.10. Nonetheless, Lemma 3.8 implies that the prediction of the binary classifiers
when d = o(k) is just slightly better than a random guess, thus it seems to indicate that the ECOC
method will still perform poorly. Moreover, most current theoretical analyses of ECOC estimate the
error of the learned multiclass hypothesis in terms of the average error of the binary classifiers. Alas,
when the number of classes is large, Lemma 3.8 shows that this average will be close to 12 .
Finally, let us briefly discuss the tightness of Lemma 3.8. Let x1 , . . . , xd+1 ? Rd be affinely inde1
.
pendent and let D be the distribution over Rd ?[d+1] defined by P(x,y)?D ((x, y) = (xi , i)) = d+1
Is is not hard to see that for every ? : [d + 1] ? {?1}, the approximation error of the class of halfspaces with respect to D? is zero. Thus, in order to ensure a large approximation error for every
distribution, the number of classes must be at least linear in the dimension, so in this sense, the
lemma is tight. Yet, this example is very simple, since each class is concentrated on a single point
and the points are linearly independent. It is possible that in real-world distributions, a large approximation error will be exhibited even when k < d.
We note that the phenomenon of a large approximation error, described in Corollaries 3.9 and 3.10,
does not reproduce in the classes L, HOvA and HAP , since these classes are symmetric.
4
Proof Techniques
Due to lack of space, the proofs for all the results stated above are provided in the appendix. In this
section we give a brief description of our main proof techniques.
Most of our proofs for the estimation error results, stated in Section 3.1, are based on a similar
method which we now describe. Let L : {?1}l ? [k] be a multiclass-to-binary reduction (e.g., a
tree), and for H ? {?1}X , denote L(H) = {x 7? L(h1 (x), . . . , hl (x)) | h1 , . . . , hl ? H}. Our
upper bounds for dG (L(H)) are mostly based on the following simple lemma.
Lemma 4.1. If VC(H) = d then dG (L(H)) = O(ld ln(ld)).
7
The technique for the lower bound on dN (L(W)) when W is the class of halfspaces in Rd is more
involved, and quite general. We consider a binary hypothesis class G ? {?1}[d]?[l] which consists
of functions having an arbitrary behaviour over [d] ? {i}, and a very uniform behaviour on other
inputs (such as mapping all other inputs to a constant). We show that L(G) N -shatters the set [d]?[l].
Since G is quite simple, this is usually not very hard to show. Finally, we show that the class of
halfspaces is richer than G, in the sense that the inputs to G can be mapped to points in Rd such that
the functions of G can be mapped to halfspaces. We conclude that dN (L(W)) ? dN (L(G)).
To prove the approximation error lower bounds stated in Section 3.2, we use the techniques of VC
theory in an unconventional way. The idea of this proof is as follows: Using a uniform convergence
argument based on the VC dimension of the binary hypothesis class, we show that there exists a
small labeled sample S whose approximation error for the hypothesis class is close to the approximation error for the distribution, for all possible label mappings. This allows us to restrict our
attention to a finite set of hypotheses, by their restriction to the sample. For these hypotheses, we
show that with high probability over the choice of label mapping, the approximation error on the
sample is high. A union bound on the finite set of possible hypotheses shows that the approximation
error on the distribution will be high, with high probability over the choice of the label mapping.
5
Implications
The first immediate implication of our results is that whenever the number of examples in the training
?
set is ?(dk),
MSVM should be preferred to OvA and TC. This is certainly true if the hypothesis
class of MSVM, L, has a zero approximation error (the realizable case), since the ERM is then
solvable with respect to L. Note that since the inclusions given in Theorem 3.5 are strict, there are
cases where the data is realizable with MSVM but not with HOvA or with respect to any tree.
In the non-realizable case, implementing the ERM is intractable for all of these methods. Nonetheless, for each method there are reasonable heuristics to approximate the ERM, which should work
well when the approximation error is small. Therefore, we believe that MSVM should be the method
of choice in this case as well due to its lower approximation error. However, variations in the optimality of algorithms for different hypothesis classes should also be taken into account in this analysis. We leave this detailed analysis of specific training heuristics for future work. Our analysis
also implies that it is highly unrecommended to use TC with a randomly selected ? or ECOC with a
random code whenever k > d. Finally, when the number of examples is much larger than dk 2 , the
analysis implies that it is better to choose the AP approach.
To conclude this section, we illustrate the relative performance of MSVM, OvA, TC, and ECOC, by
considering the simplistic case where d = 2, and each class is concentrated on a single point in R2 .
In the leftmost graph below, there are two classes in R2 , and the approximation error of all algorithms
is zero. In the middle graph, there are 9 classes ordered on the unit circle of R2 . Here, both MSVM
and OvA have a zero approximation error, but the error of TC and of ECOC with a random code will
most likely be large. In the rightmost graph, we chose random points in R2 . MSVM still has a zero
approximation error. However, OvA cannot learn the binary problem of distinguishing between the
middle point and the rest of the points and hence has a larger approximation error.
MSVM
OvA
TC/ECOC
3
3
3
3
3
7
3
7
7
Acknowledgements: Shai Shalev-Shwartz was supported by the John S. Cohen Senior Lectureship in Computer Science. Amit Daniely is a recipient of the Google Europe Fellowship in Learning
Theory, and this research is supported in part by this Google Fellowship.
8
References
E. L. Allwein, R.E. Schapire, and Y. Singer. Reducing multiclass to binary: A unifying approach
for margin classifiers. Journal of Machine Learning Research, 1:113?141, 2000.
S. Ben-David, N. Cesa-Bianchi, D. Haussler, and P. Long. Characterizations of learnability for
classes of {0, . . . , n}-valued functions. Journal of Computer and System Sciences, 50:74?86,
1995.
S. Bengio, J. Weston, and D. Grangier. Label embedding trees for large multi-class tasks. In NIPS,
2011.
A. Beygelzimer, J. Langford, and P. Ravikumar. Multiclass classification with filter trees. Preprint,
June, 2007.
K. Crammer and Y. Singer. On the algorithmic implementation of multiclass kernel-based vector
machines. Journal of Machine Learning Research, 2:265?292, 2001.
A. Daniely, S. Sabato, S. Ben-David, and S. Shalev-Shwartz. Multiclass learnability and the erm
principle. In COLT, 2011.
T. G. Dietterich and G. Bakiri. Solving multiclass learning problems via error-correcting output
codes. Journal of Artificial Intelligence Research, 2:263?286, January 1995.
Trevor Hastie and Robert Tibshirani. Classification by pairwise coupling. The Annals of Statistics,
26(1):451?471, 1998.
Ryan Rifkin and Aldebaro Klautau. In defense of one-vs-all classification. Journal of Machine
Learning Research, 5:101?141, 2004.
David E. Rumelhart, Geoffrey E. Hinton, and Ronald J. Williams. Learning internal representations
by error propagation. In David E. Rumelhart and James L. McClelland, editors, Parallel Distributed Processing ? Explorations in the Microstructure of Cognition, chapter 8, pages 318?362.
MIT Press, 1986.
G. Takacs. Convex polyhedron learning and its applications. PhD thesis, Budapest University of
Technology and Economics, 2009.
V. N. Vapnik. Statistical Learning Theory. Wiley, 1998.
J. Weston and C. Watkins. Support vector machines for multi-class pattern recognition. In Proceedings of the Seventh European Symposium on Artificial Neural Networks, April 1999.
9
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4,064 | 4,679 | Efficient Reinforcement Learning for High
Dimensional Linear Quadratic Systems
Adel Javanmard
Stanford University
Stanford, CA 94305
[email protected]
Morteza Ibrahimi
Stanford University
Stanford, CA 94305
[email protected]
Benjamin Van Roy
Stanford University
Stanford, CA 94305
[email protected]
Abstract
We study the problem of adaptive control of a high dimensional linear quadratic
(LQ) system. Previous work established the asymptotic convergence to an optimal
controller for various adaptive control schemes.
More recently, for the average
?
cost LQ problem, a regret bound of O( T ) was shown, apart form logarithmic
factors. However, this bound scales exponentially with p, the dimension of the
state space. In this work we consider the case where the matrices describing the
dynamic of the LQ system are sparse and their dimensions are large.
? We present
an adaptive control scheme that achieves a regret bound of O(p T ), apart from
logarithmic factors. In particular, our algorithm has an average cost of (1 + )
times the optimum cost after T = polylog(p)O(1/2 ). This is in comparison to
previous work on the dense dynamics where the algorithm requires time that scales
exponentially with dimension in order to achieve regret of times the optimal cost.
We believe that our result has prominent applications in the emerging area of
computational advertising, in particular targeted online advertising and advertising
in social networks.
1
Introduction
In this paper we address the problem of adaptive control of a high dimensional linear quadratic (LQ)
system. Formally, the dynamics of a linear quadratic system are given by
x(t + 1)
=
A0 x(t) + B 0 u(t) + w(t + 1),
c(t)
=
x(t)T Qx(t) + u(t)T Ru(t),
r
(1)
p
where u(t) ? R is the control (action) at time t, x(t) ? R is the state at time t, c(t) ? R is
the cost at time t, and {w(t + 1)}t?0 is a sequence of random vectors in Rp with i.i.d. standard
Normal entries. The matrices Q ? Rp?p and R ? Rr?r are positive semi-definite (PSD) matrices
that determine the cost at each step. The evolution of the system is described through the matrices
A0 ? Rp?p and B 0 ? Rp?r . Finally by high dimensional system we mean the case where p, r 1.
A celebrated fundamental theorem in control theory asserts that the above LQ system can be optimally controlled by a simple linear feedback if the pair (A0 , B 0 ) is controllable and the pair
(A0 , Q1/2 ) is observable. The optimal controller can be explicitly computed from the matrices
describing the dynamics and the cost. Throughout this paper we assume that controllability and
observability conditions hold.
When the matrix ?0 ? [A0 , B 0 ] is unknown, the task is that of adaptive control, where the system
is to be learned and controlled at the same time. Early works on the adaptive control of LQ systems
relied on the certainty equivalence principle [2]. In this scheme at each time t the unknown parameter ?0 is estimated based on the observations collected so far and the optimal controller for the
1
estimated system is applied. Such controllers are shown to converge to an optimal controller in the
case of minimum variance cost, however, in general they may converge to a suboptimal controller
[11]. Subsequently, it has been shown that introducing random exploration by adding noise to the
control signal, e.g., [14], solves the problem of converging to suboptimal estimates.
All the aforementioned work have been concerned with the asymptotic convergence of the controller
to an optimal controller. In order to achieve regret bounds, cost-biased parameter estimation [12, 8,
1], in particular the optimism in the face of uncertainty (OFU) principle [13] has been shown to be
effective. In this method a confidence set S is found such that ?0 ? S with high probability. The
b ? S with the smallest
system is then controlled using the most optimistic parameter estimates, i.e., ?
optimum cost. The asymptotic convergence of the average cost of OFU for the LQR problem was
shown in [6]. This asymptotic result was extended in [1] by providing a bound for the cumulative
regret. Assume x(0) = 0 and for a control policy ? define the average cost
T
1X
J? = limsup
E[ct ] .
(2)
T ?? T t=0
Further, define the cumulative regret as
R(T ) =
T
X
(c? (t) ? J? ) ,
(3)
t=0
where c? (t) is the cost of control policy ? at time t and J? = J(?0 ) ?
is the optimal average cost.
? T ) where O
? is hiding the
The algorithm proposed in [1] is shown to have cumulative regret O(
logarithmic factors. While no lower bound was provided
for
the
regret,
comparison
with the multi?
armed bandit problem where a lower bound of O( T ) was shown for the general case [9], suggests
that this scaling with time for the cumulative regret is optimal.
The focus of [1] was on scaling of the regret with time horizon T . However, the regret of the proposed algorithm scales poorly
? with dimension. More specifically, the analysis in [1] proves a regret
bound of R(T ) < Cpp+r+2 T . The current paper focuses on (many) applications where the state
and control dimensions are much larger than the time horizon of interest. A powerful reinforcement
learning algorithm for these applications should have regret which depends gracefully on dimension.
In general, there is little to be achieved when T < p as the number of degrees of freedom (pr + p2 )
is larger than the number of observations (T p) and any estimator can be arbitrary inaccurate. However, when there is prior knowledge about the unknown parameters A0 , B 0 , e.g., when A0 , B 0 are
sparse, accurate estimation can be feasible. In particular, [3] proved that under suitable conditions
the unknown parameters of a noise driven system (i.e., no control) whose dynamics are modeled by
linear stochastic differential equations can be estimated accurately with as few as O(log(p)) samples. However, the result of [3] is not directly applicable here since for a general feedback gain L
even if A0 and B 0 are sparse, the closed loop gain A0 ? B 0 L need not be sparse. Furthermore,
system dynamics would be correlated with past observations through the estimated gain matrix L.
Finally, there is no notion of cost in [3] while here we have to obtain bounds on cost and its scaling
with p. In this work we extend the result of [3] by showing that under suitable conditions, unknown parameters of sparse high dimensional LQ systems can be accurately estimated with as few
as O(log(p + r)) observations. Equipped with this efficient learning method,
? we show that sparse
?
T ).
high dimensional LQ systems can be adaptively controlled with regret O(p
To put this result in perspective note that even when x(t) = 0, the expected cost at time t + 1 is
?(p) due to the noise. Therefore, the cumulative cost at time T is bounded as ?(pT ). Comparing
this to our regret bound, we see that for T = polylog(p)O( 12 ), the cumulative cost of our algorithm
is bounded by (1 + ) times the optimum cumulative cost. In other words, our algorithm performs
close to optimal after polylog(p) steps. This is in contrast with the result of [1] where the algorithm
needs ?(p2p ) steps in order to achieve regret of times the optimal cost.
Sparse high dimensional LQ systems appear in many engineering applications. Here we are particularly motivated by an emerging field of applications in marketing and advertising. The use of
dynamical optimal control models in advertising has a history of at least four decades, cf. [17, 10]
for a survey. In these models, often a partial differential equation is used to describe how advertising
expenditure over time translates into sales. The basic problem is to find the advertising expenditure that maximizes the net profit. The focus of these works is to model the temporal dynamics of
2
the advertising expenditure (the control variable) and the variables of interest (sales, goodwill level,
etc.). There also exists a rich literature studying the spatial interdependence of consumers? and
firms? behavior to devise marketing schemes [7]. In these models space can be generalized beyond
geographies to include notions like demographies and psychometry.
Combination of spatial interdependence and temporal dynamics models for optimal advertising was
also considered [16, 15]. A simple temporal dynamics model is extended in [15] by allowing state
and control variables to have spatial dependence and introducing a diffusive component in the controlled PDE which describes the spatial dynamics. The controlled PDE is then showed to be equivalent to an abstract linear control system of the form
dx(t)
= Ax(t) + Bu(t).
dt
(4)
Both [15] and [7] are concerned with the optimal control and the interactions are either dictated
by the model or assumed known. Our work deals with a discrete and noisy version of (4) where
the dynamics is to be estimated but is known to be sparse. In the model considered in [15] the
state variable x lives in an infinite dimensional space. Spatial models in marketing [7] usually
consider state variables which have a large number of dimensions, e.g., number of zip codes in the
US (? 50K). High dimensional state space and control is a recurring theme in these applications.
In particular, with the modern social networks customers are classified in a highly granular way, potentially with each customer representing his own class. With the number of classes and complexity
of their interactions, its unlikely that we could formulate an effective model a priori for how classes
interact. Further, the nature of these interactions change over time with the changing landscape of
Internet services and information available to customers. This makes it important to efficiently learn
from real-time data about the nature of these interactions.
Notation: We bundle the unknown parameters into one variable ?0 = [A0 , B 0 ] ? Rp?q where
q = p + r and call it the interaction matrix. For v ? Rn , M ? Rm?n and p ? 1, we denote by kvkp
the standard p-norm and by kM kp the corresponding operator norm. For 1 ? i ? m, Mi represents
the ith row of matrix M . For S ? [m], J ? [n], MSJ is the submatrix of M formed by the rows in
S and columns in J. For a set S denote by |S| its cardinality. For an integer n denote by [n] the set
{1, . . . , n}.
2
Algorithm
Our algorithm employs the Optimism in the Face of Uncertainty (OFU) principle in an episodic
fashion. At the beginning of episode i the algorithm constructs a confidence set ?(i) which is
guaranteed to include the unknown parameter ?0 with high probability. The algorithm then chooses
e (i) ? ?(i) that has the smallest expected cost as the estimated parameter for episode i and applies
?
the optimal control for the estimated parameter during episode i.
The confidence set is constructed using observations from the last episode only but the length of
episodes are chosen to increase geometrically allowing for more accurate estimates and shrinkage
of the confidence set by a constant factor at each episode. The details of each step and the pseudo
code for the algorithm follows.
Constructing confidence set: Define ?i to be the start of episode i with ?0 = 0. Let L(i) be the
controller that has been chosen for episode i. For t ? [?i , ?i+1 ) the system is controlled by u(t) =
?L(i) x(t) and the system dynamics can be written as x(t + 1) = (A0 ? B 0 L(i) )x(t) + w(t + 1). At
b is obtained by solving the following convex
the beginning of episode i + 1, first an initial estimate ?
optimization problem for each row ?u ? Rq separately:
b (i+1) ? argmin L(?u ) + ?k?u k1 ,
?
u
(5)
where
?i+1 ?1
X
1
e (i) x(t)}2 ,
{xu (t + 1) ? ?u L
L(?u ) =
2??i+1 t=?
i
3
??i+1 = ?i+1 ? ?i ,
(6)
A LGORITHM: Reinforcement learning algorithm for LQ systems.
Input: Precision , failure probability 4?, initial (?, Cmin , ?) identifiable controller L(0) , `(?0 , )
e (i) , confidence sets ?(i) and controllers L(i)
Output: Series of estimates ?
(0)
1: Let `0 = max(1, maxj?[r] kLj k2 ), and
4 ? 103 k 2 `20
1
k
4kq
n0 =
+
log(
),
2
?(1 ? ?)Cmin
2
(1 ? ?)2
?
k
4kq
4 ? 103 k 2 `(?0 , )2 1
+
log(
n1 =
).
2
(1 ? ?)Cmin
2
(1 ? ?)2
?
Pi
Let ??0 = n0 , ??i = 4i (1 + i/ log(q/?))n1 for i ? 1, and ?i = j=0 ??j .
2: for i = 0, 1, 2, . . . do
3:
Apply the control u(t) = ?L(i) x(t) until ?i+1 ? 1 and observe the trace {x(t)}?i ?t<?i+1 .
b (i+1) from (5) and construct the confidence set ?(i+1) .
4:
Calculate the estimate ?
(i+1)
e
e (i+1) ).
5:
Calculate ?
from (9) and set L(i+1) ? L(?
e (i) = [I, ?L(i)T ]T . The estimator ?
b u is known as the LASSO estimator. The first term
and L
in the cost function is the normalized negative log likelihood which measures the fidelity to the
observations while the second term imposes the sparsity constraint on ?u . ? is the regularization
parameter.
For ?(1) , ?(2) ? Rp?q define the distance d(?(1) , ?(2) ) as
(2)
d(?(1) , ?(2) ) = max k?(1)
u ? ?u k2 ,
u?[p]
(7)
where ?u is the uth row of the matrix ?. It is worth noting that for k-sparse matrices with k
constant, this distance does not scale with p or q. In particular, ?
if the absolute value of the elements
of ?(1) and ?(2) are bounded by ?max then d(?(1) , ?(2) ) ? 2 k?max .
b (i) the algorithm constructs the confidence set for episode i as
Having the estimator ?
b (i) ) ? 2?i },
?(i) = {? ? Rp?q | d(?, ?
(8)
where > 0 is an input parameter to the algorithm. For any fixed ? > 0, by choosing ?i judiciously
we ensure that with probability at least 1 ? ?, ?0 ? ?(i) , for all i ? 1. (see Theorem 3.2).
Design of the controller: Let J(?) be the minimum expected cost if the interaction matrix is
? = [A, B] and denote by L(?) the optimal controller that achieves the expected cost J(?). The
e (i) ?
algorithm implements OFU principle by choosing, at the beginning of episode i, an estimate ?
(i)
? such that
e (i) ? argmin J(?).
?
(9)
???(i)
e (i) is then applied during episode i, i.e., u(t) =
The optimal control corresponding to ?
e (i) )x(t) for t ? [?i , ?i+1 ). Recall that for ? = [A, B], the optimal controller is given through
?L(?
the following relations
K(?) = Q + AT K(?)A ? AT K(?)B(B T K(?)B + R)?1 B T K(?)A , (Riccati equation)
L(?) = (B T K(?)B + R)?1 B T K(?)A .
The pseudo code for the algorithm is summarized in the table.
3
Main Results
In this section we present performance guarantees in terms of cumulative regret and learning accuracy for the presented algorithm. In order to state the theorems, we first need to present some
assumptions on the system.
4
e = [I, ?LT ]T ? Rq?p and let ? ? Rp?p be a solution to
Given ? ? Rp?q and L ? Rr?p , define L
the following Lyapunov equation
e L
e T ?T = I.
? ? ?L?
(10)
0
0
If the closed loop system (A ? B L) is stable then the solution to the above equation exists and the
state vector x(t) has a Normal stationary distribution with covariance ?.
We proceed by introducing an identifiable regulator.
e =
Definition 3.1. For a k-sparse matrix ?0 = [A0 , B 0 ] ? Rp?q and L ? Rr?p , define L
e L
e T where ? is the solution of Eq. (10) with ? = ?0 . De[I, ?LT ]T ? Rq?p and let H = L?
fine L to be (?, Cmin , ?) identifiable (with respect to ?0 ) if it satisfies the following conditions for
all S ? [q], |S| ? k.
(1) kA0 ? B 0 Lk2 ? ? < 1,
(2) ?min (HSS ) ? Cmin ,
?1
(3) kHS c S HSS
k? ? 1 ? ?.
The first condition simply states that if the system is controlled using the regulator L then the closed
loop autonomous system is asymptotically stable. The second and third conditions are similar to
what is referred to in the sparse signal recovery literature as the mutual incoherence or irreprepresentable conditions. Various examples and results exist for the matrix families that satisfy these
conditions [18]. Let S be the set of indices of the nonzero entries in a specific row of ?0 . The
second condition states that the corresponding entries in the extended state variable y = [xT , uT ] are
sufficiently distinguishable from each other. In other words, if the trajectories corresponding to this
group of state variables are observed, non of them can be well approximated as a linear combination
of the others. The third condition can be thought of as a quantification of the first vs. higher order
dependencies. Consider entry j in the extended state variable. Then, the dynamic of yj is directly
influenced by entries yS . However they are also influenced indirectly by other entries of y. The third
condition roughly states that the indirect influences are sufficiently weaker than the direct influences.
There exists a vast literature on the applicability of these conditions and scenarios in which they are
known to hold. These conditions are almost necessary for the successful recovery by `1 relaxation.
For a discussion on these and other similar conditions imposed for sparse signal recovery we refer
the reader to [19] and [20] and the references therein.
Define ?min = mini?[p],j?[q],?0ij 6=0 |?0ij |. Our first result states that the system can be learned
efficiently from its trajectory observations when it is controlled by an identifiable regulator.
Theorem 3.2. Consider the LQ system of Eq. (1) and assume ?0 = [A0 , B 0 ] is k-sparse. Let
u(t) = ?Lx(t) where L is a (?, Cmin , ?) identifiable regulator with respect to ?0 and define
` = max(1, maxj?[r] kLj k2 ). Let n denote the number of samples of the trajectory that is observed.
3
For any 0 < < min(?min , 2` , 1??
), there exists ? such that, if
1
k
4kq
4 ? 103 k 2 `2
+
log(
),
(11)
n? 2
2
? (1 ? ?)Cmin
2
(1 ? ?)2
?
b of Eq. (5) satisfies d(?,
b ?0 ) ? with probability
then the `1 -regularized least squares solution ?
p
larger than 1 ? ?. In particular, this is achieved by taking ? = 6` log(4q/?)/(n?2 (1 ? ?)) .
Our second result states that equipped with an efficient learning algorithm, the LQ system of Eq. (1)
?
3
?
T log 2 (1/?)) under suitable assumptions.
can be controlled with regret O(p
Define an -neighborhood of ?0 as N (?0 ) = {? ? Rp?q | d(?0 , ?) ? }. Our assumption asserts
the identifiably of L(?) for ? close to ?0 .
Assumption: There exist , C > 0 such that for all ? ? N (?0 ), L(?) is identifiable w.r.t. ?0 and
?L (?0 , ) =
sup
??N
?K (?0 , ) =
kL(?)k2 ? C,
(?0 )
sup
??N (?0 )
Also define
`(?0 , ) =
sup
max(1, max kLj (?)k2 ) .
j?[r]
??N (?0 )
Note that `(?0 , ) ? max(C, 1), since maxj?[r] kLj (?)k2 ? kL(?)k2 .
5
kK(?)k2 ? C.
Theorem 3.3. Consider the LQ system of Eq. (1). For some constants , Cmin and 0 < ?, ? < 1,
assume that an initial (?, Cmin , ?) identifiable regulator L(0) is given. Further, assume that for any
? ? N (?0 ), L(?) is (?, Cmin , ?) identifiable. Then, with probability at least 1 ? ? the cumulative
regret of A LGORITHM (cf. the table) is bounded as
?
3
?
R(T ) ? O(p
T log 2 (1/?)) ,
(12)
? is hiding the logarithmic factors.
where O
4
4.1
Analysis
Proof of Theorem 3.2
To prove theorem 3.2 we first state a set of sufficient conditions for the solution of the `1 -regularized
least squares to be within some distance, as defined by d(?, ?), of the true parameter. Subsequently,
we prove that these conditions hold with high probability.
Define X = [x(0), x(1), . . . , x(n ? 1)] ? Rp?n and let W = [w(1), . . . , w(n)] ? Rp?n be the
matrix containing the Gaussian noise realization. Further let the Wu denote the uth row of W .
Define the normalized gradient and Hessian of the likelihood function (6) as
T
b = ??L(?0 ) = 1 LXW
e
G
u
u ,
n
T eT
b = ?2 L(?0 ) = 1 LXX
e
H
L .
u
n
(13)
The following proposition, a proof of which can be found in [20], provides a set of sufficient conditions for the accuracy of the `1 -regularized least squares solution.
Proposition 4.1. Let S be the support of ?0u with |S| < k, and H be defined per Definition 3.1.
Assume there exist 0 < ? < 1 and Cmin > 0 such that
?min (HS,S ) ? Cmin ,
?1
kHS c ,S HS,S
k? ? 1 ? ? .
(14)
For any 0 < < ?min if the following conditions hold
??
,
3
? Cmin
? ,
?
12 k
b S C S ? HS C S k?
kH
Cmin
? ?,
4k
? Cmin
? ,
? HSS k? ?
12 k
b S k? ?
kG
b ??
kGk
b SS
kH
(15)
(16)
b u , ?0u ) ? .
the `1 -regularized least square solution (5) satisfies d(?
In the sequel, we prove that the conditions in Proposition 4.1 hold with high probability given that the
assumptions of Theorem 3.2 are satisfied. A few lemmas are in order proofs of which are deferred
to the Appendix.
b concentrates in infinity norm around its mean of zero.
The first lemma states that G
Lemma 4.2. Assume ? = kA0 ? B 0 Lk2 < 1 and let ` = max(1, maxi?[r] kLi k2 ). Then, for any
S ? [q] and 0 < < 2`
n(1 ? ?)2
b
P kGS k? > ? 2|S| exp ?
.
(17)
4`2
To prove the conditions in Eq. (16) we first bound in the following lemma the absolute deviations
b from their mean H, i.e., |H
b ij ? Hij |.
of the elements of H
3
Lemma 4.3. Let i, j ? [q], ? = kA0 ? B 0 Lk2 < 1, and 0 < < 1??
< n . Then,
3 2
b ij ? Hij | > ) ? 2 exp ? n(1 ? ?) .
P(|H
24`2
b JS ? HJS k? for J, S ? [q].
The following corollary of Lemma 4.3 bounds kH
6
(18)
3|S|
3
Corollary 4.4. Let J, S ? [q], ? = kA0 ? B 0 Lk2 < 1, < 1??
, and n > 1??
. Then,
3 2
b JS ? HJS k? > ) ? 2|J||S| exp ? n(1 ? ?) .
P(kH
2
24|S| `2
The proof of Corollary 4.4 is by applying union bound as
b JS ? HJS k? > ) ? |J||S| max P(|H
b ij ? Hij | > /|S|).
P(kH
i?J,j?S
(19)
(20)
Proof of Theorem 3.2. We show that the conditions given by Proposition 4.1 hold. The conditions
in Eq. (14) are true by the assumption of identifiability of L with respect to ?0 . In order to make the
b imply the second constraint on G,
b we assume that ??/3 ? Cmin /(4k) ? ?,
first constraint on G
b ? > ??/3) ? ?/2 if
which is ensured to hold if ? ? Cmin /(6k). By Lemma 4.2, P(kGk
2
4q
36`
log( ) .
?2 =
(21)
n(1 ? ?)?2
?
Requiring ? ? Cmin /(6k), we obtain
4q
362 k 2 `2
log( ) .
(22)
n? 2 2 2
? Cmin (1 ? ?)
?
?
b can also be aggregated as kH
b [q],S ?H[q],S k? ? ?Cmin /(12 k) . By Corollary
The conditions on H
?
b
4.4, P(kH[q]S ? H[q]S k? > ?Cmin /(12 k)) ? ?/2 if
3456 k 3 `2
4kq
log(
).
(23)
2
?2 (1 ? ?)3 Cmin
?
Merging the conditions in Eq. (22) and (23) we conclude that the conditions in Proposition 4.1 hold
with probability at least 1 ? ? if
k
4 ? 103 k 2 `2
1
4kq
+
).
(24)
n? 2
log(
2
2
2
? (1 ? ?)Cmin
(1 ? ?)
?
Which finishes the proof of Theorem 3.2.
n?
4.2
Proof of Theorem 3.3
The high-level idea of the proof is similar to the proof of main Theorem in [1]. First, we give a
decomposition for the gap between the cost obtained by the algorithm and the optimal cost. We then
upper bound each term of the decomposition separately.
4.2.1
Cost Decomposition
Writing the Bellman optimality equations [5, 4] for average cost dynamic programming, we get
e t ) + x(t)T K(?
e t )x(t) = min x(t)T Qx(t) + uT Ru + E z(t + 1)T K(?
e t )z(t + 1)|Ft ,
J(?
u
e t = [A,
e B]
e is the estimate used at time t, z(t + 1) = A
et x(t) + B
et u + w(t + 1), and Ft
where ?
is the ?-field generated by the variables {(z? , x? )}t? =0 . Notice that the left-hand side is the average
cost occurred with initial state x(t) [5, 4]. Therefore,
e t ) + x(t)T K(?
e t )x(t) = x(t)T Qx(t) + u(t)T Ru(t)
J(?
et x(t) + B
et u(t) + w(t + 1))T K(?
e t )(A
et x(t) + B
et u(t) + w(t + 1))|Ft
+ E (A
et x(t) + B
et u(t))T K(?
e t )(A
et x(t) + B
et u(t))|Ft
= x(t)T Qx(t) + u(t)T Ru(t) + E (A
e t )w(t + 1)|Ft ]
+ E w(t + 1)T K(?
e t )x(t + 1)|Ft
= x(t)T Qx(t) + u(t)T Ru(t) + E x(t + 1)T K(?
et x(t) + B
et u(t))T K(?
e t )(A
et x(t) + B
et u(t))
+ (A
e t )(A0 x(t) + B 0 u(t)) .
? (A0 x(t) + B 0 u(t))T K(?
7
Consequently
T
X
T
X
e t ) + C1 + C2 + C3 ,
x(t) Qx(t) + u(t) Ru(t) =
J(?
T
T
t=0
(25)
t=0
where
C1 =
T
X
e t )x(t) ? E x(t + 1)T K(?
e t+1 )x(t + 1) Ft ,
x(t)T K(?
(26)
t=0
C2 = ?
T
X
e t ) ? K(?
e t+1 ))x(t + 1) Ft ,
E x(t + 1)T (K(?
(27)
t=0
C3 = ?
T
X
et x(t) + B
et u(t))T K(?
e t )(A
et x(t) + B
et u(t))
(A
t=0
e t )(A0 x(t) + B 0 u(t)) .
? (A0 x(t) + B 0 u(t))T K(?
4.2.2
(28)
Good events
We proceed by defining the following two events in the probability space under which we can bound
the terms C1 , C2 , C3 . We then provide a lower bound on the probability of these events.
p
E1 = {?0 ? ?(i) , for i ? 1}, E2 = {kw(t)k ? 2 p log(T /?), for 1 ? t ? T + 1}.
4.2.3
Technical lemmas
The following lemmas establish upper bounds on C1 , C2 , C3 .
Lemma 4.5. Under the event E1 ? E2 , the following holds with probability at least 1 ? ?.
r
?
128 C ?
T
1
C1 ?
T p log( ) log( ) .
2
(1 ? ?)
?
?
Lemma 4.6. Under the event E1 ? E2 , the following holds.
8C
T
C2 ?
p log( ) log T .
2
(1 ? ?)
?
Lemma 4.7. Under the event E1 ? E2 , the following holds with probability at least 1 ? ?.
s
r
C 52
?
pT
k2 1 + C
4kq
|C3 | ? 800
?
? log( ) log(
) ? p log T T .
1+
k
2
1??
(1 ? ?)
Cmin
?
?
Lemma 4.8. The following holds true.
P(E1 ) ? 1 ? ?,
Therefore, P(E1 ? E2 ) ? 1 ? 2?.
P(E2 ) ? 1 ? ?.
(29)
(30)
(31)
(32)
We are now in position to prove Theorem 3.3.
Proof (Theorem 3.3). Using cost decomposition (Eq. (25)), under the event E1 ? E2 , we have
T
T
X
X
e t ) + C1 + C2 + C3
(x(t)T Qx(t) + u(t)T Ru(t)) =
J(?
t=0
t=0
? T J(?0 ) + C1 + C2 + C3 ,
e t by the algorithm (cf. Eq (9)) and the fact that
where the last inequality stems from the choice of ?
?0 ? ?t , for all t under the event E1 . Hence, R(T ) ? C1 + C2 + C3 . Now using the bounds on
C1 , C2 , C3 , we get the desired result.
Acknowledgments
The authors thank the anonymous reviewers for their insightful comments. A.J. is supported by a
Caroline and Fabian Pease Stanford Graduate Fellowship.
8
References
[1] Y. Abbasi-Yadkori and C. Szepesv?ari. Regret bounds for the adaptive control of linear quadratic
systems. Proceeding of the 24th Annual Conference on Learning Theory, pages 1?26, 2011.
[2] Y. Bar-Shalom and E. Tse. Dual effect, certainty equivalence, and separation in stochastic
control. Automatic Control, IEEE Transactions on, 19(5):494?500, 1974.
[3] J. Bento, M. Ibrahimi, and A. Montanari. Learning networks of stochastic differential equations. Advances in Neural Information Processing Systems 23, pages 172?180, 2010.
[4] D. Bertsekas. Dynamic Programming: Deterministic and Stochastic Models. Prentice-Hall,
1987.
[5] D. P. Bertsekas. Dynamic Programming and Optimal Control. Athena Scientific, 3rd edition,
2007.
[6] S. Bittanti and M. Campi. Adaptive control of linear time invariant systems: the bet on the best
principle. Communications in Information and Systems, 6(4):299?320, 2006.
[7] E. Bradlow, B. Bronnenberg, G. Russell, N. Arora, D. Bell, S. Duvvuri, F. Hofstede, C. Sismeiro, R. Thomadsen, and S. Yang. Spatial models in marketing. Marketing Letters,
16(3):267?278, 2005.
[8] M. Campi. Achieving optimality in adaptive control: the bet on the best approach. In Decision
and Control, 1997., Proceedings of the 36th IEEE Conference on, volume 5, pages 4671?4676.
IEEE, 1997.
[9] V. Dani, T. Hayes, and S. Kakade. Stochastic linear optimization under bandit feedback. In
Proceedings of the 21st Annual Conference on Learning Theory (COLT), 2008.
[10] G. Feichtinger, R. Hartl, and S. Sethi. Dynamic optimal control models in advertising: recent
developments. Management Science, pages 195?226, 1994.
[11] L. Guo and H. Chen. The a? strom-wittenmark self-tuning regulator revisited and els-based
adaptive trackers. Automatic Control, IEEE Transactions on, 36(7):802?812, 1991.
[12] P. Kumar and A. Becker. A new family of optimal adaptive controllers for markov chains.
Automatic Control, IEEE Transactions on, 27(1):137?146, 1982.
[13] T. Lai and H. Robbins. Asymptotically efficient adaptive allocation rules. Advances in applied
mathematics, 6(1):4?22, 1985.
[14] T. Lai and C. Wei. Least squares estimates in stochastic regression models with applications to
identification and control of dynamic systems. The Annals of Statistics, 10(1):154?166, 1982.
[15] C. Marinelli and S. Savin. Optimal distributed dynamic advertising. Journal of Optimization
Theory and Applications, 137(3):569?591, 2008.
[16] T. Seidman, S. Sethi, and N. Derzko. Dynamics and optimization of a distributed salesadvertising model. Journal of Optimization Theory and Applications, 52(3):443?462, 1987.
[17] S. Sethi. Dynamic optimal control models in advertising: a survey. SIAM review, pages 685?
725, 1977.
[18] J. Tropp. Just relax: Convex programming methods for identifying sparse signals in noise.
Information Theory, IEEE Transactions on, 52(3):1030?1051, 2006.
[19] M. Wainwright. Sharp thresholds for high-dimensional and noisy sparsity recovery usingconstrained quadratic programming (lasso). Information Theory, IEEE Transactions on,
55(5):2183?2202, 2009.
[20] P. Zhao and B. Yu. On model selection consistency of Lasso. The Journal of Machine Learning
Research, 7:2541?2563, 2006.
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4,065 | 468 | Operators and curried functions:
Training and analysis of simple recurrent networks
Anthony Bloesch,
Janet Wiles
Depts of Psychology and Computer Science,
University of Queensland
QLD 4072 Australia.
[email protected]
Dept of Computer Science,
University of Queensland,
QLD 4072 Australia
[email protected]
Abstract
We present a framework for programming tbe bidden unit representations of
simple recurrent networks based on the use of hint units (additional targets at
the output layer). We present two ways of analysing a network trained within
this framework: Input patterns act as operators on the information encoded by
the context units; symmetrically, patterns of activation over tbe context units
act as curried functions of the input sequences. Simulations demonstrate that a
network can learn to represent three different functions simultaneously and
canonical discriminant analysis is used to investigate bow operators and curried
functions are represented in the space of bidden unit activations.
1 INTRODUCTION
Many recent papers have contributed to the understanding of recurrent networks and their
potential for modelling sequential pbenomena (see for example Giles, Sun, Chen, Lee, &
Chen, 1990; Elman, 1989; 1990; Jordan, 1986; Cleeremans, Servan-Schreiber &
McClelland, 1989; Williams & Zipser, 1988). Of particular interest in these papers is
the development of recurrent architectures and learning algorithms able to solve complex
problems. The perspective of the work we present bere has many similarities with these
stUdies, however, we focus on programming a recurrent network for a specific task, and
hence provide appropriate sequences of inputs to learn the temporal component.
The function computed by a neural network is conventionally represented by its weights.
During training, the task of a network is to learn a set of weights that causes tbe
appropriate action (or set of context-specific actions) for each input pattern. However, in
325
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Wiles and Bloesch
a network with recurrent connections, patterns of activation are also part of the function
computed by a network. After training (when the weights have been fixed) each input
pattern has a specific effect on the pattern of activation across the hidden and output units
which is modulated by the current state of those units. That is, each input pattern is a
context sensitive operator on the state of the system.
To illustrate this idea, we present a task in which many sequences of the form, {F, argl,
..., argn} are input to a network, which is required to output the value of each function,
F(argl, ... , argn). The task is interesting since it illustrates how more than one function
can be computed by the same network and how the function selected can be specified by
the inputs. Viewing all the inputs (both function patterns, F, and argument patterns,
argi) as operators allows us to analyse the effect of each input on the state of the network
(the pattern of activation in the hidden and context units). From this perspective, the
weights in the network can be viewed as an interpreter which has been programmed to
carry out the operations specified by each input pattern.
We use the term programming intentionally, to convey the idea that the actions of each
input pattern playa specific role in the processing of a sequence. In the simulations
described in this paper, we use the simple recurrent network (SRN) proposed by Elman
(1990). The art of programming enters the simulations in the use of extra target units,
called hints, that are provided at the output layer. At each step in learning a sequence,
hints specify all the information that the network must preserve in the hidden unit
representation (the state of the system) in order to calculate outputs later in the sequence
(for a discussion of the use of hints in training a recurrent network see Rumelhart,
Hinton & Williams, 1986).
2 SIMULATIONS
Three different boolean functions and their arguments were specified as sub-sequences of
patterns over the inputs to an SRN. The network was required to apply the function
specified by the first pattern in each sequence to each of the subsequent arguments in
turn. The functions provided were boolean functions of the current input and previous
output, AND, OR and XOR (Le., exclusive-or) and the arguments were arbitrary length
strings of O's and 1'so The context units were not reset between sub-sequences. An SRN
with 3 input, 5 hidden, 5 context, 1 output and 5 hint units was trained using
backpropagation with a momentum term. The 5 hint units at the output layer provided
information about the boolean functions during training (via the backpropagation of
errors), but not during testing. The network was trained on three data sets each
containing 700 (ten times the number of weights in the network) randomly generated
patterns, forming function and arguments sequences of average length 0.5, 2 and 4
arguments respectively. The network was trained for one thousand iterations on each
training set.
2.1 RESULTS AND GENERALISATION
After training, the network correctly computed every pattern in the three training sets
(using a closest match criterion for scoring the output) and also in a test set of sequences
generated using the same statistics. Generalisation test data consisting of all possible
sequences composed of each function and eight arguments, and long sequences each of 50
arguments also produced the correct output for every pattern in every sequence. To test
Operators and curried functions: Training and analysis of simple recurrent networks
c::
c::
u
8.
8
0
Co)
~
Co)
'8
0
?
Co)
"0
~.
First canonical component
.!:l
~
~AI
flJa
r-:-
,.
jj ?
~. L
?
ffiQ
rr
,.
~
-!.
First canonical component
lb.
la.
Figure la. The hidden unit patterns for the training data, projected onto the first two
canonical components. These components separate the patterns into 3 distinct regions
corresponding to the initial pattern (AND, OR or XOR) in each sequence. lb. The first
and third canonical components further separate the hidden unit patterns into 6 regions
which have been marked in the diagrams above by the corresponding output classes AI,
AO, Rl, RO, Xl and XO. These regions are effectively the computational states of the
network.
1~G
O'U
Figure 2. Finite state machine to compute the three-function task.
Another way of considering sub-sequences in the input stream is to describe all the
inputs as functions, not over the other inputs, as above, but as functions of the state (for
which we use the term operators). Using this terminology, a sub-sequence is a
composition of operators which act on the current state,
G(S(t) =argt ? ... ? arg2 ? argJo S(O),
where (f ? g) (x)
=f(g(x)), and S(O) is the initial state of the network.
A consequence of
describing the input patterns as operators is that even the 0 and 1 data bits can be seen as
operators that transform the internal state (see Box 1).
327
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Wiles and Bloesch
[].
ru-
.
?
3a.
First canonical component
First canonical component
3d.
First canonical component
.
?
First canonical component
3e.
First canonical component
Figure 3. State transitions caused by each input pattern, projected onto the ftrst and third
canonical components of the hidden unit patterns (generated by the training data as in
Figure 1). 3a-c. Transitions caused by the AND, OR and XOR input patterns
respectively. From every point in the hidden unit space, the input patterns for AND, OR
and XOR transform the hidden units to values corresponding to a point in the regions
marked AI, RO and XO respectively. 3d-e. Transitions caused by the 0 and I input
patterns respectively. The 0 and I inputs are context sensitive operators. The 0 input
causes changes in the hidden unit patterns corresponding to transitions from the state Al
to AO, but does not cause transitions from the other 5 regions. Conversely. a I input
does not cause the hidden unit patterns to change from the regions AI, AO or RI, but
causes transitions from the regions RO, Xl and XO.
Operators and curried functions: Training and analysis of simple recurrent networks
Input
operators
Patterns on
the input units
AND
OR
XOR
1
011
110
101
111
o
000
Effect on information
encoded in the state
cf-'AND
cf-. OR
cf-'XOR
x(t) -. x(t-1)
1
NOT(x(t-1?
x(t) -.
0
x(t-1)
x(t-1)
if cf= AND
if cf= OR
if cf= XOR
if cf= AND
if cf= OR
if cf= XOR
Box 1. Operators for the 5 input patterns. The operation performed by each input
pattern is described in terms of the effect it has on information encoded by the hidden
unit patterns. The first and second columns specify the input operators and their
corresponding input patterns. The third column specifies the effect that each input in a
sub-sequence has on information encoded in the state, represented as cf, for current
function, and x(t) for the last output.
For each input pattern, we plotted all the transitions in hidden unit space resulting from
that input projected onto the canonical components used in Figure 1. Figures 3a to 3e
show transitions for each of the five input operators. For the three "function" inputs,
OR, AND, and XOR, the effect is to collapse the hidden unit patterns to a single region
- a particular state. These are relatively context insensitive operations. For the two
"argument" inputs, 0 and 1, the effect is sensitive to the context in which the input
occurs (i.e., the previous state of the hidden units). A similar analysis of the states
themselves focuses on the hidden unit patterns and the information that they must encode
in order to compute the three-function task. At each timestep the weights in the network
construct a pattern of activation over the hidden units that reduces the structured
arguments of a complex function of several arguments by a simpler function of one less
argument. This can be represented as follows:
G(F, arg1, ... argn) -.
-.
-.
F(arg1, ... argn)
Fargl(arg2, ... argn)
F arg l arg 2(arg3, ... argn).
This process of replacing structured arguments by a corresponding sequence of simple
ones is known as currying the input sequence (for a review of curried functions, see Bird
and Wadler, 1988). Using this terminology, the pattern of activation in the hidden units
is a curried function of the entire input sequence up to that time step. The network
combines the previous hidden unit patterns (preserved in the context units) with the
current input patterns to compute the next curried function in the sequence. Since there
are 6 states required by the network, there are 6 classes of equivalent curried functions.
Figure 4 shows the transition diagrams for each of the 6 equivalence classes of curried
functions from the same simulation shown in Figures 1 and 3.
329
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Wiles and Bloesch
?
First canonical component
4a.
First canonical component
4d.
?
First canonical component
4b.
First canonical component
4c.
First canonical component
4e.
First canonical component
4f.
Figure 4. State transitions for each hidden unit pattern, grouped into classes of curried
functions, projected onto the frrst and third canonical components. 4a-f. Transitions from
AI, RI, Xl, AO, RO and XO respectively. Each pattern of activation corresponds to a
curried function of the input sequence up to that item in the sequence.
Operators and curried fUnctions: Training and analysis of simple recurrent networks
how often the network finds a good solution, five simulations were completed with the
above parameters, all started with different sets of random weights, and randomly
generated training patterns. Three simulations learnt the training set perfectly (the other
two simulations appeared to be converging, but slowly: worst case error less than 1%).
On the test data, the results were also good (worst case 7% error).
2.2 ANALYSIS
The hidden unit patterns generated by the training data in the simulations described above
were analysed using canonical discriminant analysis (CDA, Kotz & Johnson, 1982). Six
output classes were specified, corresponding to one class for each output for each
function. The output classes were used to compute the first three canonical components
of the hidden unit patterns (which are 5-dimensional patterns corresponding to the 5
hidden units). The graph of the first two canonical components (see Figure 1a) shows
the hidden unit patterns separated into three tight clusters, corresponding to the sequence
type (OR, AND and XOR). The first and third canonical components (see Figure 1b)
reveals more of the structure within each class. The six classes of hidden unit patterns
are spread across six distinct regions (these correspond to the 6 states of the minimal
finite state machine, as shown in Figure 2). The first canonical component separates the
hidden unit patterns into sequence type (OR, AND, or XOR, separated across the page).
Within each region, the third canonical component separates the outputs into O's and l's
(separated down the page). Cluster analysis followed by CDA on the clusters gave
similar results.
3 DISCUSSION
In a network that is dedicated to computing a boolean function such as XOR, it seems
obvious that the information for computing the function is in the weights. The
simulations described in this paper show that this intuition does not necessarily
generalise to other networks. The three-function task requires that the network use the
first input in a sequence to select a function which is then applied to subsequent
arguments. In general, for any given network, the function that is computed over a given
sulrsequence will be specified by the interaction between the weights and the activation
pattern.
The function computed by the networks in these simulations can be described in terms of
the output of the global function, O(t) = G(argl, ... , argt), computed by the weights of
the network, which is a function of the whole input sequence. An equivalent description
can be given in terms of sulrsequences of the input stream, which specify a boolean
function over subsequent arguments, G(F, argl, ..., argt) =F(argJ, ... , argt). Both these
levels of description follow the traditional approach of separating functions and data,
where the patterns of activity can be described as either one or the other.
331
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Wiles and Bloesch
It appears to us that descriptions based on operators and curried functions provide a
promising approach for the integration of representation and process within recurrent
networks. For example, in the simulations described by Elman (1990), words can be
understood as denoting operators which act on the state of the recurrent network, rather
than denoting objects as they do in traditional linguistic theory. The idea of currying can
also be applied to feedback from the output layer, for example in the networks developed
by Jordan (1986), or to the product units used by Giles et al. (1990).
Acknowledgements
We thank Jeff Elman, Ian Hayes, Julie Stewart and Bill Wilson for many discussions on
these ideas, and Simon Dennis and Steven Phillips for developing the canonical
discriminant program. This work was supported by grants from the Australian Research
Council and A. Bloesch was supported by an Australian Postgraduate Research Award.
References
Bird, R., and Wadler P. (1988). Introduction to Functional Programming, Prentice Hall,
NY.
Cleeremans, A., Servan-Schreiber, D., and McClelland, J.L. (1989). Finite state
automata and simple recurrent networks, Neural Computation, 1,372-381.
Elman, J. (1989). Representation and structure in connectionist models. UCSD CRL
Technical Report 8903, August 1989.
Elman, J. (1990). Finding structure in time. Cognitive Science, 14, 179-211.
Giles, C. L., Sun, G. Z., Chen, H. H., Lee, Y. C., and Chen, D. (1990). Higher Order
Recurrent Networks. In D.S. Touretzky (ed.) Advances in Neural Information Processing
Systems 2, Morgan-Kaufmann, San Mateo, Ca., 380-387.
Jordan, M. I. (1986). Serial order: A parallel distributed processing approach. Institute
for Cognitive Science, Technical Report 8604. UCSD.
Kotz, S., and Johnson, N.L. (1982). Encyclopedia of Statistical Sciences. John Wiley
and Sons, NY.
Rumelhart, D.E., Hinton, G.E., and Williams, R.J. (1986). Learning internal
representations by error propagation. In D.E. Rumelhart & J.L. McClelland (eds.),
Parallel distributed processing: Explorations in the microstructure of cognition (Vol. 1,
pp.318-362). Cambridge, MA: MIT Press.
Williams, R. J., and Zipser, D. (1988). A Learning Algorithm for Continually Running
Fully Recurrent Neural Networks, Institute for Cognitive SCience, Technical Report
8805. UCSD.
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4,066 | 4,680 | Analyzing 3D Objects in Cluttered Images
Mohsen Hejrati
UC Irvine
Deva Ramanan
UC Irvine
[email protected]
[email protected]
Abstract
We present an approach to detecting and analyzing the 3D configuration of objects
in real-world images with heavy occlusion and clutter. We focus on the application
of finding and analyzing cars. We do so with a two-stage model; the first stage
reasons about 2D shape and appearance variation due to within-class variation
(station wagons look different than sedans) and changes in viewpoint. Rather
than using a view-based model, we describe a compositional representation that
models a large number of effective views and shapes using a small number of
local view-based templates. We use this model to propose candidate detections
and 2D estimates of shape. These estimates are then refined by our second stage,
using an explicit 3D model of shape and viewpoint. We use a morphable model
to capture 3D within-class variation, and use a weak-perspective camera model
to capture viewpoint. We learn all model parameters from 2D annotations. We
demonstrate state-of-the-art accuracy for detection, viewpoint estimation, and 3D
shape reconstruction on challenging images from the PASCAL VOC 2011 dataset.
1
Introduction
Figure 1: We describe two-stage models for detecting and analyzing the 3D shape of objects in
unconstrained images. In the first stage, our models reason about 2D appearance and shape using
variants of deformable part models (DPMs). We use global mixtures of trees with local mixtures
of gradient-based part templates (top-left). Global mixtures capture constraints on visibility and
shape (headlights are only visible in certain views at certain locations), while local mixtures capture
constraints on appearance (headlights look different in different views). Our 2D models localize
even fully-occluded landmarks, shown as hollow circles and dashed lines in (top-middle). We feed
this output to our second stage, which directly reasons about 3D shape and camera viewpoint. We
show the reconstructed 3D model and associated ground-plane (assuming its parallel to the car body)
on (top-right). The bottom row shows 3D reconstructions from four novel viewpoints.
A grand challenge in machine vision is the task of understanding 3D objects from 2D images. Classic approaches based on 3D geometric models [2] could sometimes exhibit brittle behavior on cluttered, ?in-the-wild? images. Contemporary recognition methods tend to build statistical models of
2D appearance, consisting of classifiers trained with large training sets using engineered appearance features. Successful examples include face detectors [30], pedestrian detectors [7], and general
1
object-category detectors [10]. Such methods seem to work well even in cluttered scenes, but are
usually limited to coarse 2D output, such as bounding-boxes.
Our work is an attempt to combine the two approaches, with a focus on statistical, 3D geometric
models of objects. Specifically, we focus on the practical application of detecting and analyzing
cars in cluttered, unconstrained images. We refer the reader to our results (Fig.4) for a sampling of
cluttered images that we consider. We develop a model that detects cars, estimates camera viewpoint,
and recovers 3D landmarks configurations and their visibility with state-of-the-art accuracy. It does
so by reasoning about appearance, 3D shape, and camera viewpoint through the use of 2D structured,
relational classifiers and 3D geometric subspace models.
While deformable models and pictorial structures [10, 31, 11] are known to successfully model articulation, 3D viewpoint is still not well understood. The typical solution is to ?discretize? viewpoint
and build multiple view-based models tuned for each view (frontal, side, 3/4...). One advantage of
such a ?brute-force? approach is that it is computationally efficient, at least for a small number of
views. Fine-grained 3D shape estimation may still be difficult with such a strategy. On the other
hand, it is difficult to build models that reason directly in 3D because the ?inverse-rendering? problem is hard to solve. We introduce a two-stage approach that first reasons about 2D shape and appearance variation, and then reasons explicitly about 3D shape and viewpoint given 2D correspondences
from the first stage. We show that ?inverse-rendering? is feasible by way of 2D correspondences.
2D shape and appearance: Our first stage models 2D shape and appearance using a variant of
deformable part models (DPMs) designed to produce reliable 2D landmark correspondences. Our
approach differs from traditional view-based models in that it is compositional; it ?cuts and pastes?
together different sets of local view-based templates to model a large set of global viewpoints. We
use global mixtures of trees with local mixtures of ?part? or landmark templates. Global mixtures
capture constraints on visibility and shape (headlights are only visible in certain views at certain
locations), while local mixtures capture constraints on appearance (headlights look different in different views). We use this model to efficiently generate candidate 2D detections that are refined
by our second 3D stage. One salient aspect of our 2D model is that it reports 2D locations of all
landmarks including occluded ones, each augmented with a visibility flag.
3D shape and viewpoint: Our second layer processes the 2D output of our first stage, incorporating global shape constraints arising from 3D shape variation and viewpoint. To capture viewpoint
constraints, we model landmarks as weak-perspective projections of a 3D object. To capture withinclass variation, we model the 3D shape of any object instance as a linear combination of 3D basis
shapes. We use tools from nonrigid structure-from-motion (SFM) to both learn and enforce such
models using 2D correspondences. Crucially, we make use of occlusion reports generated by our
local view-based templates to estimate morphable 3D shape and camera viewpoint.
2
Related Work
We focus most on recognition methods that deal explicitly with 3D viewpoint variation.
Voting-based methods: One approach to detection and viewpoint classification is based on bottomup geometric voting, using a Hough transform or geometric hashing. Images are first processed to
obtain a set of local feature detections. Each detection can then vote for both an object location
and viewpoint. Examples include [12] and implicit shape models [1, 26]. Our approach differs in
that we require no initial feature detection stage, and instead we reason about all possible geometric
configurations and occlusion states.
View-based models: Early successful approaches included multivew face detection [24, 17]. Recent
approaches based on view-based deformable part models include [19, 13, 10]. Our model differs in
that we use a single representation that directly generates multiple views. One can augment viewbased models to share local parts across views [27, 21, 32]. This typically requires reasoning about
topological changes in viewpoint; certain parts or features can only be visible in certain view due
to self-occlusion. One classic representation for encoding such visibility constraints is an aspect
graph [5]. [33] model such topological constraints with global mixtures with varying tree structures.
Our model is similar to such approaches, except that we use a decomposable notion of aspect; we
simultaneously reason about global and semi-local changes in visibility using local part mixtures
with global co-occurrence constraints.
2
3D models: One can also directly reason about local features and their geometric arrangement in
a 3D coordinate system [23, 25, 34]. Though such models are three-dimensional in terms of their
underlying representation, run-time inference usually proceeds in a bottom-up manner, where detected features vote for object locations. To handle non-Gaussian observation models, [18] evaluate
randomly sampled model estimates within a RANSAC search. Our approach is closely related to the
recent work of [22], which also uses a deformable part model (DPM) to capture viewpoint variation
in cars. Though they learn spatial constraints in a 3D coordinate frame, their model at run-time is
equivalent to a view-based model, where each view is modeled with a star-structured DPM. Our
model differs in that we directly reason about the location of fully-occluded landmarks, we model
an exponential number of viewpoints by using a compositional representation, and we produce continuous 3D shapes and camera viewpoints associated with each detection using only 2D training
data. Finally, we represent the space of 3D models of an object category using a set of basis shapes,
similar to the morphable models of [3]. To estimate such models from 2D data, we adapt methods
designed for tracking morphable shapes to 3D object category recognition [29, 28].
3
2D Shape and Appearance
We first describe our 2D model of shape and appearance. We write it as a scoring function with
linear parameters. Our model can be seen as an extension of the flexible mixtures-of-part model
[31], which itself augments a deformable part model (DPM) [10] to reason about local mixtures.
Our model differs its encoding of occlusion states using local mixtures, as well as the introduction of global mixtures that enforce occlusions and spatial geometry consistent with changes in 3D
viewpoint. We take care to design our model so as to allow for efficient dynamic-programming
algorithms for inference.
Let I be an image, pi = (x, y) be the pixel location for part i and ti ? {1..T } be the local mixture
component of part i. As an example, part i may correspond to a front-left headlight, and ti can
correspond to different appearances of a headlight in frontal, side, or three-quarter views. A notable
aspect of our model is that we estimate landmark locations for all parts in all views, even when they
are fully occluded. We will show that local mixture variables perform surprisingly well at modeling
complex appearances arising from occlusions.
Let i ? V where V is the set of all landmarks. We consider different relational graphs Gm =
(V, Em ) where Em connects pairs of landmarks constrained to have consistent locations and local
mixtures in global mixture m. We can loosely think of m as a ?global viewpoint?, though it will be
latently estimated from the data. We use the lack of subscript to denote the set of variables obtained
by iterating over that subscript; e.g., p = {pi : i ? V }. Given an image, we score a collection of
landmark locations and mixture variables
i
i
Xh i
X h ti ,tj
ti ,tj
S(I, p, t, m) =
?it ? ?(I, pi ) +
?ijm ? ?(pi ? pj ) + ?ijm
(1)
i?V
ij?Em
Local model: The first term scores the appearance evidence for placing a template ?iti for part i,
tuned for mixture ti , at location pi . We write ?(I, pi ) for the feature vector (e.g., HOG descriptor
[7]) extracted from pixel location pi in image I. Note that we define a template even for mixtures
ti corresponding to fully-occluded states. One may argue that no image evidence should be scored
during an occlusion; we take the view that the learning algorithm can decide for itself. It may
choose to learn a template of all zeros (essentially ignoring image evidence) or it may find gradient
features statistically correlated with occlusions (such as t-junctions). Unlike the remaining terms in
our scoring function, the local appearance model is not dependent on the global mixture/viewpoint.
We show that this independence allows our model to compose together different local mixtures to
model a single global viewpoint.
Relational model:
The second term scores relational constraints between pairs of parts. We write
?(pi ? pj ) = dx dx2 dy dy 2 , a vector of relative offsets between part i and part j. We
ti ,tj
can interpret ?ijm
as the parameters of a spring specifying the relative rest location and quadratic
spring penalty for deviating from that rest location. Notably, this spring depends on part i and j,
the local mixture components of part i and j, and the global mixture m. This dependency captures
many natural constraints due to self-occlusion; for example, if a car?s left-front wheel lies to the
right of the other front wheel (in image space), than it is likely self-occluded. Hence it is crucial that
ti ,tj
local appearance and geometry depend on each other. The last term ?ijm
defines a co-occurrence
score associated with instancing local mixture ti and tj , and global mixture m. This encodes the
3
constraint that, if the left front headlight is occluded due to self occlusion, the left front wheel is also
likely occluded.
Global model: We define different graphs Gm = (V, Em ) corresponding to different global mixtures. We can loosely think of the global variable m are capturing a coarse, quantized viewpoint. To
ensure tractability, we force all edge structures to be tree-structured. Intuitively, different relational
structures may help because occluded landmarks tend to be localized with less reliability. One may
expect occluded/unreliable parts should have fewer connections (lower degrees in Gm ) than reliable
parts. Even for a fixed global mixture m, our model can generate an exponentially-large set of appearances |V |T , where T is the number of local mixture types. We show such a model outperforms
a naive view-based model in our experiments.
3.1
Inference
Inference corresponds to maximizing (1) with respect to landmark locations p, local mixtures t, and
global mixtures m:
S ? (I) = max[max S(I, p, t, m)]
(2)
m
p,t
We optimize the above equation by enumerating all global mixtures m, and for each global mixture,
finding the optimal combination of landmark locations p and local mixtures t by dynamic programing (DP). To see that the inner maximization can be optimized by DP, let us define zi = (pi , ti ) to
denote both the discrete pixel position and discrete mixture type of part i. We can rewrite the score
from (1) for a fixed image I and global mixture m with edge structure E as:
X
X
S(z) =
?i (zi ) +
?ij (zi , zj ), (for a fixed I and m)
(3)
i?V
where ?i (zi ) =
?iti
ij?E
? ?(I, pi ) and
t ,t
t ,t
i j
i j
?ij (zi , zj ) = ?ijm
? ?(pi ? pj ) + ?ijm
From this perspective, it is clear that our model (conditioned on I and m) is a discrete, pairwise
Markov random field (MRF). When G = (V, E) is tree-structured, one can compute maxz S(z)
with dynamic programming [31].
3.2
Learning
We assume we are given training data consisting of image-landmark triplet {In , pin , oin }, where
landmarks are augmented with an additional discrete visibility flag oin . With a slight abuse of notation, we use n to denote an instance of a training image. We use oin ? {0, 1, 2} to denote visible,
self-occlusion, and other-occlusion respectively, where other occlusion corresponds to a landmark
that is occluded by another object (or the image border). We now show how to augment this training set with local mixtures labels tin , global mixtures labels mn , and global edge structures Em .
Essentially, we infer such mixture labels using probabilistic algorithms for generating local/global
clusters of 2D landmark configurations. We then use this inferred mixture labels to train the linear
parameters of the scoring function (1) using supervised, max-margin methods.
Learning local mixtures: We use the clustering algorithm described in [8, 4] to learn local part mixtures. We construct a ?local-geometric-context? vector for each part, and obtain landmark mixture
labels by grouping landmark instances with similar local geometry. Specifically, for each landmark
i and image n, we construct a K-element vector gin that defines the 2D relative location of a landmark with respect to the other K landmarks in instance n, normalized for the size of that training
instance. We construct sets of features Setij = {gin : n ? 1..N and oin = j} corresponding to
each part i and occlusion state j. We separately cluster each set of vectors using K-means, and then
interpret cluster membership as mixture label tin . This means that, for landmark i, a third of its T
local mixtures will model visible instances in the training set, a third will model self-occlusions, and
a third will capture other-occlusions.
Learning relational structure: Given local mixture labels tin , we simultaneously learn global mixtures mn and edge structure Em with a probabilistic model of zin = (pin , tin ). We find the global
mixtures and edge structure that maximizes the probability of the observed {zin } labels. Probabilistically speaking, our spatial spring model is equivalent to a Gaussian model (who?s mean and
covariance correspond to the rest location and rigidity), making estimation relatively straightforward. We first describe the special case of a single global mixture, for which the most-likely tree E
can be obtained by maximizing the mutual information of the labels using the Chow-Liu algorithm
4
[6, 15]. In our case, we find the maximum-weight spanning tree in a fully connected graph whose
edges are labeled with the mutual information (MI) between zi = (pi , ti ) and zj = (pj , tj ):
X
M I(zi , zj ) = M I(ti , tj ) +
P (ti , tj )M I(pi , pj |ti , tj )
(4)
ti ,tj
M I(ti , tj ) can be directly computed from the empirical joint frequency of mixture labels in the
training set. M I(pi , pj |ti , tj ) is the mutual information of the Gaussian random variables for the
location of landmarks i and j given a fixed pair of discrete mixture types ti , tj ; this again is readily
obtained by computing the determinant of the sample covariance of the locations of landmarks i and
j, estimated from the training data. Hence both spatial consistency and mixture consistency are used
when learning our relational structure.
Learning structure and global mixtures: To simultaneously learn global mixture labels mn and
edge structures associated with each mixture Em , we use an EM algorithm for learning mixtures of
trees [20, 15]. Specifically, Meila and Jordan [20] describe an EM algorithm that iterates between
inferring distributions over tree mixture assignments (the E-step) and estimating the tree structure
(the M-step). One can write the expected complete log-likelihood of the observed labels {z}, where
? are the model parameters (Gaussian spatial models, local mixture co-occurrences and global mixture priors) to be maximized and the global mixture assignment variables {mn } are the hidden
variables to be marginalized. Notably, the M-step makes use of the Chow-Liu algorithm. We omit
detailed equations for lack of space, but note that this is a relatively straightforward application of
[20]. We demonstrate that our latently-estimated global mixtures are crucial for high-performance
in 3D reasoning.
Learning parameters: The previous steps produces local/global mixture labels and edge structures.
Treating these as ?ground-truth?, we now define a supervised max-margin framework for learning
model parameters. To do so, let us write the landmark position labels pn , local mixtures labels
tn , and global mixture label mn collectively as yn . Given a training set of positive images with
labels {In , yn } and negative images not containing the object of interest, we define a structured
prediction objective function similar to one proposed in [31]. The scoring function in (1) is linear
in the parameters w = {?, ?, ?}, and therefore can be expressed as S(In , yn ) = w ? ?(In , yn ). We
learn a model of the form:
X
1 T
argmin
w ?w+C
?n
(5)
2
w,?i ?0
n
s.t. ?n ? positive images
?n ? negative images, ?y
w ? ?(In , yn ) ? 1 ? ?n
w ? ?(In , y) ? ?1 + ?n
The above constraint states that positive examples should score better than 1 (the margin), while
negative examples, for all configurations of part positions and mixtures, should score less than -1.
We collect negative examples from images that does not contain any cars. This form of learning
problem is known as a structural SVM, and there exist many well-tuned solvers such as the cutting
plane solver of SVMStruct in [16] and the stochastic gradient descent solver in [10]. We use the
dual coordinate-descent QP solver of [31]. We show an example of a learned model and its learned
tree structure in Fig.1.
4
3D Shape and Viewpoint
The previous section describes our 2D model of appearance and shape. We use it to propose detections with associated landmarks positions p? . In this section, we describe a 3D shape and viewpoint
model for refining p? . Consider 2D views of a single rigid object; 2D landmark positions must obey
epipolar geometry constraints. In our case, we must account for within-class shape variation as well
(e.g., sedans look different than station wagons). To do so, we make two simplifying assumptions:
(1) We assume depth variation of our objects are small compared to the distance from the camera,
which corresponds to a weak-perspective camera model. (2) We assume the 3D landmarks of all
object instances can be written as linear combinations of a few basis shapes. Let us write the set of
detected landmark positions as p? as a 2 ? K matrix where K = |V |. We now describe a procedure
for refining p? to be consistent with these two assumptions:
X
min ||p? ? R
?i Bi ||2 where p ? R2?K , R ? R2?3 , RRT = Id, Bi ? R3?K
(6)
R,?
i
5
Here, R is an orthonormal camera projection matrix and Bi is the ith basis shape, and Id is the
identity matrix. We factor out camera translations by working with mean-centered points p? and let
? directly model weak-perspective scalings.
Inference: Given 2D landmark locations p? and a known set of 3D basis shapes B i , inference
corresponds to minimizing (6). For a single basis shape (nB = 1), this problem is equivalent to
the well-known ?extrinsic orientation? problem of registering a 3D point cloud to a 2D point cloud
with known correspondence [14]. Because the squared error is linear in ai and R, we solve for the
coefficients and rotation with an iterative least-squares algorithm. We enforce the orthonormality
of R with a nonlinear optimization, initialized by the least-squares solution [14]. This means that
we can associate each detection with shape basis coefficients ?i (which allows us to reconstruct
the 3D shape) and camera viewpoint R. One could combine the reprojection error of (6) with our
original scoring function from (1) into a single objective that jointly searches over all 2D and 3D
unknowns. However inference would be exponential in K. We find a two-layer inference algorithm
to be computationally efficient but still effective.
Learning: The above inference algorithm requires the morphable 3D basis Bi at test-time. One
can estimate such a basis given training data with labeled 2D landmark positions by casting this as
nonrigid structure from motion (SFM) problem. Stack all 2D landmarks from N training images
into a 2N ? K matrix. In the noise-free case, this matrix is rank 3nB (where nB is the number of
basis shapes), since each row can be written as a linear combination of the 3D coordinates of nB
basis shapes. This means that one can use rank constraints to learn a 3D morphable basis. We use the
publically-available nonrigid SFM code [28]. By slightly modifying it to estimate ?motion? given a
known ?structure?, we can also use it to perform the previous projection step during inference.
Occlusion: A well-known limitation of SFM methods is their restricted success under heavy occlusion. Notably, our 2D appearance model provides location estimates for occluded landmarks. Many
SFM methods (including [28]) can deal with limited occlusion through the use of low-rank constraints; essentially, one can still estimate low-rank approximations of matrices with some missing
entries. We can use this property to learn models from partially-labeled training sets. Recall that
our learning formulation requires all landmarks (including occluded ones) to be labeled in training
data. Manually labeling the positions of occluded landmarks can be ambiguous. Instead, we use the
estimated shape basis and camera viewpoints to infer/correct the locations of occluded landmarks.
5
Experiments
Datasets: To evaluate our model, we focus on car detection and 3D landmark estimation in cluttered,
real-world datasets with severe occlusions. We labeled a subset of 500 images from the PASCAL
VOC 2011 dataset [9] with locations and visibility states of 20 car landmarks. Our dataset contains
723 car instances. 36% of landmarks are not visible due to self-occlusion, while 21% of landmarks
are not visible due to occlusion by another object (or truncation due to the image border). Hence
over half our landmarks are occluded, making our dataset considerably more difficult than those
typically used for landmark localization or 3D viewpoint estimation. We evenly split the images
into a train/test set. We also compare results on a more standard viewpoint dataset from [1], which
consists of 200 relatively ?clean? cars from the PASCAL VOC 2007 dataset, marked with 40 discrete
viewpoint class labels.
Implementation: We modify the publically-available code of [31] and [28] to learn our models,
setting the number of local mixtures T = 9, the number of global mixtures M = 50, and the
number of basis shapes nB = 5. We found results relatively robust to these settings. Learning our
2D deformable model takes roughly 4 hours, while learning our 3D shape model takes less than
a minute. Our model is defined at a canonical scale, so we search over an image pyramid to find
detections at multiple scales. Total run-time for a test image (including both 2D and 3D processing
over all scales) is 10 seconds.
Evaluation: Given an image, our algorithm produces multiple detections, each with 3D landmark
locations, visibility flags, and camera viewpoints. We qualitatively visualize such output in Fig.4.
To evaluate our output, we assume test images are marked with ground-truth cars, each annotated
with ground-truth 2D landmarks and visibility flags. We measure the performance of our algorithm
on four tasks. We evaluate object detection (AP) using using the PASCAL criteria of Average
Precision [9], defining a detection to be correct if its bounding box overlaps the ground truth by
50% or more. We evaluate 2D landmark localization (LP) by counting the fraction of predicted
6
80
Our model
60
Arie?Nachimson and Basri
40
Glasner et al.
Glasner et al.
Arie?Nachimson and Basri
Our Model
20
0
0
50
100
degrees
150
0
10
20
30
Median Degree Error
Figure 2: We report histograms of viewpoint label errors for the dataset of [1]. We compare to the
reported performance of [1] and [12]. Our model reduces the median error (right) by a factor of 2.
1
1
0.9
0.8
VP
0.7
AP
0.6
MV Tree
Local
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.2
MV Tree = 72.3%
0.1
Us = 72.5%
0
0.8
AP
0.7
MV star = 74.0%
0.3
US
VP
Global
DPM = 63.6%
0.4
0.9
precision
LP
precision
MV Star
LP
0
0.6
0.5
0.4
0.3
Local = 69%
0.2
Global+3D
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
recall
0.55
0
0.6
0.65
0.7
0.75
Global = 72.5%
0.1
0.8
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
recall
(a) Baseline comparison
(b) Diagnostic analysis
Figure 3: We compare our model with various view-based baselines in (a), and examine various
components of our model through a diagnostic analysis in (b). We refer the reader to the text for
a detailed analysis, but our model outperforms many state-of-the-art view-based baselines based on
trees, stars, and latent parts. We also find that modeling the effects of shape due to global changes
in 3D viewpoint is crucial for both detection and landmark localization.
landmarks that lie within .5x pixels of the ground-truth, where x is the diameter of the associated
ground-truth wheel. We evaluate landmark visibility prediction (VP) by counting the number
of landmarks whose predicted visibility state matches the ground-truth, where landmarks may be
?visible?, ?self-occluded?, or ?other-occluded?. Our 3D shape model refines only LP and VP, so
AP is determined solely by our 2D (mixtures of trees) model. To avoid conflating the evaluation
measures, we evaluate LP and VP assuming bounding-box correspondences between candidates and
ground-truth instances are provided. Finally to evaluate viewpoint classification (VC), we compare
predicted camera viewpoints with ground-truth viewpoints on the standard benchmark of [1].
Viewpoint Classification: We first present results for viewpoint classification in Fig.2 on the benchmark of [1]. Given a test instance, we run our detector, estimate the camera rotation R, and report the
reconstructed 2D landmarks generated using the estimated R. Then we produce a quantized viewpoint label by matching the reconstructions to landmark locations for a reference image (provided
in the dataset). We found this approach more reliable than directly matching 3D rotation matrices
(for which metric distances are hard to define). We produce a median error of 9 degrees, a factor of
2 improvement over state-of-the-art. This suggests our model does accurately capture viewpoints.
We next turn to a detailed analysis on our new cluttered dataset.
Baselines: We compare the performance of our overall system to several existing approaches for
multiview detection in Fig.3(a). We first compare to widely-used latent deformable part model
(DPM) of [10], trained on the exact same data as our model. A supervised DPM (MV-star) considerably improves performance from 63 to 74% AP, where supervision is provided for (view-specific)
root mixtures and part locations. This latter model is equivalent in structure to a state-of-the-art
model for car detection and viewpoint estimation [22], which trains a DPM using supervision provided by a 3D CAD model. By allowing for tree-structured relations in each view-specific global
mixture (MV-tree), we see a small drop in AP = 72.3%. Our final model is similar in term of
detection performance (AP = 72.5%), but does noticeably better than both view-based models for
landmark prediction. We correctly localize landmarks 69.5% of time, while MV-tree and MV-star
score 65.7% and 64.7%, respectively. We produce landmark visibility (VP) estimates from our multiview baselines by predicting a fixed set of visibility labels conditioned on the view-based mixture.
We should note that accurate landmark localization is crucial for estimating the 3D shape of the detected instance. We attribute our improvement to the fact that our model can model a large number
of global viewpoints by composing together different local view-based templates.
7
1
Figure 4: Sample results of our system on real images with heavy clutter and occlusion. We show
pairs of images corresponding to detections that matched to ground-truth annotations. The top image
(in the pair) shows the output of our tree model, and the bottom shows our 3D shape reconstruction,
following the notational conventions of Fig.1. Our system estimates 3D shapes of multiple cars
under heavy clutter and occlusions, even in cases where more than 50% of a car is occluded. Our
morphable 3D model adapts to the shape of the car, producing different reconstructions for SUVs
and sedans (row 2, columns 2-3). Recall that our tree model explicitly reasons about changes in
visibility due to self-occlusions versus occlusions from other objects, manifested as local mixture
templates. This allow our 3D reconstructions to model occlusions due to other objects (e.g., the rear
of the car in row 2, column 3). In some cases, the estimated 3D shape is misaligned due to extreme
shape variation of the car instance (e.g., the folding doors on the lower-right).
Diagnostics: We compare various aspects of our model in Fig.3(b). ?Local? refers to a single tree
model with local mixtures only, while ?Global? refers to our global mixtures of trees. We see a
small improvement in terms of AP, from 69% for ?Local? to 72.5% for ?Global?. However, in terms
of landmark prediction, ?Global? strongly outperforms ?Local?, 69.4% to 57.2%. We use these
predicted landmarks to estimate 3D shape below.
3D Shape: Our 3D shape model reports back a z depth value for each landmark (x, y) position.
Unfortunately, depth is hard to evaluate without ground-truth 3D annotations. Instead, we evaluate
the improvement in re-projected VP and LP due to our 3D shape model; we see a small 2% improvement in LP accuracy, from 69.4% to 71.2%. We further analyze this by looking at the improvement
in localization accuracy of ground-truth landmarks that are visible (73.3 to 74.8%), self-occluded
(70.5 to 72.5%), and other-occluded (22.5 to 23.4%). We see the largest improvement for occluded
parts, which makes intuitive sense. Local templates corresponding to occluded mixtures will be less
accurate, and so will benefit more from a 3D shape model.
Conclusion: We have described a geometric model for detecting and estimating the 3D shape of
objects in heavily cluttered, occluded, real-world images. Our model differs from typical multiview
approaches by reasoning about local changes in landmark appearance and global changes in visibility and shape, through the aid of a morphable 3D model. While our model is similar to prior
work in terms of detection performance, it produces significantly better estimates of 2D/3D landmarks and camera positions, and quantifiably improves localization of occluded landmarks. Though
we have focused on the application of analyzing cars, we believe our method could apply to other
geometrically-constrained objects.
8
References
[1] M. Arie-Nachimson and R. Basri. Constructing implicit 3d shape models for pose estimation. In ICCV,
2009.
[2] T. Binford. Survey of model-based image analysis systems. The International Journal of Robotics Research, 1(1):18?64, 1982.
[3] V. Blanz and T. Vetter. A morphable model for the synthesis of 3d faces. In Proceedings of the 26th annual conference on Computer graphics and interactive techniques, pages 187?194. ACM Press/AddisonWesley Publishing Co., 1999.
[4] L. Bourdev and J. Malik. Poselets: Body part detectors trained using 3d human pose annotations. In
Computer Vision, 2009 IEEE 12th International Conference on, pages 1365?1372. IEEE, 2009.
[5] K. Bowyer and C. Dyer. Aspect graphs: An introduction and survey of recent results. International
Journal of Imaging Systems and Technology, 2(4):315?328, 1990.
[6] C. Chow and C. Liu. Approximating discrete probability distributions with dependence trees. Information
Theory, IEEE Transactions on, 14(3):462?467, 1968.
[7] N. Dalal and B. Triggs. Histograms of oriented gradients for human detection. In CVPR, 2005.
[8] C. Desai and D. Ramanan. Detecting actions, poses, and objects with relational phraselets. ECCV, 2012.
[9] M. Everingham, L. Van Gool, C. K. I. Williams, J. Winn, and A. Zisserman.
The
PASCAL Visual Object Classes Challenge 2011 (VOC2011) Results.
http://www.pascalnetwork.org/challenges/VOC/voc2011/workshop/index.html.
[10] P. F. Felzenszwalb, R. B. Girshick, D. McAllester, and D. Ramanan. Object detection with discriminatively trained part based models. IEEE PAMI, 99(1), 5555.
[11] R. Girshick, P. Felzenszwalb, and D. McAllester. Object detection with grammar models. In NIPS, 2011.
[12] D. Glasner, M. Galun, S. Alpert, R. Basri, and G. Shakhnarovich. Viewpoint-aware object detection and
pose estimation. In ICCV, pages 1275?1282. IEEE, 2011.
[13] C. Gu and X. Ren. Discriminative mixture-of-templates for viewpoint classification. ECCV, pages 408?
421, 2010.
[14] B. Horn. Robot vision. The MIT Press, 1986.
[15] S. Ioffe and D. Forsyth. Mixtures of trees for object recognition. In CVPR, 2001.
[16] T. Joachims, T. Finley, and C. Yu. Cutting plane training of structural SVMs. Machine Learning, 2009.
[17] M. Jones and P. Viola. Fast multi-view face detection. In CVPR 2003.
[18] Y. Li, L. Gu, and T. Kanade. A robust shape model for multi-view car alignment. In CVPR, 2009.
[19] R. Lopez-Sastre, T. Tuytelaars, and S. Savarese. Deformable part models revisited: A performance evaluation for object category pose estimation. In Computer Vision Workshops (ICCV Workshops), 2011.
[20] M. Meila and M. Jordan. Learning with mixtures of trees. JMLR, 1:1?48, 2001.
[21] P. Ott and M. Everingham. Shared parts for deformable part-based models. In CVPR, 2011.
[22] B. Pepik, M. Stark, P. Gehler, and B. Scheile. Teaching geometry to deformable part models. In CVPR,
2012.
[23] S. Savarese and L. Fei-Fei. 3d generic object categorization, localization and pose estimation. In ICCV,
pages 1?8. IEEE, 2007.
[24] H. Schneiderman and T. Kanade. A statistical method for 3d object detection applied to faces and cars.
In CVPR, volume 1, pages 746?751. IEEE, 2000.
[25] M. Sun, H. Su, S. Savarese, and L. Fei-Fei. A multi-view probabilistic model for 3d object classes. In
CVPR, pages 1247?1254. IEEE, 2009.
[26] A. Thomas, V. Ferrar, B. Leibe, T. Tuytelaars, B. Schiel, and L. Van Gool. Towards multi-view object
class detection. In CVPR, volume 2, pages 1589?1596. IEEE, 2006.
[27] A. Torralba, K. Murphy, and W. Freeman. Sharing visual features for multiclass and multiview object
detection. PAMI, 29(5):854?869, 2007.
[28] L. Torresani, A. Hertzmann, and C. Bregler. Learning non-rigid 3d shape from 2d motion. Advances in
Neural Information Processing Systems, 16, 2003.
[29] L. Torresani, D. Yang, E. Alexander, and C. Bregler. Tracking and modeling non-rigid objects with rank
constraints. In CVPR, volume 1, pages I?493. IEEE, 2001.
[30] P. Viola and M. Jones. Rapid object detection using a boosted cascade of simple features. In CVPR,
volume 1, pages I?511. IEEE, 2001.
[31] Y. Yang and D. Ramanan. Articulated pose estimation with flexible mixtures-of-parts. In CVPR, 2011.
[32] L. Zhu, Y. Chen, A. Torralba, W. Freeman, and A. Yuille. Part and appearance sharing: Recursive
compositional models for multi-view multi-object detection. Pattern Recognition, 2010.
[33] X. Zhu and D. Ramanan. Face detection, pose estimation, and landmark localization in the wild. In
CVPR, 2012.
[34] M. Zia, M. Stark, B. Schiele, and K. Schindler. Revisiting 3d geometric models for accurate object shape
and pose. In ICCV Workshops, pages 569?576. IEEE, 2011.
9
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4,067 | 4,681 | A mechanistic model of early sensory
processing based on subtracting sparse
representations
Shaul Druckmann*
Tao Hu*
Dmitri B. Chklovskii
* - Equal contribution
Janelia Farm Research Campus
{druckmanns, hut, mitya}@janelia.hhmi.org
Abstract
Early stages of sensory systems face the challenge of compressing
information from numerous receptors onto a much smaller number of
projection neurons, a so called communication bottleneck. To make more
efficient use of limited bandwidth, compression may be achieved using
predictive coding, whereby predictable, or redundant, components of the
stimulus are removed. In the case of the retina, Srinivasan et al. (1982)
suggested that feedforward inhibitory connections subtracting a linear
prediction generated from nearby receptors implement such compression,
resulting in biphasic center-surround receptive fields. However, feedback
inhibitory circuits are common in early sensory circuits and furthermore
their dynamics may be nonlinear. Can such circuits implement predictive
coding as well? Here, solving the transient dynamics of nonlinear reciprocal
feedback circuits through analogy to a signal-processing algorithm called
linearized Bregman iteration we show that nonlinear predictive coding can
be implemented in an inhibitory feedback circuit. In response to a step
stimulus, interneuron activity in time constructs progressively less sparse
but more accurate representations of the stimulus, a temporally evolving
prediction. This analysis provides a powerful theoretical framework to
interpret and understand the dynamics of early sensory processing in a
variety of physiological experiments and yields novel predictions regarding
the relation between activity and stimulus statistics.
1 Introduction
Receptor neurons in early sensory systems are more numerous than the projection
neurons that transmit sensory information to higher brain areas, implying that sensory
signals must be compressed to pass through a limited bandwidth channel known as
?Barlow?s bottleneck? [1]. Since natural signals arise from physical objects, which are
contiguous in space and time, they are highly spatially and temporally correlated [2-4]. Such
signals are ideally suited for predictive coding, a compression strategy borrowed from
engineering whereby redundant, or predictable components of the signal are subtracted and
only the residual is transmitted [5].
Consider, for example, the processing of natural images in the retina. Instead of
transmitting photoreceptor signals, which are highly correlated in space and time, ganglion
cells can transmit differences in signal between nearby pixels or consecutive time points.
The seminal work of Srinivasan et al. introduced predictive coding to neuroscience,
proposing that feedforward inhibition could implement predictive coding by subtracting a
prediction for the activity of a given photoreceptor generated from the activity of nearby
receptors [6]. Indeed, the well known center surround spatial receptive fields or biphasic
temporal receptive fields of ganglion cells [7] may be viewed as evidence of predictive
coding because they effectively code such differences [6, 8-10]. Although the Srinivasan et
al. model captured the essence of predictive coding it does not reflect two important
biological facts. First, in the retina, and other early sensory systems, inhibition has a
significant feedback component [11-13]. Second, interneuron transfer functions are often
non-linear [14-16].
Here, we demonstrate that feedback circuits can be viewed as implementing predictive
coding. Surprisingly, by taking advantage of recent developments in applied mathematics and
signal processing we are able to solve the non-linear recurrent dynamics of such a circuit, for an
arbitrary number of sensory channels and interneurons, allowing us to address in detail the circuit
dynamics and consequently the temporal and stimulus dependencies. Moreover, introducing nonlinear feedback dramatically changes the nature of predictions. Instead of a static relation
between stimulus and prediction, we find that the prediction becomes both stimulus and time
dependent.
2 Model
2.1 Dynamics of the linear single-channel feedback circuit
We start by considering predictive coding in feedback circuits, where principal neurons are
reciprocally connected with inhibitory interneuron forming a negative feedback loop. Much of the
intuition can be developed from linear circuits and we start from this point. Consider a negative
feedback circuit composed of a single principal neuron, p, and a single interneuron, n (Fig. 1a).
Assuming that both types of neurons are linear first-order elements, their dynamics are given by:
!!!
!"
!"
!"
!!!
!"
= ?!!! !(!) + !!! !(!) ? !"(!) ,
= ?!!! ! ! + !!! !" ! ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
(1)
where gm is the membrane conductance (inverse of membrane resistance), Cm the membrane
capacitance, gs synaptic conductance and the subscript designates the neuron class (principal and
interneuron) and w in the second equation is the weight of the synapse from the principal neuron
to the interneuron. For simplicity, we assumed that the weight of the synapse from the interneuron
to the principal neuron is the same in magnitude but with negative sign, -w. Although we do not
necessarily expect the brain to fully reconstruct the stimulus on the receiving side, we must still
ensure that the transmitted signal is decodable. To guarantee that this is the case, the prediction
made by the interneuron must be strictly causal. In other words, there must be a delay between the
input to the interneurons, !"(!), and the output of the interneurons, !(! + !). Given that feedback
requires signals passing through a synapse, such delay is biologically plausible. When discussing
analytical solutions below, we assume that ! ? 0 to avoid clutter and do not explicitly indicate the
time dependence of the vectors p, s, and n. By rearranging the terms in Eq. 1 we obtain:
!!
!!
!"
!"
!"
!"
= ?! + !!! !!! ! ? !" ,
= ?! + !!! !!! !"
(2)
where ?=RC is the membrane time constant. Since principal neurons should be able to transmit
fast changes in the stimuli, we assume that the time constant of the principal cells is small
compared to that of the interneurons. Therefore, we can assume that the first equation reaches
equilibrium instantaneously:
! = ! ! ? !"
!!
!"
!"
= ?! + !!! !!! !" ,
(3)
where we defined ! = !!! !!! . As the purpose of interneuron integration will be to construct
stimulus representation, the integration time should be on the order of the auto-correlation time in
the stimulus. Since here we study the simplified case of the semi-infinite step-stimulus, the time
constant of the neuron should approach infinity. We assume this occurs by the interneurons having
a very large membrane resistance (or correspondingly a very small conductance) and moderate
capacitance. Therefore, the leakage term, -n, which is the only term in the second line of Eq. 3 that
doesn?t grow with the membrane resistance, can be neglected in the dynamics of interneurons. By
this assumption and substituting the first equation into the second, we find:
! = ! ! ? !"
!!
!!
!"
= !!! !!! !" ! ? !" .
(4)
Defining the effective time constant ! = !!! !!! ! we have:
! = ! ! ? !"
!
!!
!"
(5)
= !" ? ? ? ? ? ? ? ? ? ?
In response to a step stimulus: !(!) ? = ! ! !, where !(!) is the Heavyside function, the dynamics
of equation 5 are straightforward to solve, yielding:
!
!
!
!
! ! = ? ?? ! ? 1 ?? ?exp ?! ! !
? ? ?
(6)
!
! ! = !" ?? ! exp ?! ! ! ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
!
The interneuron?s activity, n(t), grows with time as it integrates the output of the principal
neuron, p(t), Fig. 1a. In turn, the principal neuron?s output, p(t), is the difference between the
incoming stimulus and the interneuron?s activity, n(t), i.e. a residual, which decays with time from
the onset of the stimulus. In the limit considered here (infinite interneuron time constant), the
interneuron?s feedback will approach the incoming stimulus and the residual will decay to zero.
To summarize, one can view the interneuron?s activity as a series of progressively more accurate
predictions of the stimulus. The principal neuron subtracts these predictions and sends the series of
residuals to higher brain areas, a more efficient approach than direct transmission (Fig. 1a).
Negative feedback circuit
a
6HQVRU\LQSXW?V
Coding
w
Principal neuron, p
p
b
Direct transmission
6HQVRU\LQSXW?V
Interneuron, n
?
?
p
Principal neuron, p
n
w
Transmission channel
Transmission channel
w
Output
Time
Output
n
p
w
Decoding
Time
p
Time
Figure 1 Schematic view of early processing in a single sensory channel in response to a
step stimulus. a. A predictive coding model consists of a coding circuit, transmission channel
and, for theoretical analysis only, a virtual decoding circuit. Coding is performed in a negative
feedback circuit containing a principal neuron, p, and an inhibitory interneuron, n. In response
to a step-stimulus (top left) the interneuron charges up with time (top right) till it reaches the
value of the stimulus. Principal neuron (middle left) transmits the difference between the
interneuron activity and the stimulus, resulting in a transient signal. b. Direct transmission.
The transient response to a step stimulus (Fig. 1a left) is consistent with electrophysiological
measurements from principal neurons in invertebrate and vertebrate retina [10, 17]. For example,
in flies, cells post-synaptic to photoreceptors (the LMCs) have graded potential response
consistent with Equation 5. In the vertebrate retina, most recordings are performed on ganglion
cells, which read out signals from bipolar cells. In response to a step-stimulus the firing rate of
ganglion cells is consistent with Equation 6 [17].
2.2 Dynamics of the linear multi-channel feedback circuit
In most sensory systems, stimuli are transmitted along multiple parallel sensory channels,
such as mitral cells in the olfactory bulb, or bipolar cells in the retina. Although a circuit could
implement predictive coding by replicating the negative feedback loop in each channel, this
solution is likely suboptimal due to the contiguous nature of objects in space, which often results
in stimuli correlated across different channels. Therefore, interneurons that combine inputs across
channels may generate an accurate prediction more rapidly. The dynamics of a multichannel linear
negative feedback circuit are given by:
! = ! ? !"
!
!!
!"
= !! !
,
(7)
where boldface lowercase letters are column vectors representing stimulus, ?! = (!! , !! , !! , ? )! , ?
activity of principal neurons, !, and interneurons, ?!, Fig. 2a. Boldface uppercase letters designate
synaptic weight matrices. Synaptic weights from principal neurons to interneurons are !! , and
synaptic weights from interneurons to principal neurons are, for simplicity, symmetric but with the
negative sign, ?!. Such symmetry was suggested for olfactory bulb, considering dendrodendritic synapses [18]. Each column of ! contains the weights of synapses from correlated
principal neurons to a given interneuron, thus defining that interneuron?s feature vector (Fig. 2b).
Linear dynamics of the feedback circuit in response to a multi-dimensional step stimulus can
be solved in the standard manner similarly to equation 6:
!
! = (!! !)!! ? 1 ?? ?exp ?!! ! !
!! ! ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
!
!
! = ! ?! ? !(!! !)!! ? 1 ?? ?exp ?!! ! !
!
!
!! !
(8)
!
provided ! ! is invertible. When the matrix ! ! is not full rank, for instance if the number of
interneurons exceeds the number of sensory channels, the solution of Equation 7 is given by:
!
! = !! (!!! )!! ? 1 ?? ?exp ?!!! !
!
!
!
! = !exp ?!!! ! ! ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
(9)
!
Recapitulating the equations in words, as above one can view the interneurons? activity as a series
of progressively more accurate stimulus predictions, ! = !". The principal neuron sends the
series of residuals of these predictions, ! = ! ? !, to higher brain areas, and the dynamics result in
the transmitted residual decreasing in time [19-22] (Fig. 2c,d).
2.3 Dynamics of the non-linear multi-channel feedback circuit
Our solution of the circuit dynamics in the previous sub-section relied on the assumption that
neurons act as linear elements, which in view of non-linearities in real neurons, represents a
drastic simplification. We now extend this analysis to the non-linear circuit. A typical neural
response non-linearity is the existence of a non-zero input threshold below which neurons do not
respond. A pair of such on- and off- neurons is described by a threshold function (Fig. 2e) that has
a ?gap? or ?deadzone? around zero activity and is not equivalent to a linear neuron:
Thresh ! =
! ? !, ! > !
0, ! ? !
! + !, ! < ?!
(10)
Accordingly, the dynamics are given by:
! = ! ? !" ? ? ? ? ? ? ?
!
!!
!"
= !! ! ? ? ? ? ? ? ? ? ,
(11)
! = Thresh! (!)
The central contribution of this paper is an analysis of predictive coding in a feedback circuit
with threshold-linear interneurons inspired by the equivalence of the network dynamics to a
signal-processing algorithm called linearized Bregman iteration [23, 24]. Before showing the
equivalence, we first describe linearized Bregman iteration. This algorithm constructs a faithful
representation of an input as a linear sum over dictionary elements while minimizing the L1-L2
norm of the representation [25]. Formally, the problem is defined as follows:
for ?! ! ? ! |! |! +
!
!!
| !| !! , min! !(!) ?!. !. !" = !.
(12)
Remarkably, this high-dimensional non-linear optimization problem can be solved by a simple
iterative scheme (see Appendix):
!!!! = !! + !!! ! ? !!!
,
(13)
!!!! = Thresh! !!!! ? ? ? ? ? ? ? ? ? ? ?
combining a linear step, which looks like gradient descent on the representation error, and a
component-wise threshold-linear step.
Eq. 11, the network dynamics, is the continuous version of linearized Bregman iteration, Eq.
13. Intuitively speaking, the dynamics of the network play the role of the iterations in the
algorithm. Having identified this equivalence, we are able to both solve and interpret the transient
non-linear dynamics (see supplementary materials for further details). The analytical solution
allows us a deeper understanding, for instance of the convergence of the algorithm. We note that if
the interneuron feature vectors span the stimulus space the steady-state activity will be zero for
any stimulus and thus non-informative. Therefore, solving the transient dynamics, as opposed to
just the steady-state activity [18, 19, 21, 26], was particularly crucial in this case.
Next, we describe in words the mathematical expressions for the response of the feedback
circuit to a step-stimulus (see Supplement for dynamics equations), Fig. 2f-g. Unlike in the linear
circuit, interneurons do not inhibit principal neurons until their internal activity crosses threshold,
Fig. 2f. Therefore, their internal activity initially grows with a rate proportional to the projection
of the sensory stimulus on their feature vectors, !! !. With time, interneurons cross threshold and
contribute to the stimulus representation, thereby constructing a more accurate representation of
the stimulus, Fig. 2f,g. The first interneuron to cross threshold is the one for which the projection
of the sensory stimulus on its feature vector, !! ! is highest. As its contribution is subtracted from
the activity of the principal neurons, the driving force on other interneurons !! (! ? !")
changes. Therefore, the order by which interneurons cross threshold depends also on the
correlation between the feature vectors, Fig. 2b,f.
n1
p1
p2
W
n2
p3
n3
n4
Output to higher brain areas
b
Stimulus
Feature vectors
0.2
0.1
0
?0.1
-5
-?
f
Internal act., n
g
External activity, a
?
Internal
activity
n
min(? /? WTs)
?
0
-??
Expanded view of early time
0
10
20
Time
30
40
Thresh-linear negative feedback circuit
0.3
0.2
0.1
0
?0.1
?0.2
?0.3
-5
0
10
20
Time
30
40
50
Linear negative feedback circ.
0.12
0.1
0.08
0.06
0.04
0.02
0
-5
50
0.4
External activity, a
e
d
Linear negative feedback circ.
h
Principal neuron activity
WT
0.3
Principal neuron activity
c
Sensory Input
Interneuron activity
a
0
10
20
30
Time
40
50
Thresh-linear negative feedback circ.
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
-5
0
10
20
Time
30
40
50
Figure 2. Predictive coding in a feedback circuit in response to a step stimulus at time
zero. a. Circuit diagram for feedback circuit. b. Stimulus (grayscale in black box left) and a
subset of interneuron?s feature vector (grayscale in boxes). c-d. Response of linear feedback
circuit to a step stimulus at time zero in interneurons (c) and principal neurons (d). e. Thresholdlinear transfer function relating internal, n, and external, a, activity of interneurons. Dashed line
shows diagonal. Firing rates cannot be negative and therefore the threshold-linear function
should be thought as combining a pair of on and off-cells. f-h. Response of interneurons (f-g)
and principal neurons to a step stimulus at time zero. f. Expanded view of internal activity of the
interneurons (only some are shown, see grayscale in boxes color coded to match b) at early
times. g. External activity of a larger subset of interneurons over a longer time period. Grayscale
boxes show the stimulus represented by the interneuron layer at various times marked by
arrows. h. Principal neuron activity as a function of time. As interneurons cross threshold they
more closely represent the stimulus and cancel out more of the principal cell activity.
Eventually, the interneuron representation (right box in g) is nearly identical to the stimulus and
the principal neurons? activity drops almost to zero.
Collectively the representation progresses from sparse to dense, but individual interneurons may
first be active then become silent. Eventually interneurons will accurately represent the input with
their activity, ! = !", and will fully subtract it from the principal cells? activity, resulting in no
further excitation to the interneurons, Fig. 2g,h.
However, this description leads to an immediate puzzle. Namely, the algorithm builds a
representation of the stimulus by the activity of interneurons. Yet, interneurons are local circuit
elements whose activity is not transmitted outside the circuit. Why would a representation be built
if it is available only locally within the neural circuit? The answer to this conundrum is found by
considering the notion of predictive coding in early sensory circuits presented in the introduction.
The interneurons serve as the predictor and the principal neurons transmit a prediction residual.
As expected by the framework of predictive coding, at each point in time, the circuit subtracts
the prediction, ! = !", which was constructed in the interneurons from previous incoming
sensory signals, from the current sensory stimulus and the principal neurons transmit the residual,
! = ! ? !, to higher brain areas. We note that initially the interneurons are silent and the principal
neurons transmit the stimulus directly. If there were no bandwidth limitation, the stimulus could
be decoded just from this initial transmission. However, the bandwidth limitation results in coarse,
or noisy, principal neuron transmission, an issue we will return to later.
3 Results
In neuroscience, the predictive coding strategy was originally suggested to allow efficient
transmission through a limited bandwidth channel (Srinivasan et al., 1982). Our main result is the
solution of the transient dynamics given in the section above. Understanding circuit dynamics in
the predictive coding framework allows us to make a prediction regarding the length of transient
activity for different types of stimuli. Recall that the time from stimulus onset to cancellation of
the stimulus depends on the rate of the interneurons? activation, which in turn is proportional to
the projection of the stimulus on the interneurons? feature vectors. Presumably, interneuron feature
vectors are adapted to the most common stimuli, e.g. natural images in the case of the retina,
therefore this type of stimulus should be relatively quickly cancelled out. In contrast, non-natural
stimuli, like white noise patterns, will be less well captured by interneuron receptive fields and
their activation will occur after a longer delay. Accordingly, it will take longer to cancel out nonnatural stimuli, leading to longer principal neuron transients.
Below, we show that the feedback circuit with threshold-linear neurons is indeed more
efficient than the existing alternatives. We first consider a scenario in which effective bandwidth
limitation is imposed through addition of noise. Secondly, we consider a more biologically
relevant model, where transmission bandwidth is set by the discreteness of Poisson neural activity.
We find that threshold linear interneurons achieve more accurate predictions when faced with
stimulus corrupted with i.i.d Gaussian noise. The intuition behind this result is that of sparse
denoising [23]. Namely, if the signal can be expressed as a sparse sum of strong activation of
dictionary elements, whereas the noise requires a large number of weakly activated elements, then
thresholding the elements will suppress the noise more than the signal, yielding denoising. We
note that this fact alone does not in itself argue for the biological plausibility of this network, but
threshold-linear dynamics are a common approximation in neural networks.
Figure 3. Predictions by negative feedback circuit. Left: Relative prediction error ( ! ? ! ! /
! ! ), where ! = !", as a function of time for a stimulus consisting of an image patch
corrupted by i.i.d Gaussian noise at every time point. Right: An image is sent through principal
neurons that transmit Poisson. The reconstruction error as a function of time following the
presentation of stimulus is shown for the full non-linear negative feedback circuit (black), for a
linear negative feedback circuit (red), for a direct transmission circuit (blue), and for a circuit
where the sparse approximation itself is transmitted instead of the residual (green). Time on the
x-axis is measured in units of the time length in which a single noisy transmission occurs. Inset
shows log-log plot.
In addition to considering transmission of stimuli corrupted by Gaussian noise, we also
studied a different model where bandwidth limitation is set by the discreteness of spiking,
modeled by a Poisson process. Although the discreteness of transmission can be overcome by
averaging over time, this comes at the cost of longer perceptual delays, or lower transmission
rates, as longer integration takes place. Therefore, we characterize transmission efficiency by
reconstruction error as a function of time, Fig. 3. We find that, for Poisson transmission, predictive
coding provides more accurate stimulus reconstruction than direct transmission for all times but
the brief interval until the first interneuron has crossed threshold (Fig. 3).
4 Discussion
By solving the dynamics of the negative feedback circuit through equivalence to
linearized Bregman iteration we have shown that the development of activity in a simplified
early sensory circuit can be viewed as implementing an efficient, non-linear, intrinsically
parallel algorithm for predictive coding. Our study maps the steps of the algorithm onto
specific neuronal substrates, providing a solid theoretical framework for understanding
physiological experiments on early sensory processing as well as experimentally testing
predictive coding ideas on a finer, more quantitative level.
Recently, sparse representations were studied in a single-layer circuit with lateral
inhibitory connections proposed as a model of a different brain area, namely primary cortical
areas. The circuit constructs the stimulus representation in the projection neurons themselves
and directly transmits it downstream [27, 28]. We believe it does not model early sensory
systems as well as the negative feedback circuit for a number of reasons. First, anatomical
data is more consistent with the reciprocally connected interneuron layer than lateral
connections between principal neurons [11, 13]. Second, direct transmission of the
representation would result in greater perceptual delays after stimulus onset since no
information is transmitted while the representation is being built up in the sub-threshold
range. In contrast, in the predictive coding model the projection neurons pass forth (a coarse
and possibly noisy version of) the input stimulus from the very beginning. We note that
adding a nonlinearity on the principal neurons would result in a delay in transmission in both
models. Although there is no biological justification for introducing a threshold to
interneurons only, the availability of an analytically solvable model justifies this abstraction.
Dynamics of a circuit with threshold on principal neurons will be explored elsewhere.
From a computational point of view there are three main advantages to
overcompleteness in the negative feedback circuit. First, the delay until subtraction of
prediction, which occurs when the first interneuron crosses threshold, will be briefer as the
number of feature vectors grows since the maximal projection of the stimulus on the
interneurons? feature vectors will be higher. Second, the larger the number of feature vectors
the fewer the number of interneurons with supra-threshold activity, which may be
energetically more efficient. Third, if stimuli come from different statistical ensembles, it
could be advantageous to have feature vectors tailored to the different stimulus ensembles,
which may result in more feature vectors, i.e., interneurons than principle neurons.
Our study considered responses to step-like stimuli. If the sensory environment changes
on slow time scales, a series of step-like responses may be taken as an approximation to the
true signal. Naturally, the extension of our framework to fully time-varying stimuli is an
important research direction.
Acknowledgements
We thank S. Baccus, A. Genkin, V. Goyal, A. Koulakov, M. Meister, G. Murphy, D.
Rinberg, and R. Wilson for valuable discussions and their input.
Appendix: Derivation of linearized Bregman iteration
Here, inspired by [22,23], we solve the following basis pursuit-like optimization problem:
!
For ?! ! ? ! |! |! + | !| !! , min! !(!) ?!. !. !" = !.
(A1)
!!
The idea behind linearized Bregman iteration, is to start with !! = 0 and, at each iteration, to seek
to update a so as to minimize the square error plus the distance from the previous value of a. Thus,
we perform the following update:
!!
!!!! = argmin! !!
!
!, !! + | ! ? !" |!
(A2)
!
!
!!
where we used a notation
!, ! for the Bregman divergence [29] between the two points a and
b induced by the convex function J. The Bregman divergence is an appropriate measure for such
problems that can handle the non-differentiable nature of the cost. It is defined by the following
!
expression: !! !, ! = ! ! ? ! ! ? !, ! ? ! , where ! ? !"(!) is an element of the
subgradient of J at the point b.
The Bregman divergence for the elastic net cost function J defined in Eq. A1 is:
!
!! (!, !! ) = !| !| ! ? ! |!! |! +
!
!!
|! |!! ?
!
!!
!! |!! ? ? !, ! ? !! ,
(A3)
where ! is a subgradient of J at ak . The condition for the minimum in Eq. A2 is:
! ! |!!!! |! +
!
!!
!!!! |!! ? !! + !! ! ? !!! ,
(A4)
where ! [.] designates a subdifferential. Consistency of the iteration scheme requires that the
update !!!! ?be a subgradient of J as well:
! ! |!!!! |! +
!
!!
!!!! |!! ? !!!! .
(A5)
By combining Eqs. A4,A5 we set:
!!!! = !! + !! ! ? !!! .
By substituting Eq. A6 into Eq. A4 and simplifying we get:
!!!! = argmin! ! !
!
+
!
!!
| ! ? !!!!! |! ,
(A6)
(A7)
which has the explicit solution:
!!!! = Thresh!" (!!!!! )
!
(A8)
!
By defining ?! = !! and expressing it in Eqs. A6,A8 with substitution ! = !" we get:
!!!! = !! + !!! ! ? !!!
(A9)
!!!! = Thresh! !!!! ? ? ? ? ? ? ? ? ? ? ?
Eq. A9 is the linearized Bregman iteration algorithm (main text Eq. 13), thereby showing that the
iterative scheme indeed finds a minimum of Eq. A2 at every time point. The sequence
convergence proof [23, 24] is beyond the scope of this paper.
References
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.
28.
29.
Barlow, H.B. and W.R. Levick, Threshold setting by the surround of cat retinal ganglion
cells. The Journal of physiology, 1976. 259(3): p. 737-57.
Dong, D.W. and J.J. Atick, Statistics of natural time-varying images. Network: Computation
in Neural Systems, 1995. 6(3): p. 345--358.
Field, D.J., Relations between the statistics of natural images and the response properties of
cortical cells. Journal of the Optical Society of America. A, Optics and image science, 1987.
4(12): p. 2379-94.
Ruderman, D.L. and W. Bialek, Statistics of natural images: Scaling in the woods. Physical
review letters, 1994. 73(6): p. 814-817.
Elias, P., Predictive coding. Information Theory, IRE Transactions on, 1955. 1(1): p. 16--24.
Srinivasan, M.V., S.B. Laughlin, and A. Dubs, Predictive coding: a fresh view of inhibition
in the retina, in Proc R Soc Lond, B, Biol Sci1982. p. 427-59.
Victor, J.D., Temporal aspects of neural coding in the retina and lateral geniculate.
Network-Computation in Neural Systems, 1999. 10(4): p. R1-R66.
Hosoya, T., S.A. Baccus, and M. Meister, Dynamic predictive coding by the retina. Nature,
2005. 436(7047): p. 71-7.
Huang, Y. and R.P.N. Rao, Predictive coding. Wiley Interdisciplinary Reviews: Cognitive
Science, 2011. 2(5): p. 580-593.
Laughlin, S., A simple coding procedure enhances a neuron's information capacity.
Zeitschrift fur Naturforschung. Section C: Biosciences, 1981. 36(9-10): p. 910-2.
Masland, R.H., The fundamental plan of the retina. Nature neuroscience, 2001. 4(9): p. 87786.
Olsen, S.R., V. Bhandawat, and R.I. Wilson, Divisive normalization in olfactory population
codes. Neuron, 2010. 66(2): p. 287-99.
Shepherd, G.M., et al., The olfactory granule cell: from classical enigma to central role in
olfactory processing. Brain research reviews, 2007. 55(2): p. 373-82.
Arevian, A.C., V. Kapoor, and N.N. Urban, Activity-dependent gating of lateral inhibition in
the mouse olfactory bulb. Nature neuroscience, 2008. 11(1): p. 80-7.
Baccus, S.A., Timing and computation in inner retinal circuitry. Annu Rev Physiol, 2007.
69: p. 271-90.
Rieke, F. and G. Schwartz, Nonlinear spatial encoding by retinal ganglion cells: when 1+1
not equal 2. Journal of General Physiology, 2011. 138(3): p. 283-290.
Shapley, R.M. and J.D. Victor, The effect of contrast on the transfer properties of cat retinal
ganglion cells. The Journal of physiology, 1978. 285: p. 275-98.
Koulakov, A.A. and D. Rinberg, Sparse Incomplete Representations: A Potential Role of
Olfactory Granule Cells. Neuron, 2011. 72(1): p. 124-136.
Lee, D.D. and H.S. Seung, Unsupervised learning by convex and conic coding. Advances in
Neural Information Processing Systems, 1997: p. 515--521.
Lochmann, T. and S. Deneve, Neural processing as causal inference. Curr Opin Neurobiol,
2011.
Olshausen, B.A. and D.J. Field, Sparse coding with an overcomplete basis set: a strategy
employed by V1? Vision research, 1997. 37(23): p. 3311-25.
Rao, R.P.N. and D.H. Ballard, Predictive coding in the visual cortex: a functional
interpretation of some extra-classical receptive-field effects. nature neuroscience, 1999. 2: p.
79--87.
Osher, S., et al., Fast linearized Bregman iteration for compressive sensing and sparse
denoising. Communications in Mathematical Sciences, 2009.
Yin, W., et al., Bregman iterative algorithms for l1-minimization with applications to
compressed sensing. SIAM Journal on Imaging Sciences, 2008. 1(1): p. 143--168.
Zou, H. and T. Hastie, Regularization and variable selection via the elastic net. Journal of
the Royal Statistical Society: Series B (Statistical Methodology), 2005. 67(2): p. 301--320.
Dayan, P., Recurrent sampling models for the Helmholtz machine. Neural computation, 1999.
11(3): p. 653-78.
Rehn, M. and F.T. Sommer, A network that uses few active neurones to code visual input
predicts the diverse shapes of cortical receptive fields. Journal of computational
neuroscience, 2007. 22(2): p. 135-46.
Rozell, C.J., et al., Sparse coding via thresholding and local competition in neural circuits.
Neural computation, 2008. 20(10): p. 2526-63.
Bregman, L.M., The relaxation method of finding the common point of convex sets and its
application to the solution of problems in convex programming* 1. USSR computational
mathematics and mathematical physics, 1967. 7(3): p. 200--217.
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4,068 | 4,682 | Multiresolution Gaussian Processes
David B. Dunson
Dept of Statistical Science, Duke University
[email protected]
Emily B. Fox
Dept of Statistics, University of Washington
[email protected]
Abstract
We propose a multiresolution Gaussian process to capture long-range, nonMarkovian dependencies while allowing for abrupt changes and non-stationarity.
The multiresolution GP hierarchically couples a collection of smooth GPs, each
defined over an element of a random nested partition. Long-range dependencies are captured by the top-level GP while the partition points define the abrupt
changes. Due to the inherent conjugacy of the GPs, one can analytically marginalize the GPs and compute the marginal likelihood of the observations given the partition tree. This property allows for efficient inference of the partition itself, for
which we employ graph-theoretic techniques. We apply the multiresolution GP to
the analysis of magnetoencephalography (MEG) recordings of brain activity.
1 Introduction
A key challenge in many time series applications is capturing long-range dependencies for which
Markov-based models are insufficient. One method of addressing this challenge is through employing a Gaussian process (GP) with an appropriate (non-band-limited) covariance function. However, GPs typically assume smoothness properties that can blur key elements of the signal if abrupt
changes occur. The Mat?ern kernel enables less smooth functions, but assumes a stationary process
that does not adapt to varying levels of smoothness. Likewise, a changepoint [21] or partition [8]
model between smooth functions fails to capture long range dependencies spanning changepoints.
Another long-memory process is the fractional ARIMA process [5, 13]. Wavelet methods have also
been proposed, including recently for smooth functions with discontinuities [2]. We take a fundamentally different approach based on GPs that allows (i) direct interpretability, (ii) local stationarity,
(iii) irregular grids of observations, and (iv) sharing information across related time series.
As a motivating application, consider magnetoencephalography (MEG) recordings of brain activity
in response to some word stimulus. Due to the low signal-to-noise-ratio (SNR) regime, multiple
trials are often recorded, presenting a functional data analysis scenario. Each trial results in a noisy
trajectory with key discontinuities (e.g., after stimulus onset). Although there are overall similarities
between the trials, there are also key differences that occur based on various physiological phenomena, as depicted in Fig. 1. We clearly see abrupt changes as well as long-range correlations. Key to
the data analysis is the ability to share information about the overall trajectory between the single
trials without forcing unrealistic smoothness assumptions on the single trials themselves.
In order to capture both long-range dependencies and potential discontinuities, we propose a multiresolution GP (mGP) that hierarchically couples a collection of smooth GPs, each defined over an
element of a nested partition set. The top-level GP captures a smooth global trajectory, while the
partition points define abrupt changes in correlation induced by the lower-level GPs. Due to the inherent conjugacy of the GPs, conditioned on the partition points the resulting function at the bottom
level is marginally GP-distributed with a partition-dependent (and thus non-stationary) covariance
function. The correlation between any two observations yi and yj generated by the mGP at locations
xi and xj is a function of the distance ||xi ? xj || and which partition sets contain both xi and xj .
In a standard regression setting, the marginal GP structure of the mGP allows us to compute the
marginal likelihood of the data conditioned on the partition, enabling efficient inference of the partition itself. We integrate over the hierarchy of GPs and only sample the partition points. For our
1
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Figure 2: mGP on a balanced, binary tree
Figure 1: For sensor 1 and word house, Left: Data from three partition: Parent function is split by A1 =
1
1
trials; Middle: Empirical correlation matrix from 20 trials; Right: {A1 , A2 }. Recursing down the tree, each
Hierarchical segmentation produced by recursive minimization of partition has a GP with mean given by its
parent function restricted to that set.
normalized cut objective, with color indicating tree level.
proposal distribution, we borrow the graph-theoretic idea of normalized cuts [22] often used in image
segmentation. Our inferences integrate over the partition tree, allowing blurring of discontinuities
and producing functions which can appear smooth when discontinuities are not present in the data.
2 Background
A GP provides a distribution on real-valued functions f : X ? ?, with the property that the function
evaluated at any finite collection of points is jointly Gaussian. The GP, denoted GP(m, c), is uniquely
defined by its mean function m and covariance function c. That is, f ? GP(m, c) if and only if for
all n ? 1 and x1 , . . . , xn , (f (x1 ), . . . , f (xn )) ? Nn (?, K), with ? = [m(x1 ), . . . , m(xn )] and
[K]ij = c(xi , xj ). The properties (e.g., continuity, smoothness, periodicity, etc.) of functions
drawn from a given GP are determined by the covariance function. The squared exponential kernel,
c(x, x? ) = d exp(??||x ? x? ||22 ), leads to smooth functions. Here, d is a scale hyperparameter and ?
is the bandwidth determining the extent of the correlation in f over X . See [18] for further details.
3 Multiresolution Gaussian Process Formulation
Our interest is in modeling a function g that (i) is locally smooth, (ii) exhibits long-range correlations
(i.e., corr(g(x), g(x? )) > 0 for ||x ? x? || relatively large), and (iii) has abrupt changes. We begin by
modeling a single function, but with a specification that readily lends itself to modeling a collection
of functions that share a common global trajectory, as explored in Sec. 4.
Generative Model Assume a set of noisy observations y = {y1 , . . . , yn }, yi ? ?, of the function
g at locations {x1 , . . . , xn }, xi ? X ? ?p :
yi = g(xi ) + ?i , ?i ? N (0, ? 2 ).
(1)
We hierarchically define g as follows. Let A = {A0 , A1 , . . . , AL?1 } be a nested partition, or tree
S
for some k. Furthermore,
partition, of X with A0 = X , X = i A?i , A?i ? A?j = ?, and A?i ? A??1
k
assume that each A?i is a contiguous subset of X . Fig. 2 depicts a balanced, binary tree partition. We
define a global parent function on A0 as f 0 ? GP(0, c0 ). This function captures the overall shape
of g and its long-range dependencies. Then, over each partition set A?i we independently draw
f ? (A?i ) ? GP(f ??1 (A?i ), c?i ).
(2)
That is, the mean of the GP is given by the parent function restricted to the current partition set. Due
to the conditional independence of these draws, f ? can have discontinuities at the partition points.
However, due to the coupling of GPs through the tree, f ? will maintain aspects of the shape of f 0 .
Finally, we set g = f L?1 . A pictorial representation of the mGP is shown in Fig. 2.
P
We can equivalently represent the mGP as an additive GP model: ?? (A?i ) ? GP(0, c?i ), g = ? ?? .
?
? 2
Covariance Function We assume a squared exponential kernel c?i = d?i exp(??
Pi?||x ?? x2 ||2 ),
?
?
?
encouraging local smoothness over each partition set Ai . We focus on di = d with ?=1 (d ) < 1
for finite variance regardless of tree depth and additionally encouraging lower levels to vary less from
their parent function, providing regularization and robustness to the choice of L.
We typically assume bandwidths ??i = ?/||A?i ||22 so that each child function is locally as smooth as
its parent. One can think of this formulation as akin to a fractal process: zooming in on any partition,
the locally defined function has the same smoothness as that of its parent over the larger partition.
Thus, lower levels encode finer-resolution details. We denote the covariance hyperparameters as ? =
{d0 , . . . , dL?1 , ?}, and omit the dependency in conditional distributions for notational simplicity.
See the Supplementary Material for discussion of other possible covariance specifications.
2
Induced Marginal GP
The conditional independencies of our mGP imply that
Z
L?1
Y
p(g | A) = p(f 0 )
p(f ? | f ??1 , A? )df 0:L?2 .
(3)
?=1
Due to the inherent conjugacy of the GPs, one can analytically marginalize the hierarchy of GPs
conditioned on the partition tree A yielding
g | A ? GP(0, c?A ),
c?A =
L?1
XX
?=0
c?i IA?i .
(4)
i
Here, IA?i (x, x? ) = 1 if x, x? ? A?i and 0 otherwise. Eq. (4) provides an interpretation of the mGP
as a (marginally) partition-dependent GP, where the partition A defines the discontinuities in the
covariance function c?A . The covariance function encodes local smoothness of g and discontinuities
at the partition points. Note that c?A defines a non-stationary covariance function.
The correlation between any two observations yi and yj at locations xi and xj generated as in Eq. (1)
is a function of how many tree levels contain both xi and xj and the distance ||xi ? xj ||. Let ri?
index the partition set such that xi ? A?r? and Lij the lowest level for which xi and xj fall into the
i
same set (i.e., the largest ? such that ri? = rj? ). Then, for xi 6= xj ,
PLij ?
PLij ?
2
? 2
?=0 d exp(??||xi ? xj ||2 /||Ari? ||2 )
?=0 cri? (xi , xj )
,
=
corr(yi , yj | A) = Q
PL?1
PL?1 ?
1
2
2
? 2 + ?=0 d?
k?{i,j} (? +
?=0 cr ? (xk , xk ))
k
(5)
where the second equality follows from assuming the previously described kernels. An example
correlation matrix is shown in Fig. 3(c). ? determines the width of the bands while d? controls the
contribution of level ?. Since d? is square summable, lower levels are less influential.
Marginal Likelihood Based on a vector of observations y = [y1 ? ? ? yn ]? at locations
x = [x1 ? ? ? xn ]? , we can restrict our attention to evaluating the GPs at x. Let f ? (x) =
[f ? (x1 ) ? ? ? f ? (xn )]? . By definition of the GP, we have
c?r (xi , xj ) xi , xj ? A?r
f ? (x) | f ??1 (x), A? ? N (f ??1 (x), K? ), [K? ]i,j =
.
(6)
0
otherwise
The level-specific covariance matrix K? is block-diagonal with structure determined by the levelspecific partition A? . Observations are generated as y | g(x) ? N (g(x), ? 2 In ). Recalling Eq. (3),
standard results yield
L?1
L?1
X
X
(7)
y | A ? N 0, ? 2 In +
K? .
g(x) | A ? N 0,
K?
?=0
?=0
This result can also be derived from the induced mGP of Eq. (4). We see that the marginal likelihood
p(y | A) has a closed form. Alternatively, one can condition on the GP at any level ?? :
L?1
X
??
??
2
(8)
y | f (x), A ? N f (x), ? In +
K? .
?=?? +1
A key advantage of the mGP is the conditional conjugacy of the latent GPs that allows us to compute
the likelihood of the data simply conditioned on the hierarchical partition A (see Eq. (7)). This fact
is fundamental to the efficiency of the partition inference procedure described in Sec. 5.
4 Multiple Trials
In many applications, such as the motivating MEG application, one has a collection of observations
of an underlying signal. To capture the common global trajectory of these trials while still allowing
for trial-specific variability, we model each as a realization from an mGP with a shared parent function f 0 . One could trivially allow for alternative structures of hierarchical sharing beyond f 0 if an
application warranted. For simplicity, and due to the motivating MEG application, we additionally
assume shared changepoints between the trials, though this assumption can also be relaxed.
3
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(a)
(b)
(c)
(d)
(e)
Figure 3: (a) Three trials and (b) all 100 trials of data generated from a 5-level mGP with a shared parent
function f 0 and partition A (randomly sampled). (c) True correlation matrix. (d) Empirical correlation matrix
from 100 trials. (e) Hierarchical segmentation produced by recursive minimization of normalized cut objective.
(j)
(j)
Generative Model For each trial y(j) = {y1 , . . . , yn }, we model
(j)
yi
(j)
= g (j) (xi ) + ?i ,
(j)
?i
? N (0, ? 2 ),
(9)
with g (j) = f L?1,(j) generated from a trial-specific GP hierarchy f 0 ? f 1,(j) ? ? ? ? ? f L?1,(j)
with shared parent f 0 . (Again, alternative structures can be considered.) From Eq. (8) with ?? = 0,
and exploiting the independence of {f ?,(j) }, independently for each j
L?1
X
(j)
0
(j)
0
2
(10)
K? .
y | f (x), A ? N y ; f (x), ? In +
?=1
Note that with our GP-based formulation, we need not assume coincident observation locations
x1 , . . . , xn between the trials. However, for simplicity of exposition, we consider shared locations.
PL?1
We compactly denote the covariance by ? = ? 2 In + ?=1
K? .
Simulated data generated from a 5-level mGP with shared f 0 and A are shown in Fig. 3. The sample
correlation matrix is also shown. Compare with the MEG data of Fig. 1. Both the qualitative structure of the raw time series as well as blockiness of the correlation matrix have striking similarities.
Posterior Global Trajectory and Predictions Based on a set of trials {y(1) , . . . , y(J) }, it is of
interest to infer the posterior of f 0 . Standard Gaussian conjugacy results imply that
?1
?1
0
(1)
(J)
?1 ?1
?1 ?1
? , K0 + J?
,
(11)
p(f (x) | y , . . . , y , A) = N K0 + J?
y
? = ??1
where y
p(y(J+1)
y(i) . Likewise, the predictive distribution of data from a new trial is
Z
| y(1) , . . . , y(J) , A) = p(y(J+1) | f 0 (x), A)p(f 0 (x) | y(1) , . . . , y(J) , A)df 0
?1
?1
?1 ?1
?1 ?1
? , ? + K0 + J?
= N K0 + J?
.
(12)
y
P
i
Marginal Likelihood Since the set of trials Y = {y(1) , . . . , y(J) } are generated from a shared
parent function f 0 , the marginal likelihood does not decompose over trials. Instead,
1 X (i)? ?1 (i) 1 ? ?1
|K0 |?1/2 |?|?J/2
?1 ?1
?
?
exp
?
y
?
y
+
y
(K
p(Y | A) =
+
J?
)
y
.
0
2 i
2
(2?)?nJ/2 |K0?1 + J??1 |1/2
(13)
See the Supplementary Material for a derivation. One can easily verify that the above simplifies to
the marginal likelihood of Eq. (7) when J = 1.
5 Inference of the Hierarchical Partition
In the formulation so far, we have assumed that the hierarchical partition A is given. A key question
is to infer the partition from the data. Assume that we have prior p(A) on the hierarchical partition.
Based on the fact that we can analytically compute p(Y | A), we can use importance sampling or
independence chain Metropolis Hastings to draw samples from the posterior p(A | Y ).
In what follows, we assume a balanced binary tree for A. See the Supplementary Material for a
discussion of how unbalanced trees can be considered via modifications to the covariance hyperparameter specification or by considering alternative priors p(A) such as the Mondrian process [20].
4
Partition Prior We consider a prior solely on the partition points {z1 , . . . , z2L?1 ?1 } rather than
taking tree level into account as well. Because of our time-series
analysis focus, we assume X ? ?.
Q
We define a distribution F on X and specify p(A) = i F (zi ). Generatively, one can think of
drawing 2L?1 ? 1 partition points from F and deterministically forming a balanced binary tree
A from these. For multidimensional X , one could use Voronoi tessellation and graph matching to
build the tree from the randomly selected zi . Such a prior allows for trivial specification of a uniform
distribution on A (simply taking F uniform on X ) or for eliciting prior information on changepoints,
such as based on physiological information for the MEG data. Eliciting such information in a leveldependent setup is not straightforward. Also, despite common deployment, taking the partition point
at level ? as uniformly distributed over the parent set A??1
yields high mass on A with small A?i .
i
This property is undesirable because it leads to trees with highly unbalanced partitions.
Our resulting inferences perform Bayesian model averaging over trees. As such, even though we
specify a prior on partitions with 2L?1 ? 1 changepoints, the resulting functions can appear to
adaptively use fewer by averaging over the uncertainty in the discontinuity location.
Partition Proposal Although stochastic tree search algorithms tend to be inefficient in general,
we can harness the well-defined correlation structure associated with a given hierarchical partition
to much more efficiently search the tree space. One can think of every observed location xi as a
node in a graph with edge weights between xi and xj defined by the magnitude of the correlation of
yi and yj . Based on this interpretation, the partition points of A correspond to graph cuts that bisect
small edge weights, as graphically depicted in Fig. 4. As such, we seek a method for hierarchically
cutting a graph. Given a cost matrix W with elements wuv defined for all pairs of nodes u, v in a set
V , the normalized cut metric [22] for partitioning V into disjoint sets A and B is given by
(14)
ncut(A, B) = cut(A, B) assoc(A, V )?1 + assoc(B, V )?1 ,
P
P
where cut(A, B) = u?A,v?B wuv and assoc(A, V ) = u?A,v?V wuv . Typically, the cut point
is selected as the minimum of the metric ncut(A, B) computed over all possible subsets A and B.
The normalized cut metric balances between the cost of edge weights cut and the connectivity of the
cut component, thus avoiding cuts that separate small sets. Fig. 1 shows an example of applying a
greedy normalized cuts algorithm (recursively minimizing ncut(A, B)) to MEG data.
Instead of deterministically selecting cut points, we employ the
cut 1
cut 2
cut 2
normalized cut objective as a proposal distribution. Let the cost
matrix W be the absolute value of the empirical correlation matrix
computed from trials {y(1) , . . . , y(J) } (see Fig. 1). Due to the
natural ordering of our locations xi ? X ? ?, the algorithm is
TIME
straightforwardly implemented. We step down the hierarchy, first
Figure 4: Illustration of cutpoints
proposing a cut of A0 into {A11 , A12 } with probability
dividing contiguous segments at
1
1
1
1 ?1
q({A1 , A2 }) ? ncut(A1 , A2 ) .
(15) points of low correlation.
At level ?, each A?i is partitioned via a normalized cut proposal based on the submatrix of W corresponding to the locations xi ? A?i . The probability of any partition A under the specified proposal
distribution is simply computed as the product of the sequence of conditional probabilities of each
cut. This procedure generates cut points only at the observed locations xi . More formally, the
partition point in X is proposed as uniformly distributed between xi and xi+1 . Extensions to multidimensional X rely on spectral clustering algorithms based on the graph Laplacian [24].
Markov Chain Monte Carlo An importance sampler draws hierarchical partitions A(m) ? q,
with the proposal distribution q defined as above, and then weights the samples by p(A(m) )/q(A(m) )
to obtain posterior draws [19]. Such an approach is naively parallelizable, and thus amenable to
efficient computations, though the effective sample size may be low if q does not adequately match
the posterior p(A | Y ). Alternatively, a straightforward independence chain Metropolis Hastings
algorithm (see Supplementary Material) is defined by iteratively proposing A? ? q which is accepted
with probability min{r(A? | A), 1} where A is a previous sample of a hierarchical partition and
r(A? | A) = p(Y | A? )p(A? )q(A)/[p(Y | A)p(A)q(A? )].
(16)
The tailoring of the proposal distribution q to this application based on normalized cuts dramatically
aids in improving the acceptance rate relative to more naive tree proposals. However, the acceptance
5
rate tends to decrease as higher posterior probability partitions A are discovered, especially for trees
with many levels and large input spaces X for which the search space is larger.
One benefit of the MCMC approach over importance sampling is the ability to include more intricate
tree proposals to increase efficiency. We choose to interleave both local and global tree proposals. At
each iteration, we first randomly select a node in the tree (i.e., a partition set A?i ) and then propose a
new sequence of cuts for all children of this node. When the root node is selected, corresponding to
A0 , the proposal is equivalent to the global proposals previously considered. We adapt the proposal
distribution for node selection to encourage more global searches at first and then shift towards a
greater balance between local and global searches as the sampling progresses. Sequential Monte
Carlo methods [4] can also be considered, with particles generated as global proposals.
Computational Complexity The per iteration complexity is O(n3 ), equivalent to a typical likelihood evaluation under a GP prior. Using dynamic programming, the cost associated with the normalized cuts proposal is O(n2 (L ? 1)). Standard techniques for more efficient GP computations are
readily applicable, as well as extensions that harness the additive block structure of the covariance.
6 Related Work
Various aspects of the mGP have similarities to other models proposed in the literature that primarily
fall into two main categories: (i) GPs defined over a partitioned input space, and (ii) collections of
GPs defined at tree nodes. The treed GP [8] captures non-stationarities by defining independent GPs
at the leaves of a Bayesian CART-partitioned input space. The related approach of [12] assumes
a Voronoi tessellation. For time series, [21] examines online inference of changepoints with GPs
modeling the data within each segment. These methods capture abrupt changes, but do not allow for
long-range dependencies spanning changepoints nor a functional data hierarchical structure, both
inherent to our multiresolution perspective. A main motivation of the treed GP is the resulting
computational speed-ups of an independently partitioned GP. A two-level hierarchical GP also aimed
at computational efficiency is considered by [16], where the top-level GP is defined at a coarser scale
and provides a piece-wise constant mean for lower-level GPs on a pre-partitioned input space.
[10, 11] consider covariance functions defined on a phylogenetic tree such that the covariance between function-valued traits depends on both their spatial distance and evolutionary time spanned
via a common ancestor. Here, the tree defines the strength and structure of sharing between a collection of functions rather than abrupt changes within the function. The Bayesian rose tree of [3]
considers a mixture of GP experts, as in [14, 17], but using Bayesian hierarchical clustering with
arbitrary branching structure in place of a Dirichlet process mixture. Such an approach is fundamentally different from the mGP: each GP is defined over the entire input space, data result from a
GP mixture, and input points are not necessarily spatially clustered. Alternatively, multiscale processes have a long history (cf. [25]): the variables define a Markov process on a typically balanced,
binary tree and higher-level nodes capture coarser level information about the process. In contrast,
the higher level nodes in the mGP share the same temporal resolution and only vary in smoothness.
At a high level, the mGP differs from previous GP-based tree models in that the nodes of our tree
represent GPs over a contiguous subset of the input space X constrained in a hierarchical fashion.
Thus, the mGP combines ideas of GP-based tree models and GP-based partition models.
As presented in Sec. 3, one can formulate an mGP as an additive GP where each GP in the sum
decomposes independently over the level-specific partition of the input space X . The additive GPs
of [6] instead focus on coping with multivariate inputs, in a similar vain to hierarchical kernel learning [1], thus addressing an inherently different task.
7 Results
7.1 Synthetic Experiments
To assess our ability to infer a hierarchical partition via the proposed MCMC sampler, we generated
100 trials of length 200 from a 5-level mGP with a shared parent function f 0 . The hyperparameters
were set to ? 2 = 0.1, ? = 10, d? = d0 exp(?0.5(? + 1)) for ? = 0, . . . , L ? 1 with d0 = 5. The
data are shown in Fig. 3, along with the empirical correlation matrix that is used as the cost matrix
for the normalized cuts proposals.
For inference, we set ? 2 = ?
? 2 /3 and d? = (?
? 2 /3) exp(?0.5?), where ?
? 2 is the average timespecific sample variance. ? was as in the simulation. The hyperparameter mismatch demonstrates
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hGP
GP
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Iteration
3000
0
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40
estimated(f0) ? f0
20
40
Log Likelihood
20
x 10
?300
?350
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?450
?500
200
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G
P
hG
P
L=
2
L=
5
L=
7
L=
10
?2.88
(a)
(b)
(c)
(d)
(e)
Figure 5: For the data of Fig. 3, (a) true and (b) MAP partitions. (c) Trace plots of log likelihood versus
MCMC iteration for 10 chains. Log likelihood under the true partition (cyan) and minimized normalized cut
partition of Fig. 3 (magenta) are also shown. (d) Errors between posterior mean f 0 and true f 0 for GP, hGP, and
mGP. (e) Predictive log likelihood of 10 heldout sequences for GP, hGP, and mGP with L = 2, 5(true), 7, 10.
some robustness to mispecification. For a uniform prior p(A), 10 independent MCMC chains were
run for 3000 iterations, thinned by 10. The first 1000 iterations used pure global tree searches; the
sampler was then tempered to uniform node proposals. The effects of this choice are apparent in
the likelihood plot of Fig. 5, which also displays the true hierarchical partition and MAP estimate.
Compare to the normalized cuts partition of Fig. 3, especially at the important level 1 cut. The full
simulation study took less than 7 minutes to run on a single 1.8 GHz Intel Core i7 processor.
To assess sensitivity to the choice of L, we compare the predictive log-likelihood of 10 heldout test
sequences under an mGP with 2, 5, 7, and 10 levels. As shown in Fig. 5(e), there is a clear gain
going from 2 to 5 levels. However, overestimating L has minimal influence on predictive likelihood
since lower tree levels capture finer details and have less overall effect. We also compare to a single
GP and a 2-level hierarchical GP (hGP) (see Sec. 7.2). For a direct comparison, both use squared
exponential kernels. Hyperparameters were set as in the mGP for the top-level GP. The total variance
was also matched with the GP taking this as noise and the hGP splitting between level 2 and noise.
In addition to better predictive performance, Fig. 5(d) shows the mGP?s improved estimation of f 0 .
7.2 MEG Analysis
We analyzed magnetoencephalography (MEG) recordings of neuronal activity collected from a helmet with gradiometers distributed over 102 locations around the head. The gradiometers measure
the spatial gradient of the magnetic activity in Teslas per meter (T/m) [9]. Since the firings of neurons in the brain only induce a weak magnetic field outside of the skull, the signal-to-noise ratio of
the MEG data is very low and typically multiple recordings, or trials, of a given task are collected.
Our MEG data was recorded while a subject viewed 20 stimuli describing concrete nouns (both
the written noun and a representative line drawing), with 20 interleaved trials per word. See the
Supplementary Material for further details on the data and our analyses presented herein.
Efficient sharing of information between the single trials is important for tasks such as word classification [7]. A key insight of [7] was the importance of capturing the time-varying correlations
between MEG sensors for performing classification. However, the formulation still necessitates a
mean model. [7] propose a 2-level hierarchical GP (hGP): a parent GP captures the common global
trajectory, as in the mGP, and each trial-specific GP is centered about the entire parent function1.
This formulation maintains global smoothness at the individual trial level. The mGP instead models the trial-specific variability with a multi-level tree of GPs defined as deviations from the parent
function over local partitions, allowing for abrupt changes relative to the smooth global trajectory.
For our analyses, we consider the words associated with the ?building? and ?tool? categories shown
in Fig. 7. Independently for each of the 10 words and 102 sensors, we trained a 5-level mGP
using 15 randomly selected trials as training data and the 5 remaining for testing. Each trial was
of length n = 340. We ran 3 independent MCMC chains for 3000 iterations with both global and
local tree searches. We discarded the first 1000 samples as burn-in and thinned by 10. The mGP
hyperparameters were set exactly as in the simulated study of Sec. 7.1 for structure learning and
then optimized over a grid to maximize the marginal likelihood of the training data.
We compare the predictive performance of the mGP in terms of MSE of heldout segments relative to
a GP and hGP, each with similarly optimized hyperparameters. The predictive mean conditioned on
data up to the heldout time is straightforwardly derived from Eq. (12). For the mGP, the calculation is
averaged over the posterior samples of A. Fig. 6 displays the MSEs decomposed by cortical region.
1
The model of [7] uses an hGP in a latent space. The mGP could be similarly deployed.
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x 10
MLE
hGP
mGP
10
Heldout Log Likelihood
10
20
w
fm
m
15
4
Visual
Frontal
Parietal
Temporal
25
Observations
20
% Decrease in MSE v. hGP
% Decrease in MSE v. GP
Visual
Frontal
Parietal
Temporal
25
(b)
(c)
(d)
Figure 6: Per-lobe comparison of mGP to (a) GP and (b) hGP: For various values of ? , % decrease in predictive
?
MSE of heldout y??:? +30 conditioned on y1:?
?1 and 15 training sequences. (c) For a visual cortex sensor and
word hammer, plots of test data, empirical mean (MLE), and hGP and mGP predictive mean for entire heldout
y? . (d) Boxplots of predictive log likelihood of heldout y? for the mGP and wavelet-based method of [15].
The results clearly indicate that the mGP consistently better captures the features of the data, and particularly for
sensors with large abrupt changes such as in the visual
cortex. The heldout trials for a visual cortex sensor are
displayed in Fig. 6(c). Relative to the hGP, the mGP
much better tracks the early dip in activity right after the
visual stimulus onset (t = 0). The posterior distribution of inferred changepoints at level 1, also broken down
by cortical region, are displayed in Fig. 7. As expected,
the visual cortex has the earliest changepoints. Similar
trends are seen in the parietal lobe that handles perception and sensory integration. The temporal lobe, which
is key in semantic processing, has changepoints occurring later. These results concur with the findings of [23]:
semantic processing starts between 250 and 600 ms and
word length (a visual feature) is decoded most accurately
very near the standard 100ms response time (?n100?).
Visual L=1
Frontal L=1
igloo
house
church
apartment
barn
hammer
saw
screwdriver
pliers
chisel
0
0.5
1
Time (sec)
0
Parietal L=1
0.5
1
Time (sec)
Temporal L=1
igloo
house
church
apartment
barn
hammer
saw
screwdriver
pliers
chisel
0
0.5
1
Time (sec)
0
0.5
1
Time (sec)
Figure 7: Inferred changepoints at level 1
aggregated over sensors within each lobe:
visual (top-left), frontal (top-right), parietal
(bottom-left), and temporal (bottom-right).
We also compare our predictive performance to that of the wavelet-based functional mixed model
(wfmm) of [15]. The wfmm has become a standard approach for functional data analysis since it
allows for spiky trajectories and efficient sharing of information between trials. One limitation, however, is the restriction to a regular grid of observations. The wfmm enables analysis in a multivariate
setting, but for a direct comparison we simply apply the wfmm to each word and sensor independently. Fig. 6(d) shows boxplots of the predictive heldout log likelihood of the test trials under the
mGP and wfmm. The results are over 5 heldout trials, 102 sensors, and 10 words. In addition to the
easier interpretability of the mGP, the predictive performance also exceeds that of the wfmm.
8 Discussion
The mGP provides a flexible framework for characterizing the dependence structure of real data,
such as the examined MEG recordings, capturing certain features more accurately than previous
models. In particular, the mGP provides a hierarchical functional data analysis framework for modeling (i) strong, locally smooth sharing of information, (ii) global long-range correlations, and (iii)
abrupt changes. The simplicity of the mGP formulation enables further theoretical analysis, for
example, combining posterior consistency results from changepoint analysis with those for GPs.
Although we focused on univariate time series analysis, our formulation is amenable to multivariate functional data analysis extensions: one can naturally accommodate hierarchical dependence
structures through partial sharing of parents in the tree, or possibly via mGP factor models.
There are many interesting questions relating to the proposed covariance function. Our fractal specification represents a particular choice to avoid over-parameterization, although alternatives could
be considered. For hyperparameter inference, we anticipate that joint sampling with the partition
would mix poorly, and consider it a topic for future exploration. Another interesting topic is to
explore proposals for more general tree structures. We believe that the proposed mGP represents a
powerful, broadly applicable new framework for non-stationary analyses, especially in a functional
data analysis setting, and sets the foundation for many interesting possible extensions.
Acknowledgments
The authors thank Alona Fyshe, Gustavo Sudre and Tom Mitchell for their help with data acquisition, preprocessing, and useful suggestions. This work was supported in part by AFOSR Grant FA9550-12-1-0453 and the
National Institute of Environmental Health Sciences (NIEHS) of the NIH under Grant R01 ES017240.
8
References
[1] F. Bach. High-dimensional non-linear variable selection through hierarchical kernel learning. Technical
Report 0909.0844v1, arXiv, 2009.
[2] J. Beran and Y. Shumeyko. On asymptotically optimal wavelet estimation of trend functions under longrange dependence. Bernoulli, 18(1):137?176, 2012.
[3] C. Blundell, Y. W. Teh, and K. A. Heller. Bayesian rose trees. In Proc. Uncertainty in Artificial Intelligence, pages 65?72, 2010.
[4] P. Del Moral, A. Doucet, and A. Jasra. Sequential Monte Carlo samplers. Journal of the Royal Statistical
Society, Series B, 68(3):411?436, 2006.
[5] F. X. Diebold and G. D. Rudebusch. Long memory and persistence in aggregate output. Journal of
Monetary Economics, 24:189?209, 1989.
[6] D. Duvenaud, H. Nickisch, and C. E. Rasmussen. Additive Gaussian processes. In Advances in Neural
Information Processing Systems, volume 24, pages 226?234, 2011.
[7] A. Y. Fyshe, E. B. Fox, D. B. Dunson, and T. Mitchell. Hierarchical latent dictionaries for models of
brain activation. In Proc. International Conference on Artificial Intelligence and Statistics, pages 409?
421, 2012.
[8] R. .B. Gramacy and H. K. H. Lee. Bayesian treed Gaussian process models with an application to computer modeling. Journal of the American Statistical Association, 103(483):1119?1130, 2008.
[9] P. Hansen, M. Kringelbach, and R. Salmelin. MEG: An Introduction to Methods. Oxford University
Press, USA, 2010. ISBN 0195307232.
[10] R. Henao and J. E. Lucas. Efficient hierarchical clustering for continuous data. Technical Report
1204.4708v1, arXiv, 2012.
[11] N. S. Jones and J. Moriarty. Evolutionary inference for function-valued traits: Gaussian process regression
on phylogenies. Technical Report 1004.4668v2, arXiv, 2011.
[12] H. M. Kim, B. K. Mallick, and C. C. Holmes. Analyzing nonstationary spatial data using piecewise
Gaussian processes. Journal of the American Statistical Association, 100(470):653?668, 2005.
[13] P. S. Kokoszka and M. S. Taqqu. Parameter estimation for infinite variance fractional ARIMA. The
Annals of Statistics, 24(5):1880?1913, 1996.
[14] E. Meeds and S. Osindero. An alternative mixture of Gaussian process experts. In Advances in Neural
Information Processing Systems, volume 18, pages 883?890, 2006.
[15] J. S. Morris and R. J. Carroll. Wavelet-based functional mixed models. Journal of the Royal Statistical
Society, Series B, 68(2):179?199, 2006.
[16] S. Park and S. Choi. Hierarchical Gaussian process regression. In Proc. Asian Conference on Machine
Learning, pages 95?110, 2010.
[17] C. E. Rasmussen and Z. Ghahramani. Infinite mixtures of Gaussian process experts. In Advances in
Neural Information Processing Systems, volume 2, pages 881?888, 2002.
[18] C. E. Rasmussen and C. K. .I. Williams. Gaussian Processes for Machine Learning. MIT Press, 2006.
[19] C. P. Robert and G. Casella. Monte Carlo Statistical Methods. Springer, 2005.
[20] D. M. Roy and Y. W. Teh. The Mondrian process. In Advances in Neural Information Processing Systems,
volume 21, pages 1377?1384, 2009.
[21] Y. Saatci, R. Turner, and C. E. Rasmussen. Gausssian process change point models. In Proc. International
Conference on Machine Learning, pages 927?934, 2010.
[22] J. Shi and J. Malik. Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis
and Machine Intelligence, 22(8):888?905, 2000.
[23] G. Sudre, D. Pomerleaum, M. Palatucci, L. Wehbe, A. Fyshe, R. Salmelin, and T. Mitchell. Tracking
neural coding of perceptual and semantic features of concrete nouns. Neuroimage, 62(1):451?463, 2012.
[24] U. von Luxburg. A tutorial on spectral clustering. Statistics and Computing, 17(4):395?416, 2007.
[25] A. S. Willsky. Multiresolution Markov models for signal and image processing. Proceedings of the IEEE,
90(8):1396?1458, 2002.
9
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4,069 | 4,683 | Multimodal Learning with Deep Boltzmann Machines
Nitish Srivastava
Department of Computer Science
University of Toronto
[email protected]
Ruslan Salakhutdinov
Department of Statistics and Computer Science
University of Toronto
[email protected]
Abstract
A Deep Boltzmann Machine is described for learning a generative model of data
that consists of multiple and diverse input modalities. The model can be used
to extract a unified representation that fuses modalities together. We find that
this representation is useful for classification and information retrieval tasks. The
model works by learning a probability density over the space of multimodal inputs.
It uses states of latent variables as representations of the input. The model can
extract this representation even when some modalities are absent by sampling
from the conditional distribution over them and filling them in. Our experimental
results on bi-modal data consisting of images and text show that the Multimodal
DBM can learn a good generative model of the joint space of image and text
inputs that is useful for information retrieval from both unimodal and multimodal
queries. We further demonstrate that this model significantly outperforms SVMs
and LDA on discriminative tasks. Finally, we compare our model to other deep
learning methods, including autoencoders and deep belief networks, and show that
it achieves noticeable gains.
1
Introduction
Information in the real world comes through multiple input channels. Images are associated with
captions and tags, videos contain visual and audio signals, sensory perception includes simultaneous
inputs from visual, auditory, motor and haptic pathways. Each modality is characterized by very
distinct statistical properties which make it difficult to ignore the fact that they come from different
input channels. Useful representations can be learned about such data by fusing the modalities into a
joint representation that captures the real-world ?concept? that the data corresponds to. For example,
we would like a probabilistic model to correlate the occurrence of the words ?beautiful sunset? and
the visual properties of an image of a beautiful sunset and represent them jointly, so that the model
assigns high probability to one conditioned on the other.
Before we describe our model in detail, it is useful to note why such a model is required. Different
modalities typically carry different kinds of information. For example, people often caption an image
to say things that may not be obvious from the image itself, such as the name of the person, place,
or a particular object in the picture. Unless we do multimodal learning, it would not be possible to
discover a lot of useful information about the world (for example, ?what do beautiful sunsets look
like??). We cannot afford to have discriminative models for each such concept and must extract this
information from unlabeled data.
In a multimodal setting, data consists of multiple input modalities, each modality having a different
kind of representation and correlational structure. For example, text is usually represented as discrete sparse word count vectors, whereas an image is represented using pixel intensities or outputs
of feature extractors which are real-valued and dense. This makes it much harder to discover relationships across modalities than relationships among features in the same modality. There is a lot
of structure in the input but it is difficult to discover the highly non-linear relationships that exist
1
Figure 1: Left: Examples of text generated from a DBM by sampling from P (vtxt |vimg , ?). Right: Examples of images retrieved using features generated from a DBM by sampling from P (vimg |vtxt , ?).
between low-level features across different modalities. Moreover, these observations are typically
very noisy and may have missing values.
A good multimodal learning model must satisfy certain properties. The joint representation must be
such that similarity in the representation space implies similarity of the corresponding ?concepts?. It
is also desirable that the joint representation be easy to obtain even in the absence of some modalities.
It should also be possible to fill-in missing modalities given the observed ones. In addition, it is also
desirable that extracted representation be useful for discriminative tasks.
Our proposed multimodal Deep Boltzmann Machine (DBM) model satisfies the above desiderata.
DBMs are undirected graphical models with bipartite connections between adjacent layers of hidden units [1]. The key idea is to learn a joint density model over the space of multimodal inputs.
Missing modalities can then be filled-in by sampling from the conditional distributions over them
given the observed ones. For example, we use a large collection of user-tagged images to learn a
joint distribution over images and text P (vimg , vtxt |?). By drawing samples from P (vtxt |vimg , ?)
and from P (vimg |vtxt , ?) we can fill-in missing data, thereby doing image annotation and image
retrieval respectively, as shown in Fig. 1.
There have been several approaches to learning from multimodal data. In particular, Huiskes et
al. [2] showed that using captions, or tags, in addition to standard low-level image features significantly improves classification accuracy of Support Vector Machines (SVM) and Linear Discriminant
Analysis (LDA) models. A similar approach of Guillaumin et al. [3], based on multiple kernel learning framework, further demonstrated that an additional text modality can improve the accuracy of
SVMs on various object recognition tasks. However, all of these approaches are discriminative by
nature and cannot make use of large amounts of unlabeled data or deal easily with missing input
modalities.
On the generative side, Xing et al. [4] used dual-wing harmoniums to build a joint model of images and text, which can be viewed as a linear RBM model with Gaussian hidden units together
with Gaussian and Poisson visible units. However, various data modalities will typically have very
different statistical properties which makes it difficult to model them using shallow models. Most
similar to our work is the recent approach of Ngiam et al. [5] that used a deep autoencoder for speech
and vision fusion. There are, however, several crucial differences. First, in this work we focus on
integrating together very different data modalities: sparse word count vectors and real-valued dense
image features. Second, we develop a Deep Boltzmann Machine as a generative model as opposed
to unrolling the network and fine-tuning it as an autoencoder. While both approaches have lead
to interesting results in several domains, using a generative model is important for applications we
consider in this paper, as it allows our model to naturally handle missing data modalities.
2
2
Background: RBMs and Their Generalizations
Restricted Boltzmann Machines (RBMs) have been used effectively in modeling distributions over
binary-valued data. Recent work on Boltzmann machine models and their generalizations to exponential family distributions have allowed these models to be successfully used in many application
domains. In particular, the Replicated Softmax model [6] has been shown to be effective in modeling sparse word count vectors, whereas Gaussian RBMs have been used for modeling real-valued
inputs for speech and vision tasks. In this section we briefly review these models, as they will serve
as our building blocks for the multimodal model.
2.1 Restricted Boltzmann Machines
A Restricted Boltzmann Machine is an undirected graphical model with stochastic visible units
v ? {0, 1}D and stochastic hidden units h ? {0, 1}F , with each visible unit connected to each
hidden unit. The model defines the following energy function E : {0, 1}D+F ? R:
D X
F
D
F
X
X
X
E(v, h; ?) = ?
vi Wij hj ?
bi vi ?
aj hj
i=1 j=1
i=1
j=1
where ? = {a, b, W} are the model parameters. The joint distribution over the visible and hidden
units is defined by:
1
exp (?E(v, h; ?)).
(1)
P (v, h; ?) =
Z(?)
2.2 Gaussian RBM
Consider modeling visible real-valued units v ? RD , and let h ? {0, 1}F be binary stochastic
hidden units. The energy of the state {v, h} of the Gaussian RBM is defined as follows:
D
D F
F
X
X
(vi ? bi )2 X X vi
E(v, h; ?) =
?
W
h
?
aj hj ,
(2)
ij j
2?i2
?
i=1
i=1 j=1 i
j=1
where ? = {a, b, W, ?} are the model parameters.
2.3 Replicated Softmax Model
The Replicated Softmax Model is useful for modeling sparse count data, such as word count vectors
in a document. Let v ? NK be a vector of visible units where vk is the number of times word k
occurs in the document with the vocabulary of size K. Let h ? {0, 1}F be binary stochastic hidden
topic features. The energy of the state {v, h} is defined as follows
K X
F
K
F
X
X
X
E(v, h; ?) = ?
vk Wkj hj ?
bk v k ? M
aj hj
(3)
k=1 j=1
j=1
k=1
P
where ? = {a, b, W} are the model parameters and M = k vk is the total number of words in
a document. We note that this replicated softmax model can also be interpreted as an RBM model
that uses a single visible multinomial unit with support {1, ..., K} which is sampled M times.
For all of the above models, exact maximum likelihood learning is intractable. In practice, efficient
learning is performed using Contrastive Divergence (CD) [7].
3
Multimodal Deep Boltzmann Machine
A Deep Boltzmann Machine (DBM) is a network of symmetrically coupled stochastic binary units.
It contains a set of visible units v ? {0, 1}D , and a sequence 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 hidden units
in adjacent layers. Let us first consider a DBM with two hidden layers. The energy of the joint
configuration {v, h} is defined as (ignoring bias terms):
E(v, h; ?) = ?v> W(1) h(1) ? h(1)> W(2) h(2) ,
where h = {h(1) , h(2) } represent the set of hidden units, and ? = {W(1) , W(2) } are the model parameters, representing visible-to-hidden and hidden-to-hidden symmetric interaction terms. Similar
to RBMs, this binary-binary DBM can be easily extended to modeling dense real-valued or sparse
count data, which we discuss next.
3
Multimodal DBM
Image-specific DBM
Text-specific DBM
Figure 2: Left: Image-specific two-layer DBM that uses a Gaussian model to model the distribution over realvalued image features. Middle: Text-specific two-layer DBM that uses a Replicated Softmax model to model
its distribution over the word count vectors. Right: A Multimodal DBM that models the joint distribution over
image and text inputs.
We illustrate the construction of a multimodal DBM using an image-text bi-modal DBM as our
running example. Let vm ? RD denote an image input and vt ? NK denote a text input. Consider
modeling each data modality using separate two-layer DBMs (Fig. 2). The image-specific two-layer
DBM assigns probability to vm that is given by (ignoring bias terms on the hidden units for clarity):
X
P (vm ; ?) =
P (vm , h(1) , h(2) ; ?) =
(4)
h(1) ,h(2)
?
1
=
Z(?)
X
exp ??
D
X
(vmi ? bi )2
2?i2
i=1
h(1) ,h(2)
+
F1
D X
X
vmi
i=1 j=1
?i
(1) (1)
Wij hj +
F2
F1 X
X
?
(1)
(2) (2)
hj Wjl hl ? .
j=1 l=1
Note that we borrow the visible-hidden interaction term from the Gaussian RBM (Eq. 2) and the
hidden-hidden one from the Binary RBM (Eq. 1). Similarly, the text-specific DBM will use terms
from the Replicated Softmax model for the visible-hidden interactions (Eq. 3) and the hidden-hidden
ones from the Binary RBM (Eq. 1).
To form a multimodal DBM, we combine the two models by adding an additional layer of binary
hidden units on top of them. The resulting graphical model is shown in Fig. 2, right panel. The joint
distribution over the multi-modal input can be written as:
X
X
X
(2)
(1)
(2)
m
t
(2)
(3)
(1)
(2)
P (v , v ; ?)=
P (hm , ht , h )
P (vm , hm , hm )
P (vt , ht , ht ) .
(2)
(2)
(1)
hm ,ht ,h(3)
3.1
(1)
hm
ht
Approximate Learning and Inference
Exact maximum likelihood learning in this model is intractable, but efficient approximate learning
can be carried out by using mean-field inference to estimate data-dependent expectations, and an
MCMC based stochastic approximation procedure to approximate the model?s expected sufficient
statistics [1]. In particular, during the inference step, we approximate the true posterior P (h|v; ?),
where v = {vm , vt }, with a fully factorized approximating distribution over the five sets of hidden
(1)
(2)
(1)
(2)
units {hm , hm , ht , ht , h(3) }:
Y
Y
Y
F1
F2
F1
F2
F3
Y
Y
(1)
(2)
(1)
(2)
(3)
Q(h|v; ?) =
q(hmj |v)
q(hml |v)
q(htj |v)
q(htl |v)
q(hk |v),
(5)
j=1
(1)
(2)
l=1
(1)
j=1
(2)
l=1
k=1
(l)
(l)
where ? = {?m , ?m , ?t , ?t , ?(3) } are the mean-field parameters with q(hi = 1) = ?i for
l = 1, 2, 3.
Learning proceeds by finding the value of ? that maximizes the variational 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 an MCMC-based stochastic approximation [1, 8, 9].
To initialize the model parameters to good values, we use a greedy layer-wise pretraining strategy
by learning a stack of modified RBMs (for details see [1]).
4
(a) RBM
(b) Multimodal DBN
(c) Multimodal DBM
Figure 3: Different ways of combining multimodal inputs
3.2
Salient Features
A Multimodal DBM can be viewed as a composition of unimodal undirected pathways. Each pathway can be pretrained separately in a completely unsupervised fashion, which allows us to leverage
a large supply of unlabeled data. Any number of pathways each with any number of layers could
potentially be used. The type of the lower-level RBMs in each pathway could be different, accounting for different input distributions, as long as the final hidden representations at the end of each
pathway are of the same type.
The intuition behind our model is as follows. Each data modality has very different statistical properties which make it difficult for a single hidden layer model (such as Fig. 3a) to directly find correlations across modalities. In our model, this difference is bridged by putting layers of hidden units
between the modalities. The idea is illustrated in Fig. 3c, which is just a different way of displaying Fig. 2. Compared to the simple RBM (Fig. 3a), where the hidden layer h directly models the
(1)
distribution over vt and vm , the first layer of hidden units hm in a DBM has an easier task to
(2)
perform - that of modeling the distribution over vm and hm . Each layer of hidden units in the
DBM contributes a small part to the overall task of modeling the distribution over vm and vt . In the
process, each layer learns successively higher-level representations and removes modality-specific
correlations. Therefore, the middle layer in the network can be seen as a (relatively) ?modality-free?
representation of the input as opposed to the input layers which were ?modality-full?.
Another way of using a deep model to combine multimodal inputs is to use a Multimodal Deep
Belief Network (DBN) (Fig. 3b) which consists of an RBM followed by directed belief networks
leading out to each modality. We emphasize that there is an important distinction between this model
and the DBM model of Fig. 3c. In a DBN model the responsibility of the multimodal modeling falls
entirely on the joint layer. In the DBM, on the other hand, this responsibility is spread out over the
entire network. The modality fusion process is distributed across all hidden units in all layers. From
the generative perspective, states of low-level hidden units in one pathway can influence the states
of hidden units in other pathways through the higher-level layers, which is not the case for DBNs.
3.3
Modeling Tasks
Generating Missing Modalities: As argued in the introduction, many real-world applications will
often have one or more modalities missing. The Multimodal DBM can be used to generate such
missing data modalities by clamping the observed modalities at the inputs and sampling the hidden
modalities from the conditional distribution by running the standard alternating Gibbs sampler [1].
For example, consider generating text conditioned on a given image1 vm . The observed modality
vm is clamped at the inputs and all hidden units are initialized randomly. P (vt |vm ) is a multinomial distribution over the vocabulary. Alternating Gibbs sampling can be used to sample words
from P (vt |vm ). Fig. 1 shows examples of words that have high probability under the conditional
distributions.
Inferring Joint Representations: The model can also be used to generate a fused representation
that multiple data modalities. This fused representation is inferred by clamping the observed modalities and doing alternating Gibbs sampling to sample from P (h(3) |vm , vt ) (if both modalities are
present) or from P (h(3) |vm ) (if text is missing). A faster alternative, which we adopt in our experimental results, is to use variational inference (see Sec. 3.1) to approximate posterior Q(h(3) |vm , vt )
or Q(h(3) |vm ). The activation probabilities of hidden units h(3) constitute the joint representation
of the inputs.
1
Generating image features conditioned on text can be done in a similar way.
5
This representation can then be used to do information retrieval for multimodal or unimodal queries.
Each data point in the database (whether missing some modalities or not) can be mapped to this
latent space. Queries can also be mapped to this space and an appropriate distance metric can be
used to retrieve results that are close to the query.
Discriminative Tasks: Classifiers such as SVMs can be trained with these fused representations as
inputs. Alternatively, the model can be used to initialize a feed forward network which can then be
finetuned [1]. In our experiments, logistic regression was used to classify the fused representations.
Unlike finetuning, this ensures that all learned representations that we compare (DBNs, DBMs and
Deep Autoencoders) use the same discriminative model.
4
4.1
Experiments
Dataset and Feature Extraction
The MIR Flickr Data set [10] was used in our experiments. The data set consists of 1 million images
retrieved from the social photography website Flickr along with their user assigned tags. Among the
1 million images, 25,000 have been annotated for 24 topics including object categories such as, bird,
tree, people and scene categories, such as indoor, sky and night. A stricter labeling was done for
14 of these classes where an image was annotated with a category only if that category was salient.
This leads to a total of 38 classes where each image may belong to several classes. The unlabeled
975,000 images were used only for pretraining. We use 15,000 images for training and 10,000 for
testing, following Huiskes et al. [2]. Mean Average Precision (MAP) is used as the performance
metric. Results are averaged over 5 random splits of training and test sets.
Each text input was represented using a vocabulary of the 2000 most frequent tags. The average
number of tags associated with an image is 5.15 with a standard deviation of 5.13. There are 128,501
images which do not have any tags, out of which 4,551 are in the labeled set. Hence about 18% of the
labeled data has images but is missing text. Images were represented by 3857-dimensional features,
that were extracted by concatenating Pyramid Histogram of Words (PHOW) features [11], Gist [12]
and MPEG-7 descriptors [13] (EHD, HTD, CSD, CLD, SCD). Each dimension was mean-centered
and normalized to unit variance. PHOW features are bags of image words obtained by extracting
dense SIFT features over multiple scales and clustering them. We used publicly available code
( [14, 15]) for extracting these features.
4.2
Model Architecture and Learning
The image pathway consists of a Gaussian RBM with 3857 visible units followed by 2 layers of
1024 hidden units. The text pathway consists of a Replicated Softmax Model with 2000 visible
units followed by 2 layers of 1024 hidden units. The joint layer contains 2048 hidden units. Each
layer of weights was pretrained using PCD for initializing the DBM model. When learning the
DBM model, all word count vectors were scaled so that they sum to 5. This avoids running separate
Markov chains for each word count to get the model distribution?s sufficient statistics.
Each pathway was pretrained using a stack of modified RBMs. Each Gaussian unit has unit variance that was kept fixed. For discriminative tasks, we perform 1-vs-all classification using logistic
regression on the joint hidden layer representation. We further split the 15K training set into 10K
for training and 5K for validation.
4.3
Classification Tasks
Multimodal Inputs: Our first set of experiments, evaluate the DBM as a discriminative model
for multimodal data. For each model that we trained, the fused representation of the data was
extracted and feed to a separate logistic regression for each of the 38 topics. The text input layer
in the DBM was left unclamped when the text was missing. Fig. 4 summarizes the Mean Average
Precision (MAP) and precision@50 (precision at top 50 predictions) obtained by different models.
Linear Discriminant Analysis (LDA) and Support Vector Machines (SVMs) [2] were trained using
the labeled data on concatenated image and text features that did not include SIFT-based features.
Hence, to make a fair comparison, our model was first trained using only labeled data with a similar
set of features (i.e., excluding our SIFT-based features). We call this model DBM-Lab. Fig. 4
shows that the DBM-Lab model already outperforms its competitor SVM and LDA models. DBMLab achieves a MAP of 0.526, compared to 0.475 and 0.492, achieved by SVM and LDA models.
6
Multimodal Inputs
Model
MAP
Prec@50
Random
LDA [2]
SVM [2]
DBM-Lab
DBM-Unlab
DBN
Autoencoder (based on [5])
DBM
0.124
0.492
0.475
0.526
0.585
0.599
0.600
0.609
0.124
0.754
0.758
0.791
0.836
0.867
0.875
0.873
Unimodal Inputs
Model
MAP
Prec@50
Image-SVM [2]
Image-DBN
Image-DBM
DBM-ZeroText
DBM-GenText
0.375
0.463
0.469
0.522
0.531
0.801
0.803
0.827
0.832
Figure 4: Classification Results. Left: Mean Average Precision (MAP) and precision@50 obtained by different models. Right: MAP using representations from different layers of multimodal DBMs and DBNs.
To measure the effect of using unlabeled data, a DBM was trained using all the unlabeled examples
that had both modalities present. We call this model DBM-Unlab. The only difference between the
DBM-Unlab and DBM-Lab models is that DBM-Unlab used unlabeled data during its pretraining
stage. The input features for both models remained the same. Not surprisingly, the DBM-Unlab
model significantly improved upon DBM-Lab achieving a MAP of 0.585. Our third model, DBM,
was trained using additional SIFT-based features. Adding these features improves the MAP to 0.609.
We compared our model to two other deep learning models: Multimodal Deep Belief Network
(DBN) and a deep Autoencoder model [5]. These models were trained with the same number of
layers and hidden units as the DBM. The DBN achieves a MAP of 0.599 and the autoencoder
gets 0.600. Their performance was comparable but slightly worse than that of the DBM. In terms
of precision@50, the autoencoder performs marginally better than the rest. We also note that the
Multiple Kernel Learning approach proposed in Guillaumin et. al. [3] achieves a MAP of 0.623 on
the same dataset. However, they used a much larger set of image features (37,152 dimensions).
Unimodal Inputs: Next, we evaluate the ability of the model to improve classification of unimodal
inputs by filling in other modalities. For multimodal models, the text input was only used during
training. At test time, all models were given only image inputs.
Fig. 4 compares the Multimodal DBM model with an SVM over image features alone (ImageSVM) [2], a DBN over image features (Image-DBN) and a DBM over image features (ImageDBM). All deep models had the same depth and same number of hidden units in each layer. Results
are reported for two different settings for the Multimodal DBM at test time. In one case (DBMZeroText), the state of the joint hidden layer was inferred keeping the missing text input clamped at
zero. In the other case (DBM-GenText), the text input was not clamped and the model was allowed
to update the state of the text input layer when performing mean-field updates. In doing so, the
model effectively filled-in the missing text modality (some examples of which are shown in Fig. 1).
These two settings helped to ascertain the contribution to the improvement that comes from filling
in the missing modality.
The DBM-GenText model performs better than all other models, showing that the DBM is able to
generate meaningful text that serves as a plausible proxy for missing data. Interestingly, the DBMZeroText model does better than any unimodal model. This suggests that learning multimodal features helps even when some modalities are absent at test time. Having multiple modalities probably
regularizes the model and makes it learn much better features. Moreover, this means that we do not
need to learn separate models to handle each possible combination of missing data modalities. One
joint model can be deployed at test time and used for any situation that may arise.
Each layer of the DBM provides a different representation of the input. Fig. 4, right panel, shows
the MAP obtained by using each of these representations for classification using logistic regression.
The input layers, shown at the extreme ends, are not very good at representing useful features. As
we go deeper into the model from either input layer towards the middle, the internal representations
get better. The joint layer in the middle serves as the most useful feature representation. Observe
that the performance of any DBM layer is always better than the corresponding DBN layer, though
they get close at the joint layer.
7
(a) Multimodal Queries
(b) Unimodal Queries
Figure 5: Precision-Recall curves for Retrieval Tasks
Figure 6: Retrieval Results for Multimodal Queries from the DBM model
4.4
Retrieval Tasks
Multimodal Queries: The next set of experiments was designed to evaluate the quality of the
learned joint representations. A database of images was created by randomly selecting 5000 imagetext pairs from the test set. We also randomly selected a disjoint set of 1000 images to be used as
queries. Each query contained both image and text modalities. Binary relevance labels were created
by assuming that if any of the 38 class labels overlapped between a query and a data point, then that
data point is relevant to the query.
Fig. 5a shows the precision-recall curves for the DBM, DBN, and Autoencoder models (averaged
over all queries). For each model, all queries and all points in the database were mapped to the joint
hidden representation under that model. Cosine similarity function was used to match queries to data
points. The DBM model performs the best among the compared models achieving a MAP of 0.622.
The autoencoder and DBN models perform worse with a MAP of 0.612 and 0.609 respectively.
Fig. 6 shows some examples of multimodal queries and the top 4 retrieved results. Note that even
though there is little overlap in terms of text, the model is able to perform well.
Unimodal Queries: The DBM model can also be used to query for unimodal inputs by filling in
the missing modality. Fig. 5b shows the precision-recall curves for the DBM model along with
other unimodal models, where each model received the same image queries as input. By effectively
inferring the missing text, the DBM model was able to achieve far better results than any unimodal
method (MAP of 0.614 as compared to 0.587 for an Image-DBM and 0.578 for an Image-DBN).
5
Conclusion
We proposed a Deep Boltzmann Machine model for learning multimodal data representations. Large
amounts of unlabeled data can be effectively utilized by the model. Pathways for each modality can
be pretrained independently and ?plugged in? together for doing joint training. The model fuses
multiple data modalities into a unified representation. This representation captures features that are
useful for classification and retrieval. It also works nicely when some modalities are absent and
improves upon models trained on only the observed modalities.
Acknowledgments: This research was supported by OGS, NSERC and by Early Researcher Award.
8
References
[1] R. R. Salakhutdinov and G. E. Hinton. Deep Boltzmann machines. In Proceedings of the
International Conference on Artificial Intelligence and Statistics, volume 12, 2009.
[2] Mark J. Huiskes, Bart Thomee, and Michael S. Lew. New trends and ideas in visual concept
detection: the MIR flickr retrieval evaluation initiative. In Multimedia Information Retrieval,
pages 527?536, 2010.
[3] M. Guillaumin, J. Verbeek, and C. Schmid. Multimodal semi-supervised learning for image
classification. In Computer Vision and Pattern Recognition (CVPR), 2010 IEEE Conference
on, pages 902 ?909, june 2010.
[4] Eric P. Xing, Rong Yan, and Alexander G. Hauptmann. Mining associated text and images
with dual-wing harmoniums. In UAI, pages 633?641. AUAI Press, 2005.
[5] Jiquan Ngiam, Aditya Khosla, Mingyu Kim, Juhan Nam, Honglak Lee, and Andrew Y. Ng.
Multimodal deep learning. In International Conference on Machine Learning (ICML), Bellevue, USA, June 2011.
[6] Ruslan Salakhutdinov and Geoffrey E. Hinton. Replicated softmax: an undirected topic model.
In NIPS, pages 1607?1614. Curran Associates, Inc., 2009.
[7] Geoffrey E. Hinton. Training products of experts by minimizing contrastive divergence. Neural
Computation, 14(8):1711?1800, 2002.
[8] T. Tieleman. Training restricted Boltzmann machines using approximations to the likelihood
gradient. In ICML. ACM, 2008.
[9] L. Younes. On the convergence of Markovian stochastic algorithms with rapidly decreasing
ergodicity rates, March 17 2000.
[10] Mark J. Huiskes and Michael S. Lew. The MIR Flickr retrieval evaluation. In MIR ?08:
Proceedings of the 2008 ACM International Conference on Multimedia Information Retrieval,
New York, NY, USA, 2008. ACM.
[11] A Bosch, Andrew Zisserman, and X Munoz. Image classification using random forests and
ferns. IEEE 11th International Conference on Computer Vision (2007), 23:1?8, 2007.
[12] Aude Oliva and Antonio Torralba. Modeling the shape of the scene: A holistic representation
of the spatial envelope. International Journal of Computer Vision, 42:145?175, 2001.
[13] B.S. Manjunath, J.-R. Ohm, V.V. Vasudevan, and A. Yamada. Color and texture descriptors.
Circuits and Systems for Video Technology, IEEE Transactions on, 11(6):703 ?715, 2001.
[14] A. Vedaldi and B. Fulkerson. VLFeat: An open and portable library of computer vision algorithms, 2008.
[15] Muhammet Bastan, Hayati Cam, Ugur Gudukbay, and Ozgur Ulusoy. Bilvideo-7: An mpeg-7compatible video indexing and retrieval system. IEEE Multimedia, 17:62?73, 2010.
9
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4,070 | 4,684 | Generalization Bounds for Domain Adaptation
Chao Zhang1 , Lei Zhang2 , Jieping Ye1,3
Center for Evolutionary Medicine and Informatics, The Biodesign Institute,
and 3 Computer Science and Engineering, Arizona State University, Tempe, USA
{czhan117,jieping.ye}@asu.edu
2
School of Computer Science and Technology,
Nanjing University of Science and Technology, Nanjing, P.R. China
[email protected]
1
Abstract
In this paper, we provide a new framework to study the generalization bound of
the learning process for domain adaptation. We consider two kinds of representative domain adaptation settings: one is domain adaptation with multiple sources
and the other is domain adaptation combining source and target data. In particular, we use the integral probability metric to measure the difference between two
domains. Then, we develop the specific Hoeffding-type deviation inequality and
symmetrization inequality for either kind of domain adaptation to achieve the corresponding generalization bound based on the uniform entropy number. By using
the resultant generalization bound, we analyze the asymptotic convergence and the
rate of convergence of the learning process for domain adaptation. Meanwhile, we
discuss the factors that affect the asymptotic behavior of the learning process. The
numerical experiments support our results.
1
Introduction
In statistical learning theory, one of the major concerns is to obtain the generalization bound of a
learning process, which measures the probability that a function, chosen from a function class by an
algorithm, has a sufficiently small error (cf. [1,2]). Generalization bounds have been widely used to
study the consistency of the learning process [3], the asymptotic convergence of empirical process
[4] and the learnability of learning models [5]. Generally, there are three essential aspects to obtain
generalization bounds of a specific learning process: complexity measures of function classes, deviation (or concentration) inequalities and symmetrization inequalities related to the learning process
(cf. [3, 4, 6, 7]).
It is noteworthy that the aforementioned results of statistical learning theory are all built under the
assumption that training and test data are drawn from the same distribution (or briefly called the
assumption of same distribution). This assumption may not be valid in many practical applications
such as speech recognition [8] and natural language processing [9] in which training and test data
may have different distributions. Domain adaptation has recently been proposed to handle this
situation and it is aimed to apply a learning model, trained by using the samples drawn from a
certain domain (source domain), to the samples drawn from another domain (target domain) with a
different distribution (cf. [10, 11, 12, 13]).
This paper is mainly concerned with two variants of domain adaptation. In the first variant, the
learner receives training data from several source domains, known as domain adaptation with multiple sources (cf. [14, 15, 16, 17]). In the second variant, the learner minimizes a convex combination
1
of empirical source and target risk, termed as domain adaptation combining source and target data
(cf. [13, 18])1 .
1.1
Overview of Main Results
In this paper, we present a new framework to study generalization bounds of the learning processes
for domain adaptation with multiple sources and domain adaptation combining source and target
data, respectively. Based on the resultant bounds, we then study the asymptotic behavior of the
learning process for the two kinds of domain adaptation, respectively. There are three major aspects
in the framework: the quantity that measures the difference between two domains, the deviation
inequalities and the symmetrization inequalities that are both designed for the situation of domain
adaptation2 .
Generally, in order to obtain the generalization bounds of a learning process, it is necessary to obtain
the corresponding deviation (or concentration) inequalities. For either kind of domain adaptation,
we use a martingale method to develop the related Hoeffding-type deviation inequality. Moreover,
in the situation of domain adaptation, since the source domain differs from the target domain, the
desired symmetrization inequality for domain adaptation should incorporate some quantity to reflect the difference. We then obtain the related symmetrization inequality incorporating the integral
probability metric that measures the difference between the distributions of the source and the target
domains.
Next, we present the generalization bounds based on the uniform entropy number for both kinds of
domain adaptation. Finally, based on the resultant bounds, we give a rigorous theoretic analysis to
the asymptotic convergence and the rate of convergence of the learning processes for both types of
domain adaptation. Meanwhile, we give a comparison with the related results under the assumption
of same distribution. We also present numerical experiments to support our results.
1.2
Organization of the Paper
The rest of this paper is organized as follows. Section 2 introduces the problems studied in this
paper. Section 3 introduces the integral probability metric that measures the difference between the
distributions of two domains. We introduce the uniform entropy number for the situation of multiple
sources in Section 4. In Section 5, we present the generalization bounds for domain adaptation with
multiple sources, and then analyze the asymptotic behavior of the learning process for this type of
domain adaptation. The last section concludes the paper. In the supplement (part A), we discuss the
relationship between the integral probability metric DF (S, T ) and the other quantities proposed in
the existing works including the H-divergence and the discrepancy distance. Proofs of main results
of this paper are provided in the supplement (part B). We study domain adaptation combining source
and target data in the supplement (part C) and then give a comparison with the existing works on the
theoretical analysis of domain adaptation in the supplement (part D).
2
Problem Setup
We denote Z (Sk ) := X (Sk ) ?Y (Sk ) ? RI ?RJ (1 ? k ? K) and Z (T ) := X (T ) ?Y (T ) ? RI ?RJ
as the k-th source domain and the target domain, respectively. Set L = I + J. Let D(Sk ) and
D(T ) stand for the distributions of the input spaces X (Sk ) (1 ? k ? K) and X (T ) , respectively.
(S )
(T )
Denote g? k : X (Sk ) ? Y (Sk ) and g? : X (T ) ? Y (T ) as the labeling functions of Z (Sk )
(T )
(1 ? k ? K) and Z , respectively. In the situation of domain adaptation with multiple sources,
(S )
the distributions D(Sk ) (1 ? k ? K) and D(T ) differ from each other, or g? k (1 ? k ? K) and
(T )
g? differ from each other, or both of the cases occur. There are sufficient amounts of i.i.d. samples
1
Due to the page limitation, the discussion on domain adaptation combining source and target data is provided in the supplement (part C).
2
Due to the page limitation, we only present the generalization bounds for domain adaptation with multiple
sources and the discussions of the corresponding deviation inequalities and symmetrization inequalities are
provided in the supplement (part B) along with the proofs of main results.
2
(k)
(Sk )
k
k
= {zn }N
(1 ? k ? K) but little or no labeled
ZN
1
n=1 drawn from each source domain Z
(T )
samples drawn from the target domain Z .
PK
Given w = (w1 , ? ? ? , wK ) ? [0, 1]K with k=1 wk = 1, let gw ? G be the function that minimizes
the empirical risk
E(S)
w (` ? g) =
K
X
k=1
(S )
wk ENkk (` ? g) =
Nk
K
X
wk X
(k)
`(g(x(k)
n ), yn )
Nk n=1
(1)
k=1
k K
over G with respect to sample sets {ZN
1 }k=1 , and it is expected that gw will perform well on the
target expected risk:
Z
(T )
E (` ? g) := `(g(x(T ) ), y(T ) )dP(z(T ) ), g ? G,
(2)
(T )
i.e., gw approximates the labeling g?
as precisely as possible.
In the learning process of domain adaptation with multiple sources, we are mainly interested in the
following two types of quantities:
(S)
? E(T ) (` ? gw ) ? Ew (` ? gw ), which corresponds to the estimation of the expected risk in
the target domain Z (T ) from a weighted combination of the empirical risks in the multiple
sources {Z (Sk ) }K
k=1 ;
? E(T ) (` ? gw ) ? E(T ) (` ? ge? ), which corresponds to the performance of the algorithm for
domain adaptation with multiple sources,
where ge? ? G is the function that minimizes the expected risk E(T ) (` ? g) over G.
Recalling (1) and (2), since
E(S)
e? ) ? E(S)
w (` ? g
w (` ? gw ) ? 0,
we have
E(T ) (` ? gw ) =E(T ) (` ? gw ) ? E(T ) (` ? ge? ) + E(T ) (` ? ge? )
(T )
?E(S)
e? ) ? E(S)
(` ? gw ) ? ET (` ? ge? ) + ET (` ? ge? )
w (` ? g
w (` ? gw ) + E
(T )
(T )
?2 sup E (` ? g) ? E(S)
(` ? ge? ),
(3)
w (` ? g) + E
g?G
and thus
0 ? E(T ) (` ? gw ) ? E(T ) (` ? ge? ) ? 2 sup E(T ) (` ? g) ? E(S)
w (` ? g) .
g?G
This shows that the asymptotic behaviors of the aforementioned two quantities, when the sample
numbers N1 , ? ? ? , NK go to infinity, can both be described by the supremum
sup E(T ) (` ? g) ? E(S)
(4)
w (` ? g) ,
g?G
which is the so-called generalization bound of the learning process for domain adaptation with
multiple sources.
For convenience, we define the loss function class
F := {z 7? `(g(x), y) : g ? G},
(5)
k K
and call F as the function class in the rest of this paper. By (1) and (2), given sample sets {ZN
1 }k=1
(Sk ) K
drawn from the multiple sources {Z
}k=1 respectively, we briefly denote for any f ? F,
Z
Nk
K
X
wk X
E(T ) f := f (z(T ) )dP(z(T ) ) ; E(S)
f
:=
f (z(k)
(6)
w
n ).
Nk n=1
k=1
Thus, we can equivalently rewrite the generalization bound (4) for domain adaptation with multiple
sources as
sup E(T ) f ? E(S)
(7)
w f .
f ?F
3
3
Integral Probability Metric
As shown in some prior works (e.g. [13, 16, 17, 18, 19, 20]), one of major challenges in the
theoretical analysis of domain adaptation is how to measure the distance between the source domain
Z (S) and the target domain Z (T ) . Recall that, if Z (S) differs from Z (T ) , there are three possibilities:
(S)
(T )
D(S) differs from D(T ) , or g? differs from g? , or both of them occur. Therefore, it is necessary
to consider two kinds of distances: the distance between D(S) and D(T ) and the distance between
(S)
(T )
g? and g? .
In [13, 18], the H-divergence was introduced to derive the generalization bounds based on the VC
dimension under the condition of ??-close?. Mansour et al. [20] obtained the generalization bounds
based on the Rademacher complexity by using the discrepancy distance. Both quantities are aimed
to measure the difference between two distributions D(S) and D(T ) . Moreover, Mansour et al. [17]
used the R?enyi divergence to measure the distance between two distributions. In this paper, we
use the following quantity to measure the difference of the distributions between the source and the
target domains:
Definition 3.1 Given two domains Z (S) , Z (T ) ? RL , let z(S) and z(T ) be the random variables
taking value from Z (S) and Z (T ) , respectively. Let F ? RZ be a function class. We define
DF (S, T ) := sup |E(S) f ? E(T ) f |,
(8)
f ?F
where the expectations E(S) and E(T ) are taken with respect to the distributions Z (S) and Z (T ) ,
respectively.
The quantity DF (S, T ) is termed as the integral probability metric that plays an important role in
probability theory for measuring the difference between the two probability distributions (cf. [23,
24, 25, 26]). Recently, Sriperumbudur et al. [27] gave the further investigation and proposed the
empirical methods to compute the integral probability metric in practice. As mentioned by M?uller
[page 432, 25], the quantity DF (S, T ) is a semimetric and it is a metric if and only if the function
class F separates the set of all signed measures with ?(Z) = 0. Namely, according to Definition
3.1, given a non-trivial function class F, the quantity DF (S, T ) is equal to zero if the domains Z (S)
and Z (T ) have the same distribution.
In the supplement (part A), we discuss the relationship between the quantity DF (S, T ) and other
quantities proposed in the previous works, and then show that the quantity DF (S, T ) can be bounded
by the summation of the discrepancy distance and another quantity, which measure the difference
between the input-space distributions D(S) and D(T ) and the difference between the labeling func(S)
(T )
tions g? and g? , respectively.
4
The Uniform Entropy Number
Generally, the generalization bound of a certain learning process is achieved by incorporating the
complexity measure of the function class, e.g., the covering number, the VC dimension and the
Rademacher complexity. The results of this paper are based on the uniform entropy number that is
derived from the concept of the covering number and we refer to [22] for more details.
The covering number of a function class F is defined as follows:
Definition 4.1 Let F be a function class and d is a metric on F. For any ? > 0, the covering
number of F at radius ? with respect to the metric d, denoted by N (F, ?, d) is the minimum size of
a cover of radius ?.
In some classical results of statistical learning theory, the covering number is applied by letting
d be the distribution-dependent metric. For example, as shown in Theorem 2.3 of [22], one can
set d as the norm `1 (ZN
1 ) and then derive the generalization bound of the i.i.d. learning process
by incorporating the expectation of the covering number, i.e., EN (F, ?, `1 (ZN
1 )). However, in
the situation of domain adaptation, we only know the information of source domain, while the
expectation EN (F, ?, `1 (ZN
1 )) is dependent on both distributions of source and target domains
4
because z = (x, y). Therefore, the covering number is no longer applicable to our framework to
obtain the generalization bounds for domain adaptation. By contrast, uniform entropy number is
distribution-free and thus we choose it as the complexity measure of function class to derive the
generalization bounds for domain adaptation.
For clarity of presentation, we give a useful notation for the following discussion. For any 1 ? k ?
(k)
N
(k)
(Sk )
k
k
k
K, given a sample set ZN
:= {zn }N
, we denote Z0 1 k := {z0 n }N
1
n=1 drawn from Z
n=1 as
(k)
the ghost-sample set drawn from Z (Sk ) such that the ghost sample z0 n has the same distribution as
(k)
2Nk
0 Nk
k
zn for any 1 ? n ? Nk and any 1 ? k ? K. Denoting Z1 := {ZN
1 , Z 1 }. Moreover, given
P
K
any f ? F and any w = (w1 , ? ? ? , wK ) ? [0, 1]K with k=1 wk = 1, we introduce a variant of the
`1 norm:
Nk
K
X
wk X
0 (k)
kf k`w ({Z2Nk }K ) :=
|f (z(k)
(9)
n )| + |f (z n )| .
1
1
k=1
Nk n=1
k=1
It is noteworthy that the variant `w
1 of the `1 norm is still a norm on the functional space, which
can be easily verified by using the definition of norm, so we omit it here. In the situation of domain
2Nk K
}k=1 ), we then define the uniform
adaptation with multiple sources, setting the metric d as `w
1 ({Z1
2Nk K
entropy number of F with respect to the metric `w
}k=1 ) as
1 ({Z1
ln N1w F, ?, 2
5
K
X
Nk :=
sup
2Nk K
}k=1
{Z1
k=1
2Nk K
ln N F, ?, `w
({Z
}
)
.
1
k=1
1
(10)
Domain Adaptation with Multiple Sources
In this section, we present the generalization bound for domain adaptation with multiple sources.
Based on the resultant bound, we then analyze the asymptotic convergence and the rate of convergence of the learning process for such kind of domain adaptation.
5.1
Generalization Bounds for Domain Adaptation with Multiple Sources
Based on the aforementioned uniform entropy number, the generalization bound for domain adaptation with multiple sources is presented in the following theorem:
Theorem 5.1 Assume that F is a function class consisting of the bounded functions with the range
PK
[a, b]. Let w = (w1 , ? ? ? , wK ) ? [0, 1]K with k=1 wk = 1. Then, given an arbitrary ? >
8(b?a)2
QK
(w)
DF (S, T ), we have for any
k=1 Nk ? (? 0 )2 and any > 0, with probability at least 1 ? ,
? 21
?
?
?
(w)
(T )
sup E(S)
f
?
E
f
?
D
(S,
T
)
+
?
w
F
?
f ?F
ln N1w
0
F, ? /8, 2
PK
Nk ? ln(/8)
k=1 Nk
P
Q
k=1
QK
32(b?a)2
K
k=1
2(
wk
i6=k
?
?
? , (11)
?
Ni )
(w)
where ? 0 = ? ? DF (S, T ) and
(w)
DF (S, T ) :=
K
X
wk DF (Sk , T ).
(12)
k=1
(S)
In the above theorem, we show that the generalization bound supf ?F |E(T ) f ? Ew f | can be
bounded by the right-hand side of (11). Compared to the classical result under the assumption
of same distribution (cf. Theorem 2.3 and Definition 2.5 of [22]): with probability at least 1 ? ,
?
! 21 ?
ln
N
F,
?,
N
?
ln(/8)
1
?
(13)
sup EN f ? Ef ? O ?
N
f ?F
5
with EN f being the empirical risk with respect to the sample set ZN
1 , there is a discrepancy quantity
(w)
DF (S, T ) that is determined by the two factors: the choice of w and the quantities DF (Sk , T )
(1 ? k ? K). The two results will coincide if any source domain and the target domain match, i.e.,
DF (Sk , T ) = 0 holds for any 1 ? k ? K.
In order to prove this result, we develop the related Hoeffding-type deviation inequality and the
symmetrization inequality for domain adaptation with multiple sources, respectively. The detailed
proof is provided in the supplement (part B). By using the resultant bound (11), we can analyze the
asymptotic behavior of the learning process for domain adaptation with multiple sources.
5.2
Asymptotic Convergence
In statistical learning theory, it is well-known that the complexity of the function class is the main
factor to the asymptotic convergence of the learning process under the assumption of same distribution (cf. [3, 4, 22]).
Theorem 5.1 can directly lead to the concerning theorem showing that the asymptotic convergence
of the learning process for domain adaptation with multiple sources:
Theorem 5.2 Assume that F is a function class consisting of bounded functions with the range
PK
[a, b]. Let w = (w1 , ? ? ? , wK ) ? [0, 1]K with k=1 wk = 1. If the following condition holds:
PK
ln N1w F, ? 0 /8, 2 k=1 Nk
lim
< +?
(14)
QK
N1 ,??? ,NK ?+?
k=1 Nk
P
Q
K
k=1
32(b?a)2
with ? 0 = ? ?
(w)
DF (S, T ),
2(
wk
i6=k
Ni )
(w)
DF (S, T ),
then we have for any ? >
o
n
>
?
= 0.
f
lim
Pr sup E(T ) f ? E(S)
w
N1 ,??? ,NK ?+?
(15)
f ?F
As shown in Theorem 5.2, if the choice of w ? [0, 1]K and the uniform entropy number
PK
PK
ln N1w F, ? 0 /8, 2 k=1 Nk satisfy the condition (14) with k=1 wk = 1, the probability of the
(S)
(w)
event that ?supf ?F E(T ) f ? Ew f > ?? will converge to zero for any ? > DF (S, T ), when
the sample numbers N1 , ? ? ? , NK of multiple sources go to infinity, respectively. This is partially in
accordance with the classical result of the asymptotic convergence of the learning process under the
assumption of same distribution (cf. Theorem 2.3 and Definition 2.5 of [22]): the probability of the
event that ?supf ?F Ef ? EN f > ?? will converge to zero for any ? > 0, if the uniform entropy
number ln N1 (F, ?, N ) satisfies the following:
ln N1 (F, ?, N )
lim
< +?.
(16)
N ?+?
N
Note that in the learning process of domain adaptation with multiple sources, the uniform convergence of the empirical risk on the source domains to the expected risk on the target domain may not
(w)
hold, because the limit (15) does not hold for any ? > 0 but for any ? > DF (S, T ). By contrast,
the limit (15) holds for all ? > 0 in the learning process under the assumption of same distribution,
if the condition (16) is satisfied.
By Cauchy-Schwarz inequality, setting wk = PKNk N (1 ? k ? K) minimizes the second term of
k
k=1
the right-hand side of (11) and then we arrive at
(T )
sup E(S)
f
w f ?E
f ?F
PK
?
?
PK
? 12
Nk DF (Sk , T ) ? (ln N1w (F, ? 0 /8, 2 k=1 Nk ) ? ln(/8) ?
+?
? ,
PK
PK
k=1 Nk
k=1 Nk
k=1
(17)
32(b?a)2
PKNk
(1 ? k ? K) can result in the fastest rate of convergence
which implies that setting wk =
k=1 Nk
and our numerical experiments presented in the next section also support this point (cf. Fig. 1).
6
6
Numerical Experiments
We have performed the numerical experiments to verify the theoretic analysis of the asymptotic
convergence of the learning process for domain adaptation with multiple sources. Without loss of
generality, we only consider the case of K = 2, i.e., there are two source domains and one target
domain. The experiment data are generated in the following way:
For the target domain Z (T ) = X (T ) ?Y (T ) ? R100 ?R, we consider X (T ) as a Gaussian distribution
(T )
(T )
T
N (0, 1) and draw {xn }N
randomly and independently. Let ? ? R100
n=1 (NT = 4000) from X
be a random vector of a Gaussian distribution N (1, 5), and let the random vector R ? R100 be a
noise term with R ? N (0, 0.5). For any 1 ? n ? NT , we randomly draw ? and R from N (1, 5)
(T )
and N (0, 0.01) respectively, and then generate yn ? Y as follows:
)
yn(T ) = hx(T
n , ?i + R.
(T )
(18)
(T )
(T )
T
The derived {(xn , yn )}N
and will be
n=1 (NT = 4000) are the samples of the target domain Z
used as the test data.
(1)
(1)
1
In the similar way, we derive the sample set {(xn , yn )}N
n=1 (N1 = 2000) of the source domain
(S1 )
(1)
(1)
100
Z
= X ? Y ? R ? R: for any 1 ? n ? N1 ,
yn(1) = hx(1)
n , ?i + R,
(19)
(1)
where xn ? N (0.5, 1), ? ? N (1, 5) and R ? N (0, 0.5).
(2)
(2)
2
For the source domain Z (S2 ) = X (2) ? Y (2) ? R100 ? R, the samples {(xn , yn )}N
n=1 (N2 =
2000) are generated in the following way: for any 1 ? n ? N2 ,
yn(2) = hx(2)
n , ?i + R,
where
(2)
xn
(20)
? N (2, 5), ? ? N (1, 5) and R ? N (0, 0.5).
In this experiment, we use the method of Least Square Regression to minimize the empirical risk
E(S)
w (` ? g) =
N1
N2
w X
(1 ? w) X
(1)
(2)
`(g(x(1)
),
y
)
+
`(g(x(2)
n
n
n ), yn )
N1 n=1
N2 n=1
(21)
for different combination coefficients w ? {0.1, 0.3, 0.5, 0.9} and then compute the discrepancy
(T )
(S)
|Ew f ? ENT f | for each N1 + N2 . The initial N1 and N2 both equal to 200. Each test is repeated
30 times and the final result is the average of the 30 results. After each test, we increment both N1
and N2 by 200 until N1 = N2 = 2000. The experiment results are shown in Fig. 1.
(S)
(T )
From Fig. 1, we can observe that for any choice of w, the curve of |Ew f ? ENT f | is decreasing
when N1 + N2 increases, which is in accordance with the results presented in Theorems 5.1 & 5.2.
(S)
(T )
Moreover, when w = 0.5, the discrepancy |Ew f ? ENT f | has the fastest rate of convergence, and
the rate becomes slower as w is further away from 0.5. In this experiment, we set N1 = N2 that
implies that N2 /(N1 + N2 ) = 0.5. Recalling (17), we have shown that w = N2 /(N1 + N2 ) will
provide the fastest rate of convergence and this proposition is supported by the experiment results
shown in Fig. 1.
7
Conclusion
In this paper, we present a new framework to study the generalization bounds of the learning process
for domain adaptation. We use the integral probability metric to measure the difference between the
distributions of two domains. Then, we use a martingale method to develop the specific deviation
inequality and the symmetrization inequality incorporating the integral probability metric. Next, we
utilize the resultant deviation inequality and symmetrization inequality to derive the generalization
bound based on the uniform entropy number. By using the resultant generalization bound, we analyze the asymptotic convergence and the rate of convergence of the learning process for domain
adaptation.
7
0.5
w=0.1
w=0.3
w=0.5
w=0.9
(T )
0.4
(S )
|E w f ?E N T f |
0.45
0.35
0.3
0.25
500
1000
1500
2000
2500
3000
3500
4000
N1 + N2
Figure 1: Domain Adaptation with Multiple Sources
We point out that the asymptotic convergence of the learning process is determined by the complexity of the function class F measured by the uniform entropy number. This is partially in accordance
with the classical result of the asymptotic convergence of the learning process under the assumption
of same distribution (cf. Theorem 2.3 and Definition 2.5 of [22]). Moreover, the rate of convergence
of this learning process is equal to that of the learning process under the assumption of same distribution. The numerical experiments support our results. Finally, we give a comparison with the
previous works [13, 14, 15, 16, 17, 18, 20] (cf. supplement, part D).
It is noteworthy that by Theorem 2.18 of [22], the generalization bound (11) can lead to the result
based on the fat-shattering dimension. According to Theorem 2.6.4 of [4], the bound based on the
VC dimension can also be obtained from the result (11). In our future work, we will attempt to
find a new distance between two distributions and develop the generalization bounds based on other
complexity measures, e.g., Rademacher complexities, and analyze other theoretical properties of
domain adaptation.
Acknowledgments
This research is sponsored in part by NSF (IIS-0953662, CCF-1025177), NIH (LM010730), and
ONR (N00014-1-1-0108).
References
[1] V.N. Vapnik (1999). An overview of statistical learning theory. IEEE Transactions on Neural Networks
10(5):988-999.
[2] O. Bousquet, S. Boucheron, and G. Lugosi (2004). Introduction to Statistical Learning Theory. In O.
Bousquet et al. (ed.), Advanced Lectures on Machine Learning, 169-207.
[3] V.N. Vapnik (1998). Statistical Learning Theory. New York: John Wiley and Sons.
[4] A. Blumer, A. Ehrenfeucht, D. Haussler, and M. K. Warmuth (1989). Learnability and the VapnikChervonenkis dimension. Journal of the ACM 36(4):929-965.
[5] A. van der Vaart, and J. Wellner (2000). Weak Convergence and Empirical Processes With Applications to
Statistics (Hardcover). Springer.
[6] P.L. Bartlett, O. Bousquet, and S. Mendelson (2005). Local Rademacher Complexities. Annals of Statistics
33:1497-1537.
[7] Z. Hussain, and J. Shawe-Taylor (2011). Improved Loss Bounds for Multiple Kernel Learning. Journal of
Machine Learning Research - Proceedings Track 15:370-377.
8
[8] J. Jiang, and C. Zhai (2007). Instance Weighting for Domain Adaptation in NLP. Proceedings of the 45th
Annual Meeting of the Association of Computational Linguistics (ACL), 264-271.
[9] J. Blitzer, M. Dredze, and F. Pereira (2007). Biographies, bollywood, boomboxes and blenders: Domain
adaptation for sentiment classification. Proceedings of the 45th Annual Meeting of the Association of Computational Linguistics (ACL), 440-447.
[10] S. Bickel, M. Br?uckner, and T. Scheffer (2007). Discriminative learning for differing training and test
distributions. Proceedings of the 24th international conference on Machine learning (ICML), 81-88.
[11] P. Wu, and T.G. Dietterich (2004). Improving SVM accuracy by training on auxiliary data sources. Proceedings of the twenty-first international conference on Machine learning (ICML), 871-878.
[12] J. Blitzer, R. McDonald, and F. Pereira (2006). Domain adaptation with structural correspondence learning.
Conference on Empirical Methods in Natural Language Processing (EMNLP), 120-128.
[13] S. Ben-David, J. Blitzer, K. Crammer, A. Kulesza, F. Pereira, and J. Wortman (2010). A Theory of
Learning from Different Domains. Machine Learning 79:151-175.
[14] K. Crammer, M. Kearns, and J. Wortman (2006). Learning from Multiple Sources. Advances in Neural
Information Processing Systems (NIPS).
[15] K. Crammer, M. Kearns, and J. Wortman (2008). Learning from Multiple Sources. Journal of Machine
Learning Research 9:1757-1774.
[16] Y. Mansour, M. Mohri, and A. Rostamizadeh (2008). Domain adaptation with multiple sources. Advances
in Neural Information Processing Systems (NIPS), 1041-1048.
[17] Y. Mansour, M. Mohri, and A. Rostamizadeh (2009). Multiple Source Adaptation and The R?enyi Divergence. Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence (UAI).
[18] J. Blitzer, K. Crammer, A. Kulesza, F. Pereira, and J. Wortman (2007). Learning Bounds for Domain
Adaptation. Advances in Neural Information Processing Systems (NIPS).
[19] S. Ben-David, J. Blitzer, K. Crammer, and F. Pereira, F (2006). Analysis of Representations for Domain
Adaptation. Advances in Neural Information Processing Systems (NIPS), 137-144.
[20] Y. Mansour, M. Mohri, and A. Rostamizadeh (2009). Domain Adaptation: Learning Bounds and Algorithms. Conference on Learning Theory (COLT).
[21] W. Hoeffding (1963). Probability Inequalities for Sums of Bounded Random Variables. Journal of the
American Statistical Association 58(301):13-30.
[22] S. Mendelson (2003). A Few Notes on Statistical Learning Theory. Lecture Notes in Computer Science
2600:1-40.
[23] V.M. Zolotarev (1984). Probability Metrics. Theory of Probability and its Application 28(1):278-302.
[24] S.T. Rachev (1991). Probability Metrics and the Stability of Stochastic Models. John Wiley and Sons.
[25] A. M?uller (1997). Integral Probability Metrics and Their Generating Classes of Functions. Advances in
Applied Probability 29(2):429-443.
[26] M.D. Reid and R.C. Williamson (2011). Information, Divergence and Risk for Binary Experiments. Journal of Machine Learning Research 12:731-817.
[27] B.K. Sriperumbudur, A. Gretton, K. Fukumizu, G.R.G. Lanckriet and B. Sch?olkopf (2009). A Note on
Integral Probability Metrics and ?-Divergences. CoRR abs/0901.2698.
9
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4,071 | 4,685 | Regularized Off-Policy TD-Learning
Bo Liu, Sridhar Mahadevan
Computer Science Department
University of Massachusetts
Amherst, MA 01003
{boliu, mahadeva}@cs.umass.edu
Ji Liu
Computer Science Department
University of Wisconsin
Madison, WI 53706
[email protected]
Abstract
We present a novel l1 regularized off-policy convergent TD-learning method
(termed RO-TD), which is able to learn sparse representations of value functions
with low computational complexity. The algorithmic framework underlying ROTD integrates two key ideas: off-policy convergent gradient TD methods, such
as TDC, and a convex-concave saddle-point formulation of non-smooth convex
optimization, which enables first-order solvers and feature selection using online
convex regularization. A detailed theoretical and experimental analysis of RO-TD
is presented. A variety of experiments are presented to illustrate the off-policy
convergence, sparse feature selection capability and low computational cost of the
RO-TD algorithm.
1
Introduction
Temporal-difference (TD) learning is a widely used method in reinforcement learning (RL). Although TD converges when samples are drawn ?on-policy? by sampling from the Markov chain
underlying a policy in a Markov decision process (MDP), it can be shown to be divergent when
samples are drawn ?off-policy?. Off-policy methods are of wider applications since they are able to
learn while executing an exploratory policy, learn from demonstrations, and learn multiple tasks in
parallel [2]. Sutton et al. [20] introduced convergent off-policy temporal difference learning algorithms, such as TDC, whose computation time scales linearly with the number of samples and the
number of features. Recently, a linear off-policy actor-critic algorithm based on the same framework
was proposed in [2].
Regularizing reinforcement learning algorithms leads to more robust methods that can scale up to
large problems with many potentially irrelevant features. LARS-TD [7] introduced a popular approach of combining l1 regularization using Least Angle Regression (LARS) with the least-squares
TD (LSTD) framework. Another approach was introduced in [5] (LCP-TD) based on the Linear
Complementary Problem (LCP) formulation, an optimization approach between linear programming and quadratic programming. LCP-TD uses ?warm-starts?, which helps significantly reduce
the burden of l1 regularization. A theoretical analysis of l1 regularization was given in [4], including
error bound analysis with finite samples in the on-policy setting. Another approach integrating the
Dantzig Selector with LSTD was proposed in [3], overcoming some of the drawbacks of LARS-TD.
An approximate linear programming approach for finding l1 regularized solutions of the Bellman
equation was presented in [17]. All of these approaches are second-order methods, requiring complexity approximately cubic in the number of (active) features. Another approach to feature selection is to greedily add new features, proposed recently in [15]. Regularized first-order reinforcement
learning approaches have recently been investigated in the on-policy setting as well, wherein convergence of l1 regularized temporal difference learning is discussed in [16] and mirror descent [6] is
used in [11].
1
In this paper, the off-policy TD learning problem is formulated from the stochastic optimization
perspective. A novel objective function is proposed based on the linear equation formulation of
the TDC algorithm. The optimization problem underlying off-policy TD methods, such as TDC,
is reformulated as a convex-concave saddle-point stochastic approximation problem, which is both
convex and incrementally solvable. A detailed theoretical and experimental study of the RO-TD
algorithm is presented.
Here is a brief roadmap to the rest of the paper. Section 2 reviews the basics of MDPs, RL and recent
work on off-policy convergent TD methods, such as the TDC algorithm. Section 3 introduces the
proximal gradient method and the convex-concave saddle-point formulation of non-smooth convex
optimization. Section 4 presents the new RO-TD algorithm. Convergence analysis of RO-TD is
presented in Section 5. Finally, in Section 6, experimental results are presented to demonstrate the
effectiveness of RO-TD.
2
Reinforcement Learning and the TDC Algorithm
a
A Markov Decision Process (MDP) is defined by the tuple (S, A, Pss
0 , R, ?), comprised of a set of
states S, a set of (possibly state-dependent) actions A (As ), a dynamical system model comprised of
a
0
the transition kernel Pss
0 specifying the probability of transition to state s from state s under action
a, a reward model R, and 0 ? ? < 1 is a discount factor. A policy ? : S ? A is a deterministic
mapping from states to actions. Associated with each policy ? is a value function V ? , which is the
fixed point of the Bellman equation:
V ? (s) = T ? V ? (s) = R? (s) + ?P ? V ? (s)
where R? is the expected immediate reward function (treated here as a column vector) and P ? is
the state transition function under fixed policy ?, and T ? is known as the Bellman operator. In what
follows, we often drop the dependence of V ? , T ? , R? on ?, for notational simplicity. In linear value
function approximation, a value function is assumed to lie in the linear span of a basis function
matrix ? of dimension |S| ? d, where d is the number of linear independent features. Hence,
V ? V? = ??. The vector space of all value functions is a P
normed inner product space, where the
?length? of any value function f is measured as ||f ||2? = s ?(s)f 2 (s) = f 0 ?f weighted by ?,
where ? is defined in Figure 1. For the t-th sample, ?t ,?0t , ?t and ?t are defined in Figure 1. TD
learning uses the following update rule ?t+1 = ?t + ?t ?t ?t , where ?t is the stepsize. However,
TD is only guaranteed to converge in the on-policy setting, although in many off-policy situations,
it still has satisfactory performance [21]. TD with gradient correction (TDC) [20] aims to minimize
the mean-square projected Bellman error (MSPBE) in order to guarantee off-policy convergence.
MSPBE is defined as
2
MSPBE(?) = k?? ? ?T (??)k? = (?T ?(T ?? ? ??))T (?T ??)?1 ?T ?(T ?? ? ??)
(1)
To avoid computing the inverse matrix (?T ??)?1 and to avoid the double sampling problem [19]
in (1), an auxiliary variable w is defined
w
=
(?T ??)?1 ?T ?(T ?? ? ??)
(2)
The two time-scale gradient descent learning method TDC [20] is defined below
?t+1 = ?t + ?t ?t ?t ? ?t ??t 0 (?Tt wt ), wt+1 = wt + ?t (?t ? ?Tt wt )?t
(3)
where ??t ??t 0 (?Tt wt ) is the term for correction of gradient descent direction, and ?t = ??t , ? > 1.
3
Proximal Gradient and Saddle-Point First-Order Algorithms
We now introduce some background material from convex optimization. The proximal mapping
associated with a convex function h is defined as:1
1
2
proxh (x) = arg min(h(u) + ku ? xk )
u
2
1
The proximal mapping can be shown to be the resolvent of the subdifferential of the function h.
2
(4)
? ? is a diagonal matrix whose entries ?(s) are given by a positive probability distribution
over states. ? = ?(?T ??)?1 ?T ? is the weighted least-squares projection operator.
1
? A square root of A is a matrix B satisfying B 2 = A and B is denoted as A 2 . Note that
1
A 2 may not be unique.
? [?, ?] is a row vector, and [?; ?] is a column vector.
? For the t-th sample, ?t (the t-th row of ?), ?0t (the t-th row of ?0 ) are the feature vectors
corresponding to st , s0t , respectively. ?t is the coefficient vector for t-th sample in first0
order TD learning methods, and ?t = (rt + ??tT ?t ) ? ?Tt ?t is the temporal difference
error. Also, xt = [wt ; ?t ], ?t is a stepsize, ?t = ??t , ? > 0.
P
1
1
? m, n are conjugate numbers if m
+ n1 = 1, m ? 1, n ? 1. ||x||m = ( j |xj |m ) m is
the m-norm of vector x.
? ? is l1 regularization parameter, ? is the eligibility trace factor, N is the sample size, d
is the number of basis functions, p is the number of active basis functions.
Figure 1: Notation used in this paper.
In the case of h(x) = ?kxk1 (? > 0), which is particularly important for sparse feature selection,
the proximal operator turns out to be the soft-thresholding operator S? (?), which is an entry-wise
shrinkage operator:
proxh (x)i = S? (xi ) = max(xi ? ?, 0) ? max(?xi ? ?, 0)
(5)
where i is the index, and ? is a threshold. With this background, we now introduce the proximal
gradient method. If the optimization problem is
x? = arg min (f (x) + h(x))
(6)
x?X
wherein f (x) is a convex and differentiable loss function and the regularization term h(x) is convex,
possibly non-differentiable and computing proxh is not expensive, then computation of (6) can be
carried out using the proximal gradient method:
xt+1 = prox?t h (xt ? ?t ?f (xt ))
(7)
where ?t > 0 is a (decaying) stepsize, a constant or it can be determined by line search.
3.1
Convex-concave Saddle-Point First Order Algorithms
The key novel contribution of our paper is a convex-concave saddle-point formulation for regularized off-policy TD learning. A convex-concave saddle-point problem is formulated as follows. Let
x ? X, y ? Y , where X, Y are both nonempty bounded closed convex sets, and f (x) : X ? R
be a convex function. If there exists a function ?(?, ?) such that f (x) can be represented as
f (x) := supy?Y ?(x, y), then the pair (?, Y ) is referred as the saddle-point representation of f .
The optimization problem of minimizing f over X is converted into an equivalent convex-concave
saddle-point problem SadV al = inf x?X supy?Y ?(x, y) of ? on X ?Y . If f is non-smooth yet convex and well structured, which is not suitable for many existing optimization approaches requiring
smoothness, its saddle-point representation ? is often smooth and convex. Thus, convex-concave
saddle-point problems are, therefore, usually better suited for first-order methods [6]. A comprehensive overview on extending convex minimization to convex-concave saddle-point problems with
unified variational inequalities is presented in [1]. As an example, consider f (x) = ||Ax ? b||m
which admits a bilinear minimax representation
f (x) := kAx ? bkm = max y T (Ax ? b)
(8)
kykn ?1
where m, n are conjugate numbers. Using the approach in [13], Equation (8) can be solved as
xt+1 = xt ? ?t AT yt , yt+1 = ?n (yt + ?t (Axt ? b))
(9)
where ?n is the projection operator of y onto the unit ln -ball kykn ? 1,which is defined as
?n (y) = min(1, 1/kykn )y, n = 2, 3, ? ? ? , ?? (yi ) = min(1, 1/|yi |)yi
(10)
and ?? is an entrywise operator.
3
4
Regularized Off-policy Convergent TD-Learning
We now describe a novel algorithm, regularized off-policy convergent TD-learning (RO-TD), which
combines off-policy convergence and scalability to large feature spaces. The objective function
is proposed based on the linear equation formulation of the TDC algorithm. Then the objective
function is represented via its dual minimax problem. The RO-TD algorithm is proposed based on
the primal-dual subgradient saddle-point algorithm, and inspired by related methods in [12],[13].
4.1
Objective Function of Off-policy TD Learning
In this subsection, we describe the objective function of the regularized off-policy RL problem. We
now first formulate the two updates of ?t , wt into a single iteration by rearranging the two equations
in (3) as xt+1 = xt ? ?t (At xt ? bt ), where xt = [wt ; ?t ],
T
??t ?t T ??t (?t ? ??0 t )
?rt ?t
At =
, bt =
(11)
T
rt ?t
??0 t ?t T ?t (?t ? ??0 t )
Following [20], the TDC algorithm solution follows from the linear equation Ax = b, where
A = E[At ], b = E[bt ], x = [w; ?]
(12)
There are some issues regarding the objective function, which arise from the online convex optimization and reinforcement learning perspectives, respectively. The first concern is that the objective
function should be convex and stochastically solvable. Note that A, At are neither PSD nor symmetric, and it is not straightforward to formulate a convex objective function based on them. The second
concern is that since we do not have knowledge of A, the objective function should be separable so
that it is stochastically solvable based on At , bt . The other concern regards the sampling condition
in temporal difference learning: double-sampling. As pointed out in [19], double-sampling is a
necessary condition to obtain an unbiased estimator if the objective function is the Bellman residual or its derivatives (such as projected Bellman residual), wherein the product of Bellman error or
projected Bellman error metrics are involved. To overcome this sampling condition constraint, the
product of TD errors should be avoided in the computation of gradients. Consequently, based on the
linear equation formulation in (12) and the requirement on the objective function discussed above,
we propose the regularized loss function as
L(x) = kAx ? bkm + h(x)
(13)
Here we also enumerate some intuitive objective functions and give a brief analysis on the reasons
why they are not suitable for regularized off-policy first-order TD learning. One intuitive idea is
to add a sparsity penalty on MSPBE, i.e., L(?) = MSPBE(?)+?k?k1 . Because of the l1 penalty
term, the solution to ?L = 0 does not have an analytical form and is thus difficult to compute.
The second intuition is to use the online least squares formulation of the linear equation Ax = b.
1
However, since A is not symmetric and positive semi-definite (PSD), A 2 does not exist and thus
1
1
Ax = b cannot be reformulated as minx?X ||A 2 x ? A? 2 b||22 . Another possible idea is to attempt
to find an objective function whose gradient is exactly At xt ? bt and thus the regularized gradient
is prox?t h(xt ) (At xt ? bt ). However, since At is not symmetric, this gradient does not explicitly
correspond to any kind of optimization problem, not to mention a convex one2 .
4.2
RO-TD Algorithm Design
In this section, the problem of (13) is formulated as a convex-concave saddle-point problem, and the
RO-TD algorithm is proposed. Analogous to (8), the regularized loss function can be formulated as
kAx ? bkm + h(x) = max y T (Ax ? b) + h(x)
kykn ?1
(14)
Similar to (9), Equation (14) can be solved via an iteration procedure as follows, where xt = [wt ; ?t ].
xt+ 21 = xt ? ?t ATt yt
yt+ 12 = yt + ?t (At xt ? bt )
,
xt+1 = prox?t h (xt+ 12 ) ,
yt+1 = ?n (yt+ 12 )
(15)
2
Note that the A matrix in GTD2?s linear equation representation is symmetric, yet is not PSD, so it cannot
be formulated as a convex problem.
4
The averaging step, which plays a crucial role in stochastic optimization convergence, generates the
approximate saddle-points [6, 12]
X t
?1 Xt
X t
?1 Xt
x
?t =
?i
?i xi , y?t =
?i
?i yi
(16)
i=0
i=0
i=0
i=0
Due to the computation of At in (15) at each iteration, the computation cost appears to be O(N d2 ),
where N, d are defined in Figure 1. However, the computation cost is actually O(N d) with a linear
algebraic trick by computing not At but ytT At , At xt ? bt . Denoting yt = [y1,t ; y2,t ], where y1,t ; y2,t
are column vectors of equal length, we have
h
i
T
T
T
T
T
ytT At = ??Tt (y1,t
(17)
?t ) + ??Tt (y2,t
?0t ) (?t ? ??0t ) (?y1,t
+ y2,t
)?t
At xt ? bt can be computed according to Equation (3) as follows:
At xt ? bt = ??(?t ? ?Tt wt )?t ; ?(?Tt wt )?t 0 ? ?t ?t
(18)
Both (17) and (18) are of linear computation complexity. Now we are ready to present the RO-TD
algorithm:
Algorithm 1 RO-TD
Let ? be some fixed policy of an MDP M , and let the sample set S = {si , ri , si 0 }N
i=1 . Let ? be
some fixed basis.
1: repeat
0
2:
Compute ?t , ?t 0 and TD error ?t = (rt + ??tT ?t ) ? ?Tt ?t
T
3:
Compute yt At , At xt ? bt in Equation (17) and (18).
4:
Compute xt+1 , yt+1 as in Equation (15)
5:
Set t ? t + 1;
6: until t = N ;
7: Compute x
?N , y?N as in Equation (16) with t = N
There are some design details of the algorithm to be elaborated. First, the regularization term h(x)
can be any kind of convex regularization, such as ridge regression or sparsity penalty ?||x||1 . In case
of h(x) = ?||x||1 , prox?t h (?) = S?t ? (?). In real applications the sparsification requirement on ?
and auxiliary variable w may be different, i.e., h(x) = ?1 k?k1 + ?2 kwk1 , ?1 6= ?2 , one can simply
replace the uniform soft thresholding S?t ? by two separate soft thresholding operations S?t ?1 , S?t ?2
and thus the third equation in (15) is replaced by the following,
h
i
xt+ 21 = wt+ 12 ; ?t+ 12 , ?t+1 = S?t ?1 (?t+ 21 ), wt+1 = S?t ?2 (wt+ 12 )
(19)
Another concern is the choice of conjugate numbers (m, n). For ease of computing ?n , we use
(2, 2)(l2 fit), (+?, 1)(uniform fit) or (1, +?). m = n = 2 is used in the experiments below.
4.3
RO-GQ(?) Design
GQ(?)[10] is a generalization of the TDC algorithm with eligibility traces and off-policy learning
of temporally abstract predictions, where the gradient update changes from Equation (3) to
?t+1 = ?t + ?t [?t et ? ?(1 ? ?)wt T et ??t+1 ], wt+1 = wt + ?t (?t et ? wT ?t ?t )
(20)
t
The central element is to extend the MSPBE function to the case where it incorporates eligibility
traces. The objective function and corresponding linear equation component At , bt can be written
as follows:
L(?) = ||?? ? ?T ?? ??||2?
(21)
"
#
T
T
??t ?t
?et (?t ? ? ??t+1 )
?rt et
At =
, bt =
(22)
T
T
rt et
?
?
?(1 ? ?)?t+1 e
et (?t ? ? ?t+1 )
t
Similar to Equation (17) and (18), the computation of ytT At , At xt ? bt is
T
T ?
T
T
?t+1 ) (?t ? ? ??t+1 )T (?y1,t
+ y2,t
)et
??Tt (y1,t
?t ) + ?(1 ? ?)eTt (y2,t
ytT At =
(23)
At xt ? bt =
??(?t et ? ?Tt wt ?t ); ?(1 ? ?)(eTt wt )??t+1 ? ?t et
??
?
where eligibility traces et , and ?t , T are defined in [10]. Algorithm 2, RO-GQ(?), extends the
RO-TD algorithm to include eligibility traces.
5
Algorithm 2 RO-GQ(?)
Let ? and ? be as defined in Algorithm 1. Starting from s0 .
1: repeat
2:
Compute ?t , ??t+1 and TD error ?t = (rt + ? ??Tt+1 ?t ) ? ?Tt ?t
3:
Compute ytT At , At xt ? bt in Equation (23).
4:
Compute xt+1 , yt+1 as in Equation (15)
5:
Choose action at , and get st+1
6:
Set t ? t + 1;
7: until st is an absorbing state;
8: Compute x
?t , y?t as in Equation (16)
4.4
Extension
It is also worth noting that there exists another formulation of the loss function different from Equation (13) with the following convex-concave formulation as in [14, 6],
min
x
1
2
kAx ? bk2 + ?kxk1
2
=
=
?
(bT y ? y T y)
2
?
min max
xT u + y T (Ax ? b) ? y T y
x kuk? ?1,y
2
max
kAT yk? ?1
(24)
which can be solved iteratively without the proximal gradient step as follows, which serves as a
counterpart of Equation (15),
?t
xt+1 = xt ? ?t ?(ut + At T yt ) , yt+1 = yt + (At xt ? bt ? ?yt )
?
?t
(25)
ut+ 12 = ut + xt , ut+1 = ?? (ut+ 21 )
?
5
Convergence Analysis of RO-TD
Assumption 1 (MDP)[20]: The underlying Markov Reward Process (MRP) M = (S, P, R, ?) is finite and mixing, with stationary distribution ?. Assume that ? a scalar Rmax such that V ar[rt |st ] ?
Rmax holds w.p.1.
Assumption 2 (Basis Function)[20]: ? is a full column rank matrix, namely, ? comprises a linear
independent set of basis functions w.r.t all sample states in sample set S. Also, assume the fea0
0
tures (?t , ?t ) have uniformly bounded second moments. Finally, if (st , at , st ) is an i.i.d sequence,
0
?t, k?t k? < +?, k?t k? < +?.
Assumption 3 (Subgradient Boundedness)[12]: Assume for the bilinear convex-concave loss
function defined in (14), the sets X, Y are closed compact sets. Then the subgradient ytT At and
At xt ? bt in RO-TD
are uniformly bounded, i.e., there exists a constant L such that
algorithm
kAt xt ? bt k ? L,
ytT At
? L.
Proposition 1: The approximate saddle-point x
?t of RO-TD converges w.p.1 to the global minimizer
of the following,
x? = arg minkAx ? bkm + ?kxk1
(26)
x?X
Proof Sketch: See the supplementary material for details.
6
Empirical Results
We now demonstrate the effectiveness of the RO-TD algorithm against other algorithms across a
number of benchmark domains. LARS-TD [7], which is a popular second-order sparse reinforcement learning algorithm, is used as the baseline algorithm for feature selection and TDC is used as
the off-policy convergent RL baseline algorithm, respectively.
6
||Ax?b||2 and yT(Ax?b)
Comparison of MSPBE
14
100
90
TD
80
TDC
MSPBE Minimization
0.12
0.11
Inverted
Tabular
Dependent
12
10
70
0.1
yT(Ax?b)
0.09
8
MSPBE
MSPBE
RO?TD
60
||Ax?b||2
50
6
40
30
0.08
0.07
0.06
0.05
4
20
0.04
2
10
0
0
0.03
10
20
30
40
50
60
70
80
90
Sweeps
100
0
0
20
40
60
80
100
120
140
160
180
200
0.02
0
20
40
60
80
100
120
140
160
180
200
Sweeps
Sweeps
Figure 2: Illustrative examples of the convergence of RO-TD using the Star and Random-walk
MDPs.
6.1
MSPBE Minimization and Off-Policy Convergence
This experiment aims to show the minimization of MSPBE and off-policy convergence of the ROTD algorithm. The 7 state star MDP is a well known counterexample where TD diverges monotonically and TDC converges. It consists of 7 states and the reward w.r.t any transition is zero. Because
of this, the star MDP is unsuitable for LSTD-based algorithms, including LARS-TD since ?T R = 0
always holds. The random-walk problem is a standard Markov chain with 5 states and two absorbing state at two ends. Three sets of different bases ? are used in [20], which are tabular features,
inverted features and dependent features respectively. An identical experiment setting to [20] is used
for these two domains. The regularization term h(x) is set to 0 to make a fair comparison with TD
and TDC. ? = 0.01, ? = 10 for TD, TDC and RO-TD. The comparison with TD, TDC and RO-TD
is shown in the left subfigure of Figure 2, where TDC and RO-TD have almost identical MSPBE
over iterations. The middle subfigure shows the value of ytT (Axt ? b) and kAxt ? bk2 , wherein
kAxt ? bk2 is always greater than the value of ytT (Axt ? b). Note that for this problem, the Slater
condition is satisfied so there is no duality gap between the two curves. As the result shows, TDC
and RO-TD perform equally well, which illustrates the off-policy convergence of the RO-TD algorithm. The result of random-walk chain is averaged over 50 runs. The rightmost subfigure of Figure
2 shows that RO-TD is able to reduce MSPBE over successive iterations w.r.t three different basis
functions.
6.2
Feature Selection
In this section, we use the mountain car example with a variety of bases to show the feature selection
capability of RO-TD. The Mountain car MDPis an optimal control problem with a continuous twodimensional state space. The steep discontinuity in the value function makes learning difficult for
bases with global support. To make a fair comparison, we use the same basis function setting as in
[7], where two dimensional grids of 2, 4, 8, 16, 32 RBFs are used so that there are totally 1365 basis
functions. For LARS-TD, 500 samples are used. For RO-TD and TDC, 3000 samples are used by
executing 15 episodes with 200 steps for each episode, stepsize ?t = 0.001, and ?1 = 0.01, ?2 =
0.2. We use the result of LARS-TD and l2 LSTD reported in [7]. As the result shows in Table 1,
RO-TD is able to perform feature selection successfully, whereas TDC and TD failed. It is worth
noting that comparing the performance of RO-TD and LARS-TD is not the focus of this paper since
LARS-TD is not convergent off-policy and RO-TD?s performance can be further optimized using
the mirror-descent approach with the Mirror-Prox algorithm [6] which incorporates mirror descent
with an extragradient [9], as discussed below.
Algorithm
Success(20/20)
Steps
LARS-TD
100%
142.25 ? 9.74
RO-TD
100%
147.40 ? 13.31
l2 LSTD
0%
-
TDC
0%
-
TD
0%
-
Table 1: Comparison of TD, LARS-TD, RO-TD, l2 LSTD, TDC and TD
7
Experiment\Method
Experiment 1
Experiment 2
RO-GQ(?)
6.9 ? 4.82
14.7 ? 10.70
GQ(?)
11.3 ? 9.58
27.2 ? 6.52
LARS-TD
-
Table 2: Comparison of RO-GQ(?), GQ(?), and LARS-TD on Triple-Link Inverted Pendulum Task
showing minimum number of learning episodes.
6.3
High-dimensional Under-actuated Systems
The triple-link inverted pendulum [18] is a highly nonlinear under-actuated system with 8dimensional state space and discrete action space. The state space consists of the angles and angular
velocity of each arm as well as the position and velocity of the car. The discrete action space is
{0, 5Newton, ?5Newton}. The goal is to learn a policy that can balance the arms for Nx steps
within some minimum number of learning episodes. The allowed maximum number of episodes
is 300. The pendulum initiates from zero equilibrium state and the first action is randomly chosen
to push the pendulum away from initial state. We test the performance of RO-GQ(?), GQ(?) and
LARS-TD. Two experiments are conducted with Nx = 10, 000 and 100, 000, respectively. Fourier
basis [8] with order 2 is used, resulting in 6561 basis functions. Table 2 shows the results of this
experiment, where RO-GQ(?) performs better than other approaches, especially in Experiment 2,
which is a harder task. LARS-TD failed in this domain, which is mainly not due to LARS-TD itself
but the quality of samples collected via random walk.
To sum up, RO-GQ(?) tends to outperform GQ(?) in all aspects, and is able to outperform LARSTD based policy iteration in high dimensional domains, as well as in selected smaller MDPs where
LARS-TD diverges (e.g., the star MDP). It is worth noting that the computation cost of LARS-TD
is O(N dp3 ), where that for RO-TD is O(N d). If p is linear or sublinear w.r.t d, RO-TD has a
significant advantage over LARS-TD. However, compared with LARS-TD, RO-TD requires fine
tuning the parameters of ?t , ?1 , ?2 and is usually not as sample efficient as LARS-TD. We also find
that tuning the sparsity parameter ?2 generates an interpolation between GQ(?) and TD learning,
where a large ?2 helps eliminate the correction term of TDC update and make the update direction
more similar to the TD update.
7
Conclusions
This paper presents a novel unified framework for designing regularized off-policy convergent RL
algorithms combining a convex-concave saddle-point problem formulation for RL with stochastic
first-order methods. A detailed experimental analysis reveals that the proposed RO-TD algorithm
is both off-policy convergent and is robust to noisy features. There are many interesting future
directions for this research. One direction for future work is to extend the subgradient saddlepoint solver to a more generalized mirror descent framework. Mirror descent is a generalization of
subgradient descent with non-Euclidean distance [1], and has many advantages over gradient descent
in high-dimensional spaces. In [6], two algorithms to solve the bilinear saddle-point formulation are
proposed based on mirror descent and the extragradient [9], such as the Mirror-Prox algorithm. [6]
also points out that the Mirror-Prox algorithm may be further optimized via randomization. To scale
to larger MDPs, it is possible to design SMDP-based mirror-descent methods as well.
Acknowledgments
This material is based upon work supported by the Air Force Office of Scientific Research (AFOSR)
under grant FA9550-10-1-0383, and the National Science Foundation under Grant Nos. NSF CCF1025120, IIS-0534999, IIS-0803288, and IIS-1216467 Any opinions, findings, and conclusions or
recommendations expressed in this material are those of the authors and do not necessarily reflect
the views of the AFOSR or the NSF. We thank M. F. Duarte for helpful discussions.
8
References
[1] A. Ben-Tal and A. Nemirovski. Non-Euclidean restricted memory level method for large-scale
convex optimization. Mathematical Programming, 102(3):407?456, 2005.
[2] T. Degris, M. White, and R. S. Sutton. Linear off-policy actor-critic. In International Conference on Machine Learning, 2012.
[3] M. Geist, B. Scherrer, A. Lazaric, and M. Ghavamzadeh. A Dantzig Selector Approach to
Temporal Difference Learning. In International Conference on Machine Learning, 2012.
[4] M. Ghavamzadeh, A. Lazaric, R. Munos, and M. Hoffman. Finite-Sample Analysis of LassoTD . In Proceedings of the 28th International Conference on Machine Learning, 2011.
[5] J. Johns, C. Painter-Wakefield, and R. Parr. Linear complementarity for regularized policy
evaluation and improvement. In Proceedings of the International Conference on Neural Information Processing Systems, 2010.
[6] A. Juditsky and A. Nemirovski. Optimization for Machine Learning, chapter First-Order Methods for Nonsmooth Convex Large-Scale Optimization. MIT Press, 2011.
[7] J. Zico Kolter and A. Y. Ng. Regularization and feature selection in least-squares temporal
difference learning. In Proceedings of 27 th International Conference on Machine Learning,
2009.
[8] G. Konidaris, S. Osentoski, and PS Thomas. Value function approximation in reinforcement
learning using the fourier basis. In Proceedings of the Twenty-Fifth Conference on Artificial
Intelligence, 2011.
[9] G. M. Korpelevich. The extragradient method for finding saddle points and other problems.
1976.
[10] H.R. Maei and R.S. Sutton. GQ (?): A general gradient algorithm for temporal-difference
prediction learning with eligibility traces. In Proceedings of the Third Conference on Artificial
General Intelligence, pages 91?96, 2010.
[11] S. Mahadevan and B. Liu. Sparse Q-learning with Mirror Descent. In Proceedings of the
Conference on Uncertainty in AI, 2012.
[12] A. Nedi?c and A. Ozdaglar. Subgradient methods for saddle-point problems. Journal of optimization theory and applications, 142(1):205?228, 2009.
[13] A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro. Robust stochastic approximation approach
to stochastic programming. SIAM Journal on Optimization, 19:1574?1609, 2009.
[14] Y. Nesterov.
Gradient methods for minimizing composite objective function.
In
www.optimization-online.org, 2007.
[15] C. Painter-Wakefield and R. Parr. Greedy algorithms for sparse reinforcement learning. In
International Conference on Machine Learning, 2012.
[16] C. Painter-Wakefield and R. Parr. L1 regularized linear temporal difference learning. Technical
report, Duke CS Technical Report TR-2012-01, 2012.
[17] M. Petrik, G. Taylor, R. Parr, and S. Zilberstein. Feature selection using regularization in approximate linear programs for Markov decision processes. In Proceedings of the International
Conference on Machine learning (ICML), 2010.
[18] J. Si and Y. Wang. Online learning control by association and reinforcement. IEEE Transactions on Neural Networks, 12:264?276, 2001.
[19] R. Sutton and A. G. Barto. Reinforcement Learning: An Introduction. MIT Press, 1998.
[20] R.S. Sutton, H.R. Maei, D. Precup, S. Bhatnagar, D. Silver, C. Szepesv?ari, and E. Wiewiora.
Fast gradient-descent methods for temporal-difference learning with linear function approximation. In International Conference on Machine Learning, pages 993?1000, 2009.
[21] J. Zico Kolter. The Fixed Points of Off-Policy TD. In Advances in Neural Information Processing Systems 24, pages 2169?2177, 2011.
9
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4,072 | 4,686 | Image Denoising and Inpainting with Deep Neural
Networks
Junyuan Xie, Linli Xu, Enhong Chen1
School of Computer Science and Technology
University of Science and Technology of China
[email protected], [email protected], [email protected]
Abstract
We present a novel approach to low-level vision problems that combines sparse
coding and deep networks pre-trained with denoising auto-encoder (DA). We propose an alternative training scheme that successfully adapts DA, originally designed for unsupervised feature learning, to the tasks of image denoising and blind
inpainting. Our method?s performance in the image denoising task is comparable
to that of KSVD which is a widely used sparse coding technique. More importantly, in blind image inpainting task, the proposed method provides solutions to
some complex problems that have not been tackled before. Specifically, we can
automatically remove complex patterns like superimposed text from an image,
rather than simple patterns like pixels missing at random. Moreover, the proposed
method does not need the information regarding the region that requires inpainting to be given a priori. Experimental results demonstrate the effectiveness of the
proposed method in the tasks of image denoising and blind inpainting. We also
show that our new training scheme for DA is more effective and can improve the
performance of unsupervised feature learning.
1
Introduction
Observed image signals are often corrupted by acquisition channel or artificial editing. The goal of
image restoration techniques is to restore the original image from a noisy observation of it. Image
denoising and inpainting are common image restoration problems that are both useful by themselves
and important preprocessing steps of many other applications. Image denoising problems arise when
an image is corrupted by additive white Gaussian noise which is common result of many acquisition
channels, whereas image inpainting problems occur when some pixel values are missing or when
we want to remove more sophisticated patterns, like superimposed text or other objects, from the
image. This paper focuses on image denoising and blind inpainting.
Various methods have been proposed for image denoising. One approach is to transfer image signals
to an alternative domain where they can be more easily separated from the noise [1, 2, 3]. For
example, Bayes Least Squares with a Gaussian Scale-Mixture (BLS-GSM), which was proposed by
Portilla et al, is based on the transformation to wavelet domain [2].
Another approach is to capture image statistics directly in the image domain. Following this strategy,
A family of models exploiting the (linear) sparse coding technique have drawn increasing attention
recently [4, 5, 6, 7, 8, 9]. Sparse coding methods reconstruct images from a sparse linear combination
of an over-complete dictionary. In recent research, the dictionary is learned from data instead of hand
crafted as before. This learning step improves the performance of sparse coding significantly. One
example of these methods is the KSVD sparse coding algorithm proposed in [6].
1
Corresponding author.
1
Image inpainting methods can be divided into two categories: non-blind inpainting and blind inpainting. In non-blind inpainting, the regions that need to be filled in are provided to the algorithm
a priori, whereas in blind inpainting, no information about the locations of the corrupted pixels is
given and the algorithm must automatically identify the pixels that require inpainting. The stateof-the-art non-blind inpainting algorithms can perform very well on removing text, doodle, or even
very large objects [10, 11, 12]. Some image denoising methods, after modification, can also be applied to non-blind image inpainting with state-of-the-art results [7]. Blind inpainting, however, is a
much harder problem. To the best of our knowledge, existing algorithms can only address i.i.d. or
simply structured impulse noise [13, 14, 15].
Although sparse coding models perform well in practice, they share a shallow linear structure. Recent research suggests, however, that non-linear, deep models can achieve superior performance in
various real world problems. One typical category of deep models are multi-layer neural networks.
In [16], Jain et al. proposed to denoise images with convolutional neural networks. In this paper, we propose to combine the advantageous ?sparse? and ?deep? principles of sparse coding and
deep networks to solve the image denoising and blind inpainting problems. The sparse variants of
deep neural network are expected to perform especially well in vision problems because they have
a similar structure to human visual cortex [17].
Deep neural networks with many hidden layers were generally considered hard to train before a new
training scheme was proposed which is to adopt greedy layer-wise pre-training to give better initialization of network parameters before traditional back-propagation training [18, 19]. There exist
several methods for pre-training, including Restricted Boltzmann Machine (RBM) and Denoising
Auto-encoder (DA) [20, 21].
We employ DA to perform pre-training in our method because it naturally lends itself to denoising
and inpainting tasks. DA is a two-layer neural network that tries to reconstruct the original input
from a noisy version of it. The structure of a DA is shown in Fig.1a. A series of DAs can be stacked
to form a deep network called Stacked Denoising Auto-encoders (SDA) by using the hidden layer
activation of the previous layer as input of the next layer.
SDA is widely used for unsupervised pre-training and feature learning [21]. In these settings, only
the clean data is provided while the noisy version of it is generated during training by adding random
Gaussian or Salt-and-Pepper noise to the clean data. After training of one layer, only the clean data
is passed on to the network to produce the clean training data for the next layer while the noisy
data is discarded. The noisy training data for the next layer is similarly constructed by randomly
corrupting the generated clean training data.
For the image denoising and inpainting tasks, however, the choices of clean and noisy input are
natural: they are set to be the desired image after denoising or inpainting and the observed noisy
image respectively. Therefore, we propose a new training scheme that trains the DA to reconstruct
the clean image from the corresponding noisy observation. After training of the first layer, the
hidden layer activations of both the noisy input and the clean input are calculated to serve as the
training data of the second layer. Our experiments on the image denoising and inpainting tasks
demonstrate that SDA is able to learn features that adapt to specific noises from white Gaussian
noise to superimposed text.
Inspired by SDA?s ability to learn noise specific features in denoising tasks, we argue that in unsupervised feature learning problems the type of noise used can also affect the performance. Specifically,
instead of corrupting the input with arbitrarily chosen noise, more sophisticated corruption process
that agrees to the true noise distribution in the data can improve the quality of the learned features.
For example, when learning audio features, the variations of noise on different frequencies are usually different and sometimes correlated. Hence instead of corrupting the training data with simple
i.i.d. Gaussian noise, Gaussian noise with more realistic parameters that are either estimated from
data or suggested by theory should be a better choice.
2
Model Description
In this section, we first introduce the problem formulation and some basic notations. Then we briefly
give preliminaries about Denoising Auto-encoder (DA), which is a fundamental building block of
our proposed method.
2
(a) Denoising auto-encoder (DA) architecture
(b) Stacked sparse denoising auto-encoder architecture
Figure 1: Model architectures.
2.1
Problem Formulation
Assuming x is the observed noisy image and y is the original noise free image, we can formulate
the image corruption process as:
x = ?(y).
(1)
where ? : Rn ? Rn is an arbitrary stochastic corrupting process that corrupts the input. Then, the
denoising task?s learning objective becomes:
f = argmin Ey kf (x) ? yk22
(2)
f
From this formulation, we can see that the task here is to find a function f that best approximates
? ?1 . We can now treat the image denoising and inpainting problems in a unified framework by
choosing appropriate ? in different situations.
2.2
Denoising Auto-encoder
Let yi be the original data for i = 1, 2, ..., N and xi be the corrupted version of corresponding yi .
DA is defined as shown in Fig.1a:
h(xi ) = ?(Wxi + b)
(3)
0
0
y
?(xi ) = ?(W h(xi ) + b )
(4)
where ?(x) = (1 + exp(?x))?1 is the sigmoid activation function which is applied element-wise to
vectors, hi is the hidden layer activation, y
?(xi ) is an approximation of yi and ? = {W, b, W0 , b0 }
represents the weights and biases. DA can be trained with various optimization methods to minimize
the reconstruction loss:
N
X
kyi ? y
?(xi )k.
(5)
? = argmin
?
i=1
After finish training a DA, we can move on to training the next layer by using the hidden layer
activation of the first layer as the input of the next layer. This is called Stacked denoising autoencoder (SDA) [21].
2.3
Stacked Sparse Denoising Auto-encoders
In this section, we will describe the structure and optimization objective of the proposed model
Stacked Sparse Denoising Auto-encoders (SSDA). Due to the fact that directly processing the entire image is intractable, we instead draw overlapping patches from the image as our data objects.
In the training phase, the model is supplied with both the corrupted noisy image patches xi , for
i = 1, 2, ..., N , and the original patches yi . After training, SSDA will be able to reconstruct the
corresponding clean image given any noisy observation.
To combine the virtues of sparse coding and neural networks and avoid over-fitting, we train a DA
to minimize the reconstruction loss regularized by a sparsity-inducing term:
N
1 X1
?
L1 (X, Y; ?) =
kyi ? y
?(xi )k22 + ? KL(?
?k?) + (kWk2F + kW0 k2F )
(6)
N i=1 2
2
3
Method
SSDA
BLS-GSM
KSVD
25/PSNR=20.17
30.52 ? 1.02
30.49 ? 1.17
30.96 ? 0.77
Standard deviation ?
50/PSNR=14.16 100/PSNR=8.13
27.37 ? 1.10
24.18 ? 1.39
27.28 ? 1.44
24.37 ? 1.36
27.34 ? 1.11
23.50 ? 1.15
Table 1: Comparison of the denoising performance. Performance is measured by Peak Signal to
Noise Ratio (PSNR). Results are averaged over testing set.
where
KL(?
?k?) =
|?|
?
X
? log
j=1
?
(1 ? ?)
+ (1 ? ?) log
,
??j
1 ? ??j
?? =
N
1 X
h(xi ).
N i
and h(?) and y
?(?) are defined in (3), (4) respectively. Here ?? is the average activation of the hidden
layer. We regularize the hidden layer representation to be sparse by choosing small ? so that the KLdivergence term will encourage the mean activation of hidden units to be small. Hence the hidden
units will be zero most of the time and achieve sparsity.
After training of the first DA, we use h(yi ) and h(xi ) as the clean and noisy input respectively
for the second DA. This is different from the approach described in [21], where xi is discarded and
?(h(yi )) is used as the noisy input. We point out that our method is more natural in that, since h(yi )
lies in a different space from yi , the meaning of applying ?(?) to h(yi ) is not clear.
We then initialize a deep network with the weights obtained from K stacked DAs. The network has
one input layer, one output and 2K ? 1 hidden layers as shown in Fig.1b. The entire network is then
trained using the standard back-propagation algorithm to minimize the following objective:
L2 (X, Y; ?) =
N
2K
?X
1 X1
kyi ? y(xi )k22 +
(kWj k2F ).
N i=1 2
2 j=1
(7)
Here we removed the sparsity regularization because the pre-trained weights will serve as regularization to the network [18].
In both of the pre-training and fine-tuning stages, the loss functions are optimized with L-BFGS
algorithm (a Quasi-Newton method) which, according to [22], can achieve fastest convergence in
our settings.
3
Experiments
We narrow our focus down to denoising and inpainting of grey-scale images, but there is no difficulty
in generalizing to colored images. We use a set of natural images collected from the web1 as our
training set and standard testing images2 as the testing set. We create noisy images from clean
training and testing images by applying the function (1) to them. Image patches are then extracted
from both clean and noisy images to train SSDAs. We employ Peak Signal to Noise Ratio (PSNR)
to quantify denoising results: 10 log10 (2552 /?e2 ), where ?e2 is the mean squared error. PSNR is one
of the standard indicators used for evaluating image denoising results.
3.1
Denoising White Gaussian Noise
We first corrupt images with additive white Gaussian noise of various standard deviations. For the
proposed method, one SSDA model is trained for each noise level. We evaluate different hyperparameter combinations and report the best result. We set K to 2 for all cases because adding more
layers may slightly improve the performance but require much more training time. In the meantime,
we try different patch sizes and find that higher noise level generally requires larger patch size. The
1
http://decsai.ugr.es/cvg/dbimagenes/
Widely used images commonly referred to as Lena, Barbara, Boat, Pepper, etc. in the image processing
community.
2
4
Figure 2: Visual comparison of denoising results. Results of images corrupted by white Gaussian
noise with standard deviation ? = 50 are shown. The last row zooms in on the outlined region of
the original image.
5
dimension of hidden layers is generally set to be a constant factor times the dimension of the input3 .
SSDA is not very sensitive to the weights of the regularization terms. For Bayes Least SquaresGaussian Scale Mixture (BLS-GSM) and KSVD method, we use the fully trained and optimized
toolbox obtained from the corresponding authors [2, 7]. All three models are tuned to specific
noise level of each input. The comparison of quantitative results are shown in Tab.1. Numerical
results showed that differences between the three algorithms are statistical insignificant. A visual
comparison is shown in Fig.2. We find that SSDA gives clearer boundary and restores more texture
details than KSVD and BLS-GSM although the PSNR scores are close. This indicates that although
the reconstruction errors averaged over all pixels are the same, SSDA is better at denoising complex
regions.
3.2
Image Inpainting
Figure 3: Visual comparison of inpainting results.
For the image inpainting task, we test our model on the text removal problem. Both the training
and testing set compose of images with super-imposed text of various fonts and sizes from 18-pix to
36-pix. Due to the lack of comparable blind inpainting algorithms, We compare our method to the
non-blind KSVD inpainting algorithm [7], which significantly simplifies the problem by requiring
the knowledge of which pixels are corrupted and require inpainting. A visual comparison is shown
in Fig.3. We find that SSDA is able to eliminate text of small fonts completely while text of larger
fonts is dimmed. The proposed method, being blind, generates results comparable to KSVD?s even
though KSVD is a non-blind algorithm. Non-blind inpainting is a well developed technology that
works decently on the removal of small objects. Blind inpainting, however, is much harder since it
demands automatic identification of the patterns that requires inpainting, which, by itself is a very
challenging problem. To the best of our knowledge, former methods are only capable of removing
i.i.d. or simply structured impulse noise [13, 14, 15]. SSDA?s capability of blind inpainting of
complex patterns is one of this paper?s major contributions.
6
Training noise
Gaussian
Salt-and-Pepper
Image background
Gaussian
91.42%
90.05%
84.88%
Testing noise
Salt-and-Pepper Image background
82.95%
86.45%
90.14%
81.77%
74.47%
86.87%
Table 2: Comparison of classification results. Highest accuracy in each column is shown in bold
font.
3.3
Hidden Layer Feature Analysis
Traditionally when training denoising auto-encoders, the noisy training data is usually generated
with arbitrarily selected simple noise distribution regardless of the characteristics of the specific
training data [21]. However, we propose that this process deserves more attention. In real world
problems, the clean training data is in fact usually subject to noise. Hence, if we estimate the
distribution of noise and exaggerate it to generate noisy training data, the resulting DA will learn to
be more robust to noise in the input data and produce better features.
Inspired by SSDA?s ability to learn different features when trained on denoising different noise patterns, we argue that training denoising auto-encoders with noise patterns that fit to specific situations
can also improve the performance of unsupervised feature learning. We demonstrate this by a comparison of classification performance with different sets of features learned on the MNIST dataset.
We train DAs with different types of noise and then apply them to handwritten digits corrupted by
the type of noise they are trained on as well as other types of noise. We compare the quality of the
learned features by feeding them to SVMs and comparing the corresponding classification accuracy.
The results are shown in Tab.2. We find that the highest classification accuracy on each type of
noise is achieved by the DA trained to remove that type of noise. This is not surprising since more
information is utilized, however it indicates that instead of arbitrarily corrupting input with noise
that follows simple distribution and feeding it to DA, more sophisticated methods that corrupt input
in more realistic ways can achieve better performance.
4
Discussion
4.1
Prior vs. Learned Structure
Unlike models relying on structural priors, our method?s denoising ability comes from learning.
Some models, for example BLS-GSM, have carefully designed structures that can give surprisingly
good results with random parameter settings [23]. However, randomly initialized SSDA obviously
can not produce any meaningful results. Therefore SSDA?s ability to denoise and inpaint images is
mostly the result of training. Whereas models that rely on structural priors usually have very limited
scope of applications, our model can be adapted to other tasks more conveniently. With some modifications, it is possible to denoise audio signals or complete missing data (as a data preprocessing
step) with SSDA.
4.2
Advantages and Limitations
Traditionally, for complicated inpainting tasks, an inpainting mask that tells the algorithm which
pixels correspond to noise and require inpainting is supplied a priori. However, in various situations
this is time consuming or sometimes even impossible. Our approach, being blind, has significant
advantages in such circumstances. This makes our method a suitable choice for fully automatic and
noise pattern specific image processing.
The limitation of our method is also obvious: SSDA strongly relies on supervised training. In our
experiment, we find that SSDA can generalize to unseen, but similar noise patterns. Generally speaking, however, SSDA can remove only the noise patterns it has seen in the training data. Therefore,
3
We set this factor to 5. The other hyper-parameters are: ? = 10?4 , ? = 10?2 , ? = 0.05.
7
SSDA would only be suitable in circumstances where the scope of denoising tasks is narrow, such
as reconstructing images corrupted by a certain procedure.
5
Conclusion
In this paper, we present a novel approach to image denoising and blind inpainting that combines
sparse coding and deep neural networks pre-trained with denoising auto-encoders. We propose a
new training scheme for DA that makes it possible to denoise and inpaint images within a unified
framework. In the experiments, our method achieves performance comparable to traditional linear
sparse coding algorithm on the simple task of denoising additive white Gaussian noise. Moreover,
our non-linear approach successfully tackles the much harder problem of blind inpainting of complex patterns which, to the best of our knowledge, has not been addressed before. We also show
that the proposed training scheme is able to improve DA?s performance in the tasks of unsupervised
feature learning.
In our future work, we would like to explore the possibility of adapting the proposed approach to various other applications such as denoising and inpainting of audio and video, image super-resolution
and missing data completion. It is also meaningful to investigate into the effects of different hyperparameter settings on the learned features.
6
Acknowledgement
Research supported by grants from the National Natural Science Foundation of China (No.
61003135 & No. 61073110), NSFC Major Program (No. 71090401/71090400), the Fundamental
Research Funds for the Central Universities (WK0110000022), the National Major Special Science
& Technology Projects (No. 2011ZX04016-071), and Research Fund for the Doctoral Program of
Higher Education of China (20093402110017, 20113402110024).
References
[1] J. Xu, K. Zhang, M. Xu, and Z. Zhou. An adaptive threshold method for image denoising based
on wavelet domain. Proceedings of SPIE, the International Society for Optical Engineering,
7495:165, 2009.
[2] J. Portilla, V. Strela, M.J. Wainwright, and E.P. Simoncelli. Image denoising using scale
mixtures of Gaussians in the wavelet domain. Image Processing, IEEE Transactions on,
12(11):1338?1351, 2003.
[3] F. Luisier, T. Blu, and M. Unser. A new SURE approach to image denoising: Interscale
orthonormal wavelet thresholding. IEEE Transactions on Image Processing, 16(3):593?606,
2007.
[4] B.A. Olshausen and D.J. Field. Sparse coding with an overcomplete basis set: A strategy
employed by V1? Vision research, 37(23):3311?3325, 1997.
[5] K. Kreutz-Delgado, J.F. Murray, B.D. Rao, K. Engan, T.W. Lee, and T.J. Sejnowski. Dictionary
learning algorithms for sparse representation. Neural computation, 15(2):349?396, 2003.
[6] M. Elad and M. Aharon. Image denoising via sparse and redundant representations over
learned dictionaries. IEEE Transactions on Image Processing, 15(12):3736?3745, 2006.
[7] J. Mairal, M. Elad, and G. Sapiro. Sparse representation for color image restoration. IEEE
Transactions on Image Processing, 17(1):53?69, 2008.
[8] X. Lu, H. Yuan, P. Yan, Y. Yuan, L. Li, and X. Li. Image denoising via improved sparse coding.
Proceedings of the British Machine Vision Conference, pages 74?1, 2011.
[9] J. Mairal, F. Bach, J. Ponce, and G. Sapiro. Online dictionary learning for sparse coding.
Proceedings of the 26th Annual International Conference on Machine Learning, pages 689?
696, 2009.
[10] A. Criminisi, P. P?erez, and K. Toyama. Region filling and object removal by exemplar-based
image inpainting. IEEE Transactions on Image Processing, 13(9):1200?1212, 2004.
8
[11] M. Bertalmio, G. Sapiro, V. Caselles, and C. Ballester. Image inpainting. Proceedings of
the 27th annual conference on Computer graphics and interactive techniques, pages 417?424,
2000.
[12] A. Telea. An image inpainting technique based on the fast marching method. Journal of
graphics tools., 9(1):23?34, 2004.
[13] B. Dong, H. Ji, J. Li, Z. Shen, and Y. Xu. Wavelet frame based blind image inpainting. Applied
and Computational Harmonic Analysis, 2011.
[14] Y. Wang, A. Szlam, and G. Lerman. Robust locally linear analysis with applications to image
denoising and blind inpainting. preprint, 2011.
[15] M. Yan. Restoration of images corrupted by impulse noise using blind inpainting and l0 norm.
preprint, 2011.
[16] V. Jain and H.S. Seung. Natural image denoising with convolutional networks. Advances in
Neural Information Processing Systems, 21:769?776, 2008.
[17] H. Lee, C. Ekanadham, and A. Ng. Sparse deep belief net model for visual area V2. Advances
in Neural Information Processing Systems 20, pages 873?880, 2008.
[18] D. Erhan, Y. Bengio, A. Courville, P.A. Manzagol, P. Vincent, and S. Bengio. Why does
unsupervised pre-training help deep learning? The Journal of Machine Learning Research,
11:625?660, 2010.
R in Machine Learn[19] Y. Bengio. Learning deep architectures for AI. Foundations and Trends
ing, 2(1):1?127, 2009.
[20] R. Salakhutdinov and G.E. Hinton. Deep boltzmann machines. Proceedings of the international conference on artificial intelligence and statistics, 5(2):448?455, 2009.
[21] P. Vincent, H. Larochelle, I. Lajoie, Y. Bengio, and P.A. Manzagol. Stacked denoising autoencoders: Learning useful representations in a deep network with a local denoising criterion. The
Journal of Machine Learning Research, 11:3371?3408, 2010.
[22] Q.V. Le, A. Coates, B. Prochnow, and A.Y. Ng. On optimization methods for deep learning.
Learning, pages 265?272, 2011.
[23] S. Roth and M.J. Adviser-Black. High-order markov random fields for low-level vision. Brown
University Press, 2007.
9
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4,073 | 4,687 | Large Scale Distributed Deep Networks
Jeffrey Dean, Greg S. Corrado, Rajat Monga, Kai Chen,
Matthieu Devin, Quoc V. Le, Mark Z. Mao, Marc?Aurelio Ranzato,
Andrew Senior, Paul Tucker, Ke Yang, Andrew Y. Ng
{jeff, gcorrado}@google.com
Google Inc., Mountain View, CA
Abstract
Recent work in unsupervised feature learning and deep learning has shown that being able to train large models can dramatically improve performance. In this paper,
we consider the problem of training a deep network with billions of parameters
using tens of thousands of CPU cores. We have developed a software framework
called DistBelief that can utilize computing clusters with thousands of machines to
train large models. Within this framework, we have developed two algorithms for
large-scale distributed training: (i) Downpour SGD, an asynchronous stochastic
gradient descent procedure supporting a large number of model replicas, and (ii)
Sandblaster, a framework that supports a variety of distributed batch optimization
procedures, including a distributed implementation of L-BFGS. Downpour SGD
and Sandblaster L-BFGS both increase the scale and speed of deep network training. We have successfully used our system to train a deep network 30x larger than
previously reported in the literature, and achieves state-of-the-art performance on
ImageNet, a visual object recognition task with 16 million images and 21k categories. We show that these same techniques dramatically accelerate the training
of a more modestly- sized deep network for a commercial speech recognition service. Although we focus on and report performance of these methods as applied
to training large neural networks, the underlying algorithms are applicable to any
gradient-based machine learning algorithm.
1
Introduction
Deep learning and unsupervised feature learning have shown great promise in many practical applications. State-of-the-art performance has been reported in several domains, ranging from speech
recognition [1, 2], visual object recognition [3, 4], to text processing [5, 6].
It has also been observed that increasing the scale of deep learning, with respect to the number
of training examples, the number of model parameters, or both, can drastically improve ultimate
classification accuracy [3, 4, 7]. These results have led to a surge of interest in scaling up the
training and inference algorithms used for these models [8] and in improving applicable optimization
procedures [7, 9]. The use of GPUs [1, 2, 3, 8] is a significant advance in recent years that makes
the training of modestly sized deep networks practical. A known limitation of the GPU approach is
that the training speed-up is small when the model does not fit in GPU memory (typically less than
6 gigabytes). To use a GPU effectively, researchers often reduce the size of the data or parameters
so that CPU-to-GPU transfers are not a significant bottleneck. While data and parameter reduction
work well for small problems (e.g. acoustic modeling for speech recognition), they are less attractive
for problems with a large number of examples and dimensions (e.g., high-resolution images).
In this paper, we describe an alternative approach: using large-scale clusters of machines to distribute
training and inference in deep networks. We have developed a software framework called DistBelief that enables model parallelism within a machine (via multithreading) and across machines (via
1
message passing), with the details of parallelism, synchronization and communication managed by
the framework. In addition to supporting model parallelism, the DistBelief framework also supports
data parallelism, where multiple replicas of a model are used to optimize a single objective. Within
this framework, we have designed and implemented two novel methods for large-scale distributed
training: (i) Downpour SGD, an asynchronous stochastic gradient descent procedure which leverages adaptive learning rates and supports a large number of model replicas, and (ii) Sandblaster
L-BFGS, a distributed implementation of L-BFGS that uses both data and model parallelism.1 Both
Downpour SGD and Sandblaster L-BFGS enjoy significant speed gains compared to more conventional implementations of SGD and L-BFGS.
Our experiments reveal several surprising results about large-scale nonconvex optimization. Firstly,
asynchronous SGD, rarely applied to nonconvex problems, works very well for training deep networks, particularly when combined with Adagrad [10] adaptive learning rates. Secondly, we show
that given sufficient resources, L-BFGS is competitive with or faster than many variants of SGD.
With regard to specific applications in deep learning, we report two main findings: that our distributed optimization approach can both greatly accelerate the training of modestly sized models,
and that it can also train models that are larger than could be contemplated otherwise. To illustrate
the first point, we show that we can use a cluster of machines to train a modestly sized speech model
to the same classification accuracy in less than 1/10th the time required on a GPU. To illustrate the
second point, we trained a large neural network of more than 1 billion parameters and used this
network to drastically improve on state-of-the-art performance on the ImageNet dataset, one of the
largest datasets in computer vision.
2
Previous work
In recent years commercial and academic machine learning data sets have grown at an unprecedented pace. In response, a great many authors have explored scaling up machine learning algorithms through parallelization and distribution [11, 12, 13, 14, 15, 16, 17]. Much of this research has
focused on linear, convex models, where distributed gradient computation is the natural first step.
Within this area, some groups have relaxed synchronization requirements, exploring delayed gradient updates for convex problems [12, 17]. In parallel, other groups working on problems with sparse
gradients (problems where only a tiny fraction of the coordinates of the gradient vector are non-zero
for any given training example) have explored lock-less asynchronous stochastic gradient descent
on shared-memory architectures (i.e. single machines) [5, 18]. We are interested in an approach
that captures the best of both worlds, allowing the use of a cluster of machines asynchronously
computing gradients, but without requiring that the problem be either convex or sparse.
In the context of deep learning, most work has focused on training relatively small models on a single
machine (e.g., Theano [19]). Suggestions for scaling up deep learning include the use of a farm of
GPUs to train a collection of many small models and subsequently averaging their predictions [20],
or modifying standard deep networks to make them inherently more parallelizable [21]. Our focus
is scaling deep learning techniques in the direction of training very large models, those with a few
billion parameters, but without introducing restrictions on the form of the model. In special cases
where one layer dominates computation, some authors have considered distributing computation in
that one layer and replicating computation in the remaining layers [5]. But in the general case where
many layers of the model are computationally intensive, full model parallelism in a spirit similar
to [22] is required. To be successful, however, we believe that model parallelism must be combined
with clever distributed optimization techniques that leverage data parallelism.
We considered a number of existing large-scale computational tools for application to our problem, MapReduce [23] and GraphLab [24] being notable examples. We concluded that MapReduce, designed for parallel data processing, was ill-suited for the iterative computations inherent in
deep network training; whereas GraphLab, designed for general (unstructured) graph computations,
would not exploit computing efficiencies available in the structured graphs typically found in deep
networks.
1
We implemented L-BFGS within the Sandblaster framework, but the general approach is also suitable for
a variety of other batch optimization methods.
2
Machine 1
Machine 2
Machine 3
Machine 4
Figure 1: An example of model parallelism in DistBelief. A five layer deep neural network with
local connectivity is shown here, partitioned across four machines (blue rectangles). Only those
nodes with edges that cross partition boundaries (thick lines) will need to have their state transmitted
between machines. Even in cases where a node has multiple edges crossing a partition boundary,
its state is only sent to the machine on the other side of that boundary once. Within each partition,
computation for individual nodes will the parallelized across all available CPU cores.
3
Model parallelism
To facilitate the training of very large deep networks, we have developed a software framework,
DistBelief, that supports distributed computation in neural networks and layered graphical models.
The user defines the computation that takes place at each node in each layer of the model, and the
messages that should be passed during the upward and downward phases of computation.2 For
large models, the user may partition the model across several machines (Figure 1), so that responsibility for the computation for different nodes is assigned to different machines. The framework
automatically parallelizes computation in each machine using all available cores, and manages communication, synchronization and data transfer between machines during both training and inference.
The performance benefits of distributing a deep network across multiple machines depends on the
connectivity structure and computational needs of the model. Models with a large number of parameters or high computational demands typically benefit from access to more CPUs and memory, up
to the point where communication costs dominate. We have successfully run large models with up
to 144 partitions in the DistBelief framework with significant speedups, while more modestly sized
models show decent speedups for up to 8 or 16 partitions. (See Section 5, under the heading Model
Parallelism Benchmarks, for experimental results.) Obviously, models with local connectivity structures tend to be more amenable to extensive distribution than fully-connected structures, given their
lower communication requirements. The typical cause of less-than-ideal speedups is variance in
processing times across the different machines, leading to many machines waiting for the single
slowest machine to finish a given phase of computation. Nonetheless, for our largest models, we can
efficiently use 32 machines where each machine achieves an average CPU utilization of 16 cores, for
a total of 512 CPU cores training a single large neural network. When combined with the distributed
optimization algorithms described in the next section, which utilize multiple replicas of the entire
neural network, it is possible to use tens of thousands of CPU cores for training a single model,
leading to significant reductions in overall training times.
4
Distributed optimization algorithms
Parallelizing computation within the DistBelief framework allows us to instantiate and run neural
networks considerably larger than have been previously reported. But in order to train such large
models in a reasonable amount of time, we need to parallelize computation not only within a single
2
In the case of a neural network ?upward? and ?downward? might equally well be called ?feedforward? and
?backprop?, while for a Hidden Markov Model, they might be more familiar as ?forward? and ?backward?.
3
Parameter Server
w
w? = w - ??w
Parameter Server
Coordinator
(small messages)
?w
Model
Replicas
Model
Replicas
Data
Shards
Data
Figure 2: Left: Downpour SGD. Model replicas asynchronously fetch parameters w and push gradients ?w to the parameter server. Right: Sandblaster L-BFGS. A single ?coordinator? sends small
messages to replicas and the parameter server to orchestrate batch optimization.
instance of the model, but to distribute training across multiple model instances. In this section we
describe this second level of parallelism, where we employ a set of DistBelief model instances, or
replicas, to simultaneously solve a single optimization problem.
We present a comparison of two large-scale distributed optimization procedures: Downpour SGD,
an online method, and Sandblaster L-BFGS, a batch method. Both methods leverage the concept
of a centralized sharded parameter server, which model replicas use to share their parameters. Both
methods take advantage of the distributed computation DistBelief allows within each individual
replica. But most importantly, both methods are designed to tolerate variance in the processing
speed of different model replicas, and even the wholesale failure of model replicas which may be
taken offline or restarted at random.
In a sense, these two optimization algorithms implement an intelligent version of data parallelism.
Both approaches allow us to simultaneously process distinct training examples in each of the many
model replicas, and periodically combine their results to optimize our objective function.
4.1
Downpour SGD
Stochastic gradient descent (SGD) is perhaps the most commonly used optimization procedure for
training deep neural networks [25, 26, 3]. Unfortunately, the traditional formulation of SGD is
inherently sequential, making it impractical to apply to very large data sets where the time required
to move through the data in an entirely serial fashion is prohibitive.
To apply SGD to large data sets, we introduce Downpour SGD, a variant of asynchronous stochastic gradient descent that uses multiple replicas of a single DistBelief model. The basic approach is
as follows: We divide the training data into a number of subsets and run a copy of the model on
each of these subsets. The models communicate updates through a centralized parameter server,
which keeps the current state of all parameters for the model, sharded across many machines (e.g.,
if we have 10 parameter server shards, each shard is responsible for storing and applying updates
to 1/10th of the model parameters) (Figure 2). This approach is asynchronous in two distinct aspects: the model replicas run independently of each other, and the parameter server shards also run
independently of one another.
In the simplest implementation, before processing each mini-batch, a model replica asks the parameter server service for an updated copy of its model parameters. Because DistBelief models
are themselves partitioned across multiple machines, each machine needs to communicate with just
the subset of parameter server shards that hold the model parameters relevant to its partition. After
receiving an updated copy of its parameters, the DistBelief model replica processes a mini-batch of
data to compute a parameter gradient, and sends the gradient to the parameter server, which then
applies the gradient to the current value of the model parameters.
It is possible to reduce the communication overhead of Downpour SGD by limiting each model
replica to request updated parameters only every nf etch steps and send updated gradient values only
every npush steps (where nf etch might not be equal to npush ). In fact, the process of fetching
4
parameters, pushing gradients, and processing training data can be carried out in three only weakly
synchronized threads (see the Appendix for pseudocode). In the experiments reported below we
fixed nf etch = npush = 1 for simplicity and ease of comparison to traditional SGD.
Downpour SGD is more robust to machines failures than standard (synchronous) SGD. For synchronous SGD, if one machine fails, the entire training process is delayed; whereas for asynchronous
SGD, if one machine in a model replica fails, the other model replicas continue processing their
training data and updating the model parameters via the parameter servers. On the other hand, the
multiple forms of asynchronous processing in Downpour SGD introduce a great deal of additional
stochasticity in the optimization procedure. Most obviously, a model replica is almost certainly
computing its gradients based on a set of parameters that are slightly out of date, in that some other
model replica will likely have updated the parameters on the parameter server in the meantime. But
there are several other sources of stochasticity beyond this: Because the parameter server shards act
independently, there is no guarantee that at any given moment the parameters on each shard of the
parameter server have undergone the same number of updates, or that the updates were applied in
the same order. Moreover, because the model replicas are permitted to fetch parameters and push
gradients in separate threads, there may be additional subtle inconsistencies in the timestamps of
parameters. There is little theoretical grounding for the safety of these operations for nonconvex
problems, but in practice we found relaxing consistency requirements to be remarkably effective.
One technique that we have found to greatly increase the robustness of Downpour SGD is the use
of the Adagrad [10] adaptive learning rate procedure. Rather than using a single fixed learning
rate on the parameter sever (? in Figure 2), Adagrad uses a separate adaptive learning rate for each
parameter. Let ?i,K be the
learning rate of the i-th parameter at iteration K and ?wi,K its gradient,
qP
K
2
then we set: ?i,K = ?/
j=1 ?wi,j . Because these learning rates are computed only from the
summed squared gradients of each parameter, Adagrad is easily implemented locally within each
parameter server shard. The value of ?, the constant scaling factor for all learning rates, is generally
larger (perhaps by an order of magnitude) than the best fixed learning rate used without Adagrad.
The use of Adagrad extends the maximum number of model replicas that can productively work
simultaneously, and combined with a practice of ?warmstarting? model training with only a single
model replica before unleashing the other replicas, it has virtually eliminated stability concerns in
training deep networks using Downpour SGD (see results in Section 5).
4.2
Sandblaster L-BFGS
Batch methods have been shown to work well in training small deep networks [7]. To apply these
methods to large models and large datasets, we introduce the Sandblaster batch optimization framework and discuss an implementation of L-BFGS using this framework.
A key idea in Sandblaster is distributed parameter storage and manipulation. The core of the optimization algorithm (e.g L-BFGS) resides in a coordinator process (Figure 2), which does not have
direct access to the model parameters. Instead, the coordinator issues commands drawn from a
small set of operations (e.g., dot product, scaling, coefficient-wise addition, multiplication) that can
be performed by each parameter server shard independently, with the results being stored locally
on the same shard. Additional information, e.g the history cache for L-BFGS, is also stored on the
parameter server shard on which it was computed. This allows running large models (billions of
parameters) without incurring the overhead of sending all the parameters and gradients to a single
central server. (See the Appendix for pseudocode.)
In typical parallelized implementations of L-BFGS, data is distributed to many machines and each
machine is responsible for computing the gradient on a specific subset of data examples. The gradients are sent back to a central server (or aggregated via a tree [16]). Many such methods wait for
the slowest machine, and therefore do not scale well to large shared clusters. To account for this
problem, we employ the following load balancing scheme: The coordinator assigns each of the N
model replicas a small portion of work, much smaller than 1/Nth of the total size of a batch, and
assigns replicas new portions whenever they are free. With this approach, faster model replicas do
more work than slower replicas. To further manage slow model replicas at the end of a batch, the
coordinator schedules multiple copies of the outstanding portions and uses the result from whichever
model replica finishes first. This scheme is similar to the use of ?backup tasks? in the MapReduce
framework [23]. Prefetching of data, along with supporting data affinity by assigning sequential
5
portions of data to the same worker makes data access a non-issue. In contrast with Downpour
SGD, which requires relatively high frequency, high bandwidth parameter synchronization with the
parameter server, Sandblaster workers only fetch parameters at the beginning of each batch (when
they have been updated by the coordinator), and only send the gradients every few completed portions (to protect against replica failures and restarts).
5
Experiments
We evaluated our optimization algorithms by applying them to training models for two different deep
learning problems: object recognition in still images and acoustic processing for speech recognition.
The speech recognition task was to classify the central region (or frame) in a short snippet of audio as
one of several thousand acoustic states. We used a deep network with five layers: four hidden layer
with sigmoidal activations and 2560 nodes each, and a softmax output layer with 8192 nodes. The
input representation was 11 consecutive overlapping 25 ms frames of speech, each represented by
40 log-energy values. The network was fully-connected layer-to-layer, for a total of approximately
42 million model parameters. We trained on a data set of 1.1 billion weakly labeled examples,
and evaluated on a hold out test set. See [27] for similar deep network configurations and training
procedures.
For visual object recognition we trained a larger neural network with locally-connected receptive
fields on the ImageNet data set of 16 million images, each of which we scaled to 100x100 pixels [28].
The network had three stages, each composed of filtering, pooling and local contrast normalization,
where each node in the filtering layer was connected to a 10x10 patch in the layer below. Our
infrastructure allows many nodes to connect to the same input patch, and we ran experiments varying
the number of identically connected nodes from 8 to 36. The output layer consisted of 21 thousand
one-vs-all logistic classifier nodes, one for each of the ImageNet object categories. See [29] for
similar deep network configurations and training procedures.
Model parallelism benchmarks: To explore the scaling behavior of DistBelief model parallelism
(Section 3), we measured the mean time to process a single mini-batch for simple SGD training as
a function of the number of partitions (machines) used in a single model instance. In Figure 3 we
quantify the impact of parallelizing across N machines by reporting the average training speed-up:
the ratio of the time taken using only a single machine to the time taken using N. Speedups for
inference steps in these models are similar and are not shown here.
The moderately sized speech model runs fastest on 8 machines, computing 2.2? faster than using a
single machine. (Models were configured to use no more than 20 cores per machine.) Partitioning
Training speed?up
15
Speech: 42M parameters
Images: 80M parameters
Images: 330M parameters
Images: 1.7B parameters
10
5
0
1
16
32
64
Machines per model instance
128
Figure 3: Training speed-up for four different deep networks as a function of machines allocated
to a single DistBelief model instance. Models with more parameters benefit more from the use of
additional machines than do models with fewer parameters.
6
Accuracy on Training Set
Accuracy on Test Set
25
Average Frame Accuracy (%)
Average Frame Accuracy (%)
25
20
15
10
SGD [1]
DownpourSGD [20]
DownpourSGD [200] w/Adagrad
Sandblaster L?BFGS [2000]
5
0
0
20
40
60
80
100
20
15
10
5
0
0
120
Time (hours)
SGD [1]
GPU [1]
DownpourSGD [20]
DownpourSGD [20] w/Adagrad
DownpourSGD [200] w/Adagrad
Sandblaster L?BFGS [2000]
20
40
60
80
100
120
Time (hours)
Figure 4: Left: Training accuracy (on a portion of the training set) for different optimization methods. Right: Classification accuracy on the hold out test set as a function of training time. Downpour
and Sandblaster experiments initialized using the same ?10 hour warmstart of simple SGD.
the model on more than 8 machines actually slows training, as network overhead starts to dominate
in the fully-connected network structure and there is less work for each machine to perform with
more partitions.
In contrast, the much larger, locally-connected image models can benefit from using many more
machines per model replica. The largest model, with 1.7 billion parameters benefits the most, giving
a speedup of more than 12? using 81 machines. For these large models using more machines
continues to increase speed, but with diminishing returns.
Optimization method comparisons: To evaluate the proposed distributed optimization procedures, we ran the speech model described above in a variety of configurations. We consider two
baseline optimization procedures: training a DistBelief model (on 8 partitions) using conventional
(single replica) SGD, and training the identical model on a GPU using CUDA [27]. The three distributed optimization methods we compare to these baseline methods are: Downpour SGD with a
fixed learning rate, Downpour SGD with Adagrad learning rates, and Sandblaster L-BFGS.
Figure 4 shows classification performance as a function of training time for each of these methods
on both the training and test sets. Our goal is to obtain the maximum test set accuracy in the
minimum amount of training time, regardless of resource requirements. Conventional single replica
SGD (black curves) is the slowest to train. Downpour SGD with 20 model replicas (blue curves)
shows a significant improvement. Downpour SGD with 20 replicas plus Adagrad (orange curve)
is modestly faster. Sandblaster L-BFGS using 2000 model replicas (green curves) is considerably
faster yet again. The fastest, however, is Downpour SGD plus Adagrad with 200 model replicas (red
curves). Given access to sufficient CPU resourses, both Sandblaster L-BFGS and Downpour SGD
with Adagrad can train models substantially faster than a high performance GPU.
Though we did not confine the above experiments to a fixed resource budget, it is interesting to
consider how the various methods trade off resource consumption for performance. We analyze
this by arbitrarily choosing a fixed test set accuracy (16%), and measuring the time each method
took to reach that accuracy as a function of machines and utilized CPU cores, Figure 5. One of the
four points on each traces corresponds to a training configuration shown in Figure 4, the other three
points are alternative configurations.
In this plot, points closer to the origin are preferable in that they take less time while using fewer resources. In this regard Downpour SGD using Adagrad appears to be the best trade-off: For any fixed
budget of machines or cores, Downpour SGD with Adagrad takes less time to reach the accuracy
target than either Downpour SGD with a fixed learning rate or Sandblaster L-BFGS. For any allotted
training time to reach the accuracy target, Downpour SGD with Adagrad used few resources than
Sandblaster L-BFGS, and in many cases Downpour SGD with a fixed learning rate could not even
reach the target within the deadline. The Sandblaster L-BFGS system does show promise in terms
7
Time to 16% accuracy
Time to 16% accuracy
80
80
Downpour SGD
Downpour SGD w/Adagrad
Sandblaster L?BFGS
GPU
70
60
Time (hours)
Time (hours)
60
50
40
50
40
30
30
20
20
10
1
Downpour SGD
Downpour SGD w/Adagrad
Sandblaster L?BFGS
GPU (CUDA cores)
70
1000
2000
3000
4000
5000
10
1
6000
Machines
2000
4000
6000
8000
10000
12000
Cores
Figure 5: Time to reach a fixed accuracy (16%) for different optimization strategies as a function of
number of the machines (left) and cores (right).
of its scaling with additional cores, suggesting that it may ultimately produce the fastest training
times if used with an extremely large resource budget (e.g., 30k cores).
Application to ImageNet: The previous experiments demonstrate that our techniques can accelerate the training of neural networks with tens of millions of parameters. However, the more significant advantage of our cluster-based approach to distributed optimization is its ability to scale to
models that are much larger than can be comfortably fit on single machine, let alone a single GPU.
As a first step toward exploring the capabilities of very large neural networks, we used Downpour
SGD to train the 1.7 billion parameter image model described above on the ImageNet object classification task. As detailed in [29], this network achieved a cross-validated classification accuracy of
over 15%, a relative improvement over 60% from the best performance we are aware of on the 21k
category ImageNet classification task.
6
Conclusions
In this paper we introduced DistBelief, a framework for parallel distributed training of deep networks. Within this framework, we discovered several effective distributed optimization strategies.
We found that Downpour SGD, a highly asynchronous variant of SGD works surprisingly well for
training nonconvex deep learning models. Sandblaster L-BFGS, a distributed implementation of
L-BFGS, can be competitive with SGD, and its more efficient use of network bandwidth enables it
to scale to a larger number of concurrent cores for training a single model. That said, the combination of Downpour SGD with the Adagrad adaptive learning rate procedure emerges as the clearly
dominant method when working with a computational budget of 2000 CPU cores or less.
Adagrad was not originally designed to be used with asynchronous SGD, and neither method is
typically applied to nonconvex problems. It is surprising, therefore, that they work so well together,
and on highly nonlinear deep networks. We conjecture that Adagrad automatically stabilizes volatile
parameters in the face of the flurry of asynchronous updates, and naturally adjusts learning rates to
the demands of different layers in the deep network.
Our experiments show that our new large-scale training methods can use a cluster of machines to
train even modestly sized deep networks significantly faster than a GPU, and without the GPU?s
limitation on the maximum size of the model. To demonstrate the value of being able to train larger
models, we have trained a model with over 1 billion parameters to achieve better than state-of-the-art
performance on the ImageNet object recognition challenge.
Acknowledgments
The authors would like to thank Samy Bengio, Tom Dean, John Duchi, Yuval Netzer, Patrick Nguyen, Yoram
Singer, Sebastian Thrun, and Vincent Vanhoucke for their indispensable advice, support, and comments.
8
References
[1] G. Dahl, D. Yu, L. Deng, and A. Acero. Context-dependent pre-trained deep neural networks for large
vocabulary speech recognition. IEEE Transactions on Audio, Speech, and Language Processing, 2012.
[2] G. Hinton, L. Deng, D. Yu, G. Dahl, A. Mohamed, N. Jaitly, A. Senior, V. Vanhoucke, P. Nguyen,
T. Sainath, and B. Kingsbury. Deep neural networks for acoustic modeling in speech recognition. IEEE
Signal Processing Magazine, 2012.
[3] D. C. Ciresan, U. Meier, L. M. Gambardella, and J. Schmidhuber. Deep big simple neural nets excel on
handwritten digit recognition. CoRR, 2010.
[4] A. Coates, H. Lee, and A. Y. Ng. An analysis of single-layer networks in unsupervised feature learning.
In AISTATS 14, 2011.
[5] Y. Bengio, R. Ducharme, P. Vincent, and C. Jauvin. A neural probabilistic language model. Journal of
Machine Learning Research, 3:1137?1155, 2003.
[6] R. Collobert and J. Weston. A unified architecture for natural language processing: Deep neural networks
with multitask learning. In ICML, 2008.
[7] Q.V. Le, J. Ngiam, A. Coates, A. Lahiri, B. Prochnow, and A.Y. Ng. On optimization methods for deep
learning. In ICML, 2011.
[8] R. Raina, A. Madhavan, and A. Y. Ng. Large-scale deep unsupervised learning using graphics processors.
In ICML, 2009.
[9] J. Martens. Deep learning via hessian-free optimization. In ICML, 2010.
[10] J. C. Duchi, E. Hazan, and Y. Singer. Adaptive subgradient methods for online learning and stochastic
optimization. Journal of Machine Learning Research, 12:2121?2159, 2011.
[11] Q. Shi, J. Petterson, G. Dror, J. Langford, A. Smola, A. Strehl, and V. Vishwanathan. Hash kernels. In
AISTATS, 2009.
[12] J. Langford, A. Smola, and M. Zinkevich. Slow learners are fast. In NIPS, 2009.
[13] G. Mann, R. McDonald, M. Mohri, N. Silberman, and D. Walker. Efficient large-scale distributed training
of conditional maximum entropy models. In NIPS, 2009.
[14] R. McDonald, K. Hall, and G. Mann. Distributed training strategies for the structured perceptron. In
NAACL, 2010.
[15] M. Zinkevich, M. Weimer, A. Smola, and L. Li. Parallelized stochastic gradient descent. In NIPS, 2010.
[16] A. Agarwal, O. Chapelle, M. Dudik, and J. Langford. A reliable effective terascale linear learning system.
In AISTATS, 2011.
[17] A. Agarwal and J. Duchi. Distributed delayed stochastic optimization. In NIPS, 2011.
[18] F. Niu, B. Retcht, C. Re, and S. J. Wright. Hogwild! A lock-free approach to parallelizing stochastic
gradient descent. In NIPS, 2011.
[19] J. Bergstra, O. Breuleux, F. Bastien, P. Lamblin, R. Pascanu, G. Desjardins, J. Turian, D. Warde-Farley,
and Y. Bengio. Theano: a CPU and GPU math expression compiler. In SciPy, 2010.
[20] D. Ciresan, U. Meier, and J. Schmidhuber. Multi-column deep neural networks for image classification.
Technical report, IDSIA, 2012.
[21] L. Deng, D. Yu, and J. Platt. Scalable stacking and learning for building deep architectures. In ICASSP,
2012.
[22] A. Krizhevsky. Learning multiple layers of features from tiny images. Technical report, U. Toronto, 2009.
[23] J. Dean and S. Ghemawat. Map-Reduce: simplified data processing on large clusters. CACM, 2008.
[24] Y. Low, J. Gonzalez, A. Kyrola, D. Bickson, C. Guestrin, and J. Hellerstein. Distributed GraphLab: A
framework for machine learning in the cloud. In VLDB, 2012.
[25] L. Bottou. Stochastic gradient learning in neural networks. In Proceedings of Neuro-N??mes 91, 1991.
[26] Y. LeCun, L. Bottou, G. Orr, and K. Muller. Efficient backprop. In Neural Networks: Tricks of the trade.
Springer, 1998.
[27] V. Vanhoucke, A. Senior, and M. Z. Mao. Improving the speed of neural networks on cpus. In Deep
Learning and Unsupervised Feature Learning Workshop, NIPS 2011, 2011.
[28] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei. ImageNet: A Large-Scale Hierarchical
Image Database. In CVPR, 2009.
[29] Q.V. Le, M.A. Ranzato, R. Monga, M. Devin, K. Chen, G.S. Corrado, J. Dean, and A.Y. Ng. Building
high-level features using large scale unsupervised learning. In ICML, 2012.
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4,074 | 4,688 | Semi-Crowdsourced Clustering: Generalizing Crowd
Labeling by Robust Distance Metric Learning
Jinfeng Yi? , Rong Jin? , Anil K. Jain? , Shaili Jain\ , Tianbao Yang?
?
Michigan State University, East Lansing, MI 48824, USA
\
Yale University, New Haven, CT 06520, USA
?
Machine Learning Lab, GE Global Research, San Ramon, CA 94583, USA
{yijinfen, rongjin, jain}@cse.msu.edu, [email protected], [email protected]
Abstract
One of the main challenges in data clustering is to define an appropriate similarity measure between two objects. Crowdclustering addresses this challenge by
defining the pairwise similarity based on the manual annotations obtained through
crowdsourcing. Despite its encouraging results, a key limitation of crowdclustering is that it can only cluster objects when their manual annotations are available. To address this limitation, we propose a new approach for clustering, called
semi-crowdsourced clustering that effectively combines the low-level features of
objects with the manual annotations of a subset of the objects obtained via crowdsourcing. The key idea is to learn an appropriate similarity measure, based on the
low-level features of objects and from the manual annotations of only a small portion of the data to be clustered. One difficulty in learning the pairwise similarity
measure is that there is a significant amount of noise and inter-worker variations in
the manual annotations obtained via crowdsourcing. We address this difficulty by
developing a metric learning algorithm based on the matrix completion method.
Our empirical study with two real-world image data sets shows that the proposed
algorithm outperforms state-of-the-art distance metric learning algorithms in both
clustering accuracy and computational efficiency.
1
Introduction
Crowdsourcing provides an easy and relatively inexpensive way to utilize human capabilities to
solve difficult computational learning problems (e.g. image annotation in ESP game [17]). It divides
a large task into a number of small-scale tasks, often referred to as Human Intelligence Tasks (HITs),
and asks a human worker to solve each individual HIT. It then combines the partial solutions obtained from individual HITs to form the final solution. In the past, crowdsourcing has been explored
for a number of machine learning tasks (e.g., classification and clustering) [21, 10, 19].
Crowdclustering [10] exploits the crowdsourcing paradigm for data clustering. The key idea is to
first obtain manual annotations of objects through crowdsourcing. The annotations can either be
in the form of grouping objects based on their perceived similarities [10] or the keyword assignments to individual objects (e.g., images) by human workers [25]. A pairwise similarity matrix is
then computed from the acquired annotations, and is used to cluster objects. Unlike the conventional clustering techniques where the similarity measure is defined based on the features of objects, in
crowdclustering, the pairwise similarities are derived from the manual annotations, which better capture the underlying inter-object similarity. Studies [10] have shown that crowdclustering performs
significantly better than the conventional clustering methods, given a sufficiently large number of
manual annotations for all the objects to be clustered.
1
Figure 1: The proposed framework for semi-crowdsourced clustering. The given N objects
(o1 , o2 , . . . , oN ) need to be clustered, but only a small subset of the N objects (o01 , o02 , ? ? ? o0n ) have
been annotated by crowdsourcing, n N .
Despite the encouraging results obtained via crowdclustering, a main shortcoming of crowdclustering is that it can only cluster objects for which manual annotations are available, significantly
limiting its application to large scale clustering problems. For instance, when clustering hundreds of
thousands of objects, it is not feasible to have each object manually annotated by multiple workers.
To address this limitation, we study the problem of semi-crowdsourced clustering, where given
the annotations obtained through crowdsourcing for a small subset of the objects, the objective is to
cluster the entire collection of objects. Figure 1 depicts the proposed framework. Given a set of N
objects to be clustered, the objective is to learn a pairwise similarity measure from the crowdsourced
labels of n objects (n N ) and the object feature vector x. Note that the available crowdclustering
algorithms [10, 25] expect that all N objects be labeled by crowdsourcing.
The key to semi-crowdsourced clustering is to define an appropriate similarity measure for the subset
of objects that do not have manual annotations (i.e., N ? n objects). To this end, we propose to learn
a similarity function, based on the object features, from the pairwise similarities derived from the
manual annotations for the subset of n objects; we then apply the learned similarity function to
compute the similarity between any two objects, and perform data clustering based on the computed
similarities. In this study, for computational simplicity, we restrict ourselves to a linear similarity
function, i.e. given two objects oi and oj and their feature representation xi and xj , respectively,
their similarity sim(Oi , Oj ) is given by sim(Oi , Oj ) = x>
i M xj , where M 0 is the learned
distance metric.
Learning a linear similarity function from given pairwise similarities (sometimes referred to as pairwise constraints when similarities are binary) is known as distance metric learning, which has been
studied extensively in the literature [24]. The key challenge of distance metric learning in semicrowdsourced clustering arises due to the noise in the pairwise similarities obtained from manual
annotations. According to [25], large disagreements are often observed among human workers in
specifying pairwise similarities. As a result, pairwise similarities based on the majority vote among
human workers often disagree with the true cluster assignments of objects. As an example, the authors in [25] show that for the Scenes data set [8], more than 80% of the pairwise labels obtained
from human workers are inconsistent with the true cluster assignment. This large noise in the pairwise similarities due to crowdsourcing could seriously misguide the distance metric learning and
lead to a poor prediction performance, as already demonstrated in [12] as well as in our empirical
study.
We propose a metric learning algorithm that explicitly addresses the presence of noise in pairwise
similarities obtained via crowdsourcing. The proposed algorithm uses the matrix completion technique [3] to rectify the noisy pairwise similarities, and regression analysis to efficiently learn a
2
Figure 2: The proposed framework of learning a distance metric from noisy manual annotations
distance metric from the restored pairwise similarities. More specifically, the proposed algorithm
for clustering N objects consists of three components: (i) filtering noisy pairwise similarities for n
objects by only keeping object pairs whose pairwise similarities are agreed by many workers (not
majority of the workers). The result of the filtering step is a partially observed n ? n similarity
matrix (n N ) with most of its entries removed/unobserved; (ii) recovering the n ? n similarity
matrix from the partially observed entries by using the matrix completion algorithm; (iii) applying a
regression algorithm to learn a distance metric from the recovered similarity matrix, and clustering
the N ? N pairwise similarities based on the learned distance metric. Figure 2 shows the basic steps
of the proposed algorithm.
Compared to the existing approaches of distance metric learning [24], the proposed algorithm has
the following three advantages: (i) by exploring the matrix completion technique, the proposed
algorithm is robust to a large amount of noise in the pairwise similarities; (ii) by utilizing regression analysis, the proposed algorithm is computationally efficient and does not have to handle the
positive semi-definite constraint, a key computational bottleneck for most distance metric learning
algorithms; (iii) the learned distance metric, with high probability, is close to the optimal metric
learned from the perfect or true similarities (i.e. similarity of 1 when two objects are in the same
cluster and 0, otherwise) for arbitrarily large n.
We finally note that in addition to distance metric learning, both kernel learning [16] and constrained clustering [2] can be applied to generalize the information in the manual annotations acquired by crowdsourcing. In this work, we focus on distance metric learning. The related work,
as well as the discussion on exploring kernel learning and constrained clustering techniques for
semi-crowdsourced clustering can be found in Section 4.
2
Semi-Crowdsourced Clustering by Robust Distance Metric Learning
We first present the problem and a general framework for semi-crowdsourced clustering. We then
describe the proposed algorithm for learning distance metric from a small set of noisy pairwise
similarities that are derived from manual annotations.
2.1
Problem Definition and Framework
Let D = {O1 , . . . , ON } be the set of N objects to be clustered, and let X = (x1 , . . . , xN ) be their
feature representation, where xi ? Rd is a vector of d dimensions. We randomly sample a subset
b = {O
b1 , . . . , O
bn }, and obtain their manual
of n N objects from the collection D, denoted by D
annotations by crowdsourcing. Let m be the number of HITs used by crowsourcing. Given the
manual annotations collected from the k-th HIT, we define a similarity matrix Ak ? Rn?n such that
bi and O
bj share common annotations (i.e. share common annotated keywords
Aki,j = 1 if objects O
or assigned to the same cluster by the worker), zero if they don?t, and ?1 if either of the two objects
is not annotated by the kth HIT (i.e. unlabeled pair). Note that we only consider a binary similarity
measure in this study because our goal is to perfectly reconstruct the ideal pairwise similarities based
on the true cluster assignments (i.e. 1 when both objects are assigned to the same cluster and zero,
otherwise). The objective of semi-crowdsourced clustering is to cluster all the N objects in D based
b
on the features in X and the m ? m similarity matrices {Ak }m
k=1 for the objects in D. Throughout
1
this paper, we assume that the number of clusters, denoted by r, is given a priori .
1
We may relax this requirement by estimating the number of clusters via some heuristic, e.g. considering
the number of clusters as the rank of the completed matrix A.
3
b to the entire collection of objects D, we
To generalize the pairwise similarities from the subset D
propose to first learn a distance metric from the similarity matrices {Ak }m
k=1 , and then compute the
pairwise similarity for all the N objects in D using the learned distance metric. The challenge is how
to learn an appropriate distance metric from a set of similarity matrices {Ak }m
k=1 . A straightforward
approach is to combine multiple similarity matrices into a single similarity matrix by computing
e ? Rn?n be the average similarity matrix. We have
their average. More specifically, let A
m
X
1
ei,j = Pm
A
I(Aki,j ? 0)Ai,j
k ? 0)
I(A
i,j
k=1
k=1
bi , O
bj ) is not labeled by the kth HIT (i.e. either object O
bi
where Aki,j < 0 indicates that the pair (O
bj is not annotated by the kth worker) and I(z) is an indicator function that outputs 1 when z is
or O
e The main problem with this
true and zero, otherwise. We then learn a distance metric M from A.
simple strategy is that due to the large disagreements among workers in determining the pairwise
similarities, the average similarities do not correlate well with the true cluster assignments. In the
next subsection, we develop an efficient and robust algorithm that learns a distance metric from a set
of noisy similarity matrices.
2.2
Learning a Distance Metric from a Set of Noisy Similarity Matrices
As illustrated in Figure 2, the proposed algorithm consists of three steps, i.e. filtering step, matrix
completion step and distance metric learning step. For the first two steps, namely the data preprocessing steps, we follow the idea proposed in [25].
Filtering step. To filter out the uncertain object pairs, we introduce two thresholds d0 and d1 (1 ?
? Since any similarity measure smaller than d0
d1 > d0 ? 0) into the average similarity matrix A.
indicates that most workers put the corresponding object pair into different clusters, we simply set
it as 0. Similarly, we set the similarity measure larger than d1 as 1. For object pairs with similarity
measure in the range between d0 and and d1 , they are treated as uncertain object pairs and are
discarded (i.e. marked as unobserved) from the similarity matrix. The resulting partially observed
similarity matrix A is given by
?
? 1
A?i,j ? [d1 , 1]
Ai,j =
(1)
0
A?i,j ? [0, d0 ]
?
unobserved Otherwise
We also define ? as the set of observed entries in Ai,j
? = {(i, j) ? [N ] ? [N ] : A?ij ? 0, A?ij ?
/ (d0 , d1 )}
Matrix completion step. Since A is constructed from the partial clustering results generated by
different workers, we expect some of the binary similarity measures in A to be incorrect. We introduce the matrix E ? Rn?n to capture the incorrect entries in A. If A? is the perfect similarity
matrix, we have P? (A? + E) = P? (A), where P? outputs a matrix with [P? (B)]i,j = Bi,j if
(i, j) ? ? and zero, otherwise. With appropriately chosen thresholds d0 and d1 , we expect most
of the observed entries in A to be correct and as a result, E to be a sparse matrix. To reconstruct
the perfect similarity matrix A? from A, following the matrix completion theory [3], we solve the
following optimization problem
b ? + C|E|1 s. t. P? (A
b + E) = P? (A),
min |A|
(2)
b
A,E
P
where |A|? is the nuclear norm of matrix A and |E|1 = i,j |Ei,j | is the `1 norm of E. Using the
b is of low rank [14], under the two assumptions made in [25],
facts that E is a sparse matrix and A
?
b where A
b is the optimal solution for (2). For completeness,
with a high probability, we have A = A,
we include in the supplementary document the theoretical result for the problem in (2)
Distance metric learning step. This step learns a distance metric from the completed similarity
b A common problem shared by most distance metric learning algorithms is their high
matrix A.
computational cost due to the constraint that a distance metric has to be positive semi-definite. In this
study, we develop an efficient algorithm for distance metric learning that does not have to deal with
4
the positive semi-definite constraint. Our algorithm is based on the key observation that with a high
b is positive semi-definite. This is because according
probability, the completed similarity matrix A
b = Y Y > , where Y ? {0, 1}n?r is
to Theorem 1 of [25], with a probability at least 1 ? n?3 , A
the true cluster assignment. This property guarantees the resulting distance metric to be positive
semi-definite.
The proposed distance metric learning algorithm is based on a standard regression algorithm [15].
b the optimal distance metric M is given by a regression problem
Given the similarity matrix A,
n
X
b )=
bi,j )2 = |X
b >M X
b ? A|
b2
bj ? A
L(M
min
(b
x>
(3)
i Mx
F
M ?Rd?d
i,j=1
bi and X
b = (b
bi is the feature vector for the sampled object O
bn ). The optimal
where x
x1 , . . . , x
c
solution to (3), denoted by M , is given by
c = (X
bX
b > )?1 X
bA
bX
b > (X
bX
b > )?1
M
(4)
c 0 if A
b 0.
where Z ?1 is pseudo inverse of Z. It is straightforward to verify M
b because of the
Directly using the solution in (4) could result in the overfitting of similarity matrix A
bX
b > . We address this challenge by a smoothing technique, i.e.
potential singularity of X
cs = (X
bX
b > + ?mI)?1 X
bA
bX
b > (X
bX
b > + ?mI)?1
M
(5)
where I is the identity matrix of size d ? d and ? > 0 is a smoothing parameter used to address the
overfitting and the curse of dimensionality. Note that the computation in (5) can be simplified by
cs in terms of the singular values and singular vectors of X.
b We omit the details due to
expressing M
the space constraints.
cs . Let A(Oi , Oj ) be the perfect similarity that outputs
We now state the theoretical property of M
1 when Oi and Oj belong to the same cluster and zero, otherwise. It is straightforward to see that
A(Oi , Oj ) = yi> yj , where yi ? {0, 1}r is the cluster assignment for object Oi . To learn an ideal
distance metric from the perfect similarity measure A(Oi , Oj ), we generalize the regression problem
in (3) as follows
2
min
L(M ) = Exi ,xj (x>
(6)
i M xj ? A(Oi , Oj ))
M ?Rd?d
?1
?1
>
The solution to (6) is given by M = CX
BB > CX
, where CX = Exi [xi x>
i ] and B = Exi [xi yi ].
?1
>
Let Ms be the smoothed version of the ideal distance metric M , i.e. M = (CX +?I) BB (CX +
cs and
?I)?1 . The following theorem shows that with a high probability, the difference between M
Ms is small if both ? and n are not too small.
Theorem 1. Assume |x|2 ? 1 for the feature representation of any object. Assume the conditions
in Theorem 1 of [25] hold. Then, with a probability 1 ? 3n?3 , we have
cs |2 = O ln?n
|Ms ? M
?2 n
where |Z|2 stands for the spectral norm of matrix Z.
The detailed proof can be found in the supplementary materials. Given the learned distance metric
cs , we construct a similarity matrix S = X > M
cs X and then apply a spectral clustering algorithM
m [18] to compute the final data partition for N objects.
3
Experiments
In this section, we demonstrate empirically that the proposed semi-crowdsourced clustering algorithm is both effective and efficient.
5
3.1
Data Sets, Baselines, and Parameter Settings
Data Sets. Two real-world image data sets are used in our experiments: (i) ImageNet data set is
a subset of the larger ImageNet database [6]. The subset contains 6, 408 images belonging to 7
categories: tractor, horse cart, bench, blackberry, violin, saxophone, and hammer. (ii) PASCAL 07
data set is a subset of the PASCAL Visual Object Classes Challenge 2007 database [7]. The subset
contains 2, 989 images belonging to five classes: car, dog, chair, cat and bird. We choose these
specific image categories because they yield relatively low classification performance in ImageNet
competition and PASCAL VOC Challenge, indicating that it could be difficult to cluster these images using low level features without side information. The image features for these datasets were
downloaded from the homepages of the ImageNet database 2 and the research group of Learning
and Recognition in Vision (LEAR) 3 , respectively.
To perform crowdlabeling, we follow [25], and ask human workers to annotate images with keywords of their choice in each HIT. A total of 249 and 332 workers were employed using the Amazon?s Mechanical Turk [13] to annotate images from ImageNet and PASCAL datasets, respectively.
On average, each image is annotated by five different workers, with three keywords from each
individual worker. For every HIT, the pairwise similarity between two images (i.e. Aki,j used in
Section 2.1) is set to 1 if the two images share at least one common annotated keyword and zero,
otherwise 4 .
Baselines. Two baseline methods are used as reference points in our study: (a) the Base method
that clusters images directly using image features without distance metric learning, and (b) the Raw
e without filtering
method that runs the proposed algorithm against the average similarity matrix A
and matrix completion steps. The comparison to the Base method allows us to examine the effect of
distance metric learning in semi-crowdsourced clustering, and the comparison to the Raw method
reveals the effect of filtering and matrix completion steps in distance metric learning.
We compare the proposed algorithm for distance metric learning to the following five state-of-the-art
distance metric learning algorithms: (a) GDM, the global distance metric learning algorithm [23],
(b) RCA, the relevant component analysis [1], (c) DCA, the discriminative component analysis [11],
(d) ITML, the information theoretic metric learning algorithm [5], and (e) LMNN, the large margin
nearest neighbor classifier [20]. Some of the other state-of-the-art distance metric learning algorithms (e.g. the neighborhood components analysis (NCA) [9]) were excluded from the comparison
because they can only work with class assignments, instead of pairwise similarities, and therefore
are not applicable in our case. The code for the baseline algorithms was provided by their respective
authors (In LMNN, Principal Component Analysis (PCA) is used at first to reduce the data to lower
dimensions). For a fair comparison, all distance metric learning algorithms are applied to the pairb the n ? n pairwise similarity matrix reconstructed by the matrix
wise constraints derived from A,
completion algorithm. We refer to the proposed distance metric learning algorithm as Regression
based Distance Metric Learning, or RDML for short, and the proposed semi-crowdsourced clustering algorithm as Semi-Crowd.
Parameter Settings. Two criteria are used in determining the values for d0 and d1 in (1). First,
d0 (d1 ) should be small (large) enough to ensure that most of the retained pairwise similarities are
consistent with the cluster assignments. Second, d0 (d1 ) should be large (small) enough to obtain a
sufficient number of observed entries in the partially observed matrix A. For both data sets, we set
d0 to 0 and d1 to 0.8. We follow the heuristic proposed in [25] to determine the parameter C in (2),
which is selected to generate balanced clustering results. Parameter ? in (5) is set to 1. We varied ?
from 0.5 to 5 and found that the clustering results essentially remain unchanged.
Evaluation. Normalized mutual information (NMI for short) [4] is used to measure the coherence
between the inferred clustering and the ground truth categorization. The number of sampled images
is varied from 100, 300, 600 to 1, 000. All the experiments are performed on a PC with Intel Xeon
2.40 GHz processor and 16.0 GB of main memory. Each experiment is repeated five times, and the
performance averaged over the five trials is reported.
2
http://www.image-net.org/download-features
http://lear.inrialpes.fr/people/guillaumin/data.php
4
We tried several other similarity measures (e.g. cosine similarity measure and tf.idf weighting) and found
that none of them yielded better performance than the simple similarity measure used in this work
3
6
(a) ImageNet data set
(b) PASCAL 07 data set
Figure 3: NMI vs. no. of sampled images (n) used in crowdlabeling.
(a)
Two images incorrectly placed in different clusters by the Base method (similarity 0.16) but correctly grouped into the same
cluster by the proposed method (similarity
0.66).
(b)
Two images incorrectly placed in different clusters by the Base method (similarity 0.31) but correctly grouped into
the same cluster by the proposed method
(similarity 0.85)
(c)
Two images incorrectly grouped into the
same cluster by the Base method (similarity
0.72) but correctly clustered to different clusters by the proposed method (similarity 0.22)
Figure 4: Sample image pairs that are incorrectly clustered by the Base method but correctly clustered by the
cs ).
proposed method (the similarity of our method is based on the normalized distance metric M
3.2
Experimental Results
First, we examine the effect of distance metric learning algorithm on semi-crowdsourced clustering.
Figure 3 compares the clustering performance with six different metric learning algorithms with
that of the Base method that does not learn a distance metric. We observed that four of the distance
metric learning algorithms (i.e. GDM, ITML, LMNN and the proposed RDML) outperform the
Base method, while RCA and DCA fail to improve the clustering performance of Base. We conjecture that the failure of RCA and DCA methods is due to their sensitivity to the noisy pairwise
similarities. In fact, RCA and DCA can yield better performance than the Base method if all the
pairwise similarities are consistent with the cluster assignments. Compared to all the baseline distance metric learning algorithms, RDML, the proposed distance metric learning algorithm, yields
the best clustering results for both the data sets and for all values of n (i.e. the number of annotated images) considered here. Furthermore, the performance of RDML gradually stabilizes as the
number of sampled images increases. This is consistent with our theoretical analysis in Theorem 1,
and implies that only a modest number of annotated images is needed by the proposed algorithm to
learn an appropriate distance metric. This observation is particularly useful for crowdclustering as
it is expensive to reliably label a very large number of images. Figure 4 shows some example image
pairs for which the Base method fails to make correct cluster assignments, but the proposed RDML
method successfully corrects these mistakes with the learned distance metric.
Our next experiment evaluates the impact of filtering and matrix completion steps. In Figure 3,
we compare the clustering results of the proposed algorithm for semi-crowdsourced clustering
(i.e. Filtering+Matrix-Completion+RDML) to the Raw method that runs the proposed distance metric algorithm RDML without the filtering and matrix completion steps. Based on these experiments,
we can make the following observations: (i) the proposed distance metric learning algorithms performs better than the Raw method, particularly when the number of annotated images is small; (ii)
the gap between the proposed semi-crowdsourced clustering method and the Raw method decreases
as the sample size increases. These results indicate the importance of filtering and matrix completion steps for the crowdsourced data in semi-crowdsourced clustering. Finally, it is interesting to
observe that the Raw method still outperforms all the baseline methods, which further verifies the
effectiveness of the proposed algorithm for distance metric learning.
Finally, we evaluate the computational efficiency of the proposed distance metric learning algorithm.
Table 1 shows that the proposed distance metric learning algorithm is significantly more efficient
than the baseline approaches evaluated here. The last row of Table 1 indicates the run time for the
7
Table 1: CPU time (in seconds) for learning the distance metrics.
CPU time (s)
Sample sizes (n)
RDML (proposed)
GDM [23]
LMNN [20]
ITML [5]
DCA [11]
RCA [1]
Matrix Completion
100
4.2
11384
59.8
2128
8.5
9.7
12.4
ImageNet Data Set
300
600
6.3
8.0
14706 18140
157
330
2376
2692
9.2
14.5
13.5
18.6
74.2
536
1,000
11.2
25155
629
3081
20.7
23.6
1916
PASCAL 07 Data Set
100
300
600
1,000
27.4
34.2
41.7
47.3
26346 36795 44237 53468
55.1
124
277
527
5311
5721
6104
6653
51.2
64.1
72.7
82.3
71.4
92.7
103
122
12.8
86.6
615
1873
matrix completion step. Since all the distance metric learning algorithms are applied to the similarity
matrix recovered by the matrix completion algorithm, the computational cost of matrix completion is
shared by all distance metric learning algorithms used in our evaluation. We observe that the matrix
completion step, particularly for large sample sizes, is computationally demanding, a problem that
will be investigated in our future work.
4
Related Work and Discussion
Crowdclustering was first proposed in [10]. It divided the task of clustering a collection of images
into a number of human intelligence tasks (or HITs). In each HIT, a small subset of images are
randomly sampled from the collection, and a worker is asked to cluster the subset of images into multiple groups. By using a large number of HITs, the authors ensure that every image in the
collection is included in at least one HIT. In [25], the authors extend the definition of HITs for
crowdclustering by asking workers to annotate images by keywords and then derive pairwise similarities between images based on the commonality of annotated keywords. A major limitation of
both these studies, as pointed out earlier, is that they can only cluster images that have been manually
annotated. Although the matrix completion technique was first proposed for crowdclustering in [25],
it had a different goal from this work. In [25], matrix completion was used to estimate the similarity
matrix, while the proposed approach uses matrix completion to estimate a distance metric, so that
crowdsourced labels can be generalized to cluster those images which were not annotated during
crowdsourcing.
Our work is closely related to distance metric learning that learns a distance metric consistent with
a given subset of pairwise similarities/constraints [24]. Although many studies on distance metric
learning have been reported, only a few address the challenge of learning a reliable distance metric
from noisy pairwise constraints [12, 22]. One limitation of these earlier studies is that they can
only work with a relatively small number (typically less than 30%) of noisy pairwise constraints.
In contrast, in semi-crowdsourced clustering, we expect that a significantly larger percentage of
pairwise similarities are inconsistent with the true cluster assignments (as many as 80% [25]).
One limitation of distance metric learning is that it is restricted to a linear similarity function. Kernel
learning generalizes distance metric learning to a nonlinear similarity function by mapping each
data point to a high dimensional space through a kernel function [16]. We plan to learn a kernel
based similarity function from a subset of manually annotated objects. Besides distance metric
learning, an alternative approach to incorporate the manual annotations into the clustering process
is constrained clustering (or semi-supervised clustering) [2]. Compared to distance metric learning,
constrained clustering can be computationally more expensive. Unlike distance metric learning that
learns a distance metric from pairwise constraints only once and applies the learned distance metric
to cluster any set of objects, a constrained clustering algorithm has to be rerun whenever a new
set of objects needs to be clustered. To exploit the strength of constrained clustering algorithms, we
plan to explore hybrid approaches that effectively combine distance metric learning with constrained
clustering approaches for more accurate and efficient semi-crowdsourced clustering.
Acknowledgments
This work was supported in part by National Science Foundation (IIS-0643494) and Office of Navy
Research (Award nos. N00014-12-1-0431, N00014-11-1-0100, N00014-12-1-0522, and N0001409-1-0663).
8
References
[1] Aharon Bar-Hillel, Tomer Hertz, Noam Shental, and Daphna Weinshall. Learning a Mahalanobis metric from equivalence constraints. JMLR, 2005.
[2] Sugato Basu, Ian Davidson, and Kiri Wagstaff. Constrained Clustering: Advances in Algorithms, Theory, and Applications. Chapman & Hall/CRC, 2008.
[3] Emmanuel J. Cand`es and Terence Tao. The power of convex relaxation: near-optimal matrix
completion. IEEE Transactions on Information Theory, 56(5):2053?2080, 2010.
[4] Thomas M. Cover and Joy A. Thomas. Elements of Information Theory (2nd ed.). Wiley, 2006.
[5] J.V. Davis, B. Kulis, P. Jain, S. Sra, and I.S. Dhillon. Information-theoretic metric learning. In
ICML, pages 209?216, 2007.
[6] J. Deng, W. Dong, R. Socher, L.J. Li, K. Li, and L. Fei-Fei. Imagenet: A large-scale hierarchical image database. In CVPR, 2009.
[7] M. Everingham, L. Van Gool, C. K. I. Williams, J. Winn, and A. Zisserman. The
PASCAL Visual Object Classes Challenge 2007 (VOC2007) Results. http://www.pascalnetwork.org/challenges/VOC/voc2007/workshop/index.html.
[8] L. Fei-Fei and P. Perona. A bayesian hierarchical model for learning natural scene categories.
In CVPR, pages 524?531, 2005.
[9] Jacob Goldberger, Sam T. Roweis, Geoffrey E. Hinton, and Ruslan Salakhutdinov. Neighbourhood components analysis. In NIPS, 2004.
[10] R. Gomes, P. Welinder, A. Krause, and P. Perona. Crowdclustering. In NIPS, 2011.
[11] S.C.H. Hoi, W. Liu, M.R. Lyu, and W.Y. Ma. Learning distance metrics with contextual constraints for image retrieval. In CVPR, pages 2072?2078, 2006.
[12] Kaizhu Huang, Rong Jin, Zenglin Xu, and Cheng-Lin Liu. Robust metric learning by smooth
optimization. In UAI, 2010.
[13] Panagiotis G. Ipeirotis. Analyzing the amazon mechanical turk marketplace. ACM Crossroads,
17(2):16?21, 2010.
[14] Ali Jalali, Yudong Chen, Sujay Sanghavi, and Huan Xu. Clustering partially observed graphs
via convex optimization. In ICML, pages 1001?1008, 2011.
[15] D.C. Montgomery, E.A. Peck, and G.G. Vining. Introduction to Linear Regression Analysis,
volume 49. John Wiley & Sons, 2007.
[16] Bernhard Scholkopf and Alexander J. Smola. Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press, Cambridge, MA, USA, 2001.
[17] L. Seneviratne and E. Izquierdo. Image annotation through gaming. In Proceedings of the 2nd
K-Space PhD Jamboree Workshop, 2008.
[18] Jianbo Shi and Jitendra Malik. Normalized cuts and image segmentation. PAMI, 2000.
[19] Omer Tamuz, Ce Liu, Serge Belongie, Ohad Shamir, and Adam Kalai. Adaptively learning the
crowd kernel. In ICML, 2011.
[20] K.Q. Weinberger, J. Blitzer, and L.K. Saul. Distance metric learning for large margin nearest
neighbor classification. In NIPS, 2006.
[21] P. Welinder, S. Branson, S. Belongie, and P. Perona. The multidimensional wisdom of crowds.
In NIPS, 2010.
[22] Lei Wu, Steven C. H. Hoi, Rong Jin, Jianke Zhu, and Nenghai Yu. Distance metric learning
from uncertain side information for automated photo tagging. ACM TIST, 2011.
[23] E.P. Xing, A.Y. Ng, M.I. Jordan, and S. Russell. Distance metric learning, with application to
clustering with side-information. In NIPS, 2002.
[24] Liu Yang and Rong Jin. Distance metric learning: A comprehensive survey. Technical report,
Department of Computer Science and Engineering, Michigan State University, 2006.
[25] Jinfeng Yi, Rong Jin, Anil K. Jain, and Shaili Jain. Crowdclustering with sparse pairwise
labels: A matrix completion approach. In AAAI Workshop on Human Computation, 2012.
9
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4,075 | 4,689 | Selecting Diverse Features via Spectral
Regularization
Abhimanyu Das?
Microsoft Research
Mountain View
[email protected]
Anirban Dasgupta
Yahoo! Labs
Sunnyvale
[email protected]
Ravi Kumar?
Google
Mountain View
[email protected]
Abstract
We study the problem of diverse feature selection in linear regression: selecting
a small subset of diverse features that can predict a given objective. Diversity is
useful for several reasons such as interpretability, robustness to noise, etc. We propose several spectral regularizers that capture a notion of diversity of features and
show that these are all submodular set functions. These regularizers, when added
to the objective function for linear regression, result in approximately submodular functions, which can then be maximized by efficient greedy and local search
algorithms, with provable guarantees. We compare our algorithms to traditional
greedy and `1 -regularization schemes and show that we obtain a more diverse set
of features that result in the regression problem being stable under perturbations.
1
Introduction
Feature selection is a key component in many machine learning settings. The process involves
choosing a small subset of features in order to build a model to approximate the target concept
well. Feature selection offers several advantages in practice. This includes reducing the dimension
of the data and hence the space requirements, enhancing the interpretability of the learned model,
mitigating over-fitting, decreasing generalization error, etc.
In this paper we focus on feature selection for linear regression, which is the prediction model of
choice for many practitioners. The goal is to obtain a linear model using a subset of k features (where
k is user-specified), to minimize the prediction error or, equivalently, maximize the squared multiple
correlation [16]. In general, feature selection techniques can be categorized into two approaches.
In the first, features are greedily selected one by one up to the pre-specified budget k; the Forward
or Backward greedy methods[19] fall into this type. In the second, the feature selection process is
intimately coupled with the regression objective itself by adding a (usually convex) regularizer. For
example, the Lasso [20] uses the `1 -norm of the coefficients as a regularizer to promote sparsity.
In this work we consider the feature selection problem of choosing the best set of features for predicting a specified target, coupled with the desire to choose as ?diverse? features as possible; our
goal will be to construct a regularizer that can capture diversity. Diversity among the chosen features
is a useful property for many reasons. Firstly, it increases the interpretability of the chosen features,
since we are assured that they not redundant and are more representative of the feature space covered by the entire dataset (see e.g. [7]). Secondly, as we show, the right notion of diversity can also
make the feature selection task resistant to noise in the data. Thirdly, it is well known that correlated
features can slow down the convergence of algorithms such as the stochastic gradient (e.g., [2]); by
demanding diversity, one can potentially obviate this slowdown.
?
This work was done while the author was at Yahoo! Labs.
1
Unfortunately, the traditional greedy and `1 -relaxation approaches to feature-selection do not explictly address feature diversity1 . In this paper, we address this problem of diverse feature selection
using an approach that falls between that of greedy methods and convex-regularization methods. In
particular, we construct regularizers that capture a notion of diversity ? unlike regularizers such as
Lasso, our regularizers are set functions as opposed to functions of the regression coefficient vector.
Our objective function are thus a combination of the linear regression objective and the regularizer. We then design provable approximation algorithms for such objectives using a combination of
greedy and local search techniques. While there is no unique way to define feature diversity, we take
a spectral approach. By defining diversity to be a carefully chosen function of the spectrum of the
chosen features, we tap into notions of submodularity and consequently into the rich literature for
maximizing submodular functions [5, 9, 14].
Our contributions are as follows: (i) We formulate an optimization problem for diverse feature
selection and construct a family of submodular spectral regularizers that capture diversity notions.
(ii) We use a novel approach of combining the diversity regularizers with the optimization objective
to obtain (approximately) submodular maximization problems, and optimize them using greedy
and local search algorithms with provable guarantees. (iii) We validate the performance of our
algorithms using experiments on real and synthetic data sets.
2
Related work
Feature selection and the closely related problems of sparse approximation/recovery have been extensively studied using two broad classes of methods: greedy [5, 19, 21, 11, 24] and convex relaxation [20, 25, 3, 22, 8]. None of these methods, however, takes feature diversity into the account
during selection. The (greedy) methods in our paper are inspired by those of Das and Kempe [5],
who provide prediction error bounds using a notion of approximate submodularity. However, they
do not incorporate any notion of feature diversity; they also require monotonicity, which does not
hold for several regularizers we construct. A related convex relaxation based approach is that of
Grave et al. [12], who address the unstable behavior of Lasso in the presence of correlated features
and propose adding a trace norm regularizer to the error objective. The focus is to select groups
of correlated variables together instead of selecting only one variable from each group. Our goal is
different: select variables that are relatively uncorrelated with each other.
Previous work on diverse feature selection includes greedy heuristics for trading-off informationtheoretic feature relevance and feature redundancy criteria when selecting features [7, 23]. However
the heuristics presented do not carry any theoretical guarantees.
There has been some work on selecting a diverse set of features to maximize the mutual information
or the entropy of a set of variables [13, 17]. But, the problem definition in these works does not
specify a target prediction vector or variable; the goal instead is to select diverse features regardless
of whether the features are relevant for predicting a particular target variable. On the other hand, our
work requires us to simultaneously optimize for both feature selection and diversity objectives.
If we consider orthogonality as a loose proxy for diversity, methods such Principal Component
Analysis and Singular Value Decomposition [15] become relevant. However, these methods do not
return elements from the original set of features and instead output linear combinations of the feature
vectors; this might not be desirable for many applications.
3
Preliminaries
For any symmetric positive semidefinite n ? n matrix A, we denote its eigenvalues by ?min (A) =
?1 (A) ? ? ? ? ? ?n (A) = ?max (A). We use det(A)
= ?ni=1 ?i (A) to denote the determinant of A.
pP
2
Recall the vector and matrix two-norms: kxk2 =
i |xi | and kAk2 = ?max (A).
Let X = {X1 , . . . , Xn } be the set of feature vectors (or random variables) where each Xi ? Rm and
let Z ? Rm be the target vector. By appropriate normalization, we can assume kXi k2 = 1 = kZk2 .
We wish to predict Z using linear regression on a small subset of X. The matrix of inner products (or
1
discussed in the supplementary material at http://cs.usc.edu/?abhimand/nips12supplementary.pdf
2
covariances) between the Xi and Xj is denoted by C, with entries Ci,j = Cov(Xi , Xj ). Similarly,
we use b to denote the inner products between Z and the Xi ?s, with bi = Cov(Z, Xi ).
For a n-dimensional Gaussian random vector v with covariance matrix C, we use H(v) =
1
n
2 log((2?e) det(C)) to denote the differential entropy of v.
For a set S ? X, if Z 0 (S) is the optimal linear predictor of Z using the vectors in S, then the
2
squared multiple correlation [6, 16] is defined as RZ
(S) = 1 ? k(Z ? Z 0 (S))k22 . This is a widely
used goodness-of-fit measure; it captures the length of the projection of Z on the subspace spanned
by the vectors in S.
Definition 1 (Diverse feature selection) Given k > 0, find a set S ? X satisfying
?
2
argmax g(S) = RZ
(S) + ?f (S),
(1)
S:|S|?k
where ? > 0 is the regularization constant and f (S) is some ?diversity-promoting? regularizer.
Note that diversity is not a uniquely-defined notion, however, we call a regularizer f to be diversitypromoting if the following two conditions are satisfied: for a fixed k, f (S) is maximized when S is
an orthogonal set of vectors and is minimized when S has the lowest rank, where |S| ? k.
For convenience, we do not distinguish between the index set S and the variables {Xi | i ? S}. We
use CS to denote the submatrix of C with row and column set S, and bS to denote the vector with
only entries bi for i ? S. Given the subset S of vectors used for prediction, the optimal regression
2
coefficients ?i are (?i )i?S = CS?1 bS (e.g., [16]) and hence RZ
(S) = bTS CS?1 bS . 2
Many of our results are phrased in terms of eigenvalues of the inner product matrix C and its submatrices. Since such matrices are positive semidefinite, their eigenvalues are real, non-negative [16].
Submodularity ratio. Das and Kempe [5] introduced the notion of submodularity ratio for a general
set function to capture how close is the function to being submodular.
Definition 2 (Submodularity ratio) Let f be a non-negative set function.
submodularity ratio of f with
P respect to a set U and a parameter k ?
x?S f (L ? {x}) ? f (L)
.
?U,k (f ) =
min
f (L ? S) ? f (L)
L?U,S:|S|?k,S?L=?
The
1 is
Thus, it captures how much f can increase by adding any subset S of size k to L, compared to the
combined benefits of adding its individual elements to L. In particular, [5] defines the submodularity
ratio for the R2 function and relates it to the smallest eigenvalue of the covariance matrix of the data.
They also show that, in practice, the submodularity ratio for R2 is often quite close to 1, and hence
a greedy algorithm is a good approximation to maximizing R2 subject to a cardinality constraint.
Theorem 3 (from [4]) Let f be a non-negative, monotone set function and let OPT be the maximum value of f value obtained by any set of size k. Then, the set S? selected by the Greedy Algorithm
? (f )
? ? (1 ? e??S,k
has the following approximation guarantee: f (S)
) ? OPT.
3.1
Robustness to perturbations
As mentioned earlier, in addition to providing better interpretability, another benefit of diverse feature selection is robustness to feature and label perturbations. Given a selected subset S, we now
obtain a connection between the robustness of the estimated regression coefficients and the spectrum
of CS , in the presence of noise. Suppose S, a subset of size k, is used to predict the target vector
Z ? Rn . Let A ? Rn?k be the vectors from X corresponding to S. Then CS = AT A and the
optimal regression coefficients are ? = CS?1 AT Z.
Now suppose the target vector is perturbed with an i.i.d. Gaussian noise, i.e., Z 0 = Z + ?, where
? ? N (0, ? 2 In ) is a random vector corresponding to measurement errors. Let the corresponding
2
We assume throughout that CS is non-singular. For some of our results, an extension to singular matrices
is possible using the Moore?Penrose generalized inverse.
3
regression coefficient vector be ?0 = CS?1 AT Z 0 . We show the following perturbation result relating
the differential entropy of the perturbation error in the regression coefficients to the spectrum of CS .
Lemma 4 H(?0 ? ?) = k log(2? 2 ?e) ?
Pk
i=1
log(?i (CS )).
Proof. Let ? = ?0 ? ? = CS?1 AT ?. Since ? ? N (0, ? 2 In?n ), we have that ? ? N (0, CS?1 AT ?
? 2 In?n ? (CS?1 AT )T ). Or, ? ? N (0, ? 2 CS?1 ). Thus, H(?) = log((2? 2 ?e)k det(CS?1 )) =
Pk
k log(2? 2 ?e) ? i=1 log(?i (CS )).
Pk
Thus the perturbation error entropy is minimized by maximizing i=1 log(?i (CS )), which motivates the smoothed differential-entropy regularizer used in Section 5.1.
We can also show (supplementary material) that the two-norm of the perturbation error in the regression coefficients is also related to the spectrum of CS : the expected noise in the
P regression
1
coefficients depends on the sum of the eigenvalues of CS?1 . This suggests the use of ? i ?i (C
as
S)
a diversity-promoting regularizer in Definition 1. Unfortunately, this regularization function is not
submodular and is thus hard to use directly. However, as seen in Sections 5.2 and 5.3, there are other
related spectral functions that are indeed submodular and can thus be used as efficient regularizers.
4
Algorithms
In this section we present a greedy and local-search based (GLS) approximation algorithm for solving (1) when f (S) is a non-negative (but not necessarily monotone) submodular function (w.l.o.g.,
? = 1). In order to give an approximation algorithm for argmaxS:|S|?k g(S), we need to follow a
sequence of steps. First we show a technical result (Theorem 5) that says that though the approximation guarantees of [5] do not carry over to the non-monotone case, we can still prove a weaker
result that relates the solution obtained by a greedy algorithm with any feasible solution, as long
as g(S) is approximately submodular and non-negative (which holds if f (S) is a non-negative submodular function). Next, we modify a local-search based algorithm for unconstrained submodular
maximization to give an approximation of argmaxS g(S) (Theorem 7). We put these together using
the framework of [9] to show (Theorem 9) a constant factor approximation for solving (1).
The greedy Forward Regression (FR) algorithm is the following.
1: S0 ? ? and U ? {X1 , . . . , Xn }.
2: In each step i + 1, select Xj ? U \ Si maximizing g(Si ? {Xj }). Set Si+1 ? Si ? {Xj } and
U ? U \ {Xj }.
3: Output Sk .
Theorem 5 For any set T such that |T | ? k, the set S selected by the greedy FR algorithm satisfies
?S,2k
2
g(S) = RZ
(S) + f (S) ? (1 ? e? 2 )g(S ? T ).
The proof is very similar to that of [5, Theorem 3.2] and is omitted due to space constraints. Next,
2
we consider the problem of unconstrained maximization of the function g(S) = RZ
(S) + f (S). For
this, we use a local search (LS) algorithm similar to [9].
1: S ? argmaxi f (Xi ) and U ? {X1 , . . . , Xn }.
2: If there exists an element x ? U \S such that f (S ?{x}) ? (1+ n2 )f (S), then set S ? S ?{x},
and go back to Step 2.
3: Output argmaxT ?{S,U \S,U } g(T ).
Notice that even though we are interested in maximizing g(S), our LS algorithm finds a local optima
using f , but then uses g to compute the maximum in the last step. To analyze the performance
guarantees of LS, we first use the following result of [9, Theorem 3.4].
Lemma 6 If f is non-negative and submodular, then for any set T ? U and any > 0, the LS
algorithm takes O( 1 n3 log n) time and outputs solution S such that (2+ 2
n )f (S)+f (U \S) ? f (T ).
Using the above, we prove an approximation guarantee for unconstrained maximization of g(S).
4
Theorem 7 The LS algorithm is a
1
4+ 4
n
approximation for solving argmaxS g(S).
Proof. Suppose the optimal solution is C ? such that g(C ? ) = OPT. Consider the set S obtained
by the LS algorithm when it terminates. We obtain g(C ? ) = f (C ? ) + R2 (C ? ) ? (2 + 2/n)f (S) +
f (U \ S) + R2 (U ) ? (2 + 2/n)g(S) + g(U \ S) + g(U ), where the second step follows from
Lemma 6 and the monotonicity of R2 and the last step follows from the non-negativity of f and R2 .
Thus, max(g(S), g(U \ S), g(U )) ? 4+14 g(C ? ).
n
We now present the greedy and local search (GLS) algorithm for solving (1) for any submodular,
non-monotone, non-negative regularizer.
1: U ? {X1 , . . . , Xn }.
2: S1 ? FR(U ), S10 ? LS(S1 ), S2 ? FR(U \ S1 ).
3: Output argmaxS?{S1 ,S 0 ,S2 } g(S).
1
Next, we prove a multiplicative approximation guarantee for the GLS algorithm.
Lemma 8 Given sets C, S1 ? U , let C 0 = C \ S1 and S2 ? U \ S1 . Then g(S1 ? C) + g(S2 ?
C 0 ) + g(S1 ? C) ? g(C).
2
(S), we obtain g(S1 ?C)+g(S2 ?
Proof. Using the submodularity of f and the monotonicity of RZ
2
2
2
(C) + f (S1 ? S2 ? C) + f (C 0 ).
(S2 ? C 0 ) + f (S1 ? C) + f (S2 ? C 0 ) ? RZ
(S1 ? C) + RZ
C 0 ) = RZ
Now, f (C 0 ) + f (S1 ? C) ? f (C) + f (?) ? f (C), or f (C 0 ) ? f (C) ? f (S1 ? C). Hence, we have
2
g(S1 ? C) + g(S2 ? C 0 ) + f (S1 ? C) ? RZ
(C) + f (C) = g(C).
Theorem 9 If f is non-negative and submodular and <
gives a
??
S,2k
1?e? 2
??
S,2k
2+(1?e? 2 )(4+4/n)
?
??
S,2k
1?e? 2
7
n
?
4 , the set S selected by the GLS algorithm
approximation for solving argmaxS:|S|?k g(S).
Proof. Let C ? be the optimal solution with g(C ? ) = OPT. Then g(S1 ) ? ?g(S1 ? C ? ), where
?S ,2k
1
? = (1 ? e? 2 ). If g(S1 ? C ? ) ? OPT, then using the LS algorithm on S1 , we get (using
Theorem 7) a solution of value at least ? g(C ? ), where ? = 4 + 4
n . Else, g(S1 ) ? ?g(S1 ?
C ? ) + ?g(S1 ? C ? ) ? ?OPT. Also, g(S2 ) ? ?g(S2 ? (C ? \ S1 )). Thus, g(S1 ) + g(S2 ) ?
?g(S1 ? C ? ) + ?g(S1 ? C ? ) ? ?OPT + ?g(S2 ? (C ? \ S1 )) ? ?g(C ? ) ? ?OPT ? ?(1 ? )OPT,
where the last inequality follows from Lemma 8. Thus, max(g(S1 ), g(S2 )) ? ?(1?)OPT
. Hence,
2
??
?
the approximation factor is max( ? , ?(1?)
).
Setting
=
,
we
get
a
-approximation.
2
??+2
??+2
When f (S) is a monotone, non-negative, submodular function, the problem becomes much easier
due to the proposition below that follows directly from the definition of the submodularity ratio.
2
Proposition 10 For any submodular set function f (S), the function g(S) = RZ
(S)+f (S) satisfies
?U,k (g) ? ?U,k (R2 ) for any U and k.
Thus, since g(S) is monotone and approximately submodular, we can directly apply [4, Theorem 3]
? (f )
to show that the greedy FR algorithm gives a (1 ? e??S,k
)-approximation.
5
Spectral regularizers for diversity
In this section we propose a number of diversity-promoting regularizers for the feature selection
problem. We then prove that our algorithms in the previous section can obtain provable guarantees
for each of the corresponding regularized feature selection problems.
Most of our analysis requires the notion of operator antitone function [1] and its connection with
submodularity that was recently obtained by Friedland and Gaubert [10].
Definition 11 (Operator antitone functions [1]) A real valued function h is operator antitone on
the interval ? ? R if for all n ? 1 and for all n ? n Hermitian matrices A and B, we have
A B =? h(B) h(A), where A B denotes that B ? A is positive semidefinite; the function
h is called operator monotone if ?h is operator antitone.
5
Theorem 12 ([10]) Let f be a real continuous function defined on an interval ? of R. If the derivative of f is operator antitone on the interior of ?, then for every
Pn n ? n Hermitian matrix C with
spectrum in ?, the set function (from 2n ?? R) tr(f (S)) = i=1 f (?i (CS )) is submodular.
We will frequently use the following lemma for proving monotonicity of set functions. The proof is
given in the supplementary material.
Lemma 13 If f is a monotone and non-negative
Pn function defined on R, then for every n ? n Hermitian matrix C, the set function tr(f (S)) = i=1 f (?i (CS )) is monotone.
5.1
Smoothed differential entropy regularizer
For any set S with the corresponding covariance matrix CS , we define the smoothed differential
P|S|
entropy regularizer as fde (S) = i=1 log2 (? + ?i (CS )) ? 3k log2 ?, where ? > 0 is the smoothing
constant. This is a smoothed version of the log-determinant function fld (S) = log(det(CS )) =
P|S|
i=1 log(?i (CS )), that is also normalized by an additive term of 3k log2 ? in order to make the
regularizer non-negative 3 .
As shown in Lemma 4, this regularizer also helps improve the robustness of the regression model to
noise since maximizing fld (S) minimizes the entropy of the perturbation error. For a multivariate
Gaussian distribution, fld (S) also equivalent (up to an additive |S| factor) to the differential entropy
of S. However, fld (S) is undefined if S is rank-deficient and might also take negative values; the
smoothed version fde (S) overcomes these issues. It is also easy to show that fde (S) is a diversitypromoting regularizer. We now show that the GLS algorithm to solve (1) with f (S) = fde (S) gives
a constant-factor approximation algorithm.
?
??
S,2k
Theorem 14 The set S? selected by the GLS algorithm gives a 1?e 7 2 multiplicative approximation guarantee for (1) using the smoothed differential entropy regularizer fde (S).
Proof. We first prove that fde (S) is non-negative and submodular. Consider the real-valued function f?(t) = log(? + t) defined on the appropriate interval of R. We will show that the derivative of
f? is operator antitone. Let A, B be k ? k Hermitian matrices, such that 0 ? A B. Let I denote
the identity matrix. Then A + ?I B + ?I. Taking inverses, (B + ?I)?1 (A + ?I)?1 . Thus,
1
is operator antitone. Since h(t) is the derivative of f?(t),
by Definition 11, the function h(t) = ?+t
a straightforward application of Theorem 12 gives us that fde (S) is submodular. By Proposition 10,
2
we obtain that g(S) is approximately submodular, with submodularity ratio at least ?S,k
? (R ) . Since
g(S) is also non-negative, we can now apply Theorem 9 to obtain the approximation guarantee.
Notice that since fde (S) is not monotone in general [13], we cannot use Theorem 3. However, in
the case when ? ? 1, a simple application of Lemma 13 shows that fde (S) becomes monotonically
increasing and we can then use Theorem 3 to obtain a tighter approximation bound.
5.2
Generalized rank regularizer
For any set S with covariance matrix CS , and constant ? such that 0 ? ? ? 1, we define the genP|S|
eralized rank regularizer as fgr (S) = i=1 ?i (CS )? . Notice that for ? = 0, fgr (S) = rank(CS ).
The rank function however, does not discriminate between a full-rank matrix and an orthogonal matrix, and hence we define fgr (S) as a generalization of the rank function. It is easy to show that
fgr (S) is diversity-promoting. We prove that fgr (S) is also monotone and submodular, and hence
obtain approximation guarantees for the greedy FR algorithm for (1) with f (S) = fgr (S).
2
? (R )
Theorem 15 The set S? selected by the greedy FR algorithm gives a (1 ? e??S,k
) multiplicative
approximation guarantee for (1) using the generalized rank regularizer fgr (S).
3
we need this regularizer to be non-negative for sets of size up to 3k, because of the use of f (S1 ? S2 ? C)
in the proof of Lemma 8
6
Proof. Consider the real-valued function f?(t) = t? defined on t ? R. It is well known [1] that
the derivative of f? is operator antitone. Thus, Theorem 12 gives us that fgr (S) is submodular.
Hence, by applying Lemma 10, we obtain that g(S) is an approximately submodular function, with
2
?
submodularity ratio at least ?S,k
? (R ) . Also, by definition f (t) is non-negative and monotone.
Thus, using Lemma 13, we get that fgr (S) and consequently g(S) is a monotonically increasing set
function. Since g(S) is non-negative, monotone, and submodular, we can now apply Theorem 3 to
2
? (R )
obtain a (1 ? e??S,k
) approximation ratio.
5.3
Spectral variance regularizer
For a set S with covariance matrix CS , we define the spectral variance regularizer as
P|S|
? i=1 (?i (CS ) ? 1)2 . This regularizes the variance of the eigenvalues of the matrix (recall that
for an orthogonal matrix, all the eigenvalues are equal to 1) and can be shown to be diversitypromoting. For non-negativity, we add a constant 9k 2 term4 to the regularizer and define fsv (S) =
P|S|
9k 2 ? i=1 (?i (CS ) ? 1)2 . As with fde (S), we can show (proof relegated to the supplementary material) that fsv (S) is submodular, but it is not monotonically increasing in general. Hence, appealing
to Theorem 9, we obtain the following.
??
S,2k
?
Theorem 16 The set S? selected by the GLS algorithm gives a 1?e 7 2
tion guarantee for (1) using the spectral variance regularizer fsv (S).
6
multiplicative approxima-
Experiments and results
In this section we conduct experiments in different settings to validate the robustness of our spectral
regularizers. We compare our approach against two baselines: Lasso and greedy FR. We use two
different datasets for the experiments, the mnist data (http://yann.lecun.com/exdb/
mnist/) and a simulation data (for which, results are presented in the supplementary material).
The way we synthesize a regression problem out of the mnist dataset is as follows. Each image
is regarded as a feature vector (of size 784) consisting of the pixel intensities. The target vector for
the regression problem consists of the vector of labels. We only sample 1000 images out of the set,
and thus have a regression problem with X ? R1000?784 and Z ? R1000 . We then preprocess the
columns of matrix X and the target vector Z to have unit `2 -length.
We use two baselines: lasso and no-reg, the greedy FR with no regularizer. We also use four
different spectral regularizers: ld (Section 5.1, with ? = 1), ld-0.1 (Section 5.1, with ? = 0.1),
sv (Section 5.3), and gr (Section 5.2). We considered two different types of perturbations: perturbing Z and X. In order to perturb Z, we first sample a random vector ? ? R1000 , ?i ? N (0, 1). We
?
, where ? is varied in [0, 1]5 . If S is the set of features selected, then the
then create Z 0 = Z + ? k?k
unperturbed regression coefficients are defined as ? = CS?1 XST Z, and the perturbed coefficients as
?0 = CS?1 XST Z 0 . The error that we measure is k? ? ?0 k2 . Similarly, in order to perturb X, we first
sample E ? R1000?784 . Let E?i denote the ith column of E. Then, we create X 0 , the perturbed
E?i
0
version of X columnwise as X?i
= X?i + ? kE
. Here again, the perturbed regression coefficients
?i k
?1
T
are ?0 = CS0 XS0 y where CS0 = (XS0 )T XS0 and the error is measured as k? ? ?0 k2 . For our
experiments, we apply each random perturbation 5 times and then take the average error. Note that
the differential entropy of ???0 is directly given by Lemma 4; we will directly measure the quantity
on the RHS of the equation of Lemma 4.
Results. Figure 1 summarizes the result for the mnist data. For clarity of presentation, we have
only shown the results of greedy FR for monotone regularizers (ld and gr) and GLS for nonmonotone (ld-0.1, sv). We also show the results only for ? = 0.1; results for other values
of ? are similar. The way we decided on the regularization parameters ? is as follows. First we
run the lasso using a regularization path approach, and obtain a set of solutions for a range of
4
as before, we need this regularizer to be non-negative for sets of size up to 3k due to the proof of Lemma 8
Strictly speaking, normalizing ? makes it non-Gaussian, but for a high dimensional vector k?k is highly
concentrated.
5
7
0.12
0.7
1
lasso
no?reg
logdet
logdet?0.1
spec?variance
gen?rank
0.6
0.95
0.9
Regularizer value(logdet)
lasso
no?reg
logdet
logdet?0.1
spec?variance
gen?rank
0.1
0.5
Error in beta
Error in beta
0.08
0.06
0.4
0.3
0.04
0.2
0.85
0.8
0.75
0.7
0.65
0.02
0.1
0.6
0
10
20
30
40
50
60
70
80
0
10
90
20
30
50
60
70
80
0.55
10
90
lasso
no?reg
logdet
20
30
Number of features selected
1
0.9
0.99
0.8
0.7
0.6
40
50
60
70
80
90
70
80
90
Number of features selected
1
0.95
Regularizer value(gen?rank)
1
Regularizer value(spec?var)
Regularizer value(logdet?0.1)
Number of features selected
40
0.98
0.97
0.96
0.95
0.9
0.85
0.8
0.75
0.7
0.5
0.94
lasso
no?reg
logdet?0.1
0.4
10
20
30
40
50
60
Number of features selected
70
80
90
0.93
10
0.65
lasso
no?reg
spec?var
20
30
40
50
60
Number of features selected
70
80
90
10
lasso
no?reg
gen?rank
20
30
40
50
60
Number of features selected
Figure 1: All plots on mnist data. (a) Error when Z is perturbed (? = 0.1). (b) Error when X is
perturbed (? = 0.1). (c) Diversity comparison for ld. (d) Diversity comparison for ld-0.1. (e)
Diversity comparison for sv. (f) Diversity comparison for gr.
.
regularization parameter values and corresponding sparsity (k) values. For the other algorithms, we
use each of this set of sparsity values as the target number of features to be selected. We chose the
regularization constant (?) to be the maximum subject to the condition that the R2 value for that
solution should be greater than that obtained by the lasso solution with corresponding sparsity.
This ensures we are not sacrificing diversity for solution quality.
Figure 1(a) shows the errors obtained when perturbing the Z vector. As is obvious from the figure, the coefficient vector obtained by lasso is very susceptible to perturbation, and the effect of
perturbation increases with the number of features used by lasso. This indicates that as lasso
starts incorporating more features, it does not ensure that the features are diverse enough so as to be
robust to perturbation. Greedy with no regularization seems more stable than lasso but still shows
an increasing trend. On the other hand, the errors obtained by perturbing is much less for any of the
regularizers, and is only very mildly increasing with k: it does not seem to matter which regularizer
we employ. Figure 1(b) shows the error obtained when perturbing the X matrix; the same story is
true here also. In both cases, using any of the regularizers we are able to pick a set of features that
are more robust to perturbation.
Figures 1(c)- 1(f) show that our features are also more diverse than the ones obtained by both lasso
and no-reg. Since there is no one definition of diversity, in each of the plots, we take one of the
definitions of diversity value corresponding to the four regularizers we use. In order to be able to
compare, the regularizer values for each k are normalized by the maximum value possible for that
k. For each of the plots we show the values of the diversity value for solutions at different levels of
sparsity. It is obvious that we get more diverse solutions than both lasso and no-reg. The lines
corresponding to lasso or no-reg show an increasing trend because of the normalization.
7
Conclusions
In this paper we proposed submodular spectral regularizers for diverse feature selection and obtained
efficient approximation algorithms using greedy and local search algorithms. These algorithms
obtain a more diverse and noise-insensitive set of features. It would be interesting to see whether we
can design convex relaxations for such approaches, and to compare our approach with related ones
e.g. CLASH [18] that presents a general framework for merging combinatorial constraints with the
L1-norm constraint for LASSO, or with Elastic-Net that provides stability to the features selected
when groups of correlated variables are present.
8
References
[1] R. Bhatia. Matrix Analysis. Springer, 1997.
[2] J. K. Bradley, A. Kyrola, D. Bickson, and C. Guestrin. Parallel coordinate descent for l1-regularized loss
minimization. In ICML, pages 321?328, 2011.
[3] E. J. Candes, J. Romberg, and T. Tao. Stable signal recovery from incomplete and inaccurate measurements. CPAM, 59:1207?1223, 2005.
[4] A. Das. Subset Selection Algorithms for Prediction. PhD thesis, University of Southern California, 2011.
[5] A. Das and D. Kempe. Submodular meets spectral: Greedy algorithms for subset selection, sparse approximation and dictionary selection. In ICML, pages 1057?1064, 2011.
[6] G. Diekhoff. Statistics for the Social and Behavioral Sciences. Wm. C. Brown Publishers, 2002.
[7] C. Ding and H. Peng. Minimum redundancy feature selection from microarray gene expression data. In
J. Bioinform. Comput. Biol., pages 523?529, 2003.
[8] D. Donoho. For most large underdetermined systems of linear equations, the minimal 11-norm nearsolution approximates the sparsest near-solution. CPAM, 59:1207?1223, 2005.
[9] U. Feige, V. S. Mirrokni, and J. Vondrak. Maximizing non-monotone submodular functions. SIAM J.
Comput, 40(4):1133?1153, 2011.
[10] S. Friedland and S. Gaubert. Submodular spectral functions of principal submatrices of a Hermitian
matrix, extensions and applications. Linear Algebra and its Applications, 2011.
[11] A. Gilbert, S. Muthukrishnan, and M. Strauss. Approximation of functions over redundant dictionaries
using coherence. In SODA, 2003.
[12] E. Grave, G. Obozinski, and F. R. Bach. Trace Lasso: a trace norm regularization for correlated designs.
In NIPS, 2011.
[13] C. Guestrin, A. Krause, and A. Singh. Near-optimal sensor placements in Gaussian processes. In ICML,
2005.
[14] A. Gupta, A. Roth, G. Schoenebeck, and K. Talwar. Constrained non-monotone submodular maximization: Offline and secretary algorithms. In WINE, pages 246?257, 2010.
[15] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, 1999.
[16] R. A. Johnson and D. W. Wichern. Applied Multivariate Statistical Analysis. Prentice Hall, 2002.
[17] C.-W. Ko, J. Lee, and M. Queyranne. An exact algorithm for maximum entropy sampling. OR, 43(4):684?
691, 1995.
[18] A. Kyrillidis and V. Cevher. Combinatorial selection and least absolute shrinkage via the clash algorithm.
In Information Theory Proceedings (ISIT), 2012 IEEE International Symposium on, pages 2216 ?2220,
july 2012.
[19] A. Miller. Subset Selection in Regression. Chapman and Hall, second edition, 2002.
[20] R. Tibshirani. Regression shrinkage and selection via the Lasso. JRSS, 58:267?288, 1996.
[21] J. Tropp. Greed is good: Algorithmic results for sparse approximation. IEEE Trans. Information Theory,
50:2231?2242, 2004.
[22] J. Tropp. Just relax: Convex programming methods for identifying sparse signals. IEEE TOIT, 51:1030?
1051, 2006.
[23] L. Yu. Redundancy based feature selection for microarray data. In SIGKDD, pages 737?742, 2004.
[24] T. Zhang. Adaptive forward-backward greedy algorithm for sparse learning with linear models. In NIPS,
2008.
[25] S. Zhou. Thresholding procedures for high dimensional variable selection and statistical estimation. In
NIPS, 2009.
9
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4,076 | 469 | Statistical Reliability of a Blowfly
Movement-Sensitive Neuron
Rob de Ruyter van Steveninck ..
Biophysics Group,
Rijksuniversiteit Groningen,
Groningen, The Netherlands
William Bialek
NEe Research Institute
4 Independence Way,
Princeton, N J 08540
Abstract
We develop a model-independent method for characterizing the reliability
of neural responses to brief stimuli. This approach allows us to measure
the discriminability of similar stimuli, based on the real-time response of a
single neuron. Neurophysiological data were obtained from a movementsensitive neuron (HI) in the visual system of the blowfly Calliphom erythrocephala. Furthermore, recordings were made from blowfly photoreceptor cells to quantify the signal to noise ratios in the peripheral visual
system. As photoreceptors form the input to the visual system, the reliability of their signals ultimately determines the reliability of any visual
discrimination task. For the case of movement detection, this limit can
be computed, and compared to the HI neuron's reliability. Under favorable conditions, the performance of the HI neuron closely approaches the
theoretical limit, which means that under these conditions the nervous
system adds little noise in the process of computing movement from the
correlations of signals in the photoreceptor array.
1
INTRODUCTION
In the 1940s and 50s, several investigators realized that understanding the reliability of computation in the nervous system posed significant theoretical challenges.
Attempts to perform reliable computations with the available electronic computers
?present address: University Hospital Groningen, Dept. of Audiology, POB 30.001, NL
9700RB Groningen, The Netherlands
27
28
de Ruyter van Steveninck and Bialek
certainly posed serious practical problems, and the possibility that the problems of
natural and artificial computing are related was explored. Guided by the practical
problems of electronic computing, von Neumann (1956) formulated the theoretical problem of "reliable computation with unreliable components". Many authors
seem to take as self-evident the claim that this is a problem faced by the nervous
system as well, and indeed the possibility that the brain may implement novel solutions to this problem has been at least a partial stimulus for much recent research.
The qualitative picture adopted in this approach is of the nervous system as a
highly interconnected network of rather noisy cells, in which meaningful signals are
represented only by large numbers of neural firing events averaged over numerous
redundant neurons. Neurophysiological experiments seem to support this view: If
the same stimulus is presented repeatedly to a sensory system, the responses of an
individual afferent neuron differ for each presentation. This apparently has led to
a widespread belief that neurons are inherently noisy, and ideas of redundancy and
averaging pervade much of the literature. Significant objections to this view have
been raised, however (c/. Bullock 1970).
As emphasized by Bullock (Ioc.cit), the issue of reliability of the nervous system is a
quantitative one. Thus, the first problem that should be overcome is to find a way
for its measurement. This paper focuses on a restricted, but basic question, namely
the reliability of a single neuron, much in the spirit of previous work (cf. Barlow and
Levick 1969, Levick et al. 1983, Tolhurst at al. 1983, Parker and Hawken 1985).
Here the methods of analysis used by these authors are extended in an attempt to
describe the neuron's reliability in a way that is as model-independent as possible.
The second-conceptually more difficult-problem, is summarized cogently in Bullock's words, "how reliable is reliable?". Just quantifying reliability is not enough,
and the qualitative question of whether redundancy, averaging, multiplexing, or yet
more exotic solutions to von Neumann's problem are relevant to the operation of
the nervous system hinges on a quantitative comparison of reliability at the level
of single cells with the reliability for the whole system. Broadly speaking, there
are two ways to make such a comparison: one can compare the performance of
the single cell either with the output or with the input of the whole system. As
to the first possibility, if a single cell responds to a certain stimulus as reliably as
the animal does in a behavioral experiment, it is difficult to imagine why multiple
redundant neurons should be used to encode the same stimulus. Alternatively, if
the reliability of a single neuron were to approach the limits set by the sensory
periphery, there seems to be little purpose for the nervous system to use functional
duplicates of such a cell, and the key theoretical problem would be to understand
how such optimal processing is implemented. Here we will use the latter approach.
We first quantify the reliability of response of HI, a wide-field movement-sensitive
neuron in the blowfly visual system. The method consists essentially of a direct
application of signal detection theory to trains of neural impulses generated by
brief stimuli, using methods familiar from psychophysics to quantify discriminability. Next we characterize signal transfer and noise in the sensory periphery-the
photoreceptor cells of the compound eye-and we compare the reliability of information coded in HI with the total amount of sensory information available at the
input.
Statistical Reliability of a Blowfly Movement-Sensitive Neuron
2
PREPARATION, STIMULATION AND RECORDING
Experiments were performed on female wild-type blowfly Calliphora erythrocephala.
Spikes from HI were recorded extracellularly with a tungsten microelectrode, their
arrival times being digitized with 50 {ts resolution. The fly watched a binary
random-bar pattern (bar width 0.029? visual angle, total size (30.5?)2) displayed on
a CRT. Movement steps of 16 different sizes (integer multiples of 0.12?) were generated by custom-built electronics, and presented at 200 ms intervals in the neuron's
preferred direction. The effective duration of the experiment was 11 hours, during
which time about 10 6 spikes were recorded over 12552 presentations of the 16-step
stimulus sequence.
Photoreceptor cells were recorded intracellularly while stimulated by a spatially
homogeneous field, generated on the same CRT that was used for the HI experiments. The CRT's intensity was modulated by a binary pseudo-random waveform,
time sampled at 1 ms. The responses to 100 stimulus periods were averaged, and
the cell's transfer function was obtained by computing the ratio of the Fourier transform of the averaged response to that of the stimulus signal. The cell's noise power
spectrum was obtained by averaging the power spectra of the 100 traces of the
individual responses with the average response subtracted.
3
3.1
DATA ANALYSIS
REPRESENTATION OF STIMULUS AND RESPONSE
A single movement stimulus consisted of a sudden small displacement of a wide-field
pattern. Steps of varying sizes were presented at regular time-intervals, long enough
to ensure that responses to successive stimuli were independent. In the analysis we
consider the stimulus to be a point event in time, parametrized by its step size Ct.
The neuron's signal is treated as a stochastic point process, the parameters of which
depend on the stimulus. Its statistical behavior is described by the conditional
probability P(rICt) of finding a response r, given that a step of size Ct was presented.
From the experimental data we estimate P(rICt) for each step size separately. To
2 ms.
represent a single response r, time is divided in discrete bins of width ~t
Then r is described by a firing pattern, which is just a vector q [qO, ql, .. ] of binary
digits qk(k = 0, n - 1), where qk = 1 and qk = 0 respectively signify the presence
or the absence of a spike in time bin k (cf. Eckhorn and Popel 1974). No response
is found within a latency time t'at=15 ms after stimulus presentation; spikes fired
within this interval are due to spontaneous activity and are excluded from analysis,
so k = 0 corresponds to 15 ms after stimulus presentation.
=
=
The probability distribution of firing patterns, P(qICt), is estimated by counting the
number of occurrences of each realization of q for a large number of presentations of
Ct. This distribution is described by a tree which results from ordering all recorded
firing patterns according to their binary representation, earlier times corresponding
to more-significant bits. Graphical representations of two such trees are shown ill
Fig. 1. In constructing a tree we thus perform two operations on the raw spike
data: first, individual response patterns are represented in discrete time bins ~t,
and second, a permutation is performed on the set of discretized patterns to order
29
30
de Ruyter van Steveninck and Bialek
them according to their binary representation. No additional assumptions are made
about the way the signal is encoded by the neuron. This approach should therefore
be quite powerful in revealing any subtle" hidden code" that the neuron might use.
As the number of branches in the tree grows exponentially with the number of time
bins n, many presentations are needed to describe the tree over a reasonable time
interval, and here we use n 13.
.
=
3.2
COMPUTATION OF DISCRIMINABILITY
To quantify the performance of the neuron, we compute the discriminability of two
nearly equal stimuli al and a2, based on the difference in neural response statistics
described by P{rlaI) and P{rl(2). The probability of correct decisions is maximized
if one uses a maximum likelihood decision rule, so that in the case of equal prior
probabilities the outcome is al if P{robslal) > P{robsl(2), and vice versa. On
average, the probability of correctly identifying step al is then:
Pc{aJ)
=L
P{rlat} . H[P(rlat) - P{rl(2)],
(1)
{r}
where H{.) is the Heaviside step function and the summation is over the set of all
possible responses {r}. An interchange of indices 1 and 2 in this expression yields
the formula for correct identification of a2. The probability of making correct
judgements over an entire experiment in which al and a2 are equiprobable is then
simply Pc(al, (2) = [Pc{at) + Pc(a2)]/2, which from now on will be referred to as
Pc.
This analysis is essentially that for a "two-alternative forced-choice" psychophysical
experiment. For convenience we convert Pc into the discriminability parameter d',
familiar from psychophysics (Green and Swets 1966), which is the signal-to-noise
ratio (difference in mean divided by the standard deviation) in the equivalent equalvariance Gaussian decision problem.
=
Using the firing-pattern representation, r
q, and computing d' for successive
subvectors of q with elements m
0, .. , k and k
0, .. , n - 1, we compute Pc for
different values of k and from that obtain d'{k), the discrimillability as a function
of time.
=
3.3
=
THEORETICAL LIMITS TO DISCRIMINATION
For the simple stimuli used here it is relatively easy to determine the theoretical limit
to discrimination based on the photoreceptor signal quality. For the computation
of this limit we use Reichardt's (1957) correlation model of movement detection.
This model has been very successful in describing a wide variety of phenomena
in biological movement detection, both in fly (Reichardt and Poggio 1976), and
in humans (van Santen and Sperling 1984). Also, correlation-like operations can
be proved to be optimal for the extraction of movement information at low signal
to noise ratio (Bialek 1990). The measured signal transfer of the photoreceptors,
combined with the known geometry of the stimulus and the optics of the visual
system determine the signal input to the model. The noise input is taken directly
Statistical Reliability of a Blowfly Movement-Sensitive Neuron
1.0
1.0
0.2G?
0.36?
0.8
~
0.8
0.6
~
:sIV
-------
0.6
~
,Q
0
,Q
b. 0.4
b. 0.4
0.2
0.2
0.0
20
30
time (ms)
40
0.0
--~
20
30
40
time (ms)
Figure 1: Representation of the firing pattern distributions for steps of 0.24? and
0.36?. Here only 11 time bins are shown.
from the measured photoreceptor noise power spectrum. Details of this computation
are given in de Ruyter van Steveninck (1986).
3.4
ERROR ANALYSIS AND DATA REQUIREMENTS
The effects of the approximation due to time-discretization can be assessed by varying the binwidth. It turns out that the results do not change appreciably if the bins
are made smaller than 2 ms. Furthermore, if the analysis is to make sense, stationarity is required, i.e. the probability distribution from which responses to a certain
stimulus are drawn should be invariant over the course of the experiment. Finally,
the distributions, being computed from a finite sample of responses, are subject to
statistical error. The statistical error in the final result was estimated by partitioning the data and working out the values of Pc for these partitions separately. The
statistical variations in Pc were of the order of 0.01 in the most interesting region
of values of Pc, i.e. from 0.6 to 0.9. This results in a typical statistical error of 0.05
in d'. In addition, this analysis revealed no significant trends with time, so we may
assume stationarity of the preparation.
4
4.1
RESULTS
STEP SIZE DISCRIMINATION BY THE HI NEURON
Although 16 different step sizes were used, we limit the presentation here to steps
of 0.24? and 0.36?; binary trees representing the two firing-pattern distributions are
shown in Fig. 1. The first time bin describes the probabilities of two possible events:
either a spike was fired (black) or not (white), and these two probabilities add up
to unity. The second time bin describes the four possible combinations of finding
31
32
de Ruyter van Steveninck and Bialek
15r-----~----~----~----~
3. 0 r---..----......----.....-----r-----r--r--,
1
1
/
/
/
/
10
2.0
1
/
I
/
/
I
/
predICted
5
/
-
1
1 --
1.0
predicted
and shifted
I
I
50
100
150
observation window (ms)
200
20
30
40
time (ms)
Figure 2: Left: Discrimination performance of an ideal movement detector. See
text for further details. Right: comparison of the theoretical and the measured
values of d'et). Fat line: measured performance of Hl. Thin solid line: predicted
performance, taken from the left figure. Dashed line: the same curve shifted by 5
ms to account for latency time in the pathway from photoreceptor to HI. This time
interval was determined independently with powerful movement stimuli.
or not finding a spike in bin 2 combined with finding or not finding a spike in bin
1, and so on. The figure shows that the probability of firing a spike in time bin 1
is slightly higher for the larger step. From above we compute Pc, the probability of
correct identification, in a task where the choice is between step sizes of 0.240 and
0.36 0 with equal prior probabilities. The decision rule is simple: if a spike is fired in
bin 1, choose the larger, otherwise choose the smaller step. In the same fashion we
apply this procedure to the following time bin, with four response categories and so
on. The value of d' computed from Pc for this step size pair as a function of time
is given by the fat line at the right in Fig. 2.
4.2
LIMITS SET BY PHOTORECEPTOR SIGNALS
Figure 2 (left) shows the limit to movement detection computed for an array of 2650
Reichardt correlators stimulated with a step size difference of 0.12 0 , conforming to
the experimental conditions. Comparing the performance of HI to this result (the
fat and the dashed lines in Fig. 2, right), we see that the neuron follows the limit
set by the sensory periphery from about 18 to 28 ms after stimulus presentation.
So, for this time window the randomness of HI's response is determined primarily
by photoreceptor noise. Up to about 20 Hz, the photoreceptor signal-to-noise ratio
closely approached the limit set by the random arrival of photons at the photoreceptors at a rate of about 104 effective conversions/so Hence most of the randomness
in the spike train was caused by photon shot noise.
Statistical Reliability of a Blowfly Movement-Sensitive Neuron
5
DISCUSSION
The approach presented here gives us estimates for the reliability of a single neuron
in a well-defined, though restricted experimental context. In addition the theoretical
limits to the reliability of movement-detection are computed. Comparing these two
results we find that HI in these conditions uses essentially all of the movement
information available over a 10 ms time interval. Further analysis shows that this
information is essentially contained in the time of firing of the first spike. The
plateau in the measured d'(t) between 28 and 34 ms presumably results from effects
of refractoriness, and the subsequent slight rise is due to firing of a second spike.
Thus, a step size difference of 0.12? can be discriminated with d' close to unity,
using the timing information of just one spike from one neuron. For the blowfly
visual system this angular difference is of the order of one-tenth of the photoreceptor
spacing, well within the hyperacuity regime (cf. Parker and Hawken 1985).
It should not be too surprising that the neuron performs well only over a short
time interval and does not reach the values for d' computed from the model at large
delays (Fig. 2, left): The experimental stimulus is not very natural, and in real-life
conditions the fly is likely to see movement changing continuously. (Methods for analyzing responses to continuous movement are treated in de Ruyter van Steveninck
and Bialek 1988, and in Bialek et al. 1991.) In such circumstances it might be
better not to wait very long to get an accurate estimate of the stimulus at one
point in time, but rather to update rough estimates as fast as possible. This would
favor a coding principle where successive spikes code independent events, which
may explain that the plateau in the measured d'{t) starts at about the point where
the computed d'{t) has maximal slope. Such a view is supported by behavioral
evidence: A chasing fly tracks the leading fly with a delay of about 30 ms (Land
and Collett 1974), corresponding to the time at which the measured d'{t) levels off.
In conclusion we can say that in the experiment, for a limited time window the
neuron effectively uses all information available at the sensory periphery. Peripheral noise is in turn determined by photon shot noise so that the reliability of HI's
output is set by the physics of its inputs. There is no neuro-anatomical or neurophysiological evidence for massive redundancy in arthropod nervous systems. More
specifically, for the fly visual system, it is known that HI is unique in its combination of visual field and preferred direction of movement (Hausen 1982), and from
the results presented here we may begin to understand why: It just makes little
sense to use functional duplicates of any neuron that performs almost perfectly
when compared to the noise levels inherently present in the stimulus. It remains to
be seen to what extent this conclusion can be generalized, but one should at least
be cautious in interpreting the variability of response of a single neuron in terms of
noise generated by the nervous system itself.
References
Barlow HB, Levick WR (1969) Three factors limiting the reliable detection of light
by retinal ganglion cells of the cat. J Physiol 200:1-24.
Bialek W (1990) Theoretical physics meets experimental neurobiology. In Jen E
(ed.) 1989 Lectures in Complex Systems, SFI Studies in the Sciences of Complexity,
33
34
de Ruyter van Steveninck and Bialek
Lect. Vol. II, pp. 513-595. Addison-Wesley, Menlo Park CA.
Bialek W, Rieke F, de Ruyter van Steveninck RR, Warland D (1991) Reading a
neural code. Science 252:1854-1857.
Bullock TH (1970) The reliability of neurons. J Gen Physiol 55:565-584.
Eckhorn R, Popel B (1974) Rigorous and extended application of information theory
to the afferent visual system of the cat. I Basic concepts. Kybernetik 16:191-200.
Green DM, Swets JA (1966) Signal detection theory and psychophysics. Wiley, New
York.
Hausen K (1982) Motion sensitive interneurons in the optomotor system of the fly.
I. The horizontal cells: Structure and signals. Bioi Cybern 45:143-156.
Land MF, Collett TS (1974) Chasing behaviour of houseflies (Fannia canicularis).
A description and analysis. J Comp Physiol 89:331-357.
Levick WR, Thibos LN, Cohn TE, Catanzaro D, Barlow HB (1983) Performance
of cat retinal ganglion cells at low light levels. J Gen Physiol 82:405-426.
Neumann J von (1956) Probabilistic logics and the synthesis of reliable organisms
from unreliable components. In Shannon CE and McCarthy J (eds.) Automata
Studies, Princeton University Press, Princeton NJ, 43-98.
Parker A, Hawken M (1985) Capabilities of monkey cortical cells in spatialresolution tasks. J Opt Soc Am A2:1101-1114.
Reichardt W (1957) Autokorrelations-Auswertung als Funktionsprinzip des Zentralnervensystems. Z Naturf 12b:448-457.
Reichardt W, Poggio T (1976) Visual control of orientation behaviour in the fly,
Part I. A quantitative analysis. Q Rev Biophys 9:311-375.
de Ruyter van Steveninck RR (1986) Real-time performance of a movement-sensitive
neuron in the blowfly visual system. Thesis, Rijksuniversiteit Groningen, the
Netherlands.
de Ruyter van Steveninck RR, Bialek W (1988) Real-time performance of a
movement-sensitive neuron in the blowfly visual system: coding and information
transfer in short spike sequences. Proc R Soc Lond B 234: 379-414.
van Santen JPH, Sperling G (1984) Temporal covariance model of human motion
perception. J Opt Soc Am A1:451-473.
Tolhurst DJ, Movshon JA, Dean AF (1983) The statistical reliability of signals in
single neurons in cat and monkey visual cortex. Vision Res 23: 775-785.
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4,077 | 4,690 | Towards a learning-theoretic analysis of
spike-timing dependent plasticity
David Balduzzi
MPI for Intelligent Systems, T?ubingen, Germany
ETH Zurich, Switzerland
[email protected]
Michel Besserve
MPI for Intelligent Systems and MPI for Biological Cybernetics
T?ubingen, Germany
[email protected]
Abstract
This paper suggests a learning-theoretic perspective on how synaptic plasticity
benefits global brain functioning. We introduce a model, the selectron, that (i)
arises as the fast time constant limit of leaky integrate-and-fire neurons equipped
with spiking timing dependent plasticity (STDP) and (ii) is amenable to theoretical
analysis. We show that the selectron encodes reward estimates into spikes and that
an error bound on spikes is controlled by a spiking margin and the sum of synaptic
weights. Moreover, the efficacy of spikes (their usefulness to other reward maximizing selectrons) also depends on total synaptic strength. Finally, based on our
analysis, we propose a regularized version of STDP, and show the regularization
improves the robustness of neuronal learning when faced with multiple stimuli.
1
Introduction
Finding principles underlying learning in neural networks is an important problem for both artificial
and biological networks. An elegant suggestion is that global objective functions may be optimized
during learning [1]. For biological networks however, the currently known neural plasticity mechanisms use a very restricted set of data ? largely consisting of spikes and diffuse neuromodulatory
signals. How a global optimization procedure could be implemented at the neuronal (cellular) level
is thus a difficult problem.
A successful approach to this question has been Rosenblatt?s perceptron [2] and its extension to
multilayer perceptrons via backpropagation [3]. Similarly, (restricted) Boltzmann machines, constructed from simple stochastic units, have provided a remarkably powerful approach to organizing
distributed optimization across many layers [4]. By contrast, although there has been significant
progress in developing and understanding more biologically realistic models of neuronal learning [5?10], these do not match the performance of simpler, more analytically and computationally
tractable models in learning tasks.
Overview. This paper constructs a bridge from biologically realistic to analytically tractable models. The selectron is a model derived from leaky integrate and fire neurons equipped with spiketiming dependent plasticity that is amenable to learning-theoretic analysis. Our aim is to extract
some of the principles implicit in STDP by thoroughly investigating a limit case.
Section ?2 introduces the selectron. We state a constrained reward maximization problem which
implies that selectrons encode empirical reward estimates into spikes. Our first result, section ?3,
1
is that the selectron arises as the fast time constant limit of well-established models of neuronal
spiking and plasticity, suggesting that cortical neurons may also be encoding reward estimates into
their spiketrains.
Two important questions immediately arise. First, what guarantees can be provided on spikes being
reliable predictors of global (neuromodulatory) outcomes? Second, what guarantees can be provided
on the usefulness of spikes to other neurons? Sections ?4 and ?5 answer these questions by providing
an upper bound on a suitably defined 0/1 loss and a lower bound on the efficacy of a selectron?s
spikes, measured in terms of its contribution to the expected reward of a downstream selectron.
Both bounds are controlled by the sum of synaptic weights kwk1 , thereby justifying the constraint
introduced in ?2. Finally, motivated by our analysis, ?6 introduces a regularized STDP rule and
shows that it learns more robustly than classical STDP. ?7 concludes the paper. Proofs of theorems
are provided in the supplementary material.
Related work. Spike-timing dependent plasticity and its implications for the neural code have
been intensively studied in recent years. The work closest in spirit to our own is Seung?s ?hedonistic?
synapses, which seek to increase average reward [6]. Our work provides guarantees on the finite
sample behavior of a discrete-time analog of hedonistic neurons. Another related line of research
derives from the information bottleneck method [9, 11] which provides an alternate constraint to the
one considered here. An information-theoretic perspective on synaptic homeostasis and metabolic
cost, complementing the results in this paper, can be found in [12, 13]. Simulations combining
synaptic renormalization with burst-STDP can be found in [14].
Important aspects of plasticity that we have not considered here are properties specific to continuoustime models, such as STDP?s behavior as a temporal filter [15], and also issues related to convergence [8, 10].
The learning-theoretic properties of neural networks have been intensively studied, mostly focusing
on perceptrons, see for example [16]. A non-biologically motivated ?large-margin? analog of the
perceptron was proposed in [17].
2
The selectron
We introduce the selectron, which can be considered a biologically motivated adaptation of the
perceptron, see ?3. The mechanism governing whether or not the selectron spikes is a Heaviside
function acting on a weighted sum of synaptic inputs; our contribution is to propose a new reward
function and corresponding learning rule.
Let us establish some notation. Let X denote the set of N -dimensional {0, 1}-valued vectors forming synaptic inputs to a selectron, and Y = {0, 1} the set of outputs. A selectron spikes according
to
?
1 if z > 0
y = fw (x) := H (w| x #) , where H(z) :=
(1)
0 else
is the Heaviside function and w is a [0, 1] ? R valued N -vector specifying the selectron?s synaptic
weights. Let P (x) denote the probability of input x arising.
To model the neuromodulatory system we introduce random variable ? : X ! { 1, 0, +1}, where
positive values correspond to desirable outcomes, negative to undesirable and zero to neutral. Let
P (?|x) denote the probability of the release of neuromodulatory signal subsequent to input x.
Definition 1. Define reward function
R(x, fw , ?) =
?(x)
|{z}
neuromodulators
? (w| x #) ? fw (x) =
| {z } | {z }
margin
selectivity
?
?(x) ? (w| x
0
#)
if y = 1
else.
(2)
The reward consists in three components. The first term is the neuromodulatory signal, which acts as
a supervisor. The second term is the total current w| x minus the threshold #. It is analogous to the
margin in support vector machines or boosting algorithms, see section ?4 for a precise formulation.
2
The third term gates rewards according to whether or not the selectron spikes. The reward is thus
selected1 : neuromodulatory signals are ignored by the selectron?s reward function when it does not
spike, enabling specialization.
Constrained reward maximization. The selectron solves the following optimization problem:
bn :=
maximize: R
w
n
X
i=1
?(x(i) ) ? (w| x(i)
#) ? fw (x(i) )
(3)
subject to: kwk1 ? ! for some ! > 0.
Remark 1 (spikes encode rewards).
Optimization problem (3) ensures that selectrons spike for inputs that, on the basis of their empirical
sample, reliably lead to neuromodulatory rewards. Thus, spikes encode expectations about rewards.
The constraint is motivated by the discussion after Theorem 1 and the analysis in ?4 and ?5. We
postpone discussion of how to impose the constraint to ?6, and focus on reward maximization here.
The reward maximization problem cannot be solved analytically in general. However, it is possible
to use an iterative approach. Although fw (x) is not continuous, the reward function is a continuous
function of w and is differentiable everywhere except for the ?corner? where w| x # = 0. We
therefore apply gradient ascent by computing the derivative of (3) with respect to synaptic weights
to obtain online learning rule
?
? ? ?(x) if xj = 1 and y = 1
wj = ? ? ?(x) ? xj ? fw (x) =
(4)
0
else
where update factor ? controls the learning rate.
The learning rule is selective: regardless of the neuromodulatory signal, synapse wjk is updated
only if there is both an input xj = 1 and output spike y = fw (x) = 1.
The selectron is not guaranteed to find a global optimum. It is prone to initial condition dependent
local optima because rewards depend on output spikes in learning rule (4). Although this is an
undesirable property for an isolated learner, it is less important, and perhaps even advantageous, in
large populations where it encourages specialization.
Remark 2 (unsupervised setting).
Define the unsupervised setting by ?(x) = 1 for all x. The reward function reduces to R(x, fw ) =
(w| x #) ? fw (x). Without the constraint synapses will saturate. Imposing the constraint yields a
more interesting solution where the selectron finds a weight vector summing to ! which balances (i)
frequent spikes and (ii) high margins.
Theorem 1 (Controlling the frequency of spikes).
Assuming synaptic inputs are i.i.d. Bernoulli variables with P (spike) = p, then
?
?2
?
?
? ! ?2
kwk1
P fw (x) = 1 ? p ?
?p?
.
#
#
The Bernoulli regime is the discrete-time analog of the homogeneous Poisson setting used to prove
convergence of reward-modulated STDP in [8]. Interestingly, in this setting the constraint provides
a lever for controlling (lower bounding) rewards per spike
n
o
reward per spike =
b
R
P (fw (x) = 1)
c1 ?
b
R
.
!2
If inputs are not Bernoulli i.i.d., then P (y = 1) and ! still covary, although the precise relationship is
more difficult to quantify. Although i.i.d. inputs are unrealistic, note that recent neurophysiological
evidence suggests neuronal firing ? even of nearby neurons ? is uncorrelated [18].
1
The name ?selectron? was chosen to emphasize this selective aspect.
3
3
Relation to leaky integrate-and-fire neurons equipped with STDP
The literature contains an enormous variety of neuronal models, which vary dramatically in sophistication and the extent to which they incorporate the the details of the underlying biochemical
processes. Similarly, there is a large menagerie of models of synaptic plasticity [19]. We consider
two well-established models: Gerstner?s Spike Response Model (SRM) which generalizes leaky
integrate-and-fire neurons [20] and the original spike-timing dependent plasticity learning rule proposed by Song et al [5], and show that the selectron arises in the fast time constant limit of the two
models.
First let us recall the SRM. Suppose neuron nk last outputted a spike at time tk and receives input
spikes at times tj from neuron nj . Neuron nk spikes or according to the Heaviside function applied
to the membrane potential Mw :
X
fw (t) = H (Mw (t) #) where Mw (t) = ?(t tk ) +
wjk ? ?(t tj ) at time t tk .
tj ?t
Input and output spikes add
? ?t t?
?t t?
j
j
?(t tj ) = K e ?m
e ?s
and ?(t
?
?
?
tk t
tk ) = # K 1 e ? m
? ?t t?
k
K2 e ?m
e
?
tk t
?s
??
to the membrane potential for tj ? t and tk ? t respectively. Here ?m and ?s are the membrane and
synapse time constants.
The original STDP update rule [5] is
wjk =
8
<
:
?+ ? e
?t
? ?e
j tk
?+
?t
?
k tj
?
?
if tj ? tk
else
(5)
where ?+ and ? are time constants. STDP potentiates input synapses that spike prior to output
spikes and depotentiates input synapses that spike subsequent to output spikes.
Theorem 2 (the selectron is the fast time constant limit of SRM + STDP).
In the fast time constant limit, lim?? ! 0, the SRM transforms into a selectron with
?
?
X
fw (t) = H Mw (t) #
where Mw =
wjk ? tk (t).
{j|tj
tk }
Moreover, STDP transforms into learning rule (4) in the unsupervised setting with ?(x) = 1 for all
x. Finally, STDP arises as gradient ascent on a reward function whose limit is the unsupervised
setting of reward function (2).
Theorem 2 shows that STDP implicitly maximizes a time-discounted analog of the reward function
in (3). We expect many models of reward-modulated synaptic plasticity to be analytically tractable
in the fast time constant limit. An important property shared by STDP and the selectron is that
synaptic (de)potentiation is gated by output spikes, see ?A.1 for a comparison with the perceptron
which does not gate synaptic learning
4
An error bound
Maximizing reward function (3) implies that selectrons encode reward estimates into their spikes.
Indeed, it recursively justifies incorporating spikes into the reward function via the margin (w| x
#), which only makes sense if upstream spikes predict reward. However, in a large system where
estimates pile on top of each other there is a tendency to overfit, leading to poor generalizations [21].
It is therefore crucial to provide guarantees on the quality of spikes as estimators.
Boosting algorithms, where the outputs of many weak learners are aggregated into a classifier [22],
are remarkably resistant to overfitting as the number of learners increases [23]. Cortical learning may
be analogous to boosting: individual neurons have access to a tiny fraction of the total brain state,
and so are weak learners; and in the fast time constant limit, neurons are essentially aggregators.
4
We sharpen the analogy using the selectron. As a first step towards understanding how the cortex
combats overfitting, we adapt a theorem developed to explain the effectiveness of boosting [24]. The
goal is to show how the margin and constraint on synaptic weights improve error bounds.
Definition 2. A selectron incurs a 0/1 loss if a spike is followed by negative neuromodulatory
feedback
?
1 if y = 1 and ?(x) = 1
l(x, fw , ?) =
=
(6)
fw (x)??(x)
0 else.
The 0/1 loss fails to take the estimates (spikes) of other selectrons into account and is difficult to
optimize, so we also introduce the hinge loss:
?
?
?
x if x 0
h? (x, fw , ?) := ? (w| x #) ? ?(x) ? fw (x), where (x)+ :=
(7)
0 else.
+
Note that l ? h? for all ? 1. Parameter ? controls the saturation point, beyond which the size of
the margin makes no difference to h? .
An alternate 0/1 loss2 penalizes a selectron if it (i) fires when it shouldn?t, i.e. when ?(x) = 1
or (ii) does not fire when it should, i.e. when ?(x) = 1. However, since the cortex contains
many neurons and spiking is metabolically expensive [25], we propose a conservative loss that only
penalizes errors of commission (?first, do no harm?) and does not penalize specialization.
Theorem 3 (spike error bound).
Suppose each selectron has ? N synapses. For any selectron nk , let S k = {nk } [ {nj : nj ! nk }
denote a 2-layer feedforward subnetwork. For all ? 1, with probability at least 1
,
p
?
? 1 X ? (i)
8(N + 1) log(n + 1) + 1
p
E l(x, fw , ?) ?
h x , fw , ?(x(i) ) +! ? 2B ?
n
n
|
{z
}
|
{z
}
i
|
{z
}
hinge loss
0/1 loss
+ 2B ?
capacity term
s
2 log 2
n
| {z }
where B = ? + !
#.
confidence term
Remark 3 (theoretical justification for maximizing margin and constraining kwk1 ).
The theorem shows how subsets of distributed systems can avoid overfitting. First, it demonstrates
the importance of maximizing the margin (i.e. the empirical reward). Second, it shows the capacity
term depends on the number of synapses N and the constraint ! on synaptic weights, rather than
the capacity of S k ? which can be very large.
The hinge loss is difficult to optimize directly since gating with output spikes fw (x) renders it
discontinuous. However, in the Bernoulli regime, Theorem 1 implies the bound in Theorem 3 can
be rewritten as
?
?
!2
E l(x, fw , ?) ? p? 2
#
bn x(i) , fw , ?(x(i) ) + ! ? capacity term + confidence term
R
(8)
and so ! again provides the lever required to control the 0/1 loss. The constraint kwk1 ? ! is best
imposed offline, see ?6.
5
A bound on the efficacy of inter-neuronal communication
Even if a neuron?s spikes perfectly predict positive neuromodulatory signals, the spikes only matter
to the extent they affect other neurons in cortex. Spikes are produced for neurons by neurons. It is
therefore crucial to provide guarantees on the usefulness of spikes.
In this section we quantify the effect of one selectron?s spikes on another selectron?s expected reward. We demonstrate a lower bound on efficacy and discuss its consequences.
2
See ?A.5 for an error bound.
5
Definition 3. The efficacy of spikes from selectron nj on selectron nk is
Rk
E[Rk |xj = 1]
:=
xj
1
E[Rk |xj = 0]
,
0
i.e. the expected contribution of spikes from selectron nj to selectron nk ?s expected reward, relative
to not spiking. The notation is intended to suggest an analogy with differentiation ? the infinitesimal
difference made by spikes on a single synapse.
Efficacy is zero if E[Rk |xj = 1] = E[Rk |xj = 0]. In other words, if spikes from nj make no
difference to the expected reward of nk .
The following theorem relies on the assumption that the average contribution of neuromodulators is
higher after nj spikes than after it does not spike (i.e. upstream spikes predict reward), see ?A.6 for
precise statement. When the assumption is false the synapse wjk should be pruned.
Theorem 4 (spike efficacy bound).
Let pj := E[Y j ] denote the frequency of spikes from neuron nj . The efficacy of nj ?s spikes on nk is
lower bounded by
h
i
h
i
j k
j |
k
j |
2E
Y
Y
?
(wC
)
x
#
E
Y
?
(wC
)
x
#
k
j k
R
wj ? E[Y Y ]
c2 ?
+
(9)
xj
pj
pj (1 pj )
1 pj
|{z}
|
{z
}
|
{z
} |
{z
}
efficacy
wj -weighted co-spike frequency
co-spike frequency
nk spike frequency
where c2 is described in ?A.6 and wijC := wi if i 6= j and 0 if i = j.
The efficacy guarantee is interpreted as follows. First, the guarantee improves as co-spiking by nj
and nk increases. However, the denominators imply that increasing the frequency of nj ?s spikes
worsens the guarantee, insofar as nj is not correlated with nk . Similarly, from the third term,
increasing nk ?s spikes worsens the guarantee if they do not correlate with nj .
An immediate corollary of Theorem 4 is that Hebbian learning rules, such as STDP and the selectron
learning rule (4), improve the efficacy of spikes. However, it also shows that naively increasing the
frequency of spikes carries a cost. Neurons therefore face a tradeoff. In fact, in the Bernoulli regime,
Theorem 1 implies (9) can be rewritten as
h
i p ? ! 2 ? (! #)
Rk
wj
2
c2 ?
? E[Y j Y k ] +
E Y j Y k ? (wjC )| x #
,
(10)
xj
p
p(1 p)
(1 p)#2
so the constraint ! on synaptic strength can be used as a lever to improve guarantees on efficacy.
Remark 4 (efficacy improved by pruning weak synapses).
The 1st term in (9) suggests that pruning weak synapses increases the efficacy of spikes, and so may
aid learning in populations of selectrons or neurons.
6
Experiments
Cortical neurons are constantly exposed to different input patterns as organisms engage in different
activities. It is therefore important that what neurons learn is robust to changing inputs [26, 27]. In
this section, as proof of principle, we investigate a simple tweak of classical STDP involving offline
regularization. We show that it improves robustness when neurons are exposed to more than one
pattern.
Observe that regularizing optimization problem (3) yields
maximize:
w
learning rule:
n
X
R x(i) , fw , ?(x(i) )
i=1
2
wj = ? ? ?(x) ? xj ? fw (x)
(kwk1
!)2
? kwk1
(11)
! ? wj
(12)
incorporates synaptic renormalization directly into the update. However, (12) requires continuously
re-evaluating the sum of synaptic weights. We therefore decouple learning into an online reward
maximization phase and an offline regularization phase which resets the synaptic weights.
6
A similar decoupling may occur in cortex. It has recently been proposed that a function of NREM
sleep may be to regulate synaptic weights [28]. Indeed, neurophysiological evidence suggests that
average cortical firing rates increase during wakefulness and decrease during sleep, possibly reflecting synaptic strengths [29, 30]. Experimental evidence also points to a net increase in dendritic
spines (synapses) during waking and a net decrease during sleep [31].
Setup. We trained a neuron on a random input pattern for 10s to 87% accuracy with regularized
STDP. See ?A.7 for details on the structure of inputs. We then performed 700 trials (350 classical
and 350 regularized) exposing the neuron to a new pattern for 20 seconds and observed performance
under classical and regularized STDP.
SRM neurons with classical STDP. We used Gerstner?s SRM model, recall ?3, with parameters
chosen to exactly coincide with [32]: ?m = 10, ?s = 2.5, K = 2.2, K1 = 2, K2 = 4 and
# = 14 #synapses. STDP was implemented via (5) with parameters ?+ = 0.03125, ?+ = 16.8,
? = 0.85?+ and ? = 33.7 also taken from [32]. Synaptic weights were clipped to fall in [0, 1].
Regularized STDP consists of a small tweak of classical STDP in the online phase, and an additional offline regularization phase:
? Online. In the online phase, reduce the depotentiation bias from 0.85?+ in the classical
implementation to ? = 0.75?+ .
? Offline. In the offline phase, modify synapses once per second according to
?
? 32 wj ? (! s) if ! < s
wj =
(13)
? (! s)
else,
where s is output spikes per second, ! = 5Hz is the target rate and update factor
The offline update rule is firing rate, and not spike, dependent.
= 0.6.
Classical STDP has a depotentiation bias to prevent runaway potentiation feedback loops leading to
seizures [5]. Since synapses are frequently renormalized offline we incorporate a weak exploratory
(potentiation) bias during the online phase which helps avoid local minima.3 This is in line with
experimental evidence showing increased cortical activity during waking [30].
Since computing the sum of synaptic weights is non-physiological, we draw on Theorem 1 and
use the neuron?s firing rate when responding to uncorrelated inputs as a proxy for kwk1 . Thus,
in the offline phase, synapses receive inputs generated as in the online phase but without repeated
patterns. Note that (12) has a larger pruning effect on stronger synapses, discouraging specialization.
Motivated by Remark 4, we introduce bias ( 32 wj ) in the offline phase to ensure weaker synapses
are downscaled more than strong synapses. For example, a synapse with wi = 0.5 is downscaled
by twice as much as a synapse with weight wj = 1.0.
Regularized STDP alternates between 2 seconds online and 4 seconds offline, which suffices to
renormalize synaptic strengths. The frequency of the offline phase could be reduced by decreasing the update factors ?? , presenting stimuli less frequently (than 7 times per second), or adding
inhibitory neurons to the system.
Results. A summary of results is presented in the table below: accuracy quantifies the fraction
of spikes that co-occur with each pattern. Regularized STDP outperforms classical STDP on both
patterns on average. It should be noted that regularized neurons were not only online for 20 seconds
but also offline ? and exposed to Poisson noise ? for 40 seconds. Interestingly, exposure to Poisson
noise improves performance.
Algorithm
Accuracy
Pattern 1 Pattern 2
Classical
54%
39%
Regularized
59%
48%
3
The input stream contains a repeated pattern, so there is a potentiation bias in practice even though the net
integral of STDP in the online phase is negative.
7
TRIALS
0
60
30
0
100
100
80
80
ACCURACY ON #1
TRIALS
ACCURACY ON #1
160
100
60
40
20
0
20 40 60 80
ACCURACY ON #2
100 0
60
40
20
0
80 140
TRIALS
(a) Classical STDP
20 40 60 80
ACCURACY ON #2
100 0
40 70
TRIALS
(b) Regularized STDP
Figure 1: Accuracy after 20 seconds of exposure to a novel pattern.
Fig. 1 provides a more detailed analysis. Each panel shows a 2D-histogram (darker shades of gray
correspond to more trials) plotting accuracies on both patterns simultaneously, and two 1D histograms plotting accuracies on the two patterns separately. The 1D histogram for regularized STDP
shows a unimodal distribution for pattern #2, with most of the mass over accuracies of 50-90%. For
pattern #1, which has been ?unlearned? for twice as long as the training period, most of the mass is
over accuracies of 50% to 90%, with a significant fraction ?unlearnt?. By contrast, classical STDP
exhibits extremely brittle behavior. It completely unlearns the original pattern in about half the trials,
and also fails to learn the new pattern in most of the trials.
Thus, as suggested by our analysis, introducing a regularization both improves the robustness of
STDP and enables an exploratory bias by preventing runaway feedback leading to epileptic seizures.
7
Discussion
The selectron provides a bridge between a particular model of spiking neurons ? the Spike Response Model [20] with the original spike-timing dependent plasticity rule [5] ? and models that
are amenable to learning-theoretic analysis. Our hope is that the selectron and related models lead
to an improved understanding of the principles underlying learning in cortex. It remains to be seen
whether other STDP-based models also have tractable discrete-time analogs.
The selectron is an interesting model in its own right: it embeds reward estimates into spikes and
maximizes a margin that improves error bounds. It imposes a constraint on synaptic weights that:
concentrates rewards/spike, tightens error bounds and improves guarantees on spiking efficacy. Although the analysis does not apply directly to continuous-time models, experiments show that a
tweak inspired by our analysis improves the performance of a more realistic model. An important avenue for future research is investigating the role of feedback in cortex, specifically NMDA
synapses, which may have interesting learning-theoretic implications.
Acknowledgements. We thank Timoth?ee Masquelier for generously sharing his source code [32]
and Samory Kpotufe for useful discussions.
References
[1] Friston K, Kilner J, Harrison L: A free energy principle for the brain. J. Phys. Paris 2006, 100:70?87.
[2] Rosenblatt F: The perceptron: a probabilistic model for information storage and organization in the
brain. Psychol Rev 1958, 65(6):386?408.
[3] Rumelhart DE, Hinton GE, Williams RJ: Learning representations by back-propagating errors. Nature 1986, 323:533?536.
[4] Hinton G, Osindero S, Teh YW: A Fast Learning Algorithm for Deep Belief Nets. Neural Computation
2006, 18:1527?1554.
8
[5] Song S, Miller KD, Abbott LF: Competitive Hebbian learning through spike-timing-dependent
synaptic plasticity. Nature Neuroscience 2000, 3(9).
[6] Seung HS: Learning in Spiking Neural Networks by Reinforcement of Stochastic Synaptic Transmission. Neuron 2003, 40(1063-1073).
[7] Bohte SM, Mozer MC: Reducing spike train variability: A computational theory of spike-timing
dependent plasticity. In Advances in Neural Information Processing Systems (NIPS) 2005.
[8] Legenstein R, Maass W: A criterion for the convergence of learning with spike timing dependent
plasticity. In Advances in Neural Information Processing Systems (NIPS) 2006.
[9] Buesing L, Maass W: Simplified rules and theoretical analysis for information bottleneck optimization and PCA with spiking neurons. In Adv in Neural Information Processing Systems (NIPS) 2007.
[10] Legenstein R, Pecevski D, Maass W: Theoretical analysis of learning with reward-modulated spiketiming-dependent plasticity. In Advances in Neural Information Processing Systems (NIPS) 2008.
[11] Tishby N, Pereira F, Bialek W: The information bottleneck method. In Proc. of the 37-th Annual Allerton Conference on Communication, Control and Computing. Edited by Hajek B, Sreenivas R 1999.
[12] Balduzzi D, Tononi G: What can neurons do for their brain? Communicate selectivity with spikes.
To appear in Theory in Biosciences 2012.
[13] Balduzzi D, Ortega PA, Besserve M: Metabolic cost as an organizing principle for cooperative learning. Under review, 2012.
[14] Nere A, Olcese U, Balduzzi D, Tononi G: A neuromorphic architecture for object recognition and
motion anticipation using burst-STDP. PLoS One 2012, 7(5):e36958.
[15] Schmiedt J, Albers C, Pawelzik K: Spike timing-dependent plasticity as dynamic filter. In Advances
in Neural Information Processing Systems (NIPS) 2010.
[16] Anthony M, Bartlett PL: Neural Network Learning: Theoretical Foundations. Cambridge Univ Press
1999.
[17] Freund Y, Schapire RE: Large Margin Classification Using the Perceptron Algorithm. Machine Learning 1999, 37(3):277?296.
[18] Ecker AS, Berens P, Keliris GA, Bethge M, Logothetis NK, Tolias AS: Decorrelated neuronal firing in
cortical microcircuits. Science 2010, 327(5965):584?7.
[19] Dan Y, Poo MM: Spike timing-dependent plasticity of neural circuits. Neuron 2004, 44:23?30.
[20] Gerstner W: Time structure of the activity in neural network models. Phys. Rev. E 1995, 51:738?758.
[21] Geman S, Bienenstock E, Doursat R: Neural Networks and the Bias/Variance Dilemma. Neural Comp
1992, 4:1?58.
[22] Freund Y, Schapire RE: Experiments with a New Boosting Algorithm. In Machine Learning: Proceedings of the Thirteenth International Conference 1996.
[23] Schapire RE, Freund Y, Bartlett P, Lee WS: Boosting the Margin: A New Explanation for the Effectiveness of Voting Methods. The Annals of Statistics 1998, 26(5).
[24] Boucheron S, Bousquet O, Lugosi G: Theory of classification: A survey of some recent advances.
ESAIM: PS 2005, 9:323?375.
[25] Hasenstaub A, Otte S, Callaway E, Sejnowski TJ: Metabolic cost as a unifying principle governing
neuronal biophysics. Proc Natl Acad Sci U S A 2010, 107(27):12329?34.
[26] Fusi S, Drew P, Abbott L: Cascade Models of Synaptically Stored Memories. Neuron 2005, 45:599?
611.
[27] Fusi S, Abbott L: Limits on the memory storage capacity of bounded synapses. Nature Neuroscience
2007, 10(4):485?493.
[28] Tononi G, Cirelli C: Sleep function and synaptic homeostasis. Sleep Med Rev 2006, 10:49?62.
[29] Vyazovskiy VV, Cirelli C, Pfister-Genskow M, Faraguna U, Tononi G: Molecular and electrophysiological evidence for net synaptic potentiation in wake and depression in sleep. Nat Neurosci 2008,
11(2):200?8.
[30] Vyazovskiy VV, Olcese U, Lazimy Y, Faraguna U, Esser SK, Williams JC, Cirelli C, Tononi G: Cortical
firing and sleep homeostasis. Neuron 2009, 63(6):865?78.
[31] Maret S, Faraguna U, Nelson AB, Cirelli C, Tononi G: Sleep and waking modulate spine turnover in
the adolescent mouse cortex. Nat Neurosci 2011, 14(11):1418?1420.
[32] Masquelier T, Guyonneau, R and Thorpe SJ: Competitive STDP-Based Spike Pattern Learning. Neural
Computation 2009, 21(5):1259?1276.
[33] Roelfsema PR, van Ooyen A: Attention-gated reinforcement learning of internal representations for
classification. Neural Comput 2005, 17(10):2176?2214.
9
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4,078 | 4,691 | Shifting Weights: Adapting Object Detectors from
Image to Video
Kevin Tang1
Vignesh Ramanathan2
Li Fei-Fei1
Daphne Koller1
1
Computer Science Department, Stanford University, Stanford, CA 94305
2
Department of Electrical Engineering, Stanford University, Stanford, CA 94305
{kdtang,vigneshr,feifeili,koller}@cs.stanford.edu
Abstract
Typical object detectors trained on images perform poorly on video, as there is a
clear distinction in domain between the two types of data. In this paper, we tackle
the problem of adapting object detectors learned from images to work well on
videos. We treat the problem as one of unsupervised domain adaptation, in which
we are given labeled data from the source domain (image), but only unlabeled
data from the target domain (video). Our approach, self-paced domain adaptation,
seeks to iteratively adapt the detector by re-training the detector with automatically discovered target domain examples, starting with the easiest first. At each
iteration, the algorithm adapts by considering an increased number of target domain examples, and a decreased number of source domain examples. To discover
target domain examples from the vast amount of video data, we introduce a simple, robust approach that scores trajectory tracks instead of bounding boxes. We
also show how rich and expressive features specific to the target domain can be
incorporated under the same framework. We show promising results on the 2011
TRECVID Multimedia Event Detection [1] and LabelMe Video [2] datasets that
illustrate the benefit of our approach to adapt object detectors to video.
1
Introduction
Following recent advances in learning algorithms and robust feature representations, tasks in video
understanding have shifted from classifying simple motions and actions [3, 4] to detecting complex
events and activities in Internet videos [1,5,6]. Detecting complex events is a difficult task, requiring
probabilistic models that can understand the semantics of what is occuring in the video. Because
many events are characterized by key objects and their interactions, it is imperative to have robust
object detectors that can provide accurate detections. In this paper, we focus on the problem of
detecting objects in complex Internet videos. It is difficult to obtain labeled objects in these types of
videos because of the large number of frames, and the fact that objects may not appear in many of
them. Thus, a common approach is to train object detectors from labeled images, which are widely
available. However, as seen in Figure 1, the domain of images and videos is quite different, as it is
often the case that images of objects are taken in controlled settings that differ greatly from where
they appear in real-world situations, as seen in video. Thus, we cannot typically expect a detector
trained on images to work well in videos.
To adapt object detectors from image to video, we take an incremental, self-paced approach to learn
from the large amounts of unlabeled video data available. We make the assumption that within
our unlabeled video data, there exist instances of our target object. However, we do not assume
that every video has an instance of the object, due to the noise present in Internet videos. We start
by introducing a simple, robust method for discovering examples in the video data using KanadeLucas-Tomasi (KLT) feature tracks [8,9]. Building on the discovered examples, we introduce a novel
formulation for unsupervised domain adaptation that adapts parameters of the detector from image
1
Skateboard
Sewing Machine
Sandwich
ImageNet (Image domain)
TRECVID MED (Video domain)
Figure 1: Images of the ?Skateboard?, ?Sewing machine?, and ?Sandwich? classes taken from (top
row) ImageNet [7] and (bottom row) TRECVID MED [1] illustrating differences in domain.
to video. This is done by iteratively including examples from the video data into the training set,
while removing examples from the image data based on the difficulty of the examples. We define
easy examples as ones with labels that can be predicted confidently (e.g., high likelihood, large
distance from margin), and thus are more likely to be correct. In addition, it is common to have
discriminative features that are only available in the target domain, which we term target features.
For example, in the video domain, there are contextual features in the spatial and temporal vicinity
of our detected object that we can take advantage of when performing detection. Our approach is
able to incorporate the learning of parameters for these target features into a single objective.
2
Related Work
Most relevant are works that also deal with adapting detectors to video [10?13], but these works
typically deal with a constrained set of videos and limited object classes. The work of [14] deals
with a similar problem, but they adapt detectors from video to image. Our overall method is also
similar to [15], in which we adopt an incremental approach to learn object category models.
Our setting is closely related to the domain adaptation problem, which has been studied extensively
in vision settings. Several previous approaches focus on learning feature transformations between
domains [16?18]. More similar to our method are approaches based on optimizing Support Vector
Machine (SVM) related objectives [19?24] or joint cost functions [25], that treat the features as fixed
and seek to adapt parameters of the classifier from source to target domain. However, with the exception of [18, 25], previous works deal with supervised or semi-supervised domain adaptation, which
require labeled data in the target domain to generate associations between the source and target domains. In our setting, unsupervised domain adaptation, the target domain examples are unlabeled,
and we must simultaneously discover and label examples in addition to learning parameters.
The objective we optimize to learn our detector draws inspiration from [26?28], in which we include
and exclude the loss of certain examples using binary-valued indicator variables. Although our formulation is similar to [27, 28], our method is iterative and anneals weights that govern the number
of examples to use, which is similar to the idea of self-paced learning [26], where a single weight
is decreased to eventually include the loss of all examples in the objective. However, our method is
different from [26] in that we have three sets of weights that govern the source examples, target examples, and target features. The weights are annealed in different directions, giving us the flexibility
to iteratively include examples from the target domain, exclude examples from the source domain,
and include parameters for the target features. In addition, our objective is able to incorporate target
features, which is novel and not considered in [26?28].
Previous works have also considered ideas similar to our target features [29?32]. The work of [29]
considers feature augmentation, but only with observed features common to both domains. Unobserved features in the context of clustering are investigated in [31], but in their setting all examples
are assumed to have the same unobserved features. In [30, 32], features or modalities unseen in
the training data are used to help in testing. However, both works assume there exists relationships
between the seen and unseen features, whereas our target features are completely unrestricted.
3
Our Approach
We begin by providing an overview of our approach to adapting object detectors, as illustrated in
Figure 2, and then elaborate on each of the steps. We assume that we are given a large amount of
unlabeled video data with positive instances of our object class within some of these videos.
2
Step 1
Train detector using:
image positives + negatives
IM+
IM+
IM+
IM+
Step 2
Discover top K video positives +
negatives from unlabeled
videos using detector VID+
IMIM-
IM-
IM-
VID- VID+
VID+
VIDVID+VID-
Unlabeled videos
VIDIMVID+ IM+
VIDIM+
IM+ IM- IMVIDVID+
IM+ VID+
IM-
VID+
VID+
Step 3
Re-train detector using:
image positives + negatives
video positives + negatives
VIDVIDVID-
Step 4
Update annealed weights
to include/exclude more
examples and features
Figure 2: Overview of our algorithm. We start by initializing our detector using image positives and
negatives (Step 1). We then proceed to enter a loop in which we discover the top K video positives
and negatives (Step 2), re-train our detector using these (Step 3), and then update the annealed
parameters of the algorithm (Step 4).
We initialize our detector (Step 1 of Figure 2) by training a classifier on the labeled image positives
and negatives, which we denote by our dataset (hx1 , y1 i, ..., hxn , yn i) with binary class labels yi 2
{ 1, 1}. We consider a common method of learning weights w of a linear classifier:
!
n
X
w = arg min r(w) + C
Loss(xi , yi ; w)
(1)
w
i=1
where r(?) is a regularizer over the weights, Loss(?) is a loss function over the training example,
and C controls the tradeoff between the two.
Our goal then is to discover the top K positive and negative examples from the unlabeled videos,
and to use these examples to help re-train our detector. We do not attempt to discover all instances,
but simply a sufficient quantity to help adapt our detector to the video domain. To discover the
top K video positives and negatives (Step 2 of Figure 2), we utilize the strong prior of temporal
continuity and score trajectory tracks instead of bounding boxes, which we describe in Section 3.1.
Given the discovered examples, we optimize a novel objective inspired by self-paced learning [26]
that simultaneously selects easy examples and trains a new detector (Step 3 of Figure 2). Using this
new detector, we repeat this process of example discovery and detector training until convergence,
as illustrated in Figure 2.
3.1 Discovering Examples in Video
In this step of the algorithm, we are given weights w of an object detector that can be used to
score bounding boxes in video frames. A naive approach would run our detector on frames of
video, taking the highest scoring and lowest scoring bounding boxes as the top K video positives
and negatives. Although reasonable, this method doesn?t take advantage of temporal continuity in
videos. An object that appears in one frame of a video is certain to appear close in neighboring
frames as well. Previous works have shown this intuition to yield good results [10, 12, 13].
Track-based scoring Our key idea is to score trajectory tracks, rather than bounding boxes, as
illustrated in Figure 3. We obtain tracks by running a KLT tracker on our videos, which tracks a
sparse set of features over large periods of time. Because of the large number of unlabeled videos
we have, we elect to extract KLT tracks rather than computing dense tracks using optical flow. Note
that these tracks follow features, and so they may not correspond to centered locations of objects.
For each track, we consider the set of all bounding box placements B around it that intersect with
the track. Each box placement bi 2 B is associated with a relative coordinate (bxi , byi ) as well as a
score bsi . The relative coordinate (bxi , byi ) is the point within the box (relative to the top-left corner of
the box) that intersects the track. Using this coordinate, we can compute the position of bi at every
point in time along the track. Note that the number of bounding boxes in B is only dependent on
the dimensions of the detector and the scales we search over. The score bsi is computed by pooling
scores of the bounding box along multiple points of the track in time. We use average pooling in
our experiments to be robust to noisy scores. Finally, we associate the track with the bounding box
bmax with the highest score, and use the score bsmax as the score of the track.
After scoring each track in our unlabeled videos, we select the top and bottom few scoring tracks,
and extract bounding boxes from each using the associated box coordinates (bxmax , bymax ) to get our
top K video positives and negatives. The boxes are extracted by sampling frames along the track.
Advantages Compared to the naive approach without tracks, this approach allows us to recover
from false detections with high scores, which are common for weak detectors, as it is less likely that
3
Box coordinates
5
Frame 3
Average detector scores
along track in time
4
11
Frame 2
Detector
scores
7 10
Frame 3
6
Frame 2
6
2
12
Frame 1
3
3
11
Frame 1
Track
Best box score
and coordinates
Figure 3: For a given KLT track, we consider all bounding box placements that intersect with it,
denoted by the colored rectangular boxes. The purple cross denotes the intersection coordinates
(bxi , byi ) for each box. For each box, we average the scores at each point along the track, and take the
box with the maximum score as the score and associated bounding box coordinates for this track.
there will be multiple false detections with high scores along a KLT track. Similarly, if the detection
scores are consistently high along many points of a track, we can be more confident of the object?s
presence along the track. Hence, we can obtain novel examples of the object from various points
of the track that had low scores, since we know the trajectory should correspond to the object. The
same intuitions hold for true detections with low scores and obtaining negative examples.
3.2
Self-Paced Domain Adaptation
In this step of the algorithm, we are given the discovered top K video positives and negatives, which we denote by the dataset (hz1 , h1 i, ..., hzk , hk i). Together with our original dataset
(hx1 , y1 i, ..., hxn , yn i), we would like to learn a new detector.
A simple method would be to re-train our detector with both datasets using Equation 1. However,
we typically aren?t certain that the labels h are correct, especially in the first iteration when our
detector is trained solely from the image examples. Ideally, we would like to re-train with a set of
easier examples whose labels we are confident of first, and then re-discover video examples with
this new detector. We would also like to stop learning from examples we are unsure of in the image
domain, as they may be the examples most affected by the differences in domain. By repeating
this process, we can avoid bad examples and iteratively refine our set of top K video positives and
negatives before having to train with all of them.
Formulating this intuition, our algorithm selects easier examples to learn from in the discovered
video examples, and simultaneously selects harder examples in the image examples to stop learning
from. An example is difficult if it has a large loss, as we are not confident in its correct label. The
number of examples selected from the video examples and image examples are governed by weights
that will be annealed over iterations (Step 4 of Figure 2).
Basic approach We start by introducing our approach without target features. We introduce binary
variables v1 , ..., vn for the source domain (image) examples, and binary variables u1 , ..., uk for the
target domain (video) examples. A value of 0 indicates that an example is difficult, and so we would
like to remove its loss from consideration in the objective function. To prevent the algorithm from
assigning all examples to be difficult, we introduce parameters K source and K target that control the
number of examples considered from the source and target domain, respectively.
(wt+1 , v t+1 , ut+1 ) = arg min r(w) + C
w,v,u
n
?X
vi Loss(xi , yi ; w) +
i=1
1
K source
n
X
vi
i=1
1
K target
k
X
j=1
uj
!
k
X
j=1
uj Loss(zj , hj ; w)
?
(2)
If K target is large, the algorithm prefers to consider only easy target examples with a small Loss(?),
and the same is true for K source . In the annealing of the weights for the algorithm (Step 4 of
Figure 2), we decrease K target and increase K source to iteratively include more examples from the
target domain and decrease examples from the source domain.
4
Similar to self-paced learning [26], we obtain a tight relaxation when allowing the binary variables
v and u to take on any value in the interval [0, 1]. With the choice of r(?) and Loss(?) convex in w,
the problem becomes a bi-convex problem, and can be solved by alternating between (1) solving
for w given v and u, and (2) solving for v and u given w. We refer the reader to [26] for further
intuitions on the binary variables and annealed weights.
Leveraging target features Often, the target domain we are adapting to has additional features we
can take advantage of. At the start, when we?ve only learned from a few examples in our target
domain, we do not wish to rely on these rich and expressive features, as they can easily cause us
to overfit. However, as we iteratively adapt to the target domain and build more confidence in our
detector, we can start utilizing these target features to help with detection. The inclusion of these
features is naturally self-paced as well, and can be easily integrated into our framework.
We assume there are a set of features that are shared between the source and target domains as
= [ shared
shared , and a set of target domain-only features as target :
target ]. The weights
w we want to learn can now be divided into wshared and wtarget : w = [wshared wtarget ]. Since
the source data doesn?t have target features, we initialize those features to be 0 so that wtarget
doesn?t affect the loss on the source data. The new objective function is formulated as:
(wt+1 , v t+1 , ut+1 ) = arg min r(w) + C
w,v,u
n
?X
vi Loss(xi , yi ; w) +
i=1
+
1
K f eat
||wtarget ||1
k
X
uj Loss(zj , hj ; w)
j=1
1
K source
n
X
i=1
vi
1
K target
k
X
j=1
uj
!
?
(3)
This is similar to Equation 2, with the addition of the L1 norm term K f1eat ||wtarget ||1 . To anneal
the weights for target features, we increase K f eat to iteratively reduce the L1 norm on the target
features so that wtarget can become non-zero. Intuitively, we are forcing the weights w to only use
shared features first, and to consider more target features when we have a better model of the target
domain. The optimization can be solved in the same manner as Equation 2. We can also approximate
the L1 norm term for all target features to be effectively binary, forcing K f eat to be 0 initially and
switching to 1 at a particular iteration. This amounts to only considering target features after a
certain iteration, and is done in our experiments for more tractable learning.
4
Experiments
We present experimental results for adapting object detectors on the 2011 TRECVID Multimedia
Event Detection (MED) dataset [1] and LabelMe Video [2] dataset. For both, we select a set of
objects which are known to appear in the videos. We used images from ImageNet [7] for the labeled
image data, as there are a large number of diverse categories on ImageNet that correspond well with
the objects that appear in the videos. We evaluate the detection performance of our models with the
measure used in the PASCAL Visual Object Classes challenge [33], and report average precision
(AP) scores for each class. The detection scores are computed on annotated video frames from the
respective video datasets that are disjoint from the unlabeled videos used in the adapting stage.
4.1
Implementation Details
In our experiments, we use object detectors that are rectangular filters over Histogram-of-Gradient
(HOG) features [34]. We use L2 regularization for r(?) and hinge loss for Loss(?), which corresponds to the standard linear SVM formulation. For target features, we use contextual spatial
features. The spatial features are taken to be HOG features bordering the object with dimensions
half the size of the object bounding box. As described previously, we approximate the L1 norm term
to be binary to enable fast training using LIBLINEAR [35] when optimizing for w. This also further
decreases the number of model parameters needed to be searched over.
To isolate the effects of adaptation and better analyze our method, we restrict our experiments to
the setting in which we fix the video negatives, and focus our problem on adapting from the labeled
image positives to the unlabeled video positives. This scenario is realistic and commonly seen, as
we can easily obtain video negatives by sampling from a set of unlabeled or weakly-labeled videos.
5
Table 1: Average Precision (AP) values for detection on the TRECVID MED dataset
Object
InitialBL VideoPosBL Our method(nt) Our method(full) Gopalan et al. [18] (PLS) Gopalan et al. [18] (SVM)
Skateboard
4.29%
2.89%
10.44%
10.44%
0.04%
0.94%
Animal
0.41%
0.40%
0.39%
3.76%
0.16%
0.24%
Tire
11.22%
11.04%
15.54%
15.54%
0.60%
15.52%
Vehicle
4.03%
4.08%
3.57%
3.57%
3.33%
3.16%
Sandwich
10.07%
9.85%
9.45%
12.49%
0.21%
6.68%
Sewing machine 9.76%
9.71%
10.35%
10.35%
0.12%
3.81%
Mean AP
6.63%
6.33%
8.29%
9.36%
0.74%
5.06%
Table 2: Average Precision (AP) values for detection on the LabelMe Video dataset
Object InitialBL VideoPosBL Our method(nt) Our method(full) Gopalan et al. [18] (PLS) Gopalan et al. [18] (SVM)
Car
2.60%
2.13%
2.15%
9.18%
0.34%
1.00%
Boat
0.22%
0.22%
0.22%
0.22%
0.05%
0.32%
Bicycle
19.85%
19.76%
20.27%
20.27%
0.21%
16.32%
Dog
1.74%
2.42%
2.47%
4.75%
0.18%
1.48%
Keyboard 0.41%
0.67%
0.59%
0.59%
0.13%
0.09%
Mean AP 4.96%
5.04%
5.14%
7.00%
0.18%
3.84%
Model parameters In our experiments, we fix the total number of iterations to 5 for tractable training time. For the K target and K source weights, we set values for the first and final iterations, and
linearly interpolate values for the remaining iterations in between. For the K target weight, we estimate the weights so that we start by considering only the video examples that have no loss, and
end with all video examples considered. For the K source weight, we vary the ending weight so
that differing numbers of source examples are left for training at the final iteration. For the target
features, we set the algorithm to allow target features at the midpoint of total iterations. Based on
the number of KLT tracks extracted, we set the top K examples to be between 100 and 500.
Model selection The free model parameters that can be varied are the number of top K examples to
discover, the ending K source weight, and whether or not to use target features. In our results, we perform model selection by comparing the distribution of scores on the discovered video positives. The
distributions are compared between the initial models from iteration 1 for different model parameters to select K and K source , and between the final iteration 5 models for different model parameters
to determine the use of target features. This allows us to evaluate the strength of the initial model
trained on the image positives and video negatives, as well as our final adapted model. We select the
model with the distributions indicating the highest confidence in its classification boundary.
4.2
Baseline Comparisons
InitialBL This baseline is the intial detector trained only on image positives and video negatives.
VideoPosBL This baseline uses the intial detector to discover the top K video positives from the
unlabeled video, then trains with all these examples without iterating. Thus, it incorporates our idea
of discovering video positives by scoring tracks and re-training, but does not use self-paced domain
adaptation for learning weights. It can also be thought of as our method run for one iteration.
Our method(nt) This baseline uses our full method with the exception of target features.
Gopalan et al. This is a state-of-the-art method for unsupervised domain adaptation [18] that models the domain shift in feature space. Since we are not given labels in the target domain, most
previous methods for domain adaptation cannot be applied to our setting. This method samples subspaces along the geodesic between the source and target domains on the Grassman manifold. Using
projections of both source and target data onto the common subspaces, they learn a discriminative
classifier using partial least squares (PLS) with available labels from either domains. We ran their
code using their suggested parameter settings to obtain results for their method on our task. We also
show results for their method using a linear SVM as the classifier to allow for fair comparisons.
4.3
TRECVID MED
The 2011 TRECVID MED dataset [1] consists of a collection of Internet videos collected by the
Linguistic Data Consortium from various Internet video hosting sites. There are a total of 15 complex events, and videos are labeled with either an event class or no label, where an absence of label
indicates the video belongs to no event class. We select 6 object classes to learn object detectors for
because they are commonly present in selected events: ?Skateboard?, ?Animal?, ?Tire?, ?Vehicle?,
?Sandwich?, and ?Sewing machine?. These objects appear respectively in the events ?Attempting a
6
New examples (Iterations 1,2)
New examples (Iterations 4,5)
Removed examples
New examples (Iterations 1,2)
New examples (Iterations 4,5)
Removed examples
Car
Sandwich
Figure 4: Discovered top K video positives using our method for ?Sandwich? and ?Car?. After
sets of iterations, we show samples of newly discovered video positives (red boxes) that were not
in the set of top K of previous iterations (left, middle columns). We also show bad examples that
were removed from the top K over all iterations (right column). As our model adapts, it is able to
iteratively refine its set of top K video positives. Figure best viewed magnified and in color.
Sandwich
Animal
Tire
Car
Figure 5: Detections for ?Sandwich?, ?Tire?, ?Animal?, and ?Car?. Green boxes detections from
our method, red boxes detections from ?InitialBL?, blue boxes detections from ?VideoPosBL?, and
magenta boxes detections from Gopalan et al.(SVM). Figure best viewed magnified and in color.
board trick?, ?Feeding an animal?, ?Changing a vehicle tire?, ?Getting a vehicle unstuck?, ?Making
a sandwich?, and ?Working on a sewing project?. The video negatives were randomly sampled from
the videos that were labeled with no event class.
To test our algorithm, we manually annotated approximately 200 frames with bounding boxes of
positive examples for each object, resulting in 1234 annotated frames total from over 500 videos,
giving us a diverse set of situations the objects can appear in. For each object, we use 20 videos
from the associated event as unlabeled video training data. Results are given in Table 1.
4.4
LabelMe Video
LabelMe Video [2] is a database of real-world videos that contains a large set of annotations including object category, shape, motion, and activity information. We use the database of videos that was
introduced in the original paper [2]. There are a large number of objects that are annotated in this
database, and we select the most frequently occuring objects that are not scene parts, resulting in
5 objects: ?Car?, ?Boat?, ?Bicycle?, ?Dog?, and ?Keyboard?. The video negatives were randomly
sampled from the videos that were not annotated with any of these objects.
We extract more than 200 frames with positive examples for each object class, resulting in a test set
of 1137 images. For each object class, we use the remaining videos that contain the object as the
unlabeled video training data, resulting in around 9 videos per object. Results are given in Table 2.
5
Discussion
From our results in Tables 1 and 2, we can observe similar patterns for most object classes. First,
we note that the ?VideoPosBL? baseline typically performs on par with the ?InitialBL? baseline,
and rarely does it post a slight gain in performance. This shows that if we discover the top K
video positives and re-train our detector with all of them, we do not obtain consistent gains in
performance. Our method of self-paced domain adaptation is crucial in this case, as we can see
that our full method typically outperforms all other methods by significant margins. As illustrated
in Figure 4, our method is able to add new video positives from iteration to iteration that are good
examples, and remove bad examples at the same time. The method of Gopalan et al. [18] performs
very poorly when used in conjunction with the PLS classifier, but becomes more competitive when
used with an SVM. However, even then their method performs much worse than our method for
nearly all object classes, as it is difficult to model the underlying domain shift in feature space. This
also serves to illustrate the difficulty of our problem, as poor adaptation can lead to results worse than
the baselines. We show visualizations of our detections compared to baseline methods in Figure 5.
Observing the visualizations of the learned weights for the ?Tire?, ?Car? and ?Sandwich? classes
in Figure 6, we see that weights trained with our method exhibit more clearly defined structure
7
InitialBL
Tire
Our Method
Car
InitialBL
Sandwich
Our Method
InitialBL
Our Method
Figure 6: Visualizations of the positive HOG weights learned for three classes for the ?InitialBL?
baseline and our method. The spatial context weights are 0 for ?InitialBL? because it does not
consider target features, resulting in a black border. Figure best viewed magnified and in color.
than the ?InitialBL? baseline. The target features also help performance significantly. By capturing
interesting patterns in the spatial context, difficult objects can become easier to detect in the target
domain. For the ?Sandwich? class, we can see circular weights in the spatial context surrounding
the sandwich, suggesting that sandwiches typically appear on plates, and for ?Car?, we can clearly
distinguish weights for the road beneath the car object. We observe an average AP gain of 3.93%
for classes that choose models with target features versus no target features. Note that we chose to
use simple spatial context as target features in our models, as they are fast to implement and easily
incorporated. However, we hypothesize that the inclusion of more complex target features such as
temporal movement could help our method achieve even better results.
We observe that for the ?Vehicle? and ?Keyboard? classes, the ?VideoPosBL? baseline performs
better than our full method. Although this is not a common occurrence, it can happen when our
method of self-paced domain adaptation replaces good video positives taken in the first iteration
with bad examples in future iterations. This situation arises when there are incorrect examples
present in the easiest of the top K video positives, causing our detector to re-train and iteratively
become worse. If we had better methods for model selection, we could also search over the number
of total iterations as a model parameter, which would include the ?VideoPosBL? model in our set of
models to select over, as it is essentially our method run for a single iteration.
6
Conclusion
In this paper we have introduced an approach for adapting detectors from image to video. To discover examples in the unlabeled video data, we classify tracks instead of bounding boxes, allowing us to leverage temporal continuity to avoid spurious detections, and to discover examples we
would?ve otherwise missed. Furthermore, we introduce a novel self-paced domain adaptation algorithm that allows our detector to iteratively adapt from source to target domain, while also considering target features unique to the target domain. Our formulation is general, and can be applied
to various other problems in domain adaptation. We?ve shown convincing results that illustrate the
benefit of our approach to adapting object detectors to video.
Possible directions for future work could include better methods for model selection. A measure
that would allow us to estimate our performance on the target domain with theoretical guarantees
would be an interesting direction. Another possible direction would be to relax the assumption of
having no labeled target domain examples, and to formulate similar methods for this scenario.
Acknowledgments. We thank Tianshi Gao for helpful discussions. We also thank Chris Baldassano, Eric Huang, Jia Deng, and Olga Russakovsky for helpful comments on the paper. This work
was supported by the Defense Advanced Research Projects Agency under Contract No. HR001108-C-0135 and by the Intelligence Advanced Research Projects Activity (IARPA) via Department
of Interior National Business Center contract number D11PC20069. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright
annotation thereon. Disclaimer: The views and conclusions contained herein are those of the authors
and should not be interpreted as necessarily representing the official policies or endorsements, either
expressed or implied, of DARPA, IARPA, DoI/NBC, or the U.S. Government.
References
[1] P. Over, G. Awad, M. Michel, J. Fiscus, W. Kraaij, and A. F. Smeaton. Trecvid 2011 ? an overview of the
goals, tasks, data, evaluation mechanisms and metrics. In TRECVID 2011. NIST, USA, 2011.
[2] J. Yuen, B. C. Russell, C. Liu, and A. Torralba. Labelme video: Building a video database with human
annotations. In ICCV, 2009.
8
[3] C. Schuldt, I. Laptev, and B. Caputo. Recognizing human actions: A local svm approach. In ICPR, 2004.
[4] L. Gorelick, M. Blank, E. Shechtman, M. Irani, and R. Basri. Actions as space-time shapes. IEEE TPAMI,
2007.
[5] J. C. Niebles, C.-W. Chen, and L. Fei-Fei. Modeling temporal structure of decomposable motion segments
for activity classification. In ECCV, 2010.
[6] K. Tang, L. Fei-Fei, and D. Koller. Learning latent temporal structure for complex event detection. In
CVPR, 2012.
[7] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei. ImageNet: A Large-Scale Hierarchical
Image Database. In CVPR, 2009.
[8] B. D. Lucas and T. Kanade. An iterative image registration technique with an application to stereo vision.
In IJCAI, 1981.
[9] C. Tomasi and T. Kanade. Detection and tracking of point features. Technical report, CMU, 1991.
[10] P. Sharma, C. Huang, and R. Nevatia. Unsupervised incremental learning for improved object detection
in a video. In CVPR, 2012.
[11] X. Wang, G. Hua, and T. X. Han. Detection by detections: Non-parametric detector adaptation for a
video. In CVPR, 2012.
[12] M. Yang, S. Zhu, F. Lv, and K. Yu. Correspondence driven adaptation for human profile recognition. In
CVPR, 2011.
[13] N. Cherniavsky, I. Laptev, J. Sivic, and A. Zisserman. Semi-supervised learning of facial attributes in
video. In ECCV 2010, 2010.
[14] A. Prest, C. Leistner, J. Civera, C. Schmid, and V. Ferrari. Learning object class detectors from weakly
annotated video. In CVPR, 2012.
[15] L.-J. Li and L. Fei-Fei. OPTIMOL: automatic Online Picture collecTion via Incremental MOdel Learning.
IJCV, 2009.
[16] K. Saenko, B. Kulis, M. Fritz, and T. Darrell. Adapting visual category models to new domains. In ECCV,
2010.
[17] B. Kulis, K. Saenko, and T. Darrell. What you saw is not what you get: Domain adaptation using
asymmetric kernel transforms. In CVPR, 2011.
[18] R. Gopalan, R. Li, and R. Chellappa. Domain adaptation for object recognition: An unsupervised approach. In ICCV, 2011.
[19] A. Bergamo and L. Torresani. Exploiting weakly-labeled web images to improve object classification: a
domain adaptation approach. In NIPS, 2010.
[20] G. Schweikert, C. Widmer, B. Sch?olkopf, and G. R?atsch. An empirical analysis of domain adaptation
algorithms for genomic sequence analysis. In NIPS, 2008.
[21] J. Yang, R. Yan, and A. G. Hauptmann. Cross-domain video concept detection using adaptive svms. In
ACM Multimedia, 2007.
[22] L. Duan, D. Xu, I. W.-H. Tsang, and J. Luo. Visual event recognition in videos by learning from web
data. In CVPR, 2010.
[23] T. Joachims. Transductive inference for text classification using support vector machines. In ICML, 1999.
[24] T. Tommasi, F. Orabona, and B. Caputo. Safety in numbers: Learning categories from few examples with
multi model knowledge transfer. In CVPR, 2010.
[25] C. Zhang, R. Hamid, and Z. Zhang. Taylor expansion based classifier adaptation: Application to person
detection. In CVPR, 2008.
[26] P. Kumar, B. Packer, and D. Koller. Self-paced learning for latent variable models. In NIPS, 2010.
[27] J. J. Lim, R. Salakhutdinov, and A. Torralba. Transfer learning by borrowing examples for multiclass
object detection. In NIPS, 2011.
[28] T. Gao and D. Koller. Discriminative learning of relaxed hierarchy for large-scale visual recognition. In
ICCV, 2011.
[29] H. D. III. Frustratingly easy domain adaptation. In ACL, 2007.
[30] B. Taskar, M.-F. Wong, and D. Koller. Learning on the test data: Leveraging ?unseen? features. In ICML,
2003.
[31] E. Krupka and N. Tishby. Generalization in clustering with unobserved features. In NIPS, 2005.
[32] C. M. Christoudias, R. Urtasun, M. Salzmann, and T. Darrell. Learning to recognize objects from unseen
modalities. In ECCV, 2010.
[33] M. Everingham, L. Van Gool, C. K. I. Williams, J. Winn, and A. Zisserman. The pascal visual object
classes (voc) challenge. IJCV, 2010.
[34] N. Dalal and B. Triggs. Histograms of oriented gradients for human detection. In CVPR, 2005.
[35] R.-E. Fan, K.-W. Chang, C.-J. Hsieh, X.-R. Wang, and C.-J. Lin. LIBLINEAR: A library for large linear
classification. JMLR, 2008.
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4,079 | 4,692 | Submodular-Bregman and the Lov?asz-Bregman
Divergences with Applications
Jeff Bilmes
Department of Electrical Engineering
University of Washington
[email protected]
Rishabh Iyer
Department of Electrical Engineering
University of Washington
[email protected]
Abstract
We introduce a class of discrete divergences on sets (equivalently binary vectors)
that we call the submodular-Bregman divergences. We consider two kinds, defined
either from tight modular upper or tight modular lower bounds of a submodular
function. We show that the properties of these divergences are analogous to the
(standard continuous) Bregman divergence. We demonstrate how they generalize
many useful divergences, including the weighted Hamming distance, squared
weighted Hamming, weighted precision, recall, conditional mutual information,
and a generalized KL-divergence on sets. We also show that the generalized
Bregman divergence on the Lov?asz extension of a submodular function, which we
call the Lov?asz-Bregman divergence, is a continuous extension of a submodular
Bregman divergence. We point out a number of applications, and in particular show
that a proximal algorithm defined through the submodular Bregman divergence provides a framework for many mirror-descent style algorithms related to submodular
function optimization. We also show that a generalization of the k-means algorithm
using the Lov?asz Bregman divergence is natural in clustering scenarios where
ordering is important. A unique property of this algorithm is that computing the
mean ordering is extremely efficient unlike other order based distance measures.
1
Introduction
The Bregman divergence first appeared in the context of relaxation techniques in convex programming
([4]), and has found numerous applications as a general framework in clustering ([2]), proximal
minimization ([5]) and online learning ([27]). Many of these applications are due to the nice properties
of the Bregman divergence, and the fact that they are parameterized by a single convex function.
They also generalize a large class of divergences on vectors. Recently Bregman divergences have
also been defined between matrices ([26, 6]) and between functions ([8]).
In this paper we define a class of divergences between sets, where each divergence is parameterized by
a submodular function. This can alternatively and equivalently be seen as a divergence between binary
vectors in the same way that submodular functions are special cases of pseudo-Boolean functions
[3]. We call this the class of submodular Bregman divergences (or just submodular Bregman). We
show an interesting mathematical property of the submodular Bregman, namely that they can be
defined based on either a tight modular (linear) upper bound or alternatively a tight modular lower
bound, unlike the traditional (continuous) Bregman definable only via a tight linear lower bound.
Let V refer to a finite ground set {1, 2, . . . , |V |}. A set function f : 2V ? R is submodular if
?S, T ? V , f (S) + f (T ) ? f (S ? T ) + f (S ? T ). Submodular functions have attractive properties
that make their exact or approximate optimization efficient and often practical. Submodularity
can be seen as a discrete counterpart to convexity and concavity ([20]) and often the problems are
closely related ([1]). Indeed, as we shall see in this paper, the connections between submodularity
1
and convexity and concavity will help us formulate certain discrete divergences that are analogous
to the Bregman divergence. We in fact show a direct connection between a submodular Bregman
and a generalized Bregman divergence defined through the Lov?asz extension. Further background
on submodular functions may be found in the text [9].
An outline of the paper follows. We first define the different types of submodular Bregman in
Section 2. We also define the Lov?asz Bregman divergence, and show its relation to a version of
the submodular Bregman. Then in Section 3, we prove a number of properties of the submodular
Bregman and show how they are related to the Bregman divergence. Finally in Section 4, we show
how the submodular Bregman can be used in applications in machine learning. In particular, we show
how the proximal framework of the submodular Bregman generalizes a number of mirror-descent
style approximate submodular optimization algorithms. We also consider generalizations of the
k-means algorithm using the Lov?asz Bregman divergence, and show how they can be used in
clustering applications where ordering or ranking is important.
2
The Bregman and Submodular Bregman divergences
Notation: We use ? to refer to a convex function, f to refer to a submodular function, and f? as
f ?s Lov?asz extension. Lowercase characters x, y will refer to continuous vectors, while upper case
characters X, Y, S will refer to sets. We will also refer to the characteristic vectors of a set X as
1X ? {0, 1}V . Note that the characteristic vector of a set X, 1X is such that 1X (j) = I(j ? X),
where I(?) is the standard indicator function. We will refer to the ground set as V , and the cardinality
of the ground set as n = |V |. A divergence on vectors and sets is formally defined as follows:
Given a domain of vectors or sets S (and if sets, S = a lattice of sets L, where L is a lattice if
?X, Y ? L, X ? Y, X ? Y ? L), a function d : S ? S ? R+ is called a divergence if ?x, y ? S,
d(x, y) ? 0 and ?x ? S, d(x, x) = 0. For simplicity, we consider mostly the Boolean lattice L = 2V
but generalizations are possible as well [9].
2.1
Bregman and Generalized Bregman divergences
Recall the definition of the Bregman divergence: d? : S ? S ? R+ as:
d? (x, y) = ?(x) ? ?(y) ? h??(y), x ? yi.
(1)
For non-differentiable convex functions we can extend equation (1) to define the generalized Bregman
divergence [13, 18]. Define a subgradient map H? , which for every vector y, gives a subgradient
H? (y) = hy ? ??(y) [13], where ??(y) is the subdifferential of ? at y.
H
d? ? (x, y) = ?(x) ? ?(y) ? hH? (y), x ? yi, ?x, y ? S.
(2)
When ? is differentiable, then ??(x) = {??(x)} and H? (y) = ??(y). More generally, there may
be multiple distinct subgradients in the subdifferential, hence the generalized Bregman divergence is
parameterized both by ? and the subgradient-map H? . The generalized Bregman divergences have
also been defined in terms of ?extreme? subgradients [25, 18].
d]? (x, y) = ?(x) ? ?(y) ? ???(y) (x ? y)
and
d\? (x, y) = ?(x) ? ?(y) + ???(y) (y ? x), (3)
H
where, for a convex set C, ?C (.) , maxx?C h., xi. Clearly, we then have: d]? (x, y) ? d? ? (x, y) ?
d\? (x, y), ?H? which justifies their being called the extreme generalized Bregman divergences [13].
2.2
The Submodular Bregman divergences
In a similar spirit, we define a submodular Bregman divergence parameterized by a submodular
function and defined as the difference between the function and its modular (sometimes called linear)
bounds. Surprisingly, any submodular function has both a tight upper and lower modular bound
([15]), unlike strict convexity where only a tight first-order lower bound exists. Hence, we define
two distinct forms of submodular Bregman parameterized by a submodular function and in terms
of either its tight upper or tight lower bounds.
2
2.2.1
Lower bound form of the Submodular Bregman
Given a submodular function f , the submodular polymatroid Pf , the corresponding base polytope
Bf and the subdifferential ?f (Y ) (at a set Y ) for a submodular function f [9] are respectively:
Pf = {x : x(S) ? f (S), ?S ? V },
Bf = Pf ? {x : x(V ) = f (V )}, and
V
?f (Y ) = {y ? R : ?X ? V, f (Y ) ? y(Y ) ? f (X) ? y(X)}.
(4)
(5)
P
Note that here y(S) = j?S y(j) is a modular function. In a manner similar to the generalized
Bregman divergence ([13]), we define a discrete subgradient map for a submodular function Hf ,
which for every set Y , picks a subgradient Hf (Y ) = hY ? ?f (Y ). Then, given a submodular
function f and a subgradient-map Hf , the generalized lower bound submodular Bregman, which
H
we shall henceforth call df f , is defined as:
H
df f (X, Y ) = f (X) ? f (Y ) ? hY (X) + hY (Y ) = f (X) ? f (Y ) ? hHf (Y ), 1X ? 1Y i). (6)
We remark here that similar to the definition of the generalized Bregman divergence, this submodular
Bregman is parameterized both by the submodular function f and the subgradient map Hf .
The subdifferential corresponding to a submodular function is an unbounded polyhedron [9], with
an uncountable number of possible subgradients. Its extreme points, however, are easy to find
H
and characterize using the greedy algorithm [7]. Thus, we define a subclass of df f with Hf
chosen so that it picks an extreme points of ?f (Y ), which we will call the permutation based lower
bound submodular Bregman, henceforth referred to with d?
f . The extreme points of ?f (Y ) can
be obtained via a greedy algorithm ([7, 9]) as follows: Let ? be a permutation of V and define
Si = {?(1), ?(2), . . . , ?(i)} as its corresponding chain. We define ?Y as the set of permutations ?Y
such that their corresponding chains contain Y , meaning S|Y | = Y . Then we can define a subgradient
hY,?Y (which is an extreme point of ?f (Y )) where:
??Y ? ?Y , hY,?Y (?Y (i)) =
f (S1 )
f (Si ) ? f (Si?1 )
if i = 1
.
otherwise
(7)
In the above, hY,?Y (Y ) = f (Y ). Hence define Hf? as a subgradient map which picks a subgradient
hY,?Y , for some ?(Y ) = ?Y ? ?Y . Here we treat ? as a permutation operator which, for a given set
Y , produces a permutation ?Y ? ?Y . Hence we can rewrite Eqn. (6), with the above subgradient as
?
d?
f (X, Y ) = f (X) ? hY,?Y (X) = f (X) ? hHf (Y ), 1X i.
(8)
H
f
As can readily be seen, the d?
f are special cases of the df . Similar to the extreme generalized Bregman divergence above, we can define forms of the ?extreme? lower bound
submodular Bregman divergences d]f (X, Y ) and d\f (X, Y ). Since in the case of a submodular function ?f (Y ) is an unbounded polyhedron, we restrict C = ?f (Y ) ? Pf , and define:
d]f (X, Y ) = f (X) ? f (Y ) ? ?C (1X ? 1Y ) and d\f (X, Y ) = f (X) ? f (Y ) + ?C (1Y ? 1X ) The
extreme lower bound submodular Bregman have very nice forms as shown in the theorem below:
H
Theorem 2.1. For every hY ? ?f (Y ) ? Pf , d]f (X, Y ) ? df f (X, Y ) ? d\f (X, Y ). Similarly
\
]
for every permutation map ?, d]f (X, Y ) ? d?
f (X, Y ) ? df (X, Y ). Further df (X, Y ) = f (X) +
f (Y )?f (X ?Y )?f (X ?Y ). Similarly d\f (X, Y ) = f (X)?f (Y )?f (Y \X)?f (V )+f (V \X\Y )
H
]
The above theorem gives bounds for df f and d?
f . Further we see that df is exactly the divergence
which defines the submodularity of f . Also notice that this is unlike the generalized Bregman
divergences, where the ?extreme? forms may not be easy to obtain in general [13].
3
2.2.2
The upper bound submodular Bregman
For submodular f , [23] established properties of submodular function using which we can define the
following divergences (which we call here the Nemhauser divergences):
X
X
df] (X, Y ) , f (X) ?
f (j|X ? {j}) +
f (j|X ? Y ) ? f (Y )
(9)
j?Y \X
j?X\Y
df\ (X, Y
) , f (X) ?
X
f (j|X ? Y ? {j}) +
X
f (j|X) ? f (Y ),
(10)
j?Y \X
j?X\Y
where f (j|X) , f (X ? j) ? f (X). Similar to the approach in ([15]), we can relax the Nemhauser
divergences to obtain three modular upper bound submodular Bregmans as:
X
X
df1 (X, Y ) , f (X) ?
f (j|X ? {j}) +
f (j|?) ? f (Y ),
(11)
j?Y \X
j?X\Y
df2 (X, Y
X
) , f (X) ?
) , f (X) ?
X
f (j|X) ? f (Y ).
(12)
f (j|?) ? f (Y ).
(13)
j?Y \X
j?X\Y
df3 (X, Y
X
f (j|V ? {j}) +
f (j|V ? {j}) +
X
j?Y \X
j?X\Y
We call these the Nemhauser based upper-bound submodular Bregmans of, respectively, type-I, II
and III. Henceforth, we shall refer to them as df1 , df2 and df3 and when referring to them collectively,
we will use df1:3 . The Nemhauser divergences are analogous to the extreme divergences of the
generalized Bregman divergences since they bound the Nemhauser based submodular Bregmans. Its
not hard to observe that for a submodular function f , df3 (X, Y ) ? df1 (X, Y ) ? df] (X, Y ). Similarly
df3 (X, Y ) ? df2 (X, Y ) ? df\ (X, Y )
H
Similar to the generalized lower bound submodular Bregman df f , we define a generalized upper
bound submodular Bregman divergence dfG f in terms of any supergradient of f . Interestingly for
a submodular function, we can define a superdifferential ? f (X) at X as follows:
? f (X) = {x ? RV : ?Y ? V, f (X) ? x(X) ? f (Y ) ? x(Y )}.
(14)
Given a supergradient at X, gX ? ? f (X), we can define a divergence dfG f , as:
dfG f (X, Y ) = f (X) ? f (Y ) ? gX (X) ? gX (Y ) = f (X) ? f (Y ) ? hgX , 1X ? 1Y i
(15)
Similar to the subgradient map, we can define G f as the supergradient map, which picks a supergradient from G f (X) = gX ? ? f (X). In fact, it can be shown that all three forms of df1:3 are actually
special cases of dfG f , in that they form specific supergradient maps. Define three supergradients
1
2
3
1
gX
, gX
and gX
(with the corresponding maps G1f , G2f and G3f ) such that: gX
(j) = f (j|X ? {j})
2
3
1
3
and gX (j) = gX (j) = f (j|V ? {j}) for j ? X. Similarly let gX (j) = gX
(j) = f (j|?)
2
1
2
3
and gX (j) = f (j|X) for j ?
/ X. Then it can be shown [12] that gX , gX , gX ? ? f (X), and
correspondingly df1 , df2 and df3 are special cases of dfG f .
dfG f also subsumes an interesting class of divergences for any submodular function representable as
concave over
P modular. Consider any decomposable submodular function [24] f , representable as:
f (X) = i ?i hi (mi (X)), where the hi s are (not necessarily smooth) concave functions
and the mi s
P
cm
are vectors in Rn . Let h0i be any supergradient of hi . Then we define gX
= i ?i h0i (mi (X))mi .
Further we can define a divergence defined for a concave over modular function as:
X
dfcm (X, Y ) =
?i (hi (mi (X)) ? hi (mi (Y )) ? hi (mi (X))(mi (X) ? mi (Y ))
(16)
i
cm
Then it can be shown [12] that dfcm is also a special case of dfG f with gX = gX
when f is a
decomposable submodular function.
4
H
Table 1: Instances of weighted distances measures as special cases of df f and dfG f for w ? Rn+
Name
Hamming
Hamming
Type
H
df f
dfG f
Recall
Precision
df f
dfG f
AER(Y, X; Y )
Cond. MI
Itakura-Saito
Gen. KL
df f
d]f
dfG f
dfG f
H
H
d
w(X\Y ) + w(Y \X)
w(X\Y ) + w(Y \X)
w(X?Y )
w(Y )
w(X?Y )
w(X)
|Y |+|Y ?X|
2|Y |
1?
1?
1?
f (X)
w(X)
?w(X)
Hf (Y )/G f (X)
2 ? w 1Y
?2 ? w 1X
1
-1
w 1Y
w(Y )
X
? w 1
|X|
1Y
2|Y |
1
2
I(XX\Y ; XY \X |XX?Y )
w(Y )
w(Y )
? log w(X)
?1
w(X)
w(Y )
w(Y ) log w(X) ? w(Y ) + w(X)
H(XX )
log w(X)
?w(X) log w(X)
w
w(X)
?w(1 + log w(X))
H
Finally both df f and dfG f generalize a number of interesting distance measures like Hamming, recall,
precision, conditional mutual information, and weighted hamming. We show this in detail in [12],
and owing to lack of space briefly summarize them in Table 1. The distance measures are shown in
weighted form, but cardinality based distances are special cases with w =1
2.3
The Lov?asz Bregman divergence
The Lov?asz extension ([20]) offers a natural connection between submodularity and convexity. The
Lov?asz extension is a non-smooth convex function, and hence we can define a generalized Bregman
divergence ([13, 18]) which has a number of properties and applications analogous to the Bregman
divergence. Recall that the generalized Bregman divergence corresponding to a convex function
? is parameterized by the choice of the subgradient map H? . The Lov?asz extension of a submodular
function has a very interesting set of subgradients, which have a particularly nice structure in that
there is a very simple way of obtaining them [7].
Given a vector y, define a permutation ?y such that y[?y (1)] ? y[?y (2)] ? ? ? ? ? y[?y (n)]
and define Yk = {?y (1), ? ? ? , ?y (k)}. The Lov?asz extension ([7, 20]) is defined as: f?(y) =
Pn
?
k=1 y[?y (k)]f (?y (k)|Yk?1 ). For each point y, we can define a subdifferential ? f (y), which has
n
?
a particularly nice form [9]: for any point y ? [0, 1] , ? f (y) = ?{?f (Yi )|i = 1, 2 ? ? ? , n}. This
H?
naturally defines a generalized Bregman divergence df? f of the Lov?asz extension, parameterized by a
subgradient map Hf?, which we can define as:
H?
df? f (x, y) = f?(x) ? f?(y) ? hhy , x ? yi, for some hy = Hf?(y) ? ? f?(y).
(17)
We can also define specific subgradients of f? at y as hy,?y , with hy,?y (?y (k)) = f (Yk )?f (Yk?1 ), ?k.
These subgradients are really the extreme points of the submodular polyhedron. Then define the
Lov?asz Bregman divergence df? as the Bregman divergence of f? and the subgradient hy,?y . Similar
to d? , it can be shown [12], that d ?(x, y) = f?(x) ? hhy,? , xi. Note that if the vector y is totally
f
f
y
ordered (no two elements are equal to each other), the subgradient of f? and the corresponding
permutation ?y at y will actually be unique. When the vector is not totally ordered, we can consider
?y as a permutation operator which defines a valid and consistent total ordering for every vector y,
and we can then define the Bregman divergence in terms of it. Note also that the points with no total
ordering in the interior of the hypercube is of measure zero. Hence for simplicity we just refer to the
Lov?asz Bregman divergence as df?. The Lov?asz Bregman divergence is closely related to the lower
bound submodular Bregman, as we show below.
Theorem 2.2. The Lov?asz Bregman divergences are an extension of the lower bound submodular
Bregman, over the interior of the hypercube. Further the Lov?asz Bregman divergence can be expressed
as df?(x, y) = hx, hx,?x ? hy,?y i, and hence depends only x, the permutation ?x and the permutation
of y(?y ), but is independent of the values of y.
5
3
Properties of the submodular Bregman and Lov?asz Bregman divergences
In this section, we investigate some of the properties of the submodular Bregman and Lov?asz
Bregman divergences which make these divergences interesting for Machine Learning applications.
We only state them here ? for an elaborate discussion refer to [12]. All forms of the submodular
Bregman divergences are non-negative, and hence they are valid divergences. The lower bound
submodular Bregman is submodular in X for a given Y , while the upper bound submodular Bregman
is supermodular in Y for a given X. A direct consequence of this is that problems involving
optimization in X or Y (for example in finding the discrete representatives in a discrete k-means
like application which we consider in [12]), can be performed either exactly or approximately in
polynomial time. In addition to these the forms of the submodular Bregman divergence also satisfy
interesting properties like a characterization of equivalence classes, a form of set separation, a
generalized triangle inequality over sets and a form of both Fenchel and submodular duality. Finally
the generalized submodular Bregman divergence has an interesting alternate characterization, which
shows that they can potentially subsume a large number of discrete divergences. In particular, a
H
divergence d is of the form df f iff for any sets A, B ? V , the set function fA (X) = d(X, A) is
submodular in X and the set function d(X, A) ? d(X, B) is modular in X. Similarly a divergence d
is of the form dfG f iff, for any set A, B ? V , the set function fA (Y ) = d(A, Y ) is supermodular in
Y and the set function d(A, Y ) ? d(B, Y ) is modular in Y . These facts show that the generalized
Bregman divergences are potentially a very large class of divergences while Table 1 provides just a
few of them.
Additionally, the Lov?asz Bregman divergence also has a number of very interesting properties.
Notable amongst these is the fact that it has an interesting property related to permutations.
Theorem 3.1. [12] Given
p a submodular function whose polyhedron contains all possible extreme
points (e.g., f (X) = |X|), df?(x, y) = 0 if and only if ?x = ?y .
Hence the Lov?asz Bregman divergence can be seen as a divergence between the permutations. While
a number of distance measures capture the notion of a distance amongst orderings [17], the Lov?asz
Bregman divergences has a unique feature not present in these distance measures. The Lov?asz
Bregman divergences not only capture the distance between ?x and ?y , but also weighs it with the
value of x, thus giving preference to the values and not just the orderings. Hence it can be seen
as a divergence between a score x and a permutation ?y , and hence we shall also represent it as
df?(x, y) = df?(x||?y ) = hx, hx,?x ? hx,?y i. Correspondingly, given a collection of scores, it also
measures how confident the scores are about the ordering. For example given two scores x and y
with the same orderings such that the values of x are nearly equal (low confidence), while the values
of y have large differences, the distance to any other permutation will be more for y than x. This
property intuitively desirable in a permutation based divergence. Finally, as we shall see the Lov?asz
Bregman divergences are easily amenable to k-means style alternating minimization algorithms for
clustering ranked data, a process that is typically difficult using other permutation-based distances.
4
Applications
In this section, we show the utility of the submodular Bregman and Lov?asz Bregman divergences by
considering some practical applications in machine learning and optimization. The first application
is that of proximal algorithms which generalize several mirror descent algorithms. As a second
application, we motivate the use of the Lov?asz Bregman divergence as a natural choice in clustering
where the order is important. Due to lack of space, we only concisely describe these applications,
and for a more elaborate discussion please see [12] where we also consider a third discrete clustering
application, and provide a clustering framework for the submodular Bregman with fast algorithms
for clustering sets of binary vectors
4.1
A proximal framework for the submodular Bregman divergence
The Bregman divergence has some nice properties related to a proximal method. In particular ([5]), let
? be a convex function that is hard to optimize, but suppose the function ?(x) + ?d? (x, y) is easy to
6
optimize for a given fixed y. Then a proximal algorithm, which starts with a particular x0 and updates
at every iteration xt+1 = argmaxx ?(x) + ?d? (x, xt ), is bound to converge to the global minima.
We define a similar framework for the submodular Bregmans. Consider a set function Algorithm 1: Proximal Minimization Algorithm
F , and an underlying combinatorial constraint X 0 = ?
S. Optimizing this set function may not be while until convergence do
easy ? e.g., if S is the constraint that X be a
X t+1 := argminX?S F (X) + ?d(X, X t )
graph-cut, then this optimization problem is NP
t?t+1
hard even if F is submodular ([15]). Consider
now a divergence d(X, Y ) that can be either an upper or lower bound submodular Bregman. Note,
the combinatorial constraints S are the discrete analogs of the convex set projection in the proximal
method. We offer a proximal minimization algorithm (Algorithm 1) in a spirit similar to [5].
Furthermore, Algorithm 1 is guaranteed to monotonically decrease the function value over the
iterations [12]. Interestingly, a number of approximate optimization problems considered in the past
turn out to be special cases of the proximal framework. We show this below:
Minimizing the difference between submodular (DS) functions: Consider the case where
F (X) = f (X) ? g(X) is a difference between two submodular functions f and g. This problem
is known to be NP hard and even NP hard to approximate [22, 11]. However there are a number
of heuristic algorithms which have been shown to perform well in practice [22, 11]. Consider first:
t
t
d(X, X t ) = d?
g (X, X ) (for some appropriate schedule ?t of permutations), with ? = 1 and
V
S = 2 . Then it can be shown trivially [12] that we obtain the submodular-supermodular (sub-sup)
procedure ([22]). Moreover, we can define d(X, X t ) = df1:3 (X t , X), again with ? = 1 and S = 2V .
Then again we can show [12] that we obtain the supermodular-submodular (sup-sub) procedure [11].
t
t
Finally defining d(X, X t ) = df1:3 (X t , X) + d?
g (X, X ), we get the modular-modular (mod-mod)
procedure [11]. Further, the sup-sub and mod-mod procedures can be used with more complicated
constraints like cardinality, matroid and knapsack constraints while the mod-mod algorithm can be
extended with even combinatorial constraints like the family of cuts, spanning trees, shortest paths,
covers, matchings, etc. [11]
Submodular function minimization: Algorithm 1 also generalizes a number of approximate
submodular minimization algorithms. If F is a submodular function and the underlying constraints
S represent the family of cuts, then we obtain the cooperative cut problem ([15], [14]) and one of
the algorithms developed in ([15]) is a special case of Algorithm 1. If S = 2V above, we get a
form of the approximate submodular minimization algorithm suggested for arbitrary (non-graph
representable) submodular functions ([16]). The proximal minimization algorithm also generalizes
three submodular function minimization algorithms IMA-I, II and III, described in detail in [12]
again with ? = 1, S = 2V and d = df1 , df2 and df3 respectively. These algorithms are similar to
the greedy algorithm for submodular maximization [23]. Interestingly these algorithms provide
bounds to the lattice of minimizers of the submodular functions. It is known [1] that the sets
A = {j : f (j|?) < 0}, B = {j : f (j|V ? {j}) > 0 are such that, for every minimizer X ? ,
A ? X ? ? B. Thus the lattice formed with A and B defined as the join and meet respectively, gives
a bound on the minimizers, and we can restrict the submodular minimization algorithms to this lattice.
However using d = df3 as a regularizer (which is IMA-III) and starting with X 0 = ? and X 0 = V , we
get the sets A and B [10, 12] respectively from Algorithm 1. With versions of algorithm 1 with d = df1
and d = df2 , and starting respectively from X 0 = ? and X 0 = V , we get sets that provide a tighter
bound on the lattice of minimizers than the one obtained with A and B. Further these algorithms
also provide improved bounds in the context of monotone submodular minimization subject to
combinatorial constraints. In particular, these algorithms provide bounds which are better than ?1 ,
where ? is a parameter related to the curvature of the submodular function. Hence when the parameter
? is a constant, these bounds are constant factor guarantees, which contrasts the O(n) bounds for
most of these problems. For a more elaborate and detailed discussion related to this, refer to [10]
Submodular function maximization: If f is a submodular function, then using d(X, X v ) =
v
v
d?
f (X, X ) forms an iterative algorithm for maximizing the modular lower bound of a submodular
function. This algorithm then generalizes a algorithms number of unconstrained submodular
maximization and constrained submodular maximization, in that by an appropriate schedule of ?v
we can obtain these algorithms. Notable amongst them is a 12 approximate algorithm and a 1 ? 1e
7
http:
http:
//youtu.be/IqRhemUg14I
//youtu.be/kfEnLOmvEVc
Figure 1: Results of k-means clustering using the Lov?asz Bregman divergence (two plots on the left)
and the Euclidean distance (two plots on the right). URLs above link to videos.
approximation algorithm for unconstrained and cardinality constrained submodular maximization
respectively. For a complete list of algorithms generalized by this, refer to [10].
4.2
Clustering framework with the Lov?asz Bregman divergence
In this section we investigate a clustering framework similar to [2], using the Lov?asz Bregman
divergence and show how this is natural for a number of applications.
Recall that the Lov?asz
Bregman divergence in some sense measures the distance between the ordering of the vectors and can
be seen as a form of the ?sorting? distance. We define the clustering problem as given a set of vectors,
find a clustering into subsets of vectors with similar orderings. For example, given a set of voters and
their corresponding ranked preferences, we might want to find subsets of voters who mostly agree.
Let X = {x1 , x2 , ? ? ? , xm } represent a set of m vectors, such that ?i, xi ? [0, 1]n . We first consider
the problem of finding the representative of these vectors. Given a set of vectors X and a Lov?asz
Bregman divergence df?, a natural choice of a representative (in this case a permutation) is the point
Pn
with minimum average distance, or in other words: ? = argmin?0 i=1 df?(xi ||? 0 ). Interestingly
for the Lov?asz Bregman divergence this problem is easy and the representative permutation is exactly
the permutation of the arithmetic mean of X
Theorem P
4.1. [12] Given a submodular function
Pn f , the Lov?asz Bregman representative
n
argmin?0 i=1 df?(xi ||? 0 ) is exactly ? = ?? , ? = n1 i=1 xi
It may not suffice to encode X using a single representative, and hence we partition X into disjoint
blocks C = {C1 , ? ? ? , Ck } with each block having its own Lov?asz Bregman representative, with the
set of representatives given by M = {?1 , ?2 , ? ? ? , ?k }. Then we define an objective, which captures
Pk P
this idea of clustering vectors into subsets of similar orderings: minM,C j=1 xi ?Cj df?(xi , ?j ).
Consider then a k-means like alternating algorithm [19, 21]. It has two stages, often called the assignment and the re-estimation step. In the assignment stage, for every point xi we choose its cluster
membership Cj such that j = argminl df?(xi ||?l ). The re-estimation step involves finding the representatives for every cluster Cj , which is exactly the permutation of the mean of the vectors in Cj . We
skip the algorithm here due to space constraints, and refer the reader to [12] for a complete discussion.
We remark here that a number of distance measures capture the notion of orderings, like the
bubble-sort distance [17], etc. However for these distance measures, finding the representative may
not be easy. The Lov?asz Bregman divergence naturally captures the notion of distance between
orderings of vectors and yet, the problem of finding the representative in this case is very easy.
Finally similar to the analysis in [2, 12], we can show that the k-means algorithm using the Lov?asz
Bregman divergence will monotonically decrease the objective at every iteration, and the algorithm
converges to a local minima. [12] To demonstrate the utility of our clustering framework, we
show some results in 2 and 3 dimensions (Fig. 1), where we compare our framework
p to a k-means
algorithm using the euclidean distance. We use the submodular function f (X) = w(X), for an
arbitrary vector w ensuring unique base extreme points. The results clearly show that the Lov?asz
Bregman divergence clusters the data based on the orderings of the vectors.
Acknowledgments: We thank Stefanie Jegelka, Karthik Narayanan, Andrew Guillory, Hui Lin, John
Halloran and the rest of the submodular group at UW-EE for discussions. This material is based
upon work supported by the National Science Foundation under Grant No. (IIS-1162606), and is also
supported by a Google, a Microsoft, and an Intel research award.
8
References
[1] F. Bach. Learning with Submodular functions: A convex Optimization Perspective. Arxiv preprint, 2011.
[2] A. Banerjee, S. Meregu, I. S. Dhilon, and J. Ghosh. Clustering with Bregman divergences. JMLR,
6:1705?1749, Oct. 2005.
[3] E. Boros and P. L. Hammer. Pseudo-boolean optimization. Discrete Applied Math., 123(1?3):155 ? 225,
2002.
[4] L. Bregman. The relaxation method of finding the common point of convex sets and its application to the
solution of problems in convex programming. USSR Comput. Math and Math Physics, 7, 1967.
[5] Y. Censor and S. Zenios. Parallel optimization: Theory, algorithms, and applications. Oxford University
Press, USA, 1997.
[6] I. Dhillon and J. Tropp. Matrix nearness problems with Bregman divergences. SIAM Journal on Matrix
Analysis and Applications, 29(4):1120?1146, 2007.
[7] J. Edmonds. Submodular functions, matroids and certain polyhedra. Combinatorial structures and their
Applications, 1970.
[8] B. Frigyik, S. Srivastava, and M. Gupta. Functional Bregman divergence. In In Proc. ISIT, pages
1681?1685. IEEE, 2008.
[9] S. Fujishige. Submodular functions and optimization, volume 58. Elsevier Science, 2005.
[10] R. Iyer and J. Bilmes. A framework of mirror descent algorithms for submodular optimization. To
Appear in NIPS Workshop on Discrete Optimization in Machine Learning (DISCML) 2012- Structure and
Scalability, 2012.
[11] R. Iyer and J. Bilmes. Algorithms for approximate minimization of the difference between submodular
functions, with applications. In Proc. UAI, 2012.
[12] R. Iyer and J. Bilmes. Submodular-Bregman and the Lov?asz-Bregman Divergences with Applications:
Extended Version. 2012.
[13] R. Iyer and J. Bilmes. A unified theory on the generalized bregman divergences. Manuscript, 2012.
[14] S. Jegelka and J. Bilmes. Cooperative cuts: Graph cuts with submodular edge weights. Technical report,
Technical Report TR-189, Max Planck Institute for Biological Cybernetics, 2010.
[15] S. Jegelka and J. Bilmes. Submodularity beyond submodular energies: coupling edges in graph cuts. In
Computer Vision and Pattern Recognition (CVPR), Colorado Springs, CO, June 2011.
[16] S. Jegelka, H. Lin, and J. Bilmes. Fast approximate submodular minimization. In Proc. NIPS, 2011.
[17] M. Kendall. A new measure of rank correlation. Biometrika, 30(1/2):81?93, 1938.
[18] K. C. Kiwiel. Free-steering relaxation methods for problems with strictly convex costs and linear constraints.
Mathematics of Operations Research, 22(2):326?349, 1997.
[19] S. Lloyd. Least squares quantization in pcm. IEEE Transactions on IT, 28(2):129?137, 1982.
[20] L. Lov?asz. Submodular functions and convexity. Mathematical Programming, 1983.
[21] J. MacQueen et al. Some methods for classification and analysis of multivariate observations. In
Proceedings of the fifth Berkeley symposium on math. stats and probability, volume 1, pages 281?297.
California, USA, 1967.
[22] M. Narasimhan and J. Bilmes. A submodular-supermodular procedure with applications to discriminative
structure learning. In Uncertainty in Artificial Intelligence (UAI), Edinburgh, Scotland, July 2005.
[23] G. Nemhauser, L. Wolsey, and M. Fisher. An analysis of approximations for maximizing submodular set
functions?i. Mathematical Programming, 14(1):265?294, 1978.
[24] P. Stobbe and A. Krause. Efficient minimization of decomposable submodular functions. In Proc. Neural
Information Processing Systems (NIPS), 2010.
[25] M. Telgarsky and S. Dasgupta. Agglomerative bregman clustering. In Proc. ICML, 2012.
[26] K. Tsuda, G. Ratsch, and M. Warmuth. Matrix exponentiated gradient updates for on-line learning and
Bregman projection. JMLR, 6(1):995, 2006.
[27] M. K. Warmuth. Online learning and Bregman divergences. Tutorial at the Machine Learning Summer
School, 2006.
9
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4,080 | 4,693 | Robustness and risk-sensitivity in Markov decision
processes
Takayuki Osogami
IBM Research - Tokyo
5-6-52 Toyosu, Koto-ku, Tokyo, Japan
[email protected]
Abstract
We uncover relations between robust MDPs and risk-sensitive MDPs. The objective of a robust MDP is to minimize a function, such as the expectation of cumulative cost, for the worst case when the parameters have uncertainties. The objective of a risk-sensitive MDP is to minimize a risk measure of the cumulative cost
when the parameters are known. We show that a risk-sensitive MDP of minimizing the expected exponential utility is equivalent to a robust MDP of minimizing
the worst-case expectation with a penalty for the deviation of the uncertain parameters from their nominal values, which is measured with the Kullback-Leibler
divergence. We also show that a risk-sensitive MDP of minimizing an iterated
risk measure that is composed of certain coherent risk measures is equivalent to
a robust MDP of minimizing the worst-case expectation when the possible deviations of uncertain parameters from their nominal values are characterized with a
concave function.
1
Introduction
Robustness against uncertainties and sensitivity to risk are major issues that have been addressed
in recent development of the Markov decision process (MDP). The robust MDP [3, 4, 10, 11, 12,
20, 21] deals with uncertainties in parameters; that is, some of the parameters of the MDP are not
known exactly. The objective of a robust MDP is to minimize a function for the worst case when
the values of its parameters vary within a predefined set called an uncertainty set. The standard
objective function is the expected cumulative cost [11]. When the uncertainty set is trivial, the
robust MDP is reduced to the standard MDP [17]. The risk-sensitive MDP [5, 7, 13, 14, 19], on the
other hand, assumes that the parameters are exactly known. The objective of a risk-sensitive MDP is
to minimize the value of a risk measure, such as the expected exponential utility [5, 7, 8, 15, 18], of
the cumulative cost. When the risk measure is expectation, the risk-sensitive MDP is reduced to the
standard MDP. The robust MDP and the risk-sensitive MDP have been developed independently.
The goal of this paper is to reveal relations between these two seemingly unrelated models of MDPs.
Such unveiled relations will provide insights into the two models of MDPs. For example, it is not
always clear what it means to minimize the value of a risk measure or to minimize the worst case
expected cumulative cost under an uncertainty set. In particular, the iterated risk measure studied
in [13, 14, 19] is defined recursively, which prevents an intuitive understanding of its meaning. The
unveiled relation to a robust MDP can allow us to understand what it means to minimize the value of
an iterated risk measure in terms of uncertainties. In addition, the optimal policy for a robust MDP
is often found too conservative [3, 4, 10, 21], or it becomes intractable to find the optimal policy
particularly when the transition probabilities have uncertainties [3, 4, 10]. The unveiled relations to
a risk-sensitive MDP, for which the optimal policy can be found efficiently, can allow us to find the
optimal robust policy efficiently, avoiding that the policy is too conservative. We will explore these
possibilities.
1
The contributions of this paper can be summarized in two points. First, we prove that a risk-sensitive
MDP with the objective of minimizing the value of an iterated risk measure is equivalent to a robust MDP with the objective of minimizing the expected cumulative cost for the worst case when
the probability mass functions for the transition and cost from a state have uncertainties. More
specifically, the iterated risk measure of the risk-sensitive MDP is defined recursively with a class of
coherent risk measures [9], and it evaluates the riskiness of the sum of the value of a coherent risk
measure of immediate cost. The uncertainty set of the robust MDP is characterized by the use of a
representation of the coherent risk measure. See Section 2.
Second, we prove that a risk-sensitive MDP with the objective of minimizing an expected exponential utility is equivalent to a robust MDP whose objective is to minimize the expected cumulative
cost minus a penalty function for the worst case when the probability mass functions for the transition and cost from a state have uncertainties. More specifically, the expected exponential utility
evaluates the riskiness of the sum of the value of an entropic risk measure [6] of immediate cost.
The penalty function measures the deviation of the values of the probability mass functions from
their nominal values using the Kullback-Leibler divergence. See Section 3.
2
Robust representations of iterated coherent risk measures
Throughout this paper, we consider Markov decision processes over a finite horizon, so that there
are N decision epochs. Let Sn be the set of possible states at the n-th decision epoch for n =
0, . . . , N ? 1. Let A(s) be the set of candidate actions from the state, s. We assume that a nominal
transition probability, p0 (s0 |s, a), is associated with the transition from each state s ? Sn to each
state s0 ? Sn+1 given that the action a ? A(s) is taken at s for n = 0, . . . , N ? 1. For a robust MDP,
the corresponding true transition probability, p(s0 |s, a), has the uncertainty that will be specified in
the sequel. The random cost, C(s, a), is associated with each pair of a state, s, and an action,
a ? S(a). We assume that C(s, a) has a nominal probability distribution, but the true probability
distribution for a robust MDP has the uncertainty that will be specified in the sequel. We assume
that Si and Sj are disjoint for any j 6= i (e.g., the state space is augmented with time).
2.1
Special case of the iterated conditional tail expectation
We start by studying a robust MDP where the uncertainty is specified by the factor, ?, such that
0 < ? < 1, which determines the possible deviation from the nominal value. Specifically, for each
pair of s ? Sn and a ? A(s), the true transition probabilities are in the following uncertainty set:
X
1
0 ? p(s0 |s, a) ? p0 (s0 |s, a), ?s0 ? Sn+1 and
p(s0 |s, a) = 1.
(1)
?
0
s ?Sn+1
Throughout Section 2.1, we assume that the cost C(s, a) is deterministic and has no uncertainty.
Because the uncertainty set (1) is convex, the existing technique [11] can be used to efficiently find
the optimal policy that minimizes the expected cumulative cost for the worst case where the true
probability is chosen to maximize the expected cumulative cost within the uncertainty set:
?
min max Ep [C(?)],
(2)
?
p?Up
?
where C(?)
is the cumulative cost with a policy ?, and Ep is the expectation with respect to p, which
is chosen from the uncertainty set, Up , defined with (1).
Our key finding is that there is a risk-sensitive MDP that is equivalent to the robust MDP having
the objective (2). Specifically, consider the risk-sensitive MDP, where the transition probability is
given by p0 , and the cost C(s, a) is deterministic given s and a. This risk-sensitive MDP becomes
equivalent to the robust MDP having the objective (2) when the objective of the risk-sensitive MDP
is to find the optimal ? with respect to an iterated conditional tail expectation (ICTE) [13]:
(N ) ?
min ICTE?
[C(?)],
(3)
?
(N )
where ICTE? denotes the ICTE defined for N decision epochs with parameter ?. Specifically,
)
ICTE(N
is defined recursively with conditional tail expectation (CTE) as follows [13]:
?
h
i
h
h
ii
(N ?i+1) ?
?i) ?
C(?) ? CTE? ICTE(N
C(?)|Si , for i = 1, . . . , N,
(4)
ICTE?
?
2
(5)
(N ?i) ?
ICTE?
[C(?)|Si ]
?
ICTE(0)
? [C(?)]
?
?
where we define
? C(?).
In (4),
denotes the ICTE of C(?)
(N ?i) ?
conditioned on the state at the i-th decision epoch. When Si is random, so is ICTE?
[C(?)|Si ].
?i) ?
The right-hand side of (4) evaluates the CTE of this random ICTE(N
[
C(?)|S
].
CTE
is also
i
?
known as conditional value at risk or average value at risk and is formally defined as follows for a
random variable Y :
(1 ? ?)E[Y |Y > V? ] + (? ? ?)V?
CTE? [Y ] ?
,
(6)
1??
where V? ? min{y | FY (y) ? ?}, and FY is the cumulative distribution function of Y . For a
continuous Y , or unless there is a mass probability at V? , we have CTE? [Y ] = E[Y |Y > V? ].
The equivalence between the robust MDP with the objective (2) and the risk-sensitive MDP with the
objective (3) can be shown by the use of the following alternative definition of CTE:
CTE? [Y ] ? max Eq [Y ],
(7)
q?Q
where Q is the set of probability mass functions, q, whose support is a subset of the probability mass
function, q0 , of Y such that q(y) ? q0 (y)/? for every y in the support of q0 . Specifically, let Ci? be
?
?
the cost incurred at the i-th epoch with policy ? so that C(?)
= C0? + ? ? ? + CN
?1 . Then, by the
1
recursive definition of ICTE and the translation invariance of CTE, it can be shown that
h
i
(N ?i+1) ?
ICTE?
C(?) | Si?1
?
?
?
?
N
?1
i?1
X
X
?i) ?
Cj? | Si ? | Si?1 ?
(8)
=
Ci? + CTE? ?Ci? + ICTE(N
?
j=i+1
j=0
=
i?1
X
j=0
?
?
(N ?i) ?
Ci? + max Ep ?Ci? + ICTE?
p?Up
N
?1
X
?
?
Cj? | Si ? | Si?1 ? ,
(9)
j=i+1
where the second equality follows from (7). What (9) suggests is that the ICTE of the cumulative
?
cost given Si can be represented by the cost already accumulated C0? +? ? ?+Ci?1
plus the maximum
possible expected value of the sum of the cost incurred at Si and the ICTE of the cumulative cost to
be incurred from the (i + 1)st epoch. Induction can now be used to establish the following theorem,
which will be proved formally for the general case in Section 2.3:
Theorem 1. When the immediate cost from a state is deterministic given that state and the action
from that state, the risk-sensitive MDP with the objective (3) is equivalent to the robust MDP with
the objective (2).
Throughout, we say that a risk-sensitive MDP is equivalent to a robust MDP if the two MDPs have a
common state space, and, regardless of the values of the parameters of the MDPs, the optimal action
for one MDP coincides with that for the other for every state.
2.2
Relation between cost uncertainty and risk-sensitivity
In addition to the transition probabilities, we now assume that the probability distribution of cost has
uncertainty. Specifically, for each pair of s ? Sn and a ? A(s), the true probability mass function2 ,
f (?|s, a), for the random cost, C(s, a), is in the following uncertainty set that is characterized with
the nominal probability mass function, f0 (?|s, a):
X
1
f (x|s, a) = 1,
(10)
0 ? f (x|s, a) ? f0 (x|s, a), ?x ? X and
?
s?X (s,a)
where X (s, a) is the support of C(s, a). Because the uncertainty sets, (1) and (10), are both convex,
the existing technique [11] can still be used to efficiently find the optimal policy with respect to
?
min max Ep,f [C(?)],
(11)
?
1
2
p?Up ,f ?Uf
CTE? [Y + b] = CTE? [Y ] + b for a random Y and a deterministic b.
Continuous cost is discussed in the supplementary material.
3
y
y
y
1
q4
q3
q3
y=t /?
q2
q1
1
q4
q3
q2
q2
q1
q1
1
p1
p2 p3
p4
g = g1 + g2
1
g2
g1
1
1
t
p1
(a) g(t) = min{t/?, 1}
p2 p3
p4
t
(b) concave g
p1
p2 p3
p4
t
(c) piecewise linear g
Figure 1: An illustration of the probabilities that give the worst case expectation.
where Uf is defined analogously to Up .
Again, our key finding is that there is a risk-sensitive MDP that is equivalent to the robust MDP
having the objective (11). To define the objective of the equivalent risk-sensitive MDP, let D(s, a) ?
?
CTE? [C(s, a)] and let D(?)
be the cumulative value of D(s, a) along the sequence of (s, a) with a
policy ?. Then the objective of the equivalent risk-sensitive MDP is given by
h
i
) ?
D(?)
.
(12)
min ICTE(N
?
?
By first applying (7) to D(s, a) and following the arguments that have led to Theorem 1, we can
establish the following theorem, which will be proved formally for the general case in Section 2.3:
Theorem 2. The risk-sensitive MDP with the objective (12) is equivalent to the robust MDP with
the objective (11).
2.3
General case of coherent risk measures
The robust MDPs considered in Section 2.1 and Section 2.2 are not quite flexible, can lead to too
conservative policies depending on the value of ?, or might be too sensitive to the particular value
of ?. We now introduce a broader class of robust MDPs and equivalent risk-sensitive MDPs.
To define the broader class of robust MDPs, we study the uncertainty set of (1) and (10) in more
detail. Given a random variable that takes value vi with nominal probability pi for i = 1, . . . , m, a
step of finding the optimal robust policy calculates the maximum possible expected value:
q1 v1 + ? ? ? + qm vm
max
q
1
pi , ?i = 1, . . . m
?
q1 + ? ? ? + qm = 1.
0 ? qi ?
s.t.
(13)
Without loss of generality, let v1 > v2 > . . . > vm . Then the optimal solution to (13) can be
illustrated with Figure 1(a): for i = 1, . . . , m, the optimal solution q ? (q1 , . . . , qm ) satisfies
!
i
i
X
X
q` = g
p` ,
(14)
`=1
`=1
where g(t) = min{t/?, 1}. Relaxing the constraints in (13), we obtain the following optimization
problem, whose optimal solution is still given by (14):
max
q
s.t.
q1 v1 + ? ? ? + qm vm
!
i
i
X
X
q` ? g
p` , ?i = 1, . . . , m
`=1
(15)
`=1
0 ? qi , ?i = 1, . . . , m.
The inflexibility of (15) stems from the inflexibility of g(t) = min{t/?, 1}, which has only one
adjustable parameter, ?. When ? is small (specifically, 0 < ? ? 1 ? pm ), some of the qi s become
4
zero. This means that the corresponding optimistic cases (those resulting in small vi s) are ignored.
Otherwise (specifically, 1?pm < ? ? 1), the uncertainty set can become too small as qi ? pi /?, ?i.
This inflexibility motivates us to generalize g to a concave function such that g(0) = 0 and g(1) = 1
(see Figure 1(b)). The optimal solution to (15) with the concave g is still given by (14). With an
appropriate g, we can consider a sufficiently large uncertainty set for the pessimistic cases (e.g.,
q1 p1 ) and at the same time consider the possibility of the optimistic cases (e.g. qm > 0).
To formally define the uncertainty set for p(s0 |s, a), s ? Sn and a ? A(s), with the concave g,
let Qp/p0 (?) denote the quantile function of a random variable that takes value p(s0 |s, a)/p0 (s0 |s, a)
with probability p0 (s0 |s, a) for s0 ? Sn+1 . Analogously, let Qf /f0 (?) denote the quantile function of
a random variable that takes value f (x|s, a)/f0 (x|s, a) with probability f0 (x|s, a) for x ? X (s, a).
Then p(s0 |s, a) and f (x|s, a) are in the uncertainty set iff we have, for 0 < t < 1, that
Z 1
Z 1
(16)
Qf /f0 (u) du ? g(t).
Qp/p0 (u) du ? g(t) and
1?t
1?t
Now (7) suggests that expectation with respect to the q illustrated in Figure 1(a) is the CTE with
parameter ? with respect to the corresponding p. It can be shown that the expectation with respect
to the q illustrated in Figure 1(b) is a coherent risk measure, CRM, of the following form [9]:
Z 1
CRMH [Y ] =
CTE? [Y ] dH(?),
(17)
0
for a nondecreasing function H such that H(0) = 0 and H(1) = 1, where Y denotes a generic
random variable. Notice that (17) is a weighted average of CTE? [Y ] for varying ?s. One can
balance the weights on worse cases (higher ?) and the weights on better cases (lower ?).
?
Let K(s, a) ? CRMH [C(s, a)] and let K(?)
be the cumulative value of K(s, a) along the sequence
?
of (s, a) with a policy, ?. We define an iterated coherent risk measure (ICRM) of K(?)
as follows:
h
i
h
h
ii
(N ?i+1) ?
(N ?i) ?
ICRMH
K(?) ? CRMH ICRMH
K(?)|Si , for i = 1, . . . , N, (18)
(0) ?
?
where ICRMH [K(?)]
? K(?).
Now we are ready to prove the general results in this section.
Theorem 3. Consider the risk-sensitive MDP with the following objective:
i
h
(N ) ?
.
min ICRMH K(?)
?
This risk-sensitive MDP is equivalent to the robust MDP with the objective (11) if
Z 1
dg(t)
1
=
dH(s) for 0 < t < 1.
dt
t s
(19)
(20)
To gain an intuition behind (20), consider the g illustrated in Figure 1(c), where g1 (t) =
min{x/?1 , r1 }, g2 (t) = min{x/?2 , r2 }, and g(t) = g1 (t) + g2 (t) for 0 ? t ? 1. The expectation
with respect to the q illustrated in Figure 1(c) can be represented by r1 CRM?1 [?] + r2 CRM?2 [?]
with respect to the corresponding p. The H is thus a piecewise constant function with a step of size
ri at ?i for i = 1, 2. The slope dg(t)
dt is either 1/?1 + 1/?2 , 1/?2 , or 0, depending on the particular
value of t in such a way that (20) holds.
Proof of Theorem 3. Notice that Bellman?s optimality equations are satisfied both for the robust
MDP and for the risk-sensitive MDP under consideration. For the robust MDP, Bellman?s optimality equations are established in [11]. For our risk-sensitive MDP, note that the coherent risk measure
satisfies strong monotonicity, translation-invariance, and positive homogeneity that are used to establish Bellman?s optimality equations in [13]. A difference between the risk-sensitive MDP in [13]
?
and our risk-sensitive MDP is that the former minimizes the value of an iterated risk measure for C,
(0)
? This
while the latter minimizes the value of an iterated risk measure (specifically, ICRMH ) for K.
difference does not affect whether Bellman?s optimality equations are satisfied.
5
The equivalence between our risk-sensitive MDP and our robust MDP can thus be established by
showing that the two sets of Bellman?s optimality equations are equivalent. For s ? Sn , Bellman?s
optimality equation for our robust MDP is
?
?
X
X
v(s0 ) p(s0 |s, a)? ,
(21)
v(s) = min
max ?
x f (x|s, a) +
a?A(s) p?Up ,f ?Uf
s0 ?Sn+1
x?X (s,a)
where v(s) denotes the value function representing the worst-case expected cumulative cost from s.
For s ? Sn , Bellman?s optimality equation for our risk-sensitive MDP is
w(s)
min CRMH [CRMH [C0 (s, a)] + W (s, a)] ,
=
(22)
a?A(s)
where w(s) denotes the value function representing the value of the iterated coherent risk measure
from s, C0 (s, a) is a random variable that takes value x with probability f0 (x|s, a) for x ? X (s, a),
and W (s, a) is a random variable that takes value w(s0 ) with probability p0 (s0 |s, a) for s0 ? Sn+1 .
Note that the inner CRMH is calculated with respect to f0 (?|s, a); the outer CRMH is with respect
to p0 (?|s, a). In the following, we will show the equivalence between (21) and (22).
We first show the following equality:
X
max
x f (x|s, a)
f ?Uf
=
CRMH [C0 (s, a)].
(23)
x?X (s,a)
Let {x1 , x2 , . . .} = X (s, a) such that x1 > x2 > . . .. As we have seen with Figure 1, the maximizer,
f ? , of the left-hand side of (23) satisfies
!
i
i
X
X
f ? (x` |s, a) = g
f0 (x` |s, a)
(24)
`=1
`=1
for i = 1, 2, . . .. For brevity, let Ri ?
max
f ?Uf
X
x f (x|s, a) =
Pi
X
`=1
Z
xi
Ri?1
i
x?X (s,a)
f0 (x` |s, a). Then we have
X Z Ri Z
dg(t) =
xi
Ri
i
Ri?1
1
t
1
dH(u) dt,
u
(25)
where the first equality follows from (24), and the second equality follows from (20). Exchanging
the integrals in the last expression, we obtain
Z min{u,Ri }
X
XZ 1
1
(26)
max
x f (x|s, a) =
xi
dt dH(u)
f ?Uf
u
Ri?1
Ri?1
i
x?X (s,a)
XZ 1
1
=
(27)
xi (min{u, Ri } ? Ri?1 ) dH(u).
u
Ri?1
i
Exchanging the integral and the summation in the last expression, we obtain
Z 1P
X
i:Ri?1 ?u xi (min{u, Ri } ? Ri?1 )
max
x f (x|s, a) =
dH(u)
f ?Uf
u
0
x?X (s,a)
Z 1
=
CTEu [C0 (s, a)] dH(u),
(28)
(29)
0
which establishes (23). To understand the last equality, plug in the following expressions in (6):
? = 1 ? u, V? = xi? , and ? = 1 ? Ri? ?1 , where i? ? min{i|Ri > u}.
Finally, we show the equivalence between (21) and (22). By (23), we have
?
?
max
X
?
p?Up ,f ?Uf
x f (x|s, a) +
X
0
0
v(s ) p(s |s, a)?= CRMH [C0 (s, a)] + max
Analogously to (23), we can show
X
max
v(s0 ) p(s0 |s, a)
p?Up
v(s0 ) p(s0 |s, a). (30)
p?Up
s0 ?Sn+1
s0 ?Sn+1
x?X (s,a)
X
s0 ?Sn+1
6
= CRMH [V0 (s, a)],
(31)
where V0 (s, a) denotes the random variable that takes value v(s0 ) with probability p0 (s0 |s, a) for
s0 ? Sn+1 . By (30) and (31), we have
v(s)
=
min (CRMH [C0 (s, a)] + CRMH [V (s, a)]) ,
a?A(s)
(32)
where the first CTEH is with respect to f0 (?|s, a); the second is with respect to p0 (?|s, a). Because
f0 (?|s, a) is independent of the state at n + 1, the translation invariance3 of CRMH implies
v(s)
=
min CRMH [CRMH [C0 (s, a)] + V (s, a)],
a?A(s)
(33)
where the inner CRMH is with respect to f0 (?|s, a); the outer is with respect to p0 (?|s, a). Because
v(s0 ) = w(s0 ) = 0, ?s0 ? SN , we can show, by induction, that (33) is equivalent to (22).
3
Robust representations of expected exponential utilities
In this section, we study risk-sensitive MDPs whose objectives are defined with expected exponential
utilities. We will see that there are robust MDPs that are equivalent to these risk-sensitive MDPs.
We start by the standard risk-sensitive MDP [5, 7, 8, 15, 18] whose objective is to minimize
?
?
E[exp(? C(?))]
for ? > 0. Because ? > 0, minimizing E[exp(? C(?))]
is equivalent to mini?
?
mizing an entropic risk measure (ERM) [6, 13]: ERM? [C(?)]
? ?1 ln E[exp(? C(?))].
The key property of ERM that we exploit in this section is
ERM? [Y ]
max {Eq0 [Y ] ? ? KL(q||q0 )} ,
=
q?P(q0 )
(34)
where Y is a generic discrete random variable, q0 is the probability mass function for Y , P(q0 ) is
the set of probability mass functions whose support is contained in the support of q0 (i.e., q(y) = 0
if q0 (y) = 0 for q ? P(q0 )), Eq is the expectation with respect to q ? P(q0 ), and KL(q||q0 ) is the
Kullback-Leibler divergence [2] from q to q0 . The property (34) has been discussed in the context
of optimal control [1, 16]. See [6] for a proof of (34). Observe that the maximizer of the right-hand
side of (34) trades a large value of Eq [Y ] for the closeness of q to q0 .
?
It is now evident that the risk-sensitive MDP with the objective of minimizing E[exp(? C(?))]
is
?
equivalent to a ?robust? MDP with the objective of minimizing Eq [C(?)] ? ? KL(q||q0 ) for the
?
worst choice of q ? P(q0 ), where q0 denotes the probability mass function for C(?).
Here, the
uncertainty is in the distribution of the cumulative cost, and it is nontrivial how this uncertainty is
related to the uncertainty in the parameters, p and f , of the MDP.
?
Our goal is to explicitly relate the risk-sensitive MDP of minimizing E[exp(? C(?))]
to uncertainties
in the parameters of the MDP. For a moment, we assume that C(s, a) has no uncertainty and is
deterministic given s and a, which will be relaxed later.
?
To see the relation, we study ERM? [C(?)]
for a given ?. Let p?0 (si+1 |si ) be the nominal transition
probability from si ? Si to si+1 ? Si+1 for i = 0, . . . , N ? 1. By the translation invariance and the
recursiveness4 of ERM [13], we have
"
"N ?1
##
X
?
?
?
?
ERM? [C(?)] = C + ERM? C + ERM?
C |S1 ,
(35)
0
1
i
i=2
where the inner ERM is with respect to p?0 (?|S1 ); the outer is with respect to p?0 (?|s0 ). By (34), the
second term of the right-hand side of (35) can be represented as follows:
(
"
"N ?1
##
)
X
?
?
?
?
max ?
Ep? (?|s0 ) C1 + ERM?
Ci | S1 ? ? KL (p (?|s0 )||p0 (?|s0 )) (36)
?
p (?|s0 )?P(p0 (?|s0 ))
i=2
Thus, by induction, we can establish the following theorem5 :
3
CRMH [Y + c] = CRMH [Y ] + c for a deterministic constant c.
ERM? [Y + c] = ERM? [Y ] + c and ERM? [Y ] = ERM? [ERM? [Y |Z]], where Y and Z are generic
random variables, and c is a deterministic constant.
5
The proof is omitted, because this is a spacial case of Theorem 5.
4
7
Theorem 4. When the immediate cost from a state is deterministic given that state and the action
?
from that state, the risk-sensitive MDP with the objective of minimizing E[exp(? C(?))]
is equivalent
to the robust MDP with the following objective:
(
"N ?1 #
"N ?2
#)
X
X
min ? max ? Ep?
Ci? ? ? Ep?
KL (p? (?|Si )||p?0 (?|Si ))
,
(37)
?
?
where p ?
p ?P(p0 )
P(p?0 )
i=0
i=0
?
denotes that p (?|si ) ? Pp?0 (?|si ) , ?si ? Si , i = 0, . . . , N ? 1.
Our results in Section 2.2 motivate us to extend Theorem 4 to the case where C(s, a) has uncertainties. Let f0? (?|s) be the nominal probability mass function for the immediate cost from a state s
under a policy ?. Consider the risk-sensitive MDP with the following objective:
h
i
?
min ERM? L(?)
,
(38)
?
?
where L(?)
is the cumulative value of L(s, a) ? ERM? [C(s, a)] along the sequence of (s, a) with
a policy ?. Then we have the following theorem, which is proved in the supplementary material.
Theorem 5. The risk-sensitive MDP with the objective (38) is equivalent to the robust MDP with
the following objective, where f ? ? P(f0? ) is defined analogously to p? ? P(p?0 ) in Theorem 4:
?
?
h
i
? Ep? ,f ? PN ?1 C ?
?
hPi=0 i
i . (39)
min ?max?
P
N
?2
N
?1
?
?
?
?
? p ?P(p0 )? ?? E ?
?
p
i=0 KL (p (?|Si )||p0 (?|Si )) +
i=0 KL (f (?|Si )||f0 (?|Si ))
?
?
f ?P(f0 )
4
Conclusion
We have shown relations between risk-sensitive MDPs and robust MDPs. Because ERM is also an
iterated risk measure [13], the objectives of the risk-sensitive MDPs studied in this paper are all
with respect to some iterated risk measures. The significance of iterated risk measures is intensively
discussed in [13], but it can represent one?s preferences that cannot be represented by standard
expected exponential utility and yet allows efficient optimization and consistent decision making.
?
While the prior work [13, 14, 19] minimizes the iterated risk measure of the cumulative cost (C(?)
in Section 2), our study on the relation to a robust MDP suggests that one might want to minimize
?
the iterated risk measure of the sum of the values of risk measures for immediate costs (e.g., K(?)
in
?
Section 2.3 or L(?)
in Section 3), because the latter is related to the robustness against uncertainty
in cost. The optimal policy with respect to an iterated risk measure can be found efficiently with
dynamic programming (specifically, the computational effort that is required in addition to that of the
dynamic programming for minimizing the expected cumulative cost is in the time to calculate a risk
measure such as CTE instead of expectation at each step of the dynamic programming) [13]. This
means that the optimal policy for the robust MDP studied in this paper can be found quite efficiently.
In particular, the robust MDP in Theorem 5 might not seem to allow an efficient optimization without
the knowledge of the relation to the corresponding risk-sensitive MDP, whose optimal policy is
readily available with dynamic programming. Overall, the relation to a robust MDP can provide
strong motivation for the corresponding risk-sensitive MDP and vice versa.
For simplicity, the uncertainty sets in Section 2 are characterized by a single parameter, ?, or a single
function, g, but it is trivial to extend our results to the cases where the uncertainty sets are defined
differently depending on the particular states, actions, and other elements of the MDP. In such cases,
the objective of the corresponding risk-sensitive MDP is composed of various risk measures. The
uncertainty set in Section 3 depends only on the support of the nominal probability mass function.
The penalty for the deviation from the nominal value can be adjusted with a single parameter, ?, but
it is also trivial to extend our results to the cases, where this parameter varies depending on the particular elements of the MDP. In such cases, the objective of the corresponding risk-sensitive MDP is an
iterated risk measure composed of ERM having varying parameters. It would also be an interesting
direction to extend our results to convex risk measures, which allows robust representations.
8
References
[1] C. D. Charalambous, F. Rezaei, and A. Kyprianou. Relations between information theory,
robustness, and statistical mechanics of stochastic systems. In Proceedings of the 43rd IEEE
Conference on Decision and Control, volume 4, pages 3479?3484, 2004.
[2] T. M. Cover and J. A. Thomas. Elements of Information Theory. John Wiley & Sons, Inc.,
Hoboken, New Jersey, 2nd edition, 2006.
[3] E. Delage and S. Mannor. Percentile optimization in uncertain MDP with application to efficient exploration. In Proceedings of the 24th Annual International Conference on Machine
Learning (ICML 2007), pages 225?232, June 2007.
[4] E. Delage and S. Mannor. Percentile optimization for MDP with parameter uncertainty. Operations Research, 58(1):203?213, 2010.
[5] E. V. Denardo and U. G. Rothblum. Optimal stopping, exponential utility, and linear programming. Mathematical Programming, 16:228?244, 1979.
[6] H. F?ollmer and A. Schied. Stochastic Finance: An Introduction in Discrete Time. Walter de
Gruyter, Berlin, Germany, 3rd edition, 2010.
[7] R. Howard and J. Matheson. Risk-sensitive Markov decision processes. Management Science,
18(7):356?369, 1972.
[8] S. C. Jaquette. A utility criterion for Markov decision processes. Management Science,
23(1):43?49, 1976.
[9] S. Kusuoka. On law invariant coherent risk measures. In S. Kusuoka and T. Maruyama, editors,
Advances in Mathematical Economics, volume 3, pages 83?95. Springer, Tokyo, 2001.
[10] S. Mannor, O. Mebel, and H. Xu. Lightning does not strike twice: Robust MDPs with coupled
uncertainty. In Proceedings of the International Conference on Machine Learning (ICML
2012), pages 385?392, 2012.
[11] A. Nilim and L. El Ghaoui. Robust control of Markov decision processes with uncertain
transition matrices. Operations Research, 53(5):780?798, 2005.
[12] A. Nilim and L. E. Ghaoui. Robustness in Markov decision problems with uncertain transition
matrices. In S. Thrun, L. Saul, and B. Sch?olkopf, editors, Advances in Neural Information
Processing Systems 16. MIT Press, Cambridge, MA, 2004.
[13] T. Osogami. Iterated risk measures for risk-sensitive Markov decision processes with discounted cost. In Proceedings of the 27th Conference on Uncertainty in Artificial Intelligence
(UAI 2011), pages 567?574, July 2011.
[14] T. Osogami and T. Morimura. Time-consistency of optimization problems. In Proceedings of
the 26th Conference on Artificial Intelligence (AAAI-12), July 2012.
[15] S. D. Patek. On terminating Markov decision processes with a risk-averse objective function.
Automatica, 37(9):1379?1386, 2001.
[16] I. R. Petersen, M. R. James, and P. Dupuis. Minimax optimal control of stochastic uncertain systems with relative entropy constraints. IEEE Transactions on Automatic Control,
45(3):398?412, 2000.
[17] M. L. Puterman. Markov Decision Processes: Discrete Dynamic Programming. WileyInterscience, Hoboken, NJ, second edition, 2005.
[18] U. G. Rothblum. Multiplicative Markov decision chains. Mathematics of Operations Research,
9(1):6?24, 1984.
[19] A. Ruszczy?nski. Risk-averse dynamic programming for Markov decision processes. Mathematical Programming, 125:235?261, 2010.
[20] H. Xu and S. Mannor. The robustness-performance tradeoff in Markov decision processes. In
B. Sch?olkopf, J. Platt, and T. Hoffman, editors, Advances in Neural Information Processing
Systems 19, pages 1537?1544. MIT Press, Cambridge, MA, 2007.
[21] H. Xu and S. Mannor. Distributionally robust Markov decision processes. In J. Lafferty, C. K. I.
Williams, J. Shawe-Taylor, R. Zemel, and A. Culotta, editors, Advances in Neural Information
Processing Systems 23, pages 2505?2513. MIT Press, Cambridge, MA, 2010.
9
| 4693 |@word nd:1 c0:9 p0:17 q1:8 minus:1 recursively:3 moment:1 existing:2 com:1 si:29 yet:1 hoboken:2 readily:1 john:1 intelligence:2 mannor:5 preference:1 mathematical:3 along:3 become:2 prove:3 introduce:1 expected:17 p1:4 xz:2 mechanic:1 bellman:7 discounted:1 becomes:2 unrelated:1 mass:13 what:3 minimizes:4 q2:3 developed:1 finding:3 nj:1 every:2 concave:5 finance:1 exactly:2 qm:5 platt:1 control:5 positive:1 rothblum:2 might:3 plus:1 twice:1 studied:3 equivalence:4 suggests:3 relaxing:1 recursive:1 delage:2 petersen:1 cannot:1 risk:91 applying:1 context:1 function2:1 equivalent:21 deterministic:8 williams:1 regardless:1 economics:1 independently:1 convex:3 simplicity:1 insight:1 nominal:12 programming:9 element:3 particularly:1 ep:8 worst:11 calculate:1 culotta:1 averse:2 trade:1 intuition:1 dynamic:6 motivate:1 terminating:1 differently:1 represented:4 various:1 jersey:1 walter:1 artificial:2 zemel:1 quite:2 whose:7 supplementary:2 say:1 otherwise:1 g1:4 nondecreasing:1 seemingly:1 eq0:1 sequence:3 p4:3 matheson:1 iff:1 intuitive:1 olkopf:2 r1:2 depending:4 measured:1 eq:4 strong:2 p2:3 implies:1 direction:1 tokyo:3 koto:1 stochastic:3 exploration:1 icte:19 material:2 dupuis:1 pessimistic:1 summation:1 adjusted:1 hold:1 sufficiently:1 considered:1 exp:6 major:1 vary:1 entropic:2 omitted:1 sensitive:49 vice:1 establishes:1 weighted:1 hoffman:1 mit:3 always:1 pn:1 varying:2 broader:2 q3:3 denardo:1 june:1 stopping:1 el:1 accumulated:1 relation:12 germany:1 issue:1 overall:1 flexible:1 morimura:1 development:1 special:1 having:4 icml:2 piecewise:2 composed:3 dg:3 divergence:3 homogeneity:1 possibility:2 hpi:1 behind:1 unveiled:3 predefined:1 chain:1 integral:2 mebel:1 unless:1 taylor:1 uncertain:6 cover:1 exchanging:2 cost:36 deviation:5 subset:1 too:5 varies:1 nski:1 st:1 international:2 sensitivity:3 sequel:2 vm:3 analogously:4 again:1 aaai:1 satisfied:2 management:2 worse:1 japan:1 de:1 summarized:1 inc:1 explicitly:1 vi:2 depends:1 later:1 multiplicative:1 optimistic:2 start:2 slope:1 contribution:1 minimize:10 efficiently:6 generalize:1 iterated:20 definition:2 against:2 evaluates:3 pp:1 james:1 associated:2 proof:3 gain:1 maruyama:1 proved:3 intensively:1 knowledge:1 cj:2 mizing:1 uncover:1 higher:1 dt:4 generality:1 hand:5 maximizer:2 reveal:1 mdp:85 true:5 former:1 equality:5 q0:17 leibler:3 illustrated:5 deal:1 puterman:1 percentile:2 coincides:1 criterion:1 evident:1 meaning:1 consideration:1 common:1 qp:2 inflexibility:3 jp:1 volume:2 tail:3 discussed:3 extend:4 versa:1 cambridge:3 rd:2 automatic:1 consistency:1 pm:2 mathematics:1 shawe:1 lightning:1 f0:17 v0:2 recent:1 certain:1 seen:1 relaxed:1 maximize:1 strike:1 july:2 ii:2 stem:1 characterized:4 plug:1 calculates:1 qi:4 expectation:14 represent:1 c1:1 addition:3 want:1 addressed:1 sch:2 lafferty:1 seem:1 crm:3 affect:1 inner:3 cn:1 tradeoff:1 whether:1 expression:3 utility:9 effort:1 penalty:4 action:7 ignored:1 clear:1 reduced:2 notice:2 disjoint:1 discrete:3 key:3 v1:3 sum:4 uncertainty:41 throughout:3 p3:3 decision:20 annual:1 nontrivial:1 constraint:2 ri:17 x2:2 argument:1 min:22 optimality:7 uf:8 son:1 osogami:4 making:1 s1:3 invariant:1 ghaoui:2 erm:20 taken:1 ln:1 equation:7 studying:1 available:1 operation:3 observe:1 v2:1 appropriate:1 generic:3 alternative:1 robustness:6 thomas:1 assumes:1 denotes:8 schied:1 exploit:1 quantile:2 establish:4 objective:36 already:1 ruszczy:1 berlin:1 thrun:1 outer:3 fy:2 trivial:3 induction:3 illustration:1 mini:1 minimizing:13 balance:1 relate:1 motivates:1 policy:18 adjustable:1 takayuki:1 markov:14 howard:1 finite:1 immediate:6 pair:3 required:1 specified:3 kl:7 coherent:11 established:2 max:18 representing:2 minimax:1 mdps:17 ready:1 ollmer:1 coupled:1 sn:18 epoch:6 understanding:1 prior:1 relative:1 law:1 loss:1 interesting:1 incurred:3 consistent:1 s0:38 editor:4 pi:4 ibm:2 translation:4 qf:2 last:3 side:4 allow:3 understand:2 saul:1 calculated:1 transition:12 cumulative:20 transaction:1 sj:1 kullback:3 monotonicity:1 rezaei:1 uai:1 q4:2 automatica:1 xi:6 continuous:2 spacial:1 ku:1 robust:48 du:2 significance:1 motivation:1 edition:3 x1:2 augmented:1 xu:3 wiley:1 nilim:2 cte:17 exponential:8 candidate:1 theorem:14 showing:1 r2:2 closeness:1 intractable:1 ci:8 conditioned:1 horizon:1 entropy:1 led:1 explore:1 prevents:1 contained:1 g2:4 springer:1 determines:1 satisfies:3 dh:7 ma:3 gruyter:1 conditional:4 goal:2 specifically:11 conservative:3 called:1 invariance:3 distributionally:1 formally:4 support:6 latter:2 brevity:1 avoiding:1 wileyinterscience:1 |
4,081 | 4,694 | Neuronal Spike Generation Mechanism as an
Oversampling, Noise -shaping A-to-D Converter
Dmitri B. Chklovskii
Janelia Farm Research Campus
Howard Hughes Medical Institute
[email protected]
Daniel Soudry
Department of Electrical Engineering
Technion
[email protected]
Abstract
We test the hypothesis that the neuronal spike generation mechanism is an
analog-to-digital (AD) converter encoding rectified low-pass filtered
summed synaptic currents into a spike train linearly decodable in postsynaptic neurons. Faithful encoding of an analog waveform by a binary
signal requires that the spike generation mechanism has a sampling rate
exceeding the Nyquist rate of the analog signal. Such oversampling is
consistent with the experimental observation that the precision of the spikegeneration mechanism is an order of magnitude greater than the cut -off
frequency of low-pass filtering in dendrites. Additional improvement in the
coding accuracy may be achieved by noise-shaping, a technique used in
signal processing. If noise-shaping were used in neurons, it would reduce
coding error relative to Poisson spike generator for frequencies below
Nyquist by introducing correlations into spike times. By using experimental
data from three different classes of neurons, we demonstrate that biological
neurons utilize noise-shaping. Therefore, the spike-generation mechanism
can be viewed as an oversampling and noise-shaping AD converter.
The nature of the neural spike code remains a central problem in neuroscience [1-3]. In particular,
no consensus exists on whether information is encoded in firing rates [4, 5] or individual spike
timing [6, 7]. On the single-neuron level, evidence exists to support both points of view. On the
one hand, post-synaptic currents are low-pass-filtered by dendrites with the cut-off frequency of
approximately 30Hz [8], Figure 1B, providing ammunition for the firing rate camp: if the signal
reaching the soma is slowly varying, why would precise spike timing be necessary? On the other
hand, the ability of the spike-generation mechanism to encode harmonics of the injected current up
to about 300Hz [9, 10], Figure 1B, points at its exquisite temporal precision [11]. Yet, in view of
the slow variation of the somatic current, such precision may seem gratuitous and puzzling.
The timescale mismatch between gradual variation of the somatic current and high precision of
spike generation has been addressed previously. Existing explanations often rely on the population
nature of the neural code [10, 12]. Although this is a distinct possibility, the question remains
whether invoking population coding is necessary. Other possible explanations for the timescale
mismatch include the possibility that some synaptic currents (for example, GABAergic) may be
generated by synapses proximal to the soma and therefore not subject to low-pass filtering or that
the high frequency harmonics are so strong in the pre-synaptic spike that despite attenuation, their
trace is still present. Although in some cases, these explanations could apply, for the majority of
synaptic inputs to typical neurons there is a glaring mismatch.
The perceived mismatch between the time scales of somatic currents and the spike-generation
mechanism can be resolved naturally if one views spike trains as digitally encoding analog
somatic currents [13-15], Figure 1A. Although somatic currents vary slowly, information that
could be communicated by their analog amplitude far exceeds that of binary signals, such as all-
or-none spikes, of the same sampling rate. Therefore, faithful digital encoding requires sampling
rate of the digital signal to be much higher than the cut-off frequency of the analog signal, socalled over-sampling. Although the spike generation mechanism operates in continuous time, the
high temporal precision of the spikegeneration mechanism may be viewed as a
manifestation of oversampling, which is
needed for the digital encoding of the
analog signal. Therefore, the extra order of
magnitude in temporal precision available
to the spike-generation mechanism
relative to somatic current, Figure 1B, is
necessary to faithfully encode the
amplitude of the analog signal, thus
potentially reconciling the firing rate and
the spike timing points of view [13-15].
Figure 1. Hybrid digital-analog operation of neuronal circuits. A. Post-synaptic currents are
low-pass filtered and summed in dendrites (black) to produce a somatic current (blue). This analog
signal is converted by the spike generation mechanism into a sequence of all-or-none spikes
(green), a digital signal. Spikes propagate along an axon and are chemically transduced across
synapses (gray) into post-synatpic currents (black), whose amplitude reflects synaptic weights,
thus converting digital signal back to analog. B. Frequency response function for dendrites (blue,
adapted from [8]) and for the spike generation mechanism (green, adapted from [9]). Note one
order of magnitude gap between the cut off frequencies. C. Amplitude of the summed postsynaptic currents depends strongly on spike timing. If the blue spike arrives just 5ms later, as
shown in red, the EPSCs sum to a value already 20% less. Therefore, the extra precision of the
digital signal may be used to communicate the amplitude of the analog signal.
In signal processing, efficient AD conversion combines the principle of oversampling with that of
noise-shaping, which utilizes correlations in the digital signal to allow more accurate encoding of
the analog amplitude. This is exemplified by a family of AD converters called ???modulators
[16], of which the basic one is analogous to an integrate-and-fire (IF) neuron [13-15]. The analogy
between the basic ???modulator and the IF neuron led to the suggestion that neurons also use
noise-shaping to encode incoming analog current waveform in the digital spike train [13].
However, the hypothesis of noise-shaping AD conversion has never been tested experimentally in
biological neurons.
In this paper, by analyzing existing experimental datasets, we demonstrate that noise-shaping is
present in three different classes of neurons from vertebrates and invertebrates. This lends support
to the view that neurons act as oversampling and noise-shaping AD converters and accounts for
the mismatch between the slowly varying somatic currents and precise spike timing. Moreover, we
show that the degree of noise-shaping in biological neurons exceeds that used by basic ??
modulators or IF neurons and propose viewing more complicated models in the noise-shaping
framework. This paper is organized as follows: We review the principles of oversampling and
noise-shaping in Section 2. In Section 3, we present experimental evidence for noise-shaping AD
conversion in neurons. In Section 4 we argue that rectification of somatic currents may improve
energy efficiency and/or implement de-noising.
2 . Oversampling and noise-shaping in AD converters
To understand how oversampling can lead to more accurate encoding of the analog signal
amplitude in a digital form, we first consider a Poisson spike encoder, whose rate of spiking is
modulated by the signal amplitude, Figure 2A. Such an AD converter samples an analog signal at
discrete time points and generates a spike with a probability given by the (normalized) signal
amplitude. Because of the binary nature of spike trains, the resulting spike train encodes the signal
with a large error even when the sampling is done at Nyquist rate, i.e. the lowest rate for alias-free
sampling.
To reduce the encoding error a Poisson encoder can sample at frequencies, fs , higher than
Nyquist, fN ? hence, the term oversampling, Figure 2B. When combined with decoding by lowpass filtering (down to Nyquist) on the receiving end, this leads to a reduction of the error, which
can be estimated as follows. The number of samples over a Nyquist half-period (1/2fN) is given by
the oversampling ratio:
.
As the normalized signal amplitude,
, stays roughly constant over
the Nyquist half-period, it can be
encoded by spikes generated with a
fixed probability, x. For a Poisson
process the variance in the number of
spikes is equal to the mean,
. Therefore, the
mean relative error of the signal
decoded by averaging over the Nyquist
half-period:
,
(1)
indicating that oversampling reduces
transmission error. However, the weak
dependence of the error on the
oversampling
frequency
indicates
diminishing returns on the investment
in oversampling and motivates one to
search for other ways to lower the error.
Figure 2. Oversampling and noise-shaping in AD conversion. A. Analog somatic current (blue)
and its digital code (green). The difference between the green and the blue curves is encoding
error. B. Digital output of oversampling Poisson encoder over one Nyquist half-period. C. Error
power spectrum of a Nyquist (dark green) and oversampled (light green) Poisson encoder.
Although the total error power is the same, the fraction surviving low-pass filtering during
decoding (solid green) is smaller in oversampled case. D. Basic ?? modulator. E. Signal at the
output of the integrator. F. Digital output of the ?? modulator over one Nyquist period. G. Error
power spectrum of the ?? modulator (brown) is shifted to higher frequencies and low-pass filtered
during decoding. The remaining error power (solid brown) is smaller than for Poisson encoder.
To reduce encoding error beyond the ? power of the oversampling ratio, the principle of noiseshaping was put forward [17]. To illustrate noise-shaping consider a basic AD converter called ??
[18], Figure 2D. In the basic ?? modulator, the previous quantized signal is fed back and
subtracted from the incoming signal and then the difference is integrated in time. Rather than
quantizing the input signal, as would be done in the Poisson encoder, ?? modulator quantizes the
integral of the difference between the incoming analog signal and the previous quantized signal,
Figure 2F. One can see that, in the oversampling regime, the quantization error of the basic ??
modulator is significantly less than that of the Poisson encoder. As the variance in the number of
spikes over the Nyquist period is less than one, the mean relative error of the signal is at most,
, which is better than the Poisson encoder.
To gain additional insight and understand the origin of the term noise-shaping, we repeat the
above analysis in the Fourier domain. First, the Poisson encoder has a flat power spectrum up to
the sampling frequency, Figure 2C. Oversampling preserves the total error power but extends the
frequency range resulting in the lower error power below Nyquist. Second, a more detailed
analysis of the basic ?? modulator, where the dynamics is linearized by replacing the quantization
device with a random noise injection [19], shows that the quantization noise is effectively
differentiated. Taking the derivative in time is equivalent to multiplying the power spectrum of the
quantization noise by frequency squared. Such reduction of noise power at low frequencies is an
example of noise shaping, Figure 2G. Under the additional assumption of the white quantization
noise, such analysis yields:
,
(2)
which for R >> 1 is significantly better performance than for the Poisson encoder, Eq.(1).
As mentioned previously, the basic ?? modulator, Figure 2D, in the continuous-time regime is
nothing other than an IF neuron [13, 20, 21]. In the IF neuron, quantization is implemented by the
spike generation mechanism and the negative feedback corresponds to the after-spike reset. Note
that resetting the integrator to zero is strictly equivalent to subtraction only for continuous-time
operation. In discrete-time computer simulations, the integrator value may exceed the threshold,
and, therefore, subtraction of the threshold value rather than reset must be used. Next, motivated
by the ??-IF analogy, we look for the signs of noise-shaping AD conversion in real neurons.
3 . Experimental evidence of noise-shaping AD conversion in real neurons
In order to determine whether noise-shaping AD conversion takes place in biological neurons, we
analyzed three experimental datasets, where spike trains were generated by time-varying somatic
currents: 1) rat somatosensory cortex L5 pyramidal neurons [9], 2) mouse olfactory mitral cells
[22, 23], and 3) fruit fly olfactory receptor neurons [24]. In the first two datasets, the current was
injected through an electrode in whole-cell patch clamp mode, while in the third, the recording
was extracellular and the intrinsic somatic current could be measured because the glial
compartment included only one active neuron.
Testing the noise-shaping AD conversion hypothesis is complicated by the fact that encoded and
decoded signals are hard to measure accurately. First, as somatic current is rectified by the spikegeneration mechanism, only its super-threshold component can be encoded faithfully making it
hard to know exactly what is being encoded. Second, decoding in the dendrites is not accessible in
these single-neuron recordings.
In view of these difficulties, we start by simply computing the power spectrum of the
reconstruction error obtained by subtracting a scaled and shifted, but otherwise unaltered, spike
train from the somatic current. The scaling factor was determined by the total weight of the
decoding linear filter and the shift was optimized to maximize information capacity, see below. At
the frequencies below 20Hz the error contains significantly lower power than the input signal,
Figure 3, indicating that the spike generation mechanism may be viewed as an AD converter.
Furthermore, the error power spectrum of the biological neuron is below that of the Poisson
encoder, thus indicating the presence of noise-shaping. For dataset 3 we also plot the error power
spectrum of the IF neuron, the threshold of which is chosen to generate the same number of spikes
as the biological neuron.
4
somatic current
biological neuron error
Poisson encoder error
I&F neuron error
10
1
10
0
Spectral power, a.u.
Spectral power, a.u.
10
3
10
-1
10
-2
10
-3
10
2
10
-4
10
0
10
20
30
40
50
60
Frequency [Hz]
70
80
90
0
10
20
30
40
50
60
70
80
90
100
Frequency [Hz]
Figure 3. Evidence of noise-shaping. Power spectra of the somatic current (blue), difference
between the somatic current and the digital spike train of the biological neuron (black), of the
Poisson encoder (green) and of the IF neuron (red). Left: datset 1, right: dataset 3.
Although the simple analysis presented above indicates noise-shaping, subtracting the spike train
from the input signal, Figure 3, does not accurately quantify the error when decoding involves
additional filtering. An example of such additional encoding/decoding is predictive coding, which
will be discussed below [25]. To take such decoding filter into account, we computed a decoded
waveform by convolving the spike train with the optimal linear filter, which predicts the somatic
current from the spike train with the least mean squared error.
Our linear decoding analysis lends additional support to the noise-shaping AD conversion
hypothesis [13-15]. First, the optimal linear filter shape is similar to unitary post-synaptic currents,
Figure 4B, thus supporting the view that dendrites reconstruct the somatic current of the presynaptic neuron by low-pass filtering the spike train in accordance with the noise-shaping
principle [13]. Second, we found that linear decoding using an optimal filter accounts for 60-80%
of the somatic current variance. Naturally, such prediction works better for neurons in suprathreshold regime, i.e. with high firing rates, an issue to which we return in Section 4. To avoid
complications associated with rectification for now we focused on neurons which were in suprathreshold regime by monitoring that the relationship between predicted and actual current is close
to linear.
2
10
C
D
1
10
somatic current
biological neuron error
Poisson encoder error
Spectral power, a.u.
Spectral power, a.u.
I&F neuron error
3
10
0
10
-1
10
-2
10
-3
10
2
10
-4
0
10
20
30
40
50
60
Frequency [Hz]
70
80
90
10
0
10
20
30
40
50
60
70
80
90
100
Frequency [Hz]
Figure 4. Linear decoding of experimentally recorded spike trains. A. Waveform of somatic
current (blue), resulting spike train (black), and the linearly decoded waveform (red) from dataset
1. B. Top: Optimal linear filter for the trace in A, is representative of other datasets as well.
Bottom: Typical EPSPs have a shape similar to the decoding filter (adapted from [26]). C-D.
Power spectra of the somatic current (blue), the decdoding error of the biological neuron (black),
the Poisson encoder (green), and IF neuron (red) for dataset 1 (C) dataset 3 (D).
Next, we analyzed the spectral distribution of the reconstruction error calculated by subtracting the
decoded spike train, i.e. convolved with the computed optimal linear filter, from the somatic
current. We found that at low frequencies the error power is significantly lower than in the input
signal, Figure 4C,D. This observation confirms that signals below the dendritic cut-off frequency
of 20-30Hz can be efficiently communicated using spike trains.
To quantify the effect of noise-shaping we computed information capacity of different encoders:
where S(f) and N(f) are the power spectra of the somatic current and encoding error
correspondingly and the sum is computed only over the frequencies for which S(f) > N(f).
Because the plots in Figure 4C,D use semi-logrithmic scale, the information capacity can be
estimated from the area between a somatic current (blue) power spectrum and an error power
spectrum. We find that the biological spike generation mechanism has higher information capacity
than the Poisson encoder and IF neurons. Therefore, neurons act as AD converters with stronger
noise-shaping than IF neurons.
We now return to the predictive nature of the spike generation mechanism. Given the causal nature
of the spike generation mechanism it is surprising that the optimal filters for all three datasets
carry most of their weight following a spike, Figure 4B. This indicates that the spike generation
mechanism is capable of making predictions, which are possible in these experiments because
somatic currents are temporally correlated. We note that these observations make delay-free
reconstruction of the signal possible, thus allowing fast operation of neural circuits [27].
The predictive nature of the encoder can be captured by a ?? modulator embedded in a predictive
coding feedback loop [28], Figure 5A. We verified by simulation that such a nested architecture
generates a similar optimal linear filter with most of its weight in the time following a spike,
Figure 5A right. Of course such
prediction is only possible for
correlated inputs implying that the
shape of the optimal linear filter
depends on the statistics of the
inputs. The role of predictive coding
is to reduce the dynamic range of the
signal that enters ??, thus avoiding
overloading. A possible biological
implementation for such integrating
feedback
could
be
Ca2+
2+
concentration and Ca dependent
potassium channels [25, 29].
Figure 5. Enhanced ?? modulators. A. ?? modulator combined with predictive coder. In such
device, the optimal decoding filter computed for correlated inputs has most of its weight following
a spike, similar to experimental measurements, Figure 4B. B. Second-order ?? modulator
possesses stronger noise-shaping properties. Because such circuit contains an internal state
variable it generates a non-periodic spike train in response to a constant input. Bottom trace shows
a typical result of a simulation. Black ? spikes, blue ? input current.
4 . Possible reasons for current rectification: energy efficiency and de-noising
We have shown that at high firing rates biological neurons encode somatic current into a linearly
decodable spike train. However, at low firing rates linear decoding cannot faithfully reproduce the
somatic current because of rectification in the spike generation mechanism. If the objective of
spike generation is faithful AD conversion, why would such rectification exist? We see two
potential reasons: energy efficiency and de-noising.
It is widely believed that minimizing metabolic costs is an important consideration in brain design
and operation [30, 31]. Moreover, spikes are known to consume a significant fraction of the
metabolic budget [30, 32] placing a premium on their total number. Thus, we can postulate that
neuronal spike trains find a trade-off between the mean squared error in the decoded spike train
relative to the input signal and the total number of spikes, as expressed by the following cost
function over a time interval T:
,
(3)
where x is the analog input signal, s is the binary spike sequence composed of zeros and ones, and
is the linear filter.
To demonstrate how solving Eq.(3) would lead to thresholding, let us consider a simplified
version taken over a Nyquist period, during which the input signal stays constant:
(4)
where and normalized by w. Minimizing such a cost function reduces to choosing the lowest
lying parabola for a given
, Figure 6A. Therefore, thresholding is a natural outcome of
minimizing a cost function combining the decoding error and the energy cost, Eq.(3).
In addition to energy efficiency, there may be a computational reason for thresholding somatic
current in neurons. To illustrate this point, we note that the cost function in Eq. (3) for continuous
variables, st, may be viewed as a non-negative version of the L1-norm regularized linear
regression called LASSO [33], which is commonly used for de-noising of sparse and Laplacian
signals [34]. Such cost function can be minimized by iteratively applying a gradient descent and a
shrinkage steps [35], which is equivalent to thresholding (one-sided in case of non-negative
variables), Figure 6B,C. Therefore, neurons may be encoding a de-noised input signal.
Figure 6. Possible reasons for rectification in neurons. A. Cost function combining encoding
error squared with metabolic expense vs. input signal for different values of the spike number N,
Eq.(4). Note that the optimal number of spikes jumps from zero to one as a function of input. B.
Estimating most probable ?clean? signal value for continuous non-negative Laplacian signal and
Gaussian noise, Eq.(3) (while setting w = 1). The parabolas (red) illustrate the quadratic loglikelihood term in (3) for different values of the measurement, s, while the linear function (blue)
reflects the linear log-prior term in (3). C. The minimum of the combined cost function in B is at
zero if s ???, and grows linearly with s, if s >??.
5 . Di scu ssi on
In this paper, we demonstrated that the neuronal spike-generation mechanism can be viewed as an
oversampling and noise-shaping AD converter, which encodes a rectified low-pass filtered
somatic current as a digital spike train. Rectification by the spike generation mechanism may
subserve both energy efficiency and de-noising. As the degree of noise-shaping in biological
neurons exceeds that in IF neurons, or basic ??, we suggest that neurons should be modeled by
more advanced ?? modulators, e.g. Figure 5B. Interestingly, ?? modulators can be also viewed as
coders with error prediction feedback [19].
Many publications studied various aspects of spike generation in neurons yet we believe that the
framework [13-15] we adopt is different and discuss its relationship to some of the studies. Our
framework is different from previous proposals to cast neurons as predictors [36, 37] because a
different quantity is being predicted. The possibility of perfect decoding from a spike train with
infinite temporal precision has been proven in [38]. Here, we are concerned with a more practical
issue of how reconstruction error scales with the over-sampling ratio. Also, we consider linear
decoding which sets our work apart from [39]. Finally, previous experiments addressing noiseshaping [40] studied the power spectrum of the spike train rather than that of the encoding error.
Our work is aimed at understanding biological and computational principles of spike-generation
and decoding and is not meant as a substitute for the existing phenomenological spike-generation
models [41], which allow efficient fitting of parameters and prediction of spike trains [42]. Yet,
the theoretical framework [13-15] we adopt may assist in building better models of spike
generation for a given somatic current waveform. First, having interpreted spike generation as AD
conversion, we can draw on the rich experience in signal processing to attack the problem.
Second, this framework suggests a natural metric to compare the performance of different spike
generation models in the high firing rate regime: a mean squared error between the injected
current waveform and the filtered version of the spike train produced by a model provided the total
number of spikes is the same as in the experimental data. The AD conversion framework adds
justification to the previously proposed spike distance obtained by subtracting low-pass filtered
spike trains [43].
As the framework [13-15] we adopt relies on viewing neuronal computation as an analog-digital
hybrid, which requires AD and DA conversion at every step, one may wonder about the reason for
such a hybrid scheme. Starting with the early days of computers, the analog mode is known to be
advantageous for computation. For example, performing addition of many variables in one step is
possible in the analog mode simply by Kirchhoff law, but would require hundreds of logical gates
in the digital mode [44]. However, the analog mode is vulnerable to noise build-up over many
stages of computation and is inferior in precisely communicating information over long distances
under limited energy budget [30, 31]. While early analog computers were displaced by their digital
counterparts, evolution combined analog and digital modes into a computational hybrid [44], thus
necessitating efficient AD and DA conversion, which was the focus of the present study.
We are grateful to L. Abbott, S. Druckmann, D. Golomb, T. Hu, J. Magee, N. Spruston, B.
Theilman for helpful discussions and comments on the manuscript, to X.-J. Wang, D. McCormick,
K. Nagel, R. Wilson, K. Padmanabhan, N. Urban, S. Tripathy, H. Koendgen, and M. Giugliano
for sharing their data. The work of D.S. was partially supported by the Intel Collaborative
Research Institute for Computational Intelligence (ICRI-CI).
R e f e re n c e s
1.
Ferster, D. and N. Spruston, Cracking the neural code. Science, 1995. 270: p. 756-7.
2.
Panzeri, S., et al., Sensory neural codes using multiplexed temporal scales. Trends
Neurosci, 2010. 33(3): p. 111-20.
3.
Stevens, C.F. and A. Zador, Neural coding: The enigma of the brain. Curr Biol,
1995. 5(12): p. 1370-1.
4.
Shadlen, M.N. and W.T. Newsome, The variable discharge of cortical neurons:
implications for connectivity, computation, and information coding. J Neurosci,
1998. 18(10): p. 3870-96.
5.
Shadlen, M.N. and W.T. Newsome, Noise, neural codes and cortical organization.
Curr Opin Neurobiol, 1994. 4(4): p. 569-79.
6.
Singer, W. and C.M. Gray, Visual feature integration and the temporal correlation
hypothesis. Annu Rev Neurosci, 1995. 18: p. 555-86.
7.
Meister, M., Multineuronal codes in retinal signaling. Proc Natl Acad Sci U S A,
1996. 93(2): p. 609-14.
8.
Cook, E.P., et al., Dendrite-to-soma input/output function of continuous timevarying signals in hippocampal CA1 pyramidal neurons. J Neurophysiol, 2007.
98(5): p. 2943-55.
9.
Kondgen, H., et al., The dynamical response properties of neocortical neurons to
temporally modulated noisy inputs in vitro. Cereb Cortex, 2008. 18(9): p. 2086-97.
10.
Tchumatchenko, T., et al., Ultrafast population encoding by cortical neurons. J
Neurosci, 2011. 31(34): p. 12171-9.
11.
Mainen, Z.F. and T.J. Sejnowski, Reliability of spike timing in neocortical neurons.
Science, 1995. 268(5216): p. 1503-6.
12.
Mar, D.J., et al., Noise shaping in populations of coupled model neurons. Proc Natl
Acad Sci U S A, 1999. 96(18): p. 10450-5.
13.
Shin, J., Adaptive noise shaping neural spike encoding and decoding.
Neurocomputing, 2001. 38-40: p. 369-381.
14.
Shin, J., The noise shaping neural coding hypothesis: a brief history and
physiological implications. Neurocomputing, 2002. 44: p. 167-175.
15.
Shin, J.H., Adaptation in spiking neurons based on the noise shaping neural coding
hypothesis. Neural Networks, 2001. 14(6-7): p. 907-919.
16.
Schreier, R. and G.C. Temes, Understanding delta-sigma data converters2005,
Piscataway, NJ: IEEE Press, Wiley. xii, 446 p.
17.
Candy, J.C., A use of limit cycle oscillations to obtain robust analog-to-digital
converters. IEEE Trans. Commun, 1974. COM-22: p. 298-305.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
Inose, H., Y. Yasuda, and J. Murakami, A telemetring system code modulation - ???
modulation. IRE Trans. Space Elect. Telemetry, 1962. SET-8: p. 204-209.
Spang, H.A. and P.M. Schultheiss, Reduction of quantizing noise by use of feedback.
IRE TRans. Commun. Sys., 1962: p. 373-380.
Hovin, M., et al., Delta-Sigma modulation in single neurons, in IEEE International
Symposium on Circuits and Systems2002.
Cheung, K.F. and P.Y.H. Tang, Sigma-Delta Modulation Neural Networks. Proc.
IEEE Int Conf Neural Networkds, 1993: p. 489-493.
Padmanabhan, K. and N. Urban, Intrinsic biophysical diversity decorelates neuronal
firing while increasing information content. Nat Neurosci, 2010. 13: p. 1276-82.
Urban, N. and S. Tripathy, Neuroscience: Circuits drive cell diversity. Nature, 2012.
488(7411): p. 289-90.
Nagel, K.I. and R.I. Wilson, personal communication.
Shin, J., C. Koch, and R. Douglas, Adaptive neural coding dependent on the timevarying statistics of the somatic input current. Neural Comp, 1999. 11: p. 1893-913.
Magee, J.C. and E.P. Cook, Somatic EPSP amplitude is independent of synapse
location in hippocampal pyramidal neurons. Nat Neurosci, 2000. 3(9): p. 895-903.
Thorpe, S., D. Fize, and C. Marlot, Speed of processing in the human visual system.
Nature, 1996. 381(6582): p. 520-2.
Tewksbury, S.K. and R.W. Hallock, Oversample, linear predictive and noiseshaping coders of order N>1. IEEE Trans Circuits & Sys, 1978. CAS25: p. 436-47.
Wang, X.J., et al., Adaptation and temporal decorrelation by single neurons in the
primary visual cortex. J Neurophysiol, 2003. 89(6): p. 3279-93.
Attwell, D. and S.B. Laughlin, An energy budget for signaling in the grey matter of
the brain. J Cereb Blood Flow Metab, 2001. 21(10): p. 1133-45.
Laughlin, S.B. and T.J. Sejnowski, Communication in neuronal networks. Science,
2003. 301(5641): p. 1870-4.
Lennie, P., The cost of cortical computation. Curr Biol, 2003. 13(6): p. 493-7.
Tibshirani, R., Regression shrinkage and selection via the Lasso. Journal of the
Royal Statistical Society Series B-Methodological, 1996. 58(1): p. 267-288.
Chen, S.S.B., D.L. Donoho, and M.A. Saunders, Atomic decomposition by basis
pursuit. Siam Journal on Scientific Computing, 1998. 20(1): p. 33-61.
Elad, M., et al., Wide-angle view at iterated shrinkage algorithms. P SOc Photo-Opt
Ins, 2007. 6701: p. 70102.
Deneve, S., Bayesian spiking neurons I: inference. Neural Comp, 2008. 20: p. 91.
Yu, A.J., Optimal Change-Detection and Spinking Neurons, in NIPS, B. Scholkopf,
J. Platt, and T. Hofmann, Editors. 2006.
Lazar, A. and L. Toth, Perfect Recovery and Sensitivity Analysis of Time Encoded
Bandlimited Signals. IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS,
2004. 51(10).
Pfister, J.P., P. Dayan, and M. Lengyel, Synapses with short-term plasticity are
optimal estimators of presynaptic membrane potentials. Nat Neurosci, 2010. 13(10):
p. 1271-5.
Chacron, M.J., et al., Experimental and theoretical demonstration of noise shaping
by interspike interval correlations. Fluctuations and Noise in Biological,
Biophysical, and Biomedical Systems III, 2005. 5841: p. 150-163.
Pillow, J., Likelihood-based approaches to modeling the neural code, in Bayesian
Brain: Probabilistic Approaches to Neural Coding, K. Doya, et al., Editors. 2007,
MIT Press.
Jolivet, R., et al., A benchmark test for a quantitative assessment of simple neuron
models. J Neurosci Methods, 2008. 169(2): p. 417-24.
van Rossum, M.C., A novel spike distance. Neural Comput, 2001. 13(4): p. 751-63.
Sarpeshkar, R., Analog versus digital: extrapolating from electronics to
neurobiology. Neural Computation, 1998. 10(7): p. 1601-38.
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4,082 | 4,695 | Coding efficiency and detectability of rate fluctuations
with non-Poisson neuronal firing
Shinsuke Koyama?
Department of Statistical Modeling
The Institute of Statistical Mathematics
10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan
[email protected]
Abstract
Statistical features of neuronal spike trains are known to be non-Poisson. Here, we
investigate the extent to which the non-Poissonian feature affects the efficiency of
transmitting information on fluctuating firing rates. For this purpose, we introduce
the Kullback-Leibler (KL) divergence as a measure of the efficiency of information encoding, and assume that spike trains are generated by time-rescaled renewal
processes. We show that the KL divergence determines the lower bound of the degree of rate fluctuations below which the temporal variation of the firing rates is
undetectable from sparse data. We also show that the KL divergence, as well as the
lower bound, depends not only on the variability of spikes in terms of the coefficient of variation, but also significantly on the higher-order moments of interspike
interval (ISI) distributions. We examine three specific models that are commonly
used for describing the stochastic nature of spikes (the gamma, inverse Gaussian
(IG) and lognormal ISI distributions), and find that the time-rescaled renewal process with the IG distribution achieves the largest KL divergence, followed by the
lognormal and gamma distributions.
1 Introduction
Characterizing the statistical features of spike time sequences in the brain is important for understanding how the brain represents information about stimuli or actions in the sequences of spikes.
Although the spike trains recorded from in vivo cortical neurons are known to be highly irregular [20, 24], a recent non-stationary analysis has revealed that individual neurons signal with nonPoisson firing, the characteristics of which are strongly correlated with the function of the cortical
area [21].
This raises the question of what the neural coding advantages of non-Poisson spiking are. It could be
that the precise timing of spikes carries additional information about the stimuli or actions [6, 15]. It
is also possible that the efficiency of transmitting fluctuating rates might be enhanced by non-Poisson
firing [5, 17]. Here, we explore the latter possibility.
In the problem of estimating firing rates, there is a minimum degree of rate fluctuation below which
a rate estimator cannot detect the temporal variation of the firing rate [23]. If, for instance, the degree
of temporal variation of the rate is on the same order as that of the noise, a constant rate might be
chosen as the most likely estimate for a given spike train. It is, therefore, interesting to see how the
minimum degree of rate fluctuation depends on the non-Poissonian feature of spike trains.
In this study, we investigate the extent to which the non-Poissonian feature of spike trains affects
the encoding efficiency of rate fluctuations. In addition, we address the question of how the de?
http://skoyama.blogspot.jp
1
tectability of rate fluctuations depends on the encoding efficiency. For this purpose, we introduce
the Kullback-Leibler (KL) divergence to measure the encoding efficiency, and assume that spike sequences are generated by time-rescaled renewal processes. With the aid of analytical and numerical
studies, we suggest that the lower bound of detectable rate fluctuations, below which the empirical
Bayes decoder cannot detect the rate fluctuations, is uniquely determined by the KL divergence. By
examining three specific models (the time-rescaled renewal process with the gamma, inverse Gaussian (IG) and lognormal interspike interval (ISI) distributions), it is shown that the KL divergence,
as well as the lower bound, depends not only on the first- and second-order moments, but also significantly on the higher-order moments of the ISI distributions. We also find that among the three
ISI distributions, the IG distribution achieves the highest efficiency of coding information on rate
fluctuations.
2 Encoding rate fluctuations using time-rescaled renewal processes
Definitions of time-rescaled renewal processes and KL divergence
We introduce time-rescaled renewal processes for a model of neuronal spike trains constructed
in the
!?
following way. Let f ? (y) be a family of ISI distributions with the unit mean (i.e., 0 yf? (y)dy =
1), where ? controls the shape of the distribution, and ?(t) be a fluctuating firing rate. A sequence of
spikes {ti } := {t1 , t2 , . . . , tn } is generated in the following steps: (i) Derive ISIs {y 1 , y2 , . . . , yn }
independently from f ? (y), and arrange the ISIs sequentially to form a spike train of the unit rate;
"
ith spike is given by summing the previous ISIs as s i = ij=1 yj . (ii) Transform {s 1 , s2 , . . . , sn }
to {t , t , . . . , tn } according to t i = ??1 (si ), where ??1 (si ) is the inverse of the function ?(t) =
!t 1 2
?(u)du. This transformation ensures that the instantaneous firing rate of {t i } corresponds to ?(t),
0
while the shape of the ISI distribution f ? (y), which characterizes the firing irregularity, is unchanged
in time. This is in agreement with the empirical fact that the degree of irregularity in neuronal firing
is generally maintained in cortical processing [21, 22], while the firing rate ?(t) changes in time.
The probability density of the occurrence of spikes at {t i } is, then, given by
p? ({ti }|{?(t)}) =
where t0 = 0.
n
#
i=1
?(ti )f? (?(ti ) ? ?(ti?1 )).
(1)
We next introduce the KL divergence for measuring the encoding efficiency of fluctuating rates. For
this purpose, we assume that ?(t) is ergodic with a stationary distribution p(?), the mean of which
is given by ?:
$ ?
$
1 T
"?#? :=
?p(?)d? = lim
?(t)dt = ?.
(2)
T ?? T 0
0
Consider a probability density of a renewal process that has the same ISI density f ? (x) and the
constant rate ?:
n
#
?f? (?(ti ? ti?1 )).
(3)
p? ({ti }|?) =
n=1
The KL divergence between p ? ({ti }|{?(t)}) and p? ({ti }|?) is, then, defined as
$ $
$ T
?
%
1 T T
D? (?(t)||?) := lim
???
p? ({ti }|{?(t)})
T ??
T 0 t1
tn?1
n=0
? log
p? ({ti }|{?(t)})
dt1 dt2 ? ? ? dtn .
p? ({ti }|?)
(4)
Since it is defined as the entropy of a renewal process with the fluctuating rate ?(t) relative to that
with the constant rate ?, D ? (?(t)||?) can be interpreted as the amount of information on the rate
fluctuations encoded into spike trains. Note that a similar quantity has been introduced in [3], where
the quantity was computed only under a Poisson model.
2
Substituting Eqs. (1) and (3) into Eq. (4) and further assuming ergodicity of spike trains, the KL
divergence can be expressed as
1
p? ({ti }|{?(t)})
log
? t0
p? ({ti }|?)
n
1 %&
log ?(ti ) + log f? (?(ti ) ? ?(ti?1 ))
= lim
n?? tn ? t0
i=1
'
? log ? ? log f? (?(ti ? ti?1 )) .
D? (?(t)||?)
=
lim
n?? tn
(5)
This expression can be used for computing the KL divergence numerically by simulating a large
number of spikes n % 1.
Three ISI distributions and their KL divergence
In order to examine the behavior of the KL divergence, we use the three specific ISI distributions
for f? (y) (the gamma, inverse Gaussian (IG) and lognormal distributions), which have been used to
describe(
the stochastic nature of ISIs [9, 10, 14]. These distributions and their coefficient of variation
(CV = V ar(X)/E(X)) are given by
?
(6)
gamma : f? (y) = ?? y ??1 e??y /?(?), CV = 1/ ?,
*
+
)
?
?
?(y ? 1)2
IG : f? (y) =
exp ?
(7)
, CV = 1/ ?,
2?y 3
2y
*
+
?
(log y + ?2 )2
1
exp ?
(8)
lognormal : f? (y) = ?
, CV = e? ? 1,
2?
y 2??
!?
where ?(?) = 0 x??1 e?x dx is the gamma function. Figure 1a illustrates the shape of the three
distributions with three different values of C V .
The KL divergence for the three models is analytically solvable when the rate fluctuation has a long
time scale relative to the mean ISI. Here, we show the derivation for the gamma distribution. (The
derivations for the IG and lognormal distributions are essentially the same.) Inserting Eq. (6) into
Eq. (5) leads to
D? (?(t)||?)
! tn
n
%
&
1
log ?(ti ) + (? ? 1) log[?(ti ) ? ?(ti?1 )]
n?? tn ? t0
i=1
'
? (? ? 1) log(ti ? ti?1 ) ? ?? log ?,
lim
(9)
n
?(t)dt ? ? and tn ?t
? ? as n ? ?. By introducing the ?averaged?
0
?(ti )??(ti?1 )
?
? i + log(ti ?
firing rate in the ith ISI: ?i :=
, we obtain log[?(ti ) ? ?(ti?1 )] = log ?
ti ?ti?1
? i is
ti?1 ). Assuming that the time scale of the rate fluctuation is longer than the mean ISI so that ?
approximated to ?(t i ), Eq. (9) becomes
where we used
1
tn ?t0
=
t0
D? (?(t)||?)
n
%
1
log ?(ti ) ? ?? log ?
n?? tn ? t0
i=1
,
$
1 T%
= ? lim
?(t ? ti ) log ?(t)dt ? ? log ? .
T ?? T 0
i
= ? lim
(10)
The fluctuation in the apparent spike count is given by the variance to mean ratio as represented
by the Fano factor [8]. For the renewal process in which ISIs are drawn from a given distribution
function, it is proven that the Fano factor is related to the ISI variability with F ? C V2 [4]. Thus, for
a long range time scale in which a serial correlation of spikes is negligible, the spike train in Eq. (10)
can be approximated to
n
%
i=1
?(t ? ti ) ? ?(t) +
3
(
?(t)/??(t),
(11)
where ?(t) is a fluctuating process such that "?(t)# = 0 and "?(t)?(t % )# = ?(t ? t% ). Using this, the
first term on the rhs of (10) can be evaluated as
$
$
1 T
1 T(
lim
?(t) log ?(t)dt + lim
?(t)/? log ?(t)?(t)dt = "? log ?#? , (12)
T ?? T 0
T ?? T 0
where the second term on the lhs has vanished due to a property of stochastic integrals. Therefore,
the KL divergence of the gamma distribution is obtained as
&
'
D? (?(t)||?) = ? "? log ?#? ? ? log ? .
(13)
In the same way, the KL divergence for the IG and lognormal distributions are, respectively, derived
as
1
?
?+1
D? (?(t)||?) =
log ? ? "? log ?#? +
"(? ? ?)2 #? ,
(14)
2
2
2?
and
?
log ?
1
(log ?)2 ?
"? log ?#? +
"?(log ?)2 #? .
2?
?
2?
See the supplementary material for the details of their derivations.
D? (?(t)||?) =
(15)
Results
We compute the KL divergence for the three models, in which the rate fluctuates according to the
Ornstein-Uhlenbeck process. Formally, the rate process is given by ?(t) = [x(t)] + , where [?]+ is
the rectification function:
,
x, x > 0
[x]+ =
(16)
0, otherwise
and x(t) is derived from the Ornstein-Uhlenbeck process:
dx(t)
x(t) ? ?
=?
+?
dt
?
)
2
?(t),
?
(17)
where ?(t) is the Gaussian white noise.
Figure 1b depicts the KL divergence as a function of ? for C V =0.6, 1 and 1.5. The analytical results
(the solid lines) are in good agreement with the numerical results (the error bars). The KL divergence
for the three models increases as ? is increased and as C V is decreased, which is rather obvious
since larger ? and smaller C V imply lower noise entropy of spike trains. One nontrivial result is
that, even if the three models share the same values of ? and C V , the KL divergence of each model
significantly differs from that of the others: the IG distribution achieves the largest KL divergence,
followed by the lognormal and gamma distributions. The difference in the KL divergence among
the three models becomes larger as C V grows larger. Since the three models share the same firing
rate ?(t) and CV , it can be concluded that the higher-order (more than second-order) moments of
ISI distributions strongly affect the KL divergence.
In order to confirm this result for another rate process, we examine a sinusoidal rate process, ?(t) =
? + ? sin t/? , and observe the same behavior as the Ornstein-Uhlenbeck rate process (Figure 1c).
3 Decoding fluctuating rates using the empirical Bayes method
In this section, we show that the KL divergence (4) determines the lower bound of the degree of rate
fluctuation below which the empirical Bayes estimator cannot detect rate fluctuations.
The empirical Bayes method
We consider decoding a fluctuation rate ?(t) from a given spike train {t i } := {t1 . . . , tn } in an
observation interval [0, T ] by the empirical Bayes method. Let x(t) ? R be a latent variable that
4
(a)
CV=0.6
CV=1
0.15
0.4
CV=0.6
KL divergence
KL divergence
gamma
IG
lognormal
(c)
(b)
0.1
CV=1
0.05
0
0.1
0.2
?
CV=0.6
0.3
0.2
CV=1
0.1
CV=1.5
0
CV=1.5
0
0.3
CV=1.5
0
0.2
0.4
?
0.6
0.8
Figure 1: (a) The gamma (blue), IG (green) and lognormal (red) ISI distribution functions for
CV =0.6, 1 and 1.5. (b) The KL divergence as a function of ? for C V =0.6, 1 and 1.5, when the rate
fluctuates according to the Ornstein-Uhlenbeck process (17) with ? = 1 and ? = 10. The blue, green
and red indicate the KL divergence for the gamma, IG and lognormal distribution, respectively. The
lines represent the theoretical values obtained by Eqs. (13), (14) and (15), and the error bars represent the average and standard deviation numerically computed according to Eq. (5) with n = 50, 000
and 10 trials. (c) The KL divergence for the sinusoidally modulated rate, ?(t) = ? + ? sin t/? , with
? = 1 and ? = 10.
is transformed from ?(t) via the log-link function x(t) = log ?(t). For the inference of ?(t) from
{ti }, we use a prior distribution of x(t), such that the large gradient of x(t) is controlled by
.
02 1
$ T/
dx(t)
1
p? ({x(t)}) ? exp ? 2
dt ,
(18)
2? 0
dt
where the hyperparameter ? controls the roughness of the latent process x(t): with the small ?, the
model requires a constant latent process, and vice versa. By inverting the conditional probability
distribution with the Bayes? theorem, the posterior distribution of {x(t)} is obtained as
p?,? ({x(t)}|{ti }) =
p? ({ti }|{x(t)})p? ({x(t)})
.
p?,? ({ti })
(19)
The hyperparameters, ? and ?, which represent the roughness of the latent process and the shape of
the ISI density function, can be determined by maximizing the marginal likelihood [16] defined by
$
p?,? ({ti }) = p? ({ti }|{x(t)})p? ({x(t)})D{x(t)},
(20)
!
where D{x(t)} represents the integration over all possible latent process paths. Under a set of
hyperparameters ?? and ?
? that are determined by the marginal likelihood maximization, we can
determine the maximum a posteriori (MAP) estimate of the latent process x
?(t). The method for
implementing the empirical Bayes analysis is summarized in the Appendix.
Detectability of rate fluctuations
We first examine the gamma distribution (6). For synthetic spike trains (n = 1, 000) generated
by the time-rescaled renewal process with the gamma ISI distribution, in which the rate fluctuates
according to the Ornstein-Uhlenbeck process (17) with ? = 1 and ? = 10, we attempt to decode
?(t) using the empirical Bayes decoder. Depending on the amplitude of the rate fluctuation ? and
CV of f? (y), the empirical Bayes decoder provides qualitatively two distinct rate estimations: (I)
a fluctuating rate estimation (?
? > 0) for large ? and small C V , or (II) a constant rate estimation
(?
? = 0) for small ? and large C V (Figure 2a). When ? is increased or C V is decreased, the
5
empirical Bayes estimator exhibits a phase transition corresponding to the switch of the most likely
rate estimation from (II) to (I) (Figure 2b). Note that below the critical point of this phase transition,
the empirical Bayes method provides a constant rate as the most likely estimation even if the true
rate process fluctuates. The critical point, thus, gives the lower bound for the degree of detectable
rate fluctuations. It is also confirmed, using numerical simulations, that the phase transition occurs
not only with the gamma distribution, but also with the IG and lognormal distributions (Figure 2c,d).
For the time-rescaled renewal process with the gamma ISI distribution, we could analytically derive
the formula that the lower bound satisfies as:
D? (?(t)||?) =
4 max?
?(0)
!?
,
?(u)e??u du
0
(21)
where ?(u) is the correlation function of ?(t). (See supplementary material for the derivation.)
Eq. (21) is in good agreement with the simulation result for the entire parameter space (the solid line
in Figure 2a).
The expression of Eq. (21) itself does not depend on the gamma distribution. We investigated if this
formula is also applicable to the IG and lognormal distributions, and found that the theoretical lower
bounds (the solid lines in Figure 2c,d) indeed do correspond to those obtained by the numerical
simulations; this result implies that Eq. (21) is applicable to more general time-rescaled renewal
processes.
Figure 2e compares the lower bounds among the three distributions. The lower bound of the IG
distribution is the lowest, followed by the lognormal and gamma distributions, which is expected
from the result in Figure 1b, as the lower bound is identically determined by the KL divergence via
Eq. (21).
We also examined the sinusoidally modulated rate, ?(t) = ? + ? sin t/? ; the qualitative result
remains the same (Figure 2f-h).
4 Discussion
In this study, we first examined the extent to which spike trains derived from time-rescaled renewal
processes encode information on fluctuating rates. The encoding efficiency is measured by the KL
divergence between two renewal processes with fluctuating and constant rates. We showed that the
KL divergence significantly differs among the gamma, IG and lognormal ISI distributions, even if
these three processes share the same rate fluctuation ?(t) and C V (Figure 1b). This suggests that the
higher-order moments of ISIs play an important role in encoding information on fluctuating rates.
Among the three distributions, the IG distribution achieves the largest KL divergence, followed by
the lognormal and gamma distributions. A similar result has been reported for stationary renewal
processes [12].
Since the KL divergence gives the distance between two probability distributions, Eq. (4) is naturally related to the ability to discriminate between a fluctuating rate and a constant rate. In fact,
the lower bound of the degree of rate fluctuation, below which the empirical Bayes decoder cannot
discriminate the underlying fluctuating rate from a constant rate, satisfies the formula (21). There
commonly exists a lower bound below which the underlying rate fluctuations are undetectable, not
only in the empirical Bayes method with the above prior distribution (18), but also with other prior
distributions, and in other rate estimators such as a time-histogram. The lower bound in these methods has been derived for inhomogeneous Poisson processes as ? ? 2 /? ? O(1), where ? , ? and ?
are the time scale, amplitude and mean of the rate fluctuation, respectively [23]. Thus, Eq. (21),
or equivalently ? D ? (?(t)||?) ? O(1) is regarded as a generalization to the non-Poisson processes.
Here, the crucial step for this generalization is incorporating the KL divergence into the formula.
Note that the formula (21) was derived analytically under the assumption of the gamma ISI distribution, and then was shown to hold for the IG and lognormal ISI distributions with numerical
simulations. The analytical tractability of the gamma family lies in the fact that it is the only scale
family that admits the mean as a sufficient statistic. We conjecture, from our results with the three
specific models, that Eq. (21) is applicable to more general time-rescaled renewal processes (even
to ?non-renewal? processes), which is open to future research.
6
(a)
?>0
(II)
^
?=0
gamma
0.5
(c)
1
CV
(d)
0
IG
0.5
1
CV
0.2
0.5
1
CV
0.5
1
CV
1.5
0.5
(h)
0.2
0.3
1
CV
1.5
0.6
0.4
0
0
1.5
0.2
gamma
?
0.1
0.6
0.4
0.1
0.2
lognormal
(g)
0.6
0
0
1.5
?
?
(f)
0.2
0.1
0.1
0
0.3
?
0.2
(II)
(e)
0.3
?
?
0
1.5
0.3
0.05
?
0
?
(I)
0.2
0.1
(b) 0.1
(I)
^
^
?
0.3
0.4
0.2
0.5
1
CV
IG
1.5
0
0.5
lognormal
1
CV
1.5
Figure 2: (a) Left: the phase diagram for sequences generated by the time-rescaled renewal process
with the gamma ISI distribution. The ordinate represents the amplitude of rate fluctuation ?, and
abscissa represents CV of the gamma ISI distribution. The dots represent the result of numerical
simulations in which the empirical Bayes decoder provides a fluctuating rate estimation (?
? > 0).
Each dot is plotted if ?? > 0 in more than 20 out of 40 identical trials. The solid line represents
the theoretical lower bound obtained by the formula (21). Right: raster plots of sample spike trains
and the estimated rates. The dotted lines and the solid lines represent the underlying rates and the
estimated rates, respectively. The parameters (C V , ?) of top (?
? > 0) and bottom (?
? = 0) are
(0.6, 0.3) and (1.5, 0.15), respectively. (b) The optimal hyperparameter ?? as a function of ? for
CV = 0.6. The solid line represents the theoretical value, and the error bars represent the average
and standard deviation of ?? determined by applying the empirical Bayes algorithm to 40 trials. (c,
d) The phase diagrams for the IG and lognormal ISI distributions. (e) Comparison of the lower
bounds among the three models. (f-h) The phase diagrams for the gamma, IG and lognormal ISI
distributions, when the rate process is given by ?(t) = ? + ? sin t/? with ? = 1 and ? = 10.
A recent non-stationary analysis has revealed that individual neurons in the cortex signal with nonPoisson firing, which has empirically been characterized by measures based on the second-order
moment of ISIs, such as C V and LV [21, 22]. Our results, however, suggest that it may be important
to take into account the higher-order moments of ISIs for characterizing ?irregularity? of cortical
firing, in order to gain information on fluctuating firing rates. It has also been demonstrated that
using non-Poisson spiking models enhances the performance of neural decoding [2, 11, 19]. Our
results provide theoretical support for this as well.
Appendix: Implementation of the empirical Bayes method
Discretization
To construct a practical algorithm for performing empirical Bayes decoding, we first divide the
time axis into a set of intervals (t i?1 , ti ] (i = 1, . . . , n). We assume that the firing rate within
each interval (ti?1 , ti ] does not change drastically (which is a reasonable assumption in practice),
so that it can be approximated to a constant value ? i . Letting Ti = ti ? ti?1 be the ith ISI, the
probability density of {T i } ? {T1 , T2 , . . . , Tn }, given the rate process {? i } ? {?1 , ?2 . . . , ?n }
7
2n
is obtained from Eq. (1) as p ? ({Ti }|{?i }) = i=1 ?i f? (?i Ti ). The rate process is linked with
the latent process via xi = log ?i . With the same time-discretization, the prior distribution of the
latent 2
process {xi } ? {x1 , x2 , . . . , xn }, which corresponds to Eq. (18), is derived as p ? ({xi }) =
n
p(x1 ) i=2 p? (xi |xi?1 ), where
*
+
(xi ? xi?1 )2
1
(
exp ? 2
p? (xi |xi?1 ) =
,
(22)
? (Ti + Ti?1 )
?? 2 (Ti + Ti?1 )
and p(x1 ) is the probability density function of the initial latent rate variable.
p({Ti }|{?i }) and p? ({xi }) define a discrete-time state space model. We note that this provides a
good approximation to the original continuous-time model if the timescale of the rate fluctuation is
larger than the mean ISI.
EM algorithm
We assume that the ISI density function can be rewritten in the form of exponential family distributions with respect to the shape parameter ?:
p? (Ti |?i ) := ?i f? (?i Ti ) = exp[?S(Ti , ?i ) ? ?(?) + c(Ti , ?i )],
(23)
with an appropriate parameter representation ? i = ?(?i , ?). Here, ? is the natural parameter of
the exponential family and S(T i , ?i ) is its sufficient statistic. Suppose that the potential ?(?) is a
convex function. The expectation of S(T i , ?i ) is then given by
$
d?(?)
? = S(Ti , ?i )p? (Ti |?i )dTi =
.
(24)
d?
Since ?(?) is convex, there is one-to-one correspondence between ? and ?, and thus ? provides
alternative parametrization to ? [1]. The gamma (6), IG (7) and lognormal (8) distributions are
included in this family.
With the parameterization ?, the EM algorithm for the state space model is derived as follows.
Suppose that we have estimations ??(m) and ??(m) at the mth iteration. The estimations at the (m+1)th
iteration are given by
n
1%
"S(Ti , ?(xi ))#(m) ,
(25)
??(m+1) =
n i=1
and
n
2
??(m+1)
=
2 % "(xi ? xi?1 )2 #(m)
,
n ? 1 i=2
Ti + Ti?1
(26)
where " #(m) denotes the expectation with respect to the posterior probability of {x i }, given {Ti },
??(m) and ??(m) . The posterior probability is computed by the Laplace approximation, introduced
below. We update ?? and ?? until the estimations converge. The estimation of ? is then transformed
from ?? with Eq. (24).
Laplace approximation
We employ Laplace?s method to compute an approximate posterior distribution of {x i }. Let x =
(x1 , x2 , . . . , xn )t be the column vector of the latent process, ( ) t being the transpose of a vector.
The MAP estimate of the latent process is obtained by maximizing the log posterior distribution:
n
n
%
%
l(x) = log p(x1 ) +
log p? (xi |xi?1 ) +
log p? (Ti |xi ) + const.,
(27)
i=2
i=1
with respect to x. We use a diffuse prior for p(x 1 ) so that its contribution vanishes [7]. If
p? (xi |xi?1 ) is log-concave in x i and xi?1 , and the p? (Ti |xi ) is also log-concave in x i , computing the MAP estimate is a concave optimization problem [18], which can be solved efficiently by
a Newton method. Due to the Markovian Structure of the state-space model, the Hessian matrix,
J(x) ? ??x l(x), becomes a tridiagonal matrix, which allows us to compute the Newton step in
? denote the MAP estimation of the posterior probability. The posterior probaO(n) time [13]. Let x
?
bility is then approximated to a Gaussian whose mean vector and covariance matrix are given by x
? ?1 , respectively.
and ?J(x)
8
Acknowledgments
This work was supported by JSPS KAKENHI Grant Number 24700287.
References
[1] S. Amari and H. Nagaoka. Methods of Information Geometry. Oxford University Press, 2000.
[2] R. Barbieri, M. C. Quirk, L. M. Frank, M. A. Wilson, and E. N. Brown. Construction and analysis
of non-poisson stimulus-response models of neural spiking activity. Journal of Neuroscience Methods,
105:25?37, 2001.
[3] N. Brenner, S. P. Strong, R. Koberle, and W. Bialek. Synergy in a neural code. Neural Computation,
12:1531?1552, 2000.
[4] D. R. Cox. Renewal Theory. Chapman and Hal, 1962.
[5] J. P. Cunningham, B. M. Yu, K. V. Shenoy, and M. Sahani. Inferring neural firing rates from spike trains
using Gaussian processes. In Neural Information Processing Systems, volume 20, pages 329?336, 2008.
[6] R. M. Davies, G. L. Gerstein, and S. N. Baker. Measurement of time-dependent changes in the irregularity
of neural spiking. Journal of Neurophysiology, 96:906?918, 2006.
[7] J. Durbin and S. J. Koopman. Time Series Analysis by State Space Methods. Oxford University Press,
2001.
[8] U. Fano. Ionization yield of radiations. ii. the fluctuations of the number of ions. Physical Review,
72:26?29, 1947.
[9] G. L. Gerstein and B. Mandelbrot. Random walk models for the spike activity of a single neuron. Biophysical Journal, 4:41?68, 1964.
[10] S. Ikeda and J. H. Manton. Capacity of a single spiking neuron channel. Neural Computation, 21:1714?
1748, 2009.
[11] A. L. Jacobs, G. Fridman, R. M. Douglas, N. M. Alam, P. E. Latham, G. T. Prusky, and S. Nirenberg.
Ruling out and ruling in neural codes. Proceedings of the National Academy of Sciences, 106:5936?5941,
2009.
[12] K. Kang and S. Amari. Discrimination with spike times and ISI distributions. Neural Computation,
20:1411?1426, 2008.
[13] S. Koyama and L. Paninski. Efficient computation of the maximum a posteriori path and parameter estimation in integrate-and-fire and more general state-space models. Journal of Computational Neuroscience,
29:89?105, 2009.
[14] M. W. Levine. The distribution of the intervals between neural impulses in the maintained discharges of
retinal ganglion cells. Biological Cybernetics, 65:459?467, 1991.
[15] B. N. Lundstrom and A. L. Fairhall. Decoding stimulus variance from a distributional neural code of
interspike intervals. Journal of Neuroscience, 26:9030?9037, 2006.
[16] D. J. C. MacKay. Bayesian interpolation. Neural Computation, 4:415?447, 1992.
[17] T. Omi and S. Shinomoto. Optimizing time histograms for non-Poisson spike trains. Neural Computation,
23:3125?3144, 2011.
[18] L. Paninski. Log-concavity results on gaussian process methods for supervised and unsupervised learning.
In Neural Information Processing Systems, volume 17, pages 1025?1032, 2005.
[19] J. W. Pillow, L. Paninski, V. J. Uzzell, E. P. Simoncelli, and E. J. Chichilnisky. Prediction and decoding
of retinal ganglion cell responses with a probabilistic spiking model. Journal of Neuroscience, 23:11003?
11013, 2005.
[20] M. N. Shadlen and W. T. Newsome. The variable discharge of cortical neurons: Implications for connectivity, computation, and information coding. Journal of Neuroscience, 18:3870?3896, 1998.
[21] S. Shinomoto, H. Kim, T. Shimokawa, N. Matsuno, S. Funahashi, K. Shima, I. Fujita, H. Tamura, T. Doi,
K. Kawano, N. Inaba, K. Fukushima, S. Kurkin, K. Kurata, M. Taira, K. Tsutsui, H. Komatsu, T. Ogawa,
K. Koida, J. Tanji, and K. Toyama. Relating neuronal firing patterns to functional differentiation of
cerebral cortex. PLoS Computational Biology, 5:e1000433, 2009.
[22] S. Shinomoto, K. Shima, and J. Tanji. Differences in spiking patterns among cortical neurons. Neural
Computation, 15:2823?2842, 2003.
[23] T. Shintani and S. Shinomoto. Detection limit for rate fluctuations in inhomogeneous poisson processes.
Physical Review E, 85:041139, 2012.
[24] W. R. Softky and C. Koch. The highly irregular firing of cortical cells is inconsistent with temporal
integration of random EPSPs. Journal of Neuroscience, 13:334?350, 1993.
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4,083 | 4,696 | Mixing Properties of Conditional Markov Chains
with Unbounded Feature Functions
Mathieu Sinn
IBM Research - Ireland
Mulhuddart, Dublin 15
[email protected]
Bei Chen
McMaster University
Hamilton, Ontario, Canada
[email protected]
Abstract
Conditional Markov Chains (also known as Linear-Chain Conditional Random
Fields in the literature) are a versatile class of discriminative models for the distribution of a sequence of hidden states conditional on a sequence of observable
variables. Large-sample properties of Conditional Markov Chains have been first
studied in [1]. The paper extends this work in two directions: first, mixing properties of models with unbounded feature functions are being established; second,
necessary conditions for model identifiability and the uniqueness of maximum
likelihood estimates are being given.
1
Introduction
Conditional Random Fields (CRF) are a widely popular class of discriminative models for the distribution of a set of hidden states conditional on a set of observable variables. The fundamental
assumption is that the hidden states, conditional on the observations, form a Markov random field
[2,3]. Of special importance, particularly for the modeling of sequential data, is the case where the
underlying undirected graphical model forms a simple linear chain. In the literature, this subclass
of models is often referred to as Linear-Chain Conditional Random Fields. This paper adopts the
terminology of [4] and refers to such models as Conditional Markov Chains (CMC).
Large-sample properties of CRFs and CMCs have been first studied in [1] and [5]. [1] defines CMCs
of infinite length and studies ergodic properties of the joint sequences of observations and hidden
states. The analysis relies on fundamental results from the theory of weak ergodicity [6]. The
exposition is restricted to CMCs with bounded feature functions which precludes the application,
e.g., to models with linear features and Gaussian observations. [5] considers weak consistency
and central limit theorems for models with a more general structure. Ergodicity and mixing of the
models is assumed, but no explicit conditions on the feature functions or on the distribution of the
observations are given. An analysis of model identifiability in the case of finite sequences can be
found in [7].
The present paper studies mixing properties of Conditional Markov Chains with unbounded feature
functions. The results are fundamental for analyzing the consistency of Maximum Likelihood estimates and establishing Central Limit Theorems (which are very useful for constructing statistical
hypothesis tests, e.g., for model misspecificiations and the signficance of features). The paper is organized as follows: Sec. 2 reviews the definition of infinite CMCs and some of their basic properties.
In Sec. 3 the ergodicity results from [1] are extended to models with unbounded feature functions.
Sec. 4 establishes various mixing properties. A key result is that, in order to allow for unbounded
feature functions, the observations need to follow a distribution such that Hoeffding-type concentration inequalities can be established. Furthermore, the mixing rates depend on the tail behaviour of
the distribution. In Sec. 5 the mixture properties are used to analyze model identifiability and consistency of the Maximum Likelihood estimates. Sec. 6 concludes with an outlook on open problems
for future research.
2
Conditional Markov Chains
Preliminaries. We use N, Z and R to denote the sets of natural numbers, integers and real numbers,
respectively. Let X be a metric space with the Borel sigma-field A, and Y be a finite set. Furthermore, consider a probability space (?, F, P) and let X = (Xt )t?Z , Y = (Yt )t?Z be sequences of
measurable mappings from ? into X and Y, respectively. Here,
? X is an infinite sequence of observations ranging in the domain X ,
? Y is an aligned sequence of hidden states taking values in the finite set Y.
For now, the distribution of X is arbitrary. Next we define Conditional Markov Chains, which
parameterize the conditional distribution of Y given X.
Definition. Consider a vector f of real-valued functions f : X ? Y ? Y ? R, called the feature
functions. Throughout this paper, we assume that the following condition is satisfied:
(A1) All feature functions are finite: |f (x, i, j)| < ? for all x ? X and i, j ? Y.
Associated with the feature functions is a vector ? of real-valued model-weights. The key in the
definition of Conditional Markov Chains is the matrix M (x) with the (i, j)-th component
m(x, i, j)
=
exp(?T f (x, i, j)).
In terms of statistical physics, m(x, i, j) measures the potential of the transition between the hidden
states i and j from time t?1 to t, given the observation x at time t. Next, for a sequence x = (xt )t?Z
in X and time points s, t ? Z with s ? t, introduce the vectors
?ts (x)
= M (xt )T . . . M (xs )T (1, 1, . . . , 1)T ,
? ts (x)
= M (xs+1 ) . . . M (xt ) (1, 1, . . . , 1)T ,
and write ?st (x, i) and ?st (x, j) to denote the ith respectively jth components. Intuitively, ?st (x, i)
measures the potential of the hidden state i at time t given the observations xs , . . . , xt and assuming
that at time s ? 1 all hidden states have potential equal to 1. Similarly, ?st (x, j) is the potential of
j at time s assuming equal potential of all hidden states at time t. Now let t ? Z and k ? N, and
define the distribution of the labels Yt , . . . , Yt+k conditional on X,
P(Yt = yt , . . . , Yt+k = yt+k | X)
:=
k
Y
m(Xt+i , yt+i?1 , yt+i )
i=1
? lim
t+k+n
t
(X, yt ) ?t+k
(X, yt+k )
?t?n
n??
?tt?n (X)T ? t+k+n
(X)
t
.
Note that, under assumption (A1), the limit on the right hand side is well-defined (see Theorem 2 in
[1]). Furthermore, the family of all marginal distributions obtained this way satisfies the consistency
conditions of Kolmogorov?s Extension Theorem. Hence we obtain a unique distribution for Y
conditional on X parameterized by the feature functions f and the model weights ?. Intuitively, the
distribution is obtained by conditioning the marginal distributions of Y on the finite observational
context (Xt?n , . . . , Xt+k+n ), and then letting the size of the context going to infinity.
Basic properties. We introduce the following notation: For any matrix P = (pij ) with strictly
positive entries let ?(P ) denote the mixing coefficient
?(P )
=
min
i,j,k,l
pik pjl
.
pjk pil
Note that 0 ? ?(P ) ? 1. This coefficient will play a key role in the analysis of mixing properties.
The following proposition summarizes fundamental properties of the distribution of Y conditional
on X, which directly follow from the above definition (also see Corollary 1 in [1]).
Proposition 1. Suppose that condition (A1) holds true. Then Y conditional on X forms a timeinhomogeneous Markov chain. Moreover, if X is strictly stationary, then the joint distribution
of the aligned sequences (X, Y ) is strictly stationary. The conditional transition probabilities
Pt (x, i, j) := P(Yt = j | Yt?1 = i, X = x) of Y given X = x have the following form:
Pt (x, i, j)
?tn (x, j)
.
n?? ? n (x, i)
t?1
= m(xt , i, j) lim
In particular, a lower bound for Pt (x, i, j) is given by
Pt (x, i, j) ?
m(xt , i, j) (mink?Y m(xt+1 , i, k))
,
|Y| (maxk?Y m(xt , j, k)) (maxk,l?Y m(xt+1 , k, l))
and the matrix of transition probabilities P t (x), with the (i, j)-th component given by Pt (x, i, j),
satisfies ?(P t (x)) = ?(M (xt )).
3
Ergodicity
In this section we establish conditions under which the aligned sequences (X, Y ) are jointly ergodic. Let us first recall the definition of ergodicity of X (see [8]): By X we denote the space of
sequences x = (xt )t?Z in X , and by A the corresponding product ?-field. Consider the probability
measure PX on (X , A) defined by PX (A) := P(X ? A) for A ? A. Finally, let ? denote the
operator on X which shifts sequences one position to the left: ? x = (xt+1 )t?Z . Then ergodicity of
X is formally defined as follows:
(A2) X is ergodic, that is, PX (A) = PX (? ?1 A) for every A ? A, and PX (A) ? {0, 1} for
every set A ? A satisfying A = ? ?1 A.
As a particular consequence of the invariance PX (A) = PX (? ?1 A), we obtain that X is strictly
stationary. Now we are able to formulate the key result of this section, which will be of central
importance in the later analysis. For simplicity, we state it for functions depending on the values
of X and Y only at time t. The generalization of the statement is straight-forward. In our later
analysis, we will use the theorem to show that the time average of feature functions f (Xt , Yt?1 , Yt )
converges to the expected value E[f (Xt , Yt?1 , Yt )].
Theorem 1. Suppose that conditions (A1) and (A2) hold, and g : X ? Y ? R is a function which
satisfies E[|g(Xt , Yt )|] < ?. Then
n
1X
g(Xt , Yt ) = E[g(Xt , Yt )] P-almost surely.
lim
n?? n
t=1
Proof. Consider the sequence Z = (Zt )t?N given by Zt := (? t?1 X, Yt ), where we write ? t?1 to
denote the (t ? 1)th iterate of ? . Note that Zt represents the hidden state at time t together with the
entire aligned sequence of observations ? t?1 X. In the literature, such models are known as Markov
sequences in random environments (see [9]). The key step in the proof is to show that Z
Pisn ergodic.
Then, for any function h : X ? Y ? R with E[|h(Zt )|] < ?, the time average n1 t=1 h(Zt )
converges to the expected value E[h(Zt )] P-almost surely. Applying this result to the composition
of the function g and the projection of (? t?1 X, Yt ) onto (Xt , Yt ) completes the proof. The details
of the proof that Z is ergodic can be found in an extended version of this paper, which is included
in the supplementary material.
4
Mixing properties
In this section we are going to study mixing properties of the aligned sequences (X, Y ). To establish
the results, we will assume that the distribution of the observations X satisfies conditions under
which certain concentration inequalities hold true:
Pn
(A3) Let A ? A be a measurable set, with p := P(Xt ? A) and Sn (x) := n1 t=1 1(xt ? A)
for x ? X . Then there exists a constant ? such that, for all n ? N and > 0,
P(|Sn (X) ? p| ? ) ?
exp(?? 2 n).
If X is a sequence of independent random variables, then (A3) follows by Hoeffding?s inequality. In
the dependent case, concentration inequalities of this type can be obtained by imposing Martingale
or mixing conditions on X (see [12,13]). Furthermore, we will make the following assumption,
which relates the feature functions to the tail behaviour of the distribution of X:
(A4) Let h : [0, ?) ? [0, ?) be a differentiable decreasing function with h(z) = O(z ?(1+?) )
for some ? > 0. Furthermore, let
X
F (x) :=
|?T f (x, j, k)|
j,k?Y
?1
for x ? X . Then E[h(F (Xt ))
] and E[h0 (F (Xt ))?1 ] both exist and are finite.
The following theorem establishes conditions under which the expected conditional covariances of
square-integrable functions are summable. The result is obtained by studying ergodic properties of
the transition probability matrices.
Theorem 2. Suppose that conditions (A1) - (A3) hold true, and g : X ? Y ? R is a function with
finite second moment, E[|g(Xt , Yt )|2 ] < ?. Let ?t,k (X) = Cov(g(Xt , Yt ), g(Xt+k , Yt+k ) | X)
denote the covariance of g(Xt , Yt ) and g(Xt+k , Yt+k ) conditional on X. Then, for every t ? Z:
n
X
E[|?t,k (X)|] < ?.
lim
n??
k=1
Proof. Without loss of generality we may assume that g can be written as g(x, y) = g(x)1(y = i).
Hence, using H?older?s inequality, we obtain
E[|?t,k (X)|] ? E[|g(Xt )|] E[|g(Xt+k )|] E[|Cov(1(Yt = i), 1(Yt+k = i) | X)|].
According to the assumptions, we have E[|g(Xt )|] = E[|g(Xt+k )|] < ?, so we only need to bound
the expectation of the conditional covariance. Note that
Cov(1(Yt = i), 1(Yt+k = i) | X) = P(Yt = i, Yt+k = i | X) ? P(Yt = i | X) P(Yt+k = i | X).
Recall the definition of ?(P ) before Corollary 1. Using probabilistic arguments, it is not difficult to
show that the ratio of P(Yt = i, Yt+k = i | X) to P(Yt = i | X) P(Yt+k = i | X) is greater than or
equal to ?(P t+1 (X) . . . P t+k (X)), where P t+1 (X), . . . , P t+k (X) denote the transition matrices
introduced in Proposition 1. Hence,
|Cov(1(Yt = i), 1(Yt+k = i) | X)| ? P(Yt = i, Yt+k = i | X)[1 ? ?(P t+1 (X) . . . P t+k (X))].
Now, using results from the theory of weak ergodicity (see Chapter 3 in [6]), we can establish
p
p
k
Y
1 ? ?(P t+j (x))
1 ? ?(P t+1 (x) . . . P t+k (x))
p
p
?
1 + ?(P t+j (x))
1 + ?(P t+1 (x) . . . P t+k (x))
j=1
for all x ? X . Using Bernoulli?s inequality and the fact ?(P t+j (x)) = M (xt+j ) established in
Qk
Proposition 1, we obtain ?(P t+1 (x) . . . P t+k (x)) ? 1?4 j=1 [1??(M (xt+j ))]. Consequently,
|Cov(1(Yt = i), 1(Yt+k = i) | X)|
? 4
k
Y
[1 ? ?(M (Xt+j ))].
j=1
With the notation introduced in assumption (A3), let ? > 0 and A ? X with p > 0 be such that
x ? A implies ?(M (x)) ? ?. Furthermore, let be a constant with 0 < < p. In order to
bound |Cov(1(Yt = i), 1(Yt+k = i) | X)| for a given k ? N, we distinguish two different cases: If
|Sk (X) ? p| < , then we obtain
4
k
Y
1 ? ?(M (Xt+j ))
? 4 (1 ? ?)k(p?) .
j=1
If |Sk (X) ? p| ? , then we use the trivial upper bound 1. According to assumption (A3), the
probability of the latter event is bounded by an exponential, and hence
E[|Cov(1(Yt = i), 1(Yt+k = i) | X)|] ?
4 (1 ? ?)k(p?) + exp(?? 2 k).
Obviously, the sum of all these expectations is finite, which completes the proof.
The next theorem bounds the difference between the distribution of Y conditional on X and finite
approximations of it. Introduce the following notation: For t, k ? 0 with t + k ? n let
P(0:n) (Yt = yt , . . . , Yt+k = yt+k | X = x)
:=
k
Y
n
?0t (x, yt ) ?t+k
(x, yt+k )
.
n??
?t0 (x)T ? nt (x)
m(xt+i , yt+i?1 , yt+i ) lim
i=1
(0:n)
Accordingly, write E
for expectations taken with respect to P(0:n) . As emphasized by the superscrips, these quantities can be regarded as marginal distributions of Y conditional on the observations at times t = 0, 1, . . . , n. To simplify notation, the following theorem is stated for
1-dimensional conditional marginal distributions, however, the extension to the general case is
straight-forward.
Theorem 3. Suppose that conditions (A1) - (A4) hold true. Then the limit
P(0:?) (Yt = i | X)
:=
lim P(0:n) (Yt = i | X)
n??
is well-defined P-almost surely. Moreover, there exists a measurable function C(x) of x ? X with
finite expectation, E[|C(X)|] < ?, and a function h(z) satisfying the conditions in (A4) , such that
(0:?)
P
(Yt = i | X) ? P(0:n) (Yt = i | X) ? C(? t X) h(n ? t).
Proof. Define Gn (x) := M (xt+1 ) . . . M (xn ) and write gn (x, i, j) for the (i, j)-th component
of Gn (x). Note that ? nt (x) = Gn (x)(1, 1, . . . , 1)T . According to Lemma 3.4 in [6], there exist
numbers rij (x) such that
gn (x, i, k)
n?? gn (xj, k)
lim
= rij (x)
for all k ? Y. Consequently, the ratio of ?tn (x, i) to ?tn (x, j) converges to rij (x), and hence
?0t (x, i) ?tn (x, i)
n
n?? ?t (x)T ? t (x)
0
lim
=
1
q i (x)T r i (x)
where we use the notation q i (x) = ?t0 (x)/?0t (x, i) and r i (x) denotes the vector with the kth
component rki (x). This proves the first part of the theorem. In order to prove the second part, note
that |x ? y| ? |x?1 ? y ?1 | for any x, y ? (0, 1], and hence
(0:?)
?t (X)T ? nt (X)
P
(Yt = i | X) ? P(0:n) (Yt = i | X) ? q i (X)T r i (X) ? t 0
.
?0 (X, i) ?tn (X, i)
To bound the latter expression, introduce the vectors r ni (x) and r ni (x) with the kth components
gn (x, k, l)
gn (x, k, l)
rnki (x) = min
and rnki (x) = max
.
l?Y
l?Y
gn (x, i, l)
gn (x, i, l)
It is easy to see that q i (x)T r ni (x) ? q i (x)T r i (x) ? q i (x)T r ni (x), and
q i (x)T r ni (x) ?
?t0 (x)T ? nt (x)
? q i (x)T r ni (x).
?0t (x, i) ?tn (x, i)
Hence,
?t (X)T ? nt (X)
q i (X)T r i (X) ? t 0
?
?0 (X, i) ?tn (X, i)
q i (X)T (r ni (X) ? r n (X)) .
i
Due to space limitations, we only give a sketch of the proof how the latter quantity can be bounded.
For details, see the extended version of this paper in the supplementary material. The first step is
to show the existence of a function C1 (x) with E[|C1 (X)|] < ? such that |rnki (X) ? rnki (X)| ?
C1 (? t X) (1 ? ?)n?t for some ? > 0. With the function F (x) introduced in assumption (A4), we
define C2 (x) := exp(F (x)) for x ? X and arrive at
(0:?)
P
(Yt = i | X) ? P(0:n) (Yt = i | X) ? |Y|2 C1 (? t X) C2 (Xt ) (1 ? ?)n?t .
The next step is to construct a function C3 (x) satisfying the following two conditions: (i) If
C2 (x)(1 ? ?)k ? 1, then C3 (x)h(k) ? 1. (ii) If C2 (x)(1 ? ?)k < 1, then C3 (x)h(k) ?
C2 (x) (1 ? ?)k . Since the difference between two probabilities cannot exceed 1, we obtain
(0:?)
P
(Yt = i | X) ? P(0:n) (Yt = i | X) ? |Y|2 C1 (? t X) C3 (Xt ) h(n ? t).
The last step is to show that E[|C3 (Xt )|] < ?.
The following result will play a key role in the later analysis of empirical likelihood functions.
Theorem 4. Suppose that conditions (A1) - (A4) hold, and the function g : X ? Y ? R satisfies
E[|g(Xt , Yt )|] < ?. Then
n
1 X (0:n)
lim
E
[g(Xt , Yt ) | X] = E[g(Xt , Yt )] P-almost surely.
n?? n
t=1
Proof. Without loss of generality we may assume that g can be written as g(x, y) = g(x)1(y = i).
Using the result from Theorem 3, we obtain
n
n
n
X
X
X
E(0:n) [g(Xt , Yt ) | X] ?
E(0:?) [g(Xt , Yt ) | X] ?
|g(Xt )| |C(? t X)| h(n ? t),
t=1
t=1
t=1
where h(z) is a function satisfying the conditions in assumption (A4). See the extended version of
this paper in the supplementary material for more details. Using the facts that X is ergodic and the
expectations of |g(Xt )| and |C(? t X)| are finite, we obtain
n
n
X
1 X (0:n)
E(0:?) [g(Xt , Yt ) | X] = 0.
E
[g(Xt , Yt ) | X] ?
lim
n?? n
t=1
t=1
By similar arguments to the proof of the first part of Theorem 3 one can show that the difference
|E(0:?) [g(Xt , Yt ) | X] ? E[g(Xt , Yt ) | X]| tends to 0 as t ? ?. Thus,
n
n
X
1 X (0:?)
lim
E
[g(Xt , Yt ) | X] ?
E[g(Xt , Yt ) | X] = 0.
n?? n
t=1
t=1
Now, noting that E[g(Xt , Yt ) | X] = E[g(X0 , Y0 ) | ? t X] for every t, the theorem follows by the
ergodicity of X.
5
Maximum Likelihood learning and model identifiability
In this section we apply the previous results to analyze asymptotic properties of the empirical
likelihood function. The setting is the following: Suppose that we observe finite subsequences
X n = (X0 , . . . , Xn ) and Y n = (Y0 , . . . , Yn ) of X and Y , where the distribution of Y conditional on X follows a conditional Markov chain with fixed feature functions f and unknown model
weights ?? . We assume that ?? lies in some parameter space ?, the structure of which will become important later. To emphasize the role of the model weights, we will use subscripts, e.g.,
P? and E? , to denote the conditional distributions. Our goal is to identify the unknown model
weights from the
Pnfinite samples, X n and Y n . In order to do so, introduce the shorthand notation
f (xn , y n ) =
t=1 f (xt , yt?1 , yt ) for xn = (x0 , . . . , xn ) and y n = (y0 , . . . , yn ). Consider the
normalized conditional likelihood,
X
1 T
? f (X n , Y n ) ? log
exp ?T f (X n , y n ) .
Ln (?) =
n
n+1
y n ?Y
Note that, in the context of finite Conditional Markov Chains, Ln (?) is the exact likelihood of Y n
conditional on X n . The Maximum Likelihood estimate of ?? is given by
? n := arg max Ln (?).
?
???
If Ln (?) is strictly concave, then the arg max is unique and can be found using gradient-based
search (see [14]). It is easy to see that Ln (?) is strictly concave if and only if |Y| > 1, and there
exists a sequence y n such that at least one component of f (X n , y n ) is non-zero. In the following,
we study strong consistency of the Maximum Likelihood estimates, a property which is of central
importance in large sample theory (see [15]). As we will see, a key problem is the identifiability and
uniqueness of the model weights.
5.1
Asymptotic properties of the likelihood function
In addition to the conditions (A1)-(A4) stated earlier, we will make the following assumptions:
(A5) The feature functions have finite second moments: E?? [|f (Xt , Yt?1 , Yt )|2 ] < ?.
(A6) The parameter space ? is compact.
The next theorem establishes asymptotic properties of the likelihood function Ln (?).
Theorem 5. Suppose that conditions (A1)-(A6) are satisfied. Then the following holds true:
(i) There exists a function L(?) such that limn?? Ln (?) = L(?) P?? -almost surely for
every ? ? ?. Moreover, the convergence of Ln (?) to L(?) is uniform on ?, that is,
limn?? sup??? |Ln (?) ? L(?)| = 0 P?? -almost surely.
(ii) The gradients satisfy limn?? ?Ln (?) = E?? [f (Xt , Yt?1 , Yt )] ? E? [f (Xt , Yt?1 , Yt )]
P?? -almost surely for every ? ? ?.
(iii) If the limit of the Hessian ?2 Ln (?) is finite and non-singular, then the function L(?) is
strictly concave on ?. As a consequence, the Maximum Likelihood estimates are strongly
consistent:
?n
lim ?
n??
= ??
P?? -almost surely.
Proof. The statements are obtained analogously to Lemma 4-6 and Theorem 4 in [1], using the
auxiliary results for Conditional Markov Chains with unbounded feature functions established in
Theorem 1, Theorem 2, and Theorem 4.
Next, we study the asymptotic behaviour of the Hessian ?2 Ln (?). In order to compute the derviatives, introduce the P
vectors ?1 , . . . , ?n with ?t = ? for t = 1, . . . , n. This allows us to write
n
T
?T f (X n , Y n ) =
t=1 ?t f (Xt , Yt?1 , Yt ). Now, regard the argument ? of the likelihood function as a stacked vector (?1 , . . . , ?n ). Same as in [1], this gives us the expressions
?2
Ln (?)
??t ??Tt+k
1
(0:n)
Cov?
f (Xt , Yt?1 , Yt ), f (Xt+k , Yt+k?1 , Yt+k )T | X
n
=
(0:n)
(0:n)
where Cov?
is the covariance with respect to the measure P?
Using these expressions, the Hessian of Ln (?) can be written as
?2 Ln (?)
=
?
n
X
t=1
introduced before Theorem 3.
n?1
X n?k
X
?2
?2
L
(?)
+
2
L
(?)
.
n
n
??t ??Tt
??t ??Tt+k
k=1 t=1
The following theorem establishes an expression for the limit of ?2 Ln (?). It differs from the
expression given in Lemma 7 of [1], which is incorrect.
Theorem 6. Suppose that conditions (A1) - (A5) hold. Then
lim ?2 Ln (?)
n??
?
X
= ? ?? (0) + 2
?? (k)
P?? -almost surely
k=1
where ?? (k) = E[Cov? (f (Xt , Yt?1 , Yt ), f (Xt+k , Yt+k?1 , Yt+k ) | X)] is the expectation of the
conditional covariance (with respect to P? ) between f (Xt , Yt?1 , Yt ) and f (Xt+k , Yt+k?1 , Yt+k )
given X. In particular, the limit of ?2 Ln (?) is finite.
Proof. The key step is to show the existence of a positive measurable function U? (k, x) such that
lim
n??
n?1
X n?k
X
k=1 t=1
?2
L
(?)
n
T
??t ??t+k
?
lim
n??
n?1
X
k=1
E[U? (k, X)]
with the limit on the right hand side being finite. Then the rest of the proof is straight-forward:
Theorem 4 shows that, for fixed k ? 0,
n?k
X
?2
lim
Ln (?) = ?? (k) P?? -almost surely.
n??
??t ??Tt+k
t=1
Hence, in order to establish the theorem, it suffices to show that
n?k
n?1
X
X
?2
L
(?)
lim
?? (k) ?
?
n
T
n??
??t ??t+k
t=1
k=1
P?
for all > 0. Now let > 0 be fixed. According to Theorem 2 we have k=1 |?? (k)| < ?. Hence
we can find a finite N such that
n?1
n?1
X
X
lim
|?? (k)| + lim
E[U? (k, X)] ? .
n??
k=N
n??
On the other hand, Theorem 4 shows that
n?k
N
?1
X
X
lim
?? (k) ?
n??
k=1
t=1
k=N
?2
L
(?)
n
T
??t ??t+k
=
0.
For details on how to construct U? (k, x), see the extended version of this paper.
5.2
Model identifiability
Let us conclude the analysis by investigating conditions under which the limit of the Hessian
?2 Ln (?) is non-singular. Note that ?2 Ln (?) is negative definite for every n, so also the limit
is negative definite, but not necessarily strictly negative definite. Using the result in Theorem 6, we
can establish the following statement:
Corollary 1. Suppose that assumptions (A1)-(A5) hold true. Then the following conditions are
necessary for the limit of ?2 Ln (?) to be non-singular:
(i) For each feature function f (x, i, j), there exists a set A ? X with P(Xt ? A) > 0 such
that, for every x ? A, we can find i, j, k, l ? Y with f (x, i, j) 6= f (x, k, l).
(ii) There does not exist a non-zero vector ? and a subset A ? X with P(Xt ? A) = 1 such
that ?T f (x, i, j) is constant for all x ? X and i, j ? Y.
Condition (i) essentially says: features f (x, i, j) must not be constant in i and j. Condition (ii)
says that features must not be expressible as linear combinations of each other. Clearly, features
violating condition (i) can be assigned arbitrary model weights without any effect on the conditional
distributions. If condition (ii) is violated, then there are infinitely many ways for parameterizing the
same model. In practice, some authors have found positive effects of including redundant features
(see, e.g., [16]), however, usually in the context of a learning objective with an additional penalizer.
6
Conclusions
We have established ergodicity and various mixing properties of Conditional Markov Chains with
unbounded feature functions. The main insight is that similar results to the setting with bounded
feature functions can be obtained, however, under additional assumptions on the distribution of the
observations. In particular, the proof of Theorem 2 has shown that the sequence of observations
needs to satisfy conditions under which Hoeffding-type concentration inequalities can be obtained.
The proof of Theorem 3 has reveiled an interesting interplay between mixing rates, feature functions, and the tail behaviour of the distribution of observations. By applying the mixing properties to the empirical likelihood functions we have obtained necessary conditions for the Maximum
Likelihood estimates to be strongly consistent. We see a couple of interesting problems for future
research: establishing Central Limit Theorems for Conditional Markov Chains; deriving bounds for
the asymptotic variance of Maximum Likelihood estimates; constructing tests for the significance
of features; generalizing the estimation theory to an infinite number of features; finally, finding
sufficient conditions for the model identifiability.
References
[1] Sinn, M. & Poupart, P. (2011) Asymptotic theory for linear-chain conditional random fields. In Proc. of the
14th International Conference on Artificial Intelligence and Statistics (AISTATS).
[2] Lafferty, J., McCallum, A. & Pereira, F. (2001) Conditional random fields: Probabilistic models for segmenting and labeling sequence data. In Proc. of the 18th IEEE International Conference on Machine Learning
(ICML).
[3] Sutton, C. & McCallum, A. (2006) An introduction to conditional random fields for relational learning. In:
Getoor, L. & Taskar, B. (editors), Introduction to Statistical Relational Learning. Cambridge, MA: MIT Press.
[4] Hofmann, T., Sch?olkopf, B. & Smola, A.J. (2008) Kernel methods in machine learning. The Annals of
Statstics, Vol. 36, No. 3, 1171-1220.
[5] Xiang, R. & Neville, J. (2011) Relational learning with one network: an asymptotic analysis. In Proc. of
the 14th International Conference on Artificial Intelligence and Statistics (AISTATS).
[6] Seneta, E. (2006) Non-Negative Matrices and Markov Chains. Revised Edition. New York, NY: Springer.
[7] Wainwright, M.J. & Jordan, M.I. (2008) Graphical models: exponential families, and variational inference.
R
in Machine Learning, Vol. 1, Nos. 1-2, 1-305.
Foundations and Trends
[8] Cornfeld, I.P., Fomin, S.V. & Sinai, Y.G. (1982) Ergodic Theory. Berlin, Germany: Springer.
[9] Orey, S. (1991) Markov chains with stochastically stationary transition probabilities. The Annals of Probability, Vol. 19, No. 3, 907-928.
[10] Hern?andez-Lerma, O. & Lasserre, J.B. (2003) Markov Chains and Invariant Probabilities. Basel, Switzerland: Birkh?auser.
[11] Foguel, S.R. (1969) The Ergodic Theory of Markov Processes. Princeton, NJ: Van Nostrand.
[12] Samson, P.-M. (2000) Concentration of measure inequalities for Markov chains and ?-mixing processes.
The Annals of Probability, Vol. 28, No. 1, 416-461.
[13] Kontorovich, L. & Ramanan, K. (2008) Concentration inequalities for dependent random variables via the
martingale method. The Annals of Probability, Vol. 36, No. 6, 2126-2158.
[14] Sha, F. & Pereira, F. (2003) Shallow parsing with conditional random fields. In Proc. of the Human Language Technology Conference of the North American Chapter of the Association for Computational Linguistics
(HLT-NAACL).
[15] Lehmann, E.L. (1999) Elements of Large-Sample Theory. New York, NY: Springer.
[16] Hoefel, G. & Elkan, C. (2008) Learning a two-stage SVM/CRF sequence classifier. In Proc. of the 17th
ACM International Conference on Information and Knowledge Management (CIKM).
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4,084 | 4,697 | Spectral Learning of General Weighted Automata
via Constrained Matrix Completion
Borja Balle
Universitat Polit`ecnica de Catalunya
Mehryar Mohri
Courant Institute and Google Research
[email protected]
[email protected]
Abstract
Many tasks in text and speech processing and computational biology require estimating functions mapping strings to real numbers. A broad class of such functions can be defined by weighted automata. Spectral methods based on the singular value decomposition of a Hankel matrix have been recently proposed for
learning a probability distribution represented by a weighted automaton from a
training sample drawn according to this same target distribution. In this paper, we
show how spectral methods can be extended to the problem of learning a general
weighted automaton from a sample generated by an arbitrary distribution. The
main obstruction to this approach is that, in general, some entries of the Hankel
matrix may be missing. We present a solution to this problem based on solving a
constrained matrix completion problem. Combining these two ingredients, matrix
completion and spectral method, a whole new family of algorithms for learning
general weighted automata is obtained. We present generalization bounds for a
particular algorithm in this family. The proofs rely on a joint stability analysis of
matrix completion and spectral learning.
1
Introduction
Many tasks in text and speech processing, computational biology, or learning models of the environment in reinforcement learning, require estimating a function mapping variable-length sequences to
real numbers. A broad class of such functions can be defined by weighted automata. The mathematical and algorithmic properties of weighted automata have been extensively studied in the most
general setting where they are defined in terms of an arbitrary semiring [28, 9, 23]. Weighted automata are widely used in applications ranging from natural text and speech processing [24] to
optical character recognition [12] and image processing [1]. This paper addresses the problem of
learning weighted automata from a finite set of labeled examples.
The particular instance of this problem where the objective is to learn a probabilistic automaton
from examples drawn from this same distribution has recently drawn much attention: starting with
the seminal work of Hsu et al. [19], the so-called spectral method has proven to be a valuable tool in
developing novel and theoretically-sound algorithms for learning HMMs and other related classes
of distributions [5, 30, 31, 10, 6, 4]. Spectral methods have also been applied to other probabilistic
models of practical interest, including probabilistic context-free grammars and graphical models
with hidden variables [26, 22, 16, 3, 2]. The main idea behind these algorithms is that, under
an identifiability assumption, the method of moments can be used to formulate a set of equations
relating the parameters defining the target to observable statistics. Given enough training data, these
statistics can be accurately estimated. Then, solving the corresponding approximate equations yields
a model that closely estimates the target distribution. The spectral term takes its origin from the use
of a singular value decomposition in solving those equations.
1
This paper tackles a significantly more general and more challenging problem than the specific
instance just mentioned. Indeed, in general, there seems to be a large gap separating the scenario
of learning a probabilistic automaton using data drawn according to the distribution it generates,
from that of learning an arbitrary weighted automaton from labeled data drawn from some unknown
distribution. For a start, in the former setting there is only one object to care about because the
distribution from which examples are drawn is the target machine. In contrast, the latter involves two
distinct objects: a distribution according to which strings are drawn, and a target weighted automaton
assigning labels to these strings. It is not difficult in this setting to conceive that, for a particular
target, an adversary could find a distribution over strings making the learner?s task insurmountably
difficult. In fact, this is the core idea behind the cryptography-based hardness results for learning
deterministic finite automata given by Kearns and Valiant [20] ? these same results apply to our
setting as well.
But, even in cases where the distribution ?cooperates,? there is still an obstruction in leveraging the
spectral method for learning general weighted automata. The statistics used by the spectral method
are essentially the probabilities assigned by the target distribution to each string in some fixed finite
set B. In the case where the target is a distribution, increasingly large samples yield uniformly
convergent estimates for these probabilities. Thus, it can be safely assumed that the probability of
any string from B not present in the sample is zero. When learning arbitrary weighted automata,
however, the value assigned by the target to an unseen string is unknown. Furthermore, one cannot
expect that a sample would contain the values of the target function for all the strings in B. This
observation raises the question of whether it is possible at all to apply the spectral method in a
setting with missing data, or, alternatively, whether there is a principled way to ?estimate? this
missing information and then apply the spectral method.
As it turns out, the latter approach can be naturally formulated as a constrained matrix completion
problem. When applying the spectral method, the (approximate) values of the target on B are arranged in a matrix H. Thus, the main difference between the two settings can be restated as follows:
when learning a weighted automaton representing a distribution, unknown entries of H can be filled
in with zeros, while in the general setting there is a priori no straightforward method to fill in the
missing values. We propose to use a matrix completion algorithm for solving this last problem. In
particular, since H is a Hankel matrix whose entries must satisfy some equality constraints, it turns
out that the problem of learning weighted automata under an arbitrary distribution leads to what
we call the Hankel matrix completion problem. This is essentially a constrained matrix completion
problem where entries of valid hypotheses need to satisfy a set of equalities. We give an algorithm
for solving this problem via convex optimization. Many existing approaches to matrix completion,
e.g., [14, 13, 27, 18], are also based on convex optimization. Since the set of valid hypotheses for our
constrained matrix completion problem is convex, many of these algorithms could also be modified
to deal with the Hankel matrix completion problem.
In summary, our approach leverages two recent techniques for learning a general weighted automaton: matrix completion and spectral learning. It consists of first predicting the missing entries in
H and then applying the spectral method to the resulting matrix. Altogether, this yields a family
of algorithms parametrized by the choice of the specific Hankel matrix completion algorithm used.
These algorithms are designed for learning an arbitrary weighted automaton from samples generated
by an unknown distribution over strings and labels.
We study a special instance of this family of algorithms and prove generalization guarantees for
its performance based on a stability analysis, under mild conditions on the distribution. The proof
contains two main novel ingredients: a stability analysis of an algorithm for constrained matrix
completion, and an extension of the analysis of spectral learning to an agnostic setting where data
is generated by an arbitrary distribution and labeled by a process not necessarily modeled by a
weighted automaton.
The rest of the paper is organized as follows. Section 2 introduces the main notation and definitions
used in subsequent sections. In Section 3, we describe a family of algorithms for learning general
weighted automata by combining constrained matrix completion and spectral methods. In Section 4, we give a detailed analysis of one particular algorithm in this family, including generalization
bounds.
2
2
Preliminaries
This section introduces the main notation used in this paper. Bold letters will be used for vectors v and matrices M. For vectors, kvk denotes the standard euclidean norm. For matrices, kMk denotes the operator norm. For p ? [1, +?], kMkp denotes the Schatten p-norm:
P
kMkp = ( n?1 ?np (M))1/p , where ?n (M) is the nth singular value of M. The special case
p = 2 coincides with the Frobenius norm which will be sometimes also written as kMkF . The
Moore?Penrose pseudo-inverse of a matrix M is denoted by M+ .
2.1
Functions over Strings and Hankel Matrices
We denote by ? = {a1 , . . . , ak } a finite alphabet of size k ? 1 and by the empty string. We also
write ?0 = {} ? ?. The set of all strings over ? is denoted by ?? and the length of a string x
denoted by |x|. For any n ? 0, ??n denotes the set of all strings of length at most n. Given two
sets of strings P, S ? ?? we denote by PS the set of all strings uv obtained by concatenation of a
string u ? P and a string v ? S. A set of strings P is called ?-complete when P = P 0 ?0 for some
set P 0 . P 0 is then called the root of P. A pair (P, S) with P, S ? ?? is said to form a basis of ??
if ? P ? S and P is ?-complete. We define the dimension of a basis (P, S) as the cardinality of
PS, that is |PS|.
For any basis B = (P, S), we denote by HB the vector space of functions RPS whose dimension
is the dimension of B. We will simply write H instead of HB when the basis B is clear from the
context. The Hankel matrix H ? RP?S associated to a function h ? H is the matrix whose entries
are defined by H(u, v) = h(uv) for all u ? P and v ? S. Note that the mapping h 7? H is linear.
In fact, H is isomorphic to the vector space formed by all |P| ? |S| real Hankel matrices and we can
thus write by identification
H = H ? RP?S : ?u1 , u2 ? P, ?v1 , v2 ? S, u1 v1 = u2 v2 ? H(u1 , v1 ) = H(u2 , v2 ) .
It is clear from this characterization that H is a convex set because it is a subset of a convex
space defined by equality constraints. In particular, a matrix in H contains |P||S| coefficients with
|PS| degrees of freedom, and the dependencies can be specified as a set of equalities of the form
H(u1 , v1 ) = H(u2 , v2 ) when u1 v1 = u2 v2 . We will use both characterizations of H indistinctly for
the rest of the paper. Also, note that different orderings of P and S may result in different sets of
matrices. For convenience, we will assume for all that follows an arbitrary fixed ordering, since the
choice of that order has no effect on any of our results.
Matrix norms extend naturally to norms in H. For any p ? [1, +?], the Hankel?Schatten p-norm
on H is defined as khkp = kHkp . It is straightforward to verify that khkp is a norm by the linearity
of h 7? H. In particular, this implies that the function k ? kp : H ? R is convex. In the case p = 2,
it can be seen that khk22 = hh, hiH , with the inner product on H defined by
X
hh, h0 iH =
cx h(x)h0 (x) ,
x?PS
where cx = |{(u, v) ? P ? S : x = uv}| is the number of possible decompositions of x into a prefix
in P and a suffix in S.
2.2
Weighted finite automata
A widely used class of functions mapping strings to real numbers is that of functions defined by
weighted finite automata (WFA) or in short weighted automata [23]. These functions are also known
as rational power series [28, 9]. A WFA over ? with n states can be defined as a tuple A =
h?, ?, {Aa }a?? i, where ?, ? ? Rn are the initial and final weight vectors, and Aa ? Rn?n the
transition matrix associated to each alphabet symbol a ? ?. The function fA realized by a WFA A
is defined by
fA (x) = ?> Ax1 ? ? ? Axt ? ,
for any string x = x1 ? ? ? xt ? ?? with t = |x| and xi ? ? for all i ? [1, t]. We will say that a WFA
A = h?, ?, {Aa }i is ?-bounded if k?k, k?k, kAa k ? ? for all a ? ?. This property is convenient
to bound the maximum value assigned by a WFA to any string of a given length.
3
1/2
a, 3/4
b, 6/5
a, 0
b, 2/3
1
1/2
-1
?> = [1/2 1/2] ? > = [1 ?1]
3/4 0
6/5 2/3
Aa =
Ab =
0 1/3
3/4 1
a, 1/3
b, 1
a, 0
b, 3/4
(a)
(b)
Figure 1: Example of a weighted automaton over ? = {a, b} with 2 states: (a) graph representation;
(b) algebraic representation.
WFAs can be more generally defined over an arbitrary semiring instead of the field of real numbers
and are also known as multiplicity automata (e.g., [8]). To any function f : ?? ? R, we can
?
?
associate its Hankel matrix Hf ? R? ?? with entries defined by Hf (u, v) = f (uv). These are
just the bi-infinite versions of the Hankel matrices we introduced in the case P = S = ?? . Carlyle
?
and Paz [15] and Fliess [17] gave the following characterization of the set of functions f in R?
defined by a WFA in terms of the rank of their Hankel matrix rank(Hf ).1
Theorem 1 ([15, 17]) A function f : ?? ? R can be defined by a WFA iff rank(Hf ) is finite and in
that case rank(Hf ) is the minimal number of states of any WFA A such that f = fA .
Thus, WFAs can be viewed as those functions whose Hankel matrix can be finitely ?compressed?.
Since finite sub-blocks of a Hankel matrix cannot have a larger rank than its bi-infinite extension,
this justifies the use of a low-rank-enforcing regularization in the definition of a Hankel matrix
completion.
Note that deterministic finite automata (DFA) with n states can be represented by a WFA with at
most n states. Thus, the results we present here can be directly applied to classification problems in
?? . However, specializing our results to this particular setting may yield several improvements.
2.2.1
Example
Figure 1 shows an example of a weighted automaton A = h?, ?, {Aa }i with two states defined
over the alphabet ? = {a, b}, with both its algebraic representation (Figure 1(b)) in terms of vectors
and matrices and the equivalent graph representation (Figure 1(a)) useful for a variety of WFA
algorithms [23]. Let W = {, a, b}, then B = (W?0 , W) is a ?-complete basis. The following is
the Hankel matrix of A on this basis shown with three-digit precision entries:
?
?
a
b
aa
ab
ba
bb
? 0.00 0.20 0.14 0.22 0.15 0.45 0.31?
H>
B = ?a 0.20 0.22 0.45 0.19 0.29 0.45 0.85? .
b 0.14 0.15 0.31 0.13 0.20 0.32 0.58
By Theorem 1, the Hankel matrix of A has rank at most 2. Given HB , the spectral method described
in [19] can be used to recover a WFA A? equivalent to A, in the sense that A and A? compute the
same function. In general, one may be given a sample of strings labeled using some WFA that does
not contain enough information to fully specify a Hankel matrix over a complete basis. In that case,
Theorem 1 motivates the use of a low-rank matrix completion algorithm to fill in the missing entries
in HB prior to the application of the spectral method. This is the basis of the algorithm we describe
in the following section.
3
The HMC+SM Algorithm
In this section we describe our algorithm HMC+SM for learning weighted automata. As input,
the algorithm takes a sample Z = (z1 , . . . , zm ) containing m examples zi = (xi , yi ) ? ?? ? R,
1
The construction of an equivalent WFA with the minimal number of states from a given WFA was first
given by Sch?utzenberger [29].
4
1 ? i ? m, drawn i.i.d. from some distribution D over ?? ? R. There are three parameters a user
can specify to control the behavior of the algorithm: a basis B = (P, S) of ?? , a regularization
parameter ? > 0, and the desired number of states n in the hypothesis. The output returned by
HMC+SM is a WFA AZ with n states that computes a function fAZ : ?? ? R.
The algorithm works in two stages. In the first stage, a constrained matrix completion algorithm
with input Z and regularization parameter ? is used to return a Hankel matrix HZ ? HB . In the
second stage, the spectral method is applied to HZ to compute a WFA AZ with n states. These two
steps will be described in detail in the following sections.
As will soon become apparent, HMC+SM defines in fact a whole family of algorithms. In particular,
by combining the spectral method with any algorithm for solving the Hankel matrix completion
problem, one can derive a new algorithm for learning WFAs. For concreteness, in the following,
we will only consider the Hankel matrix completion algorithm described in Section 3.1. Through
its parametrization by a number 1 ? p ? ? and a convex loss ` : R ? R ? R+ , this completion
algorithm already gives rise to a family of learning algorithms that we denote by HMCp,` +SM.
However, it is important to keep in mind that for each existing matrix completion algorithm that can
be modified to solve the Hankel matrix completion problem, a new algorithm for learning WFAs
can be obtained via the general scheme we describe below.
3.1
Hankel Matrix Completion
We now describe our Hankel matrix completion algorithm. Given a basis B = (P, S) of ?? and a
sample Z over ?? ? R, the algorithm solves a convex optimization problem and returns a matrix
HZ ? HB . We give two equivalent descriptions of this optimization, one in terms of functions
h : PS ? R, and another in terms of Hankel matrices H ? RP?S . While the former is perhaps
conceptually simpler, the latter is easier to implement within the existing frameworks of convex
optimization.
e the subsample of Z formed by examples z = (x, y) with x ? PS and by m
We will denote by Z
e
e For any p ? [1, +?] and a convex loss function ` : R ? R ? R+ , we consider the
its size |Z|.
objective function FZ defined for any h ? H by
X
b e (h) = ? khk2p + 1
FZ (h) = ? N (h) + R
`(h(x), y) ,
Z
m
e
e
(x,y)?Z
where ? > 0 is a regularization parameter. FZ is a convex function, by the convexity of k ? kp and `.
Our algorithm seeks to minimize this loss function over the finite-dimensional vector space H and
returns a function hZ satisfying
hZ ? argmin FZ (h) .
(HMC-h)
h?H
To define an equivalent optimization over the matrix version of H, we introduce the following notation. For each string x ? PS, fix a pair of coordinate vectors (ux , vx ) ? RP ? RS such that
u>
x Hvx = H(x) for any H ? H. That is, ux and vx are coordinate vectors corresponding respectively to a prefix u ? P and a suffix v ? S, and such that uv = x. Now, abusing our previous
notation, we define the following loss function over matrices:
X
b e (H) = ? kHk2p + 1
`(u>
FZ (H) = ? N (H) + R
x Hvx , y) .
Z
m
e
e
(x,y)?Z
This is a convex function defined over the space of all |P| ? |S| matrices. Optimizing FZ over the
convex set of Hankel matrices H leads to an algorithm equivalent to (HMC-h):
HZ ? argmin FZ (H) .
(HMC-H)
H?H
We note here that our approach shares some common aspects with some previous work in matrix
completion. The fact that there may not be a true underlying Hankel matrix makes it somewhat close
to the agnostic setting in [18], where matrix completion is also applied under arbitrary distributions.
Nonetheless, it is also possible to consider other learning frameworks for WFAs where algorithms
for exact matrix completion [14, 27] or noisy matrix completion [13] may be useful. Furthermore,
since most algorithms in the literature of matrix completion are based on convex optimization problems, it is likely that most of them can be adapted to solve constrained matrix completions problems
such as the one we discuss here.
5
3.2
Spectral Method for General WFA
Here, we describe how the spectral method can be applied to HZ to obtain a WFA. We use the
same notation as in [7] and a version of the spectral method working with an arbitrary basis (as in
[5, 4, 7]), in contrast to versions restricted to P = ??2 and S = ? like [19].
We first need to partition HZ into k + 1 blocks as follows. Since B is a basis, P is ?-complete
0
and admits a root P 0 . We define a block Ha ? RP ?S for each a ? ?0 , whose entries are given by
0
Ha (u, v) = HZ (ua, v), for any u ? P and v ? S. Thus, after suitably permuting the rows of HZ ,
>
>
>
we can write H>
Z = [H , Ha1 , . . . , Hak ]. We will use the following specific notation to refer to the
rows and columns of H corresponding to ? P 0 ? S: h,S ? RS with h,S (v) = H (, v) and
0
hP 0 , (u) ? RP with hP 0 , (u) = H (u, ).
Using this notation, the spectral method can be described as follows. Given the desired number
of states n, it consists of first computing the truncated SVD of H corresponding to the n largest
singular values: Un Dn Vn> . Thus, matrix Un Dn Vn> is the best rank n approximation to H with
respect to the Frobenius norm. Then, using the right singular vectors Vn of H , the next step
consists of computing a weighted automaton AZ = h?, ?, {Aa }i as follows:
?> = h>
,S Vn
? = (H Vn )+ hP 0 ,
Aa = (H Vn )+ Ha Vn .
(SM)
The fact that the spectral method is based on a singular value decomposition justifies in part the use
of a Schatten p-norm as a regularizer in (HMC-H). In particular, two very natural choices are p = 1
and p = 2. The first one corresponds to a nuclear norm regularized optimization, which is known
to enforce a low rank constraint on HZ . In a sense, this choice can be justified in view of Theorem
1 when the target is known to be generated by some WFA. On the other hand, choosing p = 2 also
has some effect on the spread of singular values, while at the same time enforcing the coefficients
in HZ ? especially those that are completely unknown ? to be small. As our analysis suggests, this
last property is important for preventing errors from accumulating on the values assigned by AZ to
long strings.
4
Generalization Bound
In this section, we study the generalization properties of HMCp,` +SM. We give a stability analysis
for a special instance of this family of algorithms and use it to derive a generalization bound. We
study the specific case where p = 2 and `(y, y 0 ) = |y ? y 0 | for all (y, y 0 ). But, much of our analysis
can be used to derive similar bounds for other instances of HMCp,` +SM. The proofs of the technical
results presented are given in the Appendix.
We first introduce some notation needed for the presentation of our main result. For any ? > 0, let
t? be the function defined by t? (x) = x for |x| ? ? and t? (x) = ? sign(x) for |x| > ?. For any
distribution D over ?? ? R, we denote by D? its marginal distribution over ?? . The probability that
a string x ? D? belongs to PS is denoted by ? = D? (PS).
We assume that the parameters B, n, and ? are fixed. Two parameters that depend on D will appear
in our bound. In order to define these parameters, we need to consider the output HZ of (HMC-H)
>
>
>
as a random variable that depends on the sample Z. Writing H>
Z = [H , Ha1 , . . . , Hak ], as in
Section 3.2, we define:
? = E m [?n (H )]
? = E m ?n (H )2 ? ?n+1 (H )2 ,
Z?D
Z?D
where ?n (M) denotes the nth singular value of matrix M. Note that these parameters may vary
with m, n, ? and B.
In contrast to previous learning results based on the spectral method, our bound holds in an agnostic
setting. That is, we do not require that the data was generated from some (probabilistic) unknown
WFA. However, in order to prove our results we do need to make two assumptions about the tails of
the distribution. First, we need to assume that there exists a bound on the magnitude of the labels
generated by the distribution.
Assumption 1 There exists a constant ? > 0 such that if (x, y) ? D, then |y| ? ? almost surely.
6
Second, we assume that the strings generated by the distribution will not be too long. In particular,
that the length of the strings generated by D? follows a distribution whose tail is slightly lighter than
sub-exponential.
Assumption 2 There exist constants c, ? > 0 such that Px?D? [|x| ? t] ? exp(?ct1+? ) holds for
all t ? 0.
We note that in the present context both assumptions are quite reasonable. Assumption 1 is equivalent to assumptions made in other contexts where a stability analysis is pursued, e.g., in the analysis
of support vector regression in [11]. Furthermore, in our context, this assumption can be relaxed
to require only that the distribution over labels be sub-Gaussian, at the expense of a more complex
proof.
Assumption 2 is required by the fact already pointed out in [19] that errors in the estimation of
operator models accumulate exponentially with the length of the string. Moreover, it is well known
that the tail of any probability distribution generated by a WFA is sub-exponential. Thus, though we
do not require D? to be generated by a WFA, we do need its distribution over lengths to have a tail
behavior similar to that of a distribution generated by a WFA. This seems to be a limitation common
to all known learnability proofs based on the spectral method.
We can now state our main result, which is a bound on the average loss R(f ) = Ez?D [`(f (x), y)]
bZ (f ) = |Z|?1 P
in terms of the empirical loss R
z?Z `(f (x), y).
Theorem 2 Let Z be a sample formed by m i.i.d. examples generated from some distribution D
satisfying Assumptions 1 and 2. Let AZ be the WFA returned by algorithm HMCp,` +SM with p = 2
and loss function `(y, y 0 ) = |y ? y 0 |. Then, for any ? > 0, the following holds with probability at
least 1 ? ? for fZ = t? ? fAZ :
r
4 2 3/2
? |P| |S|
ln m
1
b
R(fZ ) ? RZ (fZ ) + O
ln
.
? ? 3 ??
?
m1/3
The proof of this theorem is based on an algorithmic stability analysis. Thus, we will consider
two samples of size m, Z ? Dm consisting of m i.i.d. examples drawn from D, and Z 0 differing
0
0
). The
in Z 0 = (z1 , . . . , zm?1 , zm
from Z by just one point: say zm in Z = (z1 , . . . , zm ) and zm
0
new example zm is an arbitrary point the support of D. Throughout the analysis we use the shorter
notation H = HZ and H0 = HZ 0 for the Hankel matrices obtained from (HMC-H) based on
samples Z and Z 0 respectively.
The first step in the analysis is to bound the stability of the matrix completion algorithm. This is
done in the following lemma, that gives a sample-dependent and a sample-independent bound for
the stability of H.
Lemma 3 Suppose D satisfies Assumption 1. Then, the following holds:
p
1
kH ? H0 kF ? min 2? |P||S|,
.
? min{m,
e m
e 0}
The standard method for deriving generalization bounds from algorithmic stability results could be
applied here to obtain a generalization bound for our Hankel matrix completion algorithm. However,
our goal is to give a generalization bound for the full HMC+SM algorithm.
Using the bound on the Frobenius norm kH ? H0 kF , we are able to analyze the stability of ?n (H ),
?n (H )2 ? ?n+1 (H )2 , and Vn using well-known results on the stability of singular values and
singular vectors. These results are used to bound the difference between the operators of WFA AZ
and AZ 0 . The following lemma can be proven by modifying and extending some of the arguments
of [19, 4], which were given in the specific case of WFAs representing a probability distribution.
Lemma 4pLet ? = kH?H0 kF , ?
b = min{?n (H ), ?n (H0 )}, and ?b = ?n (H )2 ??n+1 (H )2 . Suppose ? ? ?b/4. Then, there exists some constant C > 0 such that the following three inequalities
7
hold:
?a ? ? : kAa ? A0a k ? C?? 3 |P|3/2 |S|1/2 /b
??
b2 ;
k? ? ?0 k ? C?? 2 |P|1/2 |S|/b
?;
k? ? ? 0 k ? C?? 3 |P|3/2 |S|1/2 /b
??
b2 .
The other half of the proof results from combining Lemmas 3 and 4 to obtain a bound for
|fZ (x) ? fZ 0 (x)|. This is a delicate step, because some of the bounds given above involve quantities
that are defined in terms of Z. Therefore, all these parameters need to be controlled in order to
ensure that the bounds do not grow too large. Furthermore, to obtain the desired bounds we need to
extend the usual tools for analyzing spectral methods to the current setting. In particular, these tools
need to be adapted to the agnostic settings where there is no underlying true WFA. The analysis is
further complicated by the fact that now the functions we are trying to learn and the distribution that
generates the data are not necessarily related.
Once all this is achieved, it remains to combine these new tools to show an algorithmic stability
result for HMCp,` +SM. In the following lemma, we first define ?bad? samples Z and show that bad
samples have a very low probability.
Lemma 5 Suppose D satisfies Assumptions 1 and 2. If Z is a large enough i.i.d. sample from
D, then with probability at least 1 ? 1/m3 the following inequalities hold simultaneously: |xi | ?
((1/c) ln(4m4 ))1/1+? for all i, ? ? 4/(? ?m), ?
b ? ?/2, and ?b ? ?/2.
After that we give two upper bounds for |fZ (x) ? fZ 0 (x)|: a tighter bound that holds for ?good?
samples Z and Z 0 and a another one that holds for all samples. These bounds are combined using
a variant of McDiarmid?s inequality for dealing with functions that do not satisfy the bounded differences assumption almost surely [21]. The rest of the proof then follows the same scheme as the
standard one for deriving generalization bounds for stable algorithms [11, 25].
5
Conclusion
We described a new algorithmic solution for learning arbitrary weighted automata from a sample of labeled strings drawn from an unknown distribution. Our approach combines an algorithm
for constrained matrix completion with the recently developed spectral learning methods for learning probabilistic automata. Using our general scheme, a broad family of algorithms for learning
weighted automata can be obtained. We gave a stability analysis of a particular algorithm in that
family and used it to prove generalization bounds that hold for all distributions satisfying two reasonable assumptions. The particular case of Schatten p-norm with p = 1, which corresponds to
a regularization with the nuclear norm, can be analyzed using similar techniques. Our results can
be further extended by deriving generalization guarantees for all algorithms in the family we introduced. An extensive and rigorous empirical comparison of all these algorithms will be an important
complement to the research we presented. Finally, learning DFAs under an arbitrary distribution
using the algorithms we presented deserves a specific study since the problem is of interest in many
applications and since it may benefit from improved learning guarantees.
Acknowledgments
Borja Balle is partially supported by an FPU fellowship (AP2008-02064) and project TIN201127479-C04-03 (BASMATI) of the Spanish Ministry of Education and Science, the EU PASCAL2
NoE (FP7-ICT-216886), and by the Generalitat de Catalunya (2009-SGR-1428). The work of
Mehryar Mohri was partly funded by the NSF grant IIS-1117591.
8
References
[1] J. Albert and J. Kari. Digital image compression. In Handbook of Weighted Automata. Springer, 2009.
[2] A. Anandkumar, D. P. Foster, D. Hsu, S. M. Kakade, and Y-K. Liu. Two SVDs suffice: Spectral decompositions for probabilistic topic modeling and latent dirichlet allocation. CoRR, abs/1204.6703, 2012.
[3] A. Anandkumar, D. Hsu, and S. M. Kakade. A method of moments for mixture models and hidden
Markov models. COLT, 2012.
[4] R. Bailly. Quadratic weighted automata: Spectral algorithm and likelihood maximization. ACML, 2011.
[5] R. Bailly, F. Denis, and L. Ralaivola. Grammatical inference as a principal component analysis problem.
ICML, 2009.
[6] B. Balle, A. Quattoni, and X. Carreras. A spectral learning algorithm for finite state transducers. ECML?
PKDD, 2011.
[7] B. Balle, A. Quattoni, and X. Carreras. Local loss optimization in operator models: A new insight into
spectral learning. ICML, 2012.
[8] A. Beimel, F. Bergadano, N.H. Bshouty, E. Kushilevitz, and S. Varricchio. Learning functions represented
as multiplicity automata. JACM, 2000.
[9] J. Berstel and C. Reutenauer. Rational Series and Their Languages. Springer, 1988.
[10] B. Boots, S. Siddiqi, and G. Gordon. Closing the learning planning loop with predictive state representations. I. J. Robotic Research, 2011.
[11] O. Bousquet and A. Elisseeff. Stability and generalization. JMLR, 2002.
[12] T. M. Breuel. The OCRopus open source OCR system. IS&T/SPIE Annual Symposium, 2008.
[13] E.J. Candes and Y. Plan. Matrix completion with noise. Proceedings of the IEEE, 2010.
[14] E.J. Candes and T. Tao. The power of convex relaxation: Near-optimal matrix completion. IEEE Transactions on Information Theory, 2010.
[15] Jack W. Carlyle and Azaria Paz. Realizations by stochastic finite automata. J. Comput. Syst. Sci., 5(1):26?
40, 1971.
[16] S. B. Cohen, K. Stratos, M. Collins, D. P. Foster, and L. Ungar. Spectral learning of latent-variable
PCFGs. ACL, 2012.
[17] M. Fliess. Matrices de Hankel. Journal de Math?ematiques Pures et Appliqu?ees, 53:197?222, 1974.
[18] R. Foygel, R. Salakhutdinov, O. Shamir, and N. Srebro. Learning with the weighted trace-norm under
arbitrary sampling distributions. NIPS, 2011.
[19] D. Hsu, S. M. Kakade, and T. Zhang. A spectral algorithm for learning hidden Markov models. COLT,
2009.
[20] M. Kearns and L. Valiant. Cryptographic limitations on learning boolean formulae and finite automata.
JACM, 1994.
[21] S. Kutin. Extensions to McDiarmid?s inequality when differences are bounded with high probability.
Technical report, TR-2002-04, University of Chicago, 2002.
[22] F.M. Luque, A. Quattoni, B. Balle, and X. Carreras. Spectral learning in non-deterministic dependency
parsing. EACL, 2012.
[23] M. Mohri. Weighted automata algorithms. In Handbook of Weighted Automata. Springer, 2009.
[24] M. Mohri, F. C. N. Pereira, and M. Riley. Speech recognition with weighted finite-state transducers. In
Handbook on Speech Processing and Speech Communication. Springer, 2008.
[25] M. Mohri, A. Rostamizadeh, and A. Talwalkar. Foundations of Machine Learning. The MIT Press, 2012.
[26] A.P. Parikh, L. Song, and E.P. Xing. A spectral algorithm for latent tree graphical models. ICML, 2011.
[27] B. Recht. A simpler approach to matrix completion. JMLR, 2011.
[28] Arto Salomaa and Matti Soittola. Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag:
New York, 1978.
[29] M.P. Sch?utzenberger. On the definition of a family of automata. Information and Control, 1961.
[30] S. M. Siddiqi, B. Boots, and G. J. Gordon. Reduced-rank hidden Markov models. AISTATS, 2010.
[31] L. Song, B. Boots, S. Siddiqi, G. Gordon, and A. Smola. Hilbert space embeddings of hidden Markov
models. ICML, 2010.
9
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4,085 | 4,698 | Efficient Spike-Coding with Multiplicative
Adaptation in a Spike Response Model
Sander M. Bohte
CWI, Life Sciences
Amsterdam, The Netherlands
[email protected]
Abstract
Neural adaptation underlies the ability of neurons to maximize encoded information over a wide dynamic range of input stimuli. Recent spiking neuron models like the adaptive Spike Response Model implement adaptation as additive
fixed-size fast spike-triggered threshold dynamics and slow spike-triggered currents. Such adaptation accurately models neural spiking behavior over a limited
dynamic input range. To extend efficient coding over large changes in dynamic input range, we propose a multiplicative adaptive Spike Response Model where the
spike-triggered adaptation dynamics are scaled multiplicatively by the adaptation
state at the time of spiking. We show that, unlike the additive adaptation model,
the firing rate in our multiplicative adaptation model saturates to a realistic maximum spike-rate regardless of input magnitude. Additionally, when simulating
variance switching experiments, the model quantitatively fits experimental data
over a wide dynamic range. Dynamic threshold models of adaptation furthermore
suggest a straightforward interpretation of neural activity in terms of dynamic differential signal encoding with shifted and weighted exponential kernels. We show
that when thus encoding rectified filtered stimulus signals, the multiplicative adaptive Spike Response Model achieves a high coding efficiency and maintains this
efficiency over changes in the dynamic signal range of several orders of magnitude, without changing model parameters.
1
Introduction
The ability of neurons to adapt their responses to greatly varying sensory signal statistics is central
to efficient neural coding [1, 2, 3, 4, 5, 6, 7]. Consequently, accurate models for the underlying
mechanisms can provide insight into the nature of neural coding itself. For this, models of neural
computation have to account for adaptation in a manner consistent with both experimental findings
and notions of efficient neural coding.
Neural computation is often reduced to a linear-nonlinear-poisson (LNP) model: input signals are
filtered, followed by a thresholding function that determines the firing probability of the neuron. In
the Generalized Linear Model (GLM) [8] a refractory response in the form of a post-spike filter is
added (figure 1). With experimental responses fitted to such LNP models, adaptation is found to
adjust both the effective gain in the thresholding function and the linear filtering function [9, 10].
Neural adaptation responds primarily to changes in local stimulus contrast or, equivalently, to the
local detection threshold [11, 12], and a number of theoretical studies account for adaptation from
the perspective of optimal contrast estimation [12, 13]. Recent work by Ozuysal & Baccus [14]
suggests that in a Linear-Nonlinear first-order Kinetics model (LNK), the gain depends on the local
contrast of the filtered and rectified input signal.
1
g(t)
-
filter
s(t)
Nonlinearity
delay
output
Linear filter
Spiking
u(t)
{ti }
input
post-spike
Figure 1: Generalized Linear Model (GLM) of neural computation.
With substantial spike-rate adaptation occurring on a time scale of just tens of milliseconds [4, 5],
adapting neurons necessarily generate at most tens of spikes in that period. From an adaptive coding
perspective, this implies that for a neuron?s adaptation to be computable by downstream neurons,
the adaptation effects have to be derivable from just the emitted spike-train. Spike-based models are
thus central when accounting for adaptation as adaptive neural coding.
In variations of adaptive integrate-and-fire neurons [15, 16, 17], adaptation can be incorporated as
a combination of two mechanisms: spike-triggered adaptation currents and a dynamical actionpotential threshold. In such models, the adaptation mechanisms together increase the distance between the reversal potential and the threshold, effectively changing the gain of the neuron. The
adaptive Spike Response Model [16, 17] in particular has been shown to be effective for modeling
neural behavior in response to input currents with limited dynamic range [17]. On longer timescales,
spike-triggered adaptation currents fit a power-law decay rather than an exponential decay, linking
to observations of long-range power-law rate-adaptation [18, 19, 20, 21, 17].
Still, in spite of its success, the additive model of adaptation in adaptive Spike Response Model
effectively changes neural gain with at most a fixed step-size, and thus cannot respond quickly to
changes in signal variance that are large compared to this step-size. In particular, Brette [22] argues
that adaptation modulation has to be multiplicative for neurons to respond with the same level of
neural activity to drastic changes in dynamic range, as is observed experimentally (e.g. [4]).
We augment the adaptive Spike Response Model with multiplicative adaptation dynamics. We show
that such a multiplicative adaptive Spike Response Model quantitatively matches neural responses in
variance switching experiments and maximizes information transfer. Furthermore, we demonstrate
that the model?s effective gain responds to changes in either mean or variance of the filtered signal,
similar to the LNK kinetic model in [14].
In the adaptive Spike Response Model, gain modulation derives from the difference between the
adapted reversal potential and the dynamic threshold. This suggests a straightforward interpretation of spike-trains in terms of threshold-based detection of discernible signal levels in the rectified
filtered input signal: adaptive spike-coding. We show how non-linear signal encoding with a multiplicative adaptive Spike Response Model maintains a high coding efficiency for stimuli that vary in
magnitude over several orders of magnitude, unlike the additive version of the adaptive Spike Response Model. The coding efficiency is further comparable to the additive adaptive Spike Response
Model when the adaptation step-size in the latter is optimized for the local dynamic range.
2
Spike-rate Adaptation in the Spike Response Model
We follow Naud et al [17] in modeling adaptation in an augmented Spike-Response Model [23]. In
the adaptive Spike Response Model (aSRM), the dynamics of the (normalized) membrane-potential
V (t) are described as a sum of integrated input current I(t) and spike-triggered currents ?(t):
Z
Z
X
V (t) = ?(t ? s)I(s)ds ? ?(t ? s)
?(s ? ti )ds,
(1)
{ti }
where {ti } denotes the set of past emitted spikes, and the kernel ?(t) is a fast exponential low-pass
filter on membrane currents:
?t
?(t) = ?0 exp
,
?m
2
with ?m determined by the membrane capacitance and conductance, and is typically on the order of
several milliseconds [23, 17] .
The dynamical threshold is computed as the sum of a resting threshold V0 and spike-triggered threshold dynamics ?(t):
X
VT (t) = V0 +
?(t ? ti ).
(2)
{ti }
Spikes are generated either deterministically when V (t) ? VT (t) becomes positive, or stochastically
following an inhomogeneous point process with conditional firing rate:
V (t) ? VT (t)
?(t|V (t), VT (t)) = ?0 exp
,
(3)
?V
where ?V determines the slope of the exponential function; small values of ?V approximate a
neuron with a deterministic threshold. Naud et al [17] report that the threshold kernel ?(t) is best
fitted with an exponentially decaying function, whereas the shape of the spike-triggered current ?(t)
depends on the type of neuron, and furthermore for longer timescales best fits a decaying power-law:
?(t ? ti ) ? (t ? ti )?? for t >> ti , with ? ? 1.
We can denote the effective neural threshold ? as the amount of input that will trigger a spike. In
the adaptive Spike Response Model this amounts toR the sum of
Pthe dynamic threshold, VT (t), and
the (filtered) spike-triggered current: ? ? VT (t) + ?(t ? s) {ti } ?(s ? ti )ds. We can move the
reset response from (1) to the dynamic threshold (2) to obtain adaptation as the effective threshold
dynamics ?(t):
Z
X
?(t) = ?0 +
?(t ? ti ) + ?(t ? s)?(s ? ti )ds ,
(4)
{ti }
where ?0 = V0 denotes the effective threshold for an inactive neuron. As the adaptation dynamics
in this model are strictly additive, we will refer to it further as the additive aSRM.
The maximum effective threshold in the additive aSRM is limited by the maximum number of spikes
that can be generated within the short time-window reported for variance adaptation. Effectively,
the refractory period determines the upper bound for the adaptation step-size, and adaptation speed
is upper-bounded by this value times the number of generated spikes.
2.1
Multiplicative Dynamic Adaptation
We propose a modification of the additive aSRM where the effective spike-triggered adaptation is
not a fixed quantity but depends on the effective adaptation at the time of spiking. We include the
multiplicative interaction in the aSRM by scaling the effective adaptation in (4) with the current
adaptation value at the time of spiking:
Z
X
?(t) = ?0 +
?(ti ) ?(t ? ti ) + ?(t ? s)?(s ? ti )ds .
(5)
{ti }
For sparse spiking and adaptation response kernels that decay fairly rapidly to zero, such multiplicative adaptive threshold dynamics are approximately similar to the effective threshold dynamics
in (4). For rapid signal variance transitions however, the multiplicative dynamics ensure that the
effective threshold adaptation can rapidly range over multiple orders of magnitude.
The key difference in adaptation dynamics for the two aSRM models is illustrated in Figure 2. For
a given spike-train, the respective adaptation magnitudes are plotted in Figure 2a , and the neural
responses to different levels of step-size current injections are shown in Figure 2b. The additive
aSRM responds to an increasing input current with a firing rate that is essentially only bounded
by the refractory response; the firing rate in the aSRM with multiplicative adaptation saturates at a
much lower value as the effective threshold catches up with the magnitude of the injected current.
2.2
Adaptive Spike-Coding
The interpretation of spike-triggered adaptation as dynamic neural gain in the Spike Response Model
suggests a straightforward application to a spike-based neural coding model. Spike-rate adaptation
3
effective multiplicative threshold
multiplicative threshold
multiplicative spike?triggered current
effective additive threshold
additive threshold
additive spike?triggered current
spikes
40
20
10
80
60
0
40
?10
20
?20
0
10
20
30
40
50
60
70
(b)
additive adaptation
multiplicative adaptation
deterministic threshold
100
spike rate
Effective Adaptation Voltage
30
120
(a)
0
0
1
time (ms)
2
3
4
5
signal
6
7
8
9
10
Figure 2: Illustration of multiplicative and additive threshold adaptation dynamics. (a) Effective
adaptation as a sum of threshold dynamics (solid lines) and spike-triggered currents (dashed lines)
given an input spike-train (black dots). Red lines correspond to additive adaptation dynamics, blue
lines to multiplicative. (b) Firing rate as a function of signal strength. Red solid line is response for
(stochastic) additive aSRM, blue solid line for the stochastic multiplicative aSRM; dotted blue line
corresponds to a deterministic version of the multiplicative aSRM.
has been extensively studied from the point of view of optimal contrast estimation or signal threshold
detection [13, 12]. In particular the notion of signal threshold detection suggests a simple model
where individual spikes signal that the neuron has detected that its internally computed value has
reached a level distinguishable from the local noise level [11].
Taking the standard Linear-Non-Linear model of neural computation, we follow Ozuysal & Baccus
[14] in assuming that it is the rectified filtered version of the stimulus signal, u(t), that is encoded
by the spikes emitted by a neuron. We then define the Linear-Non-Linear-Adaptive-Thresholding
(LNL-AT) model as greedy differential signaling: if the signal u(t) exceeds a threshold value ?(ti )
at time ti , a spike is generated communicating a scaled response kernel ?(ti )?(t?ti ) to downstream
neurons. This response kernel is then also subtracted from the signal u(t), and the dynamic threshold
is updated to account for threshold adaptation (figure 3). In such greedy differential spike-coding,
the signal u(t) is effectively approximated as a sum of shifted and weighted response kernels:
X
u
?(t) =
?(ti )?(t ? ti ).
ti <t
This adaptive spike-coding
model corresponds to the multiplicative adaptive SRM in (5), where the
R
filtered reset function ?(t ? s)?(t)ds is interpreted as a response kernel ?(t ? ti ):
Z
X
V (t) = ?(t ? s)I(s)ds ?
?(ti )?(t ? ti ),
(6)
ti <t
= u(t) ? u
?(t),
X
?(t) = ?0 +
?(ti )?(t ? ti ),
{ti }
where spikes are generated when the membrane potential V (t) exceeds the dynamic threshold ?(t).
We let the threshold kernel ?(t) fit a decaying power-law ?(t ? ti ) ? (t ? ti )?? , and, to take
advantage of temporal correlations, we model ?(t) as an exponentially decaying kernel with timeconstant ?? similar to the (average) correlation time of u(t), ?(t) = exp(?t/?? ) [24] (note that
equation (5) implies that interchanging the behavior of ?(t) and ?(t) does not change the SRM
responses). Difference based neural coding models for spike-based neural coding have been noted
in the context of probabilistic coding [25], and fast visual coding [26].
In this adaptive spike-coding model, each spike ti communicates a signal amount of magnitude
?(ti ). In particular for signal ranges where the firing rate saturates, the effective signal magnitude
per spike grows linearly with signal size. This is depicted in figure 4, for a neuron with a stochastic
4
Nonlinearity
Spiking
filter
output
Linear filter
-
delay
input
post-spike
Figure 3: The Linear-Non-Linear-Adaptive-Thresholding (LNL-AT) model.
threshold (large ?V in (3); figure 4a) and for a neuron with a deterministic threshold (small ?V
in (3); figure 4b). Plotted is the neural behavior in response to a range of step-size increases in the
signal u(t), where firing rate and effective adapted threshold are measured two seconds after the
step-size signal increase. The average firing rate shows the traditional saturation of neural response
with increasing signal size. However, the effective adapted threshold increases linearly with signal
size, paralleling the u = u signal identity.
15
rate
u=u
40
10
30
20
5
10
0
0
60
15
50
arbitrary units
rate
50
(b)
10
40
30
arbitrary units
60
20
5
rate
(a)
10
1
2
3
4
5
signal u
6
7
8
9
0
10
0
0
1
2
3
4
5
6
signal u
7
8
9
10
0
Figure 4: Effective adapted threshold ?(ti ) (right axis) and firing rate (left axis) as a function of
signal size u (a) stochastic multiplicative aSRM; (b) deterministic multiplicative aSRM.
3
Results
We demonstrate how the multiplicative aSRM quantitatively fits with key findings on adaptation in
experimental data.
3.1
Variance Switching
The neural responses to variance switching [4, 5] in sensory signals are considered central evidence
for the information maximizing effect of adaptation, and also demonstrate the fast timescale of (initial) adaptation. In these key experiments, recordings are obtained from the blowfly?s H1 neuron,
and its responses are measured to a repeated change in perceived velocity variance. Signal variance
is repeatedly scaled from ?1 to ?2 = 10 ? ?1 , with a cycle time T . As the cycle-time T is increased,
the effective time constant of adaptation grows (as measured by fitting an exponent on the initial
segment of the decaying curve). This time-constant of adaptation shows scale-free behavior: when
normalizing for the interval time T , the neural response curves overlap, and there is linear relationship between cycle-time T and effective adaptation time constant ? . As reported in [27], the additive
aSRM is only able to match these findings qualitatively for a limited change in variance.
As in [4, 5], we generated random white noise within an interval enclosed by [??i , ?i ], for different
values of the variance ?i (1 and 10 respectively). This signal was filtered with filters obtained by the
GLM-model [8] on the original data from [4]. We fed the thus filtered and rectified signal into the
multiplicative aSRM and optimized the model parameters using exhaustive line-search.
5
The optimized multiplicative aSRM exhibits both the same firing behavior and the same relationship
between normalized switching interval and normalized firing rate as the experimental data in [5]
(Figure 5b,c). Furthermore, characterizing the input-output relationship as in [5] recovers the same
overlapping response-curves after normalizing the projected velocity signal for the scaled variance.
The fitted adaptation decay time-constant ? also closely matches the experimental data [5] (Figure
5e, simulation: red circles, data: black circles). Changing the dynamic range for both ?1 and
?2 = 10 ? ?1 by a factor of 10 did not change the relationship (green dots). We also characterized
the signal versus firing rate response for three scaled versions of the same velocity signal, with
scaling factors 1, 2 and 3, similar to [4] (open markers, Figure 5f). As in [4], the adapted signalrate response curves also overlap after normalizing the signal for the scaled variance (solid markers,
Figure 5f). Multiplicative effective adaptation thus maximizes the transmitted information as in
[4, 5].
(a)
(e) 1.8
1.4
Timescale ? (s)
60
40
30
0.4
0.2
10
0
0
5
10
15
20
time (s)
25
30
40
35
(f)
2
0
?2
10
15
20
25
30
35
40
60
40
30
20
10
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized time t/T
2
10
0.7
0.8
0.9
0
1
0
0.2
2
10
1?s segment of high contrast
1?s segment of low contrast
0.4
(g) 2.8
0.6
0.8
signal s
1
10
0
0
10
10
?1
?8
?6
?4
?2
0
2
4
projected velocity s
6
8
10 ?1
?0.8
?0.6 ?0.4
?0.2
1.2
1.4
2.4
2.2
2
1.8
1.6
1.4
mean held constant
std dev held constant
1.2
?1
10
?10
1
2.6
1
10
Firing rate
5
50
1
?1
(d)
0
Cycle time T (s)
Firing rate
Normalized rate
3
?3
1
0.8
0.6
20
(c)
1.2
gain
Firing rate
50
0
experimental data
simulation
simulation x 10
1.6
70
Normalized firing rate
(b)
0
0.2
0.4
0.6
Normalized stimulus s/ ?
0.8
1
10
1
0
1
2
3
4
5
contrast
Figure 5: Variance switching. (a) variance of sensory input is switched with a fixed cycle time.
(b) The aSRM neuron adapts its firing rate after each switch. Switching responses for different
cycle times are overlapped. (c) The response curves for various cycle times overlap when time is
normalized for cycle time T . (d) Input-output relationship derived from 1-s-wide time windows in
the two signal variance conditions: left projected velocity signal s vs normalized firing rate, right,
projected velocity signal s normalized by respective variance ?. (e) Relationship between fitted
adaptation timescale ? as a function of cycle time T . Red circles simulation data; black circles
experimental data from [5]. Green dots are simulation data for switching signals multiplied by a
factor 10. (f) Simulation response to signal scaled by factors ?1 = 1, ?2 = 2, ?3 = 3 (open
markers), and responses rescaled by signal scale factor (solid markers). (g) Effective gain (1/?(t))
in the multiplicative aSRM neuron as a function of contrast, for signal u with mean held constant
and variance varied (blue line), and variance held constant and mean varied (green line). For the
experiments, resting threshold ?0 was set to 0.008, spike-triggered adaptation currents decayed with
a power-law constant of ? = 1.15, as 3.5(t ? ti + 0.7)?? and response kernels as 2.5 exp(?t/9)
(time t in ms).
6
5
4.5
(a)
(b)
spikes
signal u(t)
estimated signal u(t)
coding efficiency (%)
4
80
3.5
signal
3
2.5
2
1.5
70
60
50
40
30
1
20
0.5
10
0
500
600
700
800
900
1000
1100
time (ms)
1200
1300
1400
1500
0
multipl adaptation
add. adaptation
scaled sum adaptation
scaled fixed
Figure 6: Multiplicative Spike-Coding: (a) illustration of stimulus encoding as a sum of shifted
and weighted response kernels. Black dots denote spike-times, black solid line the signal u(t), and
magenta the approximated signal u
?(t). (b) Computed coding efficiency. Information rate Rinfo was
computed, with effective signal and noise bandwidth cutoff at 50Hz (matching the original stimulus
signal). Coding efficiency was computed by dividing Rinfo by the spike-train entropy rate S/T [28]
for a timing precision of 1 ms. Model parameters for the multiplicative aSRM are as in Figure 4.
Note that for the grey and light-grey bars refer to the left, parameters are optimized for each ? value
individually.
For adaptation to relate to contrast, loosely defined as the ratio of (local) standard deviation ? and
local average signal u
?, ?/?
u (and thus detection threshold), it should respond accordingly to changes
in not just variance but also in changes to mean (rectified) signal magnitude. Ozuysal & Baccus [14]
show that this property holds for their kinetic model of gain modulation, which also closely matches
experimental data. In the kinetic model, effective gain scales linearly with standard deviation when
all other signal statistics are held constant, and similarly with 1/?
u; in simulations, where effective
gain in computed as 1/?(t), we find that the multiplicative aSRM shares this property (Figure 5g).
3.2
H1 encoding/decoding
With multiplicative effective adaptation responding to contrast changes, we can examine the effectiveness of the corresponding neural coding model. For this, we use the original blowfly data from
Brenner et al [4], consisting of velocity stimulus profiles presented to the blowfly, where the velocity stimulus is scaled with factors of ?1 = 18? s?1 , ?2 = 2?1 = 36? s?1 , ?3 = 90? s?1 and
?4 = 180? s?1 . We examine how well multiplicative adaptive neural coding approximates the rectified filtered signal, as compared to such neural coding with the additive aSRM.
We filter each version of this velocity stimulus with the filter obtained using GLM optimization on
the velocity stimulus with variance ?1 and optimize the parameters in both aSRM models for condition ?1 , using deterministic thresholds. Adaptation was highly robust for the parameters, provided
we chose an exponential response kernel with time-constant 10ms to match the correlation time of
the filtered signal. We further tuned the resting threshold ?0 and magnitude of the power-law adaptation kernel ? so that the average firing rate matched the experimental data at least for the ?1 signal.
An example of stimulus encoding with multiplicative adaptive neural coding is shown in figure 6a.
We compare coding efficiency for the multiplicative aSRM and for the additive aSRM for a spike
precision of 1ms [28], applying the model optimized for condition ?1 to all four stimulus conditions
?1 , ?2 , ?3 , ?4 , and, for the multiplicative aSRM additionally for the conditions 50 ? ?1 , 100 ?
?1 , 500 ? ?1 . Relative coding efficiencies are plotted in figure 6b, black and white bars. We see that
the multiplicative aSRM maintains a high coding efficiency over the entire dynamic range, even for
the 500 ? ?1 stimulus condition. The dynamic range of the additive aSRM however is insufficient
to encode the wide dynamic range of the original data. Similar to the experiment in [4], we find
7
that the firing rate for the multiplicative aSRM signal encoding remains approximately stable for all
stimulus conditions, with a firing rate of 55 ? 5 spikes/s, without changing any parameters. The
firing rate for the additive aSRM increases from a (matched) firing rate of 55 spikes/s for the ?1
stimulus, to over 180 spikes/s for the ?4 stimulus.
We also compare against the additive aSRM and neural coding with a non-adaptive, fixed response
kernel SRM, with the magnitude of the response-kernel (equivalent to ?0 ) optimized for the local
variance such that for each stimulus, the firing rate for these models matches that of the multiplicative
aSRM. This is shown in the light grey (scaled additive aSRM) and dark grey (scaled non-adaptive
SRM) bars in figure 6b. The coding efficiency for multiplicative aSRM is close to that of locally
rescaled additive aSRM?s, and exceeds locally rescaled non-adaptive coding.
4
Discussion
We showed how a multiplicative model of neural adaptation in the Spike Response Model can account quantitatively for key experimental adaptation data. When interpreting the fast adaptation
component as the manifestation of a greedy signal encoding scheme, we further showed that multiplicative adaptation allows the Spike Response Model to achieve high coding efficiency for signals
with dynamic ranges that change over several orders of magnitude, without changing parameters.
Just as the H1 blowfly neuron, the multiplicative aSRM uses a near-constant firing rate for the widely
varying dynamic range in the different stimulus conditions.
The ubiquity of adaptation in neural systems and notions of synaptic facilitation and depression
suggest that gain modulation could possibly be decoded in a receiving neuron by adaptively scaling
the size of the post-synaptic response. Although Series [29] argues that a number of visual percepts
are consistent with decoding neurons being ?unaware? of presynaptic adaptation, the presence or
absence of such coupled adaptation can be considered as a form of spectral filtering [30]. As we
have shown, a key advantage of accounting for gain modulation in spike-based neural coding is
that it greatly extends the neuron?s dynamic range, and may allow for instance implicit spike-based
probabilistic computation as in [31] to scale to multiple layers.
From a biological perspective, it may seem implausible to let threshold dynamics and spike-triggered
adaptation currents scale with vast changes in dynamic range. However, as noted in [17], there is a
theoretical link between spike-triggered plasticity like spike-timing dependent plasticity and spiketriggered adaptation [32]. That is, scaling of synaptic weights could complement adaptation to
large changes in dynamic range. The multiplicative adaptive Spike Response Model also captures
only part of the first-order dynamics in the LNK model in [14], and does not account for variancedependent changes in temporal filtering (e.g. [9]). Thus, spike-based adaptation of the response
kernel could likely further improve the coding efficiency.
The multiplicative adaptive Spike Response Model provides a spike-based account for gain modulation, which can easily be reconstructed by post-synaptic neurons as a function of the received
spike-train. It thus provides an effective neuron model for dynamical spiking neural networks, resolving for instance stability problems in spiking reservoir computing approaches.
Acknowledgement. The author thanks Hao Wang for assistance with the simulations, and Jaldert
Rombouts, Kausik Lakshminarasimhan and Hao Wang for helpful suggestions.
References
[1] S B Laughlin. The role of sensory adaptation in the retina. The Journal of experimental biology, 146:39?
62, September 1989.
[2] S.M. Smirnakis, M.J. Berry, D.K. Warland, W. Bialek, and M. Meister. Adaptation of retinal processing
to image contrast and spatial scale. Nature, 386(6620):69?73, 1997.
[3] M.J. Wainwright. Visual adaptation as optimal information transmission. Vision Research, 39(23):3960?
3974, 1999.
[4] N. Brenner, W. Bialek, and R. de Ruyter van Steveninck. Adaptive rescaling maximizes information
transmission. Neuron, 26(3):695?702, 2000.
[5] A.L. Fairhall, G.D. Lewen, W. Bialek, and R.R. de Ruyter van Steveninck. Efficiency and ambiguity in
an adaptive neural code. Nature, 412(6849):787?792, 2001.
8
[6] O Schwartz and E P Simoncelli. Natural signal statistics and sensory gain control. Nature Neuroscience,
4(8):819?25, 2001.
[7] T. Hosoya, S.A. Baccus, and M. Meister. Dynamic predictive coding by the retina. Nature, 436(7047):71?
77, 2005.
[8] J.W. Pillow, L. Paninski, V.J. Uzzell, E.P. Simoncelli, and E.J. Chichilnisky. Prediction and decoding of retinal ganglion cell responses with a probabilistic spiking model. Journal of Neuroscience,
25(47):11003?13, 2005.
[9] S. Baccus and M. Meister. Fast and slow contrast adaptation in retinal circuitry. Neuron, 36(5):909?19,
2002.
[10] S. Hong, B.N. Lundstrom, and A.L. Fairhall. Intrinsic gain modulation and adaptive neural coding. PLoS
Computational Biology, 4(7), 2008.
[11] H P Snippe and J H van Hateren. Recovery from contrast adaptation matches ideal-observer predictions.
Journal of the Optical Society of America. A, Optics, image science, and vision, 20(7):1321?1330, 2003.
[12] H P Snippe, L Poot, and J H Van Hateren. Asymmetric dynamics of adaptation after onset and offset of
flicker. Journal of Vision, pages 1?12, 2004.
[13] M. DeWeese and A. Zador. Asymmetric dynamics in optimal variance adaptation. Neural Comp,
10(5):1179?1202, 1998.
[14] Y. Ozuysal and S.A. Baccus. Linking the Computational Structure of Variance Adaptation to Biophysical
Mechanisms. Neuron, 73(5):1002?1015, March 2012.
[15] R Harris, D C O?Carroll, and S B Laughlin. Contrast gain reduction in fly motion adaptation. Neuron,
28(2):595?606, 2000.
[16] R. Jolivet, A. Rauch, H.R. L?uscher, and W. Gerstner. Predicting spike timing of neocortical pyramidal
neurons by simple threshold models. Journal of computational neuroscience, 21(1):35?49, 2006.
[17] R Naud. The Dynamics of Adapting Neurons. PhD thesis, EPFL Lausanne, 2011.
[18] P.J. Drew and LF Abbott. Models and properties of power-law adaptation in neural systems. Journal of
neurophysiology, 96(2):826, 2006.
[19] Z. Xu, JR Payne, and ME Nelson. Logarithmic time course of sensory adaptation in electrosensory
afferent nerve fibers in a weakly electric fish. Journal of neurophysiology, 76(3):2020, 1996.
[20] B.N. Lundstrom, M.H. Higgs, W.J. Spain, and A.L. Fairhall. Fractional differentiation by neocortical
pyramidal neurons. Nature neuroscience, 11(11):1335?1342, 2008.
[21] B. Wark, A. Fairhall, and F. Rieke. Timescales of inference in visual adaptation. Neuron, 61(5):750?761,
2009.
[22] R. Brette. Spiking models for level-invariant encoding. Front. in Comp. Neurosc., 5, 2011.
[23] W. Gerstner and W. Kistler. Spiking Neuron Models: Single Neurons, Populations, Plasticity. Cambridge
University Press, 2002.
[24] M. Buiatti and C. van Vreeswijk. Variance normalisation: a key mechanism for temporal adaptation in
natural vision? Vision Research, 43(17):1895?1906, August 2003.
[25] S. Deneve. Bayesian spiking neurons I: inference. Neural computation, 20(1):91?117, 2008.
[26] P. Lichtsteiner, C. Posch, and T. Delbruck. A 128? 128 120 db 15 ?s latency asynchronous temporal
contrast vision sensor. Solid-State Circuits, IEEE Journal of, 43(2):566?576, 2008.
[27] C Pozzorini, R Naud, S Mensi, and W Gerstner. Multiple timescales of adaptation in single neuron
models. In Front. Comput. Neurosci. Conference Abstract: BCCN, 2010.
[28] F. Rieke, D. Warland, and W. Bialek. Spikes: exploring the neural code. The MIT Press, 1999.
[29] P. Seri`es, A.A. Stocker, and E.P. Simoncelli. Is the Homunculus ?Aware? of Sensory Adaptation? Neural
Computation, 21:3271?3304, 2009.
[30] S.M. Bohte and J.O. Rombouts. Fractionally Predictive Spiking Neurons. In Advances in Neural Information Processing Systems (NIPS) 23, pages 253?261. The MIT Press, 2010.
[31] W.J. Ma, J.M. Beck, P.E. Latham, and A. Pouget. Bayesian inference with probabilistic population codes.
Nature neuroscience, 9(11):1432?1438, November 2006.
[32] G. Hennequin, W. Gerstner, and J.P. Pfister. Stdp in adaptive neurons gives close-to-optimal information
transmission. Front. in Comp. Neurosc., 4, 2010.
9
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4,086 | 4,699 | On the Sample Complexity of Robust PCA
Matthew Coudron
Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
Cambridge, MA 02139
[email protected]
Gilad Lerman
School of Mathematics
University of Minnesota
Minneapolis, MN 55455
[email protected]
Abstract
We estimate the rate of convergence and sample complexity of a recent robust estimator for a generalized version of the inverse covariance matrix. This estimator
is used in a convex algorithm for robust subspace recovery (i.e., robust PCA). Our
model assumes a sub-Gaussian underlying distribution and an i.i.d. sample from
it. Our main result shows with high probability that the norm of the difference
between the generalized inverse covariance of the underlying distribution and its
estimator from an i.i.d. sample of size N is of order O(N ?0.5+ ) for arbitrarily small > 0 (affecting the probabilistic estimate); this rate of convergence is
close to the one of direct covariance estimation, i.e., O(N ?0.5 ). Our precise probabilistic estimate implies for some natural settings that the sample complexity
of the generalized inverse covariance estimation when using the Frobenius norm
is O(D2+? ) for arbitrarily small ? > 0 (whereas the sample complexity of direct covariance estimation with Frobenius norm is O(D2 )). These results provide
similar rates of convergence and sample complexity for the corresponding robust
subspace recovery algorithm. To the best of our knowledge, this is the only work
analyzing the sample complexity of any robust PCA algorithm.
1
Introduction
A fundamental problem in probability and statistics is to determine with overwhelming probability
the rate of convergence of the empirical covariance (or inverse covariance) of an i.i.d. sample of
increasing size N to the covariance (or inverse covariance) of the underlying random variable (see
e.g., [17, 3] and references therein). Clearly, this problem is also closely related to estimating
with high probability the sample complexity, that is, the number of samples required to obtain a
given error of approximation . In the case of a compactly supported (or even more generally subGaussian) underlying distribution, it is a classical exercise to show that this rate of convergence is
O(N ?0.5 ) (with a comparability constant depending on properties of ?, in particular D, as well as
on the threshold probability, see e.g., [17, Proposition 2.1]). The precise estimate for this rate of
convergence implies that the sample complexity of covariance estimation is O(D) when using the
spectral norm and O(D2 ) when using the Frobenius norm. The rate of convergence and sample
complexity of PCA immediately follow from these estimates (see e.g., [15]).
While such estimates are theoretically fundamental, they can be completely useless in the presence
of outliers. That is, direct covariance or inverse covariance estimation and its resulting PCA are very
sensitive to outliers. Many robust versions of covariance estimation, PCA and dimension reduction
have been developed in the last three decades (see e.g., the standard textbooks [8, 10, 14]). In the last
few years new convex algorithms with provable guarantees have been suggested for robust subspace
recovery and its corresponding dimension reduction [5, 4, 19, 20, 11, 7, 2, 1, 21, 9].
Most of these works minimize a mixture of an `1 -type norm (depending on the application) and the
nuclear norm. Their algorithmic complexity is not as competitive as PCA and their sample com1
plexity is hard to estimate due to the problem of extending the nuclear norm out-of-sample. On
the other hand, Zhang and Lerman [21] have proposed a novel M-estimator for robust PCA, which
is based on a convex relaxation of the sum of Euclidean distances to subspaces (which is originally minimized over the non-convex Grassmannian). This procedure suggests an estimator for a
generalized version of the inverse covariance matrix and uses it to robustly recover an underlying
low-dimensional subspace. This idea was extended in [9] to obtain an even more accurate method
for subspace recovery, though it does not estimate the generalized inverse covariance matrix (in particular, it has no analogous notion of singular values or their inverses). The algorithmic complexity
of the algorithms solving the convex formulations of [21] and [9] is comparable to that of full PCA.
Here we show that for the setting of sub-Gaussian distributions the sample complexity of the robust
PCA algorithm in [21] (or its generalized inverse covariance estimation) is close to that of PCA (or
to sample covariance estimation). Our analysis immediately extends to the robust PCA algorithm of
[9].
1.1
The Generalized Inverse Covariance and its Corresponding Robust PCA
Zhang and Lerman [21] formed the set
H := {Q ? RD?D : Q = QT , tr(Q) = 1},
(1.1)
as a convex relaxation for the orthoprojectors (from RD to RD ), and defined the following energy
function on H (with respect to a data set X in RD ):
X
FX (Q) :=
kQxk,
(1.2)
x?X
where k ? k denotes the Euclidean norm of a vector in RD . Their generalized empirical inverse
covariance is
? X = arg min FX (Q).
Q
(1.3)
Q?H
They showed that when replacing the term kQxk by kQxk2 in (1.2) and when Sp{X } = RD ,
then the minimization (1.3) results in a scaled version of the empirical inverse covariance matrix.
? X as a generalized empirical inverse covariance (or `1 -type
It is thus clear why we can refer to Q
version of it). We describe the absolute notion of generalized inverse covariance matrix, i.e., nonempirical, in ?1.2. Zhang and Lerman [21] did not emphasize the empirical generalized inverse
covariance, but the robust estimate of the underlying low-dimensional subspace by the span of the
bottom eigenvectors of this matrix. They rigorously proved that such a procedure robustly recovers
the underlying subspace under some conditions.
1.2
Main Result of this Paper
? X . This problem is practically
We focus on computing the sample complexity of the estimator Q
?
equivalent with estimating the rate of convergence of QX of an i.i.d. sample X to the ?generalized
inverse covariance? of the underlying distribution ?. We may assume that ? is a sub-Gaussian
probability measure on RD (see ?2.1 and the extended version of this paper). However, in order
to easily express the dependence of our probabilistic estimates on properties of the measure ?, we
assume for simplicity that ? is compactly supported and denote by R? the minimal radius among all
balls containing the support of ?, that is,
R? = min{r > 0 : supp(?) ? B(0, r)},
where B(0, r) is the ball around the origin 0 with radius r. We further assume that for some
0 < ? < 1, ? satisfies the following condition, which we refer to as the ?two-subspaces criterion?
(for ?): For any pair of (D ? 1)-dimensional subspaces of RD , L1 and L2 :
?((L1 ? L2 )c ) ? ?.
(1.4)
We note that if ? satisfies the two-subspaces criterion for any particular 0 < ? < 1, then its support
cannot be a union of two hyperplanes of RD . The use of this assumption is clarified below in ?3.2,
though it is possible that one may weaken it.
2
We first formulate the generalized inverse covariance of the underlying measure as follows:
? = arg min F (Q),
Q
(1.5)
Q?H
where
Z
F (Q) =
kQxk d?(x).
(1.6)
Let {xi }?
i=1 be a sequence of i.i.d. random variables sampled from ? (i.e., each variable has distribution ?). Let XN := {xi }N
i=1 and denote
? N := Q
? X and FN := FX .
Q
N
N
(1.7)
? and Q
? N are uniquely defined (which we denote
Our main result shows with high probability that Q
? N }N ?N converges to Q
? in the following specified rate. It uses the
by u.d. from now on) and that {Q
common notation: a ? b := max(a, b). We explain its implications in ?2.
Theorem 1.1. If ? is a compactly supported distribution satisfying the two-subspaces criterion for
? > 0, then there exists a constant ?0 ? ?0 (?, D, ) > 0 such that for any > 0 and N > 2(D ?1)
the following estimate holds:
2 ? 1 +
?
?
?
?
P Q & QN are u.d. and kQ ? QN kF ?
N 2
?0
2
2
N
?N 2
? 1 ? C0 N D exp
?
2
(1 ? ?)N ?2(D?1) , (1.8)
D ? R?2
D?1
where
C0 ? C0 (?0 , D) := 4 ? ((4?0 ) ? 2) ?
2?0 + 4((4?0 ) ? 2)R?
10 D
0
1 ? (4?2?0 )?2
! D(D+1)
2
.
(1.9)
Intuitively, ?0 represents a lower bound on the directional second derivatives of F . Therefore,
?0 should affect sample complexity because the number of random samples taken to approximate a
minimum of F should be affected by how sharply F increases about its minimum. It is an interesting
and important open problem to find lower bounds on ?0 for general ?.
2
2.1
Implication and Extensions of the Main Result
Generalization to Sub-Gaussian Measures
We can remove the assumption that the support of ? is bounded (with radius R? ) and assume instead
that ? is sub-Gaussian. In this case, instead of Hoeffding?s inequality, we apply [18, Proposition
5.10] with ai = 1 for all 1 ? i ? n. When formulating the corresponding inequality, one may
note that supp?1 p?1/2 (E? |x|p )1/p (where x represents a random variable sampled from ?) can be
regarded as a substitute for R? (see [21] for more details of a similar analysis).
2.2
Sample Complexity
The notion of sample complexity arises in the framework of Probably-Approximately-Correct
Learning of Valiant [16]. Generally speaking, the sample complexity in our setting is the minimum number of samples N required, as a function of dimension D, to achieve a good estimation
? with high probability. We recall that in this paper we use the Frobenius norm for the estimaof Q
tion error. The following calculation will show that under some assumptions on ? it suffices to use
N = ?(D? ) samples for any ? > 2 (we recall that f (x) = ?(g(x)) as x ? ? if and only if
g(x) ?
= O(f (x))). In our analysis we will have to assume that ? is a fixed constant, and ?0 goes
as 1/ D. These assumptions are placing additional restrictions on the measure ?, which we expect
to be reasonable in practice as we later clarify. We further assume that R? = O(D?0.5 ) and also
explain later why it makes sense for the setting of robust subspace recovery.
3
To bound the sample complexity we set C1 := 4 ? ((4?0 ) ? 2) and C2 := 10 ? (2?0 + 4((4?0 ) ?
2
2)R? )/(1?2?0 /(4?0 ) ? 2) so that C0 ? C1 ?(C2 ?D)D (see (1.9)). Applying this bound and (1.8)
we obtain that if ? > 2 is fixed, 1/? < < 12 and N ? D? , then
? &Q
? N are u.d. and kQ
? ?Q
? N kF ? 2 N ? 21 +
P Q
(2.1)
?0
2
?N 2
? 2 N 2(D?1) (1 ? ?)N ?2(D?1)
? 1 ? C1 (C2 ? D ? N )D exp
D ? R?2
? 1 ? C1 exp log(C2 ? D1+? )D2 ? D2?
? 2 exp (2?(D ? 1) log(D) + log(1 ? ?)(D? ? 2(D ? 1))) .
Since > 1/? the first term in the RHS of (2.1) decays exponentially as a function of D (or,
equivalently, as a function of N ? D? ). Similarly, since 0 < ? < 1 and ? > 1 the second term in
the RHS of (2.1) decays exponentially as a function of D. Furthermore, since < 12 it follows that
1
1
the error term for the minimizer, i.e., N ? 2 + ? D?(? 2 ) , decays polynomially in D. Thus, in order
to achieve low error estimation with high probability it is sufficient to take N = ?(D? ) samples for
any ? > 2. The exact guarantees on error estimation and probability of error can be manipulated by
changing the constant hidden in the ? term.
We would like to point out the expected tradeoff between the sample complexity and the rate of
convergence. If approaches 0, then the rate of convergence becomes optimal but the sample complexity deteriorates. On the other hand, if approaches 0.5, then the sample complexity becomes
optimal, but the rate of convergence deteriorates.
To motivate our assumption on R? , ? and ?0 , we recall the needle-haystack and syringe-haystack
models of [9] as a prototype for robust subspace recovery. These models assume a mixtures of outlier
and inliers components. The distribution of the outliers component is normal N (0, (?out2 /D)ID )
and the distribution of the inliers component is a mixture of N (0, (?in2 /d)PL ) (where L is a dsubspace) and N (0, (?in2 /(CD))ID ), where C 1 (the latter component has coefficient zero in
the needle-haystack model).
The underlying distribution of the syringe-haystack (or needle-haystack) model is not compactly
supported, but clearly sub-Gaussian (as discussed in ?2.1) and its standard deviation is of order
O(D?0.5 ). We also note that ? here is the coefficient of the outlier component in the needle-haystack
model, which we denote by ?0 . Indeed, the only non-zero measure that can be contained in a (D-1)dimensional subspace is the measure associated with N (0, (?in2 /d)PL ), and that has total
? weight
at most (1 ? ?0 ). It is also possible to verify explicitly that ?0 is lower bounded by 1/ D in this
case (though our argument is currently rather lengthy and will appear in the extended version of this
paper).
2.3
From Generalized Covariances to Subspace Recovery
We recall that the underlying d-dimensional subspace can be recovered from the bottom d eigen? N . Therefore, the rate of convergence of the subspace recovery (or its corresponding
vectors of Q
sample complexity) follows directly from Theorem 1.1 and the Davis-Kahan Theorem [6]. To for? and Q
? N are u.d. (recall Theorems 3.1 and 3.2).
mulate this, we assume here for simplicity that Q
Theorem 2.1. If d < D, > 0, ?0 ? ?0 (?, D, ) is the positive constant guaranteed by Theo? and Q
? N are u.d. and L
?d, L
? d,N are the subspaces spanned by the bottom d eigenvectors
rem 2.1, Q
?
? N respectively, P ? and P ?
(i.e., with lowest d eigenvalues) of Q and Q
Ld
Ld,N are the orthoprojectors
?
on these subspaces and ?D?d is the (D ? d)th eigengap of Q, then
2
1
4
?N 2
P kPL? d ? PL? d,N kF ?
N ? 2 + ? 1 ? C0 N D exp
.
(2.2)
?0 ? ?D?d
D ? R?2
2.4
Nontrivial Robustness to Noise
We remark that (2.2) implies nontrivial robustness to noise for robust PCA. Indeed, assume for example an underlying d-subspace L?d and a mixture distribution (representing noisy inliers/outliers
4
components) whose inliers component is symmetric around L?d with relatively high level of variance
? d and its outliers component is spherically symmetric with suffiin the orthogonal component of L
? d = L? . Combining this observaciently small mixture coefficient. One can show that in this case L
d
tion and (2.2), we can verify robustness to nontrivial noise when recovering L?d from i.i.d. samples
of such distributions.
2.5
Convergence Rate of the REAPER Estimator
The REAPER and S-REAPER Algorithms [9] are variants of the robust PCA algorithm of [21]. The
objective of the REAPER algorithm can be formulated as aiming to minimize the energy FX (Q)
over the set
G := {Q ? RD?D : Q = QT , tr(Q) = D ? d and Q 4 I},
(2.3)
where 4 denotes the semi-definite order. The d-dimensional subspace can then be recovered by
the bottom d eigenvectors of Q (in [9] this minimization is formulated with P = I ? Q, whose
top d eigenvectors are found). The rate of convergence of the minimizer of FX (Q) over G to the
minimizer of F (Q) over G is similar to that in Theorem 1.1. The proof of Theorem 1.1 must be
? lies on the interior of G then the
modified to deal with the boundary of the set G. If the minimizer Q
?
proof is the same. If Q is on the boundary of G we must only consider the directional derivatives
which point towards the interior of G, or tangent to the boundary. Other than that the proof is the
same.
2.6
Convergence Rate with Additional Sparsity Term
Rothman et al. [13] and Ravikumar et al. [12] have analyzed an estimator for sparse inverse covariance. This estimator minimizes over all Q 0 the energy
b N iF ? log det(Q) + ?N kQk` ,
hQ, ?
1
(2.4)
b N is the empirical covariance matrix based on sample of size N , h?, ?iF is the Frobenius
where ?
PD
inner product (i.e., sum of elementwise products) and kQk`1 = i,j=1 |Qi,j |.
Zhang and Zou [22] have suggested a similar minimization, which replaces the first two terms
in (2.4) (corresponding to ?N = 0) with
b N iF /2 ? tr(Q).
hQ2 , ?
(2.5)
b ?1 (assuming that the
Indeed, the minimizers of (2.4) when ?N = 0 and of (2.5) are both equal to ?
N
N
D
Sp({xi }i=1 ) = R so that the inverse empirical covariance exists).
b N , i.e., ?
b N = PN xi xT /N , we note that
Using the definition of ?
i
i=1
N
1 X
b
hQ , ?N iF =
kQxi k2 .
N i=1
2
(2.6)
Therefore, the minimizer of (2.5) over all Q 0 is the same up to a multiplicative constant as
the minimizer of the RHS of (2.6) over all Q 0 with tr(Q) = 1. Teng Zhang suggested to us
replacing the RHS of (2.6) with FX and modifying the original problem of (2.4) (or more precisely
its variant in [22]) with the minimization over all Q ? H of the energy
FX (Q) + ?N kQk`1 .
(2.7)
The second term enforces sparseness and we expect the first term to enforce robustness.
By choosing ?N = O(N ?0.5 ) we can obtain similar rates of convergence for the minimizer of (2.7)
as the one when ?N = 0 (see extended version of this paper), namely, rate of convergence of order
O(N ?0.5+ ) for any > 0. The dependence on D is also the same. That is, the minimum sample
size when using the Frobenius norm is O(D? ) for any ? > 2. Nevertheless, Ravikumar et al. [12]
show that under some assumptions (see e.g., Assumption 1 in [12]), the minimal sample size is
O(log(D)r2 ), where r is the maximum node degree for a graph, whose edges are the nonzero entries
of the inverse covariance. It will be interesting to generalize such estimates to the minimization
of (2.7).
5
3
Overview of the Proof of Theorem 1.1
3.1
Structure of the Proof
? and Q
? N (with high probability). In ?3.3 and
We first discuss in ?3.2 conditions for uniqueness of Q
?3.4 we explain in short the two basic components of the proof of Theorem 1.1. The first of them is
? ?Q
? N kF can be controlled from above by differences of directional derivatives of F . The
that kQ
second component is that the rate of convergence of the derivatives of {FN }?
N =1 to the derivative
of F is easily obtained by Hoeffding?s inequality. In ?3.5 we gain some intuition for the validity
of Theorem 1.1 in view of these two components and also explain why they are not sufficient to
conclude the proof. In ?3.6 we describe the construction of ?nets? of increasing precision; using
these nets we conclude the proof of Theorem 1.1 in ?3.7. Throughout this section we only provide
the global ideas of the proof, whereas in the extended version of this paper we present the details.
3.2
Uniqueness of the Minimizers
? is u.d. and that Q
? N is u.d. with overwhelming
The two-subspaces criterion for ? guarantees that Q
probability for sufficiently large N as follows.
Theorem 3.1. If ? satisfies the two-subspaces criterion for some ? > 0, then F is strictly convex.
Theorem 3.2. If ? satisfies the two-subspaces criterion for some ? > 0 and N > 2(D ? 1), then
2
N
P (FN is not strictly convex) ? 2
(1 ? ?)N ?2(D?1) .
(3.1)
D?1
3.3
From Energy Minimizers to Directional Derivatives of Energies
? F from above by differences of derivatives of energies at Q and
We control the difference kQ ? Qk
? Here Q is an arbitrary matrix in Br (Q)
? for some r > 0 (where Br (Q)
? is the ball in H with
Q.
?
? N for some
center Q and radius r w.r.t. the Frobenius norm), but we will later apply it with Q = Q
N ? N.
3.3.1
Preliminary Notation and Definitions
The ?directions? of the derivatives, which we define below, are elements in the unit sphere of the
tangent space of H, i.e.,
D := {D ? RD?D | D = DT , tr(D) = 0, kDkF = 1}.
Throughout the paper, directions in D are often determined by particular points Q1 , Q2 ? H, where
Q1 6= Q2 . We denote the direction from Q1 to Q2 by DQ1 ,Q2 , that is,
DQ1 ,Q2 :=
Q2 ? Q1
.
kQ2 ? Q1 kF
(3.2)
Directional derivatives with respect to an element of D may not exist and therefore we use directional
derivatives from the right. That is, for Q ? H and D ? D, the directional derivative (from the right)
of F at Q in the direction D is
d
?+
F (Q + tD) t=0+ .
(3.3)
D F (Q) :=
dt
3.3.2
Mathematical Statement
? F . It involves the constant
We use the above notation to formulate the desired bound on kQ ? Qk
?0 , which is also used in Theorem 1.1. The proof of this lemma clarifies the existence of ?0 , though
it does not suggest an explicit approximation for it.
Lemma 3.3. For r > 0 there exists a constant ?0 ? ?0 (r, ?, D) > 0 such that for all Q ?
? \ {Q}:
?
Br (Q)
? ? ?0 kQ ? Qk
? F
?+
F (Q) ? ?+
F (Q)
(3.4)
DQ,Q
?
DQ,Q
?
6
and consequently
?+
D?
Q,Q
3.4
? F.
F (Q) ? ?0 kQ ? Qk
(3.5)
N ?1/2 Convergence of Directional Derivatives
We formulate the following convergence rate of the directional derivatives of FN from the right:
Theorem 3.4. For Q ? H and D ? D,
?N 2
+
+
? 21
P ?D F (Q) ? ?D FN (Q) ? N
? 2 exp
.
(3.6)
D ? R?2
+
+
It will be desirable to replace ?+
D F (Q)??D FN (Q) in (3.6) with ?D F (Q), though it is impossible
in general. We will later use the following observation to implicitly obtain a result in this direction.
? then
Lemma 3.5. If Q ? H \ {Q},
?+
D?
Q,Q
3.5
F (Q) ? 0.
(3.7)
An Incomplete Idea for Proving Theorem 1.1
At this point we can outline the basic intuition behind the proof of Theorem 1.1. We assume for
? N is u.d. Suppose, for the moment, that we could use (3.6) of Theorem 3.4 with
simplicity that Q
?
Q := QN . This is actually not mathematically sound, as we will discuss shortly, but if we could do
it then we would have from (3.6) that
?N 2
+
+
? 12
?
?
.
(3.8)
P |?D ? ? F (QN ) ? ?D ? ? FN (QN )| ? N
? 2 exp
Q,QN
Q,QN
D ? R?2
? N imply that
We note that (3.7) as well as both the convexity of FN and the definition of Q
?+
D?
?
Q,Q
N
? N ) ? 0 and ?+
F (Q
D?
?
Q,Q
N
? N ) ? 0.
FN (Q
Combining (3.8) and (3.9), we obtain that
?N 2
+
? 21
?
? 2 exp
.
P ?D ? ? F (QN ) ? N
Q,QN
D ? R?2
(3.9)
(3.10)
At last, combining (3.5), (3.10) and Theorem 3.2 we can formally prove Theorem 1.1.
? N . This is because
However, as mentioned above, we cannot legally use Theorem 3.4 with Q = Q
? N is a function of the samples (random variables) {xi }N , but for our proof to be valid, Q needs
Q
i=1
to be fixed before the sampling begins.
Therefore, our new goal is to utilize the intuition described above, but modify the proof to make
it mathematically sound. This is accomplished by creating a series of ?nets? (subsets of H) of
increasing precision. Each matrix in each of the nets is determined before the sampling begins, so
it can be used in Theorem 3.4. However, the construction of the nets guarantees that the N th net
? N to be used as a substitute for Q
? N in the above
contains a matrix Q which is sufficiently close to Q
process.
3.6
The Missing Component: Adaptive Nets
We describe here a result on the existence of a sequence of nets as suggested in ?3.5. They are
constructed in several stages, which cannot fit in here (see careful explanation in the extended version
? denotes a ball in H with center Q
? and radius 2 w.r.t. the
of this paper). We recall that B2 (Q)
Frobenius norm.
Lemma 3.6. Given ? ? 2 and ? > 0, there exists a sequence of sets {Sn }?
n=1 such that ?n ? N
? and for any Q ? B2 (Q)
? with kQ ? Qk
? F > n? 21 , ?Q0 ? Sn with
Sn ? B2 (Q)
? F ? kQ ? Qk
? F,
kQ0 ? Qk
7
(3.11)
1
1
2n? 2 (? + ??1 ) ? kQ0 ? QkF ? n? 2 ??1 and
kDQ,Q
kF ? ? n
0 ? DQ,Q
?
?
?1
(3.12)
.
(3.13)
Furthermore,
|Sn | ? 2?n
1
2
10Dn
?
D(D+1)
2
.
(3.14)
? by Q
? N as
The following lemma shows that we can use SN to guarantee good approximation of Q
long as the differences of partial derivatives are well-controlled for elements of SN (it uses the fixed
constants ? and ? for SN ; see Lemma 3.6).
Lemma 3.7. If for some > 0, FN is strictly convex and
1
+
F
(Q)
? N ? 2 + ?Q ? SN ,
?D ? F (Q) ? ?+
N
D ?
Q,Q
Q,Q
(3.15)
? N is u.d. and
then Q
1
? ?Q
? N kF ? 1 + 2?0 (? + ? ) + 4R? ?? N ? 12 + .
kQ
?0
3.7
(3.16)
Completing the Proof of Theorem 1.1
Let us fix ?0 = (4?0 ) ? 2, ?0 := (1 ? 2?0 /?0 )/(2?0 + 4R? ?0 ) and N > 2(D ? 1). We note that
1 + 2?0 (?0 +
1
) + 4R? ?0 ?0 = 2.
?0
(3.17)
We rewrite (3.14) using ? := ?0 and ? := ?0 and then bound its RHS from above as follows
|SN | ? 2((4?0 ) ? 2)N
D 2 +D+1
2
2?0 + 4R? ((4?0 ) ? 2)
10D
0
1 ? (4?2?0 )?2
?
! D(D+1)
2
(3.18)
C0 D 2
N .
2
Combining (3.6) (applied to any Q ? SN ) and (3.18) we obtain that
1
P ?Q ? SN with ?+
F (Q) ? ?+
FN (Q) ? N ? 2 +
DQ,Q
DQ,Q
?
?
2
? C0 N D exp ?N 2 /(D ? R?2 ) . (3.19)
Furthermore, (3.1) and (3.19) imply that
+
? 21 +
? N is u.d.
P ?+
F
(Q)
?
?
?Q
?
S
and
Q
F
(Q)
?
N
N
N
DQ,Q
D
?
?
Q,Q
2
2
N
?N 2
?
2
(1 ? ?)N ?2(D?1) .
? 1 ? C0 N D exp
D ? R?2
D?1
(3.20)
Theorem 1.1 clearly concludes from Lemma 3.7 (applied with ? := ?0 and ? := ?0 ), (3.20)
and (3.17).
Acknowledgment
This work was supported by NSF grants DMS-09-15064 and DMS-09-56072. Part of this work was
performed when M. Coudron attended the University of Minnesota (as an undergraduate student).
We thank T. Zhang for valuable conversations and forwarding us [22].
8
References
[1] A. Agarwal, S. Negahban, and M. Wainwright. Fast global convergence of gradient methods
for high-dimensional statistical recovery. Technical Report arXiv:1104.4824, Apr 2011.
[2] A. Agarwal, S. Negahban, and M. Wainwright. Noisy matrix decomposition via convex relaxation: Optimal rates in high dimensions. In ICML, pages 1129?1136, 2011.
[3] T. T. Cai, C.-H. Zhang, and H. H. Zhou. Optimal rates of convergence for covariance matrix
estimation. Ann. Statist., 38(4):2118?2144, 2010.
[4] E. J. Cand`es, X. Li, Y. Ma, and J. Wright. Robust principal component analysis? J. ACM,
58(3):11, 2011.
[5] V. Chandrasekaran, S. Sanghavi, P. A. Parrilo, and A. S. Willsky. Rank-sparsity incoherence
for matrix decomposition. Arxiv, 02139:1?24, 2009.
[6] C. Davis and W. M. Kahan. The rotation of eigenvectors by a perturbation. iii. SIAM J. on
Numerical Analysis, 7:1?46, 1970.
[7] D. Hsu, S. Kakade, and T. Zhang. Robust matrix decomposition with sparse corruptions.
Information Theory, IEEE Transactions on, 57(11):7221 ?7234, nov. 2011.
[8] P. J. Huber and E. Ronchetti. Robust statistics. Wiley series in probability and mathematical
statistics. Probability and mathematical statistics. Wiley, 2009.
[9] G. Lerman, M. McCoy, J. A. Tropp, and T. Zhang. Robust computation of linear models, or
How to find a needle in a haystack. ArXiv e-prints, Feb. 2012.
[10] R. A. Maronna, R. D. Martin, and V. J. Yohai. Robust statistics: Theory and methods. Wiley
Series in Probability and Statistics. John Wiley & Sons Ltd., Chichester, 2006.
[11] M. McCoy and J. Tropp. Two proposals for robust PCA using semidefinite programming. Elec.
J. Stat., 5:1123?1160, 2011.
[12] P. Ravikumar, M. J. Wainwright, G. Raskutti, and B. Yu. High-dimensional covariance estimation by minimizing `1 -penalized log-determinant divergence. Electron. J. Stat., 5:935?980,
2011.
[13] A. J. Rothman, P. J. Bickel, E. Levina, and J. Zhu. Sparse permutation invariant covariance
estimation. Electron. J. Stat., 2:494?515, 2008.
[14] P. J. Rousseeuw and A. M. Leroy. Robust regression and outlier detection. Wiley Series in
Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons
Inc., New York, 1987.
[15] J. Shawe-taylor, C. Williams, N. Cristianini, and J. Kandola. On the eigenspectrum of the
Gram matrix and the generalisation error of kernel PCA. IEEE Transactions on Information
Theory, 51(1):2510?2522, 2005.
[16] L. G. Valiant. A theory of the learnable. Commun. ACM, 27(11):1134?1142, Nov. 1984.
[17] R. Vershynin. How close is the sample covariance matrix to the actual covariance matrix? to
appear.
[18] R. Vershynin. Introduction to the non-asymptotic analysis of random matrices. In Y. C. Eldar
and G. Kutyniok, editors, Compressed Sensing: Theory and Applications. Cambridge Univ
Press, to appear.
[19] H. Xu, C. Caramanis, and S. Sanghavi. Robust pca via outlier pursuit. In NIPS, pages 2496?
2504, 2010.
[20] H. Xu, C. Caramanis, and S. Sanghavi. Robust pca via outlier pursuit. Information Theory,
IEEE Transactions on, PP(99):1, 2012.
[21] T. Zhang and G. Lerman. A novel m-estimator for robust pca. Submitted, available at
arXiv:1112.4863.
[22] T. Zhang and H. Zou. Sparse precision matrix estimation via positive definite constrained
minimization of `1 penalized d-trace loss. Personal Communication, 2012.
9
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4,087 | 47 | 249
HIERARCHICAL LEARNING CONTROL AN APPROACH WITH NEURON-LIKE ASSOCIATIVE MEMORIES
E. Ersu
ISRA Systemtechnik GmbH, Schofferstr. 15, D-6100 Darmstadt, FRG
H. Tolle
TH Darmstadt, Institut fur Regelungstechnik,
Schlo~graben 1, D-6100 Darmstadt, FRG
ABSTRACT
Advances in brain theory need two complementary approaches:
Analytical investigations by in situ measurements and as well synthetic modelling supported by computer simulations to generate
suggestive hypothesis on purposeful structures in the neural
tissue. In this paper research of the second line is described:
Starting from a neurophysiologically inspired model of stimulusresponse (S-R) and/or associative memorization and a psychologically motivated ministructure for basic control tasks, pre-conditions
and conditions are studied for cooperation of such units in a
hierarchical organisation, as can be assumed to be the general
layout of macrostructures in the brain.
I. INTRODUCTION
Theoretic modelling in brain theory is a highly speculative
subject. However, it is necessary since it seems very unlikely to
get a clear picture of this very complicated device by just analyzing the available measurements on sound and/or damaged brain parts
only. As in general physics, one has to realize, that there are
different levels of modelling: in physics stretching from the atomary level over atom assemblies till up to general behavioural
models like kinematics and mechanics, in brain theory stretching
from chemical reactions over electrical spikes and neuronal cell
assembly cooperation till general human behaviour.
The research discussed in this paper is located just above the
direct study of synaptic cooperation of neuronal cell assemblies as
studied e. g. in /Amari 1988/. It takes into account the changes of
synaptic weighting, without simulating the physical details of such
changes, and makes use of a general imitation of learning situation
(stimuli) - response connections for building up trainable basic
control loops, which allow dynamic S-R memorization and which are
themsel ves elements of some more complex behavioural loops. The
general aim of this work is to make first steps in studying structures, preconditions and conditions for building up purposeful
hierarchies and by this to generate hypothesis on reasons and
@) American Institute of Physics 1988
250
meaning behind substructures in the brain like the columnar organization of the cerebral cortex (compare e. g. /Mountcastle 1978/).
The paper is organized as follows: In Chapter II a short description is given of the basic elements for building up hierarchies,
the learning control loop LERNAS and on the role of its subelement
AMS, some ~ssociati ve memory ~stem inspired by neuronal network
considerations. Chapter III starts from certain remarks on substructures in the brain and discusses the cooperation of LERNASelements in hierarchies as possible imitations of substructures.
Chapter IV specifies the steps taken in this paper in the direction
of Chapter III and Chapter V presents the results achieved by computer simulations. Finally an outlook will be given on further
investigations.
II. LERNAS AND AMS
Since the formal neuron was introduced by /McCulloch and Pitts
1943/, various kinds of neural network models have been proposed,
such as the percept ron by /Rosenblatt 1957/ the neuron equation of
/Caianello 1961/, the cerebellar model articulation controller CMAC
by /Albus 1972, 1975/ or the associative memory models by
/Fukushima 1973/, /Kohonen 1977/ and / Amari 1977/. However, the
abili ty of such systems to store information efficiently and to
perform certain pattern recognition jobs is not adequate for survival of living creatures. So they can be only substructures in the
overall brain organization; one may call them a microstructure.
Purposeful acting means a goal driven coordination of sensory information and motor actions. Al though the human brain is a very
complex far end solution of evolution, the authors speculated in
1978 that it might be a hierarchical combination of basic elements,
which would perform in an elementary way like the human brain in
total, especially since there is a high similarity in the basic
needs as well as in the neuronal tissue of human beings and relati vely simple creatures. This led to the design of the learning
control loop LERNAS in 1981 by one of the authors - /Ersu 1984/ on the basis of psychological findings. He transformed the statement of /Piaget 1970/, that the complete intelligent action needs
three elements: "1) the question, which directs possible search
actions, 2) the hypothesis, which anticipates eventual solutions,
3) the control, which selects the solution to be chosen" into the
structure shown in Fig. 1, by identifying the "question" with an
performance criterion for assessment of possible advantages/disadvantages of certain actions, the "hypothesis" with a predictive
model of environment answers and the "control" with a control strategy which selects for known situations the best action, for unknown situations some explorative action (active learning).
In detail, Fig. 1 has to be understood in the following way: The
predictive model is built up in a step by step procedure from a
characterization of the actual situation at the time instant k-T
s
251
T sampling time) and the measured response of the unknown ens
vironment at time instant (k+1)T . The actual situation consists of
s
measurements regarding the stimuli and responses of the environment
at time instant keTs plus - as far as necessary for a unique characterization - of the situation-stimuli and responses at time instants (k-1)T s , (k-2)T s ... , provided bv- the short term memory. To
reduce learning effort, the associative memory system used to store
the predictive model has the ability of local generalization, that
means making use of the trained response value not only for the
corresponding actual situation, but also in similar situations. The
assessment module generates on the basis of a given goal - a wanted
environment response - with an adequate performance criterion an
evaluation of possible actions through testing them with the predictive model, as far as this is already built up and gives meaningful answers. The result is stored in the control strategyAMS
together with its quality: real optimal action for the actual situation or only relatively optimal action, if the testing reached the
border of the known area in the predictive model of the environment. In the second case, the real action is changed in a sense of
curiosity, so that by the action the known area of the predictive
model is extended. By this, one reaches more and more i:he first
case, in which the real optimal actions are known. Since the first
guess for a good action in the optimization phase is given to the
assessment module from the control strategy AMS - not indicated in
Fig. 1 to avoid unnecessary complication - finally the planning
level gets superfluous and one gets very quick optimal reactions,
the checking with the planning level being necessary and helpful
only to find out, whether the environment has not changed, possibly. Again the associative memory system used for the control strategy is locally generalizing to reduce the necessary training
effort.
The AMS storage elements for the predictive model, and for optimized actions are a refinement and implementation for on-line
application of the neuronal network model CMAC from J. Albus - see
e. g. /Ersu, Militzer 1982/ -, but it could be any other locally
generalizing neural network model and even a storage element based
on pure mathematical considerations, as has been shown in
/Militzer, Tolle 1986/.
The important property to build up an excellent capability to
handle different tasks in an environment known only by some sensory
information - the property which qualifies LERNAS as a possible
basic structure (a "ministructure") in the nervous system of living
creatures - has been proven by its application to the control of a
number of technical processes, starting with empty memories for the
predictive model and the control strategy storage. Details on this
as well as on the mathematical equations describing LERNAS can be
found in /Ersu, Mao 1983/, /Ersu, Tolle 1984/ and /Ersu, Militzer
1984/.
252
It should be mentioned that the concept of an explicit predictive
environmental model - as used in LERNAS - is neither the only meaningful description of human job handling nor a necessary part of
our basic learning element. It suffices to use a prediction whether
a certain action is advantegeous to reach the actual goal or
whether this is not the case. More information on such a basic
element MINLERNAS, which may be used instead of LERNAS in general
(however, with the penalty of some performance degradation) are
given in /Ersa, Tolle 1988/.
III. HIERARCHIES
There are a number of reasons to believe, that the brain is
built up as a hierarchy of control loops, the higher levels having
more and more coordinative functions. A very simple example shows
the necessity in certain cases. The legs of a jumping jack can move
together, only. If one wants to move them separately, one has to
cut the connection, has to build up a separate controller for each
leg and a coordinating controller in a hierarchically higher level
to restore the possibility of coordinated movements. Actually, one
can find such an evolution in the historical development of certain
animals. In a more complex sense a multilevel hierarchy exists in
the extrapyramidal motor system. Fig. 2 from /Albus 1979/ specifies
five levels of hierarchy for motor control. It can be speculated,
that hierarchical organizations are not existing in the senso-motoric level only, but also in the levels of general abstractions and
thinking. E. g. /Dorner 1974/ supports this idea.
If one assumes out of these indications,
that hierarchies are a
fundamental element of brain structuring - the details and numbers
of hierarchy-levels not being known - one has to look for certain
substructures and groupings of substructures in the brain. In this
connection one finds as a first subdivision the cortical layers,
but then as another more detailed subdivision the columns, cell
assemblies heavily connected in the axis vertical to cortical
layers and sparsely connected horizontally. /Mountcastle 1978/
defines minicolumns, which comprise in some neural tissue roughly
100 in other neural tissue roughly 250 individual cells. In addition to these mini columns certain packages of minicolumns, consisting out of several hundreds of the minicolumns, can be located.
They are called macrocolumns by /Mountcastle 1978/. Fig. 3 gives
some abstraction, how such structures could be interpreted: each
minicolumn is considered to be a ministructure of the type LERNAS,
a number of LERNAS units - here shown in a ring structure instead
of a filled up cylindrical structure - building up a macrocolumn.
The signals between the LERNAS elements could be overlapping and
cooperating. Minicolumns being elements of macrocolumns of a higher
cortical layer - here layer j projecting to layer k - could initiate and/or coordinate this cooperation in a hierarchical sense.
Such a complex system is difficult to simulate. One has to go into
this direction in a step by step procedure. In a first step the
253
overlapping or crosstalk between the minicolumns may be suppressed
and the number of ministructures LERNAS representing the minicolumns should be reduced heavily. This motivates Fig. 4 as a fundamental blockdiagram for research on cooperation of LERNAS elements.
IV. TOPICS ADDRESSED
From Fig. 4 only the lowest level of coordination (layer 1),
that means the coordination of two subprocesses was implemented up
to now - right half of Fig. 5. This has two reasons: Firstly, a
number of fundamental questions can be posed and discussed with
such a formulation already. Secondly, it is difficult to set up
meaningful subprocesses and coordination goals for a higher order
system.
The problem discussed in the following can be understood as the
coordination of two minicolumns as described in Chapter III, but
also as the coordination of higher level subtasks, which may be
detailed themselves by ministructures and/or systems like Fig. 4.
This is indicated in the left half of Fig. 5.
Important questions regarding hierarchies of learning control loops
are:
I.
What seem to be meaningful interventions from the coordinator
onto the lower level systems?
II.
Is parallel learning in both levels possible or requires a
meaningful learning strategy that the control of subtasks has
to be learned at first before the coordination can be learned?
III. Normally one expects, that the lower level takes care of short
term requirements and the upper level of long term strategies.
Is that necessary or what happens if the upper level works on
nearly the same time horizon as the lower levels?
IV.
Furtheron one expects, that the upper level may look after
other goals than the lower level, e. g. the lower level tries
to suppress disturbances effects since the upper level tries
to minimize overall energy consumption. But can such different
strategies work without oscillations or destabilization of the
system?
Question I can be discussed by some general arguments, for questions II-IV only indications of possible answers can be given from
simulation results. This will be postponed to Chapter V.
Fig. 6 shows three possible intervention schemes from the coordinator.
By case a) an intervention into the structure or the parameters of
254
the sublevel (=local) controllers is meant. Since associative
mappings like AMS have no parameters being directly responsible for
the behaviour of the controller - as would be the case with a parametrized linear or non-linear differential equation being the description of a conventional controller - this does not make sense
for the controller built up in LERNAS. However, one could consider
the possibility to change parameters or even elements, that means
structural terms of the performance criterion, which is responsible
for the shaping of the controller. But this would require to learn
anew, which takes a too long time span in general.
By case b) a distribution of work load regarding control commands
is meant. The possible idea could be, that the coordinator gives
control inputs to hold the long range mean value required, since
the local controllers take into account fast dynamic fluctuations
only. However, this has the disadvantage that the control actions
of the upper level have to be included into the inputs to the local
controllers, extending the dimension of in-put space of these
storage devices, since otherwise the process appears to be highly
time variant for the local controllers, which is difficult to
handle for LERNAS.
So case c) seems to be the best solution. In this case the coordinator commands the set points of the local controllers, generating
by this local subgoals for the lower level controllers. Since this
requires no input space extension for the local controllers and is
in full agreement with the working conditions of single LERNAS
loops, it is a meaningful and effective approach.
Fig. 7 shows the accordingly built up structure in detail. The
control strategy of Fig. 1 is divided here in two parts the storage
element (the controller C) and the active learning AL. The elements
are explicitly characterized for the upper level only. The whole
lower level is considered by the coordinator as a single pseudoprocess to be controlled (see Fig. 4).
v.
SIMULATION RESULTS
For answering questions II and III the very simple non-linear
process shown in Fig. 8 - detailing the subprocesses SPl, SP2 and
their coupling in Fig. 7 - was used. For the comparison of bottom
up and parallel learning suitably fixed PI-controllers were used
for bottom up learning instead of LERNAS land LERNAS 2, simulating
optimally trained local controllers. Fig. 9a shows the result due
to which in the first run a certain time is required for achieving
a good set point following through coordinator assistance. However,
with the third repetition (4th run) a good performance is reached
from the first set point change on already. For parallel learning
all (and not only the coordinator AMS-memories) were empty in the
beginning. Practically the same performance was achieved as in
bottom up training - Fig. 9b -, indicating, that at least in simple
problems, as considered here, parallel learning is a real possibi-
255
lity. However - what is not illustrated here - the coordinator
sampling time must be sufficiently long, so that the local controllers can reach the defined subgoals at least qualitatively in this
time span.
For answering question III, in which respect a higher difference in
the time horizon between local controller and coordinator changes
the picture, a doubling of the sampling rate for the coordinator
was implemented. Fig. 10 give the results. They can be interpreted
as follows: Smaller sampling rates allow the coordinator to get
more information about the pseudo-sub-processes, the global goal is
reached faster. Larger sampling rates lead to a better overall
performance when the goal is reached: there is a higher amount of
averaging regarding informations about the pseudo-sub-processes.
Up to now in both levels the goal or performance criterion was the
minimization of differences between the actual plant output and the
requested plant output. The influence of different coordinator
goals - question IV - was investigated by simulating a two stage
waste water neutralization process. A detailed description of this
process set up and the simulation results shall not be given here
out of space reasons. It was found that:
?
in hierarchical systems satisfactory overall behaviour may be
reached by well defined subgoals with clearly different coordinator goals.
?
since learning is goal driven, one has to accept that implicit
wishes on closed loop behaviour are fulfilled by chance only.
Therefore important requirements have to be included in the
performance criteria explicitly.
It should be remarked finally, that one has to keep in mind, that
simulation results with one single process are indications of
possible behaviour only, not excluding that in other cases a fundamentally different behaviour can be met.
VI. OUTLOOK
As has been mentioned already in Chapter III and IV, this work
is one of many first steps of investigations regarding hierarchical
organization in the brain, its preconditions and possible behaviour.
Subjects of further research should be the self-organizing task
distribution between the processing units of each layer, and the
formation of inter layer projections in order to build up meta-tasks
composed of a sequence of frequently occuring elementary tasks.
These investigations will on the other hand show to what extent
this kind of higher-learning functions can be achieved by a hierarchy of LERNAS-type structures which model more or less low-level
basic learning behaviour.
256
VII. ACKNOWLEDGEMENTS
The work presented has been supported partly by the Stiftung
Volkswagenwerk. The detailed evaluations of Chapter IV and V have
been performed by Dipl.-Ing. M. Zoll and Dipl.-Ing. S. Gehlen. We
are very thankful for this assistance.
VIII. REFERENCES
Albus, J. S.
Theoretical and Experimental Aspects of a
Cerebellar Model, Ph.D. Thesis, Univ. of
Maryland, 1972
Albus, J. S.
A New Approach to Manipulator Control:
The Cerebellar Model Articulation Controller
(CMAC), Trans. ASmE series, G, 1975
Albus, J. S.
A Model of the Brain for Robot Control Part 3: A Comparison of the Brain and Our
Model, Byte, 1979
Amari, S. 1.
Neural Theory of Association and Concept
Formation, BioI. Cybernetics, Vol. 26, 1977
Amari, S. 1.
Mathematical Theory of Self-Organization in
Neural Nets, in: Organization of Neural
Networks, Structures and Models, ed. by
von Seelen, Shaw, Leinhos, VHC-Verlagsges.
Weinheim, W.-Germany, 1988
Caianello, E. R.
Outline of a Theory of Thought Process and
Thinking Machines, Journal of Theoretical
Biology, Vol. 1, 1961
Dorner, D.
Problemlosen als Informationsverarbeitung
Verlag H. Huber, 1974
Ersil, E.
On the Application of Associative Neural
Network Models to Technical Control Problems,
in: Localization and Orientation in Biology and
Engineering, ed. by Varju, Schnitzler,
Springer Verlag Berlin, W.-Germany, 1984
Ersu, E.
Mao, X.
Control of pH by Use of a Self-Organizing
Concept with Associative Memories, ACI'83,
Kopenhagen (Denmark), 1983
257
Ersu, E.
Militzer, J.
Software Implementation of a Neuron-Like
Associative Memory System for Control
Application, Proceedings of the 2nd lASTED
Conference on Mini- and Microcomputer Applications, MIMI'82, Davos (Switzerland), 1982
Ersu, E.
Militzer, J.
Real-Time Implementation of an Associative
Memory-Based Learning Control Scheme for NonLinear Multivariable Processes, Symposium
"Applications of Multivariable System
Techniques", Plymouth (UK), 1984
Ersu, E.
Tolle, H.
A New Concept for Learning Control Inspired by
Brain Theory, Proceed. 9th IFAC World Congress,
Budapest (Hungary), 1984
Ersu, E.
Tolle, H.
Learning Control Structures with Neuron-Like
Associative Memory Systems, in: Organization of
Neural Networks, Structures and Models, ed. by
von Seelen, Shaw, Leinhos, VCH Verlagsgesellschaft Weinheim, W.-Germany, 1988
Fukushima, K.
A Model of Associative Memory in the Brain
BioI. Cybernetics, Vol. 12, 1973
Kohonen, T.
Associative Memory, Springer Verlag Berlin,
W.-Germany, 1977
McCulloch, W. S.
Pitts, W. H.
A Logical Calculus of the Ideas, Immanent in
Nervous Activity, Bull. Math. Biophys. 9, 1943
Militzer, J.
Tolle, H.
Vertiefungen zu einem Teilbereiche der menschlichen Intelligenz imitierenden Regelungsansatz
Tagungsband-DGLR-Jahrestagung, Munchen,
W.-Germany, 1986
Mountcastle, V. B. An Organizing Principle for Cerebral Function:
The Unit Module and the Distributed System, in:
The Mindful Brain by G. M. Edelman,
V. B. Mountcastle, The MIT-Press, Cambridge,
USA, 1978
Piaget, J.
Psychologie der Intelligenz, Rascher Verlag,
4th printing, 1970
Rosenblatt, F.
The Perceptron: A Perceiving and Recognizing
Automation, Cornell Aeronautical Laboratory,
Report No. 85-460-1, 1957
258
FIGURES
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Fig. 2. The hierarchy of aotor control that exists in the extrapyramidal motor system. Basic reflexes remain even if the brain
stem is cut at A-A. Coordination of these reflexes for standing is
possible if the cut is at B-B. The sequential coordination required
for walking requires the area below c-c to be operable. Simple
tasks can be executed if the region below D-D is intact. Lengthy
tasks and complex goals require the cerebral cortex. (/Albus 1979/)
259
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260
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subdivision:2 meaningful:6 intact:1 indicating:1 iors:1 support:1 meant:2 trainable:1 handling:1 |
4,088 | 470 | Repeat Until Bored: A Pattern Selection Strategy
Paul W. Munro
Depamnent of Information Science
University of Pittsburgh
Pittsburgh, PA 15260
ABSTRACT
An alternative to the typical technique of selecting training examples
independently from a fixed distribution is fonnulated and analyzed, in
which the current example is presented repeatedly until the error for that
item is reduced to some criterion value, ~; then, another item is randomly selected. The convergence time can be dramatically increased or
decreased by this heuristic, depending on the task, and is very sensitive
to the value of ~.
1 INTRODUCTION
In order to implement the back propagation learning procedure (Werbos, 1974; Parker,
1985; Rumelhart, Hinton and Williams, 1986), several issues must be addressed. In addition to designing an appropriate network architecture and detennining appropriate values
for the learning parameters, the batch size and a scheme for selecting training examples
must be chosen. The batch size is the number of patterns presented for which the corresponding weight changes are computed before they are actually implemented; immediate
update is equivalent to a batch size of one. The principal pattern selection schemes are
independent selections from a stationary distribution (independent identically distributed,
or i.i.d.) and epochal, in which the training set is presented cyclically (here, each cycle
through the training set is called an epoch). Under Li.d. pattern selection, the learning
perfonnance is sensitive to the sequence of training examples. This observation suggests
that there may exist selection strategies that facilitate learning. Several studies have
shown the benefit of strategic pattern selection (e.g., Mozer and Bachrach, 1990; Atlas,
Cohn, and Ladner, 1990; Baum and Lang, 1991).
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lYPically, online learning is implemented by independent identically distributed pattern selection, which cannot (by definition) take advantage of useful sequencing strategy. It
seems likely, or certainly plausible, that the success of learning depends to some extent
on the order in which stimuli are presented. An extreme, though negative, example
would be to restrict learning to a portion of the available training set; i.e. to reduce the effective training set. Let sampling functions that depend on the state of the learner in a
constructive way be termed pedagogical.
Determination of a particular input may require information exogenous to the learner; that
is, just as training algorithms have been classified as supervised and unsupervised, so can
pedagogical pattern selection techniques. For example, selection may depend on the networlc's performance relative to a desired schedule. The intent of this study is to explore
an unsupervised selection procedure (even though a supervised learning rule, backpropagation, is used). The initial selection heuristic investigated was to evaluate the errors across
the entire pattern set for each iteration and to present the pattern with the highest error, of
course, this technique has a large computational overhead, but the question was whether it
would reduce the number of learning trials. The results were quite to the contrary; preliminary trials on small tasks (two and three bit parity), show that this scheme performs
very poorly with all patterns maintaining high error.
A new unsupervised selection technique is introduced here. The "Repeat-Until-Bored"
heuristic is easily implemented and simply stated: if the current training example generates a high error (Le. greater than a fixed criterion value), it is repeated; otherwise, another
one is randomly selected. This approach was motivated by casual observations of behavior in small children; they seem to repeat seemingly arbitrary tasks several times, and
then abruptly stop and move to some seemingly arbitrary alternative (Piaget, 1952). For
the following discussion, lID and RUB will denote the two selection procedures to be
compared.
2 METHODOLOGY
RUB can be implemented by adding a condition to the lID statement; in C, this is simply
old (lID) : patno = random () % numpatsi
new(RUB): if (paterror<beta) patno = random() % numpatsj
where patno identifies the selected pattern, numpats is the number of patterns in the training set, and paterror is the sum squared error on a particular pattern. Thus, an example is
presented and repeated until it has been learned by the network to some criterion level, the
squared error summed across the output units is less than a "boredom" criterion ~; then ,
another pattern is randomly selected.
The action of RUB in weight space is illustrated in Figure 1, for a two dimensional environment consisting of just two patterns. Corresponding to each pattern, there is an isocline (or equilibrium surface) , defined by the locus of weight vectors that yield the desired
response to that pattern (here, a or b). Since the delta rule drives the weight parallel to
the presented pattern, trajectories in weight space are perpendicular to the pattern's isocline. Here, RUB is compared with alternate pattern selection.
Repeat Until Bored: A Pattern Selection Strategy
wea=A
an II D trajectory
a
RUB trajectory
Figure 1. Effect of pattern selection on weight state trajectory. A linear unit can be
trained to give arbitrary responses (A and B) to given stimuli (a and b). The isoclines
(bold lines) are defined to be the set of weights that satisfy each stimulus-response
pair. Thus, the intersection is the weight state that satisfies both constraints. The
delta rule drives the weights toward the isocline that corresponds to the presented
pattern. The RUB procedure repeats a pattern until the state approaches the isocline.
The RUB procedure was tested for a broad range of ~ across several tasks. Two performance measures were used; in both cases, performance was averaged across several (201(0) trials with different initial random weights. For the parity tasks, performance was
measured as the fraction of trials for which the squared error summed over the training set
reached a sufficiently low value (usually 0.1) within a specified number of training exampIes. Since the parity task always converged for sufficiently large ~,performance was
measured as the number of trials that converged within a pre specified number of iterations
required to reduce the total squared error summed across the pattern set to a low value
(typically, 0.1). Note that each iteration of weight modification during a set of repeated
examples was explicitly counted in the performance measure, so the comparison between
lID and RUB is fair. Also, for each task, the learning rate and momentum were fixed
(ususally 0.1 and 0.9, respectively).
Consideration of RUB (see the above C implementation, for example) indicates that, for
very small values of ~, the first example will be repeated indefinitely, and the task can
therefore not be learned. At the other extreme, for ~ greater than or equal to the maximum possible squared error (2.0, in this case), perfonnance should match IID.
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3 RESULTS
3.1. PARITY
While the expected behavior for RUB on the two and three bit parity tasks (Figure 2) is
observed for low and high values of ~, there are some surprises in the intermediate range.
Rather than proceeding monotonically from zero to its lID value, the performance curve
exhibits an "up-down-up" behavior; it reaches a maximum in the range O.2<~O.25, then
plummets to zero at J3=O.25, remains there for an interval, then partially recovers at its
final (lID) level. This "dead zone" phenomenon is not as pronounced when the momentum parameter is set to zero (Figure 3).
100
100
80
80
60
60
40
40
20
20
0
.0001 .001
.01
.1
10
0
.0001 .001
.01
.1
10
Figure 2. Performance profiles for the parity task. Each point is the average
number of successful simulations out of 100 trials. A log scale is used so that
the behavior for very low values of the error cr~erion is evident. Note the
critical falloff at ~"'0.25 for both the XOR task (left) and three-bit parity (right).
100
100
80
80
60
60
40
40
20
20
0
.0001 .001
.01
.1
10
0
.0001 .001
.01
.1
1
10
Figure 3. Performance profiles with zero momemtum. For these two tasks,
the up-down-up phenomenon is still evident, but there is no "dead zone".
Left: XOR Right: Three bit parity
Repeat Until Bored: A Pattern Selection Strategy
3.2 ENCODERS
The 4-2-4 encoder shows no significant improvement over the lID for any value of RUB.
Here, perfonnance was measured both in tenns of success rate and average number of iterations to success. Even though all simulations converge for ~>.001 (Le., there is no
dead zone), the effect of ~ is reflected in another perfonnance measure: average number of
iterations to convergence (Figure 4). However, experiments with the 5-2-5 encoder task
show an effect. While backprop converges for all values of ~ (except very small values),
the perfonnance, as measured by number of pattern presentations, does show a pronounced decrement. The 8-3-8 encoder shows a significant, but less dramatic, effect.
6000
1st data value:
8691.0
CD
u
~
..
CD
?6
5-2-5
4-2-4
6
8-3-8
~
CD
>
~
0
u
4000
0
en
~
0
..
==
..
::=
as
CD
2000
CD
~
as
CD
~
0
.001
.01
.1
1
10
13
Figure 4. Encoder performance profiles. See text.
3.3 THE MESH
The mesh (Figure 5, left) is a 2-D classification task that can be solved by a strictly layered net with five hidden units. Like the encoder and unlike parity, lID is found to converge on 100% of trials; however, there is a critical value of ~ and a well-defined dead
zone (Figure 5, right). Note that the curve depicting average number of iterations to convergence decreases monotonically, interrupted at the dead zone but continuing its apparent
trend for higher values of ~.
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20 ~~----~~"~----T10+
0
0
?
0
???
?
?
0
?
'b
4b
0
0
??
~
~
c:
06"
UC'i
(I) ....
10
'--
a~
4
J!
:::J
.S
CI)
0
.0001 .001
.01
.1
1
10
~
Figure 5. The mesh task. Left: the task. Right: Performance profile. Number of
simulations that converge is plotted along the bold line (left vertical) axis. Average
number of iterations are plotted as squares (right vertical axis).
3A NONCONVERGENCE
Nonconvergence was examined in detail for three values of ~, corresponding to high perfonnance, poor perfonnance (the dead zone), and lID, for the three bit parity task. The
error for each of the eight patterns is plotted over time. For trials that do not converge
(Figure 6), the patterns interact differently, depending on the value of~. At (3=0.05 (a
"good" value of ~ for this task), the error traces for the four odd-parity patterns are strongly correlated in an irregular oscillatory mode, as are the four even-parity traces, but the
two groups are strongly anticorrelated. In the odd parity group, the error remains low for
three of the patterns (001, 010, and 100), but ranges from less than 0.1 to values greater
than 0.95 for the fourth (111). Traces for the even parity patterns correspond almost identically; i.e. not only are they correlated, but all four maintain virtually the same value.
At this point, the dead zone phenomenon has only been observed in tasks with a single
output unit. This property hints at the following explanation. Note first that each
input/output pair in the training set divides the weight space into two halves, characterized by the sign of the linear activation into the output unit; that is, whether the output
is above or below 0.5, and hence whether the magnitude of the difference between the actual and desired responses is above or below 0.5. Since ~ is the value of the squared
error, learning is repeated for (3=0.25 only for examples for which the state is on the
wrong half of weight space. Just when it is about to cross the category boundary, which
would bring the absolute value of the error below .5, RUB switches to another example,
and the state is not pushed to the other side of the boundary. This conjecture suggests
that for tasks with multiple output units, this effect might be reduced or eliminated, as
has been demonstrated in the encoder examples.
Repeat Until Bored: A Pattern Selection Strategy
~
= 0.05
~
= 0.3
1.0't:::-----"""':;;:::~-~::;:'1
1.0~----------------------~
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
......
0.0 ~-=::=...-
25500
~:=._IIIIIIIC:~
25600
o.o ......-----------~
4900
5000
~ = 2.0
1.0 -r---------------------~
Figure 6. Error traces for
individual patterns. For each
of three values of the error
criterion, the variation of the
error for each pattern is
plotted for 100 iterations of
the three-bit parity task that
did not converge. Note the
large amplitude swings for low
values (upper left), and the
small amplitude oscillations in
the "dead zone" (upper right).
0.8
0.6
0.4
0.2
0.0~---------------4
29000
4
29100
DISCUSSION
Active learning and boredom. The sequence of training examples has an undeniable effect
on learning, both in the real world and in simulated learning systems. While the RUB
procedure influences this sequence such that the learning perfonnance is either positively
or negatively affected, it is just a minimal instance of active learning; more elaborate
learning systems have explored similar notions of "boredom" (eg., Scott and Markovitch,
1989).
Nonconvergence. From Figure 6 it can be seen, for both RUB and lID, that nonconvergence does not correspond to a local minimum in weight space. In situations where the
overall error is "stuck" at a non-zero value, the error on the individual patterns continues
to change. The weight trajectory is thus "trapped" in a nonoptimal orbit, rather than a
nonoptimal equilibrium point.
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Munro
Acknowledgements
This research was supported in part by NSF grant 00-8910368 and by Siemens Corporate
Research, which kindly provided the author with financial support and a stimulating research environment during the summer of 1990. David Cohn and Rile Belew were helpful
in bringing relevant work to my attention.
References
Baum, E. and Lang, K. (1991) Constructing multi-layer neural networks by searching
input space rather than weight space. In: Advances in Neural Information Processing
Systems 3. D. S. Touretsky, ed. Morgan Kaufmann.
Cohn, D., Atlas, L., and Ladner, R. (1990) Training connectionist networks with queries
and selective sampling. In: Advances in Neural Information Processing Systems 2. D.
S. Touretsky, ed. Morgan Kaufmann.
Mozer, M. and Bachrach, J. (1990) Discovering the structure of a reactive environment by
exploration. In: Advances in Neural Information Processing Systems 2. D. S.
Touretsky, ed. Morgan Kaufmann.
Parker, D. (1985) Learning logic. TR-47. MIT Center for Computational Economics
and Statistics. Cambridge MA.
Piaget, J. (1952) The Origins of Intelligence in Children. Norton.
Rumelhart D., Hinton G., and Williams R. (1986) Learning representations by backpropagating errors. Nature 323:533-536.
Scott, P. D. and Markovitch, S. (1989) Uncertainty based selection of learning experiences. Sixth International Workshop on Machine Learning. pp.358-361
Werbos, P. (1974) Beyond regression: new tools for prediction and analysis in the behavioral sciences. Unpublished doctoral dissertation, Harvard University.
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4,089 | 4,700 | Collaborative Gaussian Processes for
Preference Learning
Jose Miguel Hern?andez-Lobato ?
Department of Engineering
University of Cambridge
Neil Houlsby ?
Department of Engineering
University of Cambridge
Zoubin Ghahramani
Department of Engineering
University of Cambridge
Ferenc Husz?ar
Department of Engineering
University of Cambridge
Abstract
We present a new model based on Gaussian processes (GPs) for learning pairwise preferences expressed by multiple users. Inference is simplified by using
a preference kernel for GPs which allows us to combine supervised GP learning of user preferences with unsupervised dimensionality reduction for multi-user
systems. The model not only exploits collaborative information from the shared
structure in user behavior, but may also incorporate user features if they are available. Approximate inference is implemented using a combination of expectation
propagation and variational Bayes. Finally, we present an efficient active learning
strategy for querying preferences. The proposed technique performs favorably on
real-world data against state-of-the-art multi-user preference learning algorithms.
1
Introduction
Preference learning is concerned with making inference from data consisting of pairs of items and
corresponding binary labels indicating user preferences. This data arises in many contexts, including
medical assistive technologies [1], graphical design [3] and recommendation systems [5]. A popular
modeling approach assumes the existence of a utility function f such that f (x) gives the utility of
an item with feature vector x; f (xi ) > f (xj ) indicates that item i is preferred to item j. Bayesian
methods can be used to learn f , for example, by modeling f independently for each user as a draw
from a Gaussian process (GP) prior [4]. However, when data from many users is available, such
methods do not leverage similarities in preferences across users. Current multi-user approaches
require that features are available for each user and assume that users with similar features have
similar preferences [2], or perform single-user learning, ignoring user features, but tie information
across users with a hierachical prior [1]. These methods are not flexible and can only address one of
two possible scenarios: a) user features are available and they are useful for prediction and b) when
this is not the case. Additionally, they involve at least solving U GP problems, where U is the total
number of users. This cost is prohibitive even for modest U . Our approach, by contrast, can address
both a) and b) by combining informative user features with collaborative information. Furthermore,
we perform scalable inference which can handle problems with large U .
Our new multi-user model is based on dimensionality reduction ideas from the field of collaborative filtering [19, 16]. Unsupervised learning of similarities in users? behavior is exploited without
requiring access to user-specific feature vectors. However, if these are available it may be desirable
?
Both authors contributed equally.
1
to incorporate them for predictions; our model can use these user-specific features as well. The
proposed method is based on a connection between preference learning and GP binary classification. We show that both problems are equivalent when a covariance function called the preference
kernel is used. This specific kernel simplifies the inference process, allowing us to implement more
complex models such as the proposed multi-user approach. Finally, in real scenarios, querying users
for preference may be costly and intrusive, so it is desirable to learn preferences using the least
data possible. With this objective, we present BALD (Bayesian active learning by disagreement), an
efficient active learning strategy for binary classification problems with GP priors.
2
Pairwise preference learning as special case of binary classification
The problem of pairwise preference learning can be recast as a special case of binary classification.
Let us consider two items i and j with corresponding feature vectors xi , xj ? X . In the pairwise
preference learning problem, we are given pairs of feature vectors xi and xj and corresponding class
labels y ? {?1, 1} such that y = 1 if the user prefers item i to item j and y = ?1 otherwise. The
task of interest is then to predict the class label for a new pair of feature vectors not seen before.
This problem can be addressed by introducing a latent preference function f : X 7? R such that
f (xi ) > f (xj ) whenever the user prefers item i to item j and f (xi ) < f (xj ) otherwise [4]. When
the evaluations of f are contaminated with Gaussian noise with zero mean and (without loss of
generality) variance 1/2, we obtain the following likelihood for f given xi , xj and y
P(y|xi , xj , f ) = ?[(f [xi ] ? f [xj ])y] ,
(1)
where ? is the standard Gaussian cumulative distribution function. The preference learning problem
can be solved by combining a GP prior on f with the likelihood function in (1) [4]. The posterior
for f can then be used to make predictions on the user preferences for new pairs of items.
Note that the likelihood (1) depends only on the difference between f (xi ) and f (xj ). Let g : X 2 7?
R be the latent function g(xi , xj ) = f (xi ) ? f (xj ). We can recast the inference problem in terms
of g and ignore f . When the evaluation of g is contaminated with standard Gaussian noise, the
likelihood for g given xi , xj and y is
P(y|xi , xj , g) = ?[g(xi , xj )y] .
(2)
Since g is obtained from f through a linear operation, the GP prior on f induces a GP prior on g. The
covariance function kpref of the GP prior on g can be computed from the covariance function k of the
GP on f as kpref ((xi , xj ), (xk , xl )) = k(xi , xk )+k(xj , xl )?k(xi , xl )?k(xj , xk ). The derivations
can be found in Section 1 of the supplementary material. We call kpref the preference kernel. The
same kernel function can be derived from a large margin classification viewpoint [6]. However, to
our knowledge, the preference kernel has not been used previously for GP-based models.
The combination of (2) with a GP prior based on the preference kernel allows us to transform the
pairwise preference learning problem into binary classification with GPs. This means that state-ofthe-art methods for GP binary classification, such as expectation propagation [14], can be applied
directly to preference learning. Furthermore, the simplified likelihood (2) allows us to implement
complex methods such as the multi-user approach which is described in the following section.
3
Multi-user preference learning
Consider I items with feature vectors xi ? X for i = 1, . . . , I. The single-user learning approach
assumes an independent latent function for the u-th user, gu : X 2 7? R. Our approach to the multiuser problem is to assume common structure in the user latent functions. In particular, we assume a
set of D shared latent functions, hd : X 2 7? R for d = 1, . . . , D, such that the user latent functions
are generated by a linear combination of these functions, namely
gu (xj , xk ) =
D
X
d=1
wu,d hd (xj , xk ) ,
(3)
here wu,d ? R is the weight given to function hd for user u. We place a GP prior over the shared
latent functions h1 , . . . , hD using the preference kernel described in the previous section. This
model allows the preferences of the different users to share some common structure represented by
the latent functions h1 , . . . , hD . This approach is similar to dimensionality reduction methods that
are commonly used for addressing collaborative filtering problems [19, 16].
2
We may extend this model further to the case in which, for each user u, there is a feature vector
uu ? U containing information that might be useful for prediction. We denote by U the set of all
the users? feature vectors, that is, U = {u1 , . . . , uU }. The user features are incorporated now by
placing a separate GP prior over the users weights. In particular, we replace the scalars wu,d in (3)
with functions wd0 (uu ) : U ? R. These weight functions describe the contribution of shared latent
function hd to the user latent function gu as a function of the user feature vector uu .
In the multi-user setting we are given a list L = {p1 , . . . , pP } with all the pairs of items evaluated
by the users, where P ? I(I ? 1)/2 (the maximum number of pairs). The data consists of L, the
sets of feature vectors for the users U (if available), the item features X = {x1 , . . . , xI }, and U
u U
sets of preference judgements, one for each user, D = {{zu,i , yu,i }M
i=1 }u=1 , where zu,i indexes the
i-th pair evaluated by user u, yi,u = 1 if this user prefers the first item in the pair to the second and
yi,u = ?1 otherwise. Mu is the number of preference judgements made by the u-th user.
3.1
Probabilistic description
To address the task of predicting preference on unseen item pairs we cast the model into a probabilistic framework. Let G be an U ?P ?user-function? matrix, where each row corresponds to a particular
user?s latent function, that is, the entry in the u-th column and i-th row is gu,i = gu (x?(i) , x?(i) ) and
?(i) and ?(i) denote respectively the first and second item in the i-th pair from L. Let H be a D ? P
?shared-function? matrix, where each row represents the shared latent functions, that is, the entry in
the d-th row and i-th column is hd,i = hd (x?(i) , x?(i) ). Finally, we introduce the U ? D weight
matrix W such that each row contains a user?s weights, that is, the entry in the u-th row and d-th
column of this matrix is wd0 (uu ). Note that G = WH represents equation (3) in matrix form. Let
T be the U ? P target matrix given by T = sign[G + E], where E is an U ? P noise matrix whose
entries are sampled i.i.d. from a standard Gaussian distribution and the function ?sign? retains only
u U
the sign of the elements in a matrix. The observations yu,i in D = {{zu,i , yu,i }M
i=1 }u=1 are mapped
(D)
(D)
to the corresponding entries of T using tu,zu,i = yu,i . Let T
and G
represent the elements of
T and G corresponding only to the available observations yu,i in D. Then, the likelihood for G(D)
given T(D) and conditional distribution for G(D) given H and W are
P(T(D) |G(D) ) =
Mu
U Y
Y
?[tu,zu,i gu,zu,i ] and P(G(D) |W, H) =
u=1 i=1
Mu
U Y
Y
?[gu,zu,i ? wu h?,zu,i ]
u=1 i=1
respectively, where wu is the u-th row in W, h?,i is the i-th column in H and ? represents a point
probability mass at zero. We now select the priors for W and H. We assume that each function
0
w10 , . . . , wD
is sampled a priori from a GP with zero mean and specific covariance function. Let
Kusers be the U ? U covariance matrix for entries in each column of matrix W. Then
P(W|U) =
D
Y
N (w?,d |0, Kusers ) ,
(4)
d=1
where w?,d is the d-th column in W. If user features are unavailable, Kusers becomes the identity
matrix. Finally, we assume that each shared latent function h1 , . . . , hD is sampled a priori from a
GP with zero mean and covariance function given by a preference kernel. Let Kitems be the P ? P
preference covariance matrix for the item pairs in L. The prior for H is then
P(H|X, L) =
D
Y
N (hj |0, Kitems ) ,
(5)
j=1
where hj is the j-th row in H. The resulting posterior for W, H and G(D) is
P(W, H, G(D) |T(D) , X, L) =
P(T(D) |G(D) )P(G(D) |W, H)P(W|U)P(H|X, L)
.
P(T(D |X, L)
(6)
Given a new item pair pP +1 , we can compute the predictive distribution for the preference of the
u-th user (1 ? u ? U ) on this pair by integrating
out the parameters H, W and G(D) as follows:
Z
P(tu,P +1 |T(D) , X, L, pP +1 ) =
P(tu,P +1 |gu,P +1 )P(gu,P +1 |wu , h?,P +1 )
P(h?,P +1 |H, X, L, pP +1 )P(H, W, G(D) |T(D) , X, L) dH dW dG(D) ,
(7)
where P(tu,P +1 |gu,P +1 ) = ?[tu,P +1 gu,P +1 ], P(gu,P +1 |wu , h?,P +1 ) = ?[gu,P +1 ? wu h?,P +1 ],
P(h?,P +1 |H, X, L, pP +1 ) =
D
Y
T ?1
N (hd,P +1 |kT? K?1
items hd , k? ? k? Kitems k? )
d=1
3
(8)
}
}
Figure 1: Toy example with 1D input. Circles
and crosses denote labelled data. The plot
shows the mean and variance of the GP predictive distribution. Maximum Entropy Sampling (MES) samples from the region of highest marginal uncertainty, ignoring the second
term in (10). BALD samples from the region
of greatest uncertainty in the latent function.
k? is the prior variance of hd (x?(P +1) , x?(P +1) ) and k? is a P -dimensional vector that contains
the prior covariances between hd (x?(P +1) , x?(P +1) ) and hd (x?(1) , x?(1) ), . . . , hd (x?(P ) , x?(P ) ).
Computing (6) or (8) is infeasible and approximations must be used. For this, we use a combination
of expectation propagation (EP) [14] and variation Bayes (VB) [7]. Empirical studies show that EP
obtains state-of-the-art performance in the related problem of GP binary classification [15].
We want to learn user preferences with the proposed model from the least amount of data possible.
Therefore we desire to query users actively about their preferences on the most informative pairs of
items [3]. Next, we describe a novel method to implement this strategy. This method exploits the
preference kernel and so may be trivially generalized to GP binary classification problems also.
4
Bayesian active learning by disagreement
The goal of active learning is to choose item pairs such that we learn the preference functions for the
users using minimal data. Information theoretic approaches to active learning are popular because
they do not require prior knowledge of loss functions or test domains. The central goal is to identify the new data point that maximizes the expected reduction in posterior entropy. For preference
learning (see Section 2), this implies choosing the new item features xi and xj that maximize
H[P(g|D)] ? E
[H[P(g|y, x , x , D)]] ,
(9)
i
j
P(y|xi ,xj ,D)
R
where D are the user preferences observed so far and H[p(x)] = ? p(x) log p(x) dx represents
the Shannon entropy. This framework, originally proposed in [10], is difficult to apply directly to
models based on GPs. In these models, entropies can be poorly defined or their computation can
be intractable. In practice, current approaches make approximations for the computation of the
posterior entropy [12, 9]. However, a second difficulty arises; if n new data points are available for
selection, with |{?1, 1}| = 2 possible values for y. Then O(2n) potentially expensive posterior
updates are required to find the maximizer of (9): one for every available feature vector and possible
class value. This is often too expensive in practice.
A solution consists in noting that (9) is equivalent to the conditioned mutual information between y
and g. Using this we can rearrange this equation to compute entropies in y space:
H[P(y|xi , xj , D)] ? EP(g|D) [H [P(y|xi , xj , g)]] .
(10)
This overcomes the previous challenges. Entropies are now evaluated in output space, which has low
dimension. Furthermore, g is now conditioned only upon D, so only O(1) updates of the posterior
distribution are required. We only need to recompute the posterior once per data point selected, not
for every possible data point under consideration. Expression (10) also provides us with an intuition
about the objective; we seek the xi and xj for which a) the model is marginally uncertain about
y (high H[P(y|xi , xj , D)]) and b) conditioned on a particular value of g the model is confident
about y (low EP(g|D) [H[P(y|xi , xj , g])]). This can be interpreted as seeking the pair xi and xj
for which the latent functions g, under the posterior, ?disagree? with each other the most about the
outcome, that is, the preference judgement. Therefore, we refer to this objective as Bayesian Active
Learning by Disagreement (BALD). This method is independent of the approach used for inference,
something which does not hold for the techniques described in [12, 8, 9]. In the following section
we show how (10) can be applied to binary classification with GPs, and hence via the preference
kernel also to any preference learning problem.
4.1
BALD in binary classification with GPs
Most approximate inference methods for the problem of binary classification with GPs produce
a Gaussian approximation to the posterior distribution of f , the latent function of interest. In
4
the binary GP classifier, the entropy of y given the corresponding value of f can be expressed
in terms of the binary entropy function, h[f ] = ?f log f ? (1 ? f ) log(1 ? f ). In particular,
H[p(y|x, f )] = h [?(f (x)]. When a Gaussian is used to approximate the posterior of f , we have
that for each x, fx = f (x) will follow a Gaussian distribution with mean ?x and variance ?x2 . The
first
D)],
can be handled
analytically in this case: H[p(y|x, D)] ?
R term in (10), that is, H[p(y|x,
h ?(fx )N (fx |?x , ?x2 )dfx = h ? ?x (?x2 + 1)?1/2 , where ? represents here the Gaussian
approximation to the posterior of fx . The second term in (10), that is, Ep(f |D) [H[p(y|x, f )]], can
be approximated as Ep(f |D) [H[p(y|x, f )]] ? C(?x2 + C 2 )?1/2 exp ??2x (2 ?x2 + C 2 )?1 , where
p
C = ? log 2/2. This result is obtained by using the Gaussian approximation to the posterior of fx
and then approximating h[?(fx )] by the squared exponential curve exp(?fx2 /? log 2) (details can
be found in Section 3 of the supplementary material).
To summarize, the BALD algorithm for active binary GP classification / preference learning first
applies any approximate inference method to obtain the posterior mean ?x and variance ?x2 of f at
each point of interest
h x. Then, it selects
i the feature vector x that maximizes the objective
h ? ?x (?x2 + 1)?1/2
? C(?x2 + C 2 )?1/2 exp ??2x (2 ?x2 + C 2 )?1 .
(11)
BALD assigns a high value to the feature vector x when the model is both uncertain about the label
(?x close to 0) and there is high uncertainty about fx (?x2 is large). The second term prevents BALD
from sampling in regions where the model knows that the label is uncertain. Figure 1 illustrates
the differences between BALD and Maximum Entropy Sampling [17] (details in the supplementary
material, Section 5). MES considers only marginal uncertainty (the first term in (11)), and hence
seeks data in an uninformative region of the plot. By contrast, BALD samples data from the region
of greatest uncertainty in the latent function.
5
Expectation propagation and variational Bayes
Approximate inference in our model is implemented using a combination of expectation propagation
(EP) [13] and variational Bayes (VB) [7]. Here, we briefly describe the method, but full details are
in Section 4 of the supplementary. We approximate the posterior (6) by the parametric distribution
"
Q(W, H, G
(D)
)=
U Y
D
Y
#"
u=1 d=1
"
Mu
N Y
Y
D Y
P
Y
w
N (wud |mw
u,d , vu,d )
#
h
N (hd,i |mhd,i , vd,i
)
d=1 i=1
#
g
N (gu,zu,j |mgu,j , vu,j
) ,
(12)
u=1 j=1
g
g
w
h
h
where mw
u,d , vu,d , md,i , vd,i , mu,j , and vu,j are free parameters to be determined by EP
and the superscripts w, h and g indicate the random variables described by these parameters.
The joint distribution P(G(D) , W, H, T(D) , X, `) can be factorized into four factors f1 , . . . , f4 ,
Q4
(D)
namely, P(G(D) , W, H, T(D) , X, `) =
, W, H), where f1 (G(D) , W, H) =
a=1 fa (G
P(T(D) |G(D) ), f2 (G(D) , W, H) = P(G(D) |W, H), f3 (G(D) , W, H) = P(W|U) and
f4 (G(D) , W, H) = P(H|X, `). EP approximates these exact factors by approximate factors
f?1 (W, H, G(D) ), . . . , f?4 (W, H, G(D) ) that have the same functional form as Q
"
f?a (G(D) , W, H) =
U Y
D
Y
#"
a,w
N (wud |m
? a,w
?u,d
)
u,d , v
u=1 d=1
"
Mu
N Y
Y
D Y
P
Y
#
a,h
N (hd,i |m
? a,h
?d,i
)
d,i , v
d=1 i=1
#
a,g
N (gu,zu,j |m
? a,g
?u,j
) s?a ,
u,j , v
(13)
u=1 j=1
a,w
a,h
a,g
where a = 1, . . . , 4 and m
? a,w
?u,d
,m
? a,h
?d,i
,m
? a,g
?u,j
and s?a are free parameters. Note that
u,j , v
u,d , v
d,i , v
Q is the normalized product of f?1 , . . . , f?4 . The first step of EP is to initialize f?1 , . . . , f?4 and Q
to be uniform. After that, EP iteratively refines of f?1 , . . . , f?4 by minimizing the Kullback-Leibler
(KL) divergence between the product of Q\a and fa and the product of Q\a and f?a , where Q\a is
the ratio between Q and f?a . However, this does not perform well for refining f?2 ; details on this
problem can be found in Section 4 of the supplementary material and in [19]. For this factor we
follow a VB approach. Instead of minimizing KL(Q\2 f2 kQ\2 f?2 ) with respect to the parameters of
f?2 , we refine this approximate factor so that the reversed version of the KL divergence is minimized,
5
that is, we minimize KL(Q\2 f?2 kQ\2 f2 ). EP iteratively refines all the approximate factors until
convergence. This method also approximates the predictive distribution (7). For this, we replace the
exact posterior in (7) with Q. Finally, EP can also approximate the normalization constant in (6)
(the model evidence) as the integral of the product of all the approximate factors f?1 , . . . , f?4 .
5.1
A sparse approximation to speed up computation
The cost of GPs is cubic in the number of function evaluations. In our case, refining f?3 has cost
O(DU 3 ), where U is the number of users, and D the number of shared latent functions. The cost of
refining f?4 is O(DP 3 ), where P is the number of observed item pairs. These costs can be reduced
by approximating Kusers and Kitems in (4) and (5). We use the FITC approximation [18]. Under this
approximation, an n?n covariance matrix K generated by the evaluation of a covariance function at
T
n locations is approximated by K0 = Q+diag(K?Q), where Q = Knn0 K?1
n0 n0 Knn0 , Kn0 n0 is the
n0 ?n0 matrix generated by the evaluation of the covariance function at all possible combinations of
only n0 < n locations or pseudo-inputs and Knn0 is the n?n0 matrix with the covariances between
all possible combinations of original locations and pseudo-inputs. These approximations allow us
to refine f?3 and f?4 in O(DU02 U ) and O(DP02 P ) operations, where U0 and P0 are the number of
pseudo-inputs for the users and for the item pairs, respectively. A detailed description of the EP
updates based on the FITC approximation is given in Section 4.4 of the supplementary material.
6
Experiments and Discussion
The performance of our collaborative preference model with the BALD active learning strategy is
evaluated in a series of experiments with simulated and real-world data. The analyzed datasets
include a) synthetic data generated from the probabilistic model assumed by the proposed multiuser method (Synthetic), b) a collection of user preferences on different movies (MovieLens), c) the
number of votes obtained by different political parties in the 2010 UK general election (Election),
d) preferences of users about different types of sushi (Sushi), and finally, e) information regarding
the concentration of heavy metals in the Swiss Jura region (Jura). Section 6 in the supplementary
material contains a detailed description of these datasets.
6.1
Comparison with other multi-user methods
Alternative models. Two versions of the proposed collaborative preference (CP) model are used.
The first version (CPU) takes into account the available user features, as described in Section 3.
The second version (CP) ignores these features by replacing Kusers in (4) with the identity matrix.
The first multi-user method we compare to is the approach of Birlitiu et al. (BI) [1]. This method
does not use user features, and captures similarities between users with a hierarchical GP model.
In particular, a common GP prior is assumed for the preference function of each user; using this
prior the model learns the full GP posterior for each user. The second multi-user method is the
technique of Bonilla et al. (BO) [2]. In this model there exists one high-dimensional function
which depends on both the features of the two items to be compared and on the features of the
user who makes the comparison. Relationships between users? behaviors are captured only via
the user features. We implement BO and BI using the preference kernel and EP for approximate
inference1 . The computational costs of BO and BI are rather high; BO has cubic complexity in
PU
the total number of observations i.e. O(( u=1 Mu )3 ), our model (CPU) has a significantly lower
cost of O(D(U 3 + P 3 )) (before further speed-up from FITC). BI does not include user features,
but learns U GPs, so has complexity O(U P 3 ); the equivalent version of our model (CP) has cost
O(N P + DP 3 ), which is lower because D << U . More details about BI and BO are given in
sections 7 and 8 of the supplementary material. Finally, we consider a single user approach (SU)
which fits a different GP classifier independently to the data of each user.
1
Although this is not the same as the original implementations (sampling-based for BI, Laplace approximation for BO), the preference kernel and EP are likely to augment the performance of these algorithms, and
provides the fairest comparison of the underlying models.
6
Table 2: Training times (s) with 100 users.
Table 1: Average test error with 100 users.
Dataset
Synthetic
Sushi
MovieLens
Election
Jura
CPU
0.162
0.171
0.182
0.199
0.159
CP
0.180
0.163
0.166
0.123
0.153
BI
0.175
0.160
0.168
0.077
0.153
BO
0.157
0.266
0.302
0.401
0.254
Dataset
CPU
CP
BI
BO
SU
Synthetic
7.793 9.498 22.524 311.574 0.927
Sushi
5.694 4.307 20.028 215.136 0.817
MovieLens 5.313 4.013 19.366 69.048 0.604
Election
13.134 12.408 20.880 120.011 0.888
Jura
3.762 2.404 15.234 88.502 0.628
SU
0.226
0.187
0.217
0.300
0.181
Table 3: Test error for each method and active learning strategy with at most 1000 users.
Dataset
Synthetic
Sushi
MovieLens
Election
Jura
CPU-B CPU-E CPU-R
0.135
0.135
0.139
0.148
0.153
0.178
0.170
0.176
0.199
0.202
0.158
0.224
0.168
0.143
0.141
CP-B
0.153
0.144
0.163
0.097
0.138
CP-E
0.160
0.151
0.170
0.093
0.138
CP-R
0.173
0.176
0.195
0.151
0.169
SU-B
0.249
0.179
0.225
0.332
0.176
SU-E
0.259
0.197
0.235
0.346
0.166
SU-R
0.268
0.212
0.248
0.338
0.197
Experimental procedure. Due to the high computational cost of BI and BO, to compare to these
methods we must subsample the datasets, keeping only 100 users. The available data were split
randomly into training and test sets of item pairs, where the training sets contain 20 pairs per user in
Sushi, MovieLens and Election, 15 pairs in Jura and 30 in Synthetic. This was repeated 25 times to
obtain statistically meaningful results. In CPU and CP, we selected the number of latent functions
D to be 20 (see Table 6.1). In general, the proposed models, CPU and CP, are robust to over-fitting
and over-estimation of D does not harm predictive performance. Note that the Synthetic dataset is
generated using D = 5 and CPU and CP still obtain very good results using D = 20. This automatic
pruning of unnecessary degrees of freedom seems to be common in methods based on variational
Bayes [11]. We selected the kernel lengthscales to be equal to the median distance between feature
vectors. This leads to good empirical performance for most methods. An exception is BO, where the
kernel hyperparameters are tuned to some held-out data using automatic relevance determination. In
our model, we can also estimate the kernel lengthscales by maximizing the EP approximation of the
model evidence, as illustrated in Section 9 of the supplementary material. This alternative approach
can be used when it is necessary to fine tune the lengthscale parameters to the data. In CPU we
use U0 = 25 pseudo inputs for approximating Kusers . These pseudo inputs are selected randomly
from the set of available data points. Similarly, in CP and CPU, we use P0 = 25 pseudo inputs for
approximating Kitems , except in the Jura and Election datasets (which contain fewer items) where
we use P0 = 15. The results obtained are not sensitive to the number of pseudo inputs used, as long
as the number is not excessively low.
Results. Average test errors are shown in Table 1. Those highlighted in bold are statistically
different to those not highlighted (calculated using a paired t test). Overall, CP and CPU outperform
SU and BO, and breaks even with BI; the final result is notable as BI learns the full mean and
covariance structure across all users, ours uses only a few latent dimensions, which provides the key
to scaling to many more users. CP outperforms CPU in all cases except in the Synthetic dataset.
In the real-world datasets, users with similar features do not seem to have similar preferences and
so correlating behavior of users with similar features is detrimental. In this case, the unsupervised
learning of similarities in user preferences is more useful for prediction than the user features. This
also explains the poor overall results obtained by BO. Finally, running times in seconds are presented
in Table 2. The entries for BO do not include the time spent by this method to tune the kernel hyperparameters. CP and CPU are faster than BO and BI. The FITC approximation imposes a large
multiplicative constant in the cost of CP and CPU so for larger datasets the gains are much larger.
6.2
Active learning on large datasets
Here we evaluate the performance of BALD, in particular, we compare CPU, CP, and SU using
BALD (-B), Maximum Entropy Sampling (-E) and random sampling (-R). We now use all the available users from each dataset, with a maximum of 1000 users. For each user the available preference
data are split randomly into training, pool and test sets with 5, 35 and 5 data points respectively in
7
Sushi
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0
2
4
6
8
0.1
0
10
MovieLens
0.35
0.3
error
0.35
error
error
Synthetic
0.35
0.2
2
num samples
4
6
8
10
num samples
Election
0.35
0.3
error
0.3
error
4
6
8
10
CPU?B
CPU?E
CPU?R
CP?B
CP?E
CP?R
SU?B
SU?E
SU?R
0.35
0.25
0.2
0.25
0.2
0.15
0.15
0.1
4
2
Jura
0.4
2
0
num samples
0.4
0.45
0.05
0
0.25
6
num samples
8
10
0.1
0
2
4
6
8
10
num samples
Figure 2: Average test error for CPU, CP and SU, using the strategies BALD (-B), entropy (-E) and
random (-R) for active learning. For clarity, the curves for CPU are included only in the Synthetic
and Election datasets. The complete plots can be found in Section 10 of the supplementary material.
Synthetic, Sushi and MovieLens, 3, 22 and 3 data points in Election and 3, 15 and 3 data points
in Jura. Each method is fitted using the training sets and its performance is then evaluated on the
corresponding test sets. After this, the most informative data point is identified in each of the pool
sets. These data points are moved into the corresponding training sets and the process repeats until
10 of these active additions to the training sets have been completed. The entire process, including the dataset splitting is repeated 25 times. Figure 2 shows the learning curve for each method.
For clarity, the curve for CPU is included only for the Synthetic and Election datasets; in the other
datasets CPU is marginally outperformed by CP (see supplementary material, Section 10). Average
errors after 10 queries from the pool set of each user are summarized in Table 3. For each model
(CPU, CP and SU), the results of the best active learning strategy are highlighted in bold. The results of the best model/active learning strategy combination are underlined. Highlighted results are
statistically significant with respect to non-highlighted results according to a paired t test. BALD
always outperforms random sampling and usually outperforms or obtains equivalent performance
to MES. In particular, BALD significantly outperforms MES in 9 cases, while MES is better than
BALD in only 2 cases.
7
Conclusions
We have proposed a multi-user model that combines collaborative filtering methods with GP binary
preference modeling. We have shown that the task of learning user preferences can be recast as a
particular case of binary classification with GPs when a covariance function called the preference
kernel is used. We have also presented BALD, a novel active learning strategy for binary classification models with GPs. The proposed multi-user model with BALD performs favorably on simulated
and real-world data against single-user methods and existing approaches for multi-user preference
learning, whilst having significantly lower computational times than competing multi-user methods.
Acknowledgements
NH is a recipient of the Google Europe Fellowship in Statistical Machine Learning, and this research
is supported in part by this Google Fellowship. JMH is supported by Infosys Labs, Infosys Limited.
8
References
[1] A. Birlutiu, P. Groot, and T. Heskes. Multi-task preference learning with an application to
hearing aid personalization. Neurocomputing, 73(79):1177 ? 1185, 2010.
[2] Edwin V. Bonilla, Shengbo Guo, and Scott Sanner. Gaussian process preference elicitation. In
Advances in Neural Information Processing Systems 23, pages 262?270, 2010.
[3] E. Brochu, N. de Freitas, and A. Ghosh. Active preference learning with discrete choice data.
Advances in Neural Information Processing Systems 20, 20:409?416, 2007.
[4] W. Chu and Z. Ghahramani. Preference learning with Gaussian processes. In Proceedings of
the 22nd international conference on Machine learning, pages 137?144, 2005.
[5] M. De Gemmis, L. Iaquinta, P. Lops, C. Musto, F. Narducci, and G. Semeraro. Preference
learning in recommender systems. In ECML/PKDD-09 Workshop on Preference Learning,
2009.
[6] J. F?urnkranz and E. H?ullermeier. Preference learning. Springer-Verlag New York Inc, 2010.
[7] Z. Ghahramani and M. J. Beal. Advanced Mean Field Method?Theory and Practice, chapter
Graphical models and variational methods, pages 161?177. 2001.
[8] B. Krishnapuram, D. Williams, Y. Xue, A. Hartemink, L. Carin, and M. Figueiredo. On semisupervised classification. In Advances in neural information processing systems 17, pages
721?728, 2004.
[9] N.D. Lawrence, M. Seeger, and R. Herbrich. Fast sparse gaussian process methods: The
informative vector machine. Advances in Neural Information Processing Systems 15, 15:609?
616, 2002.
[10] D.V. Lindley. On a measure of the information provided by an experiment. The Annals of
Mathematical Statistics, 27(4):986?1005, 1956.
[11] D. J. C. MacKay. Local minima, symmetry-breaking, and model pruning in variational free
energy minimization. Available at http://www.inference.phy.cam.ac.uk/mackay/minima.pdf,
2001.
[12] D.J.C. MacKay. Information-based objective functions for active data selection. Neural computation, 4(4):590?604, 1992.
[13] T. Minka and J. Lafferty. Expectation-propagation for the generative aspect model. In Proceedings of the Eighteenth conference on Uncertainty in artificial intelligence, pages 352?359,
2002.
[14] Tom Minka. A family of algorithms for approximate Bayesian inference. PhD thesis, MIT,
2001.
[15] Hannes Nickisch and Carl Edward Rasmussen. Approximations for binary Gaussian process
classification. The Journal of Machine Learning Research, 9:2035?2078, 2008.
[16] T. Raiko, A. Ilin, and K. Juha. Principal component analysis for large scale problems with lots
of missing values. In Joost Kok, Jacek Koronacki, Raomon Mantaras, Stan Matwin, Dunja
Mladenic, and Andrzej Skowron, editors, Machine Learning: ECML 2007, volume 4701 of
Lecture Notes in Computer Science, pages 691?698. Springer Berlin / Heidelberg, 2007.
[17] P. Sebastiani and H.P. Wynn. Maximum entropy sampling and optimal Bayesian experimental
design. Journal of the Royal Statistical Society. Series B (Statistical Methodology), 62(1):145?
157, 2000.
[18] E. Snelson and Z. Ghahramani. Sparse gaussian processes using pseudo-inputs. In Advances
in Neural Information Processing Systems 18, 2005.
[19] D. H. Stern, R. Herbrich, and T. Graepel. Matchbox: large scale online bayesian recommendations. In Proceedings of the 18th international conference on World wide web, pages 111?120,
2009.
9
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4,090 | 4,701 | Iterative Ranking from Pair-wise Comparisons
Sewoong Oh
Department of IESE
University of Illinois at Urbana Champaign
[email protected]
Sahand Negahban
Department of EECS
Massachusetts Institute of Technology
[email protected]
Devavrat Shah
Department of EECS
Massachusetts Institute of Technology
[email protected]
Abstract
The question of aggregating pairwise comparisons to obtain a global ranking over
a collection of objects has been of interest for a very long time: be it ranking
of online gamers (e.g. MSR?s TrueSkill system) and chess players, aggregating
social opinions, or deciding which product to sell based on transactions. In most
settings, in addition to obtaining ranking, finding ?scores? for each object (e.g.
player?s rating) is of interest to understanding the intensity of the preferences.
In this paper, we propose a novel iterative rank aggregation algorithm for discovering scores for objects from pairwise comparisons. The algorithm has a natural
random walk interpretation over the graph of objects with edges present between
two objects if they are compared; the scores turn out to be the stationary probability of this random walk. The algorithm is model independent. To establish
the efficacy of our method, however, we consider the popular Bradley-Terry-Luce
(BTL) model in which each object has an associated score which determines the
probabilistic outcomes of pairwise comparisons between objects. We bound the
finite sample error rates between the scores assumed by the BTL model and those
estimated by our algorithm. This, in essence, leads to order-optimal dependence
on the number of samples required to learn the scores well by our algorithm. Indeed, the experimental evaluation shows that our (model independent) algorithm
performs as well as the Maximum Likelihood Estimator of the BTL model and
outperforms a recently proposed algorithm by Ammar and Shah [1].
1
Introduction
Rank aggregation is an important task in a wide range of learning and social contexts arising in
recommendation systems, information retrieval, and sports and competitions. Given n items, we
wish to infer relevancy scores or an ordering on the items based on partial orderings provided through
many (possibly contradictory) samples. Frequently, the available data that is presented to us is in
the form of a comparison: player A defeats player B; book A is purchased when books A and
B are displayed (a bigger collection of books implies multiple pairwise comparisons); movie A is
liked more compared to movie B. From such partial preferences in the form of comparisons, we
frequently wish to deduce not only the order of the underlying objects, but also the scores associated
with the objects themselves so as to deduce the intensity of the resulting preference order. For
example, the Microsoft TrueSkill engine assigns scores to online gamers based on the outcomes of
(pairwise) games between players. Indeed, it assumes that each player has inherent ?skill? and the
1
outcomes of the games are used to learn these skill parameters which in turn lead to scores associate
with each player. In most such settings, similar model-based approaches are employed.
In this paper, we have set out with the following goal: develop an algorithm for the above stated
problem which (a) is computationally simple, (b) works with available (comparison) data only and
does not try to fit any model per se, (c) makes sense in general, and (d) if the data indeed obeys
a reasonable model, then the algorithm should do as well as the best model aware algorithm. The
main result of this paper is an affirmative answer to all these questions.
Related work. Most rating based systems rely on users to provide explicit numeric scores for their
interests. While these assumptions have led to a flurry of theoretical research for item recommendations based on matrix completion [2, 3, 4], it is widely believed that numeric scores provided
by individual users are generally inconsistent. Furthermore, in a number of learning contexts as
illustrated above, it is simply impractical to ask a user to provide explicit scores.
These observations have led to the need to develop methods that can aggregate such forms of ordering information into relevance ratings. In general, however, designing consistent aggregation
methods can be challenging due in part to possible contradictions between individual preferences.
For example, if we consider items A, B, and C, one user might prefer A to B, while another prefers
B to C, and a third user prefers C to A. Such problems have been well studied as in the work by
Condorcet [5]. In the celebrated work by Arrow [6], existence of a rank aggregation algorithm with
reasonable sets of properties (or axioms) was shown to be impossible.
In this paper, we are interested in a more restrictive setting: we have outcomes of pairwise comparisons between pairs of items, rather than a complete ordering as considered in [6]. Based on those
pairwise comparisons, we want to obtain a ranking of items along with a score for each item indicating the intensity of the preference. One reasonable way to think about our setting is to imagine that
there is a distribution over orderings or rankings or permutations of items and every time a pair of
items is compared, the outcome is generated as per this underlying distribution. With this, our question becomes even harder than the setting considered by Arrow [6] as, in that work, effectively the
entire distribution over permutations was already known! Indeed, such hurdles have not stopped the
scientific community as well as practical designers from designing such systems. Chess rating systems and the more recent MSR TrueSkill system are prime examples. Our work falls precisely into
this realm: design algorithms that work well in practice, makes sense in general, and perhaps more
importantly, have attractive theoretical properties under common comparative judgment models.
With this philosophy in mind, in recent work, Ammar and Shah [1] have presented an algorithm that
tries to achieve the goal with which we have set out. However, their algorithm requires information
about comparisons between all pairs, and for each pair it requires the exact pairwise comparison
?marginal? with respect to the underlying distribution over permutations. Indeed, in reality, not all
pairs of items can typically be compared, and the number of times each pair is compared is also very
small. Therefore, while an important step is taken in [1], it stops short of achieving the desired goal.
In somewhat related work by Braverman and Mossel [7], the authors present an algorithm that
produces an ordering based on O(n log n) pair-wise comparisons on adaptively selected pairs. They
assume that there is an underlying true ranking and one observes noisy comparison results. Each
time a pair is queried, we are given the true ordering of the pair with probability 1/2 + ? for some
? > 0 which does not depend on the items being compared. One limitation of this model is that it
does not capture the fact that in many applications, like chess matches, the outcome of a comparison
very much depends on the opponents that are competing.
Such considerations have naturally led to the study of noise models induced by parametric distributions over permutations. An important and landmark model in this class is called the Bradley-TerryLuce (BTL) model [8, 9], which is also known as the Multinomial Logit (MNL) model (cf. [10]).
It has been the backbone of many practical system designs including pricing in the airline industry
[11]. Adler et al. [12] used such models to design adaptive algorithms that select the winner from
small number of rounds. Interestingly enough, the (near-)optimal performance of their adaptive
algorithm for winner selection is matched by our non-adaptive (model independent) algorithm for
assigning scores to obtain global rankings of all players.
Our contributions. In this paper, we provide an iterative algorithm that takes the noisy comparison
answers between a subset of all possible pairs of items as input and produces scores for each item
2
as the output. The proposed algorithm has a nice intuitive explanation. Consider a graph with
nodes/vertices corresponding to the items of interest (e.g. players). Construct a random walk on this
graph where at each time, the random walk is likely to go from vertex i to vertex j if items i and j
were ever compared; and if so, the likelihood of going from i to j depends on how often i lost to j.
That is, the random walk is more likely to move to a neighbor who has more ?wins?. How frequently
this walk visits a particular node in the long run, or equivalently the stationary distribution, is the
score of the corresponding item. Thus, effectively this algorithm captures preference of the given
item versus all of the others, not just immediate neighbors: the global effect induced by transitivity
of comparisons is captured through the stationary distribution.
Such an interpretation of the stationary distribution of a Markov chain or a random walk has been
an effective measure of relative importance of a node in wide class of graph problems, popularly
known as the network centrality [13]. Notable examples of such network centralities include the
random surfer model on the web graph for the version of the PageRank [14] which computes the
relative importance of a web page, and a model of a random crawler in a peer-to-peer file-sharing
network to assign trust value to each peer in EigenTrust [15].
The computation of the stationary distribution of the Markov chain boils down to ?power iteration?
using transition matrix lending to a nice iterative algorithm. Thus, in effect, we have produced an
algorithm that (a) is computationally simple and iterative, (b) is model independent and works with
the data only, and (c) intuitively makes sense. To establish rigorous properties of the algorithm, we
analyze its performance under the BTL model described in Section 2.1.
Formally, we establish the following result: given n items, when comparison results between randomly chosen O(npoly(log n)) pairs of them are produced as per an (unknown) underlying BTL
model, the stationary distribution produced by our algorithm (asymptotically) matches the true score
(induced by the BTL model). It should be noted that ?(n log n) is a necessary number of (random)
comparisons for any algorithm to even produce a consistent ranking (due to connectivity threshold
of random bipartite graph). In that sense, we will see that up to poly(log n) factor, our algorithm
is optimal in terms of sample complexity. Indeed, the empirical experimental study shows that the
performance of our algorithm is identical to the ML estimation of the BTL model. Furthermore, it
handsomely outperforms other popular choices including the algorithm by [1].
Some remarks about our analytic technique. Our analysis boils down to studying the induced stationary distribution of the random walk or Markov chain corresponding to the algorithm. Like most
such scenarios, the only hope to obtain meaningful results for such ?random noisy? Markov chain
is to relate it to stationary distribution of a known Markov chain. Through recent concentration of
measure results for random matrices and comparison technique using Dirichlet forms for characterizing the spectrum of reversible/self-adjoint operators, along with the known expansion property
of the random graph, we obtain the eventual result. Indeed, it is the consequence of such powerful
results that lead to near-optimal analytic results.
The remainder of this paper is organized as follows. In Section 2 we will concretely introduce our
model, the problem, and our algorithm. In Section 3 we will discuss our main theoretical results.
The proofs will be presented in Section 4.
Notation. We use C, C 0 , etc. to denote generic numerical constants. We use p
ATPto denote the
2
transpose of a matrix. The Euclidean norm of a vector is denoted by kxk =
i xi , and the
T
T
operator norm of a linear operator is denoted by kAk2 = maxx x Ax/x x. Also define [n] =
{1, 2, . . . , n} to be the set of all integers from 1 to n.
2
Model, Problem Statement, and Algorithm
We now present a concrete exposition of our underlying probabilistic model and our problem. We
then present our explicit random walk approach to ranking.
2.1
Bradley-Terry-Luce model for comparative judgment
In this section we discuss our model of comparisons between various items. As alluded to above,
for the purpose of establishing analytic properties of the algorithm, we will assume comparisons are
3
governed by the BTL model of pairwise comparisons. However, the algorithm itself operates with
data generated in arbitrary manner.
To begin with, there are n items of interest, represented as [n] = {1, . . . , n}. We shall assume that
for each item i ? [n] that there is an associated weight score wi ? R+ (i.e. it?s a strictly positive
real number). Hence, we may consider the vector w ? Rn+ to be the associated weight vector of
all items. Given a pair of items i and j we will let Yijl be 1 if j is preferred over i and 0 otherwise
during the lth competition for 1 ? l ? k, where k is the total number of competitions for the pair.
Under the BTL model we assume that
wj
P(Yijl = 1) =
.
(1)
wi + wj
l
Furthermore, conditioned on the score vector w we assume the the variables Yi,j
are independent
for all i, j, and l. We further assume that given some item i we will compare item j to i with
probability d/n. In our setting d will be poly-logarithmic in n. This model is a natural one to
consider because over a population of individuals the comparisons cannot be adaptively selected.
A more realistic model might incorporate selecting various items with different distributions: for
example, the Netflix dataset demonstrates skews in the sampling distribution for different films [16].
Thus, given this model our goal is to recover the weight vector w given such pairwise comparisons.
We now discuss our method for computing the scores wi .
2.2
Random walk approach to ranking
In our setting, we will assume that aij represents the fraction of times object j has been preferred to
object i, for example the fraction of times chess player j has defeated player i. Given the notation
Pk
above we have that aij = (1/k) l=1 Yijl . Consider a random walk on a weighted directed graph
G = ([n], E, A), where a pair (i, j) ? E if and only if the pair has been compared. The weight edges
are defined as the outcome of the comparisons Aij = aij /(aij + aji ) and Aji = aji /(aij + aji ).
We let Aij = 0 if the pair has not been compared. Note that by the Strong Law of Large Numbers,
as the number k ? ? the quantity Aij converges to wj /(wi + wj ) almost surely.
A random walk can be represented by a time-independent transition matrix P , where Pij =
P(XP
t+1 = j|Xt = i). By definition, the entries of a transition matrix are non-negative and satisfy j Pij = 1. One way to define a valid transition matrix of a random walk on G is to scale
all the edge weights by 1/dmax , where we define dmax as the maximum out-degree of a node. This
ensures that each row-sum is at most one. Finally, to ensure that each row-sum is exactly one, we
add a self-loop to each node. More concretely,
1
A
if i 6= j ,
Pdmax ij
Pij =
(2)
1
1 ? dmax k6=i Aik if i = j .
The choice to construct our random walk as above is not arbitrary. In an ideal setting with infinite
samples (k ? ?) the transition matrix P would define a reversible Markov chain. Recall that
a Markov chain is reversible if it satisfies the detailed balance equation: there exists P
v ? Rn+
n
such that vi Pij = vj Pji for all i, j; and in that case, ? ? R+ defined as ?i = vi /( j vj ) is
it?s unique stationary distribution. In the ideal setting (say k ? ?), we will have Pij ? P?ij =
(1/dmax )wj /(wi + wj ). That is, the random walk will move from state i to state j with probability
equal to the chance that item j is preferred to item i. In such a setting, it is clear that v = w satisfies
the reversibility
P conditions. Therefore, under these ideal conditions it immediately follows that the
vector w/ i wi acts as a valid stationary distribution for the Markov chain defined by P? , the ideal
matrix. Hence, as long as the graph G is connected and at least one node has a self loop then we
are guaranteed that our graph has a unique stationary distribution proportional to w. If the Markov
chain is reversible then we may apply the spectral analysis of self-adjoint operators, which is crucial
in the analysis when we repeatedly apply the operator P? .
In our setting, the matrix P is a noisy version (due to finite sample error) of the ideal matrix P?
discussed above. Therefore, it naturally suggests the following algorithm as a surrogate. We estimate the probability distribution obtained by applying matrix P repeated starting from any initial
condition. Precisely, let pt (i) = P(Xt = i) denote the distribution of the random walk at time t
4
with p0 = (p0 (i)) ? Rn+ be an arbitrary starting distribution on [n]. Then,
pTt+1 = pTt P .
(3)
Regardless of the starting distribution, when the transition matrix has a unique top eigenvalue, the
random walk always converges to a unique distribution: the stationary distribution ? = limt?? pt .
In linear algebra terms, this stationary distribution ? is the top left eigenvector of P , which makes
computing ? a simple eigenvector computation. Formally, we state the algorithm, which assigns
numerical scores to each node, which we shall call Rank Centrality:
Rank Centrality
Input: G = ([n], E, A)
Output: rank {?(i)}i?[n]
1: Compute the transition matrix P according to (2);
2: Compute the stationary distribution ?.
The stationary distribution of the random walk is a fixed point of the following equation:
X
Aji
?(j) P
?(i) =
.
` Ai`
j
This suggests an alternative intuitive justification: an object receives a high rank if it has been
preferred to other high ranking objects or if it has been preferred to many objects.
One key question remains: does P have a well defined stationary distribution? As discussed earlier, when G is connected, the idealized transition matrix P? has stationary distribution with desired
properties. But due to noise, P may not be reversible and the arguments of ideal P? do not apply to
our setting. Indeed, it is the finite sample error that governs the noise. Therefore, by analyzing the
effect of this noise (and hence the finite samples), it is likely that we can obtain the error bound on
the performance of the algorithm. As an important contribution of this work, we will show that even
the iterations (cf. (3)) induced by P are close enough to those induced by P? . Subsequently, we can
guarantee that the iterative algorithm will converge to a solution that is close to the ideal stationary
distribution.
3
Main Results
Our main result, Theorem 1, provides an upper bound on estimating the stationary distribution given
the observation model presented above. The results demonstrate that even with random sampling
we can estimate the underlying score with high probability with good accuracy. The bounds are
presented as the rescaled Euclidean norm between our estimate ? and the underlying stationary distribution P? . This error metric provides us with a means to quantify the relative certainty in guessing
if one item is preferred over another. Furthermore, producing such scores are ubiquitous [17] as they
may also be used to calculate the desired rankings. After presenting our main theoretical result we
will then provide simulations demonstrating the empirical performance of our algorithm in different
contexts.
3.1
Error bound in stationary distribution recovery via Rank Centrality
The theorem below presents our main recovery theorem under the sampling assumptions described
above. It is worth noting that while the result presented below is for the specific sampling model
described above. The results can be extended to general graphs as long as the spectral gap of the
corresponding Markov chain is well behaved. We will discuss the point further in the sequel.
Theorem 1. Assume that, among n items, each pair is chosen with probability d/n and for each
chosen pair we collect the outcomes of k comparisons according to the BTL model. Then, there exists
positive universal constants C, C 0 , and C 00 such that when d ? C(log n)2 , and k d ? Cb5 log n,
the following bound on the error rate holds with probability at least 1 ? C 00 /n3 :
r
? ? ?
?
log n
0 3
? Cb
,
k?
?k
kd
P
where ?
? (i) = wi / ` w` and b ? maxi,j wi /wj .
5
Remarks. Some remarks are in order. First, the above result implies that as long as we choose
d = ?(log2 n) and k = ?(1) (i.e. large
? enough, say k = ?(log n)), the error goes to 0 (with
k = ?(log
n),
it
goes
down
at
rate
1/
log n) as n increases. Since we are sampling each of
the n2 pairs with probability d/n and then sampling them k times, we obtain O(n log3 n) (with
k = ?(log n)) comparisons in total. Due to classical results on Erdos-Renyi graphs, the induced
graph G is connected with high probability only when total number of pairs sampled scales as
?(n log n)?we need at least those many comparisons. Thus, our result can be sub-optimal only up
to log2 n (log1+ n if k = log n).
Second, the b parameter should be treated as constant. It is the dynamic range in which we are trying
to resolve the uncertainty between scores. If b were scaling with n, then it would be really easy to
differentiate scores of items that are at the two opposite end of the dynamic range; in which case one
could focus on differentiating scores of items that have their parameter values near-by. Therefore,
the interesting and challenging regime is where b is constant and not scaling.
Finally, observe the interesting consequence that under the conditions on d, since the induced distribution ? is close to ?
? , it implies connectivity of G. Thus, the analysis of our algorithm provides an
alternative proof of connectivity in an Erdos-Renyi graph (of course, by using heavy machinery!).
3.2
Experimental Results
Under the BTL model, define an error metric of a estimated ordering ? as the weighted sum of pairs
(i, j) whose ordering is incorrect:
n
X
o1/2
1
2
Dw (?) =
(w
?
w
)
I
(w
?
w
)(?
?
?
)
>
0
,
i
j
i
j
i
j
2nkwk2 i<j
where I(?) is an indicator function. This is a more natural error metric compared to the Kemeny
distance, which is an unweighted version of the above sum, since Dw (?) is less sensitive to errors
between pairs with
P similar weights. Further, assuming without loss of generality that w is normalized such that i wi = 1, the next lemma connects the error in Dw (?) to the bound provided in
Theorem 1. Hence, the same upper bound holds for Dw error. Due to space constraints, we refer to
a longer version of this paper for a proof of this lemma.
Lemma 3.1. Let ? be an ordering of n items induced by a scoring ?. Then, Dw (?) ? kw??k/kwk.
For a fixed n = 400 and a fixed b = 10, Figure. 1 illustrates how the error scales with two problem
parameters: varying the number of comparisons per pair with fixed d = 10 log n (left) and varying
the sampling probability with fixed k = 32 (right). The ML estimator directly maximizes the
likelihood assuming the BTL model [18]. If we reparameterize the problem so that ?i = log(wi )
then we obtain our estimates ?b by solving the convex program
?b ? arg min
?
k
X X
log(1 + exp(?j ? ?i )) ? Yijl (?j ? ?i ),
(i,j)?E l=1
which is pair-wise logistic regression. This choice is optimal in the asymptotic setting, however for
fixed-samples there do not exist theoretical guarantees for recovering the transformed scores ?i . The
method Count Wins scores an item by counting the number of wins divided by the total number of
comparisons [1]. Ratio Matrix assigns scores according to the top eigenvector of a matrix, whose
(i, j)-th entry is aij /aji [19]. As we see in Figure 1, the error achieved by ?
our Random Walk
approach is comparable to that of ML estimator, and vanishes at the rate of 1/ k as predicted by
our main result. Interestingly, for fixed d, both the Count Wins and Ratio Matrix algorithms have
strictly
? positive error even if we take k ? ?. The figure on the right illustrates that the error scales
as 1/ d as expected from our main result.
4
Proofs
We may now present the proof of Theorem 1. As previously alluded to the statement of Theorem 1
can be made more general. The result that we presented is a specific instance of a more general
6
0.1
0.1
Ratio Matrix
Count Wins
Rank Centrality
ML estimate
Ratio Matrix
Count Wins
Rank Centrality
ML estimate
0.01
Dw (?)
0.01
0.001
0.0001
0.001
1
10
100
0.1
k
1
d/n
Figure 1: Average error Dw (?) of orderings from four rank aggregation algorithms, averaged over 20 instances. In the figure on the right we assume that d and n are fixed while we increase k. The figure on the right
takes k = 32 fixed and lets d increase.
lemma that we state below, which shows that our algorithm enjoys convergence properties that
result in useful upper bounds. The lemma is made general and uses standard techniques of spectral
theory. The main difficulty arises in establishing that the Markov chain P satisfies certain properties
that we will discuss below. In order to show that these properties hold we must rely on the specific
model that allows us to ultimately establish error bounds that hold with high probability. In what
follows we present the lemma and omit the proofs of certain technical details to the longer version
of the paper.
4.1
Algorithm convergence
In this section, we characterize the error rate achieved by our ranking algorithm. Given the random
Markov chain P , where the randomness comes from the outcome of the comparisons, we will show
that it does not deviate too much from its expectation P? , where we recall is defined as
(
wj
1
if i 6= j ,
dmax wi +wj
?
P
Pij =
w`
1
1 ? dmax
if i = j
`6=i wi +w`
for all (i, j) ? E and P?ij = 0 otherwise.
Recall from the discussion following equation (2) that the transition matrix P used in our ranking
algorithm has been carefully chosen such that the corresponding expected transition matrix P? has
two important properties. First, the stationary distribution of P? , which we denote with ?
? is proportional to the weight vectors w. Furthermore, when the graph is connected and has self loops (which
at least one exists), this Markov chain is irreducible and aperiodic so that the stationary distribution
is unique. The next important property of P? is that it is reversible? ?
? (i)P?ij = ?
? (j)P?ji . This obser?
vation implies that the operator P is symmetric in an appropriate defined inner product space. The
symmetry of the operator P? will be crucial in applying ideas from spectral analysis to prove our
main results.
Let ? denote the fluctuation of the transition matrix around its mean, such that ? ? P ? P? . The
following lemma bounds the deviation of the Markov chain after t steps in terms of two important
quantities: the spectral radius of the fluctuation k?k2 and the spectral gap 1 ? ?max (P? ), where
?max (P? ) ? max{?2 (P? ), ??n (P? )} .
Lemma 4.1. For any Markov chain P = P? + ? with a reversible Markov chain P? , let pt be the
distribution of the Markov chain P when started with initial distribution p0 . Then,
r
pt ? ?
?
?k ?
?max
1
?
?max
t kp0 ? ?
? ?
+
k?k2
.
(4)
k?
?k
k?
?k
?
?min
1??
?
?min
where ?
? is the stationary
distribution of P? , ?
?min = mini ?
? (i), ?
?max = maxi ?
? (i), and ? =
p
?
?max (P ) + k?k2 ?
?max /?
?min .
7
The above result provides a general mechanism for establishing error bounds between an estimated
stationary distribution ? and the desired stationary distribution ?
? . It is worth noting that the result
only requires control on the quantities k?k2 and 1 ? ?. We may now state two technical lemmas
that provide control on the quantities k?k2 and 1 ? ?, respectively.
Lemma 4.2. Under the assumptions of Theorem 1, we have that the error matrix ? = P ? P?
satisfies
r
log n
k?k2 ? C
kd
for some positive universal constant C with probability at least 1 ? 3n?4
The next lemma provides our desired bound on 1 ? ?.
Lemma 4.3. Under the assumptions of Theorem 1, the spectral radius satisfies
1 ? ? ? C 0 /b2
with probability at least 1 ? n?c , for some positive universal constant C 0 and c. The constant c can
be made as large as we want by increasing the constant C in d ? C log n.
With the above results in hand we may now proceed with the proof of Theorem 1.
When there is a positive spectral gap ? < 1 the first term in (4) vanishes as t grows. The rest of the
first term is bounded and independent of t. Formally, we have
?
?
?max /?
?min ? b , k?
? k ? 1/ n , and kp0 ? ?
?k ? 2 ,
P
by the assumption that maxi,j wi /wj ? b and the fact that ?
? (i) = wi /( j wj ). Hence, the error
between the distribution at the tth iteration pt and the true stationary distribution ?
? is dominated by
the second term in equation (4). Therefore, in order to finish the proof of Theorem 1 we require
bounds on k?k2 and 1 ? ?.
p
We recall that by Lemma 4.2 we have k?k2 ? C log n/(kd) and from Lemma 4.3 that there
is a positive spectral gap 1 ? ? ? C 0 /b2 for some numerical constants C and C 0 . Given these
observations the dominant second term in equation (4) is bounded by
r
pt ? ?
?
log n
3
lim
? Cb
.
t??
k?
?k
kd
This finishes the proof of Theorem 1.
5
Discussion
In this paper, we developed a novel iterative rank aggregation algorithm for discovering scores of
objects given pairwise comparisons. The algorithm has a natural random walk interpretation over
the graph of objects with edges present between two objects if they are compared; the scores turn out
to be the stationary probability of this random walk. In lieu of recent works on network centrality
which are graph score functions primarily based on random walks, we call this algorithm Rank
Centrality. The algorithm is model independent.
We also established the efficacy of the algorithm by analyzing its performance when data is generated as per the popular Bradley-Terry-Luce (BTL) model. We have obtained an analytic bound
on the finite sample error rates between the scores assumed by the BTL model and those estimated
by our algorithm. As shown, these lead to order-optimal dependence on the number of samples required to learn the scores well by our algorithm. The experimental evaluation show that our (model
independent) algorithm performs as well as the Maximum Likelihood Estimator of the BTL model
and outperforms other known competitors included the recently proposed algorithm by Ammar and
Shah [1]. Given the simplicity of the algorithm, analytic guarantees and wide utility of the problem
of rank aggregation, we believe that this algorithm will be of great practical value.
8
References
[1] A. Ammar and D. Shah. Communication, control, and computing (allerton), 2011, 49th annual
allerton conference on. pages 776?783, September 2011.
[2] E. J. Cand`es and B. Recht. Exact matrix completion via convex optimization. Foundations of
Computational Mathematics, 9(6):717?772, 2009.
[3] R.H. Keshavan, A. Montanari, and S. Oh. Matrix completion from a few entries. Information
Theory, IEEE Transactions on, 56(6):2980 ?2998, june 2010.
[4] S. Negahban and M. J. Wainwright. Restricted strong convexity and (weighted) matrix completion: Optimal bounds with noise. Journal of Machine Learning Research, 2012. To appear;
posted at http://arxiv.org/abs/1009.2118.
[5] M. Condorcet. Essai sur l?application de l?analyse a` la probabilit?e des d?ecisions rendues a` la
pluralit?e des voix. l?Imprimerie Royale, 1785.
[6] K. J. Arrow. Social Choice and Individual Values. Yale University Press, 1963.
[7] M. Braverman and E. Mossel. Noisy sorting without resampling. In Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms, SODA ?08, pages 268?276.
Society for Industrial and Applied Mathematics, 2008.
[8] R. A. Bradley and M. E. Terry. Rank analysis of incomplete block designs: I. the method of
paired comparisons. Biometrika, 39(3/4):324?345, 1955.
[9] D. R. Luce. Individual Choice Behavior. Wiley, New York, 1959.
[10] D. McFadden. Conditional logit analysis of qualitative choice behavior. Frontiers in Econometrics, pages 105?142, 1973.
[11] K. T. Talluri and G. Van Ryzin. The Theory and Practice of Revenue Management. springer,
2005.
[12] M. Adler, P. Gemmell, M. Harchol-Balter, R. M. Karp, and C. Kenyon. Selection in the
presence of noise: the design of playoff systems. In Proceedings of the fifth annual ACMSIAM symposium on Discrete algorithms, SODA ?94, pages 564?572. Society for Industrial
and Applied Mathematics, 1994.
[13] M. E. J. Newman. Networks: An Introduction. Oxford University Press, 2010.
[14] S. Brin and L. Page. The anatomy of a large-scale hypertextual web search engine. In Seventh
International World-Wide Web Conference (WWW 1998), 1998.
[15] S. D. Kamvar, M. T. Schlosser, and H. Garcia-Molina. The eigentrust algorithm for reputation
management in p2p networks. In Proceedings of the 12th international conference on World
Wide Web, WWW ?03, pages 640?651, New York, NY, USA, 2003. ACM.
[16] R. Salakhutdinov and N. Srebro. Collaborative filtering in a non-uniform world: Learning with
the weighted trace norm. Technical Report abs/1002.2780v1, Toyota Institute of Technology,
2010.
[17] J. C. Duchi, L. Mackey, and M. I. Jordan. On the consistency of ranking algorithms. In
Proceedings of the ICML Conference, Haifa, Israel, June 2010.
[18] L. R. Ford Jr. Solution of a ranking problem from binary comparisons. The American Mathematical Monthly, 64(8):28?33, 1957.
[19] T. L. Saaty. Decision-making with the ahp: Why is the principal eigenvector necessary. European Journal of Operational Research, 145:pp. 85?91, 2003.
9
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4,091 | 4,702 | Probabilistic n-Choose-k Models for Classification
and Ranking
Kevin Swersky
Daniel Tarlow
Dept. of Computer Science
University of Toronto
[kswersky,dtarlow]@cs.toronto.edu
Richard S. Zemel
Dept. of Computer Science
University of Toronto
[email protected]
Ryan P. Adams
School of Eng. and Appl. Sciences
Harvard University
[email protected]
Brendan J. Frey
Prob. and Stat. Inf. Group
University of Toronto
[email protected]
Abstract
In categorical data there is often structure in the number of variables that take on
each label. For example, the total number of objects in an image and the number of
highly relevant documents per query in web search both tend to follow a structured
distribution. In this paper, we study a probabilistic model that explicitly includes
a prior distribution over such counts, along with a count-conditional likelihood
that defines probabilities over all subsets of a given size. When labels are binary
and the prior over counts is a Poisson-Binomial distribution, a standard logistic
regression model is recovered, but for other count distributions, such priors induce
global dependencies and combinatorics that appear to complicate learning and
inference. However, we demonstrate that simple, efficient learning procedures
can be derived for more general forms of this model. We illustrate the utility of
the formulation by exploring applications to multi-object classification, learning
to rank, and top-K classification.
1
Introduction
When models contain multiple output variables, an important potential source of structure is the
number of variables that take on a particular value. For example, if we have binary variables indicating the presence or absence of a particular object class in an image, then the number of ?present?
objects may be highly structured, such as the number of digits in a zip code. In ordinal regression
problems there may be some prior knowledge about the proportion of outputs within each level. For
instance, when modeling scores assigned to papers submitted to a conference, this structure can be
due to instructions that reviewers assign scores such that the distribution is roughly uniform.
One popular model for multiple output classification problems is logistic regression (LR), in which
the class probabilities are modeled as being conditionally independent, given the features; another
popular approach utilizes a softmax over the class outputs. Both models can be seen as possessing
a prior on the label counts: in the case of the softmax model this prior is explicit that exactly one
is active. For LR, there is an implicit factorization in which there is a specific prior on counts; this
prior is the source of computational tractability, but also imparts an inductive bias to the model. The
starting observation for our work is that we do not lose much efficiency by replacing the LR counts
prior with a general prior, which permits the specification of a variety of inductive biases.
In this paper we present a probabilistic model of multiple output classification, the n-choosek model, which incorporates a distribution over the label counts, and show that computations needed
1
for learning and inference in this model are efficient. We develop applications of this model to diverse problems. A maximum-likelihood version of the model can be used for problems such as
multi-class recognition, in which the label counts are known at training time but only a prior distribution is known at test time. The model easily extends to ordinal regression problems, such as
ranking or collaborative filtering, in which each item is assigned to one of a small number of relevance levels. We establish a connection between n-choose-k models and ranking objectives, and
prove that optimal decision theoretic predictions under the model for ?monotonic? gain functions
(to be defined later), which include standard objectives used in ranking, can be achieved by a simple
sorting operation. Other problems can be modeled via direct maximization of expected gain. An
important aim in classification and information retrieval is to optimize expected precision@K. We
show that we can efficiently optimize this objective under the model and that it yields promising
results.
Overall, the result is a class of models along with a well-developed probabilistic framework for
learning and inference that makes use of algorithms and modeling components that are not often
used in machine learning. We demonstrate that it is a simple, yet expressive probabilistic approach
that has many desirable computational properties.
2
Binary n-Choose-k Model
We begin by defining the basic model under the assumption of binary output variables. In the
following section, we will generalize to the case of ordinal variables. The model inputs are x,
and ? is defined as ? = Wx, where W are the parameters. The model output is a vector of D
binary variables y ? Y = {0, 1}D . We will use subsets c ? {1, . . . , D} of variable indices and will
represent the value assigned to a subset of variables as y c . We will also make use of the notation c?
to mean the complement {1, . . . , D}\c. The generative procedure is then defined as follows:
? Draw k from a prior distribution p(k) over counts k.
? Draw k variables to take on label 1, where the probability of choosing subset c is given by
(
P
exp{ d?c ?d }
if |c| = k
Z
k (?)
p(y c = 1, y c? = 0 | k) =
,
(1)
0
otherwise
where ? = (?1 , . . . , ?D ) are parameters
that determine
individual variable biases towards
P
P
being off or on, and Zk (?) = y| P yd =k exp{ d ?d yd }. Under this definition Z0 = 1,
d
and p(0 | 0) = 1. This has been referred to as a conditional Bernoulli distribution [1].
Logistic regression can be viewed as an instantiation of this model, with a ?prior? distribution over
count values that depends on parameters ?. This is a forced interpretation, but it is useful in understanding the implicit prior over counts that is imposed when using LR. Specifically, if p(k) is defined
k (?)
as be a particular function of ? (known as a Poisson-Binomial distribution [2]): p(k; ?) = ZZ(?)
,
P
where Z(?) = k Zk (?), then the joint probability p(y, k; ?) becomes equivalent
to a LR model in
P
the following sense. Suppose we have a joint assignment of variables y and d yd = k, and p(k; ?)
is Poisson-Binomial, then
P
Zk (?) exp{ d?c ?d } Y exp{?d yd }
p(y, k; ?) = p(k; ?)p(y | k; ?) =
=
.
(2)
Z(?)
Zk (?)
1 + exp{?d }
d
Note that the last equality factorizes Z(?) to create independence across variables, but it requires
that the ?prior? be defined in terms of parameters ?. Our interest in this paper is in the more flexible
family of models that arise after breaking the dependence of the ?prior? on ?. First, we explore
treating p(k) as a prior in the Bayesian sense, using it to express prior knowledge about label counts;
later we will explore learning p(k) using separate parameters from ?. A consequence of these
decisions is that the distribution does not factorize. At this point, we have not made it clear that
these models can be learned efficiently, but we will show in the next section that this is indeed the
case.
2.1
Maximum Likelihood Learning
Our goal in learning is to select parameters so as to maximize the probability assigned to observed
data by the model. For notational simplicity in this section, we compute partial derivatives with
2
respect to ?, then it should be clear that these can be back-propagated to a model of ?(x; W). We
note that if this relationship is linear, and the objective is convex in terms of ?, then it will also be
convex in terms of W. The log-likelihood is as follows:
log p(y; ?) = log
D
X
p(k)p(y | k; ?) = log p(y |
k=0
=
X
X
yd ; ?) + ?
(3)
d
?d yd ? log ZPd yd (?) + ?,
(4)
d
where ? is a constant that is independent of ?. As is standard, if we
P are givenn multiple sets of
binary variables, {y n }N
n=1 , we maximize the sum of log probabilities
n log p(y ; ?). The partial
derivatives take a standard log-sum-exp form, requiring expectations Ep(yd |k=Pd0 yd0 ) [yd ].
D
A naive computation of this expectation would require summing over k=P
configurations.
y
d
d
However, there are more efficient alternatives: the dynamic programming algorithms developed
in the context of Poisson-Binomial distributions are applicable, e.g., the algorithm from [3] runs
in O(Dk) time. The basic idea is to compute partial sums along a chain that lays out variables yd in sequence. An alternative formulation of the dynamic program [4] can be made to
yield an O(D log2 D) algorithm by using a divide-and-conquer algorithm that employs Fast Fourier
Transforms (FFTs). These algorithms are quite general and can also be used to compute Zk values,
incorporate prior distributions over count values, and draw a sample of y values conditional upon
some k for the same computational cost [5]. We use the FFT tree algorithm from [5] throughout,
because it is most flexible and has best worst-case complexity.
2.2
Test-time Inference
Having learned a model, we would like to make test-time predictions. In Section 4.2, we will show
that optimal decision-theoretic predictions (i.e., that maximize expected gain) can be made in several
settings by a simple sorting procedure, and this will be our primary way of using the learned model.
However, here, we consider the task of producing a distribution over labels y, given ?(x). To draw
a joint sample of y values, we can begin by drawing k from p(k), then conditional on that k, use the
dynamic programming algorithm to draw a sample conditional on k.
To compute marginals, a simple strategy is to loop over each value of k and run dynamic programming conditioned on k, and then average the results weighted by the respective prior. For priors that
only give support to a small number of k values, this is quite efficient. An alternative approach is
to draw several samples of k from p(k), then for each sampled value, run dynamic programming to
compute marginals. Averaging these marginals can then be seen as a Rao-Blackwellized estimate.
Finally, it is possible to compute exact marginals for arbitrary p(k) in a single run of an O(D log2 D)
dynamic programming algorithm, but the simpler strategies were sufficient for our needs here, so
we do not pursue that direction further.
3
Ordinal n-Choose-k Model
An extension of the binary n-choose-k model can be developed in the case of ordinal data, where we
assume that labels y can take on one of R categorical labels, and where there is an inherent ordering
to labels R > R ? 1 > . . . > 1; each label represents a relevance label in a learning-to-rank setting.
Let kr represent the number of variables y that take on label r and define k = (kR , . . . , k1 ). The idea
in the ordinal case is to define a joint model over count variables k, then to reduce the conditional
distribution of p(y | k) to be a series of binary models. The generative model is defined as follows:
? Initialize all variables y to be unlabeled.
? Sample kR , . . . , k1 jointly from p(k).
? Repeat for r = R to 1:
? Choose a set cr of kr unlabeled variables y ?r and assign them relevance label r.
Choose subsets with probability equal to the following:
(
P
exp{ d?cr ?d }
if |cr | = kr
Z
r,k (?,y ?r )
p(y ?r,cr = 1, y ?r,?cr = 0 | kr ) =
,
(5)
0
otherwise
3
where we use the notation y ?r to represent all variables that are given a relevance
label less than or equal to r. Zr,k is similar to the normalization constant Zk that
appears in the binary
P model, but it is restricted to sum over y ?r instead of the full y:
Zr,kr (?, y ?r ) = y?r |(P 1{yd =r})=kr exp {?d ? 1 {yd = r}}.
d
Note that if R = D and p(k) specifies that kr = 1 for all r, then this process defines a Plackett-Luce
(PL) [6, 7, 8] ranking model. One interpretation of this model is as a ?group? PL model, where
instead of drawing individual elements in the generative process, groups of elements are drawn simultaneously. In this work, we focus on ranking with weak labels (R < D) which is more restrictive
than modeling distributions over permutations [9], where learning would require marginalizing over
all possible permutations consistent with the given labels. In this setting, inference in the ordinal
n-choose-k model is both exact and efficient.
3.1
Maximum Likelihood Learning
P
Let kr = d 1 {yd = r}. The log likelihood of parameters ? can be written as follows:
?
?
R
X
X
X
?
log
p(k)p(y | k; ?) =
?d ? log Zr,kr (?, y ?r )? + ?.
k?K
r=1
(6)
d:yd =r
Here, we see that learning decomposes into the sum of R objectives that are of the same form as
arise in the binary n-choose-k model. As before, the only non-trivial part of the gradient computation comes from the log-sum-exp term, but the required expectations that arise can be efficiently
computed using dynamic programming. In this case, R ? 1 calls are required.
3.2
Test-time Inference
The test-time inference procedure in the ordinal model is similar to the binary case. Brute force enumeration over k becomes exponentially more expensive as R grows, but for some priors where p(k)
has sparse support, this may be feasible. To draw samples of y, the main requirement is the ability to draw a joint sample of k from p(k). In the case that p(k) is a simple distribution such as
a multinomial, this can be done easily. It is P
also possible toQefficiently draw a joint sample if the
distribution over k takes the form p(k) = 1 { r kr = D} ? r p(kr ). That is, there is an arbitrary
but independent prior over each kr value, along with a single constraint that the chosen kr values
sum to exactly D. Given a sample of k, it is straightforward to sample y using R calls to dynamic
programming. To do so, begin by using the binary algorithm to sample kR variables to take on
value R. Then remove the chosen variables from the set of possible variables, and sample kR?1
variables to take on value R ? 1. Repeat until all variables have been assigned a value.
An alternative to producing marginal probabilities at test time is trying to optimize performance under a task-specific evaluation measure. The main motivation for the ordinal model is the learning to
rank problem [10], so our main interest is in methods that do well under such task-specific evaluation measures that arise in the ranking task. In Section 4.2, we show that we can make exact optimal
decision theoretic test-time predictions under the learning-to-rank gain functions without the need
for sampling.
4
4.1
Incorporating Gain
Training to Maximize Expected Top-K Classification Gain
One of the motivating applications for this model is the top-K classification (TKC) task. We formulate this task using a gain function, parameterized by a value K and a ?scoring vector? t, which is
assumed to be of the same dimension as y. The gain function stipulates that K elements of y are
chosen, (assigning a score of zero if some other number is chosen), and assigns reward for choosing
each element of y based on t. Specifically the gain function is defined as follows:
P
P
if d yd = K
d yd t d
GK (y, t) =
(7)
0
otherwise .
The same gain can be used for Precision@K, in which case the number of nonzero values in t is
unrestricted. Here, we focus on the case where t is binary with a single nonzero entry at index d? .
4
An interesting issue is what gain function should be used to train a model when the test-time evaluation metric is TKC, or Precision@K. Maximum likelihood training of TKC in this case of a single
target class could correspond to a version of our n-choose-k model in which p(k) is a spike at k = 1;
note that in this case the n-choose-k model is equivalent to a softmax over the output classes. An
alternative is to train using the same gain function used at test-time.
Here, we consider incorporating the TKC gain at training time for binary t with one nonzero entry,
training the model to maximize expected gain. Specifically, the objective is the following:
(
)
XX
X
X
X
Ep [GK (y, t)] =
yd = K
yd td =
p(K)p(y | K)yd? (8)
p(k)p(y | k)1
k
y
d
d
y
It becomes clear that this objective is equivalent to the marginal probability of yd? under a prior
distribution that places all its mass on k = K. In Section 5.3, we empirically investigate training
under expected gain versus training under maximum likelihood
4.2
Optimal Decision-theoretic Predictions for Monotonic Gain Functions
We now turn attention to gain functions defined on rankings of items. Letting ? be a permutation,
we define a ?monotonic? gain function as follows:
Definition 1. A gain function G(?, r) is a monotonic ranking gain if:
PD
? It can be expressed as d=1 ?d f (r?d ), where ?d is a weighting (or discount) term, and
?d is the index of the item ranked in position d,
? ?d ? ?d+1 ? 0 for all d, and
? f (r) ? f (r ? 1) ? 0 for all r ? r0 .
It is straightforward to see that popular learning-to-rank scoring functions like normalized discounted cumulative gain (NDCG) and Precision@K are monotonic ranking gains.
P 2r?d ?1
1
N DCG(?, r) ? d log
, so set ?d = ? ? log (1+d)
and f (r) = 2r ? 1. We define Preci2 (1+d)
2
sion@K gain toPbe the fraction of documents in the top K produced ranks that have label R:
P @K(?, r) = d 1 {d ? K} 1 {r?d = R}, so set ?d = 1 {d ? K} and f (r) = 1 {r = R}.
The expected gain under a monotonic ranking gain and ordinal n-choose-k model is
Ep [G(?)] =
X
y 0 ?Y
p(y 0 )
D
X
?d f (y?0 d ) =
d=1
where we have defined gd =
D
X
?d
r=1
f (y?0 d )p(y?d = y?0 d ) =
0 =1
y?
d=1
PR
R
X
d
D
X
?d g?d ,
(9)
d=1
f (r)p(yd = r).
We now state four propositions and a lemma. The proofs of the propositions mostly result from
algebraic manipulation, so we leave their proof to the supplementary materials. The main theorem
will be proved afterwards.
Proposition 1. If ?i ? ?j , then p(yi = R) ? p(yj = R).
Proposition 2. If ?i ? ?j and p(yi ? r) ? p(yj ? r), then p(yi ? r ? 1) ? p(yj ? r ? 1).
Lemma 1. If ?i ? ?j , then for all r, p(yi ? r) ? p(yj ? r).
Proof. By induction. Proposition 1 is the base case, and Proposition 2 is the inductive step.
Proposition 3. If ?i ? ?j and f is defined as in Definition 1, then gi ? gj .
Proposition 4. Consider two pairs of non-negative real numbers ai , aj and bi , bj where ai ? aj
and bi ? bj . It follows that ai bi + aj bj ? ai bj + aj bi .
Theorem 1. Under an ordinal n-choose-k model, the optimal decision theoretic predictions for a
monotonic ranking gain are made by sorting ? values.
5
Figure 1: Four example images from the embedded MNIST dataset test set, along with the PoissonBinomial distribution produced by logistic regression for each image. The area marked in red has
zero probability under the data distribution, but the logistic regression model is not flexible enough
to model it.
Proof. Without loss of generality, assume that we are given a vector ? corresponding to placing the
??s in descending order and a vector g ? where ? is some arbitrary ordering of the g?s. The goal now
is to find the ordering ? ? that maximizes the objective given in (9) which is equivalently expressed
as the inner product ?T g ? .
Assume that we are given an ordering ?
? where for at least one pair i, j where i > j, we have
that ???i < ???j . Furthermore, assume that this ordering is optimal. That is, ?
? = ? ? . By Proposition 3 we have that g??i < g??j . The contributions of these elements to the overall objective is given
by ?i g??i + ?j g??j . By Proposition 4 we improve the objective by swapping ???i and ???j contradicting the assumption that ?
? is a local optimum.
If we have multiple elements that are not in sorted order, then we can repeat this argument by
considering pairs of elements until the whole vector is sorted.
5
Experiments
5.1
Modeling Varying Numbers of Objects
Our first experiment explores an issue that arises frequently in computer vision, where there are
an unknown number of objects in an image, but the number is highly structured. We developed a
multiple image dataset that simulates this scenario.1 To generate an image, we uniformly sampled
a count between 1 and 4, and then take that number of digit instances (with at most one instance
per digit class) from the MNIST dataset and embed them in a 60 ? 60 image. The x, y locations
are chosen from a 4 ? 4 uniformly spaced grid and and then a small amount of jitter is added. We
generated 10,000 images each for the training, test, and validation sets. The goal is to predict the set
of digits that appear in a given image. Examples can be seen in Figure 1.
We train a binary n-choose-k model on this dataset. The inputs to the model are features learned
from the images by a standard Restricted Boltzmann Machine with 1000 hidden units. As a baseline, we trained a logistic regression classifier on the features and achieved a test-set negative loglikelihood (NLL) of 2.84. Ideally, this model should learn that there are never more than four digits
in any image. In Figure 1, we show four test images, and the Poisson-Binomial distribution over
counts that arises from the logistic regression model. Marked in red are regions where there is zero
probability of the count value in the data distribution. Here it is clear that the implicit count prior
in LR is not powerful enough to model this data. As a comparison, we trained a binary n-choosek model where we explicitly parameterize and learn an input-dependent prior. The model learns the
correct distribution over counts and achieves a test-set NLL of 1.95. We show a visualization of the
learned likelihood and prior parameters in the supplementary material.
5.2
Ranking
A second set of experiments considers learning-to-rank applications of the n-choose-k model. We
report on comparisons to other ranking approaches, using seven datasets associated with the LETOR
3.0 benchmark [10]. Following the standard LETOR procedures, we trained over five folds, each
with distinct training, validation, and testing splits.
For each dataset, we train an ordinal n-choose-k model to maximize the likelihood of the data, where
each training example consists of a number of items, each assigned a particular relevance level; the
number of levels ranges from 2-4 across the datasets. At test time, we produce a ranking, which as
1
http://www.cs.toronto.edu/?kswersky/data/
6
0.4
0.9
0.8
NDCG@K
0.35
NDCG@K
Ordinal nCk
AdaRank!NDCG
FRank
ListNet
RankBoost
RankSVM
Regression!Reg
SmoothRank
0.3
0.25
0.2
0.7
0.6
0.5
1
2
3 4 5 6 7 8 9
NDCG Truncation Level (K)
10
0.4
1
2
(a) TD 2003
3 4 5 6 7 8 9
NDCG Truncation Level (K)
10
(b) NP 2004
Figure 2: Ranking results on two datasets from LETOR 3.0. Results for the other 5 datasets, along
with Precision@K results, appear in the supplementary material.
shown in Section 4.2 is the optimal decision theoretic prediction under a ranking gain function, by
simply sorting the items for each test query based on their ? score values. Note that this is a very
simple ranking model, in that the score assigned to each test item by the model is a linear function
of the input features, and the only hyperparameter to tune is an `2 regularization strength.
Results for two of the data sets are shown in Figure 2 (first is our best relative performance, second is
typical); the full set of results are in the supplementary material. Several publicly available baselines
are shown for comparison. As can be seen in the graphs, our approach is competitive with the stateof-the-art on all data sets, and substantially outperforms all baselines on the TD 2003 dataset. Note
that the performance of the baseline methods is quite variable and it appears that overfitting is an issue on these datasets, even for linear models. We hypothesize that proper probabilistic incorporation
of weak labels helps to mitigate this effect to some degree.
5.3
Top-K Classification
Our third and final set of experiments concern top-K classification, an important task that has gained
considerable attention recently in the ImageNet Challenge.2 Here we consider a task analogous to
that in the ImageNet Challenge, in which each image contains a single object label, but a model
is allowed to return up to K class predictions per image. A classification is deemed correct if the
appropriate class is one of the K returned classes.
We train binary n-choose-k models, experimenting with different training protocols that directly
maximize expected gain under the model, as described in Section 4.1. That is, we train on the expected top-K gain for different values of K. Note that top-1 is equivalent to softmax regression. For
each model/evaluation criterion combination, we find the `2 penalty that gives the highest validation
accuracy; the corresponding test-set results are shown in Table 1. For comparison, we also include
logistic regression, where each output is conditionally independent. We experimented on the embedded MNIST dataset where all but one label from each example was randomly removed, and on
the Caltech-101 Silhouettes dataset [11], which consists of images of binarized silhouettes from 101
different categories. In both datasets we trained the models using the pixels as inputs. We noticed
that the optimal `2 strength chosen by each method was quite high, suggesting that overfitting is an
issue in these datasets. When the `2 strength is low, the difference between the objectives becomes
more apparent. On Caltech it is clear that training for the expected gain improves the corresponding
test accuracy in this regime. On the embedded MNIST dataset, when the `2 strength is low there is
a surprising result that the top-3 and top-5 criteria outperform top-1, even when top-1 is used as the
evaluation measure. Since there are several digits actually present in the ground truth, there is no
real signal in the data that differentiates the digit labeled as the target from the other equally valid
?distractor? digits. In order to satisfy the top-1 objective for the given target, the learning algorithm
is forced to find some arbitrary criterion by which to cause the given target to be preferred over the
distractors, which is harmful for generalization purposes. This scenario does occur in datasets like
ImageNet, where multiple objects can be present in a single image. It would be interesting to repeat
these experiments on more challenging, large scale datasets, but we leave this for future work.
2
http://www.image-net.org/challenges/LSVRC/2011/
7
LR
Top 1
Top 3
Top 5
Top 1 / Top 3 / Top 5
0.606 / 0.785 / 0.812
0.621 / 0.796 / 0.831
0.614 / 0.792 / 0.834
0.602 / 0.787 / 0.834
(a) Caltech Sil. strong `2
Top 1 / Top 3 / Top 5
0.545 / 0.716 / 0.766
0.574 / 0.755 / 0.804
0.558 / 0.771 / 0.813
0.523 / 0.767 / 0.823
Top 1 / Top 3 / Top 5
0.346 / 0.647 / 0.815
0.353 / 0.659 / 0.820
0.353 / 0.671 / 0.834
0.330 / 0.659 / 0.824
Top 1 / Top 3 / Top 5
0.263 / 0.557 / 0.742
0.268 / 0.569 / 0.757
0.318 / 0.637 / 0.815
0.313 / 0.642 / 0.822
(b) Caltech Sil. weak `2 (c) EMNIST strong `2
(d) EMNIST weak `2
Table 1: Top-K classification results when various models are trained using an expected top-K gain
and then tested using some possibly different top-K criterion. The rows correspond to training
criteria, and the columns correspond to test criteria. (a) and (c) show the test accuracy when a
strong `2 regularizer is used, while (b) and (d) use a relatively weaker regularizer. Logistic regression
is included for comparison.
6
Related Work
Our work here is related to many different areas; we cannot hope to survey all related work in multilabel classification and ranking. Instead, we focus on work related to the main novelty in this paper,
the explicit modeling of structure on label counts. That is, given that we have prior knowledge of
label count structure, or are modeling a domain that exhibits such structure, the question is how can
the structure be leveraged to improve a model.
The first and most direct approach is the one that we take here: explicitly model the count structure
within the model. There are other alternative approaches that are similar in this respect. The work
of [12] considers MAP inference in the context of cardinality-based models and develops applications to named entity recognition tasks. Similarly, [13] develops an example application where a
cardinality-based term constrains the number of pixels that take on the label ?foreground? in a foreground/background image segmentation task. [14] develops models that include a penalty in the
energy function for using more labels, which can be seen as a restricted form of structure over label
cardinalities.
An alternative way of incorporating structure over counts into a model is via the gain function. The
work of Joachims [15] can be seen in this light ? the training objective is formulated so as to optimize
performance on evaluation measures that include Precision@K. A different approach to including
count information in the gain function comes from [16], which trains an image segmentation model
so as match count statistics present in the ground truth data. Finally, there are other approaches that
do not neatly fall into either category, such as the posterior regularization framework of [17] and
related works such as [18]. There, structure, including structure that encodes prior knowledge about
counts, such as there being at least one verb in most sentences, is added as a regularization term that
is used both during learning and during inference.
Overall, the main difference between our work and these others is that we work in a proper probabilistic framework, either maximizing likelihood, maximizing expected gain, and/or making proper
decision-theoretic predictions at test time. Importantly, there is no significant penalty for assuming
the proper probabilistic approach: learning is exact, and test-time prediction is efficient.
7
Discussion
We have presented a flexible probabilistic model for multiple output variables that explicitly models
structure in the number of variables taking on specific values. The model is simple, efficient, easy to
learn due to its convex objective, and widely applicable. Our theoretical contribution provides a link
between this type of ordinal model and ranking problems, bridging the gap between the two tasks,
and allowing the same model to be effective for several quite different problems. Finally, there are
many extensions. More powerful models of ? can be put into the formulation, and gradients can
easily be back-propagated. Also, while we chose to take a maximum likelihood approach in this
paper, the model is well suited to fully Bayesian inference using e.g., slice sampling. The unimodal
posterior distribution should lead to good behavior of the sampler. Beyond these extensions, we
believe the framework here to be a valuable modeling building block that has broad application to
problems in machine learning.
8
References
[1] S. X. Chen and J. S. Liu. Statistical applications of the Poisson-Binomial and conditional
Bernoulli distributions. Statistica Sinica, 7(4), 1997.
[2] X. H. Chen, A. P. Dempster, and J. S. Liu. Weighted finite population sampling to maximize
entropy. Biometrika, 81(3):457?469, 1994.
[3] M. H. Gail, J. H. Lubin, and L. V. Rubinstein. Likelihood calculations for matched case-control
studies and survival studies with tied death times. Biometrika, 68:703?707, 1981.
[4] L. Belfore. An O(n) log2(n) algorithm for computing the reliability of k-out-of-n:G and k-tol-out-of-n:G systems. IEEE Transactions on Reliability, 44(1), 1995.
[5] D. Tarlow, K. Swersky, R. Zemel, R.P. Adams, and B. Frey. Fast exact inference for recursive
cardinality models. In Uncertainty in Artificial Intelligence, 2012.
[6] R. Plackett. The analysis of permutations. Applied Statistics, pages 193?202, 1975.
[7] R.D. Luce. Individual Choice Behavior a Theoretical Analysis. Wiley, 1959.
[8] J. Guiver and E. Snelson. Bayesian inference for plackett-luce ranking models. In International
Conference on Machine Learning, 2009.
[9] J. Huang, C. Guestrin, and L. Guibas. Efficient inference for distributions on permutations. In
Advances in Neural Information Processing Systems, 2007.
[10] T. Qin, T.Y. Liu, J. Xu, and H. Li. LETOR: A benchmark collection for research on learning
to rank for information retrieval. Information Retrieval Journal, 2010.
[11] B. Marlin, K. Swersky, B. Chen, and N. de Freitas. Inductive principles for restricted Boltzmann machine learning. In Artificial Intelligence and Statistics, 2010.
[12] R. Gupta, A. Diwan, and S. Sarawagi. Efficient inference with cardinality-based clique potentials. In International Conference on Machine Learning, 2007.
[13] D. Tarlow, I. Givoni, and R. Zemel. HOP-MAP: Efficient message passing for high order
potentials. In Artificial Intelligence and Statistics, 2010.
[14] A. Delong, A. Osokin, H.N. Isack, and Y. Boykov. Fast approximate energy minimization with
label costs. International Journal of Computer Vision, 96(1):127, 2012.
[15] T. Joachims. A support vector method for multivariate performance measures. In International
Conference on Machine Learning, 2005.
[16] P. Pletscher and P. Kohli. Learning low-order models for enforcing high-order statistics. In
Artificial Intelligence and Statistics, 2012.
[17] K. Ganchev, J. Grac?a, J. Gillenwater, and B. Taskar. Posterior regularization for structured
latent variable models. Journal of Machine Learning Research, 11:2001?2049, 2010.
[18] G. Mann and A McCallum. Generalized expectation criteria with application to semisupervised classification and sequence modeling. Journal of Machine Learning Research,
11:955?984, 2010.
9
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4,092 | 4,703 | The topographic unsupervised learning of
natural sounds in the auditory cortex
Hiroki Terashima
The University of Tokyo / JSPS
Tokyo, Japan
[email protected]
Masato Okada
The University of Tokyo / RIKEN BSI
Tokyo, Japan
[email protected]
Abstract
The computational modelling of the primary auditory cortex (A1) has been less
fruitful than that of the primary visual cortex (V1) due to the less organized properties of A1. Greater disorder has recently been demonstrated for the tonotopy of
A1 that has traditionally been considered to be as ordered as the retinotopy of V1.
This disorder appears to be incongruous, given the uniformity of the neocortex;
however, we hypothesized that both A1 and V1 would adopt an efficient coding
strategy and that the disorder in A1 reflects natural sound statistics. To provide a
computational model of the tonotopic disorder in A1, we used a model that was
originally proposed for the smooth V1 map. In contrast to natural images, natural
sounds exhibit distant correlations, which were learned and reflected in the disordered map. The auditory model predicted harmonic relationships among neighbouring A1 cells; furthermore, the same mechanism used to model V1 complex
cells reproduced nonlinear responses similar to the pitch selectivity. These results
contribute to the understanding of the sensory cortices of different modalities in a
novel and integrated manner.
1 Introduction
Despite the anatomical and functional similarities between the primary auditory cortex (A1) and the
primary visual cortex (V1), the computational modelling of A1 has proven to be less fruitful than V1,
primarily because the responses of A1 cells are more disorganized. For instance, the receptive fields
of V1 cells are localized within a small portion of the field of view [1], whereas certain A1 cells have
receptive fields that are not localized, as these A1 cells demonstrate significant responses to multiple
distant frequencies [2, 3]. An additional discrepancy that has recently been discovered between these
two regions relates to their topographic structures, i.e., the retinotopy of V1 and the tonotopy of A1;
these structures had long been considered to be quite similar, but studies on a microscopic scale have
demonstrated that in mice, the tonotopy of A1 is much more disordered [4, 5] than the retinotopy
of V1 [6, 7]. This result is consistent with previous investigations involving other species [8, 9],
suggesting that the discrepancy in question constitutes a general tendency among mammals. This
disorderliness appears to pose significant difficulties for the development of computational models
of A1.
A number of computational modelling studies have emphasized the close associations between V1
cells and natural image statistics, which suggests that the V1 adopts an unsupervised, efficient coding
strategy [10]. For instance, the receptive fields of V1 simple cells were reproduced by either sparse
coding [11] or the independent component analysis [12] of natural images. This line of research
yields explanations for the two-dimensional topography, the orientation and retinotopic maps of
V1 [13, 14, 15]. Similar efforts to address A1 have been attempted by only a few studies, which
demonstrated that the efficient coding of natural, harmonic sounds, such as human voices or piano
1
recordings, can explain the basic receptive fields of A1 cells [16, 17] and their harmony-related
responses [18, 19]. However, these studies have not yet addressed the topography of A1.
In an integrated and computational manner, the present paper attempts to explain why the tonotopy
of A1 is more disordered than the retinotopy of V1. We hypothesized that V1 and A1 still share an
efficient coding strategy, and we therefore proposed that the distant correlations in natural sounds
would be responsible for the relative disorder in A1. To test this hypothesis, we first demonstrated
the significant differences between natural images and natural sounds. Natural images and natural
sounds were then each used as inputs for topographic independent component analysis, a model that
had previously been proposed for the smooth topography of V1, and maps were generated for these
images and sounds. Due to the distant correlations of natural sounds, greater disorder was observed
in the learned map that had been adapted to natural sounds than in the analogous map that had
been adapted to images. For natural sounds, this model not only predicted harmonic relationships
between neighbouring cells but also demonstrated nonlinear responses that appeared similar to the
responses of the pitch-selective cells that were recently found in A1. These results suggest that the
apparently dissimilar topographies of V1 and A1 may reflect statistical differences between natural
images and natural sounds; however, these two regions may employ a common adaptive strategy.
2
Methods
2.1
Topographic independent component analysis
Herein, we discuss an unsupervised learning model termed topographic independent component
analysis (TICA), which was originally proposed for the study of V1 topography [13, 14]. This
model comprises two layers: the first layer of N units models the linear responses of V1 simple
cells, whereas the second layer of N units models the nonlinear responses of V1 complex cells, and
the connections between the layers define a topography. Given a whitened input vector I(x) ? Rd
(here, d = N ), the input is reconstructed by the linear superposition of a basis ai ? Rd , each of
which corresponds to the first-layer units
?
I=
si ai
(1)
i
where si ? R are activity levels of the units or model neurons. Inverse filters wi to determine si can
typically be obtained, and thus si = I T wi (inner product). Using the activities of the first layer, the
activities of the second-layer units ci ? R can be defined as follows:
?
ci =
h(i, j)s2j
(2)
j
where h(i, j) is the neighbourhood function that takes the value of 1 if i and j are neighbours
and is 0 otherwise. The neighbourhood is defined by a square window (e.g., 5 ? 5) in cases of
two-dimensional topography. The learning of wi is accomplished through the minimization of the
energy function E or the negative log likelihood:
?
E = ? log L(I; {wi }) = ?
G(ci )
(3)
?
?wi ?
?
Isi ?
i
?
??
h(i, j)g(cj )?
(4)
j
?
where G(ci ) = ? ? + ci imposes sparseness on the second-layer activities (? = 0.005 for the
stability), and g(ci ) is the derivative of G(ci ). The operator ?? ? ? ? is the mean over the iterations.
2.1.1
An extension for overcomplete representation
Ma and Zhang [15] extended the TICA model to account for overcomplete representations (d < N ),
which are observed in the V1 of primates. In this extension, inverse filters cannot be uniquely defined; therefore, a set of first-layer responses si to an input is computed by minimizing the following
extended energy function:
2
0
-4
B
285
45
Frequency [Hz]
1812
1
0
Correlation
4
Angle of view [?]
A
-1
Figure 1: Local correlations in natural images and distant correlations in natural sounds. (A)
The correlation matrix of image strips (right) demonstrated only local correlations (? 6? ) in the field
of view (? 120? ). (B) The correlation matrix of the human voice spectra (right) demonstrated not
only local correlation but also off-diagonal distant correlations produced by harmonics.
2
?
?
G(ci )
si ai ? ?
E = ? log L(I; {ai }, {si }) = I ?
i
i
?
?
?
?
?
?
sj aj ? ? ?si ?
h(i, j)g(cj )?
?si ? aTi ?I ?
j
(5)
(6)
j
where ? is the relative weight of the activity sparseness, in accordance with sparse coding [11]. The
initial value of si is set equal to the inner product of I and ai . Every 256 inputs, the basis is updated
using the following gradient. In this study, we used the learning rate ? = 0.08.
??
? ?
?
?ai = ? si ?I ?
sj aj ?
(7)
j
2.2
The discontinuity index for topographic representation
To compare the degrees of disorder in topographies of different modalities, we defined a discontinuity index (DI) for each point i of the maps. Features defining a topography f (i) (e.g., a retinotopic
position or a frequency) were normalized to the range of [0, 1]. Features f (j) within the neighbourhood of the ith unit defined by h(i, j) were linearly fitted using the least squares method, and the DI
value at i was then determined using the following equation:
??
2
j h(i, j)r(j)
DI(i) =
(8)
NNB
where r(j) is the residual error of linear regression at j and NNB is the number of units within a
neighbourhood window. If the input space is a torus (see Section 3.3), another DI value is computed
using modified f values that are increased by 1 if they were initially within [0, 12 ), and the smaller
of the calculated DI values is used.
3
3.1
Results
Correlations of natural images and natural sounds
Given that V1 is supposed to adapt to natural images and that A1 is supposed to adapt to natural
sounds, the first analysis in this study simply compared statistics for natural images and natural
sounds. The natural images were taken from the van Hateren database [20] and were reduced four
times from their original size. Vertical arrays of 120 pixels each were extracted from the reduced
3
A
100
Retinotopy
50
0
0.1
0.2
0.3
Discontinuity index (DI)
Gabor position
D
Retinotopy
25
1
3.6k
E
90
Tonotopy (smoothed)
3.6k 3.6k
[Hz]
F
CF [Hz]
0
Tonotopy
CF [Hz]
Number of units
B
C
Tonotopy
90
90
200
[ms]
Figure 2: The ordered retinotopy and disordered tonotopy. (A) The topography of units adapted
to natural images. A small square indicates a unit ai (grey: 0; white: max value). (B) The distributions of DI for the two topographies. (C) The topography of spectro-temporal units that have
been adapted to natural sounds. (D-F) The retinotopy of the visual map (D) is smooth, whereas the
tonotopy of the auditory map (F) is more disordered, although global tonotopy still exists (E).
1
images, each of which covered approximately 8? ( 15
of the vertical range of the human field of
view). Figure 1A (right) illustrates the correlation matrix for these images, which is a simple structure that contains local correlations that span approximately 6? . This result was not surprising, as
distant pixels typically depict different objects.
For natural sounds, we used human narratives from the Handbook of the International Phonetic
Association [21], as efficient representations of human voices have been successful in facilitating
studies of various components of the auditory system [22, 23], including A1 [16, 17]. After these
sounds were downsampled to 4 kHz, their spectrograms were generated using the NSL toolbox
[24] to approximate peripheral auditory processing. Short-time spectra were extracted from the
spectrograms, each of which were 128 pixels wide on a logarithmic scale (24 pixels = 1 octave).
Note that the frequency range (> 5 octaves) spans approximately half of a typical mammalian
1
hearing range (? 10 octaves [25]), whereas the image pixel array spans only 15
of the field of view.
Figure 1B illustrates the correlation matrix for these sounds, which is a complex structure that incorporates distant, off-diagonal correlations. The most prominent off-diagonal correlation, which was
just 1 octave away from the main diagonal, corresponded to the second harmonic of a sound, i.e.,
frequencies at a ratio of 1:2. Similarly, other off-diagonal peaks indicated correlations due to higher
harmonics, i.e., frequencies that were related to each other by simple integral ratios. These distant
correlations represent relatively typical results for natural sounds and differ greatly from the strictly
local correlations observed for natural images.
3.2
Greater disorder for the tonotopy than the retinotopy
To test the hypothesis that V1 and A1 share a learning strategy, the TICA model was applied to
natural images and natural sounds, which exhibit different statistical profiles, as discussed above.
4
Discontinuity index (DI)
0.2
0.1
0
0.2
0.4
Strength of distant correlation
Vision-like
Audition-like
Figure 3: The correlation between discontinuity and input ?auditoriness?. When inputs only
correlated locally (pa ? 0: vision-like inputs), DI was low, and DI increased with the input ?auditoriness? pa . Three lines: the quartiles (25, 50 (bold), and 75%) obtained from 100 iterations.
Learning with natural images was accomplished in accordance with the original TICA study [13,
14]. Images from the van Hateren database were reduced four times from their original size, and
25 ? 25 pixel image patches were randomly extracted (n = 50,000). The patches were whitened and
bandpassed by applying principal component analysis, whereby we selected 400 components and
rejected certain components with low variances and the three components with the largest variances
[13]. The topography was a 20 ? 20 torus, and the neighbourhood window was 5 ? 5.
Figure 2A illustrates the visual topographic map obtained from this analysis, a small square of which
constitutes a basis vector ai . As previously observed in the original TICA study [13, 14], each unit
was localized, oriented, and bandpassed; thus, these units appeared to be organized similarly to
the receptive fields of V1 simple cells. The orientation and position of the units changed smoothly
with the coordinates that were examined, which suggested that this map evinces an ordered topography. To quantify the retinotopic discontinuity, each unit was fitted using a two-dimensional Gabor
function, and DI was calculated using the y values of the centre coordinates of the resulting Gabor
functions as the features. Figure 2B graphically indicates that the obtained DI values were quite low,
which is consistent with the smooth retinotopy illustrated in Figure 2D.
Next, another TICA model was applied to natural sounds to create an auditory topographic map that
could be compared to the visual topography. As detailed in the previous section, spectrograms of
human voices (sampled at 8 kHz) were generated using the NSL toolbox to approximate peripheral
auditory processing. Spectrogram patches of 200 ms (25 pixels) in width were randomly extracted
(n = 50,000) and vertically reduced from 128 to 25 pixels, which enabled these spectrogram patches
to be directly compared with the image patches. The sound patches were whitened, bandpassed, and
adapted using the model in the same manner as was described for the image patches.
Figure 2C shows the resulting auditory topographic map, which is composed of spectro-temporal
units of ai that are represented by small squares. The units were localized temporally and spectrally,
and some units demonstrated multiple, harmonic peaks; thus, these units appeared to reasonably
represent the typical spectro-temporal receptive fields of A1 cells [16, 3]. The frequency to which
an auditory neuron responds most significantly is called its characteristic frequency (CF) [2]. In this
analysis, the CF of a unit was defined as the frequency that demonstrated the largest absolute value
for the unit in question. Figure 2F illustrates the spatial distribution of CFs, i.e., the tonotopic map.
Within local regions, the tonotopy was not necessarily smooth, i.e., neighbouring units displayed
distant CFs. However, at a global level, a smooth tonotopy was observed (Figure 2E). Both of these
findings are consistent with established experimental results [4, 5]. The distribution of tonotopic DI
values is shown in Figure 2B, which clearly demonstrates that the tonotopy was more disordered
than the retinotopy (p < 0.0001; Wilcoxon rank test).
3.3 The topographic disorder due to distant input correlations
The previous section demonstrated that natural sounds could induce greater topographic disorder
than natural images, and this section discusses the attempts to elucidate the disorder resulting from a
specific characteristic of natural sounds, namely, distant correlations. For this purpose, we generated
artificial inputs (d = 16) with a parameter pa ? [0, 1] that regulates the degree of distant correlations.
5
B
CF difference [octave]
5.0
4.0
3.0
2.0
1:3
1:2
2:3
1.0
0
2
4
6
8
Distance of two units
10
CF difference [octave]
A
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
1
1.2
1.4
Distance of two units
Figure 4: The harmonic relationships between CFs of neighbouring units. (A) The full distribution of distance and CF difference between two units. (B) The distribution of CF differences within
neighbourhoods (the red-dotted rectangle in (A)). There were three peaks that indicate harmonic
relationships between neighbouring units. The distances were jittered to obtain the visualization.
After the inputs were initially generated from a standard normal distribution, a constant value of 4
was added at k points of each input, where k was from a uniform distribution over {3, 4, 5, 6} and
the points? coordinates x were from a normal distribution with a random centre and ? = 2. After
adding this constant value at x, we also added another at xdist = x + 5 with a probability pa that
defines its ?auditoriness?, i.e., its degree of distant correlations. For greater simplicity and to avoid
border effects, the input space was defined to be a one-dimensional torus. The topography was also
set as a one-dimensional torus of 16 units with a neighbourhood window size of 5.
Figure 3 shows the positive correlation between the input ?auditoriness? pa and the DI of the learned
topographies. In computations of DI, the feature f of a unit was considered to be its peak coordinate
with the largest absolute value, and a toric input space was used (Section 2.2). If the input only
demonstrated local correlations like visual stimuli (pa ? 0), then its learned topography was smooth
(i.e., its DI was low). The DI values generally increased as distant correlations appeared more
frequently, i.e., more ?auditoriness? of the inputs grew. Thus, the topographic disorder of auditory
maps results from distant correlations presented by natural auditory signals.
3.4
The harmonic relationship among neighbouring units
Several experiments [4, 5] have reported that the CFs of neighbouring cells can differ by up to 4
octaves, although these studies have failed to provide additional detail regarding the local spatial
patterns of the CF distributions. However, if the auditory topography is representative of natural
stimulus statistics, the topographic map is likely to possess certain additional spatial features that
reflect the statistical characteristics of natural sounds.
To enable a detailed investigation of the CF distribution, we employed a model that had been adapted
to finer frequency spectra of natural sounds, and this model was then used throughout the remainder
of the study. As the temporal structure of the auditory receptive fields was less dominant than their
spectral structure (Figure 2C), we focused solely on the spectral domain and did not attempt to address temporal information. Therefore, the inputs for the new model (n = 100,000) were short-time
frequency spectra of 128 pixels each (24 pixels = 1 octave). The data for these spectra were first
obtained from the spectrograms of human voices (8 kHz) using the method detailed in Section 3.1,
and these data were then whitened, bandpassed, and reduced to 100 dimensions prior to input into
the model. To illustrate patterns more clearly, the results shown below were obtained using the
overcomplete extension of TICA described in Section 2.1.1, which included a 14 ? 14 torus (approximately 2? overcomplete) and 3?3 windows. The CF of a unit was determined using pure-tone
inputs of 128 frequencies.
Figure 4A illustrates the full distribution of the distance and CF difference between two units in a
learned topography. The CFs of even neighbouring units differed by up to ? 4 octaves, which is
consistent with recent experimental findings [4, 5]. A closer inspection of the red-dotted rectangular
region of Figure 4A is shown in Figure 4B. The histogram in Figure 4B demonstrates several peaks
6
1-3
2-4
10-12
C
n = 66 units
(6 simulations)
1.0
0.8
0.6
0.4
0.2
90
0
0 f0
4*f0
8*f0
Frequency [Hz]
12*f0
8k
[Hz]
CF
Normalized activity
B
Harmonic composition
of MFs
A
f0 2
4
6
8
10
Lowest harmonic present
Pitch
selective
units
Figure 5: Nonlinear responses similar to pitch selectivity. (A) The spectra of MFs that share a
f0 , all of which are perceived similarly. (B) The responses of pitch-selective units to MFs. (C) The
distribution of pitch-selective units on the smoothed tonotopy in a single session.
at harmony-related CF differences, such as 0.59 (= log2 1.5), 1.0 (= log2 2; the largest peak), and
1.59 (= log2 3). These examples indicate that CFs of neighbouring units did not differ randomly,
but tended to be harmonically related. A careful inspection of published data (Figure 5d from [5])
suggests that this relationship may be discernible in those published results; however, the magnitude
of non-harmonic relationships cannot be clearly established from the inspection of this previously
published study, as the stimuli used by the relevant experiment [5] were separated by an interval of
0.25 octaves and were therefore biased towards being harmonic. Thus, this prediction of a harmonic
relationship in neighbouring CFs will need to be examined in more detailed investigations.
3.5
Nonlinear responses similar to pitch-selectivity
Psychoacoustics have long demonstrated interesting phenomena related to harmony, namely, the
perception of pitch, which represents a subjective attribute of perceived sounds. Forming a rigid
definition for the notion of pitch is difficult; however, if a tone consists of a stack of harmonics
(f0 , 2f0 , 3f0 , . . .), then its pitch is the frequency of the lowest harmonic, which is called the fundamental frequency f0 . The perception of pitch is known to remain constant even if the sound
lacks power at lower harmonics; in fact, pitch at f0 can be perceived from a sound that lacks f0 ,
a phenomenon known as ?missing fundamental? [26]. Nonlinear pitch-selective responses similar
to this perception have recently been demonstrated in certain A1 neurons [27] that localize in the
low-frequency area of the global tonotopy.
To investigate pitch-related responses, previously described complex tones [27] that consisted of
harmonics were selected as inputs for the model described in Section 3.4. For each unit, responses
were calculated to complex tones termed missing fundamental complex tones (MFs) [27]. The MFs
were composed of three consecutive harmonics sharing a single f0 ; the lowest frequency for these
consecutive harmonics varied from the fundamental frequency (f0 ) to the tenth harmonic (10f0 ), as
shown in Figure 5A. For each unit, five patterns of f0 around its CF (? 0.2 octave) were tested,
resulting in a total of 10 ? 5 = 50 variations of MFs. The activity of a unit was normalized to
its maximum response to the MFs. Pitch-selective units were defined as those that significantly
responded (normalized activity > 0.4) to all of the MFs sharing a single f0 with a lowest harmonic
from 1 to 4.
We found certain pitch-selective units in the second layer (n = 66; 6 simulations), whereas none
were found in the first layer. Figure 5B illustrates the response profiles of the pitch-selective units,
which demonstrated sustained activity for MFs with a lowest harmonic below the sixth harmonic
(6f0 ), and this result is similar to previously published data [27]. Additionally, these units were
located in a low-frequency region of the global tonotopy, as shown in Figure 5C, and this feature of
pitch-selective units is also consistent with previous findings [27]. The second layer of the TICA
model, which contained the pitch-selective units, was originally designed to represent the layer of V1
complex cells, which have nonlinear responses that can be modelled by a summation of ?energies?
of neighbouring simple cells [13, 14, 15]. Our result suggests that the mechanism underlying V1
complex cells may be similar to the organizational mechanism for A1 pitch-selective cells.
7
B
Natural
sounds
Correlation
0
Retinal position
0
No correlation
(different objects)
Retinal distance
2:3 1:2
Smooth
V1 retinotopy
Cortical position
1:3
Disordered
A1 tonotopy
CF
Natural
images
Correlation
A
1.0
[octave]
? Frequency
Cortical position
Figure 6: The suggested relationships between natural stimulus statistics and topographies.
4
Discussion
Using a single model, we have provided a computational account explaining why the tonotopy of A1
is more disordered than the retinotopy of V1. First, we demonstrated that there are significant differences between natural images and natural sounds; in particular, the latter evince distant correlations,
whereas the former do not. The topographic independent component analysis therefore generated
a disordered tonotopy for these sounds, whereas the retinotopy adapted to natural images was locally organized throughout. Detailed analyses of the TICA model predicted harmonic relationships
among neighbouring neurons; furthermore, these analyses successfully replicated pitch selectivity, a
nonlinear response of actual cells, using a mechanism that was designed to model V1 complex cells.
The results suggest that A1 and V1 may share an adaptive strategy, and the dissimilar topographies
of visual and auditory maps may therefore reflect significant differences in the natural stimuli.
Figure 6 summarizes the ways in which the organizations of V1 and A1 reflect these input differences. Natural images correlate only locally, which produces a smooth retinotopy through an
efficient coding strategy (Figure 6A). By contrast, natural sounds exhibit additional distant correlations (primarily correlations among harmonics), which produce the topographic disorganization
observed for A1 (Figure 6B). To extract the features of natural sounds in the auditory pathway, A1
must integrate multiple channels of distant frequencies [2]; for this purpose, the disordered tonotopy
can be beneficial because a neuron can easily collect information regarding distant (and often harmonically related) frequencies from other cells within its neighbourhood. Our result suggests the
existence of a common adaptive strategy underlying V1 and A1, which would be consistent with
experimental studies that exchanged the peripheral inputs of the visual and auditory systems and
suggested the sensory experiences had a dominant effect on cortical organization [28, 29, 30].
Our final result suggested that a common mechanism may underlie the complex cells of V1 and
the pitch-selective cells of A1. Additional support for this notion was provided by recent evidence
indicating that the pitch-selective cells are most commonly found in the supragranular layer [27],
and V1 complex cells display a similar tendency. It has been hypothesized that V1 complex cells
collect information from neighbouring cells that are selective to different phases of similar orientations; in an analogous way, A1 pitch-selective cells could collect information from the activities
of neighbouring cells, which in this case could be selective to different frequencies sharing a single
f0 . To the best of our knowledge, no previous studies in the literature have attempted to use this
analogy of V1 complex cells to explain A1 pitch-selective cells (however, other potential analogues
have been mentioned [31, 32]). Our results and further investigations should help us to understand
these pitch-selective cells from an integrated, computational viewpoint. Another issue that must be
addressed is what functional roles the other units in the second layer play. One possible answer to
this question may be multipeaked responses related to harmony [3], which have been explained in
part by sparse coding [18, 19]; however, this answer has not yet been confirmed by existing evidence
and must therefore be assessed in detail by further investigations.
Acknowledgement
Supported by KAKENHI (11J04424 for HT; 22650041, 20240020, 23119708 for MO).
8
References
[1] D. H. Hubel and T. N. Wiesel. Receptive fields, binocular interaction and functional architecture in the
cat?s visual cortex. The Journal of Physiology, 160(1):106?154, 1962.
[2] C. E. Schreiner, H. L. Read, and M. L. Sutter. Modular organization of frequency integration in primary
auditory cortex. Annual Review of Neuroscience, 23(1):501?529, 2000.
[3] S. C. Kadia and X. Wang. Spectral integration in A1 of awake primates: Neurons with single- and
multipeaked tuning characteristics. Journal of Neurophysiology, 89(3):1603?1622, 2003.
[4] S. Bandyopadhyay, S. A. Shamma, and P. O. Kanold. Dichotomy of functional organization in the mouse
auditory cortex. Nature Neuroscience, 13(3):361?368, 2010.
[5] G. Rothschild, I. Nelken, and A. Mizrahi. Functional organization and population dynamics in the mouse
primary auditory cortex. Nature Neuroscience, 13(3):353?360, 2010.
[6] S. L. Smith and M. H?ausser. Parallel processing of visual space by neighboring neurons in mouse visual
cortex. Nature Neuroscience, 13(9):1144?1149, 2010.
[7] V. Bonin, M. H. Histed, S. Yurgenson, and R. C. Reid. Local diversity and fine-scale organization of
receptive fields in mouse visual cortex. The Journal of Neuroscience, 31(50):18506?18521, 2011.
[8] E. F. Evans, H. F. Ross, and I. C. Whitfield. The spatial distribution of unit characteristic frequency in the
primary auditory cortex of the cat. The Journal of Physiology, 179(2):238?247, 1965.
[9] M. H. Goldstein Jr, M. Abeles, R. L. Daly, and J. McIntosh. Functional architecture in cat primary
auditory cortex: tonotopic organization. Journal of Neurophysiology, 33(1):188?197, 1970.
[10] A. Hyv?arinen, J. Hurri, and P. O. Hoyer. Natural Image Statistics: A probabilistic approach to early
computational vision. Springer-Verlag London Ltd., 2009.
[11] B. A. Olshausen and D. J. Field. Emergence of simple-cell receptive field properties by learning a sparse
code for natural images. Nature, 381(6583):607?609, 1996.
[12] A. J. Bell and T. J. Sejnowski. The ?independent components? of natural scenes are edge filters. Vision
Research, 37(23):3327?3338, 1997.
[13] A. Hyv?arinen and P. O. Hoyer. A two-layer sparse coding model learns simple and complex cell receptive
fields and topography from natural images. Vision Research, 41(18):2413?2423, 2001.
[14] A. Hyv?arinen, P. O. Hoyer, and M. Inki. Topographic independent component analysis. Neural Computation, 13(7):1527?1558, 2001.
[15] L. Ma and L. Zhang. A hierarchical generative model for overcomplete topographic representations in
natural images. In IJCNN 2007.
[16] D. J. Klein, P. Konig, and K. P. Kording. Sparse spectrotemporal coding of sounds. EURASIP Journal on
Applied Signal Processing, 2003(7):659?667, 2003.
[17] A. M. Saxe, M. Bhand, R. Mudur, B. Suresh, and A. Y. Ng. Unsupervised learning models of primary
cortical receptive fields and receptive field plasticity. In NIPS 2011.
[18] H. Terashima and H. Hosoya. Sparse codes of harmonic natural sounds and their modulatory interactions.
Network: Computation in Neural Systems, 20(4):253?267, 2009.
[19] H. Terashima, H. Hosoya, T. Tani, N. Ichinohe, and M. Okada. Sparse coding of harmonic vocalization
in monkey auditory cortex. Neurocomputing, doi:10.1016/j.neucom.2012.07.009, in press.
[20] J. H. van Hateren and A. van der Schaaf. Independent component filters of natural images compared with
simple cells in primary visual cortex. Proceedings of the Royal Society of London. Series B: Biological
Sciences, 265(1394):359?366, 1998.
[21] International Phonetic Association. Handbook of the International Phonetic Association: A Guide to the
Use of the International Phonetic Alphabet. Cambridge: Cambridge University Press, 1999.
[22] M. S. Lewicki. Efficient coding of natural sounds. Nature Neuroscience, 5(4):356?363, 2002.
[23] E. C. Smith and M. S. Lewicki. Efficient auditory coding. Nature, 439(7079):978?982, 2006.
[24] T. Chi and S. Shamma. NSL Matlab Toolbox. http://www.isr.umd.edu/Labs/NSL/
Software.htm, 2003.
[25] M. S. Osmanski and X. Wang. Measurement of absolute auditory thresholds in the common marmoset
(callithrix jacchus). Hearing Research, 277(1?2):127?133, 2011.
[26] B. C. J. Moore. An introduction to the psychology of hearing. London: Emerald Group Publishing Ltd.,
5th edition, 2003.
[27] D. Bendor and X. Wang. The neuronal representation of pitch in primate auditory cortex. Nature,
436(7054):1161?1165, 2005.
[28] M. Sur, P.E. Garraghty, and A. W. Roe. Experimentally induced visual projections into auditory thalamus
and cortex. Science, 242(4884):1437?1441, 1988.
[29] A. Angelucci, F. Clasc?a, and M. Sur. Brainstem inputs to the ferret medial geniculate nucleus and the
effect of early deafferentation on novel retinal projections to the auditory thalamus. The Journal of Comparative Neurology, 400(3):417?439, 1998.
[30] J. Sharma, A. Angelucci, and M. Sur. Induction of visual orientation modules in auditory cortex. Nature,
404(6780):841?847, 2000.
[31] B. Shechter and D. A. Depireux. Nonlinearity of coding in primary auditory cortex of the awake ferret.
Neuroscience, 165(2):612?620, 2010.
[32] C. A. Atencio, T. O. Sharpee, and C. E. Schreiner. Hierarchical computation in the canonical auditory
cortical circuit. Proceedings of the National Academy of Sciences, 106(51):21894?21899, 2009.
9
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4,093 | 4,704 | Deep Representations and Codes for Image
Auto-Annotation
Csaba Szepesv?ari
Department of Computing Science
University of Alberta
Edmonton, AB, Canada
[email protected]
Ryan Kiros
Department of Computing Science
University of Alberta
Edmonton, AB, Canada
[email protected]
Abstract
The task of image auto-annotation, namely assigning a set of relevant tags to an
image, is challenging due to the size and variability of tag vocabularies. Consequently, most existing algorithms focus on tag assignment and fix an often large
number of hand-crafted features to describe image characteristics. In this paper
we introduce a hierarchical model for learning representations of standard sized
color images from the pixel level, removing the need for engineered feature representations and subsequent feature selection for annotation. We benchmark our
model on the STL-10 recognition dataset, achieving state-of-the-art performance.
When our features are combined with TagProp (Guillaumin et al.), we compete
with or outperform existing annotation approaches that use over a dozen distinct
handcrafted image descriptors. Furthermore, using 256-bit codes and Hamming
distance for training TagProp, we exchange only a small reduction in performance
for efficient storage and fast comparisons. Self-taught learning is used in all of our
experiments and deeper architectures always outperform shallow ones.
1
Introduction
The development of successful methods for training deep architectures have influenced the development of representation learning algorithms either on top of SIFT descriptors [1, 2] or raw pixel
input [3, 4, 5] for feature extraction of full-sized images. Algorithms for pixel-based representation
learning avoid the use of any hand-crafted features, removing the difficulty of deciding which features are better suited for the desired task. Furthermore, self-taught learning [6] can be employed,
taking advantage of feature learning from image databases independent of the target dataset.
Image auto-annotation is a multi-label classification task of assigning a set of relevant, descriptive
tags to an image where tags often come from a vocabulary of hundreds to thousands of words.
Figure 1 illustrates this task. Auto-annotation is a difficult problem due to the high variability of
tags. Tags may describe objects, colors, scenes, local regions of the image (e.g. a building) or global
characteristics (e.g. whether the image is outdoors). Consequently, many of the most successful
annotation algorithms in the literature [7, 8, 9, 10, 11] have opted to focus on tag assignment and
often fix a large number of hand-crafted features for input to their algorithms. The task of feature
selection and applicability was studied by Zhang et al. [12] who utilized a group sparsity approach
for dropping features. Furthermore, they observed that feature importance varied across datasets and
some features led to redundancy, such as RGB and HSV histograms. Our main contribution in this
paper is to remove the need to compute over a dozen hand-crafted features for annotating images
and consequently remove the need for feature selection. We introduce a deep learning algorithm for
learning hierarchical representations of full-sized color images from the pixel level, which may be
seen as a generalization of the approach by Coates et al. [13] to larger images and more layers. We
first benchmark our algorithm on the STL-10 recognition dataset, achieving a classification accuracy
1
of 62.1%. For annotation, we use the TagProp discriminitve metric learning algorithm [9] which has
enjoyed state-of-the-art performance on popular annotation benchmarks. We test performance on
three datasets: Natural scenes, IAPRTC-12 and ESP-Game. When our features are combined with
TagProp, we either compete with or outperform existing methods when 15 distinct hand-crafted
features and metrics are used. This gives the advantage of focusing new research on improving tag
assignment algorithms without the need of deciding which features are best suited for the task.
Figure 1: Sample annotation results on IAPRTC-12 (top) and ESP-Game (bottom) using TagProp
when each image is represented by a 256-bit code. The first column of tags is the gold standard and
the second column are the predicted tags. Predicted tags that are italic are those that are also gold
standard.
More recently, auto-annotation algorithms have focused on scalability to large databases with hundreds of thousands to millions of images. Such approaches include that of Tsai et al. [10] who
construct visual synsets of images and Weston et al. [11] who used joint word-image embeddings.
Our second contribution proposes the use of representing an image with a 256-bit code for annotation. Torralba et al. [14] performed an extensive analysis of small codes for image retrieval
showing that even on databases with millions of images, linear search with Hamming distance can
be performed efficiently. We utilize an autoencoder with a single hidden layer on top of our learned
hierarchical representations to construct codes. Experimental results show only a small reduction
in performance is obtained compared to the original learned features. In exchange, 256-bit codes
are efficient to store and can be compared quickly with bitwise operations. To our knowledge, our
approach is the first to learn binary codes from full-sized color images without the use of handcrafted features. Such approaches often compute an initial descriptor such as GIST for representing
an image. These approaches introduce too strong of a bottleneck too early, where the bottleneck in
our pipeline comes after multiple layers of representation learning.
2
Hierarchical representation learning
In this section we describe our approach for learning a deep feature representation from the pixellevel of a color image. Our approach involves aspects of typical pipelines: pre-processing and
whitening, dictionary learning, convolutional extraction and pooling. We define a module as a pass
through each of the above operations. We first introduce our setup with high level descriptions
followed by a more detailed descriptions of each stage. Finally, we show how to stack multiple
modules on top of eachother.
Given a set of images, the learning phase of the network is as follows:
1. Extract randomly selected patches from each image and apply pre-processing.
2. Construct a dictionary using K-SVD.
3. Convolve the dictionary with larger tiles extracted across the image with a pre-defined
stride length. Re-assemble the outputs in a non-overlapping, spatially preserving grid.
4. Pool over the reassembled features with a 2 layer pyramid.
5. Repeat the above operations for as many modules as desired.
2
For extracting features of a new image, we perform steps (3) and (4) for each module.
2.1
Patch extraction and pre-processing
Let {I (1) , . . . , I (m) } be a set of m input images. For simplicity of explanation, assume I (i) ?
RnV ?nH ?3 , i = 1 . . . m, though it need not be the case that all images are of the same size. Given a
receptive field of size r ?c, we first extract np patches across all images of size r ?c?3, followed by
flatting each patch into a column vector. Let X = {x(1) , . . . , x(np ) }, x(i) ? Rn , i = 1 . . . np , n =
3rc denote the extracted patches. We first perform mean centering and unit variance scaling across
features. This corresponds to local brightness and contrast normalization, respectively.
Next
results in patches having zero mean,
Pnp we(i)follow [13] by performing ZCA1 whitening,
Pnp (i) which
(i) T
x
=
0,
and
identity
covariance,
x
(x
)
= I. A whitening matrix is computed
i=1
i=1
np
1
as W = V (Z + I)? 2 V T where C = V ZV T is an eigendecompostion of the centered covariance
matrix C = C(X) produced by mean subtraction of M = M (X). The parameter is a small
positive number having the effect of a low-pass filter.
2.2
Dictionary learning
Let S = {s(1) , . . . , s(np ) } denote the whitened patches. We are now ready to construct a set of
bases from S. We follow Bo et al. [5] and use K-SVD for learning a dictionary. K-SVD constructs a
dictionary D ? Rn?k and a sparse representation S? ? Rk?np by solving the following optimization
problem:
? 2
minimize kS ? DSk
F
subject to ||?
s(i) ||0 ? q ?i
?
D,S
(1)
where k is the desired number of bases. Optimization is done using alternation. When D is fixed, the
problem of obtaining S? can be decomposed into np subproblems of the form ks(i) ? D?
s(i) k2 subject
(i)
to ||?
s ||0 ? q ?i which can be solved approximately using batch orthogonal matching pursuit [15].
When S? is fixed, we update D by first expressing equation 1 in terms of a residual R(l) :
? 2F = kS ?
kS ? DSk
X
T
T
T
d(j) s?(j) ? d(l) s?(l) k2F = kR(l) ? d(l) s?(l) k2F
(2)
j6=l
where l ? {1, . . . , k}. A solution for d(l) , the l-th column of D, can be obtained through an SVD of
R(l) . For space considerations, we refer the reader to Rubinstein et al. [15] for more details. 1
2.3
Convolutional feature extraction
Given an image I (i) , we first partition the image into a set of tiles T (i) of size nt ? nt with a pre(i)
defined stride length s between each tile. Each patch in tile Tt is processed in the same way as
before dictionary construction (mean centering, contrast normalization, whitening) for which the
(i)
mean and whitening matrices M and W are used. Let Ttj denote the t-th tile and j-th channel with
(l)
respect to image I (i) and let Dj ? Rr?c denote the l-th basis for channel j of D. The encoding
(i)
ftl for tile t and basis l is given by:
3
o
n
X
(i)
(i)
(l)
ftl = max tanh
Ttj ? Dj , 0
(3)
j=1
where * denotes convolution and max and tanh operations are applied componentwise. Even though
it is not the associated encoding with K-SVD, this type of ?surrogate coding? was studied by Coates
1
We use Rubinstein?s implementation
?ronrubin/software.html
available
3
at
http://www.cs.technion.ac.il/
(i)
et al. [13]. Let ft denote the concatenated encodings over bases, which have a resulting dimension
of (nt ? r + 1) ? (nt ? c + 1) ? k. These are then re-assembled into spatial-preserving, nonoverlapping regions. See figure 2 for an illustration. We perform one additional localized contrast
(i)
(i)
(i)
(i)
(i)
normalization over ft of the form ft ? (ft ? ?(ft ))/max{?(?t ), ?t }. Similar types of
normalization have been shown to be critical for performance by Ranzato et al. [16] and Bo et al.
[5].
Figure 2: Left: D is convolved with each tile (large green square) with receptive field (small blue
square) over a given stride. The outputs are re-assembled in non-overlapping regions preserving
spatial structure. Right: 2 ? 2 and 1 ? 1 regions are summed (pooled) along each cross section.
2.4
Pooling
The final step of our pipeline is to perform spatial pooling over the re-assembled regions of the
(i)
encodings ft . Consider the l-th cross section corresponding to the l-th dictionary element, l ?
{1, . . . , k}. We may then pool over each of the spatial regions of this cross section by summing
over the activations of the corresponding spatial regions. This is done in the form of a 2-layer spatial
pyramid, where the base of the pyramid consists of 4 blocks of 2?2 tiling and the top of the pyramid
consisting of a single block across the whole cross section. See figure 2 for an illustration.
Once pooling is performed, the re-assembled encodings result in a shape of size 1 ? 1 ? k and
2 ? 2 ? k from each layer of the pyramid. To obtain the final feature vector, each layer is flattened
into a vector and the resulting vectors are concatinated into a single long feature vector of dimension
5k for each image I (i) . Prior to classification, these features are normalized to have zero mean and
unit variance.
2.5
Training multiple modules
What we have described up until now is how to extract features using a single module corresponding
to dictionary learning, extraction and pooling. We can now extend this framework into a deep
network by stacking multiple modules. Once the first module has been trained, we can take the
pooled features to be input to a second module. Freezing the learned dictionary from the first module,
we can then apply all the same steps a second time to the pooled representations. This type of stacked
training can be performed to as many modules as desired.
To be more specific on the input to the second module, we use an additional spatial pooling operation
on the re-assembled encodings of the first module, where we extract 256 blocks of 16 ? 16 tiling,
resulting in a representation of size 16?16?k. It is these inputs which we then pass on to the second
module. We choose to use 16 ? 16 as a trade off to aggregating too much information and increasing
memory and time complexity. As an illustration, the same operations for the second module are
used as in figure 2 except the image is replaced with the 16 ? 16 ? k pooled features. In the next
module, the number of channels is equal to the number of bases from the previous module.
3
Code construction and discriminitive metric learning
In this section we first show to learn binary codes from our learned features, followed by a review
of the TagProp algorithm [9] used for annotation.
4
3.1
Learning binary codes for annotation
Our codes are learned by adding an autoencoder with a single
hidden layer on top of the learned output representations. Let
f (i) ? Rdm denote the learned representation for image I (i) of
dimension dm using either a one or two module architecture. The
code b(i) for f (i) is computed by b(i) = round(?(f (i) )) where
?(f (i) ) = (1 + exp(W f (i) + ?))?1 , W ? Rdb ?dm , ? ? Rdb and
db is the number of bits (in our case, db = 256). Using a linear
output layer, our objective is to minimize the mean
error
P squared
1
? ?(f (i) )+
of reconstructions of the the inputs given by m
(
W
i
? ? f (i) 2 , where W
? ? Rdm ?db , ?? ? Rdm are the second layer
?)
weights and biases respectively. The objective is minimized using standard backpropagation.
Figure 3: Coding layer activaAs is, the optimization does not take into consideration the round- tion values after training the auing used in the coding layer and consequently the output is not toencoder.
adapted for this operation. We follow Salakhutdinov et al. [17]
and use additive ?deterministic? Gaussian noise with zero mean
in the coding layer that is fixed in advance for each datapoint when performing a bottom-up pass
through the network. Using unit variance was sufficient to force almost all the activations near
{0, 1}. We tried other approaches, including simple thresholding but found the Gaussian noise to be
most successful without interfering with the optimization. Figure 3 shows the coding layer activation
values after backpropagation when noise has been added.
3.2
The tag propagation (TagProp) algorithm
Let V denote a fixed vocabulary of tags and I denote a list of input images. Our goal at test time,
given a new input image i0 , is to assign a set of tags v ? V that are most relevant to the content
of i0 . TagProp operates on pairwise distances to learn a conditional distribution of words given
images. More specifically, let yiw ? {1, ?1}, i ? I, w ? V be an indicator for whether tag w is
present in image i. In TagProp, the probability that yiw = 1 is given by ?(?w xiw + ?w ), xiw =
P
?1
is the logistic function, (?w , ?w ) are word-specific
j ?ij yjw where ?(z) = (1 + exp(?z))
model parameters to be estimated and ?ij are distance-based weights also to be estimated. More
specifically, ?ij is expressed as
exp(?dh (i, j))
,
0
j 0 exp(?dh (i, j ))
dh (i, j) = hdij ,
?ij = P
h?0
(4)
where we shall call dij the base distance between images i and j. Let ? = {?w ?w ? V, ?w ?w ?
V, h} denote the list of model parameters.
The model is trained to maximize the following quasiP
likelihood of the data given by L = i,w ciw log p(yiw ), ciw = n1+ if yiw = 1 and n1? otherwise,
where n+ is the total number of positive labels of w and likewise for n? and missing labels. This
weighting allows us to take into account imbalances between label presence and absence. Combined
with the logistic word models, it accounts for much higher recall in rare tags which would normally
be less likely to be recalled in a basic k-NN setup. Optimization of L is performed using a projected
gradient method for enforcing the non-negativity constraint in h.
The choice of base distance used depends on the image representation. In the above description, the
model was derived assuming only a single base distance is computed between images. This can be
generalized to an arbitrary number of distances by letting h be a parameter vector and letting dh (i, j)
be a weighted combination of distances in h. Under this formulation, multiple descriptors of images
can be computed and weighted. The best performance of TagProp [9] was indeed obtained using this
multiple metric formulation in combination with the logistic word models. In our case, Euclidean
distance is used and Hamming distance for binary codes. Furthermore, we only consider pairwise
distances from the K nearest neighbors, where K is chosen though cross validation.
5
4
Experiments
We perform experimental evaluation of our methods on 4 datasets: one dataset, STL-10, for object
recognition to benchmark our hierarchical model and three datasets for annotation: Natural Scenes,
IAPRTC-12 and ESP-Game 2 .
For all our experiments, we use k1 = 512 first module bases, k2 = 1024 second module bases,
receptive field sizes of 6 ? 6 and 2 ? 2 and tile sizes (nt ) of 16 ? 16 and 6 ? 6. The total number
of features for the combined first and second module representation is thus 5(k1 + k2 ) = 7680.
Images are resized such that the longest side is no larger than 300 pixels with preserved aspect ratio.
The first module stride length is chosen based on the length of the longest side of the image: 4 if
the side is less than 128 pixels, 6 if less than 214 pixels and 8 otherwise. The second module stride
length is fixed at 2. For training the autoencoder, we use 10 epochs (passes over the training set)
with minibatches of size no larger than 1000. Optimization is done using Polak Ribiere conjugate
gradients with 3 linesearches per minibatch. 3
We also incorporate the use of self-taught learning [6] in our annotation experiments by utilizing
the Mirflickr dataset for dictionary learning. Mirflickr is a collection of 25000 images taken from
flickr and deemed to have high interestness rating. We randomly sampled 10000 images from this
dataset for training K-SVD on both modules. All reported results for Natural Scenes, IAPRTC-12
and ESP-Game use self-taught learning. Our code for feature learning will be made available online.
4.1
STL-10
The STL-10 dataset is a collection of
96 ? 96 images of 10 classes with images partitioned into 10 folds of 1000 Table 1: A selection of the best results obtained on the STLimages each and a test set of size 10 dataset.
8000. Alongside these labeled imMethod
Accuracy
ages is a set of 100000 unlabeled imSparse filtering [18]
53.5%
ages that may or may not come from
OMP, k = 1600 [13]
54.9%
the same distribution as the training
OMP, SC encoder, k = 1600 [13]
59.0%
data. The evaluation procedure is
Receptive field learning, 3 modules [19]
60.1%
to perform representation learning on
Video unsup features [20]
61.0%
the unlabeled data and apply the repHierarchical matching persuit [21]
64.5%
resentations to the training set, aver1st Module
56.4 %
aging test errors across all folds. We
1st + 2nd Module
62.1 %
randomly chose 10000 images from
the unlabeled set for training and use
a linear L2-SVM for classification with 5-fold cross validation for model selection.
Table 1 shows our results on STL-10. Our 2 module architecture outperforms all existing approaches
except for the recently proposed hierarchical matching pursuit (HMP). HMP uses joint layerwise
pooling and separate training for RGB and grayscale dictionaries, approaches which may also be
adapted to our method. Moreover, we hypothesize that further improvements can be made when the
receptive field learning strategies of Coates et al. [19] and Jia et al. [22] are incorporated into a third
module.
4.2
Natural scenes
The Natural Scenes dataset is a multi-label collection of 2000 images from 5 classes: desert, forest,
mountain, ocean and sunset. We follow standard protocol and report the average results of 5 metrics
using 10 fold cross validation: Hamming loss (HL), one error (OE), coverage (C), ranking loss
(RL) and average precision (AP). For space considerations, these metrics are defined in the appendix. To perform model selection with TagProp, we perform 5-fold cross validation with each of
the 10-folds to determine the value of K which minimizes Hamming loss.
2
Tags for IAPRTC-12 and ESP-Game as well as the features used by existing approaches can be found at
http://lear.inrialpes.fr/people/guillaumin/data.php
3
Rasmussen?s minimize routine is used.
6
Table 2: A selection of the best results obtained on the Natural Scenes dataset. Arrows indicate
direction of improved performance.
Method
ML-KNN [23]
ML-I2C [24]
InsDif [25]
ML-LI2C [24]
1st Module
1st Module, 256-bit
1st + 2nd Module
1st + 2nd Module, 256-bit
HL ?
0.169
0.159
0.152
0.129
0.113
0.113
0.100
0.106
OE ?
0.300
0.311
0.259
0.190
0.170
0.169
0.140
0.155
C?
0.939
0.883
0.834
0.624
0.580
0.585
0.554
0.558
RL ?
0.168
0.156
0.140
0.091
0.080
0.082
0.074
0.075
AP ?
0.803
0.804
0.830
0.881
0.895
0.894
0.910
0.903
Table 3: A selection of the best results obtained on the IAPRTC-12 dataset (left) and ESP-Game
(right) datasets.
Method
MBRM [26]
LASSO [7]
JEC [7]
GS [12]
CCD [8]
TagProp (? SD) [9]
TagProp (? ML) [9]
1st Module
1st Module, 256-bit
1st + 2nd Module
1st + 2nd Module, 256-bit
P
0.24
0.28
0.28
0.32
0.44
0.41
0.46
0.37
0.34
0.42
0.36
R
0.23
0.29
0.29
0.29
0.29
0.30
0.35
0.25
0.22
0.29
0.25
N+
223
246
250
252
251
259
266
241
236
252
244
P
0.18
0.21
0.22
0.36
0.39
0.39
0.37
0.35
0.38
0.37
R
0.19
0.24
0.25
0.24
0.24
0.27
0.20
0.20
0.22
0.23
N+
209
224
224
232
232
239
231
231
228
236
Table 2 shows the results of our method. In all five measures we obtain improvement over previous
methods. Furthermore, using 256-bit codes offers near equivalent performance. As in the case of
STL-10, improvements are made over a single module.
4.3
IAPRTC-12 and ESP-Game
IAPRTC-12 is a collection of 20000 images with a vocabulary size of |V | = 291 and an average
of 5.7 tags per image. ESP-Game is a collection of 60000 images with |V | = 268 and an average
of 4.7 tags per class. Following Guillaumin et al. [9] we apply experiments to a pre-defined subset
of 20000 images. Using standard protocol performance is evaluated using 3 measures: precision
(P), recall (R) and the number of recalled tags (N+). N + indicates the number of tags that were
recalled at least once for annotation on the test set. Annotations are made by choosing the 5 most
probable tags for each image as is done with previous evaluations. As with the natural scenes dataset,
we perform 5-fold cross validation to determine K for training TagProp.
Table 3 shows our results with IAPRTC-12 on the left and ESP-Game on the right. Our results
give comparable performance to CCD and the single distance (SD) variation of TagProp. Unfortunately, we are unable to match the recall values obtained with the multiple metric (ML) variation of
TagProp. Of importance, we outperform GS who specifically studied the use of feature selection.
Our 256-bit codes suffer a loss of performance on IAPRTC-12 but give near equivalent results on
ESP-Game. We note again that our features were learned on an entirely different dataset (Mirflickr)
in order to show their generalization capabilities.
Finally, we perform two qualitative experiments. Figure 4 shows sample unsupervised retrieval
results using the learned 256-bit codes on IAPRTC-12 and ESP-Game while figure 5 illustrates
sample annotation performance when training on one dataset and annotating the other. These results
show that our codes are able to capture high-level semantic concepts that perform well for retrieval
and transfer learning across datasets. We note however, that when annotating ESP-game when
training was done on IAPRTC-12 led to more false human annotations (such as the bottom-right
7
Figure 4: Sample 256-bit unsupervised retrieval results on ESP-Game (top) and IAPRTC-12 (bottom). A query image from the test set is used to retrieve the four nearest neighbors from the training
set.
Figure 5: Sample 256-bit annotation results when training on one dataset and annotating the other.
Top: Training on ESP-Game, annotation on IAPRTC-12. Bottom: Training on IAPRTC-12, annotation on ESP-Game.
image in figure 5). We hypothesize that this is due to a larger proportion of persons in the IAPRTC12 training set.
5
Conclusion
In this paper we introduced a hierarchical model for learning feature representations of standard
sized color images for the task of image annotation. Our results compare favorably to existing
approaches that use over a dozen handcrafted image descriptors.
Our primary goal for future work is to test the effectiveness of this approach on web-scale annotation systems with millions of images. The success of self-taught learning in this setting means
only one dictionary per module ever needs to be learned. Furthermore, our features can be used
in combination with any nearest neighbor based algorithm for annotation. It is our hope that the
successful use of binary codes for annotation will allow further research to bridge the gap between
the annotation algorithms used on small scale problems to those required for web scale tasks. We
also intend to evaluate the effectiveness of semantic hashing on large databases when much smaller
codes are used. Krizhevsky et al. [27] evaluated semantic hashing using very deep autoencoders on
tiny (32 ? 32) images. Future work also involves performing similar experiments but on standard
sized RGB images.
Acknowledgments
The authors thank Axel Soto as well as the anonymous reviewers for helpful discussion and comments. This work was funded by NSERC and the Alberta Innovates Centre for Machine Learning.
8
References
[1] T Huang. Linear spatial pyramid matching using sparse coding for image classification. CVPR, pages
1794?1801, 2009.
[2] K. Yu F. Lv T. Huang J. Wang, J. Yang and Y. Gong. Locality-constrained linear coding for image
classification. In CVPR, pages 3360?3367, 2010.
[3] R. Ranganath H Lee, R. Grosse and A.Y. Ng. Convolutional deep belief networks for scalable unsupervised learning of hierarchical representations. ICML, pages 1?8, 2009.
[4] K. Yu, Y. Lin, and J. Lafferty. Learning image representations from the pixel level via hierarchical sparse
coding. In CVPR, pages 1713?1720, 2011.
[5] L. Bo, X. Ren, and D. Fox. Hierarchical Matching Pursuit for Image Classification: Architecture and Fast
Algorithms. In NIPS, 2011.
[6] Rajat Raina, Alexis Battle, Honglak Lee, Benjamin Packer, and Andrew Y Ng. Self-taught learning,
pages 759?766. ICML, 2007.
[7] A. Makadia, V. Pavlovic, and S. Kumar. A new baseline for image annotation. In ECCV, volume 8, pages
316?329, 2008.
[8] H. Nakayama. Linear Distance Metric Learning for Large-scale Generic Image Recognition. PhD thesis,
The University of Tokyo.
[9] M. Guillaumin, T. Mensink, J. Verbeek, and C. Schmid. Tagprop: Discriminative metric learning in
nearest neighbor models for image auto-annotation. In ICCV, pages 309?316, 2009.
[10] D. Tsai, Y. Jing, Y. Liu, H.A. Rowley, S. Ioffe, and J.M. Rehg. Large-scale image annotation using visual
synset. In ICCV, pages 611?618, 2011.
[11] J. Weston, S. Bengio, and N. Usunier. Large scale image annotation: learning to rank with joint wordimage embeddings. Machine Learning, 81(1):21?35, 2010.
[12] S. Zhang, J. Huang, Y. Huang, Y. Yu, H. Li, and D.N. Metaxas. Automatic image annotation using group
sparsity. In CVPR, pages 3312?3319, 2010.
[13] A. Coates and A.Y. Ng. The importance of encoding versus training with sparse coding and vector
quantization. In ICML, 2011.
[14] A. Torralba, R. Fergus, and Y. Weiss. Small codes and large image databases for recognition. In CVPR,
pages 1?8, 2008.
[15] R. Rubinstein, M. Zibulevsky, and M. Elad. Efficient implementation of the k-SVD algorithm using batch
orthogonal matching pursuit. Technical Report, 2008.
[16] M. Ranzato K. Jarrett, K. Kavukcuoglu and Y. LeCun. What is the best multi-stage architecture for object
recognition? ICCV, 6:2146?2153, 2009.
[17] G. Hinton and R. Salakhutdinov. Discovering binary codes for documents by learning deep generative
models. Topics in Cognitive Science, 3(1):74?91, 2011.
[18] Z. Chen S. Bhaskar J. Ngiam, P. W. Koh and A.Y. Ng. Sparse filtering. NIPS, 2011.
[19] A. Coates and A.Y. Ng. Selecting receptive fields in deep networks. NIPS, 2011.
[20] W. Zou, A. Ng, and Kai. Yu. Unsupervised learning of visual invariance with temporal coherence. In
NIPS Workshop on Deep Learning and Unsupervised Feature Learning, 2011.
[21] L. Bo, X. Ren, and D. Fox. Unsupervised Feature Learning for RGB-D Based Object Recognition. In
ISER, June 2012.
[22] Y. Jia, C. Huang, and T. Darrell. Beyond spatial pyramids: Receptive field learning for pooled image
features. In CVPR, 2012.
[23] M.L. Zhang and Z.H. Zhou. ML-KNN: A lazy learning approach to multi-label learning. Pattern Recognition, 40(7):2038?2048, 2007.
[24] Y. Hu Z. Wang and L.T. Chia. Multi-label learning by image-to-class distance for scene classification and
image annotation. In CIVR, pages 105?112, 2010.
[25] M.L. Zhang and Z.H. Zhou. Multi-label learning by instance differentiation. In AAAI, number 1, pages
669?674, 2007.
[26] SL Feng, R. Manmatha, and V. Lavrenko. Multiple Bernoulli relevance models for image and video
annotation. In CVPR, pages 1002?1009, 2004.
[27] A. Krizhevsky and G.E. Hinton.
ESANN, 2011.
Using very deep autoencoders for content-based image retrieval.
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4,094 | 4,705 | Finding Exemplars from Pairwise Dissimilarities
via Simultaneous Sparse Recovery
Ehsan Elhamifar
EECS Department
University of California, Berkeley
Guillermo Sapiro
ECE, CS Department
Duke University
Ren?e Vidal
Center for Imaging Science
Johns Hopkins University
Abstract
Given pairwise dissimilarities between data points, we consider the problem of
finding a subset of data points, called representatives or exemplars, that can efficiently describe the data collection. We formulate the problem as a row-sparsity
regularized trace minimization problem that can be solved efficiently using convex programming. The solution of the proposed optimization program finds the
representatives and the probability that each data point is associated with each one
of the representatives. We obtain the range of the regularization parameter for
which the solution of the proposed optimization program changes from selecting
one representative for all data points to selecting all data points as representatives.
When data points are distributed around multiple clusters according to the dissimilarities, we show that the data points in each cluster select representatives only
from that cluster. Unlike metric-based methods, our algorithm can be applied to
dissimilarities that are asymmetric or violate the triangle inequality, i.e., it does
not require that the pairwise dissimilarities come from a metric. We demonstrate
the effectiveness of the proposed algorithm on synthetic data as well as real-world
image and text data.
1
Introduction
Finding a subset of data points, called representatives or exemplars, which can efficiently describe
the data collection, is an important problem in scientific data analysis with applications in machine learning, computer vision, information retrieval, etc. Representatives help to summarize and
visualize datasets of images, videos, text and web documents. Computational time and memory requirements of classification algorithms improve by working on representatives, which contain much
of the information of the original data collection. For example, the efficiency of the NN method
improves [1] by comparing tests samples to K representatives as opposed to all N training samples,
where typically we have K N . Representatives provide clustering of data points, and, as the
most prototypical data points, can be used for efficient synthesis/generation of new data points.
The problem of finding representative data has been well-studied in the literature [2, 3, 4, 5, 6, 7, 8].
Depending on the type of the information that should be preserved by the representatives, algorithms
can be divided into two categories. The first group of algorithms finds representatives from data that
lie in one or multiple low-dimensional subspaces and typically operate on the measurement data
vectors directly [5, 6, 7, 8, 9, 10, 11]. The Rank Revealing QR (RRQR) algorithm [6, 9] assumes
that the data come from a low-rank model and tries to find a subset of columns of the data matrix
that corresponds to the best conditioned submatrix. Randomized and greedy algorithms have also
been proposed to find a subset of the columns of a low-rank matrix [5, 8, 10]. Assuming that the
data can be expressed as a linear combination of the representatives, [7, 11] formulate the problem
of finding representatives as a joint-sparse recovery problem, [7] showing that when the data lie in a
union of low-rank models, the algorithm finds representatives from each low-rank model.
1
The second group of algorithms finds representatives by assuming that there is a natural grouping
of the data collection based on an appropriate measure of similarity between pairs of data points [2,
4, 12, 13, 14]. As a result, such algorithms typically operate on similarities/dissimilarities between
data points. The Kmedoids algorithm [2] tries to find K representatives from pairwise dissimilarities
between data points. As solving the original optimization program is, in general, NP-hard [12], an
iterative approach is employed. The performance of Kmedoids, similar to Kmeans [15], depends on
initialization and decreases as the number of representatives, K, increases. The Affinity Propagation
(AP) algorithm [4, 13, 14] tries to find representatives from pairwise similarities between data points
by using a message passing algorithm. While AP has suboptimal properties and finds approximate
solutions, it does not require initialization and has been shown to perform well in problems such as
unsupervised image categorization [16] and facility location problems [17].
In this paper, we propose an algorithm for selecting representatives of a data collection given dissimilarities between pairs of data points. We propose a row-sparsity regularized [18, 19] trace minimization program whose objective is to find a few representatives that encode well the collection of
data points according to the provided dissimilarities. The solution of the proposed optimization program finds the representatives and the probability that each data point is associated with each one of
the representatives. Instead of choosing the number of representatives, the regularization parameter
puts a trade-off between the number of representatives and the encoding cost of the data points via
the representatives based on the dissimilarities. We obtain the range of the regularization parameter
where the solution of the proposed optimization program changes from selecting one representative
for all data points to selecting each data point as a representative. When there is a clustering of
data points, defined based on their dissimilarities, we show that, for a suitable range of the regularization parameter, the algorithm finds representatives from each cluster. Moreover, data points
in each cluster select representatives only from the same cluster. Unlike metric-based methods, we
do not require that the dissimilarities come from a metric. Specifically, the dissimilarities can be
asymmetric or can violate the triangle inequality. We demonstrate the effectiveness of the proposed
algorithm on synthetic data and real-world image and text data.
2
Problem Statement
We consider the problem of finding representatives from a collection of N data points. Assume
we are given a set of nonnegative dissimilarities {dij }i,j=1,...,N between every pair of data points
i and j. The dissimilarity dij indicates how well the data point i is suited to be a representative of
the data point j. More specifically, the smaller the value of dij is, the better the data point i is a
representative of the data point j.1 Such dissimilarities can be built from measured data points, e.g.,
by using the Euclidean/geodesic distances or the inner products between data points. Dissimilarities
can also be given directly without accessing or measuring the data points, e.g., they can be subjective
measurements of the relationships between different objects. We can arrange the dissimilarities into
a matrix of the form
? >? ?
?
d1
d11 d12 ? ? ? d1N
? ? ?
..
.. ? ? RN ?N ,
D , ? ... ? = ? ...
(1)
.
. ?
>
dN 1 dN 2 ? ? ? dN N
dN
where di ? RN denotes the i-th row of D.
Remark 1 We do not require the dissimilarities to satisfy the triangle inequality. In addition, we
do not assume symmetry on the pairwise dissimilarities. D can be asymmetric, where dij 6= dji
for some pairs of data points. In other words, how well data point i represents data point j can be
different from how well j represents i. In the experiments, we will show an example of asymmetric
dissimilarities for finding representative sentences in text documents.
Given D, our goal is to select a subset of data points, called representatives or exemplars, that efficiently represent the collection of data points. We consider an optimization program that promotes
selecting a few data points that can well encode all data points via the dissimilarities. To do so, we
consider variables zij associated with dissimilarities dij and denote by the matrix of all variables as
1
dii can be set to have a nonzero value, as we will show in the experiments on the text data.
2
Representatives for ? =0.002 ?
Representatives for ? =0.005 ?
max,2
data points
representatives
1
Representatives for ? =0.1 ?max,2
Representatives for ? =0.01 ?
max,2
max,2
data points
representatives
1
data points
representatives
1
Representatives for ? =1 ?
max,2
data points
representatives
1
0.5
0.5
0.5
0.5
0
0
0
0
0
?0.5
?0.5
?0.5
?0.5
?0.5
?1
?1
?1
0
1
2
3
4
5
?1
Representatives for ? =0.007 ?max,?
2
3
4
5
?1
Representatives for ? =0.05 ?max,?
data points
representatives
1
1
0
1
2
3
4
5
0
1
2
3
4
5
?1
Representatives for ? =0.9 ?max,?
data points
representatives
1
?1
?1
Representatives for ? =0.1 ?max,?
data points
representatives
1
0.5
?1
?1
0
data points
representatives
1
0.5
0.5
0.5
0
0
0
0
?0.5
?0.5
?0.5
?0.5
?0.5
?1
0
1
2
3
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5
?1
?1
0
1
2
3
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5
?1
1
2
3
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5
?1
2
3
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5
data points
representatives
0.5
?1
0
1
1
0
?1
0
Representatives for ? =1 ?max,?
0.5
?1
data points
representatives
1
?1
0
1
2
3
4
5
?1
0
1
2
3
4
5
Figure 1: Data points (blue dots) in two clusters and the representatives (red circles) found by the proposed
optimization program in (4) for several values of ? with ?max,q defined in (6). Top: q = 2, Bottom: q = ?.
? ?
z11
z>
1
? ? ?
Z , ? ... ? = ? ...
zN 1
z>
N
?
z12
..
.
???
zN 2
???
?
z1N
.. ? ? RN ?N ,
. ?
(2)
zN N
N
where z i ? R denotes the i-th row of Z. We interpret zij as the probability that data point i be a
representative for data point j, hence zij ? [0, 1]. A data point j can have multiple representatives in
PN
which case zij > 0 for all the indices i of the representatives. As a result, we must have i=1 zij =
1, which ensures that the total probability of data point j choosing all its representatives is equal
to one. Our goal is to select a few representatives that well encode the data collection according to
the dissimilarities. To do so, we propose a row-sparsity regularized trace minimization program on
Z that consists of two terms. First, we want the representatives to encode well all data points via
dissimilarities. If the data point i is chosen to be a representative of a data point j with probability
zij , the cost of encoding j with i is dij zij ? [0, dij ]. Hence, the total cost of encoding j using all
PN
its representatives is i=1 dij zij . Second, we would like to have as few representatives as possible
for all the data points. When the data point i is a representative of some of the data points, we have
z i 6= 0, i.e., the i-th row of Z is nonzero. Having a few representatives then corresponds to having
a few nonzero rows in the matrix Z. Putting these two goals together, we consider the following
minimization program
min
N X
N
X
dij zij + ?
j=1 i=1
N
X
I(kz i kq ) s. t. zij ? 0, ?i, j;
i=1
N
X
zij = 1, ?j,
(3)
i=1
where I(?) denotes the indicator function, which is zero when its argument is zero and is one otherwise. The first term in the objective function corresponds to the total cost of encoding all data points
using the representatives and the second term corresponds to the cost associated with the number
of the representatives. The parameter ? > 0 sets the trade-off between the two terms. Since the
minimization in (3) that involves counting the number of nonzero rows of Z is, in general, NP-hard,
we consider the following standard convex relaxation
min
N X
N
X
j=1 i=1
dij zij + ?
N
X
kz i kq s. t. zij ? 0, ?i, j;
i=1
N
X
zij = 1, ?j,
(4)
i=1
where, instead of counting the number of nonzero rows of Z, we use the sum of the `q -norms of the
rows of Z. Typically, we choose q ? {2, ?} for which the optimization program (4) is convex.2
Note that the optimization program (4) can be rewritten in the matrix form as
min tr(D > Z) + ?kZk1,q
where tr(?) denotes the trace operator, kZk1,q
vector whose elements are all equal to one.
s. t. Z ? 0, 1> Z = 1> ,
(5)
PN
, i=1 kz i kq , and 1 denotes an N -dimensional
2
It is typically the case that q = ? favors having 0 and 1 elements for Z, while q = 2 allows elements that
more often take other values in [0, 1]. Note that q = 1 also imposes sparsity in the nonzero rows of Z, which
is not desirable since it promotes only a few data points to be associated with each representative.
3
Z matrix for ? =0.002 ?max,2
Z matrix for ? =0.005 ?max,2
Z matrix for ? =0.01 ?max,2
1
10
0.8
20
0.8
20
30
60
10
20
30
40
50
60
0.2
50
0
60
Z matrix for ? =0.007 ?max,?
30
40
50
60
0.2
50
0
60
Z matrix for ? =0.05 ?max,?
0.8
20
0.8
20
60
10
20
30
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50
60
0
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60
0.8
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50
0
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40
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60
1
10
0.8
0.6
30
0.4
0.2
50
0
60
0
20
40
20
20
0.6
30
10
0.2
10
1
10
0.4
0.2
0.2
Z matrix for ? =1 ?max,?
0.6
40
10
10
1
10
30
40
50
0
0.4
40
Z matrix for ? =0.9 ?max,?
0.4
0.2
50
20
30
50
60
0.6
0.4
40
50
1
10
0.6
30
40
0.2
Z matrix for ? =0.1 ?max,?
1
10
30
0.6
30
0.4
40
20
0.8
20
30
10
1
10
0.6
0.4
40
20
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30
10
1
10
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50
0.8
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40
Z matrix for ? =1 ?max,2
1
10
20
0.6
30
Z matrix for ? =0.1 ?max,2
1
10
0.4
40
10
20
30
40
50
60
0.2
50
0
60
0.2
10
20
30
40
50
60
0
Figure 2: For the data points shown in Fig. 1, the matrix Z obtained by the proposed optimization program in
(4) is shown for several values of ?, where ?max,q is defined in (6). Top: q = 2, Bottom: q = ?.
As we change the regularization parameter ? in (4), the number of representatives found by the
algorithm changes. For small values of ?, where we put more emphasis on better encoding data
points via representatives, we obtain more representatives. In the limiting case of ? ? 0 all points
are selected as representatives, each point being the representative of itself, i.e., z ii = 1 for all i.
On the other hand, for large values of ?, where we put more emphasis on the row-sparsity of Z,
we select a small number of representatives. In the limiting case of ? ? ?, we select only one
representative for all data points. Figures 1 and 2 illustrate the representatives and the matrix Z,
respectively, for several values of ?. In Section 3, we compute the range of ? for which the solution
of (4) changes from a single representative to all points being representatives. Note that, similar to
the relationship between sparse dictionary leaning [20] and Kmeans, there is a relationship between
our method and Kmedoids. A discussion of this is part of a future publication.
Once we have solved the optimization program (4), we can find the representative indices from the
nonzero rows of Z. We can also obtain the clustering of data points into K clusters associated with
K representatives by assigning each data point to its closets representative. More specifically, if
i1 , ? ? ? , iK denote the indices of the representatives, data point j is assigned to the representative
R(j) according to R(j) = argmin`?{i1 ,??? ,iK } d`j . As mentioned before, the solution Z gives
the probability that each data point is associated with each one of the representatives, which also
provides a soft clustering of data points to the representatives. In Section 3 we show that when
there is a clustering of data points based on their dissimilarities (see Definition 1), each point selects
representatives from its own cluster.
3
Theoretical Analysis
In this section, we consider the optimization program (4) and study the behavior of its solution as
a function of the regularization parameter. First, we analyze the solution of (4) for a sufficiently
large value of ?. We obtain a threshold value on ? after which the solution of (4) remains the same,
selecting only one representative data point. More specifically, we show the following result.
Theorem 1 Consider the optimization program (4). Let ` , argmini 1> di and
?
N kdi ? d` k22
kdi ? d` k1
?max,2 , max
? >
,
?max,? , max
.
i6
=
`
i6=`
2
2
1 (di ? d` )
(6)
For q ? {2, ?}, when ? ? ?max,q , the solution of the optimization program (4) is equal to Z =
e` 1> , where e` denotes the vector whose elements are all zero except its `-th element, which is
equal to 1. In other words, the solution of (4) for ? ? ?max,q corresponds to choosing only the `-th
data point as the representative of all the data points.
Note that the threshold value of the regularization parameter, for which we obtain only one representative, is different for q = 2 and q = ?. However, the two cases obtain the same representative
given by the data point for which 1> di is minimum, i.e., the data point with the smallest sum of
4
?1
?2
?
Figure 3: Data points in two clusters with dissimilarities given by pairwise Euclidean distances. For ? <
? ? max{?1 , ?2 }, in the solution of the optimization program (4), points in each cluster are represented by
representatives from the same cluster.
dissimilarities to other data points. Notice also that when the dissimilarities are the Euclidean distances between the data points, the single representative corresponds to the data point closest to the
geometric median of all data points, as shown in the right plot of Figure 1.
When the regularization parameter ? is smaller than the threshold in (6), the optimization program
in (4) can find multiple representatives for each data point. However, when there is a clustering
of data points based on their dissimilarities (see Definition 1), we expect to select representatives
from each cluster. In addition, we expect that the data points in each cluster be associated with the
representatives in that cluster only.
Definition 1 Given dissimilarities {dij }i,j=1,...,N between N data points, we say that the data partitions into n clusters {Ci }ni=1 according to the dissimilarities, if for any data point j 0 in any Cj , the
largest dissimilarity to other data points in Cj is strictly smaller than the smallest dissimilarity to
the data points in any Ci different from Cj , i.e.,
di0 j 0 ,
max di0 j 0 < min min
0
i0 ?Cj
i6=j i ?Ci
?j = 1, . . . , n, ?j 0 ? Cj .
(7)
In other words, the data partitions into clusters {Ci }ni=1 , when the interclass dissimilarity is smaller
than the intraclass dissimilarity.
Next, we show that for a suitable range of the regularization parameter that depends on the intraclass
and interclass dissimilarities, the probability that a point chooses representatives from other clusters
is zero. More precisely, we have the following result.
Theorem 2 Given dissimilarities {dij }i,j=1,...,N between N data points, assume that the data partitions into n clusters {Ci }ni=1 according to Definition 1. Let ?c be defined as
di0 j 0 ).
di0 j 0 ? max
(min min
?c , min min
0
0
0
j
i ?Cj
j ?Cj i6=j i ?Ci
(8)
Then for ? ? ?c , the optimization program (4) finds representatives in each cluster, where the data
points in every Ci select representatives only from Ci . A less tight clustering threshold ?0c ? ?c on
the regularization parameter is given by
?0c , min
min
i6=j i0 ?Ci ,j 0 ?Cj
di0 j 0 ? max 0 max
di0 j 0 .
0
i
i 6=j ?Ci
(9)
The first term in the right-hand-side of (9) shows the minimum dissimilarity between data points
in two different clusters. The second term in the right-hand-side of (9) shows the maximum, over
all clusters, of the dissimilarity between different data points in each cluster. When ?c or ?0c increase, e.g., when the intraclass dissimilarities increase or the interclass dissimilarities decrease, the
maximum possible ? for which we obtain clustering increases. As an illustrative example, consider Figure 3, where data points are distributed in two clusters according to the dissimilarities
given by the pairwise Euclidean distances of the data points. Let ?i denote the diameter of cluster i and ? be the minimum distance among pairs of data points in different clusters. Assuming
max{?1 , ?2 } < ?, for ? < ? ? max{?1 , ?2 }, the solution of the optimization program (4) is of the
Z1 0
form Z = ?
, where ? ? RN ?N is a permutation matrix corresponding to the separation
0 Z2
of the data into the two clusters.
Remark 2 The results of Theorems 1 and 2 suggest that there is a range of the regularization
parameter for which we obtain only one representative from each cluster. In other words, if
5
20
15
10
5
0
?4
?2
10
10
10
?
25
20
15
10
5
0
?4
10
0
q = 2 , ? / ? =4
30
Left cluster
Right cluster
Number of Representatives
25
30
Number of Representatives
Number of Representatives
q = ? , ? / ? =1.1
Left cluster
Right cluster
?2
10
25
20
15
10
5
0
0
?4
10
?
q = ? , ? / ? =4
Left cluster
Right cluster
?2
10
?
30
Number of Representatives
q = 2 , ? / ? =1.1
30
20
15
10
5
0
0
10
Left cluster
Right cluster
25
10
?4
?2
10
0
10
10
?
Figure 4: Number of representatives obtained by the proposed optimization program in (4) for data points in
the two clusters shown in Fig. 1 as a function of the regularization parameter ? = ??max,q with q ? {2, ?}.
Representatives for ? =0.05 ?max,?
Z matrix for ? =0.05 ?max,?
Representatives for ? =0.5 ?max,?
Z matrix for ? =0.5 ?max,?
1
data points
representatives
3
1
data points
representatives
3
20
20
0.8
0.8
40
2
60
1
0.6
60
1
80
80
0.4
100
0
40
2
0.6
0.4
100
0
0.2
0.2
120
120
?1
?1
140
?1.5
?1
?0.5
0
0.5
1
1.5
20
40
60
80
100
120
140
0
140
?1.5
?1
?0.5
0
0.5
1
1.5
20
40
60
80
100
120
140
0
Figure 5: Representatives and the probability matrix Z obtained by our proposed algorithm in (4) for q = ?.
20 random data points are added to 120 data points generated by a mixture of 3 Gaussian distributions.
?max,q (Ci ) denotes the threshold on ? after which we obtain only one representative from Ci , then
for maxi ?max,q (Ci ) ? ? < ?c , the data points in each Ci select only one representative that is in
Ci . As we will show in the experiments, such an interval often exists and can, in fact, be large.
For a sufficiently small value of ?, where we put less emphasis in the row-sparsity term in the
optimization program (4), each data point becomes a representative, i.e., zii = 1 for all i. In such
a case, each data point forms its own cluster. From the result in Theorem 2, we obtain a threshold
?min such that for ? ? ?min the solution Z is equal to the identity matrix.
Corollary 1 Let ?min,q , minj (mini6=j dij ? djj ) for q ? {2, ?}. For ? ? ?min,q , the solution
of the optimization program (4) for q ? {2, ?} is equal to the identity matrix. In other words, each
data point is the representative of itself.
4
Experiments
In this section, we evaluate the performance of the proposed algorithm on synthetic and real datasets.
As scaling of D and ? by the same value does not change the solution of (4), we always scale
dissimilarities to lie in [0, 1] by dividing the elements of D by its largest element. Unless stated
otherwise, we typically set ? = ??max,q with ? ? [0.01, 0.1], for which we obtain good results.
4.1 Experiments on Synthetic Data
We consider the synthetic dataset shown in Figure 1 that consists of data points distributed around
two clusters. We run the proposed optimization program in (4) for both q = 2 and q = ? for several
values of ?. Figures 1 and 2 show the representatives and the matrix of variables Z, respectively, for
several values of the regularization parameter. Notice that, as discussed before, for small values of ?,
we obtain more representatives and as we increase ?, the number of representatives decreases. When
the regularization parameter reaches ?max,q , computed using our theoretical analysis, we obtain
only one representative for the dataset. It is important to note that, as we showed in the theoretical
analysis, when the regularization parameter is sufficiently small, data points in each cluster only
select representatives from that cluster (see Figure 2), i.e., Z has a block-diagonal structure when
its columns are permuted according to the clusters. Moreover, as Figure 2 shows, for a sufficiently
large range of the regularization parameter, we obtain only one representative from each cluster. To
better see this, we run the optimization program with ? = ??max,q for different values of ?. The
two left-hand side plots in Figure 4 show the number of the representatives for q = 2 and q = ?,
respectively, from each of the two clusters.
As shown, when ? gets larger than ?max,q , we obtain only one representative from the right cluster
and no representative from the left cluster, i.e., as expected, we obtain one representative for all
the data points. Also, when ? gets smaller than ?min,q , all data points become representatives, as
6
ISOLET Dataset
USPS Dataset
25
20
Classification Error (%)
20
Classification Error (%)
25
Rand
Kmedoids?w
Kmedoids?b
AP
Proposed
15
10
15
10
5
5
0
Rand
Kmedoids?w
Kmedoids?b
AP
Proposed
0
4%
16%
Percentage of Selected Training Samples
20%
40%
Percentage of Selected Training Samples
Figure 6: Classification error on the USPS (left) and ISOLET (right) datasets using representatives obtained
by different algorithms. Horizontal axis shows the percentage of the selected representatives from each class
(averaged over all classes). Dashed line shows the classification error (%) using all the training samples.
expected from our theoretical result. It is also important to note that, for a sufficiently large range of
the values of ?, we select only one representative from each cluster. The two right-hand side plots
in Figure 4 show the number of the representatives when we increase the distance between the two
clusters. Notice that we obtain similar results as before except that the range of ? for which we
select one representative from each cluster has increased. This is also expected from our theoretical
analysis, since ?c in (8) increases as the distance between the two clusters increases.
Note that we also obtain similar results for larger number of clusters. For better visualization, we
have shown the results for only two clusters. Also, when there is not a clear partitioning of the data
points into clusters according to Definition 1, e.g., when there are data points distributed between
different clusters, as shown in Figure 5, we still obtain similar results to what we have discussed
in our theoretical analysis. This suggests the existence of stronger theoretical guarantees for our
proposed algorithm, which is the subject of our future work.
4.2
Experiments on Real Data
In this section, we evaluate the performance of our proposed algorithm on real image and text data.
We report the result for q = ? as it typically obtains better results than q = 2.
4.2.1
NN Classification using Representatives
First, we consider the problem of finding prototypes for classification using the nearest neighbor
(NN) algorithm [15]. Finding representatives that correspond to the modes of the data distribution
helps to significantly reduce the computational cost and memory requirements of classification algorithms, while maintaining their performance. To investigate the effectiveness of our proposed
method for finding informative prototypes for classification, we consider two datasets of USPS [21]
and ISOLET [22]. We find the representatives of the training data in each class of a dataset and use
the representatives as a reduced training set to perform NN classification on the test data. We obtain
the representatives by taking dissimilarities to be pairwise Euclidean distances between data points.
We compare our proposed algorithm with AP [4], Kmedoids [2], and random selection of data points
(Rand) as the baseline. Since Kmedoids depends on initialization, we run the algorithm 1000 times
with different random initializations and report the results corresponding to the best solution (lowest
energy) and the worst solution (highest energy) as Kmedoids-w and Kmedoids-b, respectively. To
have a fair comparison, we run all algorithms so that they obtain the same number of representatives.
Figure 6 shows the average classification errors using the NN method for the two datasets. The
classification error using all training samples of each dataset is also shown with a black dashed
line. As the results show, the classification performance using the representatives found by our
proposed algorithm is close to that of using all the training samples. Specifically, in the USPS
dataset, using representatives found by our proposed method, which consist of only 16% of the
training samples, we obtain 6.2% classification error compared to 4.7% error obtained using all the
training samples. In the ISOLET dataset, with representatives corresponding to less than half of the
training samples, we obtain very close classification performance to using all the training samples
(12.4% error compared to 11.4% error). Notice that when the number of representatives decreases,
as expected, the classification performance also decreases. However, in all cases, our proposed
algorithm as well as AP are less affected by the decrease in the number of the representatives.
7
Figure 7: Some frames of a political debate video, which consists of multiple shots, and the automatically
computed representatives (inside red rectangles) of the whole video sequence using our proposed algorithm.
4.2.2
Video Summarization using Representatives
We now evaluate our proposed algorithm for finding representative frames of video sequences. We
take a political debate video [7], downsample the frames to 80 ? 100 pixels, and convert each frame
to a grayscale image. Each data point then corresponds to an 8000-dimensional vector obtained by
vectorizing each grayscale downsampled frame. We set the dissimilarities to be the Euclidean distances between pairs of data points. Figure 7 shows some frames of the video and the representatives
computed by our method. Notice that we obtain a representative for each shot of the video. It is
worth mentioning that the computed representatives do not change for ? ? [2.68, 6.55].
4.2.3
Finding Representative Sentences in Text Documents
As we discussed earlier, our proposed algorithm can deal with dissimilarities that are not necessarily
metric, i.e., can be asymmetric or violate the triangle inequality. We consider now an example of
asymmetric dissimilarities where we find representative sentences in the text document of this paper. We compute the dissimilarities between sentences using an information theory-based criterion
as follows [4]: we treat each sentence as a ?bag of words? and compute dij (how well sentence i
represents sentence j) based on the sum of the costs of encoding every word in sentence j using the
words in sentence i. More precisely, for sentences in the text of the paper, we extract the words delimited by spaces, we remove all punctuations, and eliminate words that have less than 5 characters.
For each word in sentence j, if the word matches3 a word in sentence i, we set the encoding cost for
the word to the logarithm of the number of words in sentence i, which is the cost of encoding the
index of the matched word. Otherwise, we set the encoding cost for the word to the logarithm of the
number of the words in the text dictionary, which is the cost of encoding the index of the word in all
the text. We also compute dii using the same procedure, i.e., dii 6= 0, which penalizes selecting very
long sentences. We found that 96% of the dissimilarities are asymmetric. The four representative
sentences obtained by our algorithm summarize the paper as follows:
?Given pairwise dissimilarities between data points, we consider the problem of finding a subset of data points,
called representatives or exemplars, that can efficiently describe the data collection.
?We obtain the range of the regularization parameter for which the solution of the proposed optimization program changes from selecting one representative for all data points to selecting all data points as representatives.
?When there is a clustering of data points, defined based on their dissimilarities, we show that, for a suitable
range of the regularization parameter, the algorithm finds representatives from each cluster.
?As the results show, the classification performance using the representatives found by our proposed algorithm
is close to that of using all the training samples.
Acknowledgment
E. Elhamifar and R. Vidal are supported by grants NSF CNS-0931805, NSF ECCS-0941463, NSF
OIA-0941362, and ONR N00014-09-10839. G. Sapiro acknowledges partial support by ONR,
DARPA, NSF, NGA, and AFOSR grants.
3
We consider a word to match another word, if either word is a substring of the other.
8
References
[1] S. Garcia, J. Derrac, J. R. Cano, and F. Herrera, ?Prototype selection for nearest neighbor classification: Taxonomy and empirical study,? IEEE Transactions on Pattern Analysis and Machine Intelligence,
vol. 34, no. 3, pp. 417?435, 2012.
[2] L. Kaufman and P. Rousseeuw, ?Clustering by means of medoids,? In Y. Dodge (Ed.), Statistical Data
Analysis based on the L1 Norm (North-Holland, Amsterdam), pp. 405?416, 1987.
[3] M. Gu and S. C. Eisenstat, ?Efficient algorithms for computing a strong rank-revealing qr factorization,?
SIAM Journal on Scientific Computing, vol. 17, pp. 848?869, 1996.
[4] B. J. Frey and D. Dueck, ?Clustering by passing messages between data points,? Science, vol. 315, pp.
972?976, 2007.
[5] J. A. Tropp, ?Column subset selection, matrix factorization, and eigenvalue optimization,? ACM-SIAM
Symp. Discrete Algorithms (SODA), pp. 978?986, 2009.
[6] C. Boutsidis, M. W. Mahoney, and P. Drineas, ?An improved approximation algorithm for the column
subset selection problem,? in Proceedings of SODA, 2009, pp. 968?977.
[7] E. Elhamifar, G. Sapiro, and R. Vidal, ?See all by looking at a few: Sparse modeling for finding representative objects,? in IEEE Conference on Computer Vision and Pattern Recognition, 2012.
[8] J. Bien, Y. Xu, and M. W. Mahoney, ?CUR from a sparse optimization viewpoint,? NIPS, 2010.
[9] T. Chan, ?Rank revealing QR factorizations,? Lin. Alg. and its Appl., vol. 88-89, pp. 67?82, 1987.
[10] L. Balzano, R. Nowak, and W. Bajwa, ?Column subset selection with missing data,? in NIPS Workshop
on Low-Rank Methods for Large-Scale Machine Learning, 2010.
[11] E. Esser, M. Moller, S. Osher, G. Sapiro, and J. Xin, ?A convex model for non-negative matrix factorization and dimensionality reduction on physical space,? IEEE Transactions on Image Processing, vol. 21,
no. 7, pp. 3239?3252, 2012.
[12] M. Charikar, S. Guha, A. Tardos, and D. B. Shmoys, ?A constant-factor approximation algorithm for the
k-median problem,? Journal of Computer System Sciences, vol. 65, no. 1, pp. 129?149, 2002.
[13] B. J. Frey and D. Dueck, ?Mixture modeling by affinity propagation,? Neural Information Processing
Systems, 2006.
[14] I. E. Givoni, C. Chung, and B. J. Frey, ?Hierarchical affinity propagation,? Conference on Uncertainty in
Artificial Intelligence, 2011.
[15] R. Duda, P. Hart, and D. Stork, Pattern Classification.
Wiley-Interscience, October 2004.
[16] D. Dueck and B. J. Frey, ?Non-metric affinity propagation for unsupervised image categorization,? International Conference in Computer Vision, 2007.
[17] N. Lazic, B. J. Frey, and P. Aarabi, ?Solving the uncapacitated facility location problem using message
passing algorithms,? International Conference on Artificial Intelligence and Statistics, 2007.
[18] R. Jenatton, J. Y. Audibert, and F. Bach, ?Structured variable selection with sparsity-inducing norms,?
Journal of Machine Learning Research, vol. 12, pp. 2777?2824, 2011.
[19] J. A. Tropp., ?Algorithms for simultaneous sparse approximation. part ii: Convex relaxation,? Signal
Processing, special issue ?Sparse approximations in signal and image processing?, vol. 86, pp. 589?602,
2006.
[20] M. Aharon, M. Elad, and A. M. Bruckstein, ?K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation,? IEEE Trans. on Signal Processing, vol. 54, no. 11, pp. 4311?4322,
2006.
[21] J. J. Hull, ?A database for handwritten text recognition research,? IEEE Transactions on Pattern Analysis
and Machine Intelligence, vol. 16, no. 5, pp. 550?554, 1994.
[22] M. Fanty and R. Cole, ?Spoken letter recognition,? in Neural Information Processing Systems, 1991.
9
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4,095 | 4,706 | Locally Uniform Comparison Image Descriptor
Andrew Ziegler? Eric Christiansen David Kriegman Serge Belongie
Department of Computer Science and Engineering, University of California, San Diego
[email protected], {echristiansen, kriegman, sjb}@cs.ucsd.edu
Abstract
Keypoint matching between pairs of images using popular descriptors like SIFT
or a faster variant called SURF is at the heart of many computer vision algorithms
including recognition, mosaicing, and structure from motion. However, SIFT and
SURF do not perform well for real-time or mobile applications. As an alternative
very fast binary descriptors like BRIEF and related methods use pairwise comparisons of pixel intensities in an image patch. We present an analysis of BRIEF and
related approaches revealing that they are hashing schemes on the ordinal correlation metric Kendall?s tau. Here, we introduce Locally Uniform Comparison Image
Descriptor (LUCID), a simple description method based on linear time permutation distances between the ordering of RGB values of two image patches. LUCID
is computable in linear time with respect to the number of pixels and does not
require floating point computation.
1
Introduction
Local image descriptors have long been explored in the context of machine learning and computer
vision. There are countless applications that rely on local feature descriptors, such as visual registration, reconstruction and object recognition. One of the most widely used local feature descriptors
is SIFT which uses automatic scale selection, orientation normalization, and histograms of oriented
gradients to achieve partial affine invariance [15]. SIFT is known for its versatility and reliable
recognition performance, but these characteristics come at a high computational cost.
Recently, mobile devices and affordable reliable imaging sensors have become ubiquitous. The
wide adoption of these devices has made new real-time mobile applications of computer vision
and machine learning feasible. Examples of such applications include visual search, augmented
reality, perceptual interfaces, and wearable computing. Despite this, these devices have less computational power than typical computers and perform poorly for floating point heavy applications.
These factors have provided an impetus for new efficient discrete approaches to feature description and matching. In this work we explore current trends in feature description and provide a new
view of BRIEF and its related methods. We also present a novel feature description method that is
surprisingly simple and effective.
1.1
Background
Bay et al. proposed SURF as an approximation to SIFT, a notable shift toward real-time feature
description [1]. SURF obtains a large speed up over SIFT while retaining most of its desirable
properties and comparable recognition rates. However, SURF is not generally suited to real-time
applications without acceleration via a powerful GPU [21].
In [3] Bosch et al. proposed Ferns as a classification based approach to key point recognition. Ferns
uses sparse binary intensity comparisons between pixels in an image patch for descriptive power.
?
This work was completed while the author was at UCSD.
1
This simple scheme provides real-time performance in exchange for expensive off-line learning.
In response to the success of Ferns, Calonder et al. presented a novel binary feature descriptor
they named BRIEF [4]. Rather than training off-line, BRIEF makes use of random pixel intensity
comparisons to create a binary descriptor quickly. These descriptors can be matched an order of
magnitude faster than SIFT with the Hamming distance, even on mobile processors. As a result,
BRIEF has come into widespread use and has inspired several variants based on the approach [12,
14, 19]. However, little explanation as to why or how these types of descriptors work is given.
There is a fuzzy notion that pairwise intensity comparisons are an approximation to signed intensity
gradients. This is not the whole story, and in fact these methods are sampling in an ad hoc manner
from a rich source of discriminative information.
1.2
Related work
In this work we diverge from the current paradigm for fast feature description and explore a deterministic approach based on permutations. The study of distances between permutations began near
the inception of group theory and has continued unabated since [5, 7, 8, 9, 11, 10, 16].
A notable early use of permutation based methods in the realm of visual feature description was presented by Bhat and Nayar in [2]. They investigated the use of rank permutations of pixel intensities
for the purpose of dense stereo, the motivation being to find a robust alternative to the `2 norm. Permutations on pixel intensities offer a transformed representation of the data which is naturally less
sensitive to noise and invariant to monotonic photometric transformations. Bhat and Nayar present
a similarity measure between two rank permutations that is based on the Kolmogorov Smirnov test.
Their measure was designed to be robust to impulse noise, sometimes called salt and pepper noise,
which can greatly corrupt a rank permutation. In [20] Scherer et al. reported that though Bhat and
Nayar?s method was useful, it suffered from poor discrimination.
In [18] Mittal and Ramesh proposed an improved version of the method presented by Bhat and
Nayar. Their improvement was in a similar vein to [20], based on a modification to Kendall?s tau
[11]. The key observation made was that both Kendall?s tau metric and Bhat and Nayar?s metric are
highly sensitive to Gaussian noise. To become robust to Gaussian noise Mittal and Ramesh account
for actual intensity differences while only considering uncorrelated order changes. We choose to
explore the Hamming and Cayley distances, in part because they are naturally robust to Gaussian
noise, impulse noise is not a major issue for modern imaging devices, and they are computable in
linear time as opposed to quadratic time.
Recently there has been more research on the application of ordinal correlation methods to sparse
visual feature description. In [22] and [13] ordinal methods were applied to SIFT descriptors. In
contrast to [2] and [20] the elements of the SIFT descriptor are sorted, rather than sorting pixel
intensities themselves. Though these methods do improve the recognition performance of SIFT they
add computational cost, rather than reducing it.
1.3
Our contributions
In this paper, we introduce LUCID, a novel approach to real-time feature description based on order
permutations. We contrast LUCID with BRIEF, and provide a theoretical basis for understanding
these two methods. We prove that BRIEF is effectively a locality sensitive hashing (LSH) scheme on
Kendall?s tau. It follows from this that other descriptors based on binary intensity comparisons are
dimensionality reduction schemes on Kendall?s tau. We then explore alternative distances based on
the observation that image patch matching can be viewed as a near duplicate recognition problem.
In the next section we describe LUCID, provide a background on permutation distances and discuss
optimizations for an efficient implementation. Section 3 provides an analysis of BRIEF and compares it to LUCID. Section 4 reports on experiments that evaluate LUCID?s accuracy and run time
relative to SURF and BRIEF.
2
2
LUCID
Here we present a new method of feature description that is surprisingly simple and effective. We
call our method Locally Uniform Comparison Image Descriptor or LUCID. Our descriptors implicitly encapsulate all possible intensity comparisons in a local area of an image. They are extremely
efficient to compute and are related through the generalized Hamming distance for efficient matching [10].
2.1
Constructing a descriptor
Let p1 and p2 be n ? n image patches with
c color channels. We can compute descriptors for both patches and the Hamming distance between them in three lines of Matlab as shown in Figure 1. Here desc1 and
desc2 are the order permutation representations for p1 and p2 respectively. Let
m = cn2 , then clearly this depiction has
an O(m log m) running time. However, our
native implementation makes use of a stable comparison-free linear time sort and thus
takes O(m) time and space. Descriptor construction is depicted in Figure 1.
2.2
[~, desc1] = sort(p1(:));
[~, desc2] = sort(p2(:));
distance = sum(desc1 ~= desc2);
Permutation distances
A more detailed discussion of the following
is given in [16]. Recall the definition of a
permutation: a bijective mapping of a finite
set onto itself. This mapping ? is a member of the symmetric group Sn formed by
function composition on the set of all permutations of n labelled objects. We write Figure 1: Top: LUCID feature construction and
?(i) = j to denote the action of ? with matching method in 3 lines of Matlab. Note: ~ is
i, j ? {1, 2, ..., n}. The permutation product used to ignore the first return value of sort; and the
for ?1 , ?2 ? Sn is defined as function com- second value is the order permutation. Bottom: An
position ?1 ?2 = ?1 ??2 , the permutation that illustration of an image patch split into its RGB color
results from first applying ?2 then ?1 . Every channels, vectorized and then sorted; inducing a perpermutation ? ? Sn can be written as a prod- mutation on the indices.
uct of disjoint cycles ?1 , ?2 , ..., ?` . Cycles are permutations such that ? k (i) = i for some k ? n
Qk
where ? k = j=1 ?. We will use the notation #cycles(?) = ` to denote the number of cycles in ?.
A convenient representation for a permutation ? ? Sn is the n dimensional vector with the ith
coordinate equal to ?(i); this is the permutation vector. The convex hull of the permutation vectors
Sn ? Rn is the permutation polytope of Sn . This is an n ? 1 dimensional polytope with |Sn | = n!
vertices. The vertices are equidistant from the centroid and lie on the surface of a circumscribed n?1
dimensional sphere. The vertices corresponding to two permutations ?1 , ?2 ? Sn are connected by
an edge if they are related by a pairwise adjacent transposition. This is analogous to Kendall?s tau,
defined to be the minimum number of pairwise adjacent transpositions between two vectors, more
precisely Kd (?1 , ?2 ) = |{(i, j)|?1 (i) < ?1 (j), ?2 (i) > ?2 (j), 1 ? i, j ? n}|.
There are at least two classes of distances that can be defined between permutations [16]. Spatial
distances can be viewed as measuring the distance travelled along some path between two vertices
of the permutation polytope. Examples of spatial distances are Kendall?s tau which steps along the
edges of the polytope, the Euclidean distance which takes the straight line path, and Spearman?s
footrule which takes unit steps on the circumscribed sphere of the polytope. A disorder distance
measures the disorder between two permutations and ignores the spatial structure of the polytope.
Examples of disorder distances are the generalized Hamming distance Hd (?1 , ?2 ) = |{i|?1 (i) 6=
?2 (i)}| which is the number of elements that differ between two permutation vectors and the Cayley
3
distance Cd (?1 , ?2 ) = n ? #cycles(?2 ?1?1 ) which is the minimum number of unrestricted transpositions between ?1 and ?2 . We choose the generalized Hamming distance to relate our descriptors
because it is much simpler than the Cayley distance to compute. Hamming also lends itself to SIMD
parallel processing unlike Cayley which is inherently serial. However, if time is not a constraint
experimental results show that the Cayley distance should be preferred for accuracy.
Disorder distances are not sensitive to Gaussian noise, but are highly sensitive to impulse noise.
In contrast, Kendall?s tau is confused by Gaussian noise, but is more resilient to impulse noise
[2, 20, 18]. Impulse noise can severely corrupt these permutations since it can cause pixels in a
patch to become maximal or minimal elements changing each element in the permutation vector.
In the presence of moderate impulse noise the Cayley and Hamming distances will likely become
maximal while Kendall?s tau would be at O(1/n) its maximal distance. Generally, modern imaging
devices do not suffer from severe impulse noise, but there are other sources of impulse noise such
as occlusions and partial shadows. LUCID is used with sparse interest points and only individual
image patches would be affected by impulse noise. Since impulse noise would cause the distance to
become maximal these bad matches can be reliably identified via threshold.
Kendall?s tau is normally used in situations where multiple independent judges are ranking sets or
subsets of objects, such as top-k lists, movie preferences or surveys. In these scenarios multiple
judges are asked to rank preferences and the permutation polytope can be used as a discrete analog
to histograms to gain valuable insight into the distribution of the judges? preferences. In the context
of sparse image patch matching, the imaging sensor ideally acts as a single consistent judge; thus a
single image patch will correspond to one vertex on the permutation polytope. Ideally, for a pair of
corresponding patches in different images the permutations should be identical. Thus in our scenario
the image sensor can be viewed as one judge comparing nearly identical objects. The structure of
the permutation polytope becomes less important in this context.
Since the Cayley and Hamming distances are computed in linear time rather than quadratic time
like Kendall?s tau, they may be better suited for fast image patch matching. In section 3 we present
a proof demonstrating that BRIEF is a locality sensitive hashing scheme on Kendall?s tau metric
between vectors of pixel intensities.
2.3
An efficient implementation
Table 1: Time in milliseconds to construct 10,000 Our choice to use the Hamming distance
descriptors and to exhaustively match 5000x5000 de- is inspired by the new Streaming SIMD
Extensions (SSE) instructions. SSE is
scriptors.
a simple way to add parallelism to naDescriptor
Dimension Construction Matching tive programs through vector operations.
In our implementation we use a 128-bit
LUCID-8-Gray
64
20
240
packed comparison which gives LUCID
LUCID-16-Gray
256
30
880
16x matching parallelism for grayscale
BRIEF
256
40
2130
image patches up to 16x16, and 8x parLUCID-24-RGB
1728
50
4120
allelism for RGB image patches up to
SURF
64
450
420
147x147. Many mobile processors have
these types of instructions, but even when
they are not available it is still possible to gain parallelism. One additional bit per descriptor element
can be reserved allowing the use of binary addition and bit masks to produce a packed Hamming
distance. For descriptor lengths less than 215 , 16 bits per element are needed. This strategy supports
RGB image patches up to 105x105 pixels and yields 4x parallelism on 64-bit processors. It is also
possible to randomly sample a small subset of pixels before sorting to achieve greater speed. This
operation can be interpreted as randomly projecting the descriptors into a lower dimension.
Order permutations are fast to construct and access memory in sequential order. Since pixel intensities are represented with small positive integers they are ideal candidates for stable linear time
sorting methods like counting and radix sort. These sorting algorithms access memory in linear
order and thus with the fewest number of possible cache misses. BRIEF accesses larger portions of
memory than LUCID in a non-linear fashion and should incur more time consuming cache misses.
Therefore LUCID offers a modest improvement in terms of descriptor construction time as shown
in Table 1.
4
We investigate three versions of LUCID since they are the first three multiples of eight: LUCID-24RGB, LUCID-16-Gray, and LUCID-8-Gray which respectively are LUCID on image patches that
are 24x24 in RGB color, 16x16 grayscale and 8x8 grayscale. Before construction a 5x5 averaging
blur is applied to the entire image to remove noise that may perturb the order permutation. BRIEF
also performs pre-smoothing; Calonder et al. reported that they found a 9x9 blurring kernel to be
?necessary and sufficient? [4].
We compare the running time of LUCID to the OpenCV implementations of SURF and BRIEF with
default parameters on a 2.66GHz Intel? Core? i7.1 In Table 1 timing results for SURF, BRIEF
and the variants of LUCID are shown. BRIEF uses 48x48 image patches and produces a descriptor
with 256 dimensions which is equal to the dimension of LUCID-16-Gray. Surprisingly, LUCID-16Gray is faster to match than BRIEF; this was not expected since BRIEF has the same complexity
as LUCID to match. This might indicate that there are further optimizations that can be made for
OpenCV?s implementation.
3
Understanding BRIEF and related methods
In [4] Calonder et al. propose BRIEF, an efficient binary descriptor. BRIEF is intended to be simple
to compute and match based solely on sparse intensity comparisons. These comparisons provide for
the efficient construction of a compact descriptor. Here we discuss their method as presented in [4].
Define a test ?
1, if p(x) < p(y)
? (p; x, y) :=
(1)
0, otherwise
where p is a square image patch and p(x) is the smoothed value of the pixel with the local coordinates x = (u, v)> . This test will represent one bit in the final descriptor. To construct a BRIEF
descriptor a set of pre-defined pixel comparisons are performed. This pattern is a set of nd pixel coordinate pairs (x, y) that should be compared
P in each image patch. A descriptor is then defined to be
the nd dimensional bitstring fnd (p) := 1?i?nd 2i?1 ? (p; xi , yi ). Calonder et al. suggest that intuitively these pairwise intensity comparisons capture the signs of intensity gradients. However, this
is not precise and in the next section we prove that the reason BRIEF works is that it inadvertently
approximates Kendall?s tau.
3.1
BRIEF is LSH on Kendall?s Tau
Consider a version of BRIEF where the pixel sampling pattern consists of all m
2 pairs of pixels.
Then the Hamming distance between two of these BRIEF descriptors is equivalent to the Kendall?s
tau distance between the pixel intensities of the vectorized image patches. The original formulation
of BRIEF is LSH on the normalized Kendall?s tau correlation metric.
Proof. Let p1 , p2 be m dimensional vectorized image patches. Define Bk (i, j) := I(pk (i) < pk (j))
where I is the indicator function. For image patches containing m pixels, BRIEF chooses a pattern of
pairs P ? {(i, j)|1 ? i < j ? m}, and for two vectorized image patches p1 , p2 , it returns the score
P
(i,j)?P I(B1 (i, j) 6= B2 (i, j)). When P = {(i, j)|1 ? i < j ? m}, this is precisely Kd (p1 , p2 ). It can
be shown that BRIEF satisfies the definition of LSH as defined in [6], consider a random pair (i, j) with i < j.
Then
X 1
0 0
0 0
P [B1 (i, j) 6= B2 (i, j)] =
m I(B1 (i , j ) 6= B2 (i , j )) = KdN (p1 , p2 ).
i0 <j 0
3.2
2
The DAG of Possible Comparisons
The motivation behind BRIEF was to create a compact descriptor that could take advantage of SSE.
This was in part inspired by hashing schemes that produced binary descriptors related by the Hamming distance [4]. However, these schemes require first constructing a large descriptor and then
1
We used a stable release of OpenCV, version 2.4.3. OpenCV is open source and all versions are publicly
available at http://opencv.willowgarage.com.
5
sampling from it. BRIEF is more efficient than these methods because it skips the step of constructing the large descriptor. BRIEF is essentially a short cut and instead it immediately performs LSH.
To our knowledge, the fact that BRIEF itself is an LSH scheme has not been previously discussed
in the literature.
In this instance the large descriptor would be the
possible pairwise pixel intensity compar set of all
2
isons in a patch, which has an impractical m
=
O(m
)
dimension. This set of comparisons can
2
be modelled as a directed
acyclic graph (DAG) with m nodes, one for each pixel in the vectorized
image patch, and m
2 edges. In this model, there exists a directed edge (i, j) connecting the node
that correspond to the pixel with index i to the one with index j in the vectorized image patch if
p(i) < p(j) where p is the m dimensional vectorized image patch and i 6= j.
The topological sort of this DAG produces a unique Hamiltonian path from the sole source node to
the sole sink node. The order in which the nodes are visited on this path is equivalent to the order
permutation produced by a stable sort of the pixel intensities. Since this path is unique the order
permutation implicitly captures all O(m2 ) possible comparisons in O(m) space. This is possible
because of the transitive property of the binary comparison and the stable order in which pixels are
sorted. This is how LUCID captures all the comparative information in a patch.
In [4] Calonder et al. explored several different types of pixel sampling patterns and concluded that
random sampling works the best in practice. This makes sense since BRIEF can be interpreted as
randomly sampling edges from the DAG. Random sampling will eventually converge to a complete
representation of the DAG through the transitive property. BRIEF can alternatively be viewed as a
random projection of the adjacency matrix of the DAG.
Several variants and extensions of BRIEF have been proposed where different patterns as well as
rotation and scale normalization are considered [12, 14, 19]. It follows from the proof in section 3.1
and the DAG model that these methods are dimensionality reduction schemes on Kendall?s tau.
4
Experiments
Table 2: Recognition Rates. The FAST (FKD) and SURF (SKD) keypoint detectors are used to
find the top 500 of 1500 keypoints in the first image for each pair. Ground truth homographies are
used to warp the keypoints into the other images. The ratio of correct matches for each descriptor to
the total number of ground truth matches is defined to be the recognition rate. For each image pair
and keypoint detector the highest and second highest recognition rates are bolded with the second
highest rate prefixed by an asterisk.
Image Pair
?
Bikes 1|2
Bikes 1|4
Bikes 1|6
Wall 1|2
Wall 1|4
Wall 1|6
Light 1|2
Light 1|4
Light 1|6
Trees 1|2
Trees 1|4
Trees 1|6
Jpeg 1|2
Jpeg 1|4
Jpeg 1|6
LUCID-24-RGB
FKD
SKD
0.94
0.92
?
0.65 ? 0.61
?
0.19
0.22
?
0.54
0.47
?
0.16
0.15
?
0.04
0.03
?
0.86
0.89
?
0.71
0.75
0.56
0.57
?
0.44 ? 0.37
?
0.20
0.10
?
0.09
0.06
0.95
0.99
?
0.88
0.89
?
0.37
0.39
LUCID-16-Gray
FKD
SKD
?
0.90
0.83
0.50
0.46
0.13
0.11
0.38
0.32
0.12
0.10
0.03
0.02
?
0.90
0.91
?
0.82
0.76
0.61
0.58
0.34
0.25
0.11
0.03
0.05
0.03
?
0.97
0.99
0.94
0.93
0.37
0.35
LUCID-8-Gray
FKD SKD
0.79
0.54
0.26
0.22
0.07
0.06
0.21
0.12
0.08
0.06
0.02
0.02
0.87
0.73
0.62
0.55
0.44
0.36
0.17
0.14
0.05
0.02
0.03
0.02
0.99
0.94
0.86
0.71
0.24
0.14
BRIEF
FKD
SKD
?
0.90 ? 0.90
0.81
0.84
0.73
0.75
0.87
0.82
0.64
0.64
0.17
0.17
0.60
0.83
0.60
0.79
?
0.59
0.78
0.79
0.69
0.67
0.42
0.63
0.42
?
0.80
0.92
?
0.80
0.92
0.79
0.90
SURF
FKD SKD
0.23
0.75
0.04
0.59
0.01 ? 0.39
0.17 ? 0.56
0.11 ? 0.32
0.03 ? 0.09
0.48
0.81
0.41
0.71
0.32 ? 0.65
0.10
0.36
0.00 ? 0.16
0.00 ? 0.07
0.77 ? 0.95
0.48
0.89
0.10 ? 0.61
We use a subset of the commonly used benchmarking dataset used in [17].2 Our subset consists
of the image pairs that do not undergo extreme affine warping since neither BRIEF nor LUCID
2
The dataset is available for download at http://www.robots.ox.ac.uk/?vgg/research/
affine/
6
(a) bikes
(b) light
Figure 3: Recognition rates for LUCID-*-RGB on the bikes 1|4 and light 1|4 image pairs. The rates
are plotted as a function of the width of the descriptor patch and of the blur kernel applied. The best
100 of 300 FAST keypoints were detected in the first image of each pair. We found a blur width of
5 rarely hurts performance and often helps. Performance increases monotonically with patch size
with diminishing returns after 30x30.
account for these transformations. These image pairs are denoted by name 1|k, where k represents
the second image used in the pair, e.g. bikes 1|k indicates the pair consisting of the first image of
the bikes set to the fifth. In each experiment we detect a large number of keypoints in the first image
of a set and select the top N keypoints sorted by response. For each pair of images the keypoints
are warped into the second image using a ground truth homography. Points that are warped out of
bounds are culled before describing the points with each descriptor. Exhaustive nearest neighbor
search is used to bring the points into correspondence. The recognition rate is then recorded as the
ratio of correct matches to the number of ground truth matches.
In Table 2 we summarize the result of our comparison to BRIEF and SURF. BRIEF and LUCID
perform well in most instances, though BRIEF degrades more slowly with respect to image transformations. This robustness can be attributed to the fact that BRIEF sparsely samples pixels. Most
of the images are taken parallel to the horizon so orientation estimation does not help and in fact
degrades SURF?s performance relative to BRIEF and LUCID. LUCID performs the best on the light
set which undergo exposure changes. This makes sense since the order permutation is invariant to
monotonic intensity transformations and unlike BRIEF captures all the comparative information.
4.1
Parameter selection
LUCID has three parameters, blur kernel width, image
patch size, and the option to use color or grayscale images. Figure 3 gives plots of recognition rate as a function of blur kernel width and patch size for the medium
difficulty warps of two different image sets. These plots
indicate that LUCID performs well with a 5x5 averaging
blur kernel, and that larger patches help with diminishing
returns. Though not shown here, we find that using color
improves recognition performance with an expected slow
down.
4.2
Distance distributions
Here we examine the discriminative capability of three
distances, the Cayley distance, the generalized Hamming
distance and Kendall?s tau on pixel intensities. The Ham- Figure 2: ROC curves for descriptors on
ming distance represents LUCID which approximates the image pair bikes 1|4 for 200 keypoints.
7
(a) Cayley
(b) Hamming
(c) Kendall?s tau
Figure 4: Histograms of distances for correct matches and impostors for the bikes 1|4 image pair.
The plots show the Cayley, the Hamming distance on the order permutations, and Kendall?s tau on
pixel intensities. These plots present a good separation of correct matches and impostors. Kendall?s
tau requires O(m2 ) time to compute while the Cayley and Hamming distances run in O(m) time
making them efficient alternatives. The Hamming distance is embarrassingly parallel and lends itself
well to existing SSE instructions making it the most efficient distance.
Cayley distance. In Figure 4 we plot the distance distributions for correct matches and impostors,
focusing on the medium difficulty warp of the bikes images. We chose this image set because bikes
is a natural man-made scene and its distributions are representative for the other image sets. An
ROC curve is shown in Figure 2 to visualize these results in a different way as well as for SURF and
BRIEF. BRIEF does particularly well on this image pair because the only transformation that occurs
is blur. Interestingly, BRIEF outperforms Kendall?s Tau and the other methods that use all the pixels.
BRIEF is in essence random projection dimensionality reduction for Kendall?s tau. This indicates
that random projections may improve the performance of the Cayley and Hamming distances as
well. It is important to note that Kendall?s tau is inefficient to compute with quadratic running time
contrasted with the linear running time of the Cayley and generalized Hamming distances.
5
Conclusions and future work
In this work we have presented an analysis of BRIEF and related methods providing a theoretical
basis as to how and why they work. We introduced a new simple and effective image descriptor that
performs comparably to SURF and BRIEF. For our comparison and simplicity we made use of every
pixel in an image patch. However, given BRIEF?s superior performance to Kendall?s tau we plan to
explore sampling patterns of pixels and other dimensionality reduction techniques. In addition, we
plan to incorporate scale and rotation normalization as in [12] and [19]. This will allow an in depth
comparison of our method to descriptors like SIFT and SURF.
LUCID offers a new simplified approach for efficient feature construction and matching. We plan
to investigate approximate nearest neighbor approaches like LSH and metric trees to improve the
speed of matching. It would also be useful to find a binary representation of LUCID to allow for a
more compact descriptor and use of existing LSH schemes. It is already possible to obtain such a
representation for LUCID through a method like WTAHash [23]. WTAHash produces an embedding
for ordinal feature spaces such that transformed feature vectors are in binary form and the Hamming
distance between them closely approximates the original metric.
Finally, we hope that this new understanding of BRIEF and other binary descriptors will allow for
the creation of new efficient visual feature descriptors. Spending less time processing visual features
provides more CPU time for core functionality and application complexity enabling new real-time
applications.
6
Acknowledgements
We would like to acknowledge Brian McFee for his helpful conversations. This work was supported
by ONR MURI Grant #N00014-08-1-0638.
8
References
[1] Herbert Bay, Andreas Ess, Tinne Tuytelaars, and Luc Van Gool. Speeded-up robust features (SURF).
Comput. Vis. Image Underst., 110(3):346?359, June 2008.
[2] D.N. Bhat and S.K. Nayar. Ordinal measures for image correspondence. Pattern Analysis and Machine
Intelligence, 20(4):415?423, Apr 1998.
[3] A. Bosch, A. Zisserman, and X. Muoz. Image classification using random forests and ferns. In Computer
Vision, 2007. ICCV 2007, pages 1?8, Oct. 2007.
[4] Michael Calonder, Vincent Lepetit, Christoph Strecha, and Pascal Fua. Brief: binary robust independent
elementary features. In Proceedings of the 11th European conference on Computer vision: Part IV,
ECCV?10, pages 778?792, Berlin, Heidelberg, 2010. Springer-Verlag.
[5] A. Cayley. Lxxvii. note on the theory of permutations. Philosophical Magazine Series 3, 34(232):527?
529, 1849.
[6] Moses S. Charikar. Similarity estimation techniques from rounding algorithms. In Proceedings of the
thiry-fourth annual ACM symposium on Theory of computing, STOC ?02, pages 380?388, New York, NY,
USA, 2002.
[7] Michael Deza, Liens ecole Normale Suprieure, and Tayuan Huang. Metrics on permutations, a survey.
Journal of Combinatorics, Information and System Sciences, 1998.
[8] Persi Diaconis and R. L. Graham. Spearman?s footrule as a measure of disarray. Journal of the Royal
Statistical Society. Series B (Methodological), 39(2):pp. 262?268, 1977.
[9] M. A. Fligner and J. S. Verducci. Distance based ranking models. Journal of the Royal Statistical Society.
Series B (Methodological), 48(3):pp. 359?369, 1986.
[10] R. W. Hamming. Error detecting and error correcting codes. Bell System Technical Journal, 29(2):147?
160, 1950.
[11] M. G. Kendall. A new measure of rank correlation. Biometrika, 30(1/2):pp. 81?93, 1938.
[12] S Leutenegger, M Chli, and R Siegwart. BRISK: Binary robust invariant scalable keypoints. In Proc. of
the IEEE International Conference on Computer Vision (ICCV), 2011.
[13] Bing Li, Rong Xiao, Zhiwei Li, Rui Cai, Bao-Liang Lu, and Lei Zhang. Rank-SIFT: Learning to rank
repeatable local interest points. In Computer Vision and Pattern Recognition (CVPR), pages 1737?1744,
June 2011.
[14] Jie Liu and Xiaohui Liang. I-BRIEF: A fast feature point descriptor with more robust features. In SignalImage Technology and Internet-Based Systems (SITIS), pages 322?328, Dec. 2011.
[15] D.G. Lowe. Object recognition from local scale-invariant features. In Computer Vision, volume 2, pages
1150?1157, 1999.
[16] John I. Marden. Analyzing and Modeling Rank Data. Chapman & Hall, 1995.
[17] Krystian Mikolajczyk and Cordelia Schmid. A performance evaluation of local descriptors. IEEE Trans.
Pattern Anal. Mach. Intell., 27(10):1615?1630, October 2005.
[18] A. Mittal and V. Ramesh. An intensity-augmented ordinal measure for visual correspondence. In Computer Vision and Pattern Recognition, volume 1, pages 849?856, June 2006.
[19] Ethan Rublee, Vincent Rabaud, Kurt Konolige, and Gary Bradski. ORB: An efficient alternative to SIFT
or SURF. International Conference on Computer Vision, 95(1):2564?2571, 2011.
[20] S Scherer, P Werth, and A Pinz. The discriminatory power of ordinal measures ? towards a new coefficient. 1, 1999.
[21] Timothy B. Terriberry, Lindley M. French, and John Helmsen. GPU accelerating speeded-up robust
features. In Proceedings of the 4th International Symposium on 3D Data Processing, Visualization and
Transmission, 3DPVT ?08, pages 355?362, Atlanta, GA, USA, 2008.
[22] M. Toews and W. Wells. Sift-rank: Ordinal description for invariant feature correspondence. In Computer
Vision and Pattern Recognition, pages 172?177, June 2009.
[23] Jay Yagnik, Dennis Strelow, David A. Ross, and Ruei-Sung Lin. The power of comparative reasoning. In
ICCV, pages 2431?2438, 2011.
9
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4,096 | 4,707 | Expectation Propagation in
Gaussian Process Dynamical Systems
Marc Peter Deisenroth?
Department of Computer Science
Technische Universit?at Darmstadt, Germany
Shakir Mohamed?
Department of Computer Science
University of British Columbia, Canada
Abstract
Rich and complex time-series data, such as those generated from engineering systems, financial markets, videos, or neural recordings are now a common feature
of modern data analysis. Explaining the phenomena underlying these diverse data
sets requires flexible and accurate models. In this paper, we promote Gaussian
process dynamical systems as a rich model class that is appropriate for such an
analysis. We present a new approximate message-passing algorithm for Bayesian
state estimation and inference in Gaussian process dynamical systems, a nonparametric probabilistic generalization of commonly used state-space models. We
derive our message-passing algorithm using Expectation Propagation and provide
a unifying perspective on message passing in general state-space models. We
show that existing Gaussian filters and smoothers appear as special cases within
our inference framework, and that these existing approaches can be improved upon
using iterated message passing. Using both synthetic and real-world data, we
demonstrate that iterated message passing can improve inference in a wide range
of tasks in Bayesian state estimation, thus leading to improved predictions and
more effective decision making.
1
Introduction
The Kalman filter and its extensions [1], such as the extended and unscented Kalman filters [7],
are principled statistical models that have been widely used for some of the most challenging and
mission-critical applications in automatic control, robotics, machine learning, and economics. Indeed, wherever complex time-series are found, Kalman filters have been successfully applied for
Bayesian state estimation. However, in practice, time series often have an unknown dynamical
structure, and they are high dimensional and noisy, violating many of the assumptions made in established approaches for state estimation. In this paper, we look beyond traditional linear dynamical
systems and advance the state-of the-art in state estimation by developing novel inference algorithms
for the class of nonlinear Gaussian process dynamical systems (GPDS).
GPDSs are non-parametric generalizations of state-space models that allow for inference in time
series, using Gaussian process (GP) probability distributions over nonlinear transition and measurement dynamics. GPDSs are thus able to capture complex dynamical structure with few assumptions,
making them of broad interest. This interest has sparked the development of general approaches for
filtering and smoothing in GPDSs, such as [8, 3, 5]. In this paper, we further develop inference
algorithms for GPDSs and make the following contributions: (1) We develop an iterative local message passing framework for GPDSs based on Expectation Propagation (EP) [11, 10], which allows
for refinement of the posterior distribution and, hence, improved inference. (2) We show that the
general message-passing framework recovers the EP updates for existing dynamical systems as a
special case and expose the implicit modeling assumptions made in these models. We show that EP
in GPDSs encapsulates all GPDS forward-backward smoothers [5] as a special case and transforms
them into iterative algorithms yielding more accurate inference.
* Authors contributed equally.
1
2
Gaussian Process Dynamical Systems
Gaussian process dynamical systems are a general class of discrete-time state-space models with
xt = h(xt?1 ) + wt , wt ? N (0, Q) , h ? GP h ,
(1)
z t = g(xt ) + v t , v t ? N (0, R) , g ? GP g ,
(2)
where t = 1, . . . , T . Here, x ? RD is a latent state that evolves over time, and z ? RE , E ? D,
are measurements. We assume i.i.d. additive Gaussian system noise w and measurement noise v.
The central feature of this model class is that both the measurement function g and the transition
function h are not explicitly known or parametrically specified, but instead described by probability
distributions over these functions. The function distributions are non-parametric Gaussian processes
(GPs), and we write h ? GP h and g ? GP g , respectively.
A GP is a probability distribution p(f ) over functions f that is specified by a mean function ?f
and a covariance function kf [15]. Consider a set of training inputs X = [x1 , . . . , xn ]> and
2
corresponding training targets y = [y1 , . . . yn ]> , yi = f (xi ) + w, w ? N (0, ?w
). The posterior predictive distribution at a test input x? is Gaussian distributed N (y? | ?f (x? ), ?f2 (x? )) with
?1
?1
mean ?f (x? ) = k>
y and variance ?f2 (x? ) = k?? ? k>
k? , where k? = kf (X, x? ),
?K
?K
k?? = kf (x? , x? ), and K is the kernel matrix.
Since the GP is a non-parametric model, its use in GPDSs is desirable as it results in fewer restrictive
model assumptions, compared to dynamical systems based on parametric function approximators for
the transition and measurement functions (1)?(2). In this paper, we assume that the GP models are
trained, i.e., the training inputs and corresponding targets as well as the GP hyperparameters are
known. For both GP h and GP g in the GPDS, we used zero prior mean functions. As covariance
functions kh and kg we use squared- exponential covariance functions with automatic relevance
determination plus a noise covariance function to account for the noise in (1)?(2).
Existing work for learning GPDSs includes the Gaussian process dynamical model (GPDM) [20],
which tackles the challenging task of analyzing human motion in (high-dimensional) video sequences. More recently, variational [2] and EM-based [19] approaches for learning GPDS were proposed. Exact Bayesian inference, i.e., filtering and smoothing, in GPDSs is analytically intractable
because of the dependency of the states and measurements on previous states through the nonlinearity of the GP. We thus make use of approximations to infer the posterior distributions p(xt |Z) over
latent states xt , t = 1, . . . , T , given a set of observations Z = z 1:T . Existing approximate inference
approaches for filtering and forward-backward smoothing are based on either linearization, particle
representations, or moment matching as approximation strategies [8, 3, 5].
A principled incorporation of the posterior GP model uncertainty into inference in GPDSs is necessary, but introduces additional uncertainty. In tracking problems where the location of an object is
not directly observed, this additional source of uncertainty can eventually lead to losing track of the
latent state. In this paper, we address this problem and propose approximate message passing based
on EP for more accurate inference. We will show that forward-backward smoothing in GPDSs [5]
benefits from the iterative refinement scheme of EP, leading to more accurate posterior distributions
over the latent state and, hence, to more informative predictions and improved decision making.
3
Bayesian State Estimation using Expectation Propagation
Expectation Propagation [10, 11] is a widely-used deterministic algorithm for approximate Bayesian
inference that has been shown to be highly accurate in many problems, including sparse regression
models [17], GP classification [9], and inference in dynamical systems [13, 6, 18]. EP is derived
using a factor-graph, in which the distribution
Q over the latent state p(xt |Z) is represented as the
product of factors fi (xt ), i.e., p(xt |Z) = i fi (xt ). EP then specifies an Q
iterative message passing
algorithm in which p(xt |Z) is approximated by a distribution q(xt ) = i qi (xt ), using approximate messages qi (xt ). In EP, q and the messages qi are members of the exponential family, and
q is determined such that the the KL-divergence KL(p||q) is minimized. EP is provably robust for
log-concave messages [17] and invariant under invertible variable transformations [16]. In practice,
EP has been shown to be more accurate than competing approximate inference methods [9, 17].
In the context of the dynamical system (1)?(2), we consider factor graphs of the form of Fig. 1 with
three types of messages: forward, backward, and measurement messages, denoted by the symbols
2
qB(xt)
xt
qM(xt)
p(xt+1|xt)
qB(xt)
qC(xt+1)
xt+1
qC(xt) qB(xt+1)
xt
qM(xt)
qM(xt+1)
xt+1
qC(xt+1)
qM(xt+1)
Figure 1: Factor graph (left) and fully factored graph (right) of a general dynamical system.
Algorithm 1 Gaussian EP for Dynamical Systems
1: Init: Set all factors qi to N (0, ?I); Set q(x1 ) = p(x1 ) and marginals q(xt6=1 ) = N (0, 1010 I)
2: repeat
3:
for t = 1 to T do
4:
for all factors qi (xt ), where i = B, M, C do
5:
Compute cavity distribution q \i (xt ) = q(xt )/qi (xt ) = N (xt | ?\i , ?\i ) with
?1 ?1
?\i = (??1
,
t ? ?i )
6:
?1
?\i = ?\i (??1
t ?t ? ?i ?i )
Determine moments of fi (xt )q \i (xt ), e.g., via the derivatives of
log Zi (?\i , ?\i ) = log ? fi (xt )q \i (xt )dxt
7:
(3)
(4)
Update the posterior q(xt ) ? N (xt | ?t , ?t ) and the approximate factor qi (xt ):
?t = ?\i + ?\i ?>
m,
\i
?m := d log Zi /d? ,
\i
?t = ?\i ? ?\i (?>
m ?m ? 2?s )?
\i
?s := d log Zi /d?
\i
qi (xt ) = q(xt )/q (xt )
(5)
(6)
(7)
8:
end for
9:
end for
10: until Convergence or maximum number of iterations exceeded
B, C, M, respectively. For EP inference, we assume a fully-factored graph, using which we compute
the marginal posterior distributions p(x1 |Z), . . . , p(xT |Z), rather than the full joint distribution
p(X|Z) = p(x1 , . . . , xT |Z). Both the states xt and measurements z t are continuous variables and
the messages qi are unnormalized Gaussians, i.e., qi (xt ) = si N (xt | ?i , ?i )
3.1
Implicit Linearizations Require Explicit Consideration
Alg. 1 describes the main steps of Gaussian EP for dynamical systems. For each node xt in the
fully-factored factor graph in Fig. 1, EP computes three messages: a forward, backward, and measurement message, denoted by qB (xt ), qC (xt ), and qM (xt ), respectively. The EP algorithm updates
the marginal q(xt ) and the messages qi (xt ) in three steps. First, the cavity distribution q \i (xt ) is
computed (step 5 in Alg. 1) by removing qi (xt ) from the marginal q(xt ). Second, in the projection step, the moments of fi (xt )q \i (xt ) are computed (step 6), where fi is the true factor. In the
exponential family, the required moments can be computed using the derivatives of the log-partition
function (normalizing constant) log Zi of fi (xt )q \i (xt ) [10, 11, 12]. Third, the moments of the
marginal q(xt ) are set to the moments of fi (xt )q \i (xt ), and the message qi (xt ) is updated (step 7).
We apply this procedure repeatedly to all latent states xt , t = 1, . . . , T , until convergence.
EP does not directly fit a Gaussian approximation qi to the non-Gaussian factor fi . Instead, EP
determines the moments of qi in the context of the cavity distribution such that qi = proj[fi q \i ]/q \i ,
where proj[?] is the projection operator, returning the moments of its argument.
To update the posterior q(xt ) and the messages qi (xt ), EP computes the log-partition function log Zi
in (4) to complete the projection step. However, for nonlinear transition and measurement models
3
in (1)?(2), computing Zi involves solving integrals of the form
Z
Z
p(a) = p(a|xt )p(xt )dxt = N (a | m(xt ), S(xt ))N (xt | b, B)dxt ,
(8)
where a = z t for the measurement message, or a = xt+1 for the forward and backward messages. In nonlinear dynamical systems m(xt ) is a nonlinear measurement or transition function.
In GPDSs, m(xt ) and S(xt ) are the corresponding predictive GP means and covariances, respectively, which are nonlinearly related to xt . Because of the nonlinear dependencies between a and xt ,
solving (8) is analytically intractable. We propose to approximate p(a) by a Gaussian distribution
? This Gaussian approximation is only correct for a linear relationship a = J xt , where
? ?).
N (a | ?,
J is independent of xt . Hence, the Gaussian approximation is an implicit linearization of the functional relationship between a and xt , effectively linearizing either the transition or the measurement
models.
When computing EP updates using the derivatives ?m and ?s according to (5), it is crucial to
explicitly account for the implicit linearization assumption in the derivatives?otherwise, the EP
updates are inconsistent. For example, in the measurement and the backward message, we directly
? i ). The consis? i, ?
approximate the partition functions Zi , i ? {M, C} by Gaussians Z?i (a) = N (?
\i
tent derivatives d(log Z?i )/d?\i and d(log Z?i )/d? of Z?i with respect to the mean and covariance
of the cavity distribution q are obtained by applying the chain rule, such that
?m =
?s =
?i
??
??\i
?i
d log Z
d?\i
?i
d log Z
d?\i
=
=
?i ? ?
?i
? log Z
? i ??\i
??
?i
?i ? ?
? log Z
? i ??\i
??
= J > ? RE?D ,
?i
??
??\i
? i )?1 J > ? R1?D ,
? i )> (?
= (a ? ?
?i
?i ? log Z
?i
Z
? i ??
??
= 12 ??(log
? RD?D ,
? i )> ? ?
?i
?
??\i
= J I4 J > ? RE?E?D?D ,
(9)
(10)
(11)
where I4 ? RD?D?D?D is an identity tensor. Note that with the implicit linear model a = J xt ,
? i /??\i vanish. Although we approximate Zi by a Gaussian Z?i ,
? i /??\i and ? ?
the derivatives ? ?
? i , which also
? i and covariance matrix ?
we are still free to choose a method of computing its mean ?
? i are general functions
? i )/??\i . However, even if ?
? i and ?
influences the computation of J = ?(?
? i /??\i must equal the corresponding partial
? i /??\i and ? ?
of ?\i and ?\i , the derivatives ? ?
i
? /??\i must be set to 0. Hence, the implicit linearization
? i /??\i and ? ?
derivatives in (11), and ? ?
expressed by the Gaussian approximation Z?i must be explicitly taken into account in the derivatives
to guarantee consistent EP updates.
3.2
Messages in Gaussian Process Dynamical Systems
We now describe each of the messages needed for inference in GPDSs, and outline the approximations required to compute the partition function in (4). Updating a message requires a projection
to compute the moments of the new posterior marginal q(xt ), followed by a Gaussian division to
update the message itself. For the projection step, we compute approximate partition functions
\i
\i
Z?i , where i ? {M, B, C}. Using the derivatives d log Z?i /d?t and d log Z?i /d?t , we update the
marginal q(xt ), see (5).
Measurement Message For the measurement message in a GPDS, the partition function is
Z
Z
\M
\M
\M
\M
ZM (?t , ?t ) = fM (xt )q\M (xt )dxt ? fM (xt )N (xt | ?t , ?t )dxt ,
(12)
fM (xt ) = p(z t |xt ) = N (z t | ?g (xt ), ?g (xt )),
(13)
where fM is the true measurement factor, and ?g (xt ) and ?g (xt ) are the predictive mean and covariance of the measurement GP GP g . In (12), we made it explicit that ZM depends on the moments
\M
\M
?t and ?t of the cavity distribution q\M (xt ). The integral in (12) is of the form (8), but is
intractable since solving it corresponds to a GP prediction at uncertain inputs [14], resulting in nonGaussian predictive distributions. However, the mean and covariance of a Gaussian approximation
Z?M to ZM can be computed analytically: either using exact moment matching [14, 3], or approximately by expected linearization of the posterior GP [8]; details are given in [4]. The moments of
4
\M
\M
Z?M are also functions of the mean ?t and variance ?t of the cavity distribution. By taking the
linearization assumption of the Gaussian approximation into account explicitly (here, we implicitly
linearize GP g ) when computing the derivatives, the EP updates remain consistent, see Sec. 3.1.
Backward Message To update the backward message qC (xt ), we require the partition function
Z
Z
\C
\C
\C
\C
ZC (?t , ?t ) = fC (xt )q\C (xt )dxt ? fC (xt )N (xt | ?t , ?t )dxt ,
(14)
Z
Z
fC (xt ) = p(xt+1 |xt )q\B (xt+1 )dxt+1 = N (xt+1 | ?h (xt ), ?h (xt ))q\B (xt+1 )dxt+1 . (15)
Here, the true factor fC (xt ) in (15) takes into account the coupling between xt and xt+1 , which
was lost in assuming the full factorization in Fig. 1. The predictive mean and covariance of GP h are
denoted ?h (xt ) and ?h (xt ), respectively. Using (15) in (14) and reordering the integration yields
Z
Z
\C
\C
ZC (?t , ?t ) ? q\B (xt+1 ) p(xt+1 |xt )q\C (xt )dxt dxt+1 .
(16)
? \C ) by
? \C , ?
We approximate the inner integral in (16), which is of the form (8), by N (xt+1 | ?
\C
\C
\C
?
? \C and ?
moment matching [14], for instance. Note that ?
are functions of ?t and ?t . This
Gaussian approximation implicitly linearizes GP h . Now, (16) can be computed analytically, and
\B
? \C + ?\B ) of ZC that allows us to
? \C , ?
we obtain a Gaussian approximation Z?C = N (?t+1 | ?
t+1
update the moments of q(xt ) and the message qC (xt ).
Forward Message
partition function
Similarly, for the forward message, the projection step involves computing the
Z
Z
\B
\B
\B
\B
ZB (?t , ?t ) = fB (xt )q\B (xt )dxt = fB (xt )N (xt | ?t , ?t )dxt ,
(17)
Z
Z
fB (xt ) = p(xt |xt?1 )q\C (xt?1 )dxt?1 = N (xt | ?f (xt?1 ), ?f (xt?1 ))q\C (xt?1 )dxt?1 ,
where the true factor fB (xt ) takes into account the coupling between xt?1 and xt , see Fig. 1. Here,
the true factor fB (xt ) is of the form (8). We propose to approximate fB (xt ) directly by a Gaussian
? B ). This approximation implicitly linearizes GP h . We obtain the updated
? B, ?
qB (xt ) ? N (?
posterior q(xt ) by Gaussian multiplication, i.e., q(xt ) ? qB (xt )q\B (xt ). With this approximation
we do not update the forward message in context, i.e., the true factor fB (xt ) is directly approximated
instead of the product fB (xt )q\B (xt ), which can result in suboptimal approximation.
3.3
EP Updates for General Gaussian Smoothers
We can interpret the EP computations in the context of classical Gaussian filtering and smoothing [1]. During the forward sweep, the marginal q(xt ) = q\C (xt ) corresponds to the filter distribution p(xt |z 1:t ). Moreover, the cavity distribution q\M (xt ) corresponds to the time update
p(xt |z 1:t?1 ). In the backward sweep, the marginal q(xt ) is the smoothing distribution p(xt |Z),
incorporating the measurements of the entire time series. The mean and covariance of Z?C can be
interpreted as the mean and covariance of the time update p(xt+1 |z 1:t ).
Updating the moments of the posterior q(xt ) via the derivatives of the log-partition function recovers
exactly the standard Gaussian EP updates in dynamical systems described by Qi and Minka [13].
For example, when incorporating an updated measurement message, the moments in (5) can also be
xz\M
\M
\M
\M
zx\M
, respectively, where ?t
=
written as ?t = ?t + K(z t ? ?z ) and ?t = ?t ? K?t
\M
\M
xz\M
\M
\M ?1
\M
cov[xt , z t ] and K = ?t
(?z ) . Here, ?z = E[g(xt )] and ?z = cov[g(xt )], where
xt ? q\M (xt ). Similarly, the updated moments of q(xt ) with a new backward message via (5)
\C
\C
\C
>
correspond to the updates [13] ?t = ?t + L(?t+1 ? ?t+1 ) and ?t = ?C
t + L(?t+1 ? ?t+1 )L ,
\C
\C
\C
\C
\C
where L = cov[xt , xt+1 ](?t+1 )?1 . Here, we defined ?t+1 = E[h(xt )] and ?t+1 = cov[h(xt )],
where xt ? q\C (xt ).
5
Table 1: Performance comparison on the synthetic data set. Lower values are better.
NLLx
MAEx
NLLz
EKS
?2.04 ? 0.07
0.03 ? 2.0 ? 10?3
?0.69 ? 0.11
EP-EKS
?2.17 ? 0.04
0.03 ? 2.0 ? 10?3
?0.73 ? 0.11
GPEKS
?1.67 ? 0.22
0.04 ? 4.6 ? 10?2
?0.75 ? 0.08
EP-GPEKS
?1.87 ? 0.14
0.04 ? 4.6 ? 10?2
?0.81 ? 0.07
GPADS
+ 1.67 ? 0.37
1.79 ? 0.21
1.93 ? 0.28
EP-GPADS
?1.91 ? 0.10
0.04 ? 4 ? 10?3
?0.77 ? 0.07
The iterative message-passing algorithm in Alg. 1 provides an EP-based generalization and a unifying view of existing approaches for smoothing in dynamical systems, e.g., (Extended/Unscented/
Cubature) Kalman smoothing and the corresponding GPDS smoothers [5]. Computing the messages
via the derivatives of the approximate log-partition functions log Z?i recovers not only standard EP
updates in dynamical systems [13], but also the standard Kalman smoothing updates [1].
Using any prediction method (e.g., unscented transformation, linearization), we can compute Gaussian approximations of (8). This influences the computation of log Z?i and its derivatives with respect
to the moments of the cavity distribution, see (9)?(10). Hence, our message-passing formulation is
also general as it includes all conceivable Gaussian filters/smoothers in (GP)DSs, solely depending
on the prediction technique used.
4
Experimental Results
We evaluated our proposed EP-based message passing algorithm on three data sets: a synthetic data
set, a low-dimensional simulated mechanical system with control inputs, and a high-dimensional
motion-capture data set. We compared to existing state-of-the-art forward-backward smoothers in
GPDSs, specifically the GPEKS [8], which is based on the expected linearization of the GP models,
and the GPADS [5], which uses moment-matching. We refer to our EP generalizations of these
methods as EP-GPEKS and EP-GPADS.
In all our experiments, we evaluated the inference methods using test sequences of measurements
Z = [z 1 , . . . , z T ]. We report the negative log-likelihood of predicted measurements using the
observed test sequence (NLLz ). Whenever available, we also compared the inferred posterior distribution q(X) ? p(X|Z) of the latent states with the underlying ground truth using the average
negative log-likelihood (NLLx ) and Mean Absolute Errors (MAEx ). We terminated EP after 100
iterations or when the average norms of the differences of the means and covariances of q(X) in
two subsequent EP iterations were smaller than 10?6 .
4.1
Synthetic Data
We considered the nonlinear dynamical system
xt+1 = 4 sin(xt ) + w ,
w ? N (0, 0.12 ) ,
zt = 4 sin(xt ) + v ,
v ? N (0, 0.12 ) .
We used p(x1 ) = N (0, 1) as a prior on the initial latent state. We assumed access to the latent state
and trained the dynamics and measurement GPs using 30 randomly generated points, resulting in
a model with a substantial amount of posterior model uncertainty. The length of the test trajectory
used was T = 20 time steps.
Tab. 1 reports the quality of the inferred posterior distributions of the latent state trajectories using the
average NLLx , MAEx , and NLLz (with standard errors), averaged over 10 independent scenarios.
For this dataset, we also compared to the Extended Kalman Smoother (EKS) and an EP-iterated EKS
(EP-EKS). Both inference methods make use of the known transition and measurement mappings
h and g, respectively. Iterated forward-backward smoothing with EP (EP-EKS, EP-GPEKS, EPGPADS) improved the smoothing posteriors using a single sweep only (EKS, GPEKS, GPADS).
The GPADS performed poorly across all our evaluation criteria for two reasons: First, the GPs were
trained using few data points, resulting in posterior distributions with a high degree of uncertainty.
Second, predictive variances using moment-matching are generally conservative and increased the
uncertainty even further. This uncertainty caused the GPADS to quickly lose track of the period of
the state, as shown in Fig. 2(a). By iterating forward-backward smoothing using EP (EP-GPADS),
the posteriors p(xt |Z) were iteratively refined, and the latent state could be followed closely as
indicated by both the small blue error bars in Fig. 2(a) and all performance measures in Tab. 1. EP
smoothing typically required a small number of iterations for the inferred posterior distribution to
closely track the true state, Fig. 2(b). On average, EP required fewer than 10 iterations to converge
to a good solution in which the mean of the latent-state posterior closely matched the ground truth.
6
Average NLL per data point
5
Latent State
2
True state
Posterior state distribution (EP?GPADS)
Posterior state distribution (GPADS)
0
?5
2
4
6
8
10
12
Time step
14
16
18
20
1
GPADS
EP?GPADS
0
?1
?2
5
(a) Example trajectory distributions with 95% confidence bounds.
10
15
EP iteration
20
25
30
(b) Average NLLx as a function of the EP iteration
with twice the standard error.
Figure 2: (a) Posterior latent state distributions using EP-GPADS (blue) and the GPADS (gray). The
ground truth is shown in red (dashed). The GPADS quickly loses track of the period of the state
revealed by the large posterior uncertainty. EP with moment matching (EP-GPADS) in the GPDS
iteratively refines the GPADS posterior and can closely follow the true latent state trajectory. (b)
Average NLLx per data point in latent space with standard errors of the posterior state distributions
computed by the GPADS and the EP-GPADS as a function of EP iterations.
4.2
Pendulum Tracking
We considered a pendulum tracking problem to demonstrate GPDS inference in multidimensional settings, as well as the ability to handle control inputs. The state x of the system is
? The pendulum
given by the angle ? measured from being upright and the angular velocity ?.
used has a mass of 1 kg and a length of 1 m, and random torques u ? [?2, 2] Nm were applied for a duration 200 ms (zero-order-hold control). The system noise covariance was set to
?w = diag(0.32 , 0.12 ). The state was measured indirectly by two bearings sensors with coordinates
(x1 , y1 ) = (?2, 0) and (x
according to z = [z1 , z2 ]> + v , v ?
2 , y2 ) = (?0.5, ?0.5), respectively,
sin
??y
N 0, diag(0.12 , 0.052 ) with zi = arctan cos ??xii , i = 1, 2. We trained the GP models using 4
randomly generated trajectories of length T = 20 time steps, starting from an initial state distribution p(x1 ) = N (0, diag(? 2 /162 , 0.52 )) around the upright position. For testing, we generated 12
random trajectories starting from p(x1 ).
Tab. 2 summarizes the performance Table 2: Performance comparison on the pendulum-swing
of the various inference methods. data. Lower values are better.
Generally, the (EP-)GPADS performed better than the (EP-)GPEKS
NLLx
MAEx
NLLz
GPEKS
?0.35 ? 0.39 0.30 ? 0.02 ?2.41 ? 0.047
across all performance measures.
EP-GPEKS ?0.33 ? 0.44 0.31 ? 0.02 ?2.39 ? 0.038
This indicates that the (EP-)GPEKS
GPADS
?0.80 ? 0.06 0.30 ? 0.02 ?2.37 ? 0.042
suffered from overconfident posteriEP-GPADS ?0.85 ? 0.05 0.29 ? 0.02 ?2.40 ? 0.037
ors compared to (EP-)GPADS, which
is especially pronounced in the degrading NLLx values with increasing EP iterations and the relatively high standard errors. In about
20% of the test cases, the inference methods based on explicit linearization of the posterior mean
function (GPEKS and EP-GPEKS) ran into numerical problems typical of linearizations [5], i.e.,
overconfident posterior distributions that caused numerical problems. We excluded these runs from
the results in Tab. 2. The inference algorithms based on moment matching (GPADS and EP-GPADS)
were numerically stable as their predictions are typically more coherent due to conservative approximations of moment matching.
4.3
Motion Capture Data
We considered motion capture data (from http://mocap.cs.cmu.edu/, subject 64) containing 10 trials of golf swings recorded at 120 Hz, which we subsampled to 20 Hz. After removing
observation dimensions with no variability we were left with observations z t ? R56 , which were
then whitened as a pre-processing step. For trials 1?7 (403 data points), we used the GPDM [20]
to learn MAP estimates of the latent states xt ? R3 . These estimated latent states and their corresponding observations are used to train the GP models GP f and GP g . Trials 8?10 were used as test
7
Figure 3: Latent space posterior distribution (95% confidence ellipsoids) of a test trajectory of the
golf-swing motion capture data. The further the ellipsoids are separated the faster the movement.
data without ground truth labels. The GPDM [20] focuses on learning a GPDS; we are interested in
good approximate inference in these models.
Fig. 3 shows the latent-state posterior distribution of a single test sequence (trial 10) obtained from
the EP-GPADS. The most significant prediction errors in observed space occurred in the region
corresponding to the yellow/red ellipsoids, which is a low-dimensional embedding of the motion
when the golf player hits the ball, i.e., the periods of high acceleration (poses 3?5).
Tab. 3 summarizes the results of inference on the golf data set in all test trials: Iterating forwardbackward smoothing by means of EP improved the inferred posterior distributions over the latent
states. The posterior distributions in latent space inferred by the EP-GPEKS were tighter than the
ones inferred by the EP-GPADS. The NLLz -values suffered a bit from this overconfidence, but the
predictive performance of the EP-GPADS and EP-GPEKS were similar. Generally, inference was
more difficult in areas with fast movements (poses 3?5 in Fig. 3) where training data were sparse.
The computational demand the two Table 3: Average inference performance (NLLz , motion
inference methods for GPDSs we capture data set). Lower values are better.
presented is vastly different. Highdimensional approximate inference
Test trial GPEKS EP-GPEKS GPADS EP-GPADS
Trial 8
14.20
13.82
14.28
14.09
in the motion capture example using
Trial 9
15.63
14.71
15.19
14.84
moment matching (EP-GPADS) was
26.68
25.73
25.64
25.42
Trial 10
about two orders of magnitude slower
than approximate inference based on
linearization of the posterior GP mean (EP-GPEKS): For updating the posterior and the messages for
a single time slice, the EP-GPEKS required less than 0.5 s, the EP-GPADS took about 20 s. Hence,
numerical stability and more coherent posterior inference with the EP-GPADS trade off against
computational demands.
5
Conclusion
We have presented an approximate message passing algorithm based on EP for improved inference and Bayesian state estimation in GP dynamical systems. Our message-passing formulation
generalizes current inference methods in GPDSs to iterative forward-backward smoothing. This
generalization allows for improved predictions and comprises existing methods for inference in the
wider theory for dynamical systems as a special case. Our new inference approach makes the full
power of the GPDS model available for the study of complex time-series data. Future work includes
investigating alternatives to linearization and moment matching when computing messages, and the
more general problem of learning in Gaussian process dynamical systems.
Acknowledgements
We thank Zhikun Wang for helping with the motion capture experiment and Jan Peters for valuable discussions. The research leading to these results has received funding from the European
Community?s Seventh Framework Programme (FP7/2007?2013) under grant agreement #270327
(ComPLACS) and the Canadian Institute for Advanced Research (CIFAR).
8
References
[1] B. D. O. Anderson and J. B. Moore. Optimal Filtering. Dover Publications, 2005.
[2] A. Damianou, M. K. Titsias, and N. D. Lawrence. Variational Gaussian Process Dynamical
Systems. In Advances in Neural Information Processing Systems. 2011.
[3] M. P. Deisenroth, M. F. Huber, and U. D. Hanebeck. Analytic Moment-based Gaussian Process
Filtering. In Proceedings of the 26th International Conference on Machine Learning, pages
225?232. Omnipress, 2009.
[4] M. P. Deisenroth and S. Mohamed. Expectation Propagation in Gaussian Process Dynamical
Systems: Extended Version, 2012. http://arxiv.org/abs/1207.2940.
[5] M. P. Deisenroth, R. Turner, M. Huber, U. D. Hanebeck, and C. E. Rasmussen. Robust Filtering
and Smoothing with Gaussian Processes. IEEE Transactions on Automatic Control, 2012.
[6] T. Heskes and O. Zoeter. Expectation Propagation for Approximate Inference in Dynamic
Bayesian Networks. In Proceedings of the International Conference on Uncertainty in Artificial Intelligence, pages 216?233, 2002.
[7] S. J. Julier and J. K. Uhlmann. Unscented Filtering and Nonlinear Estimation. Proceedings of
the IEEE, 92(3):401?422, March 2004.
[8] J. Ko and D. Fox. GP-BayesFilters: Bayesian Filtering using Gaussian Process Prediction and
Observation Models. Autonomous Robots, 27(1):75?90, 2009.
[9] M. Kuss and C. E. Rasmussen. Assessing Approximate Inference for Binary Gaussian Process
Classification. Journal of Machine Learning Research, 6:1679?1704, 2005.
[10] T. P. Minka. Expectation Propagation for Approximate Bayesian Inference. In Proceedings of
the 17th Conference on Uncertainty in Artificial Intelligence, pages 362?369. Morgan Kaufman Publishers, 2001.
[11] T. P. Minka. A Family of Algorithms for Approximate Bayesian Inference. PhD thesis, Massachusetts Institute of Technology, 2001.
[12] T. P. Minka. EP: A Quick Reference. 2008.
[13] Y. Qi and T. Minka. Expectation Propagation for Signal Detection in Flat-Fading Channels. In
Proceedings of the IEEE International Symposium on Information Theory, 2003.
[14] J. Qui?nonero-Candela, A. Girard, J. Larsen, and C. E. Rasmussen. Propagation of Uncertainty in Bayesian Kernel Models?Application to Multiple-Step Ahead Forecasting. In IEEE
International Conference on Acoustics, Speech and Signal Processing, pages 701?704, 2003.
[15] C. E. Rasmussen and C. K. I. Williams. Gaussian Processes for Machine Learning. The MIT
Press, 2006.
[16] M. W. Seeger. Expectation Propagation for Exponential Families. Technical report, University
of California Berkeley, 2005.
[17] M. W. Seeger. Bayesian Inference and Optimal Design for the Sparse Linear Model. Journal
of Machine Learning Research, 9:759?813, 2008.
[18] M. Toussaint and C. Goerick. From Motor Learning to Interaction Learning in Robotics,
chapter A Bayesian View on Motor Control and Planning, pages 227?252. Springer-Verlag,
2010.
[19] R. Turner, M. P. Deisenroth, and C. E. Rasmussen. State-Space Inference and Learning with
Gaussian Processes. In Proceedings of the International Conference on Artificial Intelligence
and Statistics, volume JMLR: W&CP 9, pages 868?875, 2010.
[20] J. M. Wang, D. J. Fleet, and A. Hertzmann. Gaussian Process Dynamical Models for Human
Motion. IEEE Transactions on Pattern Analysis and Machine Intelligence, 30(2):283?298,
2008.
9
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4,097 | 4,708 | Graphical Gaussian Vector for Image Categorization
Tatsuya Harada
The University of Tokyo/JST PRESTO
7-3-1 Hongo Bunkyo-ku, Tokyo Japan
[email protected]
Yasuo Kuniyoshi
The University of Tokyo
7-3-1 Hongo Bunkyo-ku, Tokyo Japan
[email protected]
Abstract
This paper proposes a novel image representation called a Graphical Gaussian
Vector (GGV), which is a counterpart of the codebook and local feature matching
approaches. We model the distribution of local features as a Gaussian Markov
Random Field (GMRF) which can efficiently represent the spatial relationship
among local features. Using concepts of information geometry, proper parameters
and a metric from the GMRF can be obtained. Then we define a new image feature
by embedding the proper metric into the parameters, which can be directly applied
to scalable linear classifiers. We show that the GGV obtains better performance
over the state-of-the-art methods in the standard object recognition datasets and
comparable performance in the scene dataset.
1
Introduction
The Bag of Words (BoW) [7] is the de facto standard image feature for the image categorization.
In a BoW, each local feature is assigned to the nearest codeword and an image is represented by
a histogram of the quantized features. Several approaches inspired by a BoW have been proposed
in recent years [9], [23], [28], [27], [29]. While it is well established that using a large number
of codewords improves classification performance, the drawback is that assigning local features to
the nearest codeword is computationally expensive. To overcome this problem, some studies have
proposed building an efficient image representation with a smaller number of codewords [22], [24].
Finding an explicit correspondence between local features is another way of categorizing images
using a BoW [4], [12], [26], and this approach has been improved by representing a spatial layout
of local features as a graph [11], [2], [16], [8]. Explicit correspondences between features have an
advantage over a BoW as information loss in the vector quantization can be avoided. However, the
drawback with this approach is that the identification of corresponding points with minimum distortion is also computationally expensive. Therefore, the aim of our research is to build an efficient
image representation without using codewords or explicit correspondences between local features,
while still achieving high classification accuracy.
Since having a spatial layout of local features is important for an image to have semantic meaning,
it is natural that embedding spatial information into an image feature improves classification performance [18], [5], [14], [17]. Several approaches take advantage of this fact, ranging from local (e.g.,
SIFT) to global (e.g., Spatial Pyramid). Meanwhile, we will focus on the spatial layout of local
features, which is the midlevel of the spatial information.
In this paper, we model an image as a graph representing the spatial layout of local features and
define a new image feature based on this graph, where a proper metric is embedded into the feature.
We show that the new feature provides high classification accuracy, even with a linear classifier.
Specifically, we model an image as a Gaussian Markov Random Field (GMRF) whose nodes correspond to local features and consider the GMRF parameters as the image feature. Although the
GMRF is commonly used for image segmentation, it is rarely used in modern image categorization
pipelines despite being an effective way of modeling the spatial layout. In order to extract the repre1
sentative feature vector from the GMRF, the choice of coordinates for the parameters and the metric
between them needs to be carefully made. We define the proper coordinates and the metric from
an information geometry standpoint [1] and derive an optimal feature vector. The resultant feature
vector is called a Graphical Gaussian Vector.
The contributions of this study are summarized as follows: 1) A novel and efficient image feature
is developed by utilizing the GMRF as a tool for object categorization. 2) This approach is implemented by developing the Graphical Gaussian Vector feature, which is based on the GMRF and the
information geometry. 3) Using standard image categorization benchmarks, we demonstrate that
the proposed feature has better performance over the state-of-the-art methods, even though it is not
based on mainstream modules (such as codebooks and correspondence between local features). To
the best of our knowledge, this is the first image feature for the object categorization that utilizes
the expectation parameters of the GMRF with its Fisher information metric, and achieves a level of
accuracy comparable to that of the codebook and local feature matching approaches.
2
Graphical Gaussian Vector
2.1
Overview of Proposed Method
(a) Densely sampled
local features
(b) Multivariate Gaussian
Markov Random Field
(MGMRF)
x3
x4
x5
x2
x1
x6
x8
x9
Local features x i ? R d
(
x = x1T L x
p (x; ? 4 )
p (x; ?1 )
?
Parameter of MGMRF
x7
MGMRF
MGMRF
p (x; ? )
)
T T
9
MGMRF
(e) Parameter space
(c) PDF (d) Feature Vector
?4
MGMRF
p (x; ? 3 )
?1
?3
?1
p (x; ? 2 )
?3
MGMRF
p (x; ? 5 )
?2
?6
?5
MGMRF
p (x; ? 6 )
Manifold
Geodesic distance ?
?2
Figure 1: Overview of image feature extraction based on a multivariate GMRF.
In this section, we present an overview of our method. Initially, local features {xi ? Rd }M
i=1 are
extracted using a dense sampling strategy (Fig. 1(a)). We then use a multivariate GMRF to model
the spatial relationships among local features (Fig. 1(b)). The GMRF is represented as a graph
G(V, E), whose vertices V and edges E correspond to local features and the dependent relationships
between those features, respectively. Let the vector x be a concatenation of local features in V and
let ? j be a parameter of the GMRF of an image Ij , the image Ij can be represented by a probability
distribution p(x; ? j ) of the GMRF (Fig. 1(c)). We consider the parameter ?j of the GMRF to
be a feature vector of the image Ij (Fig. 1(d)). Assuming that ? is a coordinate system, the whole
probability distribution model can be considered as a manifold, where each probability distribution is
represented as a point in that space (Fig. 1(e)). However, because the space spanned by parameters
of a probability distribution is not a Euclidean space, we have to be very careful when choosing
parameters for the probability distribution and the metric among them. We make use of concepts
from the information geometry [1] and extract proper parameters and a metric from the GMRF.
Finally, we define the new image feature by embedding the metric into the extracted parameters to
build an image categorization system with a scalable linear classifier. In the following sections, we
describe this process in more detail.
2.2
Image Model and Parameters
Given M local features {xi ? Rd }M
i=1 , the aim is to model a probability distribution of the local
features representing the spatial layout of the image using the multivariate GMRF G = (V, E).
First, a vector x is built by concatenating the local features corresponding to the vertices V of the
GMRF. Let {xi }ni=1 are local features that we are focusing on, we obtain the concatenated vector as
x = (x
1 ? ? ? xn ) (e.g., Fig. 1(b), where n = 9). Note that the dimensionality of x is nd and does
not depend on the number of local features M , the image size, or the aspect ratio. However, since
all results valid for a scalar local feature are also valid for a multivariate local feature, in this section
we consider the dimensionality of local features is 1 (d = 1) for simplicity. That is dim(x) = n.
2
Let ? = E[x], P = E[(x ? ?)(x ? ?) ], and J = P ?1 . A random vector x is called a Gaussian
Markov Random Field (GMRF) with respect to G = (V, E), if and only if its density has the form
p(x) = (2?)?n/2 |J|1/2 exp(? 12 (x ? ?) J(x ? ?)) and Jij = 0 for all {i, j} ?
/ E. Because the
Gaussian distribution can be represented as an exponential family, here, we consider an exponential
family as follows:
p(x) = exp ? ?(x) ? ?(?) ,
(1)
where
? are the natural parameters, ?(x) is the sufficient statistic, and ?(?) =
log exp(? ?(x))dx is the log-normalizer. ? and ?(x) of the GMRF are obtained as [15]:
1
?i = hi , ?ii = ? Jii , ?jk = ?Jjk , (i ? V, {j, k} ? E),
(2)
2
2
(3)
?i (x) = xi , ?ii (x) = xi , ?jk (x) = xj xk , (i ? V, {j, k} ? E),
where h = J?. The expectation parameter ? = E[?(x)] is an implicit parameterization belonging
to the exponential family. The expectation parameters are obtained as [15]:
?i = ?i , ?ii = Pii + ?2i , ?jk = Pjk + ?j ?k , (i ? V, {j, k} ? E).
(4)
The natural and expectation parameters can be transformed into each other [1]. They are called
mutually dual as each is the dual coordinate system of the other. The two coordinate systems are
??i
closely related through the Fisher information matrices (FIMs) Gij (?) and G?ij (?): Gij (?) = ??
j,
i
??
?ij
?
?1
and G (?) = ??j , where G (?) = G (?). If we take the natural parameters or the expectation
parameters as a coordinate system for an exponential family, a flat structure can be realized [1].
In particular, ? is called a 1-affine coordinate system, and the space spanned by ? is called 1-flat.
Similarly, ? is called a (-1)-affine coordinate system, and the space spanned by ? is called (-1)-flat.
Those spaces are similar to a Euclidean space, but we need to be careful that the spaces spanned by
the natural or expectation parameters are different from a Euclidean space, as the metrics vary for
different parameters. We will discuss how to determine the metrics in those spaces in Sections. 2.4
and 2.5.
To summarize this section, the natural and expectation parameters are similar and interchangeable
through the FIMs. By using these parameters, we can obtain flatness similar to the Euclidean space.
Although it does not matter whether we choose natural or expectation parameters, we use expectation parameters (Eq. (4)) as feature vectors because they can be calculated directly from the mean
and covariance of local features. We will see a multivariate extension of the GMRF and its calculation in the next section.
2.3
Calculation of Expectation Parameters
In this section, we describe the calculations of the expectation parameters of the multivariate GMRF.
First, we define the graph structure of the GMRF. We use star graphs shown in Fig. 2, where
four neighbors (Fig. 2(a)) or eight neighbors (Fig. 2(b)) are usually used. While a graph having
more neighbors is obviously able to represent richer spatial information, the compact structure is
preferable for efficiency. Therefore, we employed the approximated graph structures shown in Fig.
2(c), which represents the vertical and horizontal relationships among local features, and Fig. 2(d),
which represents vertical, horizontal and, diagonal relationships.
rk + a 2 rk + a 3 rk + a 4
rk + a 2
rk + a1 rk
(a)
(b)
rk + a1 rk
(c)
(d)
Image
region J
(e)
Figure 2: Structures of the GMRF.
Next, we show a method for estimating the expectation parameters of each image. In practice,
Eq. (4) in a multivariate case can be determined by calculating the local auto-correlations of local
3
features. Here we present the detailed calculations of Eq. (4) using Fig. 2(c) as an example. Let
x(r k ) ? Rd be the local feature at a reference point rk and let ai and aj be the displacement
vectors, which are defined by
the structure of the GMRF. Then, the local auto-correlation matrices
are obtained as: Ci,j = N1J k?J x(r k +ai )x(r k +aj ) , where NJ is the number of local features
in the image region J. Especially if we define a0 = 0, C0,i = N1J k?J x(r k )x(r k + ai ) . Let a
vector concatenating local features in the vertices at the reference point rk be x
k = (x(r k ) x(r k +
a1 ) x(r k + a2 ) ), P + ?? is calculated to be:
C0,0 C0,1 C0,2
1
P + ?? =
xk xk = C1,0 C1,1 C1,2 .
(5)
NJ
C2,0 C2,1 C2,2
k?J
The expectation parameters of the GMRF depicted in Fig. 2(c) can be obtained as: ? =
(?
0 ?1 ?2 f (C0,0 ) f (C1,1 ) f (C2,2 ) g (C0,1 ) g (C0,2 )) , where f (?) returns a column vector
consisting of the elements of the upper triangular portion of the input
matrix, g(?) returns a column
vector containing all the elements of the input matrix and ?i = N1J k?J x(r k + ai ). Note that
C1,2 is omitted, because there is no edge between the vertices at rk + a1 and rk + a2 . In general,
the expectation parameters (Eq. (4)) on the star graph can be calculated by:
? = ?
,
(6)
0 ? ? ? ?n?1 f (C0,0 ) ? ? ? f (Cn?1,n?1 ) g (C0,1 ) ? ? ? g (C0,n?1 )
where n = |V| is the number of vertices. The dimensionality of ? is: nd + n(d + 1)d/2 + (n ? 1)d2 ,
where d is the dimensionality of the local feature. Also note that {Ci,j }i=j?i,j=0 can be omitted.
By scanning the image region J (Fig. 2(e)), if we have enough local features, the means {?i }n?1
i=0
and covariance matrices {Ci,i }n?1
i=0 of local features in the region J come to the vector ?0 and matrix
C0,0 , respectively. The expectation parameters (Eq. (4)) can be approximated by:
? = (?
? ? ? ? f (C0,0 ) ? ? ? f (C0,0 ) g (C0,1 ) ? ? ? g (C0,n?1 )) .
0
0
n
n
(7)
n?1
Equation (7) is calcuated more efficiently than Eq. (6) and comes to the same vector as Eq. (6).
However, in the preliminary experiment, Eq. (6) is better than Eq. (7) in terms of the classification
accuracies. In the following sections, we use Eq. (6) for the expectation parameters.
2.4
Metric
In Section 2.2, we mentioned that the metric varies depending on the parameters. We now derive
a metric between the expectation parameters [1]. Let ds represent the length of the small
lineelement connecting ? and ? + d?. d? is represented by using basis
vectors e?i : d? = i ?i e?i .
The squared distance can be calculated by: ds2 = d?, d? = i,j e?i , e?j d?i d?j , where ?, ?
is the inner product of two vectors. By applying the Taylor expansion to KL divergence between
p(x; ?) and p(x; ?+ d?), ds2 can be represented as follows: ds2 = KL[p(x; ?) : p(x; ? + d?)] =
1
1
?
?ij
d?i d?j , where G? is the FIM. By comparing these equations, it is clear
i,j G
2 d? G d? = 2
that the metric matrix consisting of the inner products of the basis vectors corresponds to the FIM:
G?ij = e?i , e?j .
(8)
Thus, the FIM is a proper metric for the feature vectors (the expectation parameters) obtained from
the GMRF.
? is an unThe Cram?er-Rao inequality gives us a better understanding of the FIM. Assuming that ?
? satisfies: Var[?
biased estimator, the variance-covariance matrix of ?
? ] ? N1 (G? )?1 . Consequently,
the FIM is considered to be the inverse of the variance of an estimator, making it natural to use the
matrix as a distance metric between the parameters.
2.5
Implementation of Graphical Gaussian Vector
At first, we build the concatenated vector as x = (x
1 ? ? ? xn ) , where each xi corresponds to
the local feature of the vertex i. By using all training data, the mean ? = E[x] and the precision
4
x1 , x2 , L , z1 , z 2 x1 x1 , x2 x2 , L z1 z1 , z 2 z 2 x1 x2 , y1 y2 , z1 z 2 x1 y1 , x1 y2 , x2 y1 , x2 y2 , L , y1 z1 , y1 z 2 , y2 z1 , y2 z 2
z = ( z1 z 2 )T
z
y = ( y1 y2 )T
y
x
x = ( x1 x2 )T
(a)
F *=
x1
x2
M
z1
z2
x1 x1
x2 x2
M
z1 z1
z2 z2
F *ij
F *i , pp
F *i , pq
F *i , pp
F * pp ,rr
F *rr , pq
x1 x2
y1 y2
z1 z 2
x1 y1
x1 y2
x2 y1
x2 y 2
M
y1 z1
y1 z 2
y2 z1
y2 z 2
*
FGG
FG*\G
G *=
*
FFGG
*
GG
F * pq ,rs
*rr , pq
F *i , pq F
-1
FFG*\G
*
G \G
*
F\GG
FF\*G \G
*
\G \G
*
FF \GG
*
\GG
F\*G \G
(b)
(c)
Figure 3: Here V = {x, y, z} and E = {{x, y}, {x, z}}. The dimensionality of the local features is
2 (x = (x1 , x2 ) , y = (y1 , y2 ) , z = (z1 , z2 ) ). A vector concatenating the local features in V is
v = (x1 , x2 , y1 , y2 , z1 , z2 ) . Using the training data, we calculate a mean ? and a precision matrix
J of v. Using ? and J, the Fisher information matrix of the full Gaussian family can be calculated
as in (b), whose rows and columns correspond to the elements of the expectation parameters. In (b),
?
?
?
?
F ? (?) can be partitioned into the submatrices FG,G
(?), FG,\G
(?), F\G,G
(?) and F\G,\G
(?). The
Fisher information matrix of the GMRF is obtained as shown in (c) using the submatrices of F ? (?).
matrix J = P ?1 , P = E[(x ? ?)(x ? ?) ] are obtained. Since the FIM G? (?) of the GMRF
is derived from the FIM of the full Gaussian family F ? (?), we now calculate F ? (?) with ? and
J. Let e?i , e?ij denote the basis vectors corresponding to ?i and Pij + ?i ?j in Eq. (4) respec?i
?j
tively.
The
elements of F ? (?) are obtained by [20]: F ?ij
= Jij (1 + ? J?) +
(?) = e , e
?i,pq
?i
?pq
, F
(?) = e , e = ?Jpi k ?k Jkq ? Jqi k ?k Jkp , F ?i,pp (?) =
k ?k Jki
k ?k Jkj
?i
?pp
e , e = ?Jpi k ?k Jkp , F ?pq,rs (?) = e?pq , e?rs = Jps Jqr + Jqs Jpr , F ?pq,rr (?) =
2
.
e?pq , e?rr = Jpr Jrq , F ?pp,rr (?) = e?pp , e?rr = 12 Jpr
Next we derive G? (?) from F ? (?). F ? (?) can be partitioned according to the graphs G and \G:
?
?
FG,G (?) FG,\G
(?)
?
F (?) =
.
(9)
?
?
F\G,G
(?) F\G,\G
(?)
A FIM of the GMRF G? (?) is obtained as the Schur complement of F ? (?) with respect to the
?
submatrix F\G,\G
(?) [15]:
?1
?
?
?
?
(?) ? FG,\G
(?) F\G,\G
(?)
F\G,G
(?).
(10)
G? (?) = FG,G
As these calculations may be complicated, we present a simple example using a GMRF with n = 3
vertices, shown in Fig. 3.
However, G? (?) is difficult to deal with as it depends on the expectation parameters. Thus, we
approximate the model space using the tangent space at the center point of all training data [20]:
N
G? (?) ? G? (? c ) where ? c = i=1 ? i , and N is the number of training images. In order to embed
the proper metric into the expectation parameters, we multiply G? (? c )1/2 by ?:
1/2
?1
?
?
?
?
? = FG,G (? c ) ? FG,\G (? c ) F\G,\G (? c )
F\G,G (? c )
?.
(11)
We call ? Graphical Gaussian Vector (GGV). This vector is used directly to build sophisticated linear
classifiers.
We have a derivation of GGV, and the algorithm for it is very simple, consisting of the following
three steps: 1) calculation of local auto-correlations of local features; 2) estimation of the expectation
parameters of the GMRF; and 3) embedding the distance metric (the Fisher information metric) into
the expectation parameters. The calculation of GGV is given in Algorithm 1. Before the calculation
of GGV, we have to estimate the FIM of GMRF by decomposing the FIM of the full Gaussian. As a
consequence, we obtain one common FIM for all expectation parameters. In practice, since using all
training data is infeasible to estimate the FIM, we use a subset of local features randomly sampled
from training data. Note that since the calculation of the FIM is done in the preprocessing stage, it
is not necessary to calculate the FIM when extracting GGVs.
5
Algorithm 1 Calculation of GGV.
Input: An image region J, and the Fisher information matrix of the GMRF G? (? c )
Output: GGV ?
1. Calculate
local auto-correlations of local features:
?i = N1J k?J x(r k + ai ), Ci,j = N1J k?J x(r k + ai )x(r k + aj )
2. Estimate the expectation parameters:
? = (?
0 ? ? ? ?n?1 f (C0,0 ) ? ? ? f (Cn?1,n?1 ) g (C0,1 ) ? ? ? g (C0,n?1 ))
3. Embed the Fisher information metric into the expectation parameters:
1/2
? = (G? (? c )) ?
3
Experiment
We tested our method on the standard object and scene datasets (Caltech101, Caltech256, and
15-Scenes). For the first experiment, we evaluated the effects of the graph structure (i.e. spatial
information) and the FIM. As baseline methods, we used Generalized Local Correlation (GLC)
[19]: ? glc = (?
0 f (C0,0 )) without the FIM, Local Auto-Correlation features (LAC) [21], [14]:
? lac = (?0 f (C0,0 ) g (C0,1 ) ? ? ? g (C0,n?1 )) without the FIM, and the Global Gaussian with
a center linear kernel (GG) [20]: ? glc with F ? (? c ). The comparison among these methods are
shown in Table 1. Two types of graph structures were utilized for the GGVs. The first is shown in
Fig. 2(c) (GGV, n = 3), which models a horizontal and vertical spatial layout of the local features.
The second is shown in Fig. 2(d) (GGV, n = 5), which adds diagonal spatial layouts of the features
to Fig. 2(c). We also compared L2 normalized GGVs (i.e., ?? = ?/||?||). To embed the global
spatial information, we used the spatial pyramid representation with a 1 ? 1 + 2 ? 2 + 3 ? 3 pyramid
structure.
Table 1: The relationships between GLC, LAC, GG, and GGV in terms of spatial information and
Fisher information metrics.
Method
GLC
LAC
GG
GGV (proposed)
Spatial information
?
?
Fisher information metric
?
?
In the second experiment, we compared GGVs with the Improved Fisher kernel (IFK) [24], [25],
which is the best image representation available at the time of writing. In this experiment, we used
the spatial pyramid representation with a 1 ? 1 + 2 ? 2 + 3 ? 1 structure. The number of components
c in GMMs is an important parameter for IFK. We tested GMMs with c = 32, 64, 128, and 256
Gaussians to compute IFKs and compared them with GGVs.
For all datasets, SIFT features were densely sampled and were described for 16 ? 16 patches. We
downsized images if their longest side was more than 300 pixels. As the aforementioned features
depend on the dimensionality of the local feature, we reduced its dimensionality using PCA and
compared performance as a function of its new dimensionality. As a linear classifier, we used the
multi-class Passive-Aggressive Algorithm (PA) [6].
3.1
Caltech101
Caltech101 is the de facto standard object-recognition dataset [10]. To evaluate the classification
performance, we followed the most commonly used methodology. Fifteen images were randomly
selected from all 102 categories for training purposes and the remaining images were used for testing. The classification score was averaged over 10 trials.
Before comparison between GGVs and the baselines, we evaluate the sensitivities of the sampling
step of local features. The sampling step is one of the important parameters of GGV, because GGV
calculates auto-correlations of the neighboring local features. In this preliminary experiment, we fix
the number of vertices is 5 (n = 5) and the dimensionality of local feature is 32. We do not use
6
Caltech101
65
60
Generalized local correlation
Local auto?correlation
Global gaussian
GGV (proposed), n=3
GGV (proposed), n=5
GGV L2 norm (proposed), n=3
GGV L2 norm (proposed), n=5
50
45
0
5
10
15
20
25
dimensionality of local feature
30
70
70
68
68
66
66
64
64
classification rate [%]
classification rate [%]
classification rate [%]
70
55
Caltech101
Caltech101
75
62
60
58
56
IFK c=32
IFK c=64
IFK c=128
IFK c=256
GGV L2 norm (proposed), n=3
GGV L2 norm (proposed), n=5
54
52
35
50
0
5
10
15
20
25
dimensionality of local feature
30
62
60
58
56
IFK c=32
IFK c=64
IFK c=128
IFK c=256
GGV L2 norm (proposed), n=3
GGV L2 norm (proposed), n=5
54
52
35
50
0
2
4
6
8
10
dimensionality of image feature
12
14
4
x 10
Figure 4: A comparison of classification accuracies of: (left) GGV, GLC, LAC and GG; (center)
GGV and IFK with respect to the dimensionality of ?local features?; (right) GGV and IFK with
respect to the dimensionality of ?image features? in the Caltech101 dataset.
the spatial pyramid. The results are as follows: 56.7 % (step = 4 pixels), 57.7 % (step = 6 pixels) ,
57.7 % (step = 8 pixels) , 57.2 % (step = 10 pixels) , 56.5 % (step = 12 pixels). There is no clear
difference between step sizes of 6 and 8 pixels. Therefore in the following experiments, we use 6
pixels sampling step for local feature extraction.
Figure 4 (left) shows the classification accuracies as a function of the dimensionality of the local
features. A large dimensionality yielded better performance, and the proposed method (GGV) outperformed the other methods (GLC, LAC, and GG). By comparing GGV with LAC, and GG with
GLC, it is clear that embedding the Fisher information metric improved the classification accuracy
significantly. By comparing GGV with GG, as well as LAC with GLC, it can also be seen that
embedding the spatial layout of local features also improved the accuracy. In a comparison between
the graph structures, the four-neighbor structure (Fig. 2(d)) performed slightly better than the twoneighbor structure (Fig. 2(c)). If we compare the regular GGVs with the L2 normalized GGVs, we
find that the L2 normalization improved the accuracy by almost 2 %.
In the second experiment, we compared the L2 normalized GGVs with IFKs. The results are shown
in Fig. 4 (center). For all the dimensionalities and numbers of components, GGVs performed better
than IFKs. Fig. 4 (right) shows the classification accuracy as a function of the dimensionality of
the image features which are converted from the results shown in Fig. 4 (center). We see that
GGVs achieved higher accuracy for a lower dimensionality of image features. The results were
also compared against those of leading methods that use a linear classifier. The performance scores
are referenced from the original papers. LLC [27] scored 65.4 % and ScSPM [28] scored 67.0 %,
whereas our method achieved 71.3 % when the dimensionality of the local feature is 32 and the
number of vertices is 5. Therefore, our method is better than the best available methods in this
dataset, despite using a linear classifier and not requiring a codebook or descriptor matching.
3.2
Caltech256
Caltech256 consists of images from 256 object categories [13]. This database is significant for
its large inter-class variability, as well as an intra-class variability greater than that found in Caltech101. To evaluate performance, we followed a commonly used methodology. Fifteen images
were randomly selected from all categories for training purposes and the remaining images were
used for testing. The classification score was averaged over 10 trials.
Figure 5 (left) shows a comparison of classification accuracies of GGV, GLC, LAC and GG. Fig. 5
(center) and (right) show comparisons of the L2 normalized GGVs and IFKs using the Caltech256
dataset with respect to the dimensionality of local features and image features, respectively. The
results show the same trends as for Caltech101. Our method is better than all baseline methods and
IFKs. [24] reported that IFK achieved 34.7% and [27] reported that LLC scored 34.4%, while GGV
obtained 33.4%. However, a fair comparison is difficult because our method used only single-scale
SIFT whereas [24] and [27] used 5-scale SIFT and 3-scale HOG, respectively. It is known that
using multi-scale local features improves classification accuracies (e.g. [3]). To be fair comparison,
we used 3-scale SIFT (patch size = 16 ? 16, 24 ? 24, 32 ? 32) for GGV with n = 5, and L2
normalization. GGV with 3-scale SIFT achieved 36.2% which is better than those leading methods.
7
Caltech256
Caltech256
Caltech256
34
32
34
34
32
32
30
30
26
24
22
Generalized local correlation
Local auto?correlation
Global gaussian
GGV (proposed), n=3
GGV (proposed), n=5
GGV L2 norm (proposed), n=3
GGV L2 norm (proposed), n=5
20
18
16
14
0
5
10
15
20
25
dimensionality of local feature
30
classification rate [%]
28
classification rate [%]
classification rate [%]
30
28
26
24
IFK c=32
IFK c=64
IFK c=128
IFK c=256
GGV L2 norm (proposed), n=3
GGV L2 norm (proposed), n=5
22
20
18
0
35
5
10
15
20
25
dimensionality of local feature
30
28
26
24
IFK c=32
IFK c=64
IFK c=128
IFK c=256
GGV L2 norm (proposed), n=3
GGV L2 norm (proposed), n=5
22
20
18
0
35
2
4
6
8
10
dimensionality of image feature
12
14
4
x 10
Figure 5: A comparison of classification accuracies of: (left) GGV, GLC, LAC and GG; (center)
GGV and IFK with respect to the dimensionality of ?local features?; (right) GGV and IFK with
respect to the dimensionality of ?image features? in the Caltech256 dataset.
3.3
15-Scenes
We experimented with 15-Scenes, a commonly used scene classification dataset [18]. We randomly
selected 100 training images for each class and used the remaining samples as test data. We calculated the mean of the classification rate for each class. This score was averaged over 10 trials, where
the training and test sets were randomly re-selected for each trial. This is the same methodology as
that used in previous studies.
15scenes
15scenes
15scenes
84
84
84
82
82
78
classification rate [%]
classification rate [%]
80
76
74
72
Generalized local correlation
Local auto?correlation
Global gaussian
GGV (proposed), n=3
GGV (proposed), n=5
GGV L2 norm (proposed), n=3
GGV L2 norm (proposed), n=5
70
68
66
64
0
5
10
15
20
25
dimensionality of local feature
30
classification rate [%]
82
80
78
76
IFK c=32
IFK c=64
IFK c=128
IFK c=256
GGV L2 norm (proposed), n=3
GGV L2 norm (proposed), n=5
74
35
72
0
5
10
15
20
25
dimensionality of local feature
30
80
78
76
IFK c=32
IFK c=64
IFK c=128
IFK c=256
GGV L2 norm (proposed), n=3
GGV L2 norm (proposed), n=5
74
35
72
0
2
4
6
8
10
dimensionality of image feature
12
14
4
x 10
Figure 6: A comparison of classification accuracies of: (left) GGV, GLC, LAC and GG; (center)
GGV and IFK with respect to the dimensionality of ?local features?; (right) GGV and IFK with
respect to the dimensionality of ?image features? in the 15-Scenes dataset.
Figure 6 (left) shows a comparison of classification accuracies of GGV, GLC, LAC and GG using
the 15-Scenes dataset. The results show similar trends as for Caltech101 and Caltech256, except
that there is no difference between the scores of the graph structures. In the second experiment,
the results with respect to the dimensionality of local features and image features are shown in
Figs. 6 (center) and (right), respectively. In contrast to the results for Caltech101 and 256, IFKs
scored slightly higher than GGVs (IFK (c = 256, d = 32): 84.0%, GGV (n = 5, d = 32 and
L2 normalized): 83.5%). As the leading method, the spatial Fisher kernel [17] reported the highest
score (88.1%). However, since [17] used 8-scale SIFT descriptors, which provide richer information
than the single-scale SIFT descriptors we used, it is difficult to make a direct comparison.
4
Conclusion
In this paper, we proposed an efficient image feature called a Graphical Gaussian Vector, which uses
neither codebook nor local feature matching. In the proposed method, spatial information about
local features and the Fisher information metric are embedded into a feature by modeling the image
as the Gaussian Markov Random Field (GMRF). Experimental results using three standard datasets
demonstrated that the proposed method offers a performance that is superior or comparable to other
state-of-the-art methods. The proposed image feature calculates the expectation parameters of the
GMRF simply and effectively while maintaining a high classification rate.
8
References
[1] S. Amari and H. Nagaoka. Methods of Information Geometry, volume 191 of Translations of mathematical monographs. American Mathematical Society, 2001.
[2] A.C. Berg, T.L. Berg, and J. Malik. Shape matching and object recognition using low distortion correspondence. In CVPR, 2005.
[3] L. Bo, X. Ren, and D. Fox. Kernel descriptors for visual recognition. In NIPS, 2010.
[4] O. Boiman, E. Shechtman, and M. Irani. In defense of nearest-neighbor based image classification. In
CVPR, 2008.
[5] Y. Cao, C. Wang, Z. Li, L. Zhang, and L. Zhang. Spatial-bag-of-features. In CVPR, 2010.
[6] K. Crammer, O. Dekel, J. Keshet, S. Shalev-Shwartz, and Y. Singer. Online passive-aggressive algorithms.
JMLR, 7:551?585, 2006.
[7] G. Csurka, C. R. Dance, L. Fan, J. Willamowski, and C. Bray. Visual categorization with bags of keypoints. In ECCV International Workshop on SLCV, 2004.
[8] O. Duchenne, A. Joulin, and J. Ponce. A graph-matching kernel for object categorization. In ICCV, 2011.
[9] J.D.R. Farquhar, S. Szedmak, H. Meng, and J. Shawe-Taylor. Improving ?bag-of-keypoints? image categorisation: Generative models and pdf-kernels. Technical report, University of Southampton, 2005.
[10] L. 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 CVPR, Workshop on GMBV, 2004.
[11] R. Fergus, P. Perona, and A. Zisserman. Object class recognition by unsupervised scale-invariant learning.
In CVPR, 2003.
[12] R. Fergus, P. Zisserman, and A. Perona. Weakly supervised scale-invariant learning of models for visual
recognition. IJCV, 71(3):273?303, 2007.
[13] G. Griffin, A. Holub, and P. Perona. Caltech-256 object category dataset. Technical Report 7694, California Institute of Technology, 2007.
[14] T. Harada, H. Nakayama, and Y. Kuniyoshi. Improving local descriptors by embedding global and local
spatial information. In ECCV, 2010.
[15] Jason K. Johnson. Convex Relaxation Methods for Graphical Models: Lagrangian and Maximum Entropy
Approaches. PhD thesis, MIT, 2008.
[16] J. Kim and K. Grauman. Asymmetric region-to-image matching for comparing images with generic object
categories. In CVPR, 2010.
[17] J. Krapac, J. Verbeek, and F. Jurie. Modeling spatial layout with fisher vectors for image categorization.
In ICCV, 2011.
[18] S. Lazebnik, C. Schmid, and J. Ponce. Beyond bags of features: Spatial pyramid matching for recognizing
natural scene categories. In CVPR, 2006.
[19] H. Nakayama, T. Harada, and Y. Kuniyoshi. Dense sampling low-level statistics of local features. In
CIVR , 2009.
[20] H. Nakayama, T. Harada, and Y. Kuniyoshi. Global gaussian approach for scene categorization using
information geometry. In CVPR, 2010.
[21] N. Otsu and T. Kurita. A new scheme for practical, flexible and intelligent vision systems. In Proc. IAPR
Workshop on Computer Vision, 1988.
[22] F. Perronnin and C. Dance. Fisher kernels on visual vocabularies for image categorization. In CVPR,
2007.
[23] F. Perronnin, C. Dance, G. Csurka, and M. Bressan. Adapted vocabularies for generic visual categorization. In ECCV, 2006.
[24] F. Perronnin, J. S?anchez, and T. Mensink. Improving the fisher kernel for large-scale image classification.
In ECCV, 2010.
[25] J. S?anchez and F. Perronnin. High-dimensional signature compression for large-scale image classification.
In CVPR, 2011.
[26] C. Wallraven, B. Caputo, and A. Graf. Recognition with local features: the kernel recipe. In ICCV, 2003.
[27] J. Wang, J. Yang, K. Yu, F. Lv, T. Huang, and Y. Gong. Locality-constrained linear coding for image
classification. In CVPR , 2010.
[28] J. Yang, K. Yu, Y. Gong, and T. Huang. Linear spatial pyramid matching using sparse coding for image
classification. In CVPR, 2009.
[29] X. Zhou, K. Yu, T. Zhang, and T. S. Huang. Image classification using super-vector coding of local image
descriptors. In ECCV, 2010.
9
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4,098 | 4,709 | No-Regret Algorithms for Unconstrained
Online Convex Optimization
Matthew Streeter
Duolingo, Inc.?
Pittsburgh, PA 15232
[email protected]
H. Brendan McMahan
Google, Inc.
Seattle, WA 98103
[email protected]
Abstract
Some of the most compelling applications of online convex optimization, including online prediction and classification, are unconstrained: the natural feasible set
is Rn . Existing algorithms fail to achieve sub-linear regret in this setting unless
constraints on the comparator point ?
x are known in advance. We present algorithms that, without such prior knowledge, offer near-optimal regret bounds with
respect to any choice of ?
x. In particular, regret with respect to ?
x = 0 is constant.
We then prove lower bounds showing that our guarantees are near-optimal in this
setting.
1
Introduction
Over the past several years, online convex optimization has emerged as a fundamental tool for solving problems in machine learning (see, e.g., [3, 12] for an introduction). The reduction from general
online convex optimization to online linear optimization means that simple and efficient (in memory
and time) algorithms can be used to tackle large-scale machine learning problems. The key theoretical techniques behind essentially all the algorithms in this field are the use of a fixed or increasing
strongly convex regularizer (for gradient descent algorithms, this is equivalent to a fixed or decreasing learning rate sequence). In this paper, we show that a fundamentally different type of algorithm
can offer significant advantages over these approaches. Our algorithms adjust their learning rates
based not just on the number of rounds, but also based on the sum of gradients seen so far. This
allows us to start with small learning rates, but effectively increase the learning rate if the problem
instance warrants it.
?
This approach produces regret bounds of the form O R T log((1 + R)T ) , where R = k?
xk2 is the
L2 norm of an arbitrary comparator. Critically, our algorithms provide this guarantee simultaneously
for all ?
x ? Rn , without any need to know R in advance. A consequence of this is that we can
guarantee at most constant regret with respect to the origin, ?
x = 0. This technique can be applied to
any online convex optimization problem where a fixed feasible set is not an essential component of
the problem. We discuss two applications of particular interest below:
Online Prediction Perhaps the single most important application of online convex optimization
is the following prediction setting: the world presents an attribute vector at ? Rn ; the prediction
algorithm produces a prediction ?(at ? xt ), where xt ? Rn represents the model parameters, and
? : R ? Y maps the linear prediction into the appropriate label space. Then, the adversary reveals
the label yt ? Y , and the prediction is penalized according to a loss function ` : Y ? Y ? R.
For appropriately chosen ? and `, this becomes a problem of online convex optimization against
functions ft (x) = `(?(at ?x), yt ). In this formulation, there are no inherent restrictions on the model
coefficients x ? Rn . The practitioner may have prior knowledge that ?small? model vectors are more
?
This work was performed while the author was at Google.
1
likely than large ones, but this is rarely best encoded as a feasible set F, which says: ?all xt ? F are
equally likely, and all other xt are ruled out.? A more general strategy is to introduce a fixed convex
regularizer: L1 and L22 penalties are common, but domain-specific choices are also possible. While
algorithms of this form have proved very effective at solving these problems, theoretical guarantees
usually require fixing a feasible set of radius R, or at least an intelligent guess of the norm of an
optimal comparator ?
x.
The Unconstrained Experts Problem and Portfolio Management In the classic problem of
predicting with expert advice (e.g., [3]), there are n experts, and on each round t the player selects
an expert (say i), and obtains reward gt,i from a bounded interval (say [?1, 1]). Typically, one uses
an algorithm that proposes a probability distribution pt on experts, so the expected reward is pt ? gt .
Our algorithms apply to an unconstrained version of this problem: there are still n experts with
payouts in [?1, 1], but rather than selecting an P
individual expert, the player can place a ?bet? of
xt,i on each expert i, and then receives reward i xt,i gt,i = xt ? gt . The bets are unconstrained
(betting a negative value corresponds to betting against the expert). In this setting, a natural goal is
the following: place bets so as to achieve as much reward as possible, subject to the constraint that
total losses are bounded by a constant (which can be set equal to some starting budget which is to be
invested). Our algorithms can satisfy constraints of this form because regret with respect to ?
x=0
(which equals total loss) is bounded by a constant.
It is useful to contrast our results in this setting to previous applications of online convex optimization to portfolio management, for example [6] and [2]. By applying algorithms for exp-concave
loss functions, they obtain log-wealth within O(log(T )) of the best constant rebalanced portfolio.
However, this approach requires a ?no-junk-bond? assumption: on each round, for each investment,
you always retain at least an ? > 0 fraction of your initial investment. While this may be realistic
(though not guaranteed!) for blue-chip stocks, it certainly is not for bets on derivatives that can
lose all their value unless a particular event occurs (e.g., a stock price crosses some threshold). Our
model allows us to handle such investments: if we play xi > 0, an outcome of gi = ?1 corresponds
exactly to losing 100% of that investment. Our results imply that if even one investment (out of
exponentially many choices) has significant returns, we will increase our wealth exponentially.
Notation and Problem Statement For the algorithms considered in this paper, it will be more
natural to consider reward-maximization rather than loss-minimization. Therefore, we consider
online linear optimization where the goal is to maximize cumulative reward given adversarially
selected linear reward functions ft (x) = gt ? x. On each round t = 1 . . . T , the algorithm selects a
point xt ? Rn , receives reward ft (xt ) = gt ? xt , and observes gt . For simplicity, we assume gt,i ?
[?1, 1], that is, kgt k? ? 1. If the real problem is against convex loss functions `t (x), they can be
converted to our framework by taking gt = ?O`t (xt ) (see pseudo-code for R EWARD -D OUBLING),
using the standard reduction from online convex optimization to online linear optimization [13].
Pt
We use the compressed summation notation g1:t = s=1 gs for both vectors and scalars. We study
the reward of our algorithms, and their regret against a fixed comparator ?
x:
T
T
X
X
Reward ?
gt ? xt
and
Regret(?
x) ? g1:T ? ?
x?
gt ? xt .
t=1
t=1
Comparison of Regret Bounds The primary contribution of this paper is to establish matching
upper and lower bounds for unconstrained online convex optimization problems, using algorithms
that require no prior information about the comparator point ?
?
x. Specifically, we?
present
an algorithm that, for any ?
x ? Rn , guarantees Regret(?
x) ? O k?
xk2 T log((1 + k?
xk2 ) T ) . To obtain
?
this guarantee, we show that it is sufficient (and necessary) that reward is ?(exp(|g1:T |/ T )) (see
Theorem 1). This shift of emphasis from regret-minimization to reward-maximization eliminates
the quantification on ?
x, and may be useful in other contexts.
Table 1 compares the bounds for R EWARD -D OUBLING (this paper) to those of two previous algorithms: online gradient descent [13] and projected exponentiated gradient descent [8, 12]. For each
Our bounds are not directly comparable
on log? bound
to the bounds cited above: a O(log(T )) regret
wealth implies wealth at least O OPT/T , whereas we guarantee wealth like O OPT? ? T . But more
importantly, the comparison classes are different.
2
Assuming kgt k2 ? 1:
Gradient Descent, ? =
?
x=0
?
R T
?R
T
R EWARD -D OUBLING
k?
xk2 ? R
?
R T
?
R T log n(1+R)T
Arbitrary ?
x
k?
xk2 T
?
xk2 )T
k?
xk2 T log n(1+k?
k?
xk1 ? R
?
R Tlog n
?
R T log n(1+R)T
Arbitrary ?
x
k?
xk1 T
?
?
xk1 ) T
k?
xk1 T log n(1+k?
Assuming kgt k? ? 1:
Exponentiated G.D.
R EWARD -D OUBLING
?
x=0
?
R T log n
Table 1: Worst-case regret bounds for various algorithms (up to constant factors). Exponentiated
G.D. uses feasible set {x : kxk1 ? R}, and R EWARD -D OUBLING uses i = n in both cases.
algorithm, we consider a fixed choice of parameter settings and then look at how regret changes as
we vary the comparator point ?
x.
Gradient descent is minimax-optimal [1] when the comparator point is contained in a hypershere
whose radius is known in advance (k?
xk2 ? R) and gradients are sparse (kgt k2 ? 1, top table).
Exponentiated gradient descent excels when gradients are dense (kgt k? ? 1, bottom table) but the
comparator point is sparse (k?
xk1 ? R for R known in advance). In both these cases, the bounds for
R EWARD -D OUBLING match those of the previous algorithms up to logarithmic factors, even when
they are tuned optimally with knowledge of R.
The advantage of R EWARD -D OUBLING shows up when the guess of R used to tune the competing algorithms turns
x = 0, R EWARD -D OUBLING offers constant regret
? out to be wrong. When ?
x can be arbitrary, only R EWARD -D OUBLING
compared to ?( T ) for the other algorithms. When ?
offers sub-linear regret (and in fact its regret bound is optimal, as shown in Theorem 8).
In order to guarantee constant origin-regret, R EWARD -D OUBLING frequently ?jumps? back to
playing the origin, which may be undesirable in some applications. In Section 4 we introduce
S MOOTH -R EWARD -D OUBLING, which achieves similar guarantees without resetting to the origin.
Related Work Our work is related, at least in spirit, to the use of a momentum term in stochastic
gradient descent for back propagation in neural networks [7, 11, 9]. These results are similar in
motivation in that they effectively yield a larger learning rate when many recent gradients point in
the same direction.
In Follow-The-Regularized-Leader terms, the exponentiated gradient descent algorithm with unnormalized weights of Kivinen and Warmuth [8] plays xt+1 = arg minx?Rn+ g1:t ? x + ?1 (x log x ? x),
which has closed-form solution xt+1 = exp(??g1:t ). Like our algorithm, this algorithm moves
away from the origin exponentially fast, but unlike our algorithm it can incur arbitrarily large regret
with respect to ?
x = 0. Theorem 9 shows that no algorithm of this form can provide bounds like the
ones proved in this paper.
Hazan and Kale [5] give regret bounds in terms of the variance
of the gt . Letting G = |g1:t | and
?
PT
the form O( V ) where V = H ? G2 /T . This result
H = t=1 gt2 , they prove regret bounds of ?
?
has some similarity to our work in?that G/ T = H ? V , and so if we hold H constant, then
when V is low, the critical ratio G/ T that appears in our bounds is large. However, they consider
the case of a known feasible set, and their algorithm (gradient descent with a constant learning rate)
cannot obtain bounds of the form we prove.
2
Reward and Regret
In this section we present a general result that converts lower bounds on reward into upper bounds
on regret, for one-dimensional online linear optimization. In the unconstrained setting, this result
will be sufficient to provide guarantees for general n-dimensional online convex optimization.
3
Theorem 1. Consider an algorithm for one-dimensional online linear optimization that, when run
on a sequence of gradients g1 , g2 , . . . , gT , with gt ? [?1, 1] for all t, guarantees
Reward ? ? exp (?|g1:T |) ? ,
(1)
where ?, ? > 0 and ? 0 are constants. Then, against any comparator ?
x ? [?R, R], we have
R
R
Regret(?
x) ?
log
? 1 + ,
(2)
?
??
letting 0 log 0 = 0 when R = 0. Further, any algorithm with the regret guarantee of Eq. (2) must
guarantee the reward of Eq. (1).
We give a proof of this theorem in the appendix. The duality between reward and regret can also be
seen as a consequence of the fact that exp(x)
? and y log y ? y are convex conjugates. The ? term
typically contains a dependence on T like 1/ T . This bound holds for all R, and so for some small
R the log term becomes negative; however, for real algorithms the term will ensure the regret
bound remains positive. The minus one can of course be dropped to simplify the bound further.
3
Gradient Descent with Increasing Learning Rates
In this section we show that allowing the learning rate of gradient descent to sometimes increase
leads to novel theoretical guarantees.
To build intuition, consider online linear optimization in one dimension, with gradients
g1 , g2 , . . . , gT , all in [?1, 1]. In this setting, the reward of unconstrained gradient descent has a
simple closed form:
Lemma 2. Consider unconstrained gradient descent in one dimension, with learning rate ?. On
PT
round t, this algorithm plays the point xt = ?g1:t?1 . Letting G = |g1:t | and H = t=1 gt2 , the
cumulative reward of the algorithm is exactly
?
Reward =
G2 ? H .
2
We give a simple direct proof in Appendix A. Perhaps surprisingly, this result implies that the reward
is totally independent of the order of the linear functions selected by the adversary. Examining the
expression in Lemma 2, we see that the optimal choice of learning rate ? depends fundamentally on
two quantities: the absolute value of the sum of gradients (G), and the sum of the squared gradients
(H). If G2 > H, we would like to use as large a learning rate as possible in order to maximize
reward. In contrast, if G2 < H, the algorithm will obtain negative reward, and the best it can do is
to cut its losses by setting ? as small as possible.
One of the motivations for this work is the observation that the state-of-the-art online gradient descent algorithms adjust their learning rates based only on the observed value of H (or its upper bound
T ); for example [4, 10]. We would like to increase reward by also accounting for G. But unlike H,
which is monotonically increasing with time, G can both increase and decrease. This makes simple
guess-and-doubling tricks fail when applied to G, and necessitates a more careful approach.
3.1
Analysis in One Dimension
In this section we analyze algorithm R EWARD -D OUBLING -1D (Algorithm 1), which consists of a
? on H = PT gt2 is known
series of epochs. We suppose for the moment that an upper bound H
t=1
in advance. In the first epoch, we run gradient descent with a small initial learning rate ? = ?1 .
? we double ? and start a
Whenever the total reward accumulated in the current epoch reaches ? H,
new epoch (returning to the origin and forgetting all previous gradients except the most recent one).
PT
Lemma 3. Applied to a sequence of gradients g1 , g2 , . . . , gT , all in [?1, 1], where H = t=1 gt2 ?
? R EWARD -D OUBLING -1D obtains reward satisfying
H,
T
X
1 ?
|g1:T |
?
Reward =
xt gt ? ?1 H
exp a ?
? ?1 H,
(3)
?
4
H
t=1
?
for a = log(2)/ 3.
4
Algorithm 1 R EWARD -D OUBLING -1D
Parameters: initial learning rate ?1 , upper
? ? PT g 2 .
bound H
t=1 t
Initialize x1 ? 0, i ? 1, and Q1 ? 0.
for t = 1, 2, . . . , T do
Play xt , and receive reward xt gt .
Qi ? Qi + xt gt .
? then
if Qi < ?i H
xt+1 ? xt + ?i gt .
else
i ? i + 1.
?i ? 2?i?1 ; Qi ? 0.
xt+1 ? 0 + ?i gt .
Algorithm 2 R EWARD -D OUBLING
Parameters: maximum origin-regret i
for 1 ? i ? n.
for i = 1, 2, . . . , n do
Let Ai be a copy of algorithm
R EWARD -D OUBLING -1D-G UESS
(see Theorem 4), with parameter i .
for t = 1, 2, . . . , T do
Play xt , with xt,i selected by Ai .
Receive gradient vector gt = ?Oft (xt ).
for i = 1, 2, . . . , n do
Feed back gt,i to Ai .
Proof. Suppose round T occurs during the k?th epoch. Because epoch i can only come to an end if
? where ?i = 2i?1 ?1 , we have
Qi ? ?i H,
!
k
k?1
X
X
i?1
? + Qk = 2k?1 ? 1 ?1 H
? + Qk .
Reward =
Qi ?
2 ?1 H
(4)
i=1
i=1
We now lower bound Qk . For i = 1, . . . , k let ti denote the round on which Qi is initialized to 0,
with t1 ? 1, and define tk+1 ? T . By construction, Qi is the total reward of a gradient descent
algorithm that is active on rounds ti through ti+1 inclusive, and that uses learning rate ?i (note that
on round ti , this algorithm gets 0 reward and we initialize Qi to 0 on that round). Thus, by Lemma
2, we have that for any i,
!
ti+1
X
?i
?i ?
2
2
(gti :ti+1 ) ?
.
Qi =
gs ? ? H
2
2
s=t
i
Applying this bound to epoch k, we have Qk ?
?
? 12 ?k H
? Substituting into (4) gives
= ?2k?2 ?1 H.
? k?1 ? 1 ? 2k?2 ) = ?1 H(2
? k?2 ? 1) .
Reward ? ?1 H(2
(5)
|
? (otherwise
?1:T . At the end of round ti+1 ? 1, we must have had Qi < ?i H
We now show that k ? |g
?
3H
epoch i + 1 would have begun earlier). Thus, again using Lemma 2,
?i
? ? ?i H
?
(gti :ti+1 ?1 )2 ? H
2
?
? Thus,
so |gti :ti+1 ?1 | ? 3H.
|g1:T | ?
k
X
|gti :ti+1 ?1 | ? k
p
? .
3H
i=1
Rearranging gives k ?
|g1:T |
?
? ,
3H
and combining with Eq. (5) proves the lemma.
We can now apply Theorem 1 to the reward (given by Eq. (3)) of R EWARD -D OUBLING -1D to show
!
? !
p
?
4Rb H
?
?
Regret(?
x) ? bR H log
? 1 + ?1 H
(6)
?1
?
for any ?
x ? [?R, R], where b = a?1 = 3/ log(2) < 2.5. When the feasible set is
? also fixed in
advance, online gradient descent with a fixed learning obtains a regret bound of O(R T ). Suppose
? = T . By choosing ?1 = 1 , we guarantee constant regret against the origin,
we use the estimate H
T
?
x = 0 (equivalently, constant total loss). Further, for any feasible set of radius R, we still have
5
?
worst-case regret of at most O(R T log((1 + R)T )), which is only modestly worse than that of
gradient descent with the optimal R known in advance.
? can be removed using a standard guess-and-doubling approach, at
The need for an upper bound H
the cost of a constant factor increase in regret (see appendix for proof).
Theorem 4. Consider algorithm R EWARD -D OUBLING -1D-G UESS, which behaves as follows. On
? i = 2i?1 , and
each era i, the algorithm runs R EWARD -D OUBLING -1D with an upper bound of H
? i is no longer an upper bound on the sum of
initial learning rate ?1i = 2?2i . An era ends when H
?
2
squared gradients seen during that era. Letting c = ?2?1
, this algorithm has regret at most
?
R
Regret ? cR H + 1 log
(2H + 2)5/2 ? 1 + .
3.2
Extension to n dimensions
To extend our results to general online convex optimization, it is sufficient to run a separate copy of
R EWARD -D OUBLING -1D-G UESS for each coordinate, as is done in R EWARD -D OUBLING (Algorithm 2). The key to the analysis of this algorithm is that overall regret is simply the sum of regret
on n one-dimensional subproblems which can be analyzed independently.
Theorem 5. Given a sequence of convex loss functions f1 , f2 , . . . , fT from Rn to R,
R EWARD -D OUBLING with i = n has regret bounded by
n
X
n
p
|?
xi | Hi + 1 log
|?
xi |(2Hi + 2)5/2 ? 1
i=1
?
n
? + ck?
xk2 H + n log
k?
xk22 (2H + 2)5/2 ? 1
?
P
P
T
T
2
2
, where Hi = t=1 gt,i
for c = ?2?1
and H = t=1 kgt k22 .
Regret(?
x) ? + c
Proof. Fix a comparator ?
x. For any coordinate i, define
Regreti =
T
X
?
xi gt,i ?
t=1
Observe that
n
X
Regreti =
i=1
T
X
T
X
xt,i gt,i .
t=1
?
x ? gt ?
t=1
T
X
xt ? gt = Regret(?
x) .
t=1
Furthermore, Regreti is simply the regret of R EWARD -D OUBLING -1D-G UESS on the gradient sequence g1,i , g2,i , . . . , gT,i . Applying the bound of Theorem 4 to each Regreti term completes the
~ be a vector whose ith component is
proof
of the first inequality. For the second inequality, let H
?
Hi + 1, and let ~x ? Rn where ~xi = |?
xi |. Using the Cauchy-Schwarz inequality, we have
n
X
p
?
~ ? k?
~ 2 = k?
|?
xi | Hi + 1 = ~x ? H
xk2 kHk
xk2 H + n .
i=1
This, together with the fact that log(|?
xi |(2Hi + 2)5/2 ) ? log(k?
xk22 (2H + 2)5/2 ), suffices to prove
second inequality.
In some applications, n is not known in advance. In this case, we can set i =
coordinate we encounter, and get the same bound up to constant factors.
4
i2
for the ith
An Epoch-Free Algorithm
In this section we analyze S MOOTH -R EWARD -D OUBLING, a simple algorithm that achieves bounds
comparable to those of Theorem 4, without guessing-and-doubling. We consider only the 1-d problem, as the technique of Theorem 5 can be applied to extend to n dimensions. Given a parameter
6
? > 0, we achieve
?
Regret ? R T
RT 3/2
? 1 + 1.76?,
(7)
?
for all T and R, which is better (by constant factors) than Theorem 4 when gt ? {?1, 1} (which
implies T = H). The bound can be worse on a problems where H < T .
log
The idea of the algorithm is to maintain the invariant that our cumulative reward, as a function of
g1:t and t, satisfies Reward ? N (g1:t , t), for some fixed function N . Because reward changes by
gt xt on round t, it suffices to guarantee that for any g ? [?1, 1],
N (g1:t , t) + gxt+1 ? N (g1:t + g, t + 1)
(8)
where xt+1 is the point the algorithm plays on round t + 1, and we assume N (0, 1) = 0.
This inequality is approximately satisfied (for small g) if we choose
?N (g1:t + g, t)
N (g1:t + g, t) ? N (g1:t , t)
N (g1:t + g, t + 1) ? N (g1:t , t)
?
?
.
?g
g
g
?
1
This suggests that if we want to maintain
reward
at least N (g1:t , t) = t (exp(|g1:t |/ t) ? 1) , we
should set xt+1 ? sign(g1:t )t?3/2 exp |g?1:tt | . The following theorem (proved in the appendix)
provides an inductive analysis of an algorithm of this form.
Theorem 6. Fix a sequence of reward functions ft (x) = gt x with gt ? [?1, 1], and let Gt = |g1:t |.
We consider S MOOTH -R EWARD -D OUBLING, which plays 0 on round 1 and whenever Gt = 0;
otherwise, it plays
xt+1 = ? sign(g1:t )B(Gt , t + 5)
(9)
with ? > 0 a learning-rate parameter and
1
G
B(G, t) = 3/2 exp ? .
(10)
t
t
xt+1 =
Then, at the end of each round t, this algorithm has
Reward(t) ? ?
1
exp
t+5
?
Gt
t+5
? 1.76?.
Two main technical
arise in the proof: first, we prove a result like Eq. (8) for N (g1:t , t) =
? challenges
(1/t) exp |g1:t |/ t . However, this Lemma only holds for t ? 6 and when the sign of g1:t doesn?t
change. We account for this by showing that a small modification to N (costing only a constant over
all rounds) suffices.
By running this algorithm independently for each coordinate using an appropriate choice of ?, one
can obtain a guarantee similar to that of Theorem 5.
5
Lower Bounds
As with our previous results, it is sufficient to show a lower bound in one dimension, as it can then
be replicated independently in each coordinate
to obtain an n dimensional bound. Note that our
?
lower bound contains the factor log(|?
x| T ), which can be negative when ?
x is small relative to T ,
hence it is important to hold ?
x fixed and consider the behavior as T ? ?. Here we give only a
proof sketch; see Appendix A for the full proof.
Theorem 7. Consider the problem of unconstrained online linear optimization in one dimension,
and an online algorithm that guarantees origin-regret at most . Then, for any fixed comparator ?
x,
and any integer T0 , there exists a gradient sequence {gt } ? [?1, 1]T of length T ? T0 for which
the algorithm?s regret satisfies
v
u
? !
u
|?
x| T
t
Regret(?
x) ? 0.336|?
x| T log
.
7
Proof. (Sketch) Assume without loss of generality that ?
x > 0. Let Q be the algorithm?s reward
when each gt is drawn independently uniformly from {?1, 1}. We have E[Q] = 0, and because the
algorithm guarantees origin-regret at most , we have Q ? ? with probability 1. Letting G = g1:T ,
it follows that for any threshold Z = Z(T ),
0 = E[Q]
= E[Q|G < Z] ? Pr[G < Z] + E[Q|G ? Z] ? Pr[G ? Z]
? ? Pr[G < Z] + E[Q|G ? Z] ? Pr[G ? Z]
> ? + E[Q|G ? Z] ? Pr[G ? Z] .
Equivalently,
E[Q|G ? Z] <
.
Pr[G ? Z]
j
k
?
?
x| and p > 0 is a
We choose Z(T ) = kT , where k = log( R T )/ log(p?1 ) . Here R = |?
constant chosen using binomial distribution lower bounds so that Pr[G ? Z] ? pk . This implies
?
E[Q|G ? Z] < p?k = exp k log p?1 ? R T .
?
This implies there
a sequence ?
with G ? Z and Q < R T . On this sequence, regret is at least
? exists?
G?
x ? Q ? R kT ? R T = ?(R kT ).
Theorem 8. Consider the problem of unconstrained online linear optimization in Rn , and consider
an online algorithm that guarantees origin-regret at most . For any radius R, and any T0 , there exists a gradient sequence gradient sequence {gt } ? ([?1, 1]n )T of length T ? T0 , and a comparator
?
x with k?
xk1 = R, for which the algorithm?s regret satisfies
v
u
? !
n
X
u
T
|?
x
|
i
|?
xi |tT log
Regret(?
x) ? 0.336
.
i=1
Proof. For each coordinate i, Theorem 7 implies that there exists a T ? T0 and a sequence of
gradients gt,i such that
v
u
? !
T
T
X
X
u
|?
xi | T
t
?
xi gt,i ?
xt,i gt,i ? 0.336|?
xi | T log
.
t=1
t=1
(The proof of Theorem 7 makes it clear that we can use the same T for all i.) Summing this
inequality across all n coordinates then gives the regret bound stated in the theorem.
The following theorem presents a stronger negative result for Follow-the-Regularized-Leader algorithms with a fixed regularizer: for any such algorithm that guarantees origin-regret at most T after
T rounds, worst-case regret with respect to any point outside [?T , T ] grows linearly with T .
Theorem 9. Consider a Follow-The-Regularized-Leader algorithm that sets
xt = arg min (g1:t?1 x + ?T (x))
x
where ?T is a convex, non-negative function with ?T (0) = 0. Let T be the maximum origin-regret
incurred by the algorithm on a sequence of T gradients. Then, for any ?
x with |?
x| > T , there exists a
x| ? T ).
sequence of T gradients such that the algorithm?s regret with respect to ?
x is at least T ?1
2 (|?
In fact, it is clear from the proof that the above result holds for any algorithm that selects xt+1 purely
as a function of g1:t (in particular, with no dependence on t).
6
Future Work
This work leaves open many interesting questions. It should be possible to apply our techniques
to problems that do have constrained feasible sets; for example, it is natural to consider the unconstrained experts problem on the positive orthant. While we believe this extension is straightforward,
handling arbitrary non-axis-aligned constraints will be more difficult. Another possibility is to develop an algorithm with bounds in terms of H rather than T that doesn?t use a guess and double
approach.
8
References
[1] Jacob Abernethy, Peter L. Bartlett, Alexander Rakhlin, and Ambuj Tewari. Optimal strategies
and minimax lower bounds for online convex games. In COLT, 2008.
[2] Amit Agarwal, Elad Hazan, Satyen Kale, and Robert E. Schapire. Algorithms for portfolio
management based on the Newton method. In ICML, 2006.
[3] Nicol`o Cesa-Bianchi and Gabor Lugosi. Prediction, Learning, and Games. Cambridge University Press, New York, NY, USA, 2006. ISBN 0521841089.
[4] John Duchi, Elad Hazan, and Yoram Singer. Adaptive subgradient methods for online learning
and stochastic optimization. In COLT, 2010.
[5] Elad Hazan and Satyen Kale. Extracting certainty from uncertainty: Regret bounded by variation in costs. In COLT, 2008.
[6] Elad Hazan and Satyen Kale. On stochastic and worst-case models for investing. In Advances
in Neural Information Processing Systems 22. 2009.
[7] Robert A. Jacobs. Increased rates of convergence through learning rate adaptation. Neural
Networks, 1987.
[8] Jyrki Kivinen and Manfred Warmuth. Exponentiated Gradient Versus Gradient Descent for
Linear Predictors. Journal of Information and Computation, 132, 1997.
[9] Todd K. Leen and Genevieve B. Orr. Optimal stochastic search and adaptive momentum. In
NIPS, 1993.
[10] H. Brendan McMahan and Matthew Streeter. Adaptive bound optimization for online convex
optimization. In COLT, 2010.
[11] Barak Pearlmutter. Gradient descent: Second order momentum and saturating error. In NIPS,
1991.
[12] Shai Shalev-Shwartz. Online learning and online convex optimization. Foundations and Trends
in Machine Learning, 4(2):107?194, 2012.
[13] Martin Zinkevich. Online convex programming and generalized infinitesimal gradient ascent.
In ICML, 2003.
9
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4,099 | 471 | CCD Neural Network Processors for Pattern
Recognition
Alice M. Chiang
Michael L. Chuang
Jeffrey R. LaFranchise
MIT Lincoln Laboratory
244 Wood Street
Lexington, MA 02173
Abstract
A CCD-based processor that we call the NNC2 is presented. The NNC2
implements a fully connected 192-input, 32-output two-layer network and
can be cascaded to form multilayer networks or used in parallel for additional input or output nodes. The device computes 1.92 x 10 9 connections/sec when clocked at 10 MHz . Network weights can be specified to six
bits of accuracy and are stored on-chip in programmable digital memories.
A neural network pattern recognition system using NNC2 and CCD image feature extractor (IFE) devices is described. Additionally, we report
a CCD output circuit that exploits inherent nonlinearities in the charge
injection process to realize an adjustable-threshold sigmoid in a chip area
of 40 x 80 J.tlU 2 .
1
INTRODUCTION
A neural network chip based on charge-coupled device (CCD) technology, the NNC2,
is presented. The NNC2 implements a fully connected two-layer net and can be cascaded to form multilayer networks. An image feature extractor (IFE) device (Chiang
and Chuang, 1991) is briefly l?eviewed. The IFE is suited for neural networks with
local connections and shared weights and can also be used for image preprocessing
tasks. A neural network pattern recognition system based on feature extraction
using IFEs and classification using NNC2s is proposed. The efficacy of neural networks with local connections and shared weights for feature extraction in character
741
742
Chiang, Chuang, and LaFranchise
recognition and phoneme recognition t.asks has been demonstrated by researchers
such as (LeCun et. al. 1989) and (Waibel d. aI., 1989), respectively. :rvlore complex
recognition tasks are likely to prove amenable to a system using locally connected
networks as a front end with outputs generated by a highly-connected classifier.
Both the IFE and the NNC2 are hybrids composed of analog and digital components. Network weights are stored digitally while neuron states and computation
results are represented in analog form. Data enter and leave the devices in digital
form for ease of integration into digital systems.
The sigmoid is used in many network models as the nonlinear neuron output function. We have designed, fabricated and tested a compact CCD sigmoidal output
circuit that is described below. The paper concludes with a discussion of strategies
for implementing networks with particularly high or low fan-in to fan-out ratios.
2
THE NNC2 AND IFE DEVICES
The NNC2 is a neural network processor that implements a fully connected twolayer net with 192 input nodes and 32 output nodes. The device is an expanded
version of a previous neural network classifier (NNC) chip (Chiang, 1990) hence the
appellation "NNC2." The NNC2 consists of a 192-stage CCD tapped delay line for
holding and shifting input values, 192 four-quadrant multipliers, and 192 32-word
local memories for weight storage. vVhen clocked at 10 l\iIHz, the NNC2 performs
1.92 x 10 9 connections/sec. The device was fabricated using a 2-J,lm minimum feature
size double-metal, double-polysilicon CCD/CMOS process. The NNC2 measures
8.8 x 9.2 mm 2 and is depicted in Figure 1.
..
-
DIGITAL _ _.~
MEMORY
. .!
t
~,!
? - ?. : - .. ' ;
..
MDAC-~~~
::
-
-
i..:..:
; .J.. 't
_.
!
;
."'! ~'f
p' ? ..l
6
...
K
k
~
....
~
._
.'
~
-
~~~~~~~~~~~~~----.~-
t-~;--!
~
~
."
??
...
?
CCDTAPPED ______~__--~~
DELAYUNE
Figure 1: Photomicrograph of the NNC2
Tests indicate that the NNC2 has an output dynamic range exceeding 42 dB.
Figure 2 shows the output of the NNC2 when the input consists of the cosine
waveforms In = 0.2cos(27r2n/192) + 0.4cos(27r3n/192) and the weights are set to
CCO Neural Network Processors for Pattern Recognition
cos(2?Tnk/192), k = ?1, ?2, ... , ?16. Due to the orthogonality of sinusoids of different frequencies, the output correlations 91e = 2:~~o fncos(2?Tnk/192) should yield
scaled impulses with amplitudes of ?0.2 and ?0.4 for k ?2 and ?3 only; this is indeed the case as the output (lower trace) in Figure 2 shows. This test demonstrates
the linearity of the weighted sum (inner product) computed by the NNC2.
=
Figure 2: Response of the NNC2 to input cosine waveforms
Locally connected, shared weight networks can be implemented using the IFE which
raster scans up to 20 sets of 7x 7 weights over an input image. At every window
position the inner product of the windowed pixels and each of the 20 sets of weights
is computed. For additonal details, see (Chiang and Chuang, 1991). The IFE and
the NNC2 share a number of common features that are described below.
2.1
MDACS
The multiplications of the inner product are performed in parallel by multiplyingD/ A-converters (MDACs), of which there are 192 in the NNC2 and 49 in the IFE.
Each MDAC produces a charge paclcet proportional to the product of an input and
a digital weight. The partial products are summed on an output line common to
all the MDACs, yielding a complete inner product every clock cycle. The design
and operation of an MDAC are described in detail in (Chiang, 1990). Using a 2-J.lm
design rule, a four-quadrant MDAC with 8-bit weights occupies an area of 200x 200
J.lm 2 .
2.2
WEIGHT STORAGE
The NNC2 and IFE feature on-chip digital storage of programmable network
weights, specified to 6 and 8 bits, respectively. The NNC2 contains 192 local memories of 32 words each, while the IFE has forty-nine 20-word memories. Individual
words can be addressed by means of a row pointer and a column pointer. Each bit
of the CCD shift register memories is equipped with a feedback enable switch that
obviates the need to refresh the volatile CCD storage medium explictly; words are
743
744
Chiang, Chuang, and LaFranchise
rewritten as they are read for use in computation, so that no cycles need be devoted
to memory refresh.
2.3
INPUT BUFFER
Inputs to the NNC2 are held in a 192-stage CCD analog floating-gate tapped delay
line. At each stage the floating gate is coupled to the input of the corresponding
MDAC, permitting inputs to be sensed nondestructively for computation. The
NNC2 delay line is composed of three 64-stage subsections (see Figure 1). This
partionning allows the NNC2 to compute either the weighted sum of 192 inputs or
three 64-point inner products. The latter capability is well-matched to Time-Delay
Neural Networks (TDNNs) that implement a moving temporal window for phoneme
recognition (Waibel et. ai., 1989). The IFE contains a similar 775-stage delay line
that holds six lines of a 128-pixel input image plus an additional seven pixels. Taps
are placed on the first seven of every 128 stages in the IFE delay line so that the
1-dimensionalline emulates a 2-dimensional window.
3
CCD SIGMOIDAL OUTPUT CIRCUIT
A sigmoidal charge-domain nonlinear detection circuit is shown in Figure 3. The circuit has a programmable input-threshold controlled by the amplitude of the transfer
gate voltage, VTG. If the incoming signal charge is below the threshold set by VTG
no charge is transferred to the output port and the incoming signal is ignored. If the
input is above threshold, the amount of charge transferred to the output port is the
difference between the charge input and the threshold level. The circuit design is
based on the ability to calculate the charge transfer efficiency from an n+ diffusion
region over a bias gate to a receiving well as a function of device parameters and
exploits the fact that under certain operating conditions a nonlinear dependence exists between the input and output charge (Thornber, 1971). The maximum output
produced can be bounded by the size and gate voltage of the receiving well. The
predicted and measured responses of the circuit for two different threshold levels
are shown in the bottom of Figure 3. The circuit has an area of 40 x 80 J1.m 2 and
can be integrated with the NNC2 or IFE chips to perform both the weighted-sum
and output-nonlinearity computations on a single device.
4
DESIGN STRATEGIES
The NNC2 uses a time-multiplexed output (TMO) structure (Figure 4a), where the
number of multipliers and the number of local memories is equal to the number
of inputs, N. The depth of each local memory is equal to the number of output
nodes, M, and the outputs are computed serially as each set of weights is read in
sequence from the memories. A 256-input, 256-output device with 64k 8-bit weights
has been designed and can be realized in a chip area of 14x 14 mm 2 . This chip is
reconfigurable so that a single such device can be used to implement multilayer networks. If a network with a large (>1000) number of input nodes is required, then
a time-multiplexed input (TMI) architecture with M multipliers may be more suitable (Figure 4b). In contrast to a TMO system that computes the M inner products
CCO Neural Network Processors for Pattern Recognition
TGGATE
CLOCKlllS....
CALCULATED
MEASURED
, --,,,
,,,
N'""3.75
E
u
......
,...". 2.50
0
T""
5 1.25
0
,"
0
I
1.25
0
2.50 3.75 5.00
al n (10 7 .-'cm2 )
1
.--
- ' T G =2.5V
---- 'TG = 0.5 V
2 3 4 5 6 7 8
INPUT VOLTAGE (V)
9 10
Figure 3: Schematic, micrograph, and test results of the sigmoid cIrcuit
xl, x2, ... , xN
(Serial Inputs)
_ _. . . .t.e- ... _----.
x1
xN
x2
??
?
?
?
?
lWE'~HTS
? ??
"'--........- -
... - -yl,.........y2, ...,
-~
yM
(Serial Outputs)
(a)
y1
y2
yM
(b)
Figure 4: (a) Time-multiplexed output ('1'1\10), (b) Time-multiplexed input (TMI)
745
746
Chiang. Chuang. and LaFranchise
sequentially (the multiplications of each inner product are performed in parallel),
a TMI structure performs N sets of At multiplications each (all M inner products
are serially computed in parallel). As each input element arrives it is broadcast to
all At multipliers. Each multiplier multiplies the input by an appropriate weight
from its N -word deep local memory and places the result in an accumulator. The
M inner products appear in the accumulators one cycle after receipt of the final ,
Nth input.
5
SUMMARY
We have presented the NNC2, a CCD chip that implements a fully connected twolayer network at the rate of 1.92 x 109 connections/second. The NNC2 may be used
in concert with IFE devices to form a CCD-based neural network pattern recogniton
system or as a co-processor to speed up neural network simulations on conventional
computers. A VME-bus board for the NNC2 is presently being constructed. A
compact CCD circuit that generates a sigmoidal output function was described,
and finally, the relative merits of time-multiplexing input or output nodes in neural
network devices were enumerated. Table 1 below is a comparison of recent neural
network chips.
MIT LINCOLN LAB
NNC2
CIT
NN
INTEL
ETANN
MITSUBISHI
NN
AT&T
NN
HITACHI
WSINN
ADAPT. SOL.
Xl
No. OF OUTPUT NODES
32
256
TWO 64
168
16 (or 256)
576
64
No. OF INPUT NODES
192
256
TWO 64
168
256 (or 16)
64
4k
SYNAPSE ACCURACY
6b ' ANALOG
1 b ' ANALOG
3b ' 6b
8b ? 9 b
9 b ' 16 b
PROGRAMMABLE
SYNAPSES
6k
64k
10 k
28 k
4k
37k
256 k
THROUGHPUT RATE
(109 Connections/s)
1.92
0.5
2
?
5.1
1.2
1.6
11.2 ' 7.5
14.5' 14.5
4.5 ' 7
125 ? 125
26.2 ? 27.5
25 MHz
DIGITAL
CHIP AREA (mm 2)
8.8 ? 9.2
CLOCK RATE
WEIGHT STORAGE
10MHz
1.5MHz
400 kHz
?
20 MHz
2.1 MHz a
DIGITALb
ANALOG
ANALOG
ANALOG
ANALOG
DIGITAL
NO
NO
NO
YESc
NO
NO
YES
211m
CCD/CMOS
211m
CCD
111m
CMOS
111m
CMOS
0.9 11m
CMOS
0.8 11m
CMOS
0.8 11m
CMOS
NIPS 91
IJCNN 90
IJCNN89
ISSCC91
ISSCC 91
IJCNN90
ISSCC 91
ON CHIP LEARNING
DESIGN RULE
,
ANALOG ? ANALOG?
ANALOG
ANALOG
REPORTED AT :
NOTE:
a - CLOCK RATE FOR WSINN IS EXTRAPOLATED BASED ON 1/STEP TIME.
b - NO DEGRADATION OBSERVED ON DIGITALLY STORED AND REFRESHED WEIGHTS.
c - A SIMPLIFIED BOLTZMANN MACHINE LEARNING ALGORITHM IS USED.
Table 1: Selected neural network chips
Acknow ledgements
This work was supported by DARPA, the Office of Naval Research, and the Department of the Air Force. The IFE and NN C2 were fabricated by Orbit Semiconductor.
CCD Neural Network Processors for Pattern Recognition
References
A. J. Agranat, C. F. Neugebauer and A. Yariv, "A CCD Based Neural Network
Integrated Circuit with 64k Analog Programmable Synapses," IlCNN, 1990 Proceedings, pp. 11-551-11-555.
Y. Arima et. al., "A 336-Neuron 28-k Synapse Self-Learning Neural Network Chip
with branch-Neuron-Unit Architecture," in ISSCC Dig. of Tech. Papers, pp. 182183, Feb. 1991.
B. E. Boser and E. Sackinger, "An Analog Neural Network Processor with Programmable Network Topology," in ISSCC Dig. of Tech. Papers, pp. 184-185, Feb.
1991.
A. M. Chiang, "A CCD Programmable Signal Processor," IEEE lour. Solid-State
Circ., vol. 25, no. 6, pp. 1510-1517, Dec. 1990.
A. M. Chiang and M. L. Chuang, "A CCD Programmable Image Processor and its
Neural Network Applications," IEEE lour. Solid-State Circ., vol. 26, no. 12, pp.
1894-1901, Dec. 1991.
D. Hammerstrom, "A VLSI Architecture for High-Performance, Low-Cost On-chip
Learning," IlCNN, 1990 Proceedings, pp. 11-537-11-543.
M. Holler et. al., "An Electrically Trainable Artificial Neural Network (ETANN)
with 10240 "Floating Gate" Synapses," IlCNN, 1989 Proceedings, pp. 11-191-11196.
Y . Le Cun et. al., "Handwritten Digit Recognition with a Back-Propagation Network," in D. S. Touretzky (ed.), Advances in Neural Information Processing Systems
2, pp. 396-404, San Mateo, CA: Morgan Kaufmann, 1989.
K. K. Thornber, "Incomplete Charge Transfer in IGFET Bucket-Brigade Shift Registers," IEEE Trans. Elect. Dev., vol. ED-18, no. 10, pp.941-950, 1971.
A. Waibel et. al., "Phoneme Recognition Using Time-Delay Neural Networks," IEEE
Trans. on Acoust., Speech, Sig. Proc., vol. 37, no. 3, pp. 329-339, March 1989.
M. Yasunaga et. al., "Design, Fabrication and Evaluation of a 5-Inch Wafer Scale
Neural Network LSI Composed of 576 Digital Neurons," IlCNN, 1990 Proceedings,
pp. 11-527-11-535.
747
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