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Boosting the Area Under the ROC Curve Philip M. Long plong@google.com Rocco A. Servedio rocco@cs.columbia.edu Abstract We show that any weak ranker that can achieve an area under the ROC curve slightly better than 1/2 (which can be achieved by random guessing) can be efficiently boosted to achieve an area under the ROC curve arbitrarily close to 1. We further show that this boosting can be performed even in the presence of independent misclassification noise, given access to a noise-tolerant weak ranker. 1 Introduction Background. Machine learning is often used to identify members of a given class from a list of candidates. This can be formulated as a ranking problem, where the algorithm takes a input a list of examples of members and non-members of the class, and outputs a function that can be used to rank candidates. The goal is to have the top of the list enriched for members of the class of interest. ROC curves [12, 3] are often used to evaluate the quality of a ranking function. A point on an ROC curve is obtained by cutting off the ranked list, and checking how many items above the cutoff are members of the target class (“true positives”), and how many are not (“false positives”). The AUC [1, 10, 3] (area under the ROC curve) is often used as a summary statistic. It is obtained by rescaling the axes so the true positives and false positives vary between 0 and 1, and, as the name implies, examining the area under the resulting curve. The AUC measures the ability of a ranker to identify regions in feature space that are unusually densely populated with members of a given class. A ranker can succeed according to this criterion even if positive examples are less dense than negative examples everywhere, but, in order to succeed, it must identify where the positive examples tend to be. This is in contrast with classification, where, if Pr[y = 1|x] is less than 1/2 everywhere, just predicting y = −1 everywhere would suffice. Our Results. It is not hard to see that an AUC of 1/2 can be achieved by random guessing (see [3]), thus it is natural to define a “weak ranker” to be an algorithm that can achieve AUC slightly above 1/2. We show that any weak ranker can be boosted to a strong ranker that achieves AUC arbitrarily close to the best possible value of 1. We also consider the standard independent classification noise model, in which the label of each example is flipped with probability η. We show that in this setting, given a noise-tolerant weak ranker (that achieves nontrivial AUC in the presence of noisy data as described above), we can boost to a strong ranker that achieves AUC at least 1 −ǫ, for any η < 1/2 and any ǫ > 0. Related work. Freund, Iyer, Schapire and Singer [4] introduced RankBoost, which performs ranking with more fine-grained control over preferences between pairs of items than we consider here. They performed an analysis that implies a bound on the AUC of the boosted ranking function in terms of a different measure of the quality of weak rankers. Cortes and Mohri [2] theoretically analyzed the “typical” relationship between the error rate of a classifier based on thresholding a scoring function and the AUC obtained through the scoring function; they also pointed out the close relationship between the loss function optimized by RankBoost and the AUC. Rudin, Cortes, Mohri, and Schapire [11] showed that, when each of two classes are equally likely, the loss function optimized by AdaBoost coincides with the loss function of RankBoost. Noise-tolerant boosting has previously been studied for classification. Kalai and Servedio [7] showed that, if data is corrupted with noise at a rate η, it is possible to boost the accuracy of any noise-tolerant weak learner arbitrarily close to 1 −η, and they showed that it is impossible to boost beyond 1 −η. In contrast, we show that, in the presence of noise at a rate arbitrarily close to 1/2, the AUC can be boosted arbitrarily close to 1. Our noise tolerant boosting algorithm uses as a subroutine the “martingale booster” for classification of Long and Servedio [9]. Methods. The key observation is that a weak ranker can be used to find a “two-sided” weak classifier (Lemma 4), which achieves accuracy slightly better than random guessing on both positive and negative examples. Two-sided weak classifiers can be boosted to obtain accuracy arbitrarily close to 1, also on both the positive examples and the negative examples; a proof of this is implicit in the analysis of [9]. Such a two-sided strong classifier is easily seen to lead to AUC close to 1. Why is it possible to boost past the AUC past the noise rate, when this is provably not possible for classification? Known approaches to noise-tolerant boosting [7, 9] force the weak learner to provide a two-sided weak hypothesis by balancing the distributions that are constructed so that both classes are equally likely. However, this balancing skews the distributions so that it is no longer the case that the event that an example is corrupted with noise is independent of the instance; randomization was used to patch this up in [7, 9], and the necessary slack was only available if the desired accuracy was coarser than the noise rate. (We note that the lower bound from [7] is proved using a construction in which the class probability of positive examples is less than the noise rate; the essence of that proof is to show that in that situation it is impossible to balance the distribution given access to noisy examples.) In contrast, having a weak ranker provides enough leverage to yield a two-sided weak classifier without needing any rebalancing. Outline. Section 2 gives some definitions. In Section 3, we analyze boosting the AUC when there is no noise in an abstract model where the weak learner is given a distribution and returns a weak ranker, and sampling issues are abstracted away. In Section 4, we consider boosting in the presence of noise in a similarly abstract model. We address sampling issues in Section 5. 2 Preliminaries Rankings and AUC. Throughout this work we let X be a domain, c : X →{−1, 1} be a classifier, and D be a probability distribution over labeled examples (x, c(x)). We say that D is nontrivial (for c) if D assigns nonzero probability to both positive and negative examples. We write D+ to denote the marginal distribution over positive examples and D−to denote the marginal distribution over negative examples, so D is a mixture of the distributions D+ and D−. As has been previously pointed out, we may view any function h : X →R as a ranking of X. Note that if h(x1) = h(x2) then the ranking does not order x1 relative to x2. Given a ranking function h : X →R, for each value θ ∈R there is a point (αθ, βθ) on the ROC curve of h, where αθ is the false positive rate and βθ is the true positive rate of the classifier obtained by thresholding h at θ: αθ = D−[h(x) ≥θ] and βθ = D+[h(x) ≥θ]. Every ROC curve contains the points (0, 0) and (1, 1) corresponding to θ = ∞and −∞respectively. Given h : X →R and D, the AUC can be defined as AUC(h; D) = Pru∈D+,v∈D−[h(u) > h(v)] + 1 2Pru∈D+,v∈D−[h(u) = h(v)]. It is well known (see e.g. [2, 6]) that the AUC as defined above is equal to the area under the ROC curve for h. Weak Rankers. Fix any distribution D. It is easy to see that any constant function h achieves AUC(h; D) = 1 2, and also that for X finite and π a random permutation of X, the expected AUC of h(π(·)) is 1 2 for any function h. This motivates the following definition: Definition 1 A weak ranker with advantage γ is an algorithm that, given any nontrivial distribution D, returns a function h : X →R that has AUC(h; D) ≥1 2 + γ. In the rest of the paper we show how boosting algorithms originally designed for classification can be adapted to convert weak rankers into “strong” rankers (that achieve AUC at least 1−ǫ) in a range of different settings. 3 From weak to strong AUC The main result of this section is a simple proof that the AUC can be boosted. We achieve this in a relatively straightforward way by using the standard AdaBoost algorithm for boosting classifiers. As in previous work [9], to keep the focus on the main ideas we will use an abstract model in which the booster successively passes distributions D1, D2, ... to a weak ranker which returns ranking functions h1, h2, .... When the original distribution D is uniform over a training set, as in the usual analysis of AdaBoost, this is easy to do. In this model we prove the following: Theorem 2 There is an algorithm AUCBoost that, given access to a weak ranker with advantage γ as an oracle, for any nontrivial distribution D, outputs a ranking function with AUC at least 1 −ǫ. The AUCBoost algorithm makes T = O( log(1/ǫ) γ2 ) many calls to the weak ranker. If D has finite support of size m, AUCBoost takes O(mT log m) time. As can be seen from the observation that it does not depend on the relative frequency of positive and negative examples, the AUC requires a learner to perform well on both positive and negative examples. When such a requirement is imposed on a base classifier, it has been called two-sided weak learning. The key to boosting the AUC is the observation (Lemma 4 below) that a weak ranker can be used to generate a two-sided weak learner. Definition 3 A γ two-sided weak learner is an algorithm that, given a nontrivial distribution D, outputs a hypothesis h that satisfies both Prx∈D+[h(x) = 1] ≥1 2 + γ and Prx∈D−[h(x) = −1] ≥ 1 2 + γ. We say that such an h has two-sided advantage γ with respect to D. Lemma 4 Let A be a weak ranking algorithm with advantage γ. Then there is a γ/4 two-sided weak learner A′ based on A that always returns classifiers with equal error rate on positive and negative examples. Proof: Algorithm A′ first runs A to get a real-valued ranking function h : X →R. Consider the ROC curve corresponding to h. Since the AUC is at least 1 2 + γ, there must be some point (u, v) on the curve such that v ≥u + γ. Recall that, by the definition of the ROC curve, this means that there is a threshold θ such that D+[h(x) ≥θ] ≥D−[h(x) ≥θ] + γ. Thus, for the classifier obtained by thresholding h at θ, the class conditional error rates p+ def = D+[h(x) < θ] and p− def = D−[h(x) ≥θ] satisfy p+ + p−≤1 −γ. This in turn means that either p+ ≤1 2 −γ 2 or p−≤1 2 −γ 2. Suppose that p−≤p+, so that p−≤1 2 −γ 2 (the other case can be handled symmetrically). Consider the randomized classifier g that behaves as follows: given input x, (a) if h(x) < θ, it flips a biased coin, and with probability ζ ≥0, predicts 1, and with probability 1 −ζ, predicts −1, and (b) if h(x) ≥θ, it predicts 1. Let g(x, r) be the output of g on input x and with randomization r and let ǫ− def = Prx∈D−,r[g(x, r) = 1] and ǫ+ def = Prx∈D+,r[g(x, r) = −1]. We have ǫ+ = (1 −ζ)p+ and ǫ−= p−+ ζ(1 −p−). Let us choose ζ so that ǫ−= ǫ+; that is, we choose ζ = p+−p− 1+p+−p−. This yields ǫ−= ǫ+ = p+ 1 + p+ −p− . (1) For any fixed value of p−the RHS of (1) increases with p+. Recalling that we have p++p−≤1−γ, the maximum of (1) is achieved at p+ = 1 −γ −p−, in which case we have (defining ǫ def = ǫ−= ǫ+) ǫ = (1−γ)−p− 1+(1−γ−p−)−p−= (1−γ)−p− 2−γ−2p−. The RHS of this expression is nonincreasing in p−, and therefore is maximized at p−is 0, when it takes the value 1 2 − γ 2(2−γ) ≤1 2 −γ 4. This completes the proof. Figure 1 gives an illustration of the proof of the previous lemma; since the y-coordinate of (a) is at least γ more than the x-coordinate and (b) lies closer to (a) than to (1, 1), the y-coordinate of (b) is at least γ/2 more than the x-coordinate, which means that the advantage is at least γ/4. We will also need the following simple lemma which shows that a classifier that is good on both the positive and the negative examples, when viewed as a ranking function, achieves a good AUC. • (b) • (a) false positive rate true positive rate 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Figure 1: The curved line represents the ROC curve for ranking function h. The lower black dot (a) corresponds to the value θ and is located at (p−, 1−p+). The straight line connecting (0, 0) and (1, 1), which corresponds to a completely random ranking, is given for reference. The dashed line (covered by the solid line for 0 ≤x ≤.16) represents the ROC curve for a ranker h′ which agrees with h on those x for which h(x) ≥θ but randomly ranks those x for which h(x) < θ. The upper black dot (b) is at the point of intersection between the ROC curve for h′ and the line y = 1−x; its coordinates are (ǫ, 1 −ǫ). The randomized classifier g is equivalent to thresholding h′ with a value θ′ corresponding to this point. Lemma 5 Let h : X →{−1, 1} and suppose that Prx∈D+[h(x) = 1] = 1 −ǫ+ and Prx∈D−[h(x) = −1] = 1 −ǫ−. Then we have AUC(h; D) = 1 −ǫ++ǫ− 2 . Proof: We have AUC(h; D) = (1 −ǫ+)(1 −ǫ−) + ǫ+(1 −ǫ−) + ǫ−(1 −ǫ+) 2 = 1 −ǫ+ + ǫ− 2 . Proof of Theorem 2: AUCBoost works by running AdaBoost on 1 2D++ 1 2D−. In round t, each copy of AdaBoost passes its reweighted distribution Dt to the weak ranker, and then uses the process of Lemma 4 to convert the resulting weak ranking function to a classifier ht with two-sided advantage γ/4. Since ht has two-sided advantage γ/4, no matter how Dt decomposes into a mixture of D+ t and D− t , it must be the case that Pr(x,y)∈Dt[ht(x) ̸= y] ≤1 2 −γ/4. The analysis of AdaBoost (see [5]) shows that T = O log(1/ǫ) γ2 rounds are sufficient for H to have error rate at most ǫ under 1 2D+ + 1 2D−. Lemma 5 now gives that the classifier H(x) is a ranking function with AUC at least 1 −ǫ. For the final assertion of the theorem, note that at each round, in order to find the value of θ that defines ht the algorithm needs to minimize the sum of the error rates on the positive and negative examples. This can be done by sorting the examples using the weak ranking function (in O(m log m) time steps) and processing the examples in the resulting order, keeping running counts of the number of errors of each type. 4 Boosting weak rankers in the presence of misclassification noise The noise model: independent misclassification noise. The model of independent misclassification noise has been widely studied in computational learning theory. In this framework there is a noise rate η < 1/2, and each example (positive or negative) drawn from distribution D has its true label c(x) independently flipped with probability η before it is given to the learner. We write Dη to denote the resulting distribution over (noise-corrupted) labeled examples (x, y). Boosting weak rankers in the presence of independent misclassification noise. We now show how the AUC can be boosted arbitrarily close to 1 even if the data given to the booster is corrupted with independent misclassification noise, using weak rankers that are able to tolerate independent misclassification noise. We note that this is in contrast with known results for boosting the accuracy of binary classifiers in the presence of noise; Kalai and Servedio [7] show that no “black-box” boosting algorithm can be guaranteed to boost the accuracy of an arbitrary noise-tolerant weak learner to accuracy 1 −η in the presence of independent misclassification noise at rate η. ... ... ... ... ... v0,1 v0,2 v1,2 v0,3 v1,3 v2,3 v0,T +1 v1,T +1 vT −1,T +1 vT,T +1 | {z } | {z } output -1 output 1 Figure 2: The branching program produced by the boosting algorithm. Each node vi,t is labeled with a weak classifier hi,t; left edges correspond to -1 and right edges to 1. As in the previous section we begin by abstracting away sampling issues and using a model in which the booster passes a distribution to a weak ranker. Sampling issues will be treated in Section 5. Definition 6 A noise-tolerant weak ranker with advantage γ is an algorithm with the following property: for any noise rate η < 1/2, given a noisy distribution Dη, the algorithm outputs a ranking function h : X →R such that AUC(h; D) ≥1 2 + γ. Our algorithm for boosting the AUC in the presence of noise uses the Basic MartiBoost algorithm (see Section 4 of [9]). This algorithm boosts any two-sided weak learner to arbitrarily high accuracy and works in a series of rounds. Before round t the space of labeled examples is partitioned into a series of bins B0,t, ..., Bt−1,t. (The original bin B0,1 consists of the entire space.) In the t-th round the algorithm first constructs distributions D0,t, ..., Dt−1,t by conditioning the original distribution D on membership in B0,t, ..., Bt−1,t respectively. It then calls a two-sided weak learner t times using each of D0,t, ..., Dt−1,t, getting weak classifiers h0,t, ..., ht−1,t respectively. Having done this, it creates t + 1 bins for the next round by assigning each element (x, y) of Bi,t to Bi,t+1 if hi,t(x) = −1 and to Bi+1,t+1 otherwise. Training proceeds in this way for a given number T of rounds, which is an input parameter of the algorithm. The output of Basic MartiBoost is a layered branching program defined as follows. There is a node vi,t for each round 1 ≤t ≤T + 1 and each index 0 ≤i < t (that is, for each bin constructed during training). An item x is routed through the branching program the same way a labeled example (x, y) would have been routed during the training phase: it starts in node v0,1, and from each node vi,t it goes to vi,t+1 if hi,t(x) = −1, and to vi+1,t+1 otherwise. When the item x arrives at a terminal node of the branching program in layer T + 1, it is at some node vj,T +1. The prediction is 1 if j ≥T/2 and is −1 if j < T/2; in other words, the prediction is according to the majority vote of the weak classifiers that were encountered along the path through the branching program that the example followed. See Figure 3. The following lemma is proved in [9]. (The crux of the proof is the observation that positive (respectively, negative) examples are routed through the branching program according to a random walk that is biased to the right (respectively, left); hence the name “martingale boosting.”) Lemma 7 ([9]) Suppose that Basic MartiBoost is provided with a hypothesis hi,t with two-sided advantage γ w.r.t. Di,t at each node vi,t. Then for T = O(log(1/ǫ)/γ2), Basic MartiBoost constructs a branching program H such that D+[H(x) = −1] ≤ǫ and D−[H(x) = 1] ≤ǫ. We now describe our noise-tolerant AUC boosting algorithm, which we call Basic MartiRank. Given access to a noise-tolerant weak ranker A with advantage γ, at each node vi,t the Basic MartiRank algorithm runs A and proceeds as described in Lemma 4 to obtain a weak classifier hi,t. Basic MartiRank runs Basic MartiBoost with T = O(log(1/ǫ)/γ2) and simply uses the resulting classifier H as its ranking function. The following theorem shows that Basic MartiRank is an effective AUC booster in the presence of independent misclassification noise: Theorem 8 Fix any η < 1/2 and any ǫ > 0. Given access to Dη and a noise-tolerant weak ranker A with advantage γ, Basic MartiRank outputs a branching program H such that AUC(H; D) ≥1−ǫ. Proof: Fix any node vi,t in the branching program. The crux of the proof is the following simple observation: for a labeled example (x, y), the route through the branching program that is taken by (x, y) is determined completely by the predictions of the base classifiers, i.e. only by x, and is unaffected by the value of y. Consequently if Di,t denotes the original noiseless distribution D conditioned on reaching vi,t, then the noisy distribution conditioned on reaching vi,t, i.e. (Dη)i,t, is simply Di,t corrupted with independent misclassification noise, i.e. (Di,t)η. So each time the noisetolerant weak ranker A is invoked at a node vi,t, it is indeed the case that the distribution that it is given is an independent misclassification noise distribution. Consequently A does construct weak rankers with AUC at least 1/2 + γ, and the conversion of Lemma 4 yields weak classifiers that have advantage γ/4 with respect to the underlying distribution Di,t. Given this, Lemma 7 implies that the final classifier H has error at most ǫ on both positive and negative examples drawn from the original distribution D, and Lemma 5 then implies that H, viewed a ranker, achieves AUC at least 1 −ǫ. In [9], a more complex variant of Basic MartiBoost, called Noise-Tolerant SMartiBoost, is presented and is shown to boost any noise-tolerant weak learning algorithm to any accuracy less than 1 −η in the presence of independent misclassification noise. In contrast, here we are using just the Basic MartiBoost algorithm itself, and can achieve any AUC value 1 −ǫ even for ǫ < η. 5 Implementing MartiRank with a distribution oracle In this section we analyze learning from random examples. Formally, we assume that the weak ranker is given access to an oracle for the noisy distribution Dη. We thus now view a noise-tolerant weak ranker with advantage γ as an algorithm A with the following property: for any noise rate η < 1/2, given access to an oracle for Dη, the algorithm outputs a ranking function h : X →R such that AUC(h; D) ≥1 2 + γ. We let mA denote the number of examples from each class that suffice for A to construct a ranking function as described above. In other words, if A is provided with a sample of draws from Dη such that each class, positive and negative, has at least mA points in the sample with that true label, then algorithm A outputs a γ-advantage weak ranking function. (Note that for simplicity we are assuming here that the weak ranker always constructs a weak ranking function with the desired advantage, i.e. we gloss over the usual confidence parameter δ; this can be handled with an entirely standard analysis.) In order to achieve a computationally efficient algorithm in this setting we must change the MartiRank algorithm somewhat; we call the new variant Sampling Martirank, or SMartiRank. We prove that SMartiRank is computationally efficient, has moderate sample complexity, and efficiently generates a high-accuracy final ranking function with respect to the underlying distribution D. Our approach follows the same general lines as [9] where an oracle implementation is presented for the MartiBoost algorithm. The main challenge in [9] is the following: for each node vi,t in the branching program, the boosting algorithm considered there must simulate a balanced version of the induced distribution Di,t which puts equal weight on positive and negative examples. If only a tiny fraction of examples drawn from D are (say) positive and reach vi,t, then it is very inefficient to simulate this balanced distribution (and in a noisy scenario, as discussed earlier, if the noise rate is high relative to the frequency of the desired class then it may in fact be impossible to simulate the balanced distribution). The solution in [9] is to “freeze” any such node and simply classify any example that reaches it as negative; the analysis argues that since only a tiny fraction of positive examples reach such nodes, this freezing only mildly degrades the accuracy of the final hypothesis. In the ranking scenario that we now consider, we do not need to construct balanced distributions, but we do need to obtain a non-negligible number of examples from each class in order to run the weak learner at a given node. So as in [9] we still freeze some nodes, but with a twist: we now freeze nodes which have the property that for some class label (positive or negative), only a tiny fraction of examples from D with that class label reach the node. With this criterion for freezing we can prove that the final classifier constructed has high accuracy both on positive and negative examples, which is what we need to achieve good AUC. We turn now to the details. Given a node vi,t and a bit b ∈{−1, 1}, let pb i,t denote D[x reaches vi,t and c(x) = b]. The SMartiRank algorithm is like Basic MartiBoost but with the following difference: for each node vi,t and each value b ∈{−1, 1}, if pb i,t < ǫ · D[c(x) = b] T (T + 1) (2) then the node vi,t is “frozen,” i.e. it is labeled with the bit 1 −b and is established as a terminal node with no outgoing edges. (If this condition holds for both values of b at a particular node vi,t then the node is frozen and either output value may be used as the label.) The following theorem establishes that if SMartiRank is given weak classifiers with two-sided advantage at each node that is not frozen, it will construct a hypothesis with small error rate on both positive and negative examples: Theorem 9 Suppose that the SMartiRank algorithm as described above is provided with a hypothesis hi,t that has two-sided advantage γ with respect to Di,t at each node vi,t that is not frozen. Then for T = O(log(1/ǫ)/γ2), the final branching program hypothesis H that SMartiRank constructs will have D+[H(x) = −1] ≤ǫ and D−[H(x) = 1] ≤ǫ. Proof: We analyze D+[h(x) = −1]; the other case is symmetric. Given an unlabeled instance x ∈X, we say that x freezes at node vi,t if x’s path through the branching program causes it to terminate at a node vi,t with t < T + 1 (i.e. at a node vi,t which was frozen by SMartiRank). We have D[x freezes and c(x) = 1] = P i,t D[x freezes at vi,t and c(x) = 1] ≤P i,t ǫ·D[c(x)=1] T (T +1) ≤ǫ 2 · D[c(x) = 1]. Consequently we have D+[x freezes] = D[x freezes and c(x) = 1] D[c(x) = 1] < ǫ 2. (3) Naturally, D+[h(x) = −1] = D+[(h(x) = −1) & (x freezes)] + D+[(h(x) = −1) & (x does not freeze)]. By (3), this is at most ǫ 2 + D+[(h(x) = −1) & (x does not freeze)]. Arguments identical to those in the last two paragraphs of the proof of Theorem 3 in [9] show that D+[(h(x) = −1) & (x does not freeze)] ≤ǫ 2, and we are done. We now describe how SMartiRank can be run given oracle access to Dη and sketch the analysis of the required sample complexity (some details are omitted because of space limits). For simplicity of presentation we shall assume that the booster is given the value p def = min{D[c(x) = −1], D[c(x) = 1]}; we note if that p is not given a priori, a standard “guess and halve” technique can be used to efficiently obtain a value that is within a multiplicative factor of two of p, which is easily seen to suffice. We also make the standard assumption (see [7, 9]) that the noise rate η is known; this assumption can similarly be removed by having the algorithm “guess and check” the value to sufficiently fine granularity. Also, the confidence can be analyzed using the standard appeal to the union bound – details are omitted. SMartiRank will replace (2) with a comparison of sample estimates of the two quantities. To allow for the fact that they are just estimates, it will be more conservative, and freeze when the estimate of pb i,t is at most ǫ 4T (T +1) times the estimate of D[c(x) = b]. We first observe that for any distribution D and any bit b, we have Pr(x,y)∼Dη[y = b] = η + (1 − 2η)Pr(x,c(x))∼D[c(x) = b], which is equivalent to D[c(x) = b] = Dη[y=b]−η 1−2η . Consequently, given an empirical estimate of Dη[y = b] that is accurate to within an additive ± p(1−2η) 10 (which can easily be obtained from O( 1 p2(1−2η)2 ) draws to Dη), it is possible to estimate D[c(x) = b] to within an additive ±p/10, and thus to estimate the RHS of (2) to within an additive ± ǫp 10T (T +1). Now in order to determine whether node vi,t should be frozen, we must compare this estimate with a similarly accurate estimate of pb i,t (arguments similar to those of, e.g., Section 6.3 of [9] can be used to show that it suffices to run the algorithm using these estimated values). We have pb i,t = D[x reaches vi,t] · D[c(x) = b | x reaches vi,t] = Dη[x reaches vi,t] · Di,t[c(x) = b] = Dη[x reaches vi,t] · Dη i,t[y = b] −η 1 −2η ! . A standard analysis (see e.g. Chapter 5 of [8]) shows that this quantity can be estimated to additive accuracy ±τ using poly(1/τ, 1/(1−2η)) many calls to Dη (briefly, if Dη[x reaches vi,t] is less than τ(1−2η) then an estimate of 0 is good enough, while if it is greater than τ(1−2η) then a τ-accurate estimate of the second multiplicand can be obtained using O( 1 τ 3(1−2η)3 ) draws from Dη, since at least a τ(1 −2η) fraction of draws will reach vi,t.) Thus for each vi,t, we can determine whether to freeze it in the execution of SMartiRank using poly(T, 1/ǫ, 1/p, 1/(1−2η)) draws from Dη. For each of the nodes that are not frozen, we must run the noise-tolerant weak ranker A using the distribution Dη i,t. As discussed at the beginning of this section, this requires that we obtain a sample from Dη i,t containing at least mA examples whose true label belongs to each class. The expected number of draws from Dη that must be made in order to receive an example from a given class is 1/p, and since vi,t is not frozen, the expected number of draws from Dη belonging to a given class that must be made in order to simulate a draw from Dη i,t belonging to that class is O(T 2/ǫ). Thus, O(T 2mA/(ǫp)) many draws from Dη are required in order to run the weak learner A at any particular node. Since there are O(T 2) many nodes overall, we have that all in all O(T 4mA/(ǫp)) many draws from Dη are required, in addition to the poly(T, 1/ǫ, 1/p, 1/(1 −2η)) draws required to identify which nodes to freeze. Recalling that T = O(log(1/ǫ)/γ2), all in all we have: Theorem 10 Let D be a nontrivial distribution over X, p = min{D[c(x) = −1], D[c(x) = 1]}, and η < 1 2. Given access to an oracle for Dη and a noise-tolerant weak ranker A with advantage γ, the SMartiRank algorithm makes mA· poly( 1 ǫ, 1 γ , 1 1−2η, 1 p) calls to Dη, and and with probability 1 −δ outputs a branching program H such that AUC(h; D) ≥1 −ǫ. Acknowledgement We are very grateful to Naoki Abe for suggesting the problem of boosting the AUC. References [1] A. P. Bradley. Use of the area under the ROC curve in the evaluation of machine learning algorithms. Pattern Recognition, 30:1145–1159, 1997. [2] C. Cortes and M. Mohri. AUC optimization vs. error rate minimzation. In NIPS 2003, 2003. [3] T. Fawcett. ROC graphs: Notes and practical considerations for researchers. Technical Report HPL-2003-4, HP, 2003. [4] Y. Freund, R. Iyer, R. E. Schapire, and Y. Singer. An efficient boosting algorithm for combining preferences. Journal of Machine Learning Research, 4(6):933–970, 2004. [5] Y. Freund and R. Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. Journal of Computer and System Sciences, 55(1):119–139, 1997. [6] J. Hanley and B. McNeil. The meaning and use of the area under a receiver operating characteristic (ROC) curve. Radiology, 143(1):29–36, 1982. [7] A. Kalai and R. Servedio. Boosting in the presence of noise. Journal of Computer & System Sciences, 71(3):266–290, 2005. Preliminary version in Proc. STOC’03. [8] M. Kearns and U. Vazirani. An introduction to computational learning theory. MIT Press, Cambridge, MA, 1994. [9] P. Long and R. Servedio. Martingale boosting. In Proceedings of the Eighteenth Annual Conference on Computational Learning Theory (COLT), pages 79–94, 2005. [10] F. Provost, T. Fawcett, and Ron Kohavi. The case against accuracy estimation for comparing induction algorithms. ICML, 1998. [11] C. Rudin, C. Cortes, M. Mohri, and R. E. Schapire. Margin-based ranking meets boosting in the middle. COLT, 2005. [12] J. A. Swets. Signal detection theory and ROC analysis in psychology and diagnostics: Collected papers. Lawrence Erlbaum Associates, 1995.
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Sparse Feature Learning for Deep Belief Networks Marc’Aurelio Ranzato1 Y-Lan Boureau2,1 Yann LeCun1 1 Courant Institute of Mathematical Sciences, New York University 2 INRIA Rocquencourt {ranzato,ylan,yann@courant.nyu.edu} Abstract Unsupervised learning algorithms aim to discover the structure hidden in the data, and to learn representations that are more suitable as input to a supervised machine than the raw input. Many unsupervised methods are based on reconstructing the input from the representation, while constraining the representation to have certain desirable properties (e.g. low dimension, sparsity, etc). Others are based on approximating density by stochastically reconstructing the input from the representation. We describe a novel and efficient algorithm to learn sparse representations, and compare it theoretically and experimentally with a similar machine trained probabilistically, namely a Restricted Boltzmann Machine. We propose a simple criterion to compare and select different unsupervised machines based on the trade-off between the reconstruction error and the information content of the representation. We demonstrate this method by extracting features from a dataset of handwritten numerals, and from a dataset of natural image patches. We show that by stacking multiple levels of such machines and by training sequentially, high-order dependencies between the input observed variables can be captured. 1 Introduction One of the main purposes of unsupervised learning is to produce good representations for data, that can be used for detection, recognition, prediction, or visualization. Good representations eliminate irrelevant variabilities of the input data, while preserving the information that is useful for the ultimate task. One cause for the recent resurgence of interest in unsupervised learning is the ability to produce deep feature hierarchies by stacking unsupervised modules on top of each other, as proposed by Hinton et al. [1], Bengio et al. [2] and our group [3, 4]. The unsupervised module at one level in the hierarchy is fed with the representation vectors produced by the level below. Higherlevel representations capture high-level dependencies between input variables, thereby improving the ability of the system to capture underlying regularities in the data. The output of the last layer in the hierarchy can be fed to a conventional supervised classifier. A natural way to design stackable unsupervised learning systems is the encoder-decoder paradigm [5]. An encoder transforms the input into the representation (also known as the code or the feature vector), and a decoder reconstructs the input (perhaps stochastically) from the representation. PCA, Auto-encoder neural nets, Restricted Boltzmann Machines (RBMs), our previous sparse energy-based model [3], and the model proposed in [6] for noisy overcomplete channels are just examples of this kind of architecture. The encoder/decoder architecture is attractive for two reasons: 1. after training, computing the code is a very fast process that merely consists in running the input through the encoder; 2. reconstructing the input with the decoder provides a way to check that the code has captured the relevant information in the data. Some learning algorithms [7] do not have a decoder and must resort to computationally expensive Markov Chain Monte Carlo (MCMC) sampling methods in order to provide reconstructions. Other learning algorithms [8, 9] lack an encoder, which makes it necessary to run an expensive optimization algorithm to find the code associated with each new input sample. In this paper we will focus only on encoder-decoder architectures. 1 In general terms, we can view an unsupervised model as defining a distribution over input vectors Y through an energy function E(Y, Z, W): P(Y |W) = Z z P(Y, z|W) = R z e−βE(Y,z,W ) R y,z e−βE(y,z,W ) (1) where Z is the code vector, W the trainable parameters of encoder and decoder, and β is an arbitrary positive constant. The energy function includes the reconstruction error, and perhaps other terms as well. For convenience, we will omit W from the notation in the following. Training the machine to model the input distribution is performed by finding the encoder and decoder parameters that minimize a loss function equal to the negative log likelihood of the training data under the model. For a single training sample Y , the loss function is L(W, Y ) = −1 β log Z z e−βE(Y,z) + 1 β log Z y,z e−βE(y,z) (2) The first term is the free energy Fβ(Y ). Assuming that the distribution over Z is rather peaked, it can be simpler to approximate this distribution over Z by its mode, which turns the marginalization over Z into a minimization: L∗(W, Y ) = E(Y, Z∗(Y )) + 1 β log Z y e−βE(y,Z∗(y)) (3) where Z∗(Y ) is the maximum likelihood value Z∗(Y ) = argminzE(Y, z), also known as the optimal code. We can then define an energy for each input point, that measures how well it is reconstructed by the model: F∞(Y ) = E(Y, Z∗(Y )) = lim β→∞−1 β log Z z e−βE(Y,z) (4) The second term in equation 2 and 3 is called the log partition function, and can be viewed as a penalty term for low energies. It ensures that the system produces low energy only for input vectors that have high probability in the (true) data distribution, and produces higher energies for all other input vectors [5]. The overall loss is the average of the above over the training set. Regardless of whether only Z∗or the whole distribution over Z is considered, the main difficulty with this framework is that it can be very hard to compute the gradient of the log partition function in equation 2 or 3 with respect to the parameters W. Efficient methods shortcut the computation by drastically and cleverly reducing the integration domain. For instance, Restricted Boltzmann Machines (RBM) [10] approximate the gradient of the log partition function in equation 2 by sampling values of Y whose energy will be pulled up using an MCMC technique. By running the MCMC for a short time, those samples are chosen in the vicinity of the training samples, thereby ensuring that the energy surface forms a ravine around the manifold of the training samples. This is the basis of the Contrastive Divergence method [10]. The role of the log partition function is merely to ensure that the energy surface is lower around training samples than anywhere else. The method proposed here eliminates the log partition function from the loss, and replaces it by a term that limits the volume of the input space over which the energy surface can take a low value. This is performed by adding a penalty term on the code rather than on the input. While this class of methods does not directly maximize the likelihood of the data, it can be seen as a crude approximation of it. To understand the method, we first note that if for each vector Y , there exists a corresponding optimal code Z∗(Y ) that makes the reconstruction error (or energy) F∞(Y ) zero (or near zero), the model can perfectly reconstruct any input vector. This makes the energy surface flat and indiscriminate. On the other hand, if Z can only take a small number of different values (low entropy code), then the energy F∞(Y ) can only be low in a limited number of places (the Y ’s that are reconstructed from this small number of Z values), and the energy cannot be flat. More generally, a convenient method through which flat energy surfaces can be avoided is to limit the maximum information content of the code. Hence, minimizing the energy F∞(Y ) together with the information content of the code is a good substitute for minimizing the log partition function. 2 A popular way to minimize the information content in the code is to make the code sparse or lowdimensional [5]. This technique is used in a number of unsupervised learning methods, including PCA, auto-encoders neural network, and sparse coding methods [6, 3, 8, 9]. In sparse methods, the code is forced to have only a few non-zero units while most code units are zero most of the time. Sparse-overcomplete representations have a number of theoretical and practical advantages, as demonstrated in a number of recent studies [6, 8, 3]. In particular, they have good robustness to noise, and provide a good tiling of the joint space of location and frequency. In addition, they are advantageous for classifiers because classification is more likely to be easier in higher dimensional spaces. This may explain why biology seems to like sparse representations [11]. In our context, the main advantage of sparsity constraints is to allow us to replace a marginalization by a minimization, and to free ourselves from the need to minimize the log partition function explicitly. In this paper we propose a new unsupervised learning algorithm called Sparse Encoding Symmetric Machine (SESM), which is based on the encoder-decoder paradigm, and which is able to produce sparse overcomplete representations efficiently without any need for filter normalization [8, 12] or code saturation [3]. As described in more details in sec. 2 and 3, we consider a loss function which is a weighted sum of the reconstruction error and a sparsity penalty, as in many other unsupervised learning algorithms [13, 14, 8]. Encoder and decoder are constrained to be symmetric, and share a set of linear filters. Although we only consider linear filters in this paper, the method allows the use of any differentiable function for encoder and decoder. We propose an iterative on-line learning algorithm which is closely related to those proposed by Olshausen and Field [8] and by us previously [3]. The first step computes the optimal code by minimizing the energy for the given input. The second step updates the parameters of the machine so as to minimize the energy. In sec. 4, we compare SESM with RBM and PCA. Following [15], we evaluate these methods by measuring the reconstruction error for a given entropy of the code. In another set of experiments, we train a classifier on the features extracted by the various methods, and measure the classification error on the MNIST dataset of handwritten numerals. Interestingly, the machine achieving the best recognition performance is the one with the best trade-off between RMSE and entropy. In sec. 5, we compare the filters learned by SESM and RBM for handwritten numerals and natural image patches. In sec.5.1.1, we describe a simple way to produce a deep belief net by stacking multiple levels of SESM modules. The representational power of this hierarchical non-linear feature extraction is demonstrated through the unsupervised discovery of the numeral class labels in the high-level code. 2 Architecture In this section we describe a Sparse Encoding Symmetric Machine (SESM) having a set of linear filters in both encoder and decoder. However, everything can be easily extended to any other choice of parameterized functions as long as these are differentiable and maintain symmetry between encoder and decoder. Let us denote with Y the input defined in RN, and with Z the code defined in RM, where M is in general greater than N (for overcomplete representations). Let the filters in encoder and decoder be the columns of matrix W ∈RN×M, and let the biases in the encoder and decoder be denoted by benc ∈RM and bdec ∈RN, respectively. Then, encoder and decoder compute: fenc(Y ) = W T Y + benc, fdec(Z) = Wl(Z) + bdec (5) where the function l is a point-wise logistic non-linearity of the form: l(x) = 1/(1 + exp(−gx)), (6) with g fixed gain. The system is characterized by an energy measuring the compatibility between pairs of input Y and latent code Z, E(Y, Z) [16]. The lower the energy, the more compatible (or likely) is the pair. We define the energy as: E(Y, Z) = αe∥Z −fenc(Y )∥2 2 + ∥Y −fdec(Z)∥2 2 (7) During training we minimize the following loss: L(W, Y ) = E(Y, Z) + αsh(Z) + αr∥W∥1 = αe∥Z −fenc(Y )∥2 2 + ∥Y −fdec(Z)∥2 2 + αsh(Z) + αr∥W∥1 (8) The first term tries to make the output of the encoder as similar as possible to the code Z. The second term is the mean-squared error between the input Y and the reconstruction provided by the decoder. 3 The third term ensures the sparsity of the code by penalizing non zero values of code units; this term acts independently on each code unit and it is defined as h(Z) = PM i=1 log(1+l2(zi)), (corresponding to a factorized Student-t prior distribution on the non linearly transformed code units [8] through the logistic of equation 6). The last term is an L1 regularization on the filters to suppress noise and favor more localized filters. The loss formulated in equation 8 combines terms that characterize also other methods. For instance, the first two terms appear in our previous model [3], but in that work, the weights of encoder and decoder were not tied and the parameters in the logistic were updated using running averages. The second and third terms are present in the “decoder-only” model proposed in [8]. The third term was used in the “encoder-only” model of [7]. Besides the alreadymentioned advantages of using an encoder-decoder architecture, we point out another good feature of this algorithm due to its symmetry. A common idiosyncrasy for sparse-overcomplete methods using both a reconstruction and a sparsity penalty in the objective function (second and third term in equation 8), is the need to normalize the basis functions in the decoder during learning [8, 12] with somewhat ad-hoc technique, otherwise some of the basis functions collapse to zero, and some blow up to infinity. Because of the sparsity penalty and the linear reconstruction, code units become tiny and are compensated by the filters in the decoder that grow without bound. Even though the overall loss decreases, training is unsuccessful. Unfortunately, simply normalizing the filters makes less clear which objective function is minimized. Some authors have proposed quite expensive methods to solve this issue: by making better approximations of the posterior distribution [15], or by using sampling techniques [17]. In this work, we propose to enforce symmetry between encoder and decoder (through weight sharing) so as to have automatic scaling of filters. Their norm cannot possibly be large because code units, produced by the encoder weights, would have large values as well, producing bad reconstructions and increasing the energy (the second term in equation 7 and 8). 3 Learning Algorithm Learning consists of determining the parameters in W, benc, and bdec that minimize the loss in equation 8. As indicated in the introduction, the energy augmented with the sparsity constraint is minimized with respect to the code to find the optimal code. No marginalization over code distribution is performed. This is akin to using the loss function in equation 3. However, the log partition function term is dropped. Instead, we rely on the code sparsity constraints to ensure that the energy surface is not flat. Since the second term in equation 8 couples both Z and W and bdec, it is not straightforward to minimize this energy with respect to both. On the other hand, once Z is given, the minimization with respect to W is a convex quadratic problem. Vice versa, if the parameters W are fixed, the optimal code Z∗that minimizes L can be computed easily through gradient descent. This suggests the following iterative on-line coordinate descent learning algorithm: 1. for a given sample Y and parameter setting, minimize the loss in equation 8 with respect to Z by gradient descent to obtain the optimal code Z∗ 2. clamping both the input Y and the optimal code Z∗found at the previous step, do one step of gradient descent to update the parameters. Unlike other methods [8, 12], no column normalization of W is required. Also, all the parameters are updated by gradient descent unlike in our previous work [3] where some parameters are updated using a moving average. After training, the system converges to a state where the decoder produces good reconstructions from a sparse code, and the optimal code is predicted by a simple feed-forward propagation through the encoder. 4 Comparative Coding Analysis In the following sections, we mainly compare SESM with RBM in order to better understand their differences in terms of maximum likelihood approximation, and in terms of coding efficiency and robustness. RBM As explained in the introduction, RBMs minimize an approximation of the negative log likelihood of the data under the model. An RBM is a binary stochastic symmetric machine defined 4 by an energy function of the form: E(Y, Z) = −ZT W T Y −bT encZ −bT decY . Although this is not obvious at first glance, this energy can be seen as a special case of the encoder-decoder architecture that pertains to binary data vectors and code vectors [5]. Training an RBM minimizes an approximation of the negative log likelihood loss function 2, averaged over the training set, through a gradient descent procedure. Instead of estimating the gradient of the log partition function, RBM training uses contrastive divergence [10], which takes random samples drawn over a limited region Ωaround the training samples. The loss becomes: L(W, Y ) = −1 β log X z e−βE(Y,z) + 1 β log X y∈Ω X z e−βE(y,z) (9) Because of the RBM architecture, given a Y , the components of Z are independent, hence the sum over configurations of Z can be done independently for each component of Z. Sampling y in the neighborhood Ωis performed with one, or a few alternated MCMC steps over Y , and Z. This means that only the energy of points around training samples is pulled up. Hence, the likelihood function takes the right shape around the training samples, but not necessarily everywhere. However, the code vector in an RBM is binary and noisy, and one may wonder whether this does not have the effect of surreptitiously limiting the information content of the code, thereby further minimizing the log partition function as a bonus. SESM RBM and SESM have almost the same architecture because they both have a symmetric encoder and decoder, and a logistic non-linearity on the top of the encoder. However, RBM is trained using (approximate) maximum likelihood, while SESM is trained by simply minimizing the average energy F∞(Y ) of equation 4 with an additional code sparsity term. SESM relies on the sparsity term to prevent flat energy surfaces, while RBM relies on an explicit contrastive term in the loss, an approximation of the log partition function. Also, the coding strategy is very different because code units are “noisy” and binary in RBM, while they are quasi-binary and sparse in SESM. Features extracted by SESM look like object parts (see next section), while features produced by RBM lack an intuitive interpretation because they aim at modeling the input distribution and they are used in a distributed representation. 4.1 Experimental Comparison In the first experiment we have trained SESM, RBM, and PCA on the first 20000 digits in the MNIST training dataset [18] in order to produce codes with 200 components. Similarly to [15] we have collected test image codes after the logistic non linearity (except for PCA which is linear), and we have measured the root mean square error (RMSE) and the entropy. SESM was run for different values of the sparsity coefficient αs in equation 8 (while all other parameters are left unchanged, see next section for details). The RMSE is defined as 1 σ q 1 P N ∥Y −fdec( ¯Z)∥2 2, where ¯Z is the uniformly quantized code produced by the encoder, P is the number of test samples, and σ is the estimated variance of units in the input Y . Assuming to encode the (quantized) code units independently and with the same distribution, the lower bound on the number of bits required to encode each of them is given by: Hc.u. = −PQ i=1 ci P M log2 ci P M , where ci is the number of counts in the i-th bin, and Q is the number of quantization levels. The number of bits per pixel is then equal to: M N Hc.u.. Unlike in [15, 12], the reconstruction is done taking the quantized code in order to measure the robustness of the code to the quantization noise. As shown in fig. 1-C, RBM is very robust to noise in the code because it is trained by sampling. The opposite is true for PCA which achieves the lowest RMSE when using high precision codes, but the highest RMSE when using a coarse quantization. SESM seems to give the best trade-off between RMSE and entropy. Fig. 1-D/F compare the features learned by SESM and RBM. Despite the similarities in the architecture, filters look quite different in general, revealing two different coding strategies: distributed for RBM, and sparse for SESM. In the second experiment, we have compared these methods by means of a supervised task in order to assess which method produces the most discriminative representation. Since we have available also the labels in the MNIST, we have used the codes (produced by these machines trained unsupervised) as input to the same linear classifier. This is run for 100 epochs to minimize the squared error between outputs and targets, and has a mild ridge regularizer. Fig. 1-A/B show the result of these experiments in addition to what can be achieved by a linear classifier trained on the raw pixel data. Note that: 1) training on features instead of raw data improves the recognition (except for PCA 5 (A) 0 1 2 0 5 10 15 20 25 30 35 40 45 ENTROPY (bits/pixel) ERROR RATE % 10 samples 0 1 2 0 2 4 6 8 10 12 14 16 18 ENTROPY (bits/pixel) ERROR RATE % 100 samples 0 1 2 3 4 5 6 7 8 9 10 ENTROPY (bits/pixel) ERROR RATE % 1000 samples RAW: train RAW: test PCA: train PCA: test RBM: train RBM: test SESM: train SESM: test (B) 0 0.2 0.4 0 5 10 15 20 25 30 35 40 45 RMSE ERROR RATE % 10 samples 0 0.2 0.4 0 2 4 6 8 10 12 14 16 18 RMSE ERROR RATE % 100 samples 0 0.2 0.4 3 4 5 6 7 8 9 10 RMSE ERROR RATE % 1000 samples (C) 0 0.5 1 1.5 2 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 RMSE Entropy (bits/pixel) Symmetric Sparse Coding − RBM − PCA PCA: quantization in 5 bins PCA: quantization in 256 bins RBM: quantization in 5 bins RBM: quantization in 256 bins Sparse Coding: quantization in 5 bins Sparse Coding: quantization in 256 bins (D) (E) (F) (G) (H) Figure 1: (A)-(B) Error rate on MNIST training (with 10, 100 and 1000 samples per class) and test set produced by a linear classifier trained on the codes produced by SESM, RBM, and PCA. The entropy and RMSE refers to a quantization into 256 bins. The comparison has been extended also to the same classifier trained on raw pixel data (showing the advantage of extracting features). The error bars refer to 1 std. dev. of the error rate for 10 random choices of training datasets (same splits for all methods). The parameter αs in eq. 8 takes values: 1, 0.5, 0.2, 0.1, 0.05. (C) Comparison between SESM, RBM, and PCA when quantizing the code into 5 and 256 bins. (D) Random selection from the 200 linear filters that were learned by SESM (αs = 0.2). (E) Some pairs of original and reconstructed digit from the code produced by the encoder in SESM (feed-forward propagation through encoder and decoder). (F) Random selection of filters learned by RBM. (G) Back-projection in image space of the filters learned in the second stage of the hierarchical feature extractor. The second stage was trained on the non linearly transformed codes produced by the first stage machine. The back-projection has been performed by using a 1-of-10 code in the second stage machine, and propagating this through the second stage decoder and first stage decoder. The filters at the second stage discover the class-prototypes (manually ordered for visual convenience) even though no class label was ever used during training. (H) Feature extraction from 8x8 natural image patches: some filters that were learned. 6 when the number of training samples is small), 2) RBM performance is competitive overall when few training samples are available, 3) the best performance is achieved by SESM for a sparsity level which trades off RMSE for entropy (overall for large training sets), 4) the method with the best RMSE is not the one with lowest error rate, 5) compared to a SESM having the same error rate RBM is more costly in terms of entropy. 5 Experiments This section describes some experiments we have done with SESM. The coefficient αe in equation 8 has always been set equal to 1, and the gain in the logistic have been set equal to 7 in order to achieve a quasi-binary coding. The parameter αs has to be set by cross-validation to a value which depends on the level of sparsity required by the specific application. 5.1 Handwritten Digits Fig. 1-B/E shows the result of training a SESM with αs is equal to 0.2. Training was performed on 20000 digits scaled between 0 and 1, by setting αr to 0.0004 (in equation 8) with a learning rate equal to 0.025 (decreased exponentially). Filters detect the strokes that can be combined to form a digit. Even if the code unit activation has a very sparse distribution, reconstructions are very good (no minimization in code space was performed). 5.1.1 Hierarchical Features A hierarchical feature extractor can be trained layer-by-layer similarly to what has been proposed in [19, 1] for training deep belief nets (DBNs). We have trained a second (higher) stage machine on the non linearly transformed codes produced by the first (lower) stage machine described in the previous example. We used just 20000 codes to produce a higher level representation with just 10 components. Since we aimed to find a 1-of-10 code we increased the sparsity level (in the second stage machine) by setting αs to 1. Despite the completely unsupervised training procedure, the feature detectors in the second stage machine look like digit prototypes as can be seen in fig. 1-G. The hierarchical unsupervised feature extractor is able to capture higher order correlations among the input pixel intensities, and to discover the highly non-linear mapping from raw pixel data to the class labels. Changing the random initialization can sometimes lead to the discover of two different shapes of “9” without a unit encoding the “4”, for instance. Nevertheless, results are qualitatively very similar to this one. For comparison, when training a DBN, prototypes are not recovered because the learned code is distributed among units. 5.2 Natural Image Patches A SESM with about the same set up was trained on a dataset of 30000 8x8 natural image patches randomly extracted from the Berkeley segmentation dataset [20]. The input images were simply scaled down to the range [0, 1.7], without even subtracting the mean. We have considered a 2 times overcomplete code with 128 units. The parameters αs, αr and the learning rate were set to 0.4, 0.025, and 0.001 respectively. Some filters are localized Gabor-like edge detectors in different positions and orientations, other are more global, and some encode the mean value (see fig. 1-H). 6 Conclusions There are two strategies to train unsupervised machines: 1) having a contrastive term in the loss function minimized during training, 2) constraining the internal representation in such a way that training samples can be better reconstructed than other points in input space. We have shown that RBM, which falls in the first class of methods, is particularly robust to channel noise, it achieves very low RMSE and good recognition rate. We have also proposed a novel symmetric sparse encoding method following the second strategy which: is particularly efficient to train, has fast inference, works without requiring any withening or even mean removal from the input, can provide the best recognition performance and trade-off between entropy/RMSE, and can be easily extended to a hierarchy discovering hidden structure in the data. We have proposed an evaluation protocol to compare different machines which is based on RMSE, entropy and, eventually, error rate when also 7 labels are available. Interestingly, the machine achieving the best performance in classification is the one with the best trade-off between reconstruction error and entropy. A future avenue of work is to understand the reasons for this “coincidence”, and deeper connections between these two strategies. Acknowledgments We wish to thank Jonathan Goodman, Geoffrey Hinton, and Yoshua Bengio for helpful discussions. This work was supported in part by NSF grant IIS-0535166 “toward category-level object recognition”, NSF ITR-0325463 “new directions in predictive learning”, and ONR grant N00014-07-1-0535 “integration and representation of high dimensional data”. References [1] G.E. Hinton and R. R Salakhutdinov. Reducing the dimensionality of data with neural networks. Science, 313(5786):504–507, 2006. [2] Y. Bengio, P. Lamblin, D. Popovici, and H. Larochelle. Greedy layer-wise training of deep networks. In NIPS, 2006. [3] M. Ranzato, C. Poultney, S. Chopra, and Y. LeCun. Efficient learning of sparse representations with an energy-based model. In NIPS 2006. MIT Press, 2006. [4] Y. Bengio and Y. LeCun. Scaling learning algorithms towars ai. In D. DeCoste L. Bottou, O. Chapelle and J. Weston, editors, Large-Scale Kernel Machines. MIT Press, 2007. [5] M. Ranzato, Y. Boureau, S. Chopra, and Y. LeCun. A unified energy-based framework for unsupervised learning. In Proc. 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Reinforcement Learning in Continuous Action Spaces through Sequential Monte Carlo Methods Alessandro Lazaric Marcello Restelli Andrea Bonarini Department of Electronics and Information Politecnico di Milano piazza Leonardo da Vinci 32, I-20133 Milan, Italy {bonarini,lazaric,restelli}@elet.polimi.it Abstract Learning in real-world domains often requires to deal with continuous state and action spaces. Although many solutions have been proposed to apply Reinforcement Learning algorithms to continuous state problems, the same techniques can be hardly extended to continuous action spaces, where, besides the computation of a good approximation of the value function, a fast method for the identification of the highest-valued action is needed. In this paper, we propose a novel actor-critic approach in which the policy of the actor is estimated through sequential Monte Carlo methods. The importance sampling step is performed on the basis of the values learned by the critic, while the resampling step modifies the actor’s policy. The proposed approach has been empirically compared to other learning algorithms into several domains; in this paper, we report results obtained in a control problem consisting of steering a boat across a river. 1 Introduction Most of the research on Reinforcement Learning (RL) [13] has studied solutions to finite Markov Decision Processes (MDPs). On the other hand, learning in real-world environments requires to deal with continuous state and action spaces. While several studies focused on problems with continuous states, little attention has been deserved to tasks involving continuous actions. Although several tasks may be (suboptimally) solved by coarsely discretizing the action variables (for instance using the tile coding approach [11, 12]), a different approach is required for problems in which high-precision control is needed and actions slightly different from the optimal one lead to very low utility values. In fact, since RL algorithms need to experience each available action several times to estimate its utility, using very fine discretizations may be too expensive for the learning process. Some approaches, although using a finite set of target actions, deal with this problem by selecting real-valued actions obtained by interpolation of the available discrete actions on the basis of their utility values [9, 14]. Despite of this capability, the learning performance of these algorithms relies on strong assumptions about the shape of the value function that are not always satisfied in highly non-linear control problems. The wire fitting algorithm [2] (later adopted also in [4]) tries to solve this problem by implementing an adaptive interpolation scheme in which a finite set of pairs ⟨action, value⟩is modified in order to better approximate the action value function. Besides having the capability of selecting any real-valued action, RL algorithms for continuous action problems should be able to efficiently find the greedy action, i.e., the action associated to the highest estimated value. Differently from the finite MDP case, a full search in a continuous action space to find the optimal action is often unfeasible. To overcome this problem, several approaches limit their search over a finite number of points. In order to keep low this number, many algorithms (e.g., tile coding and interpolation-based) need to make (often implicit) assumptions about the shape of the value function. To overcome these difficulties, several approaches have adopted the actorcritic architecture [7, 10]. The key idea of actor-critic methods is to explicitly represent the policy (stored by the actor) with a memory structure independent of the one used for the value function (stored by the critic). In a given state, the policy followed by the agent is a probability distribution over the action space, usually represented by parametric functions (e.g., Gaussians [6], neural networks [14], fuzzy systems [5]). The role of the critic is, on the basis of the estimated value function, to criticize the actions taken by the actor, which consequently modifies its policy through a stochastic gradient on its parameter space. In this way, starting from a fully exploratory policy, the actor progressively changes its policy so that actions that yield higher utility values are more frequently selected, until the learning process converges to the optimal policy. By explicitly representing the policy, actor-critic approaches can efficiently implement the action selection step even in problems with continuous action spaces. In this paper, we propose to use a Sequential Monte Carlo (SMC) method [8] to approximate the sequence of probability distributions implemented by the actor, thus obtaining a novel actor-critic algorithm called SMC-learning. Instead of a parametric function, the actor represents its stochastic policy by means of a finite set of random samples (i.e., actions) that, using simple resampling and moving mechanisms, is evolved over time according to the values stored by the critic. Actions are initially drawn from a prior distribution, and then they are resampled according to an importance sampling estimate which depends on the utility values learned by the critic. By means of the resampling and moving steps, the set of available actions gets more and more thick around actions with larger utilities, thus encouraging a detailed exploration of the most promising action-space regions, and allowing SMC-learning to find real continuous actions. It is worth pointing out that the main goal here is not an accurate approximation of the action-value function on the whole action space, but to provide an efficient way to converge to the continuous optimal policy. The main characteristics of the proposed approach are: the agent may learn to execute any continuous action, the action selection phase and the search for the action with the best estimated value are computationally efficient, no assumption on the shape of the value function is required, the algorithm is model-free, and it may learn to follow also stochastic policies (needed in multi-agent problems). In the next section, we introduce basic RL notation and briefly discuss issues about learning with continuous actions. Section 3 details the proposed learning approach (SMC-Learning), explaining how SMC methods can be used to learn in continuous action spaces. Experimental results are discussed in Section 4, and Section 5 draws conclusions and contains directions for future research. 2 Reinforcement Learning In reinforcement learning problems, an agent interacts with an unknown environment. At each time step, the agent observes the state, takes an action, and receives a reward. The goal of the agent is to learn a policy (i.e., a mapping from states to actions) that maximizes the long-term return. An RL problem can be modeled as a Markov Decision Process (MDP) defined by a quadruple ⟨S, A, T , R, γ⟩, where S is the set of states, A(s) is the set of actions available in state s, T : S ×A×S →[0, 1] is a transition distribution that specifies the probability of observing a certain state after taking a given action in a given state, R : S×A →ℜis a reward function that specifies the instantaneous reward when taking a given action in a given state, and γ ∈[0, 1) is a discount factor. The policy of an agent is characterized by a probability distribution π(a|s) that specifies the probability of taking action a in state s. The utility of taking action a in state s and following a policy π thereafter is formalized by the action-value function Qπ(s, a) = E P∞ t=1 γt−1rt|s = s1, a = a1, π , where r1 = R(s, a). RL approaches aim at learning the policy that maximizes the action-value function in each state. The optimal action-value function can be computed by solving the Bellman equation: Q∗(s, a) = R(s, a)+γ P s′ T (s, a, s′) maxa′ Q∗(s′, a′). The optimal policy can be defined as the greedy action in each state: π∗(a|s) is equal to 1/|arg maxa Q∗(s, a)| if a ∈arg maxa Q∗(s, a), and 0 otherwise. Temporal Difference (TD) algorithms [13] allows the computation of Q∗(s, a) by direct interaction with the environment. Given the tuple ⟨st, at, rt, st+1, at+1⟩(i.e., the experience performed by the agent), at each step, action values may be estimated by online algorithms, such as SARSA, whose update rule is: Q(st, at) ←(1 −α)Q(st, at) + αu(rt, at+1, st+1), (1) where α ∈[0, 1] is a learning rate and u(rt, at+1, st+1) = rt + γQ(st+1, at+1) is the target utility. Although value-function approaches have theoretical guarantees about convergence to the optimal policy and have been proved to be effective in many applications, they have several limitations: algorithms that maximize the value function cannot solve problems whose solutions are stochastic policies (e.g., multi-agent learning problems); small errors in the estimated value of an action may lead to discontinuous changes in the policy [3], thus leading to convergence problems when function approximators are considered. These problems may be overcome by adopting actor-critic methods [7] in which the action-value function and the policy are stored into two distinct representations. The actor typically represents the distribution density over the action space through a function π(a|s, θ), whose parameters θ are updated in the direction of performance improvement, as established by the critic on the basis of its approximation of the value function, which is usually computed through an on-policy TD algorithm. 3 SMC-Learning for Continuous Action Spaces SMC-learning is based on an actor-critic architecture, in which the actor stores and updates, for each state s, a density distribution πt(a|s) that specifies the agent’s policy at time instant t. At the beginning of the learning process, without any prior information about the problem, the actor usually considers a uniform distribution over the action space, thus implementing a fully exploratory policy. As the learning process progresses, the critic collects data for the estimation of the value function (in this paper, the critic estimates the action-value function), and provides the actor with information about which actions are the most promising. On the other hand, the actor changes its policy to improve its performance and to progressively reduce the exploration in order to converge to the optimal deterministic policy. Instead of using parametric functions, in SMC-learning the actor represents its evolving stochastic policy by means of Monte Carlo sampling. The idea is the following: for each state s, the set of available actions A(s) is initialized with N samples drawn from a proposal distribution π0(a|s): A(s) = {a1, a2, · · · , aN}, ai ∼π0(a|s). Each sampled action ai is associated to an importance weight wi ∈W(s) whose value is initialized to 1/N, so that the prior density can be approximated as π0(a|s) ≃ N X i=1 wi · δ(a −ai), where ai ∈A(s), wi ∈W(s), and δ is the Dirac delta measure. As the number of samples goes to infinity this representation gets equivalent to the functional description of the original probability density function. This means that the actor can approximately follow the policy specified by the density π0(a|s), by simply choosing actions at random from A(s), where the (normalized) weights are the selection probabilities. Given the continuous action-value function estimated by the critic and chosen a suitable exploration strategy (e.g., the Boltzmann exploration), it is possible to define the desired probability distribution over the continuous action space, usually referred to as the target distribution. As long as the learning process goes on, the action values estimated by the critic become more and more reliable, and the policy followed by the agent should change in order to choose more frequently actions with higher utilities. This means that, in each state, the target distribution changes according to the information collected during the learning process, and the actor must consequently adapt its approximation. In general, when no information is available about the shape of the target distribution, SMC methods can be effectively employed to approximate sequences of probability distributions by means of random samples, which are evolved over time exploiting importance sampling and resampling techniques. The idea behind importance sampling is to modify the weights of the samples to account for the differences between the target distribution p(x) and the proposal distribution π(x) used to generate the samples. By setting each weight wi proportional to the ratio p(xi)/π(xi), the discrete weighted distribution PN i=1 wi ·δ(x−xi) better approximates the target distribution. In our context, the importance sampling step is performed by the actor, which modifies the weights of the actions according to their utility values estimated by the critic. When some samples have very small or very large normalized weights, it follows that the target density significantly differs from the proposal density used to draw the samples. From a learning perspective, this means that the set of available Algorithm 1 SMC-learning algorithm for all s ∈S do Initialize A(s) by drawing N samples from π0(a|s) Initialize W(s) with uniform values: wi = 1/N end for for each time step t do Action Selection Given the current state st, the actor selects action at from A(st) according to πt(a|s) = PN i=1 wi · δ(a −ai) Critic Update Given the reward rt and the utility of next state st+1, the critic updates the action value Q(st, at) Actor Update Given the action-value function, the actor updates the importance weights if the weights have a high variance then the set A(st) is resampled end if end for actions contains a number of samples whose estimated utility is very low. To avoid this, the actor has to modify the set of available actions by resampling new actions from the current weighted approximation of the target distribution. In SMC-learning, SMC methods are included into a learning algorithm that iterates through three main steps (see Algorithm 1): the action selection performed by the actor, the update of the actionvalue function managed by the critic, and finally the update of the policy of the actor. 3.1 Action Selection One of the main issues of learning in continuous action spaces is to determine which is the best action in the current state, given the (approximated) action-value function. Actor-critic methods effectively solve this problem by explicitly storing the current policy. As previously described, in SMC-learning the actor performs the action selection step by taking one action at random among those available in the current state. The probability of extraction of each action is equal to its normalized weight Pr(ai|s) = wi. The time complexity of the action selection phase for SMC-learning is logarithmic in the number of actions samples. 3.2 Critic Update While the actor determines the policy, the critic, on the basis of the collected rewards, computes an approximation of the action-value function. Although several function approximation schemes could be adopted for this task (e.g., neural networks, regression tress, support-vector machines), we use a simple solution: the critic stores an action value, Q(s, ai), for each action available in state s (like in tabular approaches) and modifies it according to TD update rules (see Equation 1). Using on-policy algorithms, such as SARSA, the time complexity of the critic update is constant (i.e., does not depend on the number of available actions). 3.3 Actor Update The core of SMC-learning is represented by the update of the policy distribution performed by the actor. Using the importance sampling principle, the actor modifies the weights wi, thus performing a policy improvement step based on the action values computed by the critic. In this way, actions with higher estimates get more weight. Several RL schemes could be adopted to update the weights. In this paper, we focus on the Boltzmann exploration strategy [13]. The Boltzmann exploration strategy privileges the execution of actions with higher estimated utility values. The probabilities computed by the Boltzmann exploration can be used as weights for the available actions. At time instant t, the weight of action ai in state s is updated as follows: wt+1 i = wt i e ∆Qt+1(s,ai) τ PN j=1 wje ∆Qt+1(s,aj ) τ , (2) where ∆Qt+1(s, ai) = Qt+1(s, ai) −Qt(s, ai), and the parameter τ (usually referred as to temperature) specifies the exploration degree: the higher τ, the higher the exploration. Once the weights have been modified, the agent’s policy has changed. Unfortunately, it is not possible to optimally solve continuous action MDPs by exploring only a finite set of actions sampled from a prior distribution, since the optimal action may not be available. Since the prior distribution used to initialize the set of available actions significantly differs from the optimal policy distribution, after a few iterations, several actions will have negligible weights: this problem is known as the weight degeneracy phenomenon [1]. Since the number of samples should be kept low for efficiency reasons, having actions associated with very small weights means to waste learning parameters for approximating both the policy and the value function in regions of the action space that are not relevant with respect to the optimal policy. Furthermore, long learning time is spent to execute and update utility values of actions that are not likely to be optimal. Therefore, following the SMC approach, after the importance sampling phase, a resampling step may be needed in order to improve the distribution of the samples on the action domain. The degeneracy phenomenon can be measured through the effective sample size [8], which, for each state s, can be estimated by b Neff(s) = 1 X wi∈W(s) w2 i , (3) where wi is the normalized weight. b Neff(s) is always less than the number of actions contained in A(s), and low values of b Neff(s) reveal high degeneracy. In order to avoid high degeneracy, the actions are resampled whenever the ratio between the effective sample size b Neff(s) and the number of samples N falls below some given threshold σ. The goal of resampling methods is to replace samples with small weights, with new samples close to samples with large weights, so that the discrepancy between the resampled weights is reduced. The new set of samples is generated by resampling (with replacement) N times from the following discrete distribution π(a|s) = N X i=1 wi · δ(a −ai), (4) so that samples with high weights are selected many times. Among the several resampling approaches that have been proposed, here we consider the systematic resampling scheme, since it can be easily implemented, takes O(N) time, and minimizes the Monte Carlo variance (refer to [1] for more details). The new samples inherit the same action values of their parents, and the sample weights are initialized using the Boltzmann distribution. Although the resampling step reduces the degeneracy, it introduces another problem known as sample impoverishment. Since samples with large weights are replicated several times, after a few resampling steps a significant number of samples could be identical. Furthermore, we need to learn over a continuous space, and this cannot be carried out using a discrete set of fixed samples; in fact, the learning agent would not be able to achieve the optimal policy whenever the initial set of available actions in state s (A(s)) does not contain the optimal action of that state. This limitation may be overcome by means of a smoothing step, that consists of moving the samples according to a continuous approximation π′(a|s, wi) of the posterior distribution . The approximation is obtained by using a weighted mean of kernel densities: π′(a|s, wi) = 1 h N X i=1 wiK a −ai h , (5) where h > 0 is the kernel bandwidth. Typical choices for the kernel densities are Gaussian kernels and Epanechnikov kernels. However, these kernels produce over-dispersed posterior distributions, and this negatively affects the convergence speed of the learning process, especially when a few samples are used. Here, we propose to use uniform kernels: Ki(a) = U (ai−1 −ai) 2 ; (ai+1 −ai) 2 . (6) As far as boundary samples are concerned (i.e., a1 and aN), their corresponding kernel is set to K1(a) = U [(a1 −a2); (a2 −a1)/2] and KN(a) = U [(aN−1 −aN)/2; (aN −aN−1)] respectively, thus preserving the possibility to cover the whole action domain. Using these (nonoverlapped) kernel densities, each sample is moved locally within an interval which is determined by its distances from the adjacent samples, thus achieving fast convergence. current viability zone quay 80 120 140 160 180 200 0 20 40 60 80 140 100 120 160 180 200 0 20 40 60 100 δ ω Figure 1: The boat problem. Parameter Value fc 1.25 I 0.1 sMAX 2.5 sD 1.75 p 0.9 quay (200, 110) Zs width 0.2 Zv width 20 Table 1: The dynamics parameters. Alg. Param. Value All α0/δα 0.5/0.01 All γ 0.99 SARSA τ0/δτ 3.0/0.0001 SMC σ 0.95 SMC τ0/δτ 25.0/0.0005 Cont.-QL ǫ/δǫ 0.4/0.005 Table 2: The learning parameters. Besides reducing the dispersion of the samples, this resampling scheme implements, from the critic perspective, a variable resolution generalization approach. Since the resampled actions inherit the action value associated to their parent, the learned values are generalized over a region whose width depends on the distance between samples. As a result, at the beginning of the learning process, when the actions are approximately uniformly distributed, SMC-learning performs broad generalization, thus boosting the performance. On the other hand, when the learning is near convergence the available actions tend to group around the optimal action, thus automatically reducing generalization which may prevent the learning of the optimal policy (see [12]). 4 Experiments In this section, we show experimental results with the aim of analyzing the properties of SMClearning and to compare its performance with other RL approaches. Additional experiments on a mini-golf task and on the swing-up pendulum problem are reported in Appendix. 4.1 The Boat Problem To illustrate how the SMC-learning algorithm works and to assess its effectiveness with respect to approaches based on discretization, we used a variant of the boat problem introduced in [5]. The problem is to learn a controller to drive a boat from the left bank to the right bank quay of a river, with a strong non-linear current (see Figure 1). The boat’s bow coordinates, x and y, are defined in the range [0, 200] and the controller sets the desired direction U over the range [−90◦, 90◦]. The dynamics of the boat’s bow coordinates is described by the following equations: xt+1 = min(200, max(0, xt + st+1 cos(δt+1))) yt+1 = min(200, max(0, yt −st−1 sin(δt+1) −E(xt+1))) where the effect of the current is defined by E(x) = fc x 50 − x 100 2 , where fc is the force of the current, and the boat angle δt and speed st are updated according to the desired direction Ut+1 as: δt+1 = δt + IΩt+1 Ωt+1 = Ωt + ((ωt+1 −Ωt)(st+1/sMAX)) st+1 = st + (sD −st)I ωt+1 = min(max(p(Ut+1 −δt), −45◦), 45◦) where I is the system inertia, sMAX is the maximum speed allowed for the boat, sD is the speed goal, ω is the rudder angle, and p is a proportional coefficient used to compute the rudder angle in order to reach the desired direction Ut. The reward function is defined on three bank zones. The success zone Zs corresponds to the quay, the viability zone Zv is defined around the quay, and the failure zone Zf in all the other bank points. Therefore, the reward function is defined as: R(x, y) = +10 (x, y) ∈Zs D(x,y) (x, y) ∈Zv -10 (x, y) ∈Zf 0 otherwise (7) 0 2 4 6 8 10 0 20 40 60 80 100 Total Reward Episodes (x1000) Sarsa vs SMC-learning SMC-learning (5 samples) SMC-learning (10 samples) Sarsa (5 actions) Sarsa (10 actions) Sarsa (20 actions) Sarsa (40 actions) 0 2 4 6 8 10 0 20 40 60 80 100 Total Reward Episodes (x1000) QL-Continuous, Tile coding, SMC-learning SMC-learning (10 samples) QL-Continuous (40 actions) Tile coding (80 actions) Figure 2: Performance comparison between SMC-learning and SARSA (left), SMC-learning and tile coding and Continuous Q-learning (right) where D is a function that gives a reward decreasing linearly from 10 to -10 relative to the distance from the success zone. In the experiment, each state variable is discretized in 10 intervals and the parameters of the dynamics are those listed in Table 1. At each trial, the boat is positioned at random along the left bank in one of the points shown in Figure 1. In the following, we compare the results obtained with four different algorithms: SARSA with Boltzmann exploration with different discretizations of the action space, SARSA with tile coding (or CMAC) [12], Continuous Q-learning [9], and SMC-learning. The learning parameters of each algorithm are listed in Table 2. 1 Figure 2-left compares the learning performance (in terms of total reward per episode) for SARSA with 5, 10, 20, and 40 evenly distributed actions to the results obtained by SMC-learning with 5 and 10 samples. As it can be noticed, the more the number of actions available the better the performance of SARSA is. With only 5 actions (one action each 36◦), the paths that the controller can follow are quite limited and the quay is not reachable from any of the starting point. As a result, the controller learned by SARSA achieves a very poor performance. On the other hand, a finer discretization allows the boat to reach more frequently the quay, even if it takes about three times the number of episodes to converge with respect to the case with 5 actions. As it can be noticed, SMC-learning with 5 samples outperforms SARSA with 5 and 10 actions both in terms of performance and in convergence time. In fact, after few trials, SMC-learning succeeds to remove the less-valued samples and to add new samples in regions of the action space where higher rewards can be obtained. As a result, not only it can achieve better performance than SARSA, but it does not spend time exploring useless actions, thus improving also the convergence time. Nonetheless, with only 5 samples the actor stores a very roughly approximated policy, which, as a consequence of resampling, may converge to actions that do not obtain a performance as good as that of SARSA with 20 and 40 actions. By increasing the number of samples from 5 to 10, SMC-learning succeeds in realizing a better coverage of the action space, and obtains equivalent performance as SARSA with 40 actions. At the same time, while the more actions available, the more SARSA takes to converge, the convergence time of SMC-learning, as in the case with 5 samples, benefits from the initial resampling, thus taking less than one sixth of the trials needed by SARSA to converge. Figure 2-right shows the comparison of the performance of SMC-learning, SARSA with tile coding using two tilings and a resolution of 2.25◦(equivalent to 80 actions), and Continuous Q-learning with 40 actions. We omit the results with fewer actions because both tile coding and Continuous Q-learning obtain poor performance. As it can be noticed, SMC-learning outperforms both the compared algorithms. In particular, the generalization over the action space performed by tile coding negatively affects the learning performance because of the non-linearity of the dynamics of the system. In fact, when only few actions are available, two adjacent actions may have completely different effects on the dynamics and, thus, receive different rewards. Generalizing over these actions prevents the agent from learning which is the best action among those available. On the other hand, as long as the samples get closer, SMC-learning dynamically reduces its generalization over the ac1δx is the decreasing rate for parameter x, whose value after N trials is computed as x(N) = x(0) 1+δxN . tion space, so that their utility can be more accurately estimated. Similarly, Continuous Q-learning is strictly related to the actions provided by the designer and to the implicit assumption of linearity of the action-value function. As a result, although it could learn any real-valued action, it does not succeed in obtaining the same performance as SMC-learning even with the quadruple of actions. In fact, the capability of SMC-learning to move samples towards more rewarding regions of the action space allows the agent to learn more effective policies even with a very limited number of samples. 5 Conclusions In this paper, we have described a novel actor-critic algorithm to solve continuous action problems. The algorithm is based on a Sequential Monte Carlo approach that allows the actor to represent the current policy through a finite set of available actions associated to weights, which are updated using the utility values computed by the critic. Experimental results show that SMC-learning is able to identify the highest valued actions through a process of importance sampling and resampling. This allows SMC-learning to obtain better performance with respect to static solutions such as Continuous Q-learning and tile coding even with a very limited number of samples, thus improving also the convergence time. Future research activity will follow two main directions: extending SMC-learning to problems in which no good discretization of the state space is a priori known, and experimenting in continuous action multi-agent problems. References [1] M. Sanjeev Arulampalam, Simon Maskell, Neil Gordon, and Tim Clapp. A tutorial on particle filters for online nonlinear/non-gaussian bayesian tracking. IEEE Trans. on Signal Processing, 50(2):174–188, 2002. [2] Leemon C. Baird and A. Harry Klopf. Reinforcement learning with high-dimensional, continuous actions. Technical Report WL-TR-93-117, Wright-Patterson Air Force Base Ohio: Wright Laboratory, 1993. [3] D.P. Bertsekas and J.N. Tsitsiklis. Neural Dynamic Programming. Athena Scientific, Belmont, MA, 1996. [4] Chris Gaskett, David Wettergreen, and Alexander Zelinsky. Q-learning in continuous state and action spaces. In Australian Joint Conference on Artificial Intelligence, pages 417–428, 2003. [5] L. Jouffe. Fuzzy inference system learning by reinforcement methods. IEEE Trans. on Systems, Man, and Cybernetics-PART C, 28(3):338–355, 1998. [6] H. Kimura and S. Kobayashi. Reinforcement learning for continuous action using stochastic gradient ascent. In 5th Intl. Conf. on Intelligent Autonomous Systems, pages 288–295, 1998. [7] V. R. Konda and J. N. Tsitsiklis. Actor-critic algorithms. SIAM Journal on Control and Optimization, 42(4):1143–1166, 2003. [8] J. S. Liu and E. Chen. Sequential monte carlo methods for dynamical systems. Journal of American Statistical Association, 93:1032–1044, 1998. [9] Jose Del R. Millan, Daniele Posenato, and Eric Dedieu. Continuous-action q-learning. Machine Learning, 49:247–265, 2002. [10] Jan Peters and Stefen Schaal. Policy gradient methods for robotics. In Proceedings of the IEEE International Conference on Intelligent Robotics Systems (IROS), pages 2219–2225, 2006. [11] J. C. Santamaria, R. S: Sutton, and A. Ram. Experiments with reinforcement learning in problems with continuous state and action spaces. Adaptive Behavior, 6:163–217, 1998. [12] Alexander A. Sherstov and Peter Stone. Function approximation via tile coding: Automating parameter choice. In SARA 2005, LNAI, pages 194–205. Springer Verlag, 2005. [13] Richard S. Sutton and Andrew G. Barto. Reinforcement Learning: An Introduction. MIT Press, Cambridge, MA, 1998. [14] Hado van Hasselt and Marco Wiering. Reinforcement learning in continuous action spaces. In 2007 IEEE Symposium on Approximate Dynamic Programming and Reinforcement Learning, pages 272–279, 2007.
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Receding Horizon Differential Dynamic Programming Yuval Tassa ∗ Tom Erez & Bill Smart † Abstract The control of high-dimensional, continuous, non-linear dynamical systems is a key problem in reinforcement learning and control. Local, trajectory-based methods, using techniques such as Differential Dynamic Programming (DDP), are not directly subject to the curse of dimensionality, but generate only local controllers. In this paper,we introduce Receding Horizon DDP (RH-DDP), an extension to the classic DDP algorithm, which allows us to construct stable and robust controllers based on a library of local-control trajectories. We demonstrate the effectiveness of our approach on a series of high-dimensional problems using a simulated multi-link swimming robot. These experiments show that our approach effectively circumvents dimensionality issues, and is capable of dealing with problems of (at least) 24 state and 9 action dimensions. 1 Introduction We are interested in learning controllers for high-dimensional, highly non-linear dynamical systems, continuous in state, action, and time. Local, trajectory-based methods, using techniques such as Differential Dynamic Programming (DDP), are an active field of research in the Reinforcement Learning and Control communities. Local methods do not model the value function or policy over the entire state space by focusing computational effort along likely trajectories. Featuring algorithmic complexity polynomial in the dimension, local methods are not directly affected by dimensionality issues as space-filling methods. In this paper, we introduce Receding Horizon DDP (RH-DDP), a set of modifications to the classic DDP algorithm, which allows us to construct stable and robust controllers based on local-control trajectories in highly non-linear, high-dimensional domains. Our new algorithm is reminiscent of Model Predictive Control, and enables us to form a time-independent value function approximation along a trajectory. We aggregate several such trajectories into a library of locally-optimal linear controllers which we then select from, using a nearest-neighbor rule. Although we present several algorithmic contributions, a main aspect of this paper is a conceptual one. Unlike much of recent related work (below), we are not interested in learning to follow a pre-supplied reference trajectory. We define a reward function which represents a global measure of performance relative to a high level objective, such as swimming towards a target. Rather than a reward based on distance from a given desired configuration, a notion which has its roots in the control community’s definition of the problem, this global reward dispenses with a “path planning” component and requires the controller to solve the entire problem. We demonstrate the utility of our approach by learning controllers for a high-dimensional simulation of a planar, multi-link swimming robot. The swimmer is a model of an actuated chain of links in a viscous medium, with two location and velocity coordinate pairs, and an angle and angular velocity for each link. The controller must determine the applied torque, one action dimension for ∗Y. Tassa is with the Hebrew University, Jerusalem, Israel. †T. Erez and W.D. Smart are with the Washington University in St. Louis, MO, USA. 1 each articulated joint. We reward controllers that cause the swimmer to swim to a target, brake on approach and come to a stop over it. We synthesize controllers for several swimmers, with state dimensions ranging from 10 to 24 dimensions. The controllers are shown to exhibit complex locomotive behaviour in response to real-time simulated interaction with a user-controlled target. 1.1 Related work Optimal control of continuous non-linear dynamical systems is a central research goal of the RL community. Even when important ingredients such as stochasticity and on-line learning are removed, the exponential dependence of computational complexity on the dimensionality of the domain remains a major computational obstacle. Methods designed to alleviate the curse of dimensionality include adaptive discretizations of the state space [1], and various domain-specific manipulations [2] which reduce the effective dimensionality. Local trajectory-based methods such as DDP were introduced to the NIPS community in [3], where a local-global hybrid method is employed. Although DDP is used there, it is considered an aid to the global approximator, and the local controllers are constant rather than locally-linear. In this decade DDP was reintroduced by several authors. In [4] the idea of using the second order local DDP models to make locally-linear controllers is introduced. In [5] DDP was applied to the challenging high-dimensional domain of autonomous helicopter control, using a reference trajectory. In [6] a minimax variant of DDP is used to learn a controller for bipedal walking, again by designing a reference trajectory and rewarding the walker for tracking it. In [7], trajectory-based methods including DDP are examined as possible models for biological nervous systems. Local methods have also been used for purely policy-based algorithms [8, 9, 10], without explicit representation of the value function. The best known work regarding the swimming domain is that by Ijspeert and colleagues (e.g. [11]) using Central Pattern Generators. While the inherently stable domain of swimming allows for such open-loop control schemes, articulated complex behaviours such as turning and tracking necessitate full feedback control which CPGs do not provide. 2 Methods 2.1 Definition of the problem We consider the discrete-time dynamics xk+1 = F(xk, uk) with states x ∈Rn and actions u ∈Rm. In this context we assume F(xk, uk) = xk + R ∆t 0 f(x(t), uk)dt for a continuous f and a small ∆t, approximating the continuous problem and identifying with it in the ∆t →0 limit. Given some scalar reward function r(x, u) and a fixed initial state x1 (superscripts indicating the time index), we wish to find the policy which maximizes the total reward1 acquired over a finite temporal horizon: π∗(xk, k) = argmax π(·,·) [ N X i=k r(xi, π(xi, i))]. The quantity maximized on the RHS is the value function, which solves Bellman’s equation: V (x, k) = max u [r(x, u) + V (F(x, u), k+1)]. (1) Each of the functions in the sequence {V (x, k)}N k=1 describes the optimal reward-to-go of the optimization subproblem from k to N. This is a manifestation of the dynamic programming principle. If N = ∞, essentially eliminating the distinction between different time-steps, the sequence collapses to a global, time-independent value function V (x). 2.2 DDP Differential Dynamic Programming [12, 13] is an iterative improvement scheme which finds a locally-optimal trajectory emanating from a fixed starting point x1. At every iteration, an approx1We (arbitrarily) choose to use phrasing in terms of reward-maximization, rather than cost-minimization. 2 imation to the time-dependent value function is constructed along the current trajectory {xk}N k=1, which is formed by iterative application of F using the current control sequence {uk}N k=1. Every iteration is comprised of two sweeps of the trajectory: a backward and a forward sweep. In the backward sweep, we proceed backwards in time to generate local models of V in the following manner. Given quadratic models of V (xk+1, k + 1), F(xk, uk) and r(xk, uk), we can approximate the unmaximised value function, or Q-function, Q(xk, uk) = r(xk, uk) + V k+1(F(xk, uk)) (2) as a quadratic model around the present state-action pair (xk, uk): Q(xk + δx, uk + δu) ≈Q0 + Qxδx + Quδu + 1 2[δxT δuT ] " Qxx Qxu Qux Quu #h δx δu i (3) Where the coefficients Q⋆⋆are computed by equating coefficients of similar powers in the secondorder expansion of (2) Qx = rx + V k+1 x F k x Qxx = rxx + F k x V k+1 xx F k x + V k+1 x F k xx Qu = ru + V k+1 x F k u Quu = ruu + F k u V k+1 xx F k u + V k+1 x F k uu Qxu = rxu + F k x V k+1 xx F k u + V k+1 x F k xu. (4) Once the local model of Q is obtained, the maximizing δu is solved for δu∗= argmax δu [Q(xk + δx, uk + δu)] = −Q−1 uu(Qu + Quxδx) (5) and plugged back into (3) to obtain a quadratic approximation of V k: V k 0 = V k+1 0 −Qu(Quu)−1 Qu (6a) V k x = Qk+1 x −Qu(Quu)−1 Qux (6b) V k xx = Qk+1 xx −Qxu(Quu)−1Qux. (6c) This quadratic model can now serve to propagate the approximation to V k−1. Thus, equations (4), (5) and (6) iterate in the backward sweep, computing a local model of the Value function along with a modification to the policy in the form of an open-loop term −Q−1 uuQu and a feedback term −Q−1 uuQuxδx, essentially solving a local linear-quadratic problem in each step. In some senses, DDP can be viewed as dual to the Extended Kalman Filter (though employing a higher order expansion of F). In the forward sweep of the DDP iteration, both the open-loop and feedback terms are combined to create a new control sequence (ˆuk)N k=1 which results in a new nominal trajectory (ˆxk)N k=1. ˆx1 = x1 (7a) ˆuk = uk −Q−1 uuQu −Q−1 uuQux(ˆxk −xk) (7b) ˆxk+1 = F(ˆxk, ˆuk) (7c) We note that in practice the inversion in (5) must be conditioned. We use a Levenberg Marquardtlike scheme similar to the ones proposed in [14]. Similarly, the u-update in (7b) is performed with an adaptive line search scheme similar to the ones described in [15]. 2.2.1 Complexity and convergence The leading complexity term of one iteration of DDP itself, assuming the model of F as required for (4) is given, is O(Nmγ1) for computing (6) N times, with 2 < γ1 < 3, the complexity-exponent of inverting Quu. In practice, the greater part of the computational effort is devoted to the measurement of the dynamical quantities in (4) or in the propagation of collocation vectors as described below. DDP is a second order algorithm with convergence properties similar to, or better than Newton’s method performed on the full vectorial uk with an exact Nm × Nm Hessian [16]. In practice, convergence can be expected after 10-100 iterations, with the stopping criterion easily determined as the size of the policy update plummets near the minimum. 3 2.2.2 Collocation Vectors We use a new method of obtaining the quadratic model of Q (Eq. (2)), inspired by [17]2. Instead of using (4), we fit this quadratic model to samples of the value function at a cloud of collocation vectors {xk i , uk i }i=1..p, spanning the neighborhood of every state-action pair along the trajectory. We can directly measure r(xk i , uk i ) and F(xk i , uk i ) for each point in the cloud, and by using the approximated value function at the next time step, we can estimate the value of (2) at every point: q(xk i , uk i ) = r(xk i , uk i ) + V k+1(F(xk i , uk i )) Then, we can insert the values of q(xk i , uk i ) and (xk i , uk i ) on the LHS and RHS of (3) respectively, and solve this set of p linear equations for the Q⋆⋆terms. If p > (3(n + m) + (m + n)2)/2, and the cloud is in general configuration, the equations are non-singular and can be easily solved by a generic linear algebra package. There are several advantages to using such a scheme. The full nonlinear model of F is used to construct Q, rather than only a second-order approximation. Fxx, which is an n×n×n tensor need not be stored. The addition of more vectors can allow the modeling of noise, as suggested in [17]. In addition, this method allows us to more easily apply general coordinate transformations in order to represent V in some internal space, perhaps of lower dimension. The main drawback of this scheme is the additional complexity of an O(Npγ2) term for solving the p-equation linear system. Because we can choose {xk i , uk i } in way which makes the linear system sparse, we can enjoy the γ2 < γ1 of sparse methods and, at least for the experiments performed here, increase the running time only by a small factor. In the same manner that DDP is dually reminiscent of the Extended Kalman Filter, this method bears a resemblance to the test vectors propagated in the Unscented Kalman Filter [18], although we use a quadratic, rather than linear number of collocation vectors. 2.3 Receding Horizon DDP When seeking to synthesize a global controller from many local controllers, it is essential that the different local components operate synergistically. In our context this means that local models of the value function must all model the same function, which is not the case for the standard DDP solution. The local quadratic models which DDP computes around the trajectory are approximations to V (x, k), the time-dependent value function. The standard method in RL for creating a global value function is to use an exponentially discounted horizon. Here we propose a fixed-length nondiscounted Receding Horizon scheme in the spirit of Model Predictive Control [19]. Having computed a DDP solution to some problem starting from many different starting points x1, we can discard all the models computed for points xk>1 and save only the ones around the x1’s. Although in this way we could accumulate a time-independent approximation to V (x, N) only, starting each run of N-step DDP from scratch would be prohibitively expensive. We therefore propose the following: After obtaining the solution starting from x1, we save the local model at k = 1 and proceed to solve a new N-step problem starting at x2, this time initialized with the policy obtained on the previous run, shifted by one time-step, and appended with the last control unew = [u2, u3...uNuN]. Because this control sequence is very close to the optimal solution, the second-order convergence of DDP is in full effect and the algorithm converges in 1 or 2 sweeps. Again saving the model at the first time step, we iterate. We stress the that without the fast and exact convergence properties of DDP near the maximum, this algorithm would be far less effective. 2.4 Nearest Neighbor control with Trajectory Library A run of DDP computes a locally quadratic model of V and a locally linear model of u, expressed by the gain term −Q−1 uuQux. This term generalizes the open-loop policy to a tube around the trajectory, inside of which a basin-of-attraction is formed. Having lost the dependency on the time k with the receding-horizon scheme, we need some space-based method of determining which local gain model we select at a given state. The simplest choice, which we use here, is to select the nearest Euclidian neighbor. 2Our method is a specific instantiation of a more general algorithm described therein. 4 Outside of the basin-of-attraction of a single trajectory, we can expect the policy to perform very poorly and lead to numerical divergence if no constraint on the size of u is enforced. A possible solution to this problem is to fill some volume of the state space with a library of local-control trajectories [20], and consider all of them when selecting the nearest linear gain model. 3 Experiments 3.1 The swimmer dynamical system We describe a variation of the d-link swimmer dynamical system [21]. A stick or link of length l, lying in a plane at an angle θ to some direction, parallel to ˆt = ¡ cos(θ) sin(θ) ¢ and perpendicular to ˆn = ¡ −sin(θ) cos(θ) ¢ , moving with velocity ˙x in a viscous fluid, is postulated to admit a normal frictional force −knlˆn( ˙x · ˆn) and a tangential frictional force −ktlˆt( ˙x · ˆt), with kn > kt > 0. The swimmer is modeled as a chain of d such links of lengths li and masses mi, its configuration described by the generalized coordinates q = ( xcm θ ), of two center-of-mass coordinates and d angles. Letting ¯xi = xi −xcm be the positions of the link centers WRT the center of mass , the Lagrangian is L = 1 2 ˙x2 cm X i mi + 1 2 X i mi ˙¯x 2 i + 1 2 X i Ii ˙θ2 i with Ii = 1 12mil2 i the moments-of-inertia. The relationship between the relative position vectors and angles of the links is given by the d −1 equations ¯xi+1 −¯xi = 1 2li+1ˆti+1 + 1 2liˆti, which express the joining of successive links, and by the equation P i mi¯xi = 0 which comes from the (a) Time course of two angular velocities. (b) State projection. Figure 1: RH-DDP trajectories. (a) three snapshots of the receding horizon trajectory (dotted) with the current finite-horizon optimal trajectory (solid) appended, for two state dimensions. (b) Projections of the same receding-horizon trajectories onto the largest three eigenvectors of the full state covariance matrix. As described in Section 3.3, the linear regime of the reward, here applied to a 3-swimmer, compels the RH trajectories to a steady swimming gait – a limit cycle. 5 definition of the ¯xi’s relative to the center-of-mass. The function F = −1 2kn X i [li( ˙xi · ˆni)2 + 1 12l3 i ˙θ2 i ] −1 2kt X i li( ˙xi · ˆti)2 known as the dissipation function, is that function whose derivatives WRT the ˙qi’s provide the postulated frictional forces. With these in place, we can obtain ¨q from the 2+d Euler-Lagrange equations: d dt( ∂ ∂qi L) = ∂ ∂˙qi F + u with u being the external forces and torques applied to the system. By applying d −1 torques τj in action-reaction pairs at the joints ui = τi −τi−1, the isolated nature of the dynamical system is preserved. Performing the differentiations, solving for ¨q, and letting x = ¡ q ˙q ¢ be the 4+2ddimensional state variable, finally gives the dynamics ˙x = ( ˙q ¨q ) = f(x, u). 3.2 Internal coordinates The two coordinates specifying the position of the center-of-mass and the d angles are defined relative to an external coordinate system, which the controller should not have access to. We make a coordinate transformation into internal coordinates, where only the d−1 relative angles {ˆθj = θj+1 −θj}d−1 j=1 are given, and the location of the target is given relative to coordinate system fixed on one of the links. This makes the learning isotropic and independent of a specific location on the plane. The collocation method allows us to perform this transformation directly on the vector cloud without having to explicitly differentiate it, as we would have had to using classical DDP. Note also that this transformation reduces the dimension of the state (one angle less), suggesting the possibility of further dimensionality reduction. 3.3 The reward function The reward function we used was r(x, u) = −cx ||xnose||2 p ||xnose||2 + 1 −cu||u||2 (8) Where xnose = [x1x2]T is the 2-vector from some designated point on the swimmer’s body to the target (the origin in internal space), and cx and cu are positive constants. This reward is maximized when the nose is brought to rest on the target under a quadratic action-cost penalty. It should not be confused with the desired state reward of classical optimal control since values are specified only for 2 out of the 2d + 4 coordinates. The functional form of the target-reward term is designed to be linear in ||xnose|| when far from the target and quadratic when close to it (Figure 2(b)). Because (a) Swimmer (b) Reward Figure 2: (a) A 5-swimmer with the “nose” point at its tip and a ring-shaped target. (b) The functional form of the planar reward component r(xnose) = −||xnose||2/ p ||xnose||2 + 1. This form translates into a steady swimming gait at large distances with a smooth braking and stopping at the goal. 6 of the differentiation in Eq. (5), the solution is independent of V0, the constant part of the value. Therefore, in the linear regime of the reward function, the solution is independent of the distance from the target, and all the trajectories are quickly compelled to converge to a one-dimensional manifold in state-space which describes steady-state swimming (Figure 1(b)). Upon nearing the target, the swimmer must initiate a braking maneuver, and bring the nose to a standstill over the target. For targets that are near the swimmer, the behaviour must also include various turns and jerks, quite different from steady-state swimming, which maneuver the nose into contact with the target. Our experience during interaction with the controller, as detailed below, leads us to believe that the behavioral variety that would be exhibited by a hypothetical exact optimal controller for this system to be extremely large. 4 Results In order to asses the controllers we constructed a real-time interaction package3. By dragging the target with a cursor, a user can interact with controlled swimmers of 3 to 10 links with a state dimension varying from 10 to 24, respectively. Even with controllers composed of a single trajectory, the swimmers perform quite well, turning, tracking and braking on approach to the target. All of the controllers in the package control swimmers with unit link lengths and unit masses. The normal-to-tangential drag coefficient ratio was kn/kt = 25. The function F computes a single 4thorder Runge-Kutta integration step of the continuous dynamics F(xk, uk) =xk+ R t+∆t t f(xk, uk)dt with ∆t=0.05s. The receding horizon window was of 40 time-steps, or 2 seconds. When the state doesn’t gravitate to one of the basins of attraction around the trajectories, numerical divergence can occur. This effect can be initiated by the user by quickly moving the target to a “surprising” location. Because nonlinear viscosity effects are not modeled and the local controllers are also linear, exponentially diverging torques and angular velocities can be produced. When adding as few as 20 additional trajectories, divergence is almost completely avoided. Another claim which may be made is that there is no guarantee that the solutions obtained, even on the trajectories, are in fact optimal. Because DDP is a local optimization method, it is bound to stop in a local minimum. An extension of this claim is that even if the solutions are optimal, this has to do with the swimmer domain itself, which might be inherently convex in some sense and therefore an “easy” problem. While both divergence and local minima are serious issues, they can both be addressed by appealing to our panoramic motivation in the biology. Real organisms cannot apply unbounded torque. By hard-limiting the torque to large but finite values, non-divergence can be guaranteed4. Similarly, local minima exist even in the motor behaviour of the most complex organisms, famously evidenced by Fosbury’s reinvention of the high jump. Regarding the easiness or difficulty of the swimmer problem – we made the documented code available and hope that it might serve as a useful benchmark for other algorithms. 5 Conclusions The significance of this work lies at its outlining of a new kind of tradeoff in nonlinear motor control design. If biological realism is an accepted design goal, and physical and biological constraints taken into account, then the expectations we have from our controllers can be more relaxed than those of the control engineer. The unavoidable eventual failure of any specific biological organism makes the design of truly robust controllers a futile endeavor, in effect putting more weight on the mode, rather than the tail of the behavioral distribution. In return for this forfeiture of global guarantees, we gain very high performance in a small but very dense sub-manifold of the state-space. 3Available at http://alice.nc.huji.ac.il/∼tassa/ 4We actually constrain angular velocities since limiting torque would require a stiffer integrator, but theoretical non-divergence is fully guaranteed by the viscous dissipation which enforces a Lyapunov function on the entire system, once torques are limited. 7 Since we make use of biologically grounded arguments, we briefly outline the possible implications of this work to biological nervous systems. It is commonly acknowledged, due both to theoretical arguments and empirical findings, that some form of dimensionality reduction must be at work in neural control mechanisms. A common object in models which attempt to describe this reduction is the motor primitive, a hypothesized atomic motor program which is combined with other such programs in a small “alphabet”, to produce complex behaviors in a given context. Our controllers imply a different reduction: a set of complex prototypical motor programs, each of which is nearoptimal only in a small volume of the state-space, yet in that space describes the entire complexity of the solution. Giving the simplest building blocks of the model such a high degree of task specificity or context, would imply a very large number of these motor prototypes in a real nervous system, an order of magnitude analogous, in our linguistic metaphor, to that of words and concepts. References [1] Remi Munos and Andrew W. Moore. Variable Resolution Discretization for High-Accuracy Solutions of Optimal Control Problems. In International Joint Conference on Artificial Intelligence, pages 1348–1355, 1999. [2] M. Stilman, C. G. Atkeson, J. J. Kuffner, and G. Zeglin. Dynamic programming in reduced dimensional spaces: Dynamic planning for robust biped locomotion. In Proceedings of the 2005 IEEE International Conference on Robotics and Automation (ICRA 2005), pages 2399–2404, 2005. [3] Christopher G. Atkeson. Using local trajectory optimizers to speed up global optimization in dynamic programming. In NIPS, pages 663–670, 1993. [4] C. G. Atkeson and J. Morimoto. Non-parametric representation of a policies and value functions: A trajectory based approach. In Advances in Neural Information Processing Systems 15, 2003. [5] P. Abbeel, A. Coates, M. Quigley, and A. Y. Ng. An application of reinforcement learning to aerobatic helicopter flight. In Advances in Neural Information Processing Systems 19, 2007. [6] J. Morimoto and C. G. Atkeson. Minimax differential dynamic programming: An application to robust bipedwalking. In Advances in Neural Information Processing Systems 14, 2002. [7] Emanuel Todorov and Wei-Wei Li. Optimal control methods suitable for biomechanical systems. In 25th Annual Int. Conf. IEE Engineering in Medicine and Biology Society, 2003. [8] R. Munos. Policy gradient in continuous time. Journal of Machine Learning Research, 7:771–791, 2006. [9] J. Peters and S. Schaal. Reinforcement learning for parameterized motor primitives. In Proceedings of the IEEE International Joint Conference on Neural Networks (IJCNN 2006), 2006. [10] Tom Erez and William D. Smart. Bipedal walking on rough terrain using manifold control. In IEEE/RSJ International Conference on Robots and Systems (IROS), 2007. [11] A. Crespi and A. Ijspeert. AmphiBot II: An amphibious snake robot that crawls and swims using a central pattern generator. In Proceedings of the 9th International Conference on Climbing and Walking Robots (CLAWAR 2006), pages 19–27, 2006. [12] D. Q. Mayne. A second order gradient method for determining optimal trajectories for non-linear discretetime systems. International Journal of Control, 3:85–95, 1966. [13] D. H. Jacobson and D. Q. Mayne. Differential Dynamic Programming. Elsevier, 1970. [14] L.-Z. Liao and C. A. Shoemaker. Convergence in unconstrained discrete-time differential dynamic programming. IEEE Transactions on Automatic Control, 36(6):692–706, 1991. [15] S. Yakowitz. Algorithms and computational techniques in differential dynamic programming. Control and Dynamic Systems: Advances in Theory and Applications, 31:75–91, 1989. [16] L.-Z. Liao and C. A. Shoemaker. Advantages of differential dynamic programming over newton’s method for discrete-time optimal control problems. Technical Report 92-097, Cornell Theory Center, 1992. [17] E. Todorov. Iterative local dynamic programming. Manuscript under review, available at www.cogsci.ucsd.edu/∼todorov/papers/ildp.pdf, 2007. [18] S. J. Julier and J. K. Uhlmann. A new extension of the kalman filter to nonlinear systems. In Proceedings of AeroSense: The 11th Int. 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A Risk Minimization Principle for a Class of Parzen Estimators Kristiaan Pelckmans, Johan A.K. Suykens, Bart De Moor Department of Electrical Engineering (ESAT) - SCD/SISTA K.U.Leuven University Kasteelpark Arenberg 10, Leuven, Belgium Kristiaan.Pelckmans@esat.kuleuven.be Abstract This paper1 explores the use of a Maximal Average Margin (MAM) optimality principle for the design of learning algorithms. It is shown that the application of this risk minimization principle results in a class of (computationally) simple learning machines similar to the classical Parzen window classifier. A direct relation with the Rademacher complexities is established, as such facilitating analysis and providing a notion of certainty of prediction. This analysis is related to Support Vector Machines by means of a margin transformation. The power of the MAM principle is illustrated further by application to ordinal regression tasks, resulting in an O(n) algorithm able to process large datasets in reasonable time. 1 Introduction The quest for efficient machine learning techniques which (a) have favorable generalization capacities, (b) are flexible for adaptation to a specific task, and (c) are cheap to implement is a pervasive theme in literature, see e.g. [14] and references therein. This paper introduces a novel concept for designing a learning algorithm, namely the Maximal Average Margin (MAM) principle. It closely resembles the classical notion of maximal margin as lying on the basis of perceptrons, Support Vector Machines (SVMs) and boosting algorithms, see a.o. [14, 11]. It however optimizes the average margin of points to the (hypothesis) hyperplane, instead of the worst case margin as traditional. The full margin distribution was studied earlier in e.g. [13], and theoretical results were extended and incorporated in a learning algorithm in [5]. The contribution of this paper is twofold. On a methodological level, we relate (i) results in structural risk minimization, (ii) data-dependent (but dimension-independent) Rademacher complexities [8, 1, 14] and a new concept of ’certainty of prediction’, (iii) the notion of margin (as central is most state-of-the-art learning machines), and (iv) statistical estimators as Parzen windows and NadarayaWatson kernel estimators. In [10], the principle was already shown to underlie the approach of mincuts for transductive inference over a weighted undirected graph. Further, consider the modelclass consisting of all models with bounded average margin (or classes with a fixed Rademacher complexity as we will indicate lateron). The set of such classes is clearly nested, enabling structural risk minimization [8]. On a practical level, we show how the optimality principle can be used for designing a computationally fast approach to (large-scale) classification and ordinal regression tasks, much along the same 1Acknowledgements - K. Pelckmans is supported by an FWO PDM. J.A.K. Suykens and B. De Moor are a (full) professor at the Katholieke Universiteit Leuven, Belgium. Research supported by Research Council KUL: GOA AMBioRICS, CoE EF/05/006 OPTEC, IOF-SCORES4CHEM, several PhD/postdoc & fellow grants; Flemish Government: FWO: PhD/postdoc grants, projects G.0452.04, G.0499.04, G.0211.05, G.0226.06, G.0321.06, G.0302.07, (ICCoS, ANMMM, MLDM); IWT: PhD Grants, McKnow-E, Eureka-Flite+ Belgian Federal Science Policy Office: IUAP P6/04, EU: ERNSI; 1 lines as Parzen classifiers and Nadaraya-Watson estimators. It becomes clear that this result enables researchers on Parzen windows to benefit directly from recent advances in kernel machines, two fields which have evolved mostly separately. It must be emphasized that the resulting learning rules were already studied in different forms and motivated by asymptotic and geometric arguments, as e.g. the Parzen window classifier [4], the ’simple classifier’ as in [12] chap. 1, probabilistic neural networks [15], while in this paper we show how an (empirical) risk based optimality criterion underlies this approach. A number of experiments confirm the use of the resulting cheap learning rules for providing a reasonable (baseline) performance in a small time-window. The following notational conventions are used throughout the paper. Let the random vector (X, Y ) ∈Rd × {−1, 1} obey a (fixed but unknown) joint distribution PXY from a probability space (Rd ×{−1, 1}, P). Let Dn = {(Xi, Yi)}n i=1 be sampled i.i.d. according to PXY . Let y ∈Rn be defined as y = (Y1, . . . , Yn)T ∈{−1, 1}n and X = (X1, . . . , Xn)T ∈Rn×d. This paper is organized as follows. The next section illustrates the principle of maximal average margin for classification problems. Section 3 investigates the close relationship with Rademacher complexities, Section 4 develops the maximal average margin principle for ordinal regression, and Section 5 reports experimental results of application of the MAM to classification and ordinal regression tasks. 2 Maximal Average Margin for Classifiers 2.1 The Linear Case Let the class of hypotheses be defined as H = n f(·) : Rd →R, w ∈Rd ∀x ∈Rd : f(x) = wT x, ∥w∥2 = 1 o . (1) Consequently, the signed distance of a sample (X, Y ) to the hyper-plane wT x = 0, or the margin M(w) ∈R, can be defined as M(w) = Y (wT X) ∥w∥2 . (2) SVMs maximize the worst-case margin. We instead focus on the first moment of the margin distribution. Maximizing the expected (average) margin follows from solving M ∗= max w E Y (wT X) ∥w∥2 = max f∈H E [Y f(X)] . (3) Remark that the non-separable case does not require the need for slack-variables. The empirical counterpart becomes ˆ M = max w 1 n n X i=1 Yi(wT Xi) ∥w∥2 , (4) which can be written as a constrained convex problem as minw −1 n Pn i=1 Yi(wT Xi) s.t. ∥w∥2 ≤ 1. The Lagrangian with multiplier λ ≥0 becomes L(w, λ) = −1 n Pn i=1 Yi(wT Xi) + λ 2 (wT w −1). By switching the minimax problem to a maximin problem (application of Slater’s condition), the first order condition for optimality ∂L(w,λ) ∂w = 0 gives wn = 1 λn n X i=1 YiXi = 1 λnXT y, (5) where wn ∈Rd denotes the optimum to (4). The corresponding parameter λ can be found by substituting (5) in the constraint wT w = 1, or λ = 1 n ∥Pn i=1 YiXi∥2 = 1 n p yT XXT y since the optimum is obviously taking place when wT w = 1. It becomes clear that the above derivations remain valid as n →∞, resulting in the following theorem. Theorem 1 (Explicit Actual Optimum for the MAMC) The function f(x) = wT x in H maximizing the expected margin satisfies arg max w E Y (wT X) ∥w∥2 = 1 λE[XY ] ≜w∗, (6) where λ is a normalization constant such that ∥w∗∥2 = 1. 2 2.2 Kernel-based Classifier and Parzen Window It becomes straightforward to recast the resulting classifier as a kernel classifier by mapping the input data-samples X in a feature space ϕ : Rd →Rdϕ where dϕ is possibly infinite. In particular, we do not have to resort to Lagrange duality in a context of convex optimization (see e.g. [14, 9] for an overview) or functional analysis in a Reproducing Kernel Hilbert Space. Specifically, wT n ϕ(X) = 1 λn n X i=1 YiK(Xi, X), (7) where K : Rd × Rd →R is defined as the inner product such that ϕ(X)T ϕ(X′) = K(X, X′) for any X, X′. Conversely, any function K corresponds with the inner product of a valid map ϕ if the function K is positive definite. As previously, the term λ becomes λ = 1 n p yT Ωy with kernel matrix Ω∈Rn×n where Ωij = K(Xi, Xj) for all i, j = 1, . . . , n. Now the class of positive definite Mercer kernels can be used as they induce a proper mapping ϕ. A classical choice is the use of a linear kernel (or K(X, X′) = XT X′), a polynomial kernel of degree p ∈N0 (or K(X, X′) = (XT X′ +b)p), an RBF kernel (or K(X, X′) = exp(−∥X −X′∥2 2/σ)), or a dedicated kernel for a specific application (e.g. a string kernel, a Fisher kernel, see e.g. [14] and references therein). Figure 1.a depicts an example of a nonlinear classifier based on the well-known Ripley dataset, and the contourlines score the ’certainty of prediction’ as explained in the next section. The expression (7) is similar (proportional) to the classical Parzen window for classification, but differs in the use of a positive definite (Mercer) kernel K instead of the pdf κ( X−· h ) with bandwidth h > 0, and in the form of the denominator. The classical motivation of statistical kernel estimators is based on asymptotic theory in low dimensions (i.e d = O(1)), see e.g. [4], chap. 10 and references. The functional form of the optimal rule (7) is similar to the ’simple classifier’ described in [12], chap. 1. Thirdly, this estimator was also termed and empirically validated as a probabilistic neural network by [15]. The novel element from above result is the derivation of a clear (both theoretical and empirical) optimality principle of the rule, as opposed to the asymptotic results of [4] and the geometric motivations in [12, 15]. As a direct byproduct, it becomes straightforward to extend the Parzen window classifier easily with an additional intercept term or other parametric parts, or towards additive (structured) models as in [9]. 3 Analysis and Rademacher Complexities The quantity of interest in the analysis of the generalization performance is the probability of predicting a mistake (the risk R(w; PXY )), or R(w; PXY ) = PXY Y (wT ϕ(X)) ≤0 = E I(Y (wT ϕ(X)) ≤0) , (8) where I(z) equals one if z is true, and zero otherwise. 3.1 Rademacher Complexity Let {σi}n i=1 taken from the set {−1, 1}n be Bernoulli random variables with P(σ = 1) = P(σ = −1) = 1 2. The empirical Rademacher complexity is then defined [8, 1] as ˆRn(H) ≜Eσ " sup f∈H 2 n n X i=1 σif(Xi) X1, . . . , Xn # , (9) where the expectation is taken over the choice of the binary vector σ = (σ1, . . . , σn)T ∈{−1, 1}n. It is observed that the empirical Rademacher complexity defines a natural complexity measure to study the maximal average margin classifier, as both the definitions of the empirical Rademacher complexity and the maximal average margin resemble closely (see also [8]). The following result was given in [1], Lemma 22, but we give an alternative proof by exploiting the structure of the optimal estimate explicitly. Lemma 1 (Trace bound for the Empirical Rademacher Complexity for H) Let Ω∈Rn×n be defined as Ωij = K(Xi, Xj) for all i, j = 1, . . . , n, then ˆRn(H) ≤2 n p tr(Ω). (10) 3 Proof: The proof goes along the same lines as the classical bound on the empirical Rademacher complexity for kernel machines outlined in [1], Lemma 22. Specifically, once a vector σ ∈{−1, 1}n is fixed, it is immediately seen that the maxf∈H 1 n Pn i=1 σif(Xi) equals the solution as in (7) or maxw Pn i=1 σi(wT ϕ(Xi)) = σT Ωσ √ σT Ωσ = √ σT Ωσ. Now, application of the expectation operator E over the choice of the Rademacher variables gives ˆRn(H) = E 2 n √ σT Ωσ ≤2 n E σT Ωσ 1 2 = 2 n X i,j E [σiσj] K(Xi, Xj) 1 2 = 2 n n X i=1 K(Xi, Xi) ! 1 2 = 2 n p tr(Ω), (11) where the inequality is based on application of Jensen’s inequality. This proves the Lemma. □ Remark that in the case of a kernel with constant trace (as e.g. in the case of the RBF kernel where p tr(Ω) = √n), it follows from this result that also the (expected) Rademacher complexity E[ ˆRn(H)] ≤ p tr(Ω). In general, one has that E[K(X, X)] equals the trace of the integral operator TK defined on L2(PX) defined as TK(f) = R K(X, Y )f(X)dPX(X) as in [1]. Application of McDiarmid’s inequality on the variable Z = supf∈H E[Y (wT ϕ(X))] −1 n Pn i=1 Yi(wT ϕ(Xi)) gives as in [8, 1]. Lemma 2 (Deviation Inequality) Let 0 < Bϕ < ∞be a fixed constant such that supz ∥ϕ(z)∥2 = supz p K(z, z) ≤Bϕ such that |wT ϕ(z)| ≤Bφ, and let δ ∈R+ 0 be fixed. Then with probability exceeding 1 −δ, one has for any w ∈Rd that E[Y (wT ϕ(X))] ≥1 n n X i=1 Yi(wT ϕ(Xi)) −ˆRn(H) −3Bϕ s 2 ln 2 δ n . (12) Therefore it follows that one maximizes the expected margin by maximizing the empirical average margin, while controlling the empirical Rademacher complexity by choice of the model class (kernel). In the case of RBF kernels, Bϕ = 1, resulting in a reasonable tight bound. It is now illustrated how one can obtain a practical upper-bound to the ’certainty of prediction’ using f(x) = wT n x. Theorem 2 (Occurrence of Mistakes) Given an i.i.d. sample Dn = {(Xi, Yi)}n i=1, a constant B ∈R such that supz p K(z, z) ≤Bϕ, and a fixed δ ∈R+ 0 . Then, with probability exceeding 1 −δ, one has for all w ∈Rd that P Y (wT ϕ(X)) ≤0 ≤Bϕ −E[Y (wT ϕ(X))] Bϕ ≤1 − p yT Ωy nBϕ + ˆRn(H) Bϕ + 3 s 2 ln 2 δ n . (13) Proof: The proof follows directly from application of Markov’s inequality on the positive random variable Bϕ −Y (wT ϕ(X)), with expectation Bϕ −E[Y (wT ϕ(X))], estimated accurately by the sample average as in the previous theorem. □ More generally, one obtains that with probability exceeding 1 −δ that for any w ∈Rd and for any ρ such that −Bϕ < ρ < Bϕ that P Y (wT ϕ(X)) ≤−ρ ≤ Bϕ Bϕ + ρ − p yT Ωy n(Bϕ + ρ) + ˆRn(H) Bϕ + ρ + 3Bϕ Bϕ + ρ s 2 ln 2 δ n , (14) with probability exceeding 1 −δ < 1. This results in a practical assessment of the ’certainty’ of a prediction as follows. At first, note that the random variable Y (wT n ϕ(x)) for a fixed X = x can take two values: either −|wT n ϕ(x)| or |wT n ϕ(x)|. Therefore P(Y (wT n ϕ(x)) ≤0) = P(Y (wT n ϕ(x)) = 4 X1 X2 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 −0.2 0 0.2 0.4 0.6 0.8 1 Class prediction class 1 class 2 (a) X1 X2 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 −0.2 0 0.2 0.4 0.6 0.8 1 (b) Figure 1: Example of (a) the MAM classifier and (b) the SVM on the Ripley dataset. The contourlines represent the estimate of certainty of prediction (’scores’) as derived in Theorem 2 for the MAM classifier for (a), and as in Corollary 1 for the case of SVMs with g(z) = min(1, max(−1, z)) where |z| < 1 corresponds with the inner part of the margin of the SVM (b). While the contours in (a) give an overall score of the predictions, the scores given in (b) focus towards the margin of the SVM. −|wT n ϕ(x)|) ≤P(Y (wT n ϕ(x)) ≤−|wT n ϕ(x)|) as Y can only take the two values −1 or 1. Thus the event ’Y ̸= sign(wT x∗)’ for samples X = x∗occurs with probability lower than the rhs. of (13) with ρ = |wT x∗|. When asserting this for a number nv ∈N of samples X ∼PX with nv →∞, a misprediction would occur less than δnv times. In this sense, one can use the latent variable wT ϕ(x∗) as an indication of how ’certain’ the prediction is. Figure 1.a gives an example of the MAM classifier, together with the level plots indicating the certainty of prediction. Remark however that the described ’certainty of prediction’ statement differs from a conditional statement of the risk given as P(Y (wT ϕ(X)) < 0 | X = x∗). The essential difference with the probabilistic estimates based on the density estimates resulting from the Parzen window estimator is that results become independent of the data dimension, as one avoids estimating the joint distribution. 3.2 Transforming the Margin Distribution Consider the case where the assumption of a reasonable constant B such that P(∥X∥2 < B) = 1 is unrealistic. Then, a transformation of the random variable Y (wT X) can be fruitful using a monotone increasing function g : R →R with a constant B′ ϕ ≪B such that |g(z)| ≤B′ ϕ, and g(0) = 0. In the choice of a proper transformation, two counteracting effects should be traded properly. At first, a small choice of B improves the bound as e.g. described in Lemma 2. On the other hand, such a transformation would make the expected value E[g(Y (wT ϕ(X)))] smaller than E[Y (wT ϕ(X))]. Modifying Theorem 2 gives Corollary 1 (Occurrence of Mistakes, bis) Given i.i.d. samples Dn = {(Xi, Yi)}n i=1, and a fixed δ ∈R+ 0 . Let g : R →R be a monotonically increasing function with Lipschitz constant 0 < Lg < ∞, let B′ ϕ ∈R such that |g(z)| ≤B′ ϕ for all z, and g(0) = 0. Then with probability exceeding 1 −δ, one has for any ρ such that −B′ ϕ ≤ρ ≤B′ ϕ and w ∈Rd that P g(Y (wT n ϕ(X))) ≤−ρ ≤ B′ ϕ B′ϕ + ρ− 1 n Pn i=1 g(Yi(wT n ϕ(Xi))) −Lg ˆRn(H) −3B′ ϕ q 2 log( 2 δ) n B′ϕ + ρ . (15) This result follows straightforwardly from Theorem 2 using the property that ˆRn(g ◦H) ≤ Lg ˆRn(H), see e.g. [1]. When ρ = 0, one has P g(Y (wT n ϕ(X))) ≤0 ≤ 1−E[Y g(wT ϕ(X))] 1 . Similar as in the previous section, corollary 1 can be used to score the certainty of prediction by considering for each X = x∗the value of g(wT x∗) and g(−wT x∗). Figure 1.b gives an example by considering the clipping transformation g(z) = min(1, max(−1, z)) ∈[−1, 1] such that B′ ϕ = 1. 5 Note that this a-priori choice of the function g is not dependent on the (empirical) optimality criterion at hand. 3.3 Soft-margin SVMs and MAM classifiers Except the margin-based mechanisms, the MAM classifier shares other properties with the softmargin maximal margin classifier (SVM) as well. Consider the following saturation function g(z) = (1 −z)+, where (·)+ is defined as (z)+ = z if z ≥0, and zero otherwise (g(0) = 0). Application of this function to the MAM formulation of (4), one obtains for a C > 0 max w − n X i=1 1 −Yi(wT ϕ(Xi)) + s.t. wT w = C, (16) which is similar to the support vector machine (see e.g. [14]). To make this equivalence more explicit, consider the following formulation of (16) min w,ξ n X i=1 ξi s.t. wT w ≤C and Yi(wT ϕ(Xi)) ≥1 −ξi, ξi ≥0 ∀i = 1, . . . , n, (17) which is similar to the SVM. Consider the following modification min w,ξ n X i=1 ξi s.t. wT w ≤C and Yi(wT ϕ(Xi)) ≥1 −ξi ∀i = 1, . . . , n, (18) which is equivalent to (4) as in the optimum, Yi(wT ϕ(Xi)) = (1 −ξi) for all i. Thus, omission of the slack constraints ξi ≥0 in the SVM formulation results in the Parzen window classifier. 4 Maximal Average Margin for Ordinal Regression Along the same lines as [6], the maximal average margin principle can be applied to ordinal regression tasks. Let (X, Y ) ∈Rd × {1, . . . , m} with distribution PXY . The w ∈Rd maximizing P(I(wT (ϕ(X) −ϕ(X)′)(Y −Y ′) > 0)) can be found by solving for the maximal average margin between pairs as follows M ∗= max w E sign(Y −Y ′)wT (ϕ(X) −ϕ(X)′) ∥w∥2 . (19) Given n i.i.d. samples {(Xi, Yi)}n i=1, empirical risk minimization is obtained by solving min w −1 n n X i,j=1 sign(Yj −Yi)wT (ϕ(Xj) −ϕ(Xi)) s.t. ∥w∥2 ≤1. (20) The Lagrangian with multiplier λ ≥0 becomes L(w, λ) = −1 n P i,j wT sign(Yj −Yi)(ϕ(Xj) − ϕ(Xi))+ λ 2 (wT w−1). Let there be n′ couples (i, j). Let Dy ∈{−1, 0, 1}n′×n such that Dy,ki = 1 and Dy,kj = −1 if the kth couple equals (i, j). Then, by switching the minimax problem to a maximin problem, the first order condition for optimality ∂L(w,λ) ∂w = 0 gives the expression. wn = 1 λ′n P Yi<Yj(ϕ(Xj) −ϕ(Xi)) = 1 λnXDy1n′. Now the parameter λ can be found by substituting (5) in the constraint wT w = 1, or λ = 1 n q 1T n′DTy XT X Dy1n′. Now the key element is the computation of dy = Dy1n′. Note that dy(i) = n X j=1 sign(Yj −Yi) ≜ry(i), (21) with rY denoting the ranks of all Yi in y. This expression simplifies expression for wn as wn = 1 λnXdy. It is seen that using kernels as before, the resulting estimator of the order of the responses corresponding to x and x′ becomes ˆfK(x, x′) = sign (m(x) −m(x′)) , where m(x) = 1 λn n X i=1 K(Xi, x) rY (i). (22) 6 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0 20 40 60 80 100 120 τ Frequency oMAM LS−SVM oSVM oGP (a) Data (train/test) oMAM LS-SVM oSVM oGP Bank(1) (100/8.092) 0.37 0.43 0.46 0.41 Bank(1) (500/7.629) 0.49 0.51 0.55 0.50 Bank(1) (5.000/3.192) 0.56 0.56 Bank(1) (7.500/692) 0.57 Bank(2) (100/8.092) 0.81 0.84 0.87 0.80 Bank(2) (500/7.629) 0.83 0.86 0.87 0.81 Bank(2) (5.000/3.192) 0.86 0.88 Bank(2) (7.500/692) 0.88 Cpu(1) (100/20.540) 0.44 0.62 0.64 0.63 Cpu(1) (500/20.140) 0.50 0.66 0.66 0.65 Cpu(1) (5.000/15.640) 0.57 0.68 Cpu(1) (7.500/13.140) 0.60 Cpu(1) (15.000/5.640) 0.69 (b) Figure 2: Results on ordinal regression tasks using oMAM (22) of O(n), a regression on the rank-transformed responses using LS-SVMs [16] of O(n2) −O(n3), ordinal SVMs and ordinal Gaussian Processes for preferential learning of O(n4) −O(n6). The results are expressed as Kendall’s τ (with −1 ≤τ ≤1) computed on the validation datasets. Figure (a) reports the numerical results of the artificially generated data, Table (b) gives the result on a number of large scaled datasets described in [2], if the computation took less than 5 minutes. Remark that the estimator m : Rd →R equals (except for the normalization term) the NadarayaWatson kernel based on the rank-transform rY of the responses. This observation suggest the application of standard regression tools based on the rank-transformed responses as in [7]. Experiments confirm the use of the proposed ranking estimator, and also motivate the use of a more involved function approximation tools as e.g. LS-SVMs [16] based on the rank-transformed responses. 5 Illustrative Example Table 2.b provides numerical results on the 13 classification (including 100 randomizations) benchmark datasets as described in [11]. The choice of an appropriate kernel parameter was obtained by cross-validation over a range of bandwidths from σ = 1e −2 to σ = 1e15. The results illustrate that the Parzen window classifier performs in general slightly (but not significantly so) worse than the other methods, but obviously reduces the required amount of memory and computation time (i.e. O(n) versus O(n2) −O(n3)). Hence, it is advised to use the Parzen classifier as a cheap base-line method, or to use it in a context where time- or memory requirements are stringent. The first artificial dataset for testing the ordinal regression scheme is constructed as follows. The training set {(Xi, Yi)}n i=1 ⊂R5 × R with n = 100 and a validation set {(Xv i , Y v i )}nv i=1 ⊂R5 × R with nv = 250 is constructed such that Zi = (wT ∗Xi)3 + ei and Zv i = (wT ∗Xv i )3 + ev i with w∗∈N(0, 1), X, Xv ∼N(0, I5), and e, ev ∼N(0, 0.25). Now Y (and Y v) are generated preserving the order implied by {Zi}100 i=1 (and {Zv i }250 i=1) with the intervals χ2-distributed with 5 degrees of freedom. Figure 2.a shows the results of a Monte Carlo experiment relating both the O(n) proposed estimator (22), a LS-SVM regressor of O(n2) −O(n3) on the rank-transformed responses {(Xi, rY (i))}, the O(n4) −O(n6) SVM approach as proposed in [3] and the Gaussian Process approach of O(n4) −O(n6) given in [2]. The performance of the different algorithms is expressed in terms of Kendall’s τ computed on the validation data. Table 2.b reports the results on some large scale datasets as described in [2], imposing a maximal computation time of 5 minutes. Both tests suggest the competitive nature of the proposed O(n) procedure, while clearly showing the benefit of using function estimation (as e.g. LS-SVMs) based on the rank-transformed responses. 7 6 Conclusion This paper discussed the use of the MAM risk optimality principle for designing a learning machine for classification and ordinal regression. The relation with classical methods including Parzen windows and Nadaraya-Watson estimators is established, while the relation with the empirical Rademacher complexity is used to provide a measure of ’certainty of prediction’. Empirical experiments show the applicability of the O(n) algorithms on real world problems, trading performance somewhat for computational efficiency with respect to state-of-the art learning algorithms. References [1] P.L. Bartlett and S. Mendelson. Rademacher and gaussian complexities: Risk bounds and structural results. Journal of Machine Learning Research, 3:463–482, 2002. [2] W. Chu and Z. Ghahramani. Gaussian processes for ordinal regression. Journal of Machine Learning Research, 6:1019–1041, 2006. [3] W. Chu and S. S. Keerthi. New approaches to support vector ordinal regression. In in Proc. of International Conference on Machine Learning, pages 145–152. 2005. [4] L. Devroye, L. Gy¨orfi, and G. Lugosi. A Probabilistic Theory of Pattern Recognition. Springer-Verlag, 1996. [5] A. Garg and D. Roth. Margin distribution and learning algorithms. In Proceedings of the Fifteenth International Conference on Machine Learning (ICML), pages 210–217. Morgan Kaufmann Publishers, 2003. [6] R. Herbrich, T. Graepel, and K. Obermayer. Large margin rank boundaries for ordinal regression. Advances in Large Margin Classifiers, pages 115–132, 2000. MIT Press, Cambridge, MA. [7] R.L. Iman and W.J. Conover. The use of the rank transform in regression. Technometrics, 21(4):499–509, 1979. [8] V. Koltchinski. Rademacher penalties and structural risk minimization. IEEE Transactions on Information Theory, 47(5):1902–1914, 1999. [9] K. Pelckmans. Primal-Dual kernel Machines. PhD thesis, Faculty of Engineering, K.U.Leuven, May. 2005. 280 p., TR 05-95. [10] K. Pelckmans, J. Shawe-Taylor, J.A.K. Suykens, and B. De Moor. Margin based transductive graph cuts using linear programming. In Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics, (AISTATS 2007), pp. 360-367, San Juan, Puerto Rico, 2007. [11] G. R¨atsch, T. Onoda, and K.-R. M¨uller. Soft margins for adaboost. Machine Learning, 42(3):287 – 320, 2001. [12] B. Sch¨olkopf and A. Smola. Learning with Kernels. MIT Press, Cambridge, MA, 2002. [13] J. Shawe-Taylor and N. Cristianini. Further results on the margin distribution. In Proceedings of the twelfth annual conference on Computational learning theory (COLT), pages 278–285. ACM Press, 1999. [14] J. Shawe-Taylor and N. Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press, 2004. [15] D.F. Specht. Probabilistic neural networks. Neural Networks, 3:110–118, 1990. [16] J.A.K. Suykens, T. van Gestel, J. De Brabanter, B. De Moor, and J. Vandewalle. Least Squares Support Vector Machines. World Scientific, Singapore, 2002. 8
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Managing Power Consumption and Performance of Computing Systems Using Reinforcement Learning Gerald Tesauro, Rajarshi Das, Hoi Chan, Jeffrey O. Kephart, Charles Lefurgy∗, David W. Levine and Freeman Rawson∗ IBM Watson and Austin∗Research Laboratories {gtesauro,rajarshi,hychan,kephart,lefurgy,dwl,frawson}@us.ibm.com Abstract Electrical power management in large-scale IT systems such as commercial datacenters is an application area of rapidly growing interest from both an economic and ecological perspective, with billions of dollars and millions of metric tons of CO2 emissions at stake annually. Businesses want to save power without sacrificing performance. This paper presents a reinforcement learning approach to simultaneous online management of both performance and power consumption. We apply RL in a realistic laboratory testbed using a Blade cluster and dynamically varying HTTP workload running on a commercial web applications middleware platform. We embed a CPU frequency controller in the Blade servers’ firmware, and we train policies for this controller using a multi-criteria reward signal depending on both application performance and CPU power consumption. Our testbed scenario posed a number of challenges to successful use of RL, including multiple disparate reward functions, limited decision sampling rates, and pathologies arising when using multiple sensor readings as state variables. We describe innovative practical solutions to these challenges, and demonstrate clear performance improvements over both hand-designed policies as well as obvious “cookbook” RL implementations. 1 Introduction Energy consumption is a major and growing concern throughout the IT industry as well as for customers and for government regulators concerned with energy and environmental matters. To cite a prominent example, the US Congress recently mandated a study of the power efficiency of servers, including a feasibility study of an Energy Star standard for servers and data centers [16]. Growing interest in power management is also apparent in the formation of the Green Grid, a consortium of systems and other vendors dedicated to improving data center power efficiency [7]. Recent trade press articles also make it clear that computer purchasers and data center operators are eager to reduce power consumption and the heat densities being experienced with current systems. In response to these concerns, researchers are tackling intelligent power control of processors, memory chips and whole systems, using technologies such as processor throttling, frequency and voltage manipulation, low-power DRAM states, feedback control using measured power values, and packing and virtualization to reduce the number of machines that need to be powered on to run a workload. This paper presents a reinforcement learning (RL) approach to developing effective control policies for real-time management of power consumption in application servers. Such power management policies must make intelligent tradeoffs between power and performance, as running servers in low-power modes inevitably degrades the application performance. Our approach to this entails designing a multi-criteria objective function Upp taking both power and performance into account, and using it to give reward signals in reinforcement learning. We let Upp be a function of mean 1 application response time RT, and total power Pwr consumed by the servers in a decision interval. Specifically, Upp subtracts a linear power cost from a performance-based utility U(RT): Upp(RT, Pwr) = U(RT) −ǫ ∗Pwr (1) where ǫ is a tunable coefficient expressing the relative value of power and performance objectives. This approach admits other objective functions such as “performance value per watt” Upp = U(RT)/Pwr, or a simple performance-based utility Upp = U(RT) coupled with a constraint on total power. The problem of jointly managing performance and power in IT-systems was only recently studied in the literature [5, 6, 17]. Existing approaches use knowledge-intensive and labor-intensive modeling, such as developing queuing-theoretic or control-theoretic performance models. RL methods can potentially avoid such knowledge bottlenecks, by automatically learning high-quality management policies using little or no built-in system specific knowledge. Moreover, as we discuss later, RL may have the merit of properly handling complex dynamic and delayed consequences of decisions. In Section 2 we give details of our laboratory testbed, while Section 3 describes our RL approach. Results are presented in Section 4, and the final section discusses next steps in our ongoing research and ties to related work. 2 Experimental Testbed Figure 1 provides a high-level overview of our experimental testbed. In brief, a Workload Generator produces an HTTP-based workload of dynamically varying intensity that is routed to a blade cluster, i.e., a collection of blade servers contained in a single chassis. (Specifically, we use an IBM BladeCenter containing xSeries HS20 blade servers.) A commercial performance manager and our RL-based power manager strive to optimize a joint power-performance objective cooperatively as load varies, each adjusting its control parameters individually while sharing certain information with the other manager. RL techniques (described subsequently) are used to train a state-action value function which defines the power manager’s control policy. The “state” is characterized by a set of observable performance, power and load intensity metrics collected in our data collection module as detailed below. The “action” is a throttling of CPU frequency1 that is achieved by setting a “powercap” on each blade that provides an upper limit on the power that the blade may consume. Given this limit, a feedback controller embedded in the server’s firmware [11] continuously monitors the power consumption, and continuously regulates the CPU clock speed so as to keep the power consumption close to, but not over, the powercap limit. The CPU throttling affects both application performance as well as power consumption, and the goal of learning is to achieve the optimal level of throttling in any given state that maximizes cumulative discounted values of joint reward Upp. We control workload intensity by varying the number of clients nc sending HTTP requests. We varied nc in a range from 1 to 50 using a statistical time-series model of web traffic derived from observations of a highly accessed Olympics web site [14]. Clients behave according to a closed-loop model [12] with exponentially distributed think times of mean 125 msec. The commercial performance manager is WebSphere Extended Deployment (WXD)[18], a multinode webserver environment providing extensive data collection and performance management functionality. WXD manages the routing policy of the Workload Distributer as well as control parameters on individual blades, such as the maximum workload concurrency. Our data collector receives several streams of data and provides a synchronized report to the power policy evaluator on a time scale τl (typically set to 5 seconds). Data generated on much faster time scales than τl are time-averaged over the interval, otherwise the most recent values are reported. Among the aggregated data are several dozen performance metrics collected by a daemon running on the WXD data server, such as mean response time, queue length and number of CPU cycles per transaction; CPU utilization and effective frequency collected by local daemons on each blade; and current power and temperature measurements collected by the firmware on each blade, which are polled using IPMI commands sent from the BladeCenter management module. 1An alternative technique with different power/performance trade-offs is Dynamic Voltage and Frequency Scaling (DVFS). 2 Performance Manager (WebSphere XD) Power Manager Workload Generator Power Blade System Blade System Blade System Blade Chassis Workload Distributor Power Assignment Power Data Performance Data Control Policy Control Policy HTTP Requests Manager-to-manager Interactions Figure 1: Overview of testbed environment. 2.1 Utility function definition Our specific performance-based utility U(RT) in Eq. 1 is a piecewise linear function of response time RT which returns a maximum value of 1.0 when RT is less than a specified threshold RT0, and which drops linearly when RT exceeds RT0, i.e., U(RT/RT0) = ½ 1.0 if RT ≤RT0 2.0 −RT/RT0 otherwise (2) Such a utility function reflects the common assumptions in customer service level agreements that there is no incentive to improve the performance once it reaches the target threshold, and that there is always a constant incentive to improve performance if it violates the threshold. In all of our experiments, we set RT0 = 1000 msec, and we also set the power scale factor ǫ = 0.01 in Eq. 1. At this value of ǫ the power-performance tradeoff is strongly biased in favor of performance, as is commonly desired in today’s data centers. However, larger values of ǫ could be appropriate in future scenarios where power is much more costly, in which case the optimal policies would tolerate more frequent performance threshold violations in order to save more aggressively on power consumption. 2.2 Baseline Powercap Policies To assess the effectiveness of our RL-based power management policies, we compare with two different benchmark policies: “UN” (unmanaged) and “HC” (hand-crafted). The unmanaged policy always sets the powercap to a maximal value of 120W; we verified that the CPU runs at the highest frequency under all load conditions with this setting. The hand-crafted policy was created as follows. We measured power consumption on a blade server at extremely low (nc = 1) and high (nc = 50) loads, finding that in all cases the power consumption ranged between 75 and 120 watts. Given this range, we established a grid of sample points, with pκ running from 75 watts to 120 watts in increments of 5 watts, and the number of clients running from 0 to 50 in increments of 5. For each of the 10 possible settings of pκ, we held nc fixed at 50 for 45 minutes to permit WXD to adapt to the workload, and then decremented nc by 5 every 5 minutes. Finally, the models RT(pκ, nc) and Pwr(pκ, nc), were derived by linearly interpolating for the RT and Pwr between the sampled grid points. We substitute these models into our utility function Upp(RT, Pwr) to obtain an equivalent utility function U ′ depending on pκ and nc, i.e., U ′(pκ, nc) = Upp(RT(pκ, nc), Pwr(pκ, nc)). We can then choose the optimal powercap for any workload intensity nc by optimizing U ′: p∗ κ(nc) = arg maxpκ U ′(pκ, nc). 3 3 Reinforcement Learning Approach One may naturally question whether RL could be capable of learning effective control policies for systems as complex as a population of human users interacting with a commercial web application. Such systems are surely far from full observability in the MDP sense. Without even considering whether the behavior of users is “Markovian,” we note that the state of a web application may depend, for example, on the states of the underlying middleware and Java Virtual Machines (JVMs), and these states are not only unobservable, they also have complex historical dependencies on prior load and performance levels over multiple time scales. Despite such complexities, we have found in our earlier work [15, 9] that RL can in fact learn decent policies when using severely limited state descriptions, such as a single state variable representing current load intensity. The focus of our work in this paper is to examine empirically whether RL may obtain better policies by including more observable metrics in the state description. Another important question is whether current decisions have long-range effects, or if it suffices to simply learn policies that optimize immediate reward. The answer appears to vary in an interesting way: under low load conditions, the system response to a decision is fairly immediate, whereas under conditions of high queue length (which may result from poor throttling decisions), the responsiveness to decisions may become sluggish and considerably delayed. Our reinforcement learning approach leverages our recent “Hybrid RL” approach [15], which originally was applied to autonomic server allocation. Hybrid RL is a form of offline (batch) RL that entails devising an initial control policy, running the initial policy in the live system and logging a set of (state, action, reward) tuples, and then using a standard RL/function approximator combination to learn a value function V (s, a) estimating cumulative expected reward of taking action a in state s. (The term “Hybrid” refers to the fact that expert domain knowledge can be engineered into the initial policy without needing explicit engineering or interfacing into the RL module.) The learned value function V then implies a policy of selecting the action a∗in state s with highest expected value, i.e., a∗= arg maxa V (s, a). For technical reasons detailed below, we use the Sarsa(0) update rule rather than Q-Learning (note that unlike textbook Sarsa, decisions are made by an external fixed policy). Following [15], we set the discount parameter γ = 0.5; we found some preliminary evidence that this is superior to setting γ = 0.0 but haven’t been able to systematically study the effect of varying γ. We also perform standard direct gradient training of neural net weights: we train a multilayer perceptron with 12 sigmoidal hidden units, using backprop to compute the weight changes. Such an approach is appealing, as it is simple to implement and has a proven track record of success in many practical applications. There is a theoretical risk that the approach could produce value function divergence. However, we have not seen such divergence in our application. Were it to occur, it would not entail any live performance costs, since we train offline. Additionally, we note that instead of direct gradient training, we can use Baird’s residual gradient method [4], which guarantees convergence to local Bellman error minima. In practice we find that direct gradient training yields good convergence to Bellman error minima in ∼5-10K training epochs, requiring only a few CPU minutes on a 3GHz workstation. In implementing an initial policy to be used with Hybrid RL, one would generally want to exploit the best available human-designed policy, combined with sufficient randomized exploration needed by RL, in order to achieve the best possible learned policy. However, in view of the difficulty expected in designing such initial policies, it would be advantageous to be able to learn effective policies starting from simplistic initial policies. We have therefore trained our RL policies using an extremely simple performance-biased random walk policy for setting the powercap, which operates as follows: At every decision point, pκ either is increased by 1 watt with probability p+, or decreased by 1 watt with probability p−= (1 −p+). The upward bias p+ depends on the ratio r = RT/RT0 of current mean response time to response time threshold according to: p+ = r/(1 + r). Note that this rule implies an unbiased random walk when r = 1 and that p+ →1 for r ≫1, while p+ →0 when r ≪1. This simple rule seems to strike a good balance between keeping the performance near the desired threshold, while providing plenty of exploration needed by RL, as can been seen in Figure 2. Having collected training data during the execution of an initial policy, the next step of Hybrid RL is to design an (input, output) representation and functional form of the value function approximator. 4 0 10 20 30 40 50 0 500 1000 1500 2000 2500 3000 3500 4000 Clients Time (x 5 sec) 0 500 1000 1500 2000 2500 3000 Avg RT (msec) goal 70 80 90 100 110 120 Avg Power (watts) cap (a) (b) (c) Figure 2: Traces of (a) workload intensity, (b) mean response time, and (c) powercap and consumed power of the random-walk (RW) powercap policy. We have initially used the basic input representation studied in [15], in which the state s is represented using a single metric of workload intensity (number of clients nc), and the action a is a single scalar variable—the powercap pκ. This scheme robustly produces decent learned policies, with little sensitivity to exact learning algorithm parameter settings. In later experiments, we have expanded the state representation to a much larger set of 14 state variables, and find that substantial improvements in learned policies can be obtained, provided that certain data pre-processing techniques are used, as detailed below. 3.1 System-specific innovations In our research in this application domain, we have devised several innovative “tricks” enabling us to achieve substantially improved RL performance. Such tricks are worth mentioning as they are likely to be of more general use in other problem domains with similar characteristics. First, to represent and learn V , we could employ a single output unit, trained on the total utility (reward) using Q-Learning. However, we can take advantage of the fact that total utility Upp in equation 1 is a linear combination of performance utility U and power cost −ǫ ∗Pwr. Since the separate reward components are generally observable, and since these should have completely different functional forms relying on different state variables, we propose training two separate function approximators estimating future discounted reward components Vperf and Vpwr respectively. This type of “decompositional reward” problem has been studied for tabular RL in [13], where it is shown that learning the value function components using Sarsa provably converges to the correct total value function. (Note that Q-Learning cannot be used to train the value function components, as it incorrectly assumes that the optimal policy optimizes each individual component function.) Second, we devised a new type of neuronal output unit to learn Vperf. This is motivated by the shape of U, which is a piecewise linear function of RT, with constant value for low RT and linearly decreasing for large RT. This functional form is is not naturally approximated by either a linear or a sigmoidal transfer function. However, by noting that the derivative of U is a step function (changing from 0 to -1 at the threshold), and that sigmoids give a good approximation to step functions, this suggests using an output transfer function that behaves as the integral of a sigmoid function. Specifically, our transfer function has the form Y (x) = 1 −χ(x) where χ(x) = R σ(x)dx + C, where σ(x) = 1/(1 + exp(−x)) is the standard sigmoid function, and the integration constant C is chosen so that χ →0 as x →−∞. We find that this type of output unit is easily trained by standard backprop and provides quite a good approximation to the true expected rewards. We have also trained separate neural networks to estimate Vpwr using a similar hidden layer architecture and a standard linear output unit. However, we found only a slight improvement in Bellman error over a simple estimator of predicted power ∼= pκ (although this is not always a good estimate). 5 Hence for simplicity we used Vpwr = −ǫ ∗pκ in computing the overall learned policy maximizing V = Vperf + Vpwr. Thirdly, we devised a data pre-processing technique to address a specific rate limitation in our system that the powercap decision pκ as well as the number of clients nc can only be changed every 30 seconds, whereas we collect state data from the system every 5 seconds. This limitation was imposed because faster variations in effective CPU speed or in load disrupt WXD’s functionality, as its internal models estimate parameters on much slower time scales, and in particular, it assumes that CPU speed is a constant. As a result, we cannot do standard RL on the 5 second interval data, since this would presume the policy’s ability to make a new decision every 5 seconds. A simple way to address this would be to discard data points where a decision was not made (5/6 of the data), but this would make the training set much smaller, and we would lose valuable state transition information contained in the discarded samples. As an alternative, we divide the entire training set into six subsets according to line number mod-6, so that within each subset, adjacent data points are separated by 30 second intervals. We then concatenate the subsets to form one large training set, with no loss of data, where all adjacent intervals are 30 seconds long. In effect, a sweep through such a dataset replays the experiment six times, corresponding to the six different 5-second phases within the 30second decision cycle. As we shall see in the following section, such rearranged datasets result in substantially more stable policies. Finally, we realized that in the artificially constructed dataset described above, there is an inaccuracy in training on samples in the five non-decision phases: standard RL would presume that the powercap decision is held constant over the full 30 seconds until the next recorded sample, whereas we know that decision actually changes somewhere in the middle of the interval, depending on the phase. To obtain the best approximation to a constant decision over such intervals, we compute an equally weighted average ¯pκ of the recorded decisions at times {t, t+5, t+10, t+15, t+20, t+25} and train on ¯pκ as the effective decision that was made at time t. This change results in a significant reduction (∼40%) in Bellman error, and the combination of this with the mod-6 data reordering enables us to obtain substanial improvements in policy performance. 4 Results UN HC RW 2NN 15NN 15NNp 400 600 800 1000 1200 response time (msec) UN HC RW 2NN 15NN 15NNp 90 95 100 105 110 power (watts) UN HC RW 2NN 15NN 15NNp 46 48 50 52 54 56 temperature (cent) UN HC RW 2NN 15NN 15NNp 0 0.05 0.1 0.15 0.2 0.25 utility (a) (b) (c) (d) Figure 3: Comparison of mean metrics (a) response time, (b) power consumed, (c) temperature and (d) utility for six different power management policies: “UN” (unmanaged), “HC” (hand-crafted), “RW” (random walk), “2NN” (2-input neural net), “15NN” (15-input neural net, no pre-processing), “15NNp” (15-input neural net with pre-processing). While we have conducted experiments in other work involving multiple blade servers, in this section we focus on experiments involving a single blade. Fig. 3 plots various mean performance metrics in identical six-hour test runs using identical workload traces for six different power management policies: “UN” and “HC” denote the unmanaged and hand-crafted policies described in Sec. 2.2; “RW” is the random-walk policy of Sec. 3; “2NN” denotes a two-input (single state variable) neural net; “15NN” refers to a 15-input neural net without any data pre-processing as described in Sec. 3.1, and 6 “15NNp” indicates a 15-input neural net using said pre-processing. In the figure, the performance metrics plotted are: (a) mean response time, (b) mean power consumed, (c) mean temperature, and most importantly, (d) mean utility. Standard error in estimates of these mean values are quite small, as indicated by error bars which lie well within the diamond-shaped data points. Since the runs use identical workload traces, we can also assess significance of the differences in means across policies via paired T-tests; exhaustive pairwise comparisons show that in all cases, the null hypothesis of no difference in mean metrics is rejected at 1% significance level with P-value ≤10−6. We see in Fig. 3 that all RL-based policies, after what is effectively a single round of policy iteration, significantly outperform the original random walk policy which generated the training data. Using only load intensity as a state variable, 2NN achieves utility close to (but not matching) the hand-crafted policy. 15NN is disappointing in that its utility is actually worse than 2NN, for reasons that we discuss below. Comparing 15NNp with 15NN shows that pre-processing yields great improvements; 15NNp is clearly the best of the six policies. Breaking down overall utility into separate power and performance components, we note that all RL-based policies achieve greater power savings than HC at the price of somewhat higher mean response times. An additional side benefit of this is lower mean temperatures, as shown in the lower left plot; this implies both lower cooling costs as well as prolonged machine life. 0 10 20 30 40 50 60 0 500 1000 1500 2000 2500 3000 3500 4000 Clients Time (x5 sec) 0 400 800 1200 1600 Avg RT (msec) goal 70 80 90 100 110 120 Avg Power (watts) cap 0 400 800 1200 1600 Avg RT (msec) goal 70 80 90 100 110 120 Avg Power (watts) cap (a) (b) (c) (d) (e) 0 400 800 1200 1600 0 500 1000 1500 2000 2500 3000 3500 4000 Avg RT (msec) Time (x5 sec) goal 70 80 90 100 110 120 Avg Power (watts) cap 0 400 800 1200 1600 Avg RT (msec) goal 70 80 90 100 110 120 Avg Power (watts) cap 0 400 800 1200 1600 Avg RT (msec) goal 70 80 90 100 110 120 Avg Power (watts) cap (f) (g) (h) (i) (j) (k) Figure 4: Traces of the five non-random policies: (a) workload intensity; (b) UN response time; (c) UN powercap; (d) HC response time; (e) HC powercap; (f) 2NN response time; (g) 2NN powercap; (h) 15NN response time; (i) 15NN powercap; (j) 15NNp response time; (k) 15NNp powercap. Fig. 4 shows the actual traces of response time, powercap and power consumed in all experiments except the random walk, which was plotted earlier. The most salient points to note are that 15NNp exhibits the steadiest response time, keeping closest to the response time goal, and that the powercap decsions of 15NN show quite large short-term fluctuations. We attribute the latter behavior to “overreacting” to response time fluctuations above or below the target value. Such behavior may well be correct if the policy could reset every 5 seconds, as 15NN presumes. In this case, the policy could react to a response time flucutation by setting an extreme powercap value in an attempt to quickly drive the response time back to the goal value, and then backing off to a less extreme value 5 seconds later. However, such behavior would be quite poor in the actual system, in which the extreme powercap setting is held fixed for 30 seconds. 7 5 Summary and related work This paper presented a successful application of batch RL combined with nonlinear function approximation in a new and challenging domain of autonomic management of power and performance in web application servers. We addressed challenges arising both from operating in real hardware, and from limitations imposed by interoperating with commercial middleware. By training on data from a simple random-walk initial policy, we achieved high-quality management polices that outperformed the best available hand-crafted policy. Such policies save more than 10% on server power while keeping performance close to a desired target. In our ongoing and future work, we are aiming to scale the approach to an entire Blade cluster, and to achieve much greater levels of power savings. With the existing approach it appears that power savings closer to 20% could be obtained simply by using more realistic web workload profiles in which high-intensity spikes are brief, and the ratio of peak-to-mean workload is much higher than in our current traffic model. It also appears that savings of ∼30% are plausible when using multicore processors [8]. Finally, we are also aiming to learn policies for powering machines off when feasible; this offers the potential to achieve power savings of 50% or more. In order to scale our approach to larger systems, we can leverage the fact that Blade clusters usually have sets of identical machines. All servers within such a homogeneous set can be managed in an identical fashion by the performance and power managers, thereby making the size of the overall state space and the action space more tractable for RL. An important component of our future work is also to improve our current RL methodology. Beyond Hybrid RL, there has been much recent research in offline RL methods, including LSPI [10], Apprenticeship Learning [2], Differential Dynamic Programming [1], and fitted policy iteration minimizing Bellman residuals [3]. These methods are of great interest to us, as they typically have stronger theoretical guarantees than Hybrid RL, and have delivered impressive performance in applications such as helicopter aerobatics. For powering machines on and off, we are especially interested in offline model-based RL approaches: as the number of training samples that can be acquired is likely to be severely limited, it will be important to reduce sample complexity by learning explicit state-transition models. References [1] P. Abbeel, A. Coates, M. Quigley, and A. Y. Ng. An application of reinforcement learning to aerobatic helicopter flight. In Proc. of NIPS-06, 2006. [2] P. Abbeel and A. Y. Ng. Exploration and apprenticeship learning in reinforcement learning. In Proc. of ICML-05, 2005. [3] A. Antos, C. Szepesvari, and R. Munos. Learning near-optimal policies with bellman-residual minimization based fitted policy iteration and a single sample path. In Proc. of COLT-06, 2006. [4] L. Baird. Residual algorithms: Reinforcement learning with function approximation. In Proc. of ICML95, 1995. [5] Y. Chen et al. Managing server energy and operational costs in hosting centers. In Proc. of SIGMETRICS, 2005. [6] M. Femal and V. Freeh. Boosting data center performance through non-uniform power allocation. In Second Intl. Conf. on Autonomic Computing, 2005. [7] Green Grid Consortium. Green grid. http://www.thegreengrid.org, 2006. [8] J. Chen et al. Datacenter power modeling and prediction. UC Berkeley RAD Lab presentation, 2007. [9] J. O. Kephart, H. Chan, R. Das, D. Levine, G. Tesauro, F. Rawson, and C. Lefurgy. Coordinating multiple autonomic managers to achieve specified power-performance tradeoffs. In Proc. of ICAC-07, 2007. [10] M. G. Lagoudakis and R. Parr. Least-squares policy iteration. J. of Machine Learning Research, 4:1107– 1149, 2003. [11] C. Lefurgy, X. Wang, and M. Ware. Server-level power control. In Proc. of ICAC-07, 2007. [12] D. Menasce and V. A. F. Almeida. Capacity Planning for Web Performance: Metrics, Models, and Methods. Prentice Hall, 1998. [13] S. Russell and A. L. Zimdars. Q-decomposition for reinforcement learning agents. In Proc. of ICML-03, pages 656–663, 2003. [14] M. S. Squillante, D. D. Yao, and L. Zhang. Internet traffic: Periodicity, tail behavior and performance implications. In System Performance Evaluation: Methodologies and Applications, 1999. [15] G. Tesauro, N. K. Jong, R. Das, and M. N. Bennani. A hybrid reinforcement learning approach to autonomic resource allocation. In Proc. of ICAC-06, pages 65–73, 2006. [16] United States Environmental Protection Agency. Letter to Enterprise Server Manufacturers and Other Stakeholders. http://www.energystar.gov, 2006. [17] M. Wang et al. Adaptive Performance Control of Computing Systems via Distributed Cooperative Control: Application to Power Management in Computer Clusters. In Proc. of ICAC-06, 2006. [18] WebSphere Extended Deployment. http://www.ibm.com/software/webservers/appserv/extend/, 2007. 8
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A Game-Theoretic Approach to Apprenticeship Learning Umar Syed Computer Science Department Princeton University 35 Olden St Princeton, NJ 08540-5233 usyed@cs.princeton.edu Robert E. Schapire Computer Science Department Princeton University 35 Olden St Princeton, NJ 08540-5233 schapire@cs.princeton.edu Abstract We study the problem of an apprentice learning to behave in an environment with an unknown reward function by observing the behavior of an expert. We follow on the work of Abbeel and Ng [1] who considered a framework in which the true reward function is assumed to be a linear combination of a set of known and observable features. We give a new algorithm that, like theirs, is guaranteed to learn a policy that is nearly as good as the expert’s, given enough examples. However, unlike their algorithm, we show that ours may produce a policy that is substantially better than the expert’s. Moreover, our algorithm is computationally faster, is easier to implement, and can be applied even in the absence of an expert. The method is based on a game-theoretic view of the problem, which leads naturally to a direct application of the multiplicative-weights algorithm of Freund and Schapire [2] for playing repeated matrix games. In addition to our formal presentation and analysis of the new algorithm, we sketch how the method can be applied when the transition function itself is unknown, and we provide an experimental demonstration of the algorithm on a toy video-game environment. 1 Introduction When an agent is faced with the task of learning how to behave in a stochastic environment, a common approach is to model the situation using a Markov Decision Process. An MDP consists of states, actions, rewards and a transition function. Once an MDP has been provided, the usual objective is to find a policy (i.e. a mapping from states to actions) that maximizes expected cumulative reward collected by the agent. Building the MDP model is usually the most difficult part of this process. One reason is that it is often hard to correctly describe the environment’s true reward function, and yet the behavior of the agent is quite sensitive to this description. In practice, reward functions are frequently tweaked and tuned to elicit what is thought to be the desired behavior. Instead of maximizing reward, another approach often taken is to observe and follow the behavior of an expert in the same environment. Learning how to behave by observing an expert has been called apprenticeship learning, with the agent in the role of the apprentice. Abbeel and Ng [1] proposed a novel and appealing framework for apprenticeship learning. In this framework, the reward function, while unknown to the apprentice, is assumed to be equal to a linear combination of a set of known features. They argued that while it may be difficult to correctly describe the reward function, it is usually much easier to specify the features on which the reward function depends. 1 With this setting in mind, Abbeel and Ng [1] described an efficient algorithm that, given enough examples of the expert’s behavior, produces a policy that does at least as well as the expert with respect to the unknown reward function. The number of examples their algorithm requires from the expert depends only moderately on the number of features. While impressive, a drawback of their results is that the performance of the apprentice is both upperand lower-bounded by the performance of the expert. Essentially, their algorithm is an efficient method for mimicking the expert’s behavior. If the behavior of the expert is far from optimal, the same will hold for the apprentice. In this paper, we take a somewhat different approach to apprenticeship learning that addresses this issue, while also significantly improving on other aspects of Abbeel and Ng’s [1] results. We pose the problem as learning to play a two-player zero-sum game in which the apprentice chooses a policy, and the environment chooses a reward function. The goal of the apprentice is to maximize performance relative to the expert, even though the reward function may be adversarially selected by the environment with respect to this goal. A key property of our algorithm is that it is able to leverage prior beliefs about the relationship between the features and the reward function. Specifically, if it is known whether a feature is “good” (related to reward) or “bad” (inversely related to reward), then the apprentice can use that knowledge to improve its performance. As a result, our algorithm produces policies that can be significantly better than the expert’s policy with respect to the unknown reward function, while at the same time are guaranteed to be no worse. Our approach is based on a multiplicative weights algorithm for solving two-player zero-sum games due to Freund and Schapire [2]. Their algorithm is especially well-suited to solving zero-sum games in which the “game matrix” is extremely large. It turns out that our apprenticeship learning setting can be viewed as a game with this property. Our results represent a strict improvement over those of Abbeel and Ng [1] in that our algorithm is considerably simpler, provides the same lower bound on the apprentice’s performance relative to the expert, and removes the upper bound on the apprentice’s performance. Moreover, our algorithm requires less computational expense – specifically, we are able to achieve their performance guarantee after only O(ln k) iterations, instead of the O(k ln k), where k is the number of features on which the reward function depends. Additionally, our algorithm can be applied to a setting in which no examples are available from the expert. In that case, our algorithm produces a policy that is optimal in a certain conservative sense. We are also able to extend our algorithm to a situation where the MDP’s transition function θ is unknown. We conducted experiments from a small car driving simulation that illustrate some of our theoretical findings. Ratliff et al [3] formulated a related problem to apprenticeship learning, in which the goal is to find a reward function whose optimal policy is similar to the expert’s policy. Quite different from our work, mimicking the expert was an explicit goal of their approach. 2 Preliminaries Our problem setup largely parallels that outlined in Abbeel and Ng [1]. We are given an infinitehorizon Markov Decision Process in which the reward function has been replaced by a set of features. Specifically, we are given an MDP\R M = (S, A, γ, D, θ,φφφ), consisting of finite state and action sets S and A, discount factor γ, initial state distribution D, transition function θ(s, a, s′) ≜Pr(st+1 = s′ | st = s, at = a), and a set of k features defined by the function φφφ : S →Rk. The true reward function R∗is unknown. For ease of exposition, we assume that R∗(s) = w∗·φφφ(s), for some w∗∈Rk, although we also show how our analysis extends to the case when this does not hold. For any policy π in M, the value of π (with respect to the initial state distribution) is defined by V (π) ≜E " ∞ X t=0 γtR∗(st) π, θ, D # . 2 where the initial state s0 is chosen according to D, and the remaining states are chosen according to π and θ. We also define a k-length feature expectations vector, µ(π) ≜E " ∞ X t=0 γtφφφ(st) π, θ, D # . From its definition, it should be clear that “feature expectations” is a (somewhat misleading) abbreviation for “expected, cumulative, discounted feature values.” Importantly, since R∗(s) = w∗·φφφ(s), we have V (π) = w∗· µ(π), by linearity of expectation. We say that a feature expectations vector ˆµ is an ϵ-good estimate of µ(π) if ∥ˆµ −µ(π)∥∞≤ϵ. Likewise, we say that a policy ˆπ is ϵ-optimal for M if |V (ˆπ) −V (π∗)| ≤ϵ, where π∗is an optimal policy for M, i.e. π∗= arg maxπ V (π).1 We also assume that there is a policy πE, called the expert’s policy, which we are able to observe executing in M. Following Abbeel and Ng [1], our goal is to find a policy π such that V (π) ≥ V (πE) −ϵ, even though the true reward function R∗is unknown. We also have the additional goal of finding a policy when no observations from the expert’s policy are available. In that case, we find a policy that is optimal in a certain conservative sense. Like Abbeel and Ng [1], the policy we find will not necessarily be stationary, but will instead be a mixed policy. A mixed policy ψ is a distribution over Π, the set of all deterministic stationary policies in M. Because Π is finite (though extremely large), we can fix a numbering of the policies in Π, which we denote π1, . . . , π|Π|. This allows us to treat ψ as a vector, where ψ(i) is the probability assigned to πi. A mixed policy ψ is executed by randomly selecting the policy πi ∈Π at time 0 with probability ψ(i), and exclusively following πi thereafter. It should be noted that the definitions of value and feature expectations apply to mixed policies as well: V (ψ) = Ei∼ψ[V (πi)] and µ(ψ) = Ei∼ψ[µ(πi)]. Also note that mixed policies do not have any advantage over stationary policies in terms of value: if π∗is an optimal stationary policy for M, and ψ∗is an optimal mixed policy, then V (ψ∗) = V (π∗). The observations from the expert’s policy πE are in the form of m independent trajectories in M, each for simplicity of the same length H. A trajectory is just the sequence of states visited by the expert: si 0, si 1, . . . , si H for the ith trajectory. Let µE = µ(πE) be the expert’s feature expectations. We compute an estimate ˆµE of µE by averaging the observed feature values from the trajectories: ˆµE = 1 m m X i=0 H X t=0 γtφφφ(si t). 3 Review of the Projection Algorithm We compare our approach to the “projection algorithm” of Abbeel and Ng [1], which finds a policy that is at least as good as the expert’s policy with respect to the unknown reward function.2 Abbeel and Ng [1] assume that φφφ(s) ∈[0, 1]k, and that R∗(s) = w∗·φφφ(s) for some w∗∈Bk, where Bk = {w : ∥w∥1 ≤1}. Given m independent trajectories from the expert’s policy, the projection algorithm runs for T iterations. It returns a mixed policy ψ such that ∥µ(ψ) −µE∥2 ≤ϵ as long as T and m are sufficiently large. In other words, their algorithm seeks to “match” the expert’s feature expectations. The value of ψ will necessarily be close to that of the expert’s policy, since |V (ψ) −V (πE)| = |w∗· µ(ψ) −w∗· µE| ≤ ∥w∗∥2∥µ(ψ) −µE∥2 (1) ≤ ϵ where in Eq. (1) we used the Cauchy-Schwartz inequality and ∥w∗∥2 ≤∥w∗∥1 ≤1. 1Note that this is weaker than the standard definition of optimality, as the policy only needs to be optimal with respect to the initial state distribution, and not necessarily at every state simultaneously. 2Abbeel and Ng [1] actually presented two algorithms for this task. Both had the same theoretical guarantees, but the projection algorithm is simpler and was empirically shown to be slightly faster. 3 The following theorem is the main result in Abbeel and Ng [1]. However, some aspects of their analysis are not covered by this theorem, such as the complexity of each iteration of the projection algorithm, and the sensitivity of the algorithm to various approximations. These are discussed immediately below. Theorem 1 (Abbeel and Ng [1]). Given an MDP\R, and m independent trajectories from an expert’s policy πE. Suppose we execute the projection algorithm for T iterations. Let ψ be the mixed policy returned by the algorithm. Then in order for |V (ψ) −V (πE)| ≤ϵ (2) to hold with probability at least 1 −δ, it suffices that T ≥O k (ϵ(1 −γ))2 ln k ϵ(1 −γ) and m ≥ 2k (ϵ(1 −γ))2 ln 2k δ . We omit the details of the algorithm due to space constraints, but note that each iteration involves only two steps that are computationally expensive: 1. Find an optimal policy with respect to a given reward function. 2. Compute the feature expectations of a given policy. The algorithm we present in Section 5 performs these same expensive tasks in each iteration, but requires far fewer iterations — just O(ln k) rather than O(k ln k), a tremendous savings when the number of features k is large. Also, the projection algorithm has a post-processing step that requires invoking a quadratic program (QP) solver. Comparatively, the post-processing step for our algorithm is trivial. Abbeel and Ng [1] provide several refinements of the analysis in Theorem 1. In particular, suppose that each sample trajectory has length H ≥(1/(1 −γ)) ln(1/(ϵH(1 −γ))), and that an ϵP -optimal policy is found in each iteration of the projection algorithm (see Step 1 above). Also let ϵR = minw∈Bk maxs |R∗(s) −w · φφφ(s)| be the “representation error” of the features. Abbeel and Ng [1] comment at various points in their paper that ϵH, ϵP , and O(ϵR) should be added to the error bound of Theorem 1. In Section 5 we provide a unified analysis of these error terms in the context of our algorithm, and also incorporate an ϵF term that accounts for computing an ϵF -good feature expectations estimate in Step 2 above. We prove that our algorithm is sensitive to these error terms in a similar way as the projection algorithm. 4 Apprenticeship Learning via Game Playing Notice the two-sided bound in Theorem 1: the theorem guarantees that the apprentice will do almost as well as the expert, but also almost as badly. This is because the value of a policy is a linear combination of its feature expectations, and the goal of the projection algorithm is to match the expert’s feature expectations. We will take a different approach. We assume that φφφ(s) ∈[−1, 1]k, and that R∗(s) = w∗· φφφ(s) for some w∗∈Sk, where Sk = {w ∈Rk : ∥w∥1 = 1 and w ⪰0}.3 The impact of this minor change in the domains of w and φφφ is discussed further in Section 5.2. Let Ψ be the set of all mixed policies in M. Now consider the optimization v∗= max ψ∈Ψ min w∈Sk [w · µ(ψ) −w · µE] . (3) Our goal will be to find (actually, to approximate) the mixed policy ψ∗that achieves v∗. Since V (ψ) = w∗· µ(ψ) for all ψ, we have that ψ∗is the policy in Ψ that maximizes V (ψ) −V (πE) with respect to the worst-case possibility for w∗. Since w∗is unknown, maximizing for the worstcase is appropriate. 3We use ⪰to denote componentwise inequality. Likewise, we use ≻to denote strict inequality in every component. 4 We begin by noting that, because w and ψ are both distributions, Eq. (3) is in the form of a twoperson zero-sum game. Indeed, this is the motivation for redefining the domain of w as we did. The quantity v∗is typically called the game value. In this game, the “min player” specifies a reward function by choosing w, and the “max player” chooses a mixed policy ψ. The goal of the min player is to cause the max player’s policy to perform as poorly as possible relative to the expert, and the max player’s goal is just the opposite. A game is defined by its associated game matrix. In our case, the game matrix is the k × |Π| matrix G(i, j) = µj(i) −µE(i) (4) where µ(i) is the ith component of µ and we have let µj = µ(πj) be the vector of feature expectations for the jth deterministic policy πj. Now Eq. (3) can be rewritten in the form v∗= max ψ∈Ψ min w∈Sk wT Gψ. (5) In Eq. (3) and (5), the max player plays first, suggesting that the min player has an advantage. However, the well-known minmax theorem of von Neumann says that we can swap the min and max operators in Eq. (5) without affecting the game value. In other words, v∗= max ψ∈Ψ min w∈Sk wT Gψ = min w∈Sk max ψ∈Ψ wT Gψ. (6) Finding ψ∗will not be useful unless we can establish that v∗≥0, i.e. that ψ∗will do at least as well as the expert’s policy with respect to the worst-case possibility for w∗. This fact is not immediately clear, since we are restricting ourselves to mixtures of deterministic policies, while we do not assume that the expert’s policy is deterministic. However, note that in the rightmost expression in Eq. (6), the maximization over Ψ is done after w — and hence the reward function — has been fixed. So the maximum is achieved by the best policy in Ψ with respect to this fixed reward function. Note that if this is also an optimal policy, then v∗will be nonnegative. It is well-known that in any MDP there always exists a deterministic optimal policy. Hence v∗≥0. In fact, we may have v∗> 0. Suppose it happens that µ(ψ∗) ≻µ(πE). Then ψ∗will dominate πE, i.e. ψ∗will have higher value than πE regardless of the actual value of w∗, because we assumed that w∗⪰0. Essentially, by assuming that each component of the true weight vector is nonnegative, we are assuming that we have correctly specified the “sign” of each feature. This means that, other things being equal, a larger value for each feature implies a larger reward. So when v∗> 0, the mixed policy ψ∗to some extent ignores the expert, and instead exploits prior knowledge about the true reward function encoded by the features. We present experimental results that explore this aspect of our approach in Section 7. 5 The Multiplicative Weights for Apprenticeship Learning (MWAL) Algorithm In the previous section, we motivated the goal of finding the mixed policy ψ∗that achieves the maximum in Eq. (3) (or equivalently, in Eq. (5)). In this section we present an efficient algorithm for solving this optimization problem. Recall the game formulated in the previous section. In the terminology of game theory, w and ψ are called strategies for the min and max player respectively , and ψ∗is called an optimal strategy for the max player. Also, a strategy ˙w is called pure if ˙w(i) = 1 for some i. Typically, one finds an optimal strategy for a two-player zero-sum game by solving a linear program. However, the complexity of that approach scales with the size of the game matrix. In our case, the game matrix G is huge, since it has as many columns as the number of deterministic policies in the MDP\R. Freund and Schapire [2] described a multiplicative weights algorithm for finding approximately optimal strategies in games with large or even unknown game matrices. To apply their algorithm to a game matrix G, it suffices to be able to efficiently perform the following two steps: 1. Given a min player strategy w, find arg maxψ∈Ψ wT Gψ. 5 2. Given a max player strategy ψ, compute ˙wT Gψ for each pure strategy ˙w. Observe that these two steps are equivalent to the two steps of the projection algorithm from Section 3. Step 1 amounts to finding the optimal policy in a standard MDP with a known reward function. There are a huge array of techniques available for this, such as value iteration and policy iteration. Step 2 is the same as computing the feature expectations of a given policy. These can be computed exactly by solving k systems of linear equations, or they can be approximated using iterative techniques. Importantly, the complexity of both steps scales with the size of the MDP\R, and not with the size of the game matrix G. Our Multiplicative Weights for Apprenticeship Learning (MWAL) algorithm is described below. Lines 7 and 8 of the algorithm correspond to Steps 1 and 2 directly above. The algorithm is essentially the MW algorithm of Freund and Schapire [2], applied to a game matrix very similar to G.4 We have also slightly extended their results to allow the MWAL algorithm, in lines 7 and 8, to estimate the optimal policy and its feature expectations, rather than requiring that they be computed exactly. Algorithm 1 The MWAL algorithm 1: Given: An MDP\R M and an estimate of the expert’s feature expectations ˆµE. 2: Let β = 1 + q 2 ln k T −1 . 3: Define eG(i, µ) ≜((1 −γ)(µ(i) −ˆµE(i)) + 2)/4, where µ ∈Rk. 4: Initialize W(1)(i) = 1 for i = 1, . . . , k. 5: for t = 1, . . . , T do 6: Set w(t)(i) = W(t)(i) P i W(t)(i) for i = 1, . . . , k. 7: Compute an ϵP -optimal policy ˆπ(t) for M with respect to reward function R(s) = w(t)·φφφ(s). 8: Compute an ϵF -good estimate ˆµ(t) of µ(t) = µ(ˆπ(t)). 9: W(t+1)(i) = W(t)(i) · exp(ln(β) · eG(i, ˆµ(t))) for i = 1, . . . , k. 10: end for 11: Post-processing: Return the mixed policy ψ that assigns probability 1 T to ˆπ(t), for all t ∈ {1, . . . , T}. Theorem 2 below provides a performance guarantee for the mixed policy ψ returned by the MWAL algorithm, relative to the performance of the expert and the game value v∗. Its correctness is largely based on the main result in Freund and Schapire [2]. A proof is available in the supplement [4]. Theorem 2. Given an MDP\R M, and m independent trajectories from an expert’s policy πE. Suppose we execute the MWAL algorithm for T iterations. Let ψ be the mixed policy returned by the algorithm. Let ϵF and ϵP be the approximation errors from lines 7 and 8 of the algorithm. Let H ≥(1/(1 −γ)) ln(1/(ϵH(1 −γ))) be the length of each sample trajectory. Let ϵR = minw∈Sk maxs |R∗(s) −w · φφφ(s)| be the representation error of the features. Let v∗= maxψ∈Ψ minw∈Sk [w · µ(ψ) −w · µE] be the game value. Then in order for V (ψ) ≥V (πE) + v∗−ϵ (7) to hold with probability at least 1 −δ, it suffices that T ≥ 9 ln k 2(ϵ′(1 −γ))2 (8) m ≥ 2 (ϵ′(1 −γ))2 ln 2k δ (9) (10) where ϵ′ ≤ϵ −(2ϵF + ϵP + 2ϵH + 2ϵR/(1 −γ)) 3 . (11) 4Note that eG in Algorithm 1, in contrast to G in Eq. (4), depends on ˆµE instead of µE. This is because µE is unknown, and must be estimated. The other differences between eG and G are of no real consequence, and are further explained in the supplement [4]. 6 Note the differences between Theorem 1 and Theorem 2. Because v∗≥0, the guarantee of the MWAL algorithm in (7) is at least as strong as the guarantee of the projection algorithm in (2), and has the further benefit of being one-sided. Additionally, the iteration complexity of the MWAL algorithm is much lower. This not only implies a faster run time, but also implies that the mixed policy output by the MWAL algorithm consists of fewer stationary policies. And if a purely stationary policy is desired, it is not hard to show that the guarantee in (7) must hold for at least one of the stationary polices in the mixed policy (this is also true of the projection algorithm [1]). The sample complexity in the Theorem 2 is also lower, but we believe that this portion of our analysis applies to the projection algorithm as well [Abbeel, personal communication], so the MWAL algorithm does not represent an improvement in this respect. 5.1 When no expert is available Our game-playing approach can be very naturally and easily extended to the case where we do not have data from an expert. Instead of finding a policy that maximizes Eq. (3), we find a policy ψ∗ that maximizes max ψ∈Ψ min w∈Sk [w · µ(ψ)] . (12) Here ψ∗is the best policy for the worst-case possibility for w∗. The MWAL algorithm can be trivially adapted to find this policy just by setting µE = 0 (compare (12) to (3)). The following corollary follows straightforwardly from the proof of Theorem 2. Corollary 1. Given an MDP\R M. Suppose we execute the ‘no expert’ version of the MWAL algorithm for T iterations. Let ψ be the mixed policy returned by the algorithm. Let ϵF , ϵP , ϵR be defined as in Theorem 2. Let v∗= maxψ∈Ψ minw∈Sk [w · µ(ψ)]. Then V (ψ) ≥v∗−ϵ (13) if T ≥ 9 ln k 2(ϵ′(1 −γ))2 (14) where ϵ′ ≤ϵ −(2ϵF + ϵP + 2ϵR/(1 −γ)) 3 . (15) 5.2 Representation error Although the MWAL algorithm makes different assumptions about the domains of w and φφφ than the projection algorithm, these differences are of no real consequence. The same class of reward functions can be expressed under either set of assumptions by roughly doubling the number of features. Concretely, consider a feature function φφφ that satisfies the assumptions of the projection algorithm. Then for each s, if φφφ(s) = (f1, . . . , fk), define φφφ′(s) = (f1, . . . , fk, −f1, . . . , −fk, 0). Observe that φφφ′ satisfies the assumptions of the MWAL algorithm, and that minw∈Bk maxs |R∗(s) −w ·φφφ(s)| ≥ minw∈S2k+1 maxs |R∗(s) −w · φφφ′(s)|. So by only doubling the number of features, we can ensure that the representation error ϵR does not increase. Notably, employing this reduction forces the game value v∗to be zero, ensuring that the MWAL algorithm, like the projection algorithm, will mimic the expert. This obsevation provides us with some useful guidance for selecting features for the MWAL algorithm: both the original and negated version of a feature should be used if we are uncertain how that feature is correlated with reward. 6 When the transition function is unknown In the previous sections, as well as in Abbeel and Ng [1], it was assumed that the transition function θ(s, a, ·) was known. In this section we sketch how to remove this assumption. Our approach to applying the MWAL algorithm to this setting can be informally described as follows: Let M = (S, A, θ, γ, φ) be the true MDP\R for which we are missing θ. Consider the MLE estimate bθ of θ that is formed from the expert’s sample trajectories. Let Z ⊆S × A be the set of state-action pairs that are visited “most frequently” by the expert. Then after observing enough trajectories, bθ will 7 be an accurate estimate of θ on Z. We form a pessimistic estimate c MZ of M by using bθ to model the transitions in Z, and route all other transitions to a special “dead state.” Following Kearns and Singh [5], who used a very similar idea in their analyis of the E3 algorithm, we call c MZ the induced MDP\R on Z. By a straightforward application of several technical lemmas due to Kearns and Singh [5] and Abbeel and Ng [6], it is possible to show that if the number of expert trajectories m is at least O( |S|3|A| 8ϵ3 ln |S|3|A| δϵ + |S||A| ln 2|S||A| δ ), and we let Z be the set of state-action pairs visited by the expert at least O( |S|2 4ϵ2 ln |S|3|A| ϵ ) times, then using c MZ in place of M in the MWAL algorithm will add only O(ϵ) to the error bound in Theorem 2. More details are available in the supplement [4], including a precise procedure for constructing c MZ. 7 Experiments For ease of comparison, we tested the MWAL algorithm and the projection algorithm in a car driving simulator that resembled the experimental setup from Abbeel and Ng [1]. Videos of the experiments discussed below are available in the supplement [4]. In our simulator, the apprentice must navigate a car through randomly-generated traffic on a threelane highway. We define three features for this environment: a collision feature (0 if contact with another car, and 1/2 otherwise), an off-road feature (0 if on the grass, and 1/2 otherwise), and a speed feature (1/2, 3/4 and 1 for each of the three possible speeds, with higher values corresponding to higher speeds). Note that the features encode that, other things being equal, speed is good, and collisions and off-roads are bad. Fast Expert Proj MWAL Bad Expert Proj. MWAL No Expert MWAL Speed Fast Fast Fast Slow Slow Medium Medium Collisions (per sec) 1.1 1.1 0.5 2.23 2.23 0 0 Off-roads (per sec) 0 0 0 8.0 8.0 0 0 The table above displays the results of using the MWAL and projection algorithms to learn a driving policy by observing two kinds of experts: a “fast” expert (drives at the fastest speed; indifferent to collisions), and a “bad” expert (drives at the slowest speed; tries to hit cars and go off-road). In both cases, the MWAL algorithm leverages information encoded in the features to produce a policy that is significantly better than the expert’s policy. We also applied the MWAL algorithm to the “no expert” setting (see Section 5.1). In that case, it produced a policy that drives as fast as possible without risking any collisions or off-roads. Given our features, this is indeed the best policy for the worst-case choice of reward function. Acknowledgments We thank Pieter Abbeel for his helpful explanatory comments regarding his proofs. We also thank the anonymous reviewers for their suggestions for additional experiments and other improvements. This work was supported by the NSF under grant IIS-0325500. References [1] P. Abbeel, A. Ng (2004). Apprenticeship Learning via Inverse Reinforcement Learning. ICML 21. [2] Y. Freund, R. E. Schapire (1999). Adaptive Game Playing Using Multiplicative Weights. Games and Economic Behavior 29, 79–103. [3] N. Ratliff, J. Bagnell, M. Zinkevich (2006). Maximum Margin Planning. ICML 23. [4] U. Syed, R. E. Schapire (2007). “A Game-Theoretic Approach to Apprenticeship Learning — Supplement”. http://www.cs.princeton.edu/˜usyed/nips2007/. [5] M. Kearns, S. Singh (2002). Near-Optimal Reinforcement Learning in Polynomial Time. Machine Learning 49, 209–232. [6] P. Abbeel, A. Ng (2005). Exploration and Apprenticeship Learning in Reinforcement Learning. ICML 22. (Long version; available at http://www.cs.stanford.edu/˜pabbeel/) 8
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Scene Segmentation with Conditional Random Fields Learned from Partially Labeled Images Jakob Verbeek and Bill Triggs INRIA and Laboratoire Jean Kuntzmann, 655 avenue de l’Europe, 38330 Montbonnot, France Abstract Conditional Random Fields (CRFs) are an effective tool for a variety of different data segmentation and labeling tasks including visual scene interpretation, which seeks to partition images into their constituent semantic-level regions and assign appropriate class labels to each region. For accurate labeling it is important to capture the global context of the image as well as local information. We introduce a CRF based scene labeling model that incorporates both local features and features aggregated over the whole image or large sections of it. Secondly, traditional CRF learning requires fully labeled datasets which can be costly and troublesome to produce. We introduce a method for learning CRFs from datasets with many unlabeled nodes by marginalizing out the unknown labels so that the log-likelihood of the known ones can be maximized by gradient ascent. Loopy Belief Propagation is used to approximate the marginals needed for the gradient and log-likelihood calculations and the Bethe free-energy approximation to the log-likelihood is monitored to control the step size. Our experimental results show that effective models can be learned from fragmentary labelings and that incorporating top-down aggregate features significantly improves the segmentations. The resulting segmentations are compared to the state-of-the-art on three different image datasets. 1 Introduction In visual scene interpretation the goal is to assign image pixels to one of several semantic classes or scene elements, thus jointly performing segmentation and recognition. This is useful in a variety of applications ranging from keyword-based image retrieval (using the segmentation to automatically index images) to autonomous vehicle navigation [1]. Random field approaches are a popular way of modelling spatial regularities in images. Their applications range from low-level noise reduction [2] to high-level object or category recognition (this paper) and semi-automatic object segmentation [3]. Early work focused on generative modeling using Markov Random Fields, but recently Conditional Random Field (CRF) models [4] have become popular owing to their ability to directly predict the segmentation/labeling given the observed image and the ease with which arbitrary functions of the observed features can be incorporated into the training process. CRF models can be applied either at the pixel-level [5, 6, 7] or at the coarser level of super-pixels or patches [8, 9, 10]. In this paper we label images at the level of small patches, using CRF models that incorporate both purely local (single patch) feature functions and more global ‘context capturing’ feature functions that depend on aggregates of observations over the whole image or large regions. Traditional CRF training algorithms require fully-labeled training data. In practice it is difficult and time-consuming to label every pixel in an image and most of the available image interpretation datasets contain unlabeled pixels. Working at the patch level exacerbates this problem because many patches contain several different pixel-level labels. Our CRF training algorithm handles this by allowing partial and mixed labelings and optimizing the probability for the model segmentation to be consistent with the given labeling constraints. 1 The rest of the paper is organized as follows: we describe our CRF model in Section 2, present our training algorithm in Section 3, provide experimental results in Section 4, and conclude in Section 5. 2 A Conditional Random Field using Local and Global Image Features We represent images as rectangular grids of patches at a single scale, associating a hidden class label with each patch. Our CRF models incorporate 4-neighbor couplings between patch labels. The local image content of each patch is encoded using texture, color and position descriptors as in [10]. For texture we compute the 128-dimensional SIFT descriptor [11] of the patch and vector quantize it by nearest-neighbour assignement against a ks = 1000 word texton dictionary learned by k-means clustering of all patches in the training dataset. Similarly, for color we take the 36-D hue descriptor of [12] and vector quantize it against a kh = 100 word color dictionary learned from the training set. Position is encoded by overlaying the image with an m × m grid of cells (m = 8) and using the index of the cell in which the patch falls as its position feature. Each patch is thus coded by three binary vectors with respectively ks, kh and kp = m2 bits, each with a single bit set corresponding to the observed visual word. Our CRF observation functions are simple linear functions of these three vectors. Generatively, the three modalities are modelled as being independent given the patch label. The naive Bayes model of the image omits the 4-neighbor couplings and thus assumes that each patch label depends only on its three observation functions. Parameter estimation reduces to trivially counting observed visual word frequencies for each label class and feature type. On the MSRC 9class image dataset this model returns an average classification rate of 67.1% (see Section 4), so isolated appearance alone does not suffice for reliable patch labeling. In recent years models based on histograms of visual words have proven very successful for image categorization (deciding whether or not the image as a whole belongs to a given category of scenes) [13]. Motivated by this, many of our models take the global image context into account by including observation functions based on image-wide histograms of the visual words of their patches. The hope is that this will help to overcome the ambiguities that arise when patches are classified in isolation. To this end, we define a conditional model for patch labels that incorporates both local patch level features and global aggregate features. Let xi ∈{1, . . . , C} denote the label of patch i, yi denote the W-dimensional concatenated binary indicator vector of its three visual words (W = ks + hh + kp), and h denote the normalized histogram of all visual words in the image, i.e. P patches i yi normalized to sum one. The conditional probablity of the label xi is then modeled as p(xi = l|yi, h) ∝exp −PW w=1 (αwlyiw + βwlhw) , (1) where αwl, βwl are W × C matrices of coefficients to be learned. We can think of this as a multiplicative combination of a local classifier based on the patch-level observation yi and a global context or bias based on the image-wide histogram h. To account for correlations among spatially neighboring patch labels, we add couplings between the labels of neighboring patches to the single patch model (1). Let X denote the collection of all patch labels in the image and Y denote the collected patch features. Then our CRF model for the coupled patch labels is: p(X|Y ) ∝exp −E(X|Y ) , (2) E(X|Y ) = X i W X w=1 (αwxiyiw + βwxihw) + X i∼j φij(xi, xj), (3) where i ∼j denotes the set of all adjacent (4-neighbor) pairs of patches i, j. We can write E(X|Y ) without explicitly including h as an argument because h is a deterministic function of Y . We have explored two forms of pairwise potential: φij(xi, xj) = γxi,xj [xi ̸= xj], and φij(xi, xj) = (σ + τ dij) [xi ̸= xj], where [·] is one if its argument is true and zero otherwise, and dij is some similarity measure over the appearance of the patches i and j. In the first form, γxi,xj is a general symmetric weight matrix that needs to be learned. The second potential is designed to favor label transitions at image locations with high contrast. As in [3] we use dij = exp(−∥zi −zj∥2/(2λ)), with zi ∈IR3 denoting the average RGB value in the patch and λ = ⟨∥zi −zj∥2⟩, the average L2 norm between neighboring RGB values in the image. Models using the first form of potential will be denoted ‘CRFγ’ and those using the second will be denoted ‘CRFτ’, or ‘CRFσ’ if τ has been fixed to zero. A graphical representation of the model is given in Figure 1. 2 x x x x y y y y h x x x x y y y y Figure 1: Graphical representation of the model with a single image-wide aggregate feature function denoted by h. Squares denote feature functions and circles denote variable nodes xi (here connected in a 4-neighbor grid covering the image). Arrows denote single node potentials due to feature functions, and undirected edges represent pairwise potentials. The dashed lines indicate the aggregation of the single-patch observations yi into h. 3 Estimating a Conditional Random Field from Partially Labeled Images Conditional models p(X|Y ) are usually trained by maximizing the log-likelihood of correct classification of the training data, PN n=1 log p(Xn|Yn). This requires completely labeled training data, i.e. a collection of N pairs (Xn, Yn)n=1,...,N with completely known Xn. In practice this is restrictive and it is useful to develop methods that can learn from partially labeled examples – images that include either completely unlabeled patches or ones with a retricted but nontrivial set of possible labels. Formally, we will assume that an incomplete labeling X is known to belong to an associated set of admissible labelings A and we maximise the log-likelihood for the model to predict any labeling in A: L = log p(X ∈A | Y ) = log X X∈A p(X|Y ) = log X X∈A exp −E(X|Y ) −log X X exp −E(X|Y ) . (4) Note that the log-likelihood is the difference between the partition functions of the restricted and unrestricted labelings, p(X | Y, X ∈A) and p(X|Y ). For completely labeled training images this reduces trivially to the standard labeled log-likelihood, while for partially labeled ones both terms of the log-likelihood are typically intractable because the set A contains O(Ck) distinct labelings X where k is the number of unlabeled patches and C is the number of possible labels. Similarly, to find maximum likelihood parameter estimates using gradient descent we need to calculate partial derivatives with respect to each parameter θ and in general both terms are again intractable: ∂L ∂θ = X X p(X|Y ) −p(X | Y, X ∈A) ∂E(X|Y ) ∂θ . (5) However the situation is not actually much worse than the fully-labeled case. In any case we need to approximate the full partition function log(P X exp −E(X|Y )) or its derivatives and any method for doing so can also be applied to the more restricted sum log(P X∈A exp −E(X|Y )) to give a contrast-of-partition-function based approximation. Here we will use the Bethe free energy approximation for both partition functions [14]: L ≈FBethe p(X|Y ) −FBethe p(X | Y, X ∈A) . (6) The Bethe approximation is a variational method based on approximating the complete distribution p(X|Y ) as the product of its pair-wise marginals (normalized by single-node marginals) that would apply if the graph were a tree. The necessary marginals are approximated using Loopy Belief Propagation (LBP) and the log-likelihood and its gradient are then evaluated using them [14]. Here LBP is run twice (with the singleton marginals initialized from the single node potentials), once to estimate the marginals of p(X|Y ) and once for p(X | Y, X ∈A). We used standard undamped LBP with uniform initial messages without encountering any convergence problems. In practice the approximate gradient and objective were consistent enough to allow parameter estimation using standard conjugate gradient optimization with adaptive step lengths based on monitoring the Bethe free-energy. Comparison with excision of unlabeled nodes. The above training procedure requires two runs of loopy BP. A simple and often-used alternative is to discard unlabeled patches by excising nodes 3 Class and frequency Model Building 16.1% Grass 32.4% Tree 12.3% Cow 6.2% Sky 15.4% Plane 2.2% Face 4.4% Car 9.5% Bike 1.5% Per Pixel IND loc only 63.8 88.3 51.9 56.7 88.4 28.6 64.0 60.7 24.9 67.1 IND loc+glo 69.2 88.1 70.1 69.3 89.1 44.8 78.1 67.8 40.8 74.4 CRFσ loc only 75.0 88.6 72.7 70.5 94.7 55.5 83.2 81.4 69.1 80.7 CRFσ loc+glo 73.6 91.1 82.1 73.6 95.7 78.3 89.5 84.5 81.4 84.9 CRFσ loc+glo del unlabeled 84.6 91.0 76.6 70.6 91.3 43.9 77.8 71.4 30.6 78.4 CRFγ loc only 71.4 86.8 80.2 81.0 94.2 63.8 86.3 85.7 77.3 82.3 CRFγ loc+glo 74.6 88.7 82.5 82.2 93.9 61.7 88.8 82.8 76.8 83.3 CRFτ loc only 65.6 85.4 78.2 74.3 95.4 61.8 84.8 85.2 79.4 80.3 CRFτ loc+glo 75.0 88.5 82.3 81.0 94.4 60.6 88.7 82.2 76.1 83.1 Schroff et al. [15] 56.7 84.8 76.4 83.8 81.1 53.8 68.5 71.4 72.0 75.2 PLSA-MRF [10] 74.0 88.7 64.4 77.4 95.7 92.2 88.8 81.1 78.7 82.3 Table 1: Classification accuracies on the 9 MSRC classes using different models. For each class its frequency in the ground truth labeling is also given. that correspond to unlabeled or partially labeled patches from the graph. This leaves a random field with one or more completely labeled connected components whose log-likelihood p(X′|Y ′) we maximize directly using gradient based methods. Equivalently, we can use the complete model but set all of the pair-wise potentials connected to unlabeled nodes to zero: this decouples the labels of the unlabeled nodes from the rest of the field. As a result p(X|Y ) and p(X | Y, X ∈A) are equivalent for the unlabeled nodes and their contribution to the log-likelihood in Eq. (4) and the gradient in Eq. (5) vanishes. The problem with this approach is that it systematically overestimates spatial coupling strengths. Looking at the training labelings in Figure 3 and Figure 4, we see that pixels near class boundaries often remain unlabeled. Since we leave patches unlabeled if they contain unlabeled pixels, label transitions are underrepresented in the training data, which causes the strength of the pairwise couplings to be greatly overestimated. In contrast, the full CRF model provides realistic estimates because it is forced to include a (fully coupled) label transition somewhere in the unlabeled region. 4 Experimental Results This section analyzes the performance of our segmentation models in detail and compares it to other existing methods. In our first set of experiments we use the Microsoft Research Cambridge (MSRC) dataset1. This consists of 240 images of 213 × 320 pixels and their partial pixel-level labelings. The labelings assign pixels to one of nine classes: building, grass, tree, cow, sky, plane, face, car, and bike. About 30% of the pixels are unlabeled. Some sample images and labelings are shown in Figure 4. In our experiments we divide the dataset into 120 images for training and 120 for testing, reporting average results over 20 random train-test partitions. We used 20 × 20 pixel patches with centers at 10 pixel intervals. (For the patch size see the red disc in Figure 4). To obtain a labeling of the patches, pixels are assigned to the nearest patch center. Patches are allowed to have any label seen among their pixels, with unlabeled pixels being allowed to have any label. Learning and inference takes place at the patch level. To map the patch-level segmentation back to the pixel level we assign each pixel the marginal of the patch with the nearest center. (In Figure 4 the segmentations were post-processed by a applying a Gaussian filter over the pixel marginals with the scale set to half the patch spacing). The performance metrics ignore unlabeled test pixels. The relative contributions of the different components of our model are summarized in Table 1. Models that incorporate 4-neighbor spatial couplings are denoted ‘CRF’ while ones that incorporate only (local or global) patch-level potentials are denoted ‘IND’. Models that include global aggregate features are denoted ‘loc+glo’, while ones that include only on local patch-level features are denoted ‘loc only’. 1Available from http://research.microsoft.com/vision/cambridge/recognition. 4 0 2 4 6 8 10 70 75 80 85 C Accuracy only c 1 to c c to 10 local only Figure 2: Classification accuracy as a function of the aggregation fineness c, for the ‘IND’ (individual patch) classifier using a single training and test set. Aggregate features (AF) were computed in each cell of a c × c image partition. Results are given for models with no AFs (solid line), with AFs of a single c (dotted curve), with AFs on grids 1×1 up to c×c (solid curve), and with AFs on grids c × c up to 10 × 10 (dashed curve). Benefits of aggregate features. The first main conclusion is that including global aggregate features helps, for example improving the average classification rate on the MSRC dataset from 67.1% to 74.4% for the spatially uncoupled ‘IND’ model and from 80.7% to 84.9% for the ‘CRFσ’ spatial model. The idea of aggregation can be generalized to scales smaller than the complete image. We experimented with dividing the image into c × c grids for a range of values of c. In each cell of the grid we compute a separate histogram over the visual words, and for each patch in the cell we include an energy term based on this histogram in the same way as for the image-wide histogram in Eq. (1). Figure 2 shows how the performance of the individual patch classifier depends on the use of aggregate features. From the dotted curve in the figure we see that although using larger cells to aggregate features is generally more informative, even fine 10×10 subdivisions (containing only 6–12 patches per cell) provide a significant performance increase. Furthermore, including aggregates computed at several different scales does help, but the performance increment is small compared to the gain obtained with just image-wide aggregates. Therefore we included only image-wide aggregates in the subsequent experiments. Benefits of including spatial coupling. The second main conclusion from Table 1 is that including spatial couplings (pairwise CRF potentials) helps, respectively increasing the accuracy by 10.5% for ‘loc+glo’ and by 13.6% for ‘loc only’ for ‘CRFσ’ relative to ‘IND’. The improvement is particularly noticeable for rare classes when global aggregate features are not included: in this case the single node potentials are less informative and frequent classes tend to be unduly favored due to their large a priori probability. When the image-wide aggregate features are included (‘loc+glo’), the simplest pairwise potential – the ‘CRFσ’ Potts model – works better than the more general models ‘CRFγ’ and ‘CRFτ’, while if only the local features are included (‘loc only’), the class-dependent pairwise potential ‘CRFγ’ works best. The performance increment from global features is smallest for ‘CRFγ’, the model that also includes local contextual information. The overall influence of the local label transition preferences expressed in ‘CRFγ’ appears to be similar to that of the global contextual information provided by image-wide aggregate features. Benefits of training by marginalizing partial labelings. Our third main conclusion from Table 1 is that our marginalization based training method for handling missing labels is superior to the common heuristic of deleting any unlabeled patches. Learning a ‘CRFσ loc+glo’ model by removing all unlabeled patches (‘del unlabeled’ in the table) leads to an estimate σ ≈11.5, whereas the maximum likelihood estimate of (4) leads to σ ≈1.9. In particular, with ‘delete unlabeled’ training the accuracy of the model drops significantly for the classes plane and bike, both of which have a relatively small area relative to their boundaries and thus many partially labeled patches. It is interesting to note that even though σ has been severely over-estimated in the ‘delete unlabeled’ model, the CRF still improves over the individual patch classification obtained with ‘IND loc+glo’ for most classes, albeit not for bike and only marginally for plane. Recognition as function of the amount of labeling. We now consider how the performance drops as the fraction of labeled pixels decreases. We applied a morphological erosion operator to the manual annotations, where we varied the size of the disk-shaped structuring element from 0, 5, . . . , 50. 5 20 30 40 50 60 70 60 65 70 75 80 85 Percentage of pixels labeled Accuracy disc 0 disc 10 disc 20 CRFσ loc+glo IND loc+glo Figure 3: Recognition performance when learning from increasingly eroded label images (left). Example image with its original annotation, and erosions thereof with disk of size 10 and 20 (right). In this way we obtain a series of annotations that resemble increasingly sloppy manual annotations, see Figure 3. The figure also shows the recognition performance of ‘CRFσ loc+glo’ and ‘IND loc+glo’ as a function of the fraction of labeled pixels. In addition to its superior performance when trained on well labeled images, the CRF maintains its performance better as the labelling becomes sparser. Note that ‘CRFσ loc+glo’ learned from label images eroded with a disc of radius 30 (only 28% of pixels labeled) still outperforms ‘IND loc+glo’ learned from the original labeling (71% of pixels labeled). Also, the CRF actually performs better with 5 pixels of erosion than with the original labeling, presumably because ambiguities related to training patches with mixed pixel labels are reduced. Comparison with related work. Table 1 also compares our recognition results on the MSRC dataset with those reported in [15, 10]. Our CRF model clearly outperforms the approach of [15], which uses aggregate features of an optimized scale but lacks spatial coupling in a random field, giving a performance very similar to that of our ‘IND loc+glo’ model. Our CRF model also performs slightly better than our generative approach of [10], which is based on the same feature set but differs in its implementation of image-wide contextual information ([10] also used a 90%–10% training-test partition, not 50%-50% as here). Using the Sowerby dataset and a subset of the Corel dataset we also compare our model with two CRF models that operate at pixel-level. The Sowerby dataset consists of 104 images of 96 × 64 pixels of urban and rural scenes labeled with 7 different classes: sky, vegetation, road marking, road surface, building, street objects and cars. The subset of the Corel dataset contains 100 images of 180 × 120 pixels of natural scenes, also labeled with 7 classes: rhino/hippo, polar bear, water, snow, vegetation, ground, and sky. Here we used 10 × 10 pixel patches, with a spacing of respectively 2 and 5 pixels for the Sowerby and Corel datasets. The other parameters were kept as before. Table 2 compares the recognition accuracies averaged over pixels for our CRF and independent patch models to the results reported on these datasets for TextonBoost [7] and the multi-scale CRF model of [5]. In this table ‘IND’ stands for results obtained when only the single node potentials are used in the respective models, disregarding the spatial random field couplings. The total training time and test time per image are listed for the full CRF models. The results show that on these datasets our model performs comparably to pixel-level approaches while being much faster to train and test since it operates at patch-level and uses standard features as opposed to the boosting procedure of [7]. 5 Conclusion We presented several image-patch-level CRF models for semantic image labeling that incorporate both local patch-level observations and more global contextual features based on aggregates of observations at several scales. We showed that partially labeled training images could be handled by maximizing the total likelihood of the image segmentations that comply with the partial labeling, using Loopy BP and Bethe free-energy approximations for the calculations. This allowed us to learn effective CRF models from images where only a small fraction of the pixels were labeled and class transitions were not observed. Experiments on the MSRC dataset showed that including image6 Sowerby Corel Accuracy Speed Accuracy Speed IND CRF train test IND CRF train test TextonBoost [7] 85.6% 88.6% 5h 10s 68.4% 74.6% 12h 30s He et al. [5] CRF 82.4% 89.5% Gibbs Gibbs 66.9% 80.0% Gibbs Gibbs CRFσ loc+glo 86.0% 87.4% 20min 5s 66.9% 74.6 % 15min 3s Table 2: Recognition accuracy and speeds on the Corel and Sowerby dataset. wide aggregate features is very helpful, while including additional aggregates at finer scales gives relatively little further improvement. Comparative experiments showed that our patch-level CRFs have comparable performance to state-of-the-art pixel-level models while being much more efficient because the number of patches is much smaller than the number of pixels. References [1] P. Jansen, W. van der Mark, W. van den Heuvel, and F. Groen. Colour based off-road environment and terrain type classification. In Proceedings of the IEEE Conference on Intelligent Transportation Systems, pages 216–221, 2005. [2] S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6(6):712–741, 1984. [3] C. Rother, V. Kolmogorov, and A. Blake. GrabCut: interactive foreground extraction using iterated graph cuts. ACM Transactions on Graphics, 23(3):309–314, 2004. [4] J. Lafferty, A. McCallum, and F. Pereira. Conditional random fields: probabilistic models for segmenting and labeling sequence data. In Proceedings of the International Conference on Machine Learning, volume 18, pages 282–289, 2001. [5] X. He, R. Zemel, and M. Carreira-Perpi˜n´an. Multiscale conditional random fields for image labelling. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 695–702, 2004. [6] S. Kumar and M. Hebert. A hierarchical field framework for unified context-based classification. In Proceedings of the IEEE International Conference on Computer Vision, pages 1284–1291, 2005. [7] J. Shotton, J. Winn, C. Rother, and A. Criminisi. Textonboost: joint appearance, shape and context modeling for multi-class object recognition and segmentation. In Proceedings of the European Conference on Computer Vision, pages 1–15, 2006. [8] A. Quattoni, M. Collins, and T. Darrell. Conditional random fields for object recognition. In Advances in Neural Information Processing Systems, volume 17, pages 1097–1104, 2005. [9] P. Carbonetto, G. Dork´o, C. Schmid, H. K¨uck, and N. de Freitas. A semi-supervised learning approach to object recognition with spatial integration of local features and segmentation cues. In Toward CategoryLevel Object Recognition, pages 277–300, 2006. [10] J. Verbeek and B. Triggs. Region classification with Markov field aspect models. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2007. [11] D. Lowe. Distinctive image features from scale-invariant keypoints. International Journal of Computer Vision, 60(2):91–110, 2004. [12] J. van de Weijer and C. Schmid. Coloring local feature extraction. In Proceedings of the European Conference on Computer Vision, pages 334–348, 2006. [13] The 2005 PASCAL visual object classes challenge. In F. d’Alche-Buc, I. Dagan, and J. Quinonero, editors, Machine Learning Challenges: Evaluating Predictive Uncertainty, Visual Object Classification, and Recognizing Textual Entailment, First PASCAL Machine Learning Challenges Workshop. Springer, 2006. [14] J. Yedidia, W. Freeman, and Y. Weiss. Understanding belief propagation and its generalizations. Technical Report TR-2001-22, Mitsubishi Electric Research Laboratories, 2001. [15] F. Schroff, A. Criminisi, and A. Zisserman. Single-histogram class models for image segmentation. In Proceedings of the Indian Conference on Computer Vision, Graphics and Image Processing, 2006. 7 MSRC CRFσ loc+glo Labeling Sowerby CRFσ loc+glo Labeling Corel CRFσ loc+glo Labeling Figure 4: Samples from the MSRC, Sowerby, and Corel datasets with segmentation and labeling. 8
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Measuring Neural Synchrony by Message Passing Justin Dauwels Amari Research Unit RIKEN Brain Science Institute Wako-shi, Saitama, Japan justin@dauwels.com Franc¸ois Vialatte, Tomasz Rutkowski, and Andrzej Cichocki Advanced Brain Signal Processing Laboratory RIKEN Brain Science Institute Wako-shi, Saitama, Japan {fvialatte,tomek,cia}@brain.riken.jp Abstract A novel approach to measure the interdependence of two time series is proposed, referred to as “stochastic event synchrony” (SES); it quantifies the alignment of two point processes by means of the following parameters: time delay, variance of the timing jitter, fraction of “spurious” events, and average similarity of events. SES may be applied to generic one-dimensional and multi-dimensional point processes, however, the paper mainly focusses on point processes in time-frequency domain. The average event similarity is in that case described by two parameters: the average frequency offset between events in the time-frequency plane, and the variance of the frequency offset (“frequency jitter”); SES then consists of five parameters in total. Those parameters quantify the synchrony of oscillatory events, and hence, they provide an alternative to existing synchrony measures that quantify amplitude or phase synchrony. The pairwise alignment of point processes is cast as a statistical inference problem, which is solved by applying the maxproduct algorithm on a graphical model. The SES parameters are determined from the resulting pairwise alignment by maximum a posteriori (MAP) estimation. The proposed interdependence measure is applied to the problem of detecting anomalies in EEG synchrony of Mild Cognitive Impairment (MCI) patients; the results indicate that SES significantly improves the sensitivity of EEG in detecting MCI. 1 Introduction Synchrony is an important topic in neuroscience. For instance, it is hotly debated whether the synchronous firing of neurons plays a role in cognition [1] and even in consciousness [2]. The synchronous firing paradigm has also attracted substantial attention in both the experimental (e.g., [3]) and the theoretical neuroscience literature (e.g., [4]). Moreover, medical studies have reported that many neurophysiological diseases (such as Alzheimer’s disease) are often associated with abnormalities in neural synchrony [5, 6]. In this paper, we propose a novel measure to quantify the interdependence between point processes, referred to as “stochastic event synchrony” (SES); it consists of the following parameters: time delay, variance of the timing jitter, fraction of “spurious” events, and average similarity of the events. The pairwise alignment of point processes is cast as a statistical inference problem, which is solved by applying the max-product algorithm on a graphical model [7]. In the case of one-dimensional point processes, the graphical model is cycle-free and statistical inference is exact, whereas for 1 multi-dimensional point processes, exact inference becomes intractable; the max-product algorithm is then applied on a cyclic graphical model, which not necessarily yields the optimal alignment [7]. Our experiments, however, indicate that the it finds reasonable alignments in practice. The SES parameters are determined from the resulting pairwise alignments by maximum a posteriori (MAP) estimation. The proposed method may be helpful to detect mental disorders such as Alzheimer’s disease, since mental disorders are often associated with abnormal blood and neural activity flows, and changes in the synchrony of brain activity (see, e.g., [5, 6]). In this paper, we will present promising results on the early prediction of Alzheimer’s disease from EEG signals based on SES. This paper is organized as follows. In the next section, we introduce SES for the case of onedimensional point processes. In Section 3, we consider the extension to multi-dimensional point processes. In Section 4, we use our measure to detect abnormalities in the EEG synchrony of Alzheimer’s disease patients. 2 One-Dimensional Point Processes Let us consider the one-dimensional point processes (“event strings”) X and X′ in Fig. 1(a) (ignore Y and Z for now). We wish to quantify to which extent X and X′ are synchronized. Intuitively speaking, two event strings can be considered as synchronous (or “locked”) if they are identical apart from: (i) a time shift δt; (ii) small deviations in the event occurrence times (“event timing jitter”); (iii) a few event insertions and/or deletions. More precisely, for two event strings to be synchronous, the event timing jitter should be significantly smaller than the average inter-event time, and the number of deletions and insertions should comprise only a small fraction of the total number of events. This intuitive concept of synchrony is illustrated in Fig. 1(a). The event string X′ is obtained from event string X by successively shifting X over δt (resulting in Y ), slightly perturbing the event occurrence times (resulting in Z), and eventually, by adding (plus sign) and deleting (minus sign) events, resulting in X′. Adding and deleting events in Z leads to “spurious” events in X and X′ (see Fig. 1(a); spurious events are marked in red): a spurious event in X is an event that cannot be paired with an event in X′ and vice versa. The above intuitive reasoning leads to our novel measure for synchrony between two event strings, i.e., “stochastic event synchrony” (SES); for the one-dimensional case, it is defined as the triplet (δt, st, ρspur), where st is the variance of the (event) timing jitter, and ρspur is the percentage of spurious events ρspur △= nspur + n′ spur n + n′ , (1) with n and n′ the total number of events in X and X′ respectively, and nspur and n′ spur the total number of spurious events in X and X′ respectively. SES is related to the metrics (“distances”) proposed in [9]; those metrics are single numbers that quantify the synchrony between event strings. In contrast, we characterize synchrony by means of three parameters, which allows us to distinguish different types of synchrony (see [10]). We compute those three parameters by performing inference in a probabilistic model. In order to describe that model, we consider Fig. 1(b), which shows a symmetric procedure to generate X and X′. First, one generates an event string V of length ℓ, where the events Vk are mutually independent and uniformly distributed in [0, T0]. The strings Z and Z′ are generated by delaying V over −δt/2 and δt/2 respectively and by (slightly) perturbing the resulting event occurrence times (variance of timing jitter equals st/2). The sequences X and X′ are obtained from Z and Z′ by removing some of the events; more precisely, from each pair (Zk, Z′ k), either Zk or Z′ k is removed with probability ps. This procedure amounts to the statistical model: p(x, x′, b, b′, v, δt, st, ℓ) = p(x|b, v, δt, st)p(x′|b′, v, δt, st)p(b, b′|ℓ)p(v|ℓ)p(ℓ)p(δt)p(st), (2) where b and b′ are binary strings that indicate whether the events in X and X′ are spurious (Bk = 1 if Xk is spurious, Bk = 0 otherwise; likewise for B′ k); the length ℓhas a geometric prior p(ℓ) = (1 −λ)λℓwith λ ∈(0, 1), and p(v|ℓ) = T −ℓ 0 . The prior on the binary strings b and b′ is given by p(b, b′|ℓ) = (1 −ps)n+n′p2ℓ−n−n′ s = (1 −ps)n+n′p ntot spur s , (3) 2 with ntot spur = nspur + n′ spur = 2ℓ−n −n′ the total number of spurious events in X and X′, nspur = Pn k=1 bk = ℓ−n′ the number of spurious events in X, and likewise n′ spur, the number of spurious events in X′. The conditional distributions in X and X′ are equal to: p(x|b, v, δt, st) = n Y k=1 N xk −vik; −δt 2 , st 2 1−bk (4) p(x′|b′, v, δt, st) = n′ Y k=1 N x′ k −vi′ k; δt 2 , st 2 1−b′ k , (5) where Vik is the event in V that corresponds to Xk (likewise Vi′ k), and N(x; m, s) is a univariate Gaussian distribution with mean m and variance s. Since we do not wish/need to encode prior information about δt and st, we adopt improper priors p(δt) = 1 = p(st). Eventually, marginalizing (2) w.r.t. v results in the model: p(x, x′, b, b′, δt, st, ℓ) = Z p(x, x′, b, b′, v, δt, st, ℓ)dv ∝βntot spur nnon-spur Y k=1 N(x′ j′ k −xjk; δt, st), (6) with (xjk, x′ j′ k) the pairs of non-spurious events, nnon-spur = n + n′ −ℓthe total number of nonspurious event pairs, and β = ps q λ T0 ; in the example of Fig. 1(b), J = (1, 2, 3, 5, 6, 7, 8), J′ = (2, 3, 4, 5, 6, 7, 8), and nnon-spur = 7. In the following, we will denote model (6) by p(x, x′, j, j′, δt, st) instead of p(x, x′, b, b′, δt, st, ℓ), since for given x, x′, b, and b′ (and hence given n, n′, and nnon-spur), the length ℓis fully determined, i.e., ℓ= n + n′ −nnon-spur; moreover, it is more natural to describe the model in terms of J and J′ instead of B and B′ (cf. RHS of (6)). Note that B and B′ can directly be obtained from J and J′. It also noteworthy that T0, λ and ps do not need to be specified individually, since they appear in (6) only through β. The latter serves in practice as a knob to control the number of spurious events. δt X Y Z X′ (a) Asymmetric procedure T0 X V Z′ X′ 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 3 3 4 4 5 6 6 7 7 8 8 9 9 I I′ B B′ δt 2 δt 2 Z (b) Symmetric procedure Figure 1: One-dimensional stochastic event synchrony. Given event strings X and X′, we wish to determine the parameters δt and st, and the hidden variables B and B′; the parameter ρspur (cf. (1)) can obtained from the latter : ρspur △= Pn k=1 bk + Pn′ k=1 b′ k n + n′ . (7) There are various ways to solve this inference problem, but perhaps the most natural one is cyclic maximization: first one chooses initial values ˆδ(0) t and ˆs(0) t , then one alternates the following two update rules until convergence (or until the available time has elapsed): (ˆj(i+1), ˆj′(i+1)) = argmax b,b′ p(x, x′, j, j′, ˆδ(i) t , ˆs(i) t ) (8) (ˆδ(i+1) t , ˆs(i+1) t ) = argmax δt,st p(x, x′, ˆj(i+1), ˆj′(i+1), δt, st). (9) 3 The update (9) is straightforward, it amounts to the empirical mean and variance, computed over the non-spurious events. The update (8) can readily be carried out by applying the Viterbi algorithm (“dynamic programming”) on an appropriate trellis (with the pairs of non-spurious events (xjk, x′ j′ k) as states), or equivalently, by applying the max-product algorithm on a suitable factor graph [7]; the procedure is similar to dynamic time warping [8]. 3 Multi-Dimensional Point Processes In this section, we will focus on the interdependence of multi-dimensional point processes. As a concrete example, we will consider multi-dimensional point processes in time-frequency domain; the proposed algorithm, however, is not restricted to that particular situation, it is applicable to generic multi-dimensional point processes. Suppose that we are given a pair of (continuous-time) signals, e.g., EEG signals recorded from two different channels. As a first step, the time-frequency (“wavelet”) transform of each signal is approximated as a sum of (half-ellipsoid) basis functions, referred to as “bumps” (see Fig. 2 and [17]); each bump is described by five parameters: time X, frequency F, width ∆X, height ∆F, and amplitude W. The resulting bump models Y = ((X1, F1, ∆X1, ∆F1, W1), . . . , (Xn, Fn, ∆Xn, ∆Fn, Wn)) and Y ′ = ((X′ 1, F ′ 1, ∆X′ 1, ∆F ′ 1, W ′ 1), . . . , (X′ n′, F ′ n′, ∆X′ n′, ∆F ′ n′, W ′ n′)), representing the most prominent oscillatory activity, are thus 5-dimensional point processes. Our extension of stochastic event synchrony to multi-dimensional point processes (and bump models in particular) is derived from the following observation (see Fig. 3): bumps in one time-frequency map may not be present in the other map (“spurious” bumps); other bumps are present in both maps (“non-spurious bumps”), but appear at slightly different positions on the maps. The black lines in Fig. 3 connect the centers of non-spurious bumps, and hence, visualize the offset between pairs of non-spurious bumps. We quantify the interdependence between two bump models by five parameters, i.e., the parameters ρspur, δt, and st introduced in Section 2, in addition to: • δf: the average frequency offset between non-spurious bumps, • sf: the variance of the frequency offset between non-spurious bumps. We determine the alignment of two bump models in addition to the 5 above parameters by an inference algorithm similar to the one of Section 2, as we will explain in the following; we will use the notation θ = (δt, st, δf, sf). Model (6) may naturally be extended in time-frequency domain as: p(y, y′, j, j′, θ) ∝βntot spur nnon-spur Y k=1 N x′ k′ −xk ∆xk + ∆x′ k′ ; δt, st N f ′ k′ −fk ∆fk + ∆f ′ k′ ; δf, sf · p(δt)p st p(δf)p sf , (10) where the offset x′ k′ −xk in time and offset f ′ k′ −fk in frequency are normalized by the width and height respectively of the bumps; we will elaborate on the priors on the parameters θ later on. In principle, one may determine the sequences J and J′ and the parameters θ by cyclic maximization along the lines of (8) and (9). In the multi-dimensional case, however, the update (8) is no longer tractable: one needs to allow permutations of events, the indices jk and j′ k′ are no longer necessarily monotonically increasing, and as a consequence, the state space becomes drastically larger. As a result, the Viterbi algorithm (or equivalently, the max-product algorithm applied on cycle-free factor graph of model (10)) becomes impractical. We solve this problem by applying the max-product algorithm on a cyclic factor graph of the system at hand, which will amount to a suboptimal but practical procedure to obtain pairwise alignments of multi-dimensional point processes (and bump models in particular). To this end, we introduce a representation of model (10) that is naturally represented by a cyclic graph: for each pair of events Yk and Y ′ k′, we introduce a binary variable Ckk′ that equals one if Yk and Y ′ k′ form pair of nonspurious events and is zero otherwise. Since each event in Y associated to at most one event in Y ′, we have the constraints: n′ X k′=1 C1k′ △= S1 ∈{0, 1}, n′ X k′=1 C2k′ △= S2 ∈{0, 1}, . . ., n′ X k′=1 Cnk′ △= Sn ∈{0, 1}, (11) 4 and similarly, each event in Y ′ is associated to at most one event in Y , which is expressed by a similar set of constraints. The sequences S and S′ are related to the sequences B and B′ (cf. Section 2): Bk = 1 −Sk and B′ k = 1 −S′ k. In this representation, the global statistical model (10) can be cast as: p(y, y′, b, b′, c, θ) ∝ n Y k=1 (βδ[bk −1] + δ[bk]) n′ Y k′=1 (βδ[b′ k −1] + δ[b′ k]) · n Y k=1 n′ Y k′=1 N x′ k′ −xk ∆xk + ∆x′ k′ ; δt, st N f ′ k′ −fk ∆fk + ∆f ′ k′ ; δf, sf !ckk′ p(δt)p st p(δf)p sf · n Y k=1 δ[bk + n′ X k′=1 ckk′ −1] n′ Y k′=1 δ[b′ k′ + n X k=1 ckk′ −1] . (12) Since we do not need to encode prior information about δt and δf, we choose improper priors p(δt) = 1 = p(δf). On the other hand, we have prior knowledge about st and sf. Indeed, we expect a bump in one time-frequency map to appear in the other map at about the same frequency, but there may be some timing offset between both bumps. For example, bump nr. 1 in Fig. 3(a) (t = 10.7s) should be paired with bump nr. 3 (t = 10.9s) and not with nr. 2 (t = 10.8s), since the former is much closer in frequency than the latter. As a consequence, we a priori expect smaller values for sf than for st. We encode this prior information by means of conjugate priors for st and sf, i.e., scaled inverse chi-square distributions. A factor graph of model (14) is shown in Fig. 4 (each edge represents a variable, each node corresponds to a factor of (14), as indicated by the arrows at the right hand side; we refer to [7] for an introduction to factor graphs). We omitted the edges for the (observed) variables Xk, X′ k′, Fk, F ′ k′, ∆Xk, ∆X′ k′, ∆Fk, and ∆F ′ k′ in order not to clutter the figure. Time-frequency map Time-frequency map ↓ ↓ Bump model Bump model ⇔ Figure 2: Two-dimensional stochastic event synchrony. We determine the alignment C = (C11, C12, . . . , Cnn′) and the parameters θ = (δt, st, δf, sf) by maximum a posteriori (MAP) estimation: (ˆc, ˆθ) = argmax c,θ p(y, y′, c, θ), (13) where p(y, y′, c, θ) is obtained from (14) by marginalizing over b and b′: p(y, y′, c, θ) ∝ n Y k=1 βδ n′ X k′=1 ckk′ + δ n′ X k′=1 ckk′ −1 n′ Y k′=1 βδ n X k=1 ckk′ + δ n X k=1 ckk′ −1 · n Y k=1 n′ Y k′=1 N x′ k′ −xk ∆xk + ∆x′ k′ ; δt, st N f ′ k′ −fk ∆fk + ∆f ′ k′ ; δf, sf !ckk′ p(δt)p st p(δf)p sf . (14) 1 2 3 00 5 10 15 20 5 10 15 20 25 30 t [s] f [Hz] (a) Bump models of two EEG channels. 00 5 10 15 20 5 10 15 20 25 30 t [s] f [Hz] (b) Non-spurious bumps (ρspur = 27%); the black lines connect the centers of non-spurious bumps. Figure 3: Spurious and non-spurious activity. 5 ttt p(δt, st, δf, sf) = p(δt)p(st)p(δf)p(sf) δ[bn + Pn′ k′=1 cnk′ −1] δ[bn] + βδ[bn −1] N x′ n′−xn ∆xn+∆x′ n′ ; δt, st N f ′ n′−fn ∆fn+∆f ′ n′ ; δf, sf !cnn′ µ↑′ µ↑ µ↓′ µ↓ ˆθ(k) ˆθ(k) = = = = = = = = = = ¯Σ ¯Σ ¯Σ ¯Σ ¯Σ ¯Σ θ = (δt, st, δf, sf) N N N N N N N N N β β β β β β . . . . . . . . . . . . . . . . . . . . . . . . C11 C12 C1n′ C21 C22 C2n′ Cn1 Cn2 Cnn′ B1 B2 Bn B′ 1 B′ 2 B′ n′ Figure 4: Factor graph of model (14). From ˆc, we obtain the estimate ˆρspur as: ˆρspur = Pn k=1 ˆbk + Pn′ k=1 ˆb′ k′ n + n′ = n + n′ −2 Pn k=1 Pn′ k′=1 ˆckk′ n + n′ . (15) The MAP estimate (13) is intractable, and we try to obtain (13) by cyclic maximization: first, the parameters θ are initialized: ˆδ(0) t = 0 = δ(0) f , ˆs(0) t = ˆs0,t, and ˆs(0) f = s0,f, then one alternates the following two update rules until convergence (or until the available time has elapsed): ˆc(i+1) = argmax c p(y, y′, c, ˆθ(i)) (16) ˆθ(i+1) = argmax θ p(y, y′, ˆc(i+1), θ). (17) The estimate ˆθ(i+1) (17) is available in closed-form; indeed, it is easily verified that the point estimates ˆδ(i+1) t and ˆδ(i+1) f are the (sample) mean of the timing and frequency offset respectively, computed over all pairs of non-spurious events. The estimates ˆs(i+1) t and ˆs(i+1) f are obtained similarly. Update (16), i.e., finding the optimal pairwise alignment C for given values ˆθ(i) of the parameters θ, is less straightforward: it involves an intractable combinatorial optimization problem. We attempt to solve that problem by applying the max-product algorithm to the (cyclic) factor graph depicted in Fig. 4 [7]. Let us first point out that, since the alignment C is computed for given θ = ˆθ(i), the (upward) messages along the edges θ are the point estimate ˆθ(i) (cf. (16)); equivalently, for the purpose of computing (16), one may remove the θ edges and the two bottom nodes in Fig. 4; the N-nodes then become leaf nodes. The other messages in the graph are iteratively updated according to the generic max-product update rule [7]. The resulting inference algorithm for computing (16) is summarized in Table 1. The messages µ↑(ckk′) and µ↑′(ckk′) propagate upward along the edges ckk′ towards the ¯Σ-nodes connected to the edges Bk and B′ k′ respectively (see Fig. 4, left hand side); the messages µ↓(ckk′) and µ↓′(ckk′) propagate downward along the edges ckk′ from the ¯Σ-nodes connected to the edges Bk and B′ k′ respectively. After initialization (18) of the messages µ↑(ckk′) and µ↑′(ckk′) (k = 1, 2, ..., n; k′ = 1, 2, ..., n′), one alternatively updates (i) the messages µ↓(ckk′) (19) and µ↓′(ckk′) (20), (ii) the messages µ↑(ckk′) (21) and µ↑′(ckk′) (22), until convergence; it is noteworthy that, although the max-product algorithm is not guaranteed to converge on cyclic graphs, we observed in our experiments (see Section 4) that alternating the updates (19)–(22) always converged to a fixed point. At last, one computes the marginals p(ckk′) (23), and from the latter, one may determine the decisions ˆckk′ by greedy decimation. 4 Diagnosis of MCI from EEG We analyzed rest eyes-closed EEG data recorded from 21 sites on the scalp based on the 10–20 system. The sampling frequency was 200 Hz, and the signals were bandpass filtered between 4 6 Initialization µ↑(ckk′) = µ↑′(ckk′) ∝ N x′ k′ −xk ∆xk + ∆x′ k′ ; δt, st N f ′ k′ −fk ∆fk + ∆f ′ k′ ; δf, sf !ckk′ (18) Iteratively compute messages until convergence A. Downward messages: µ↓(ckk′ = 0) µ↓(ckk′ = 1) ∝ max (β, maxℓ′̸=k′ µ↑(ckℓ′ = 1)/µ↑(ckℓ′ = 0)) 1 (19) µ↓′(ckk′ = 0) µ↓′(ckk′ = 1) ∝ max (β, maxℓ̸=k µ↑′(cℓk′ = 1)/µ↑′(cℓk′ = 0)) 1 (20) B. Upward messages: µ↑(ckk′) ∝µ↓′(ckk′) N x′ k′ −xk ∆xk + ∆x′ k′ ; δt, st N f ′ k′ −fk ∆fk + ∆f ′ k′ ; δf, sf !ckk′ (21) µ↑′(ckk′) ∝µ↓(ckk′) N x′ k′ −xk ∆xk + ∆x′ k′ ; δt, st N f ′ k′ −fk ∆fk + ∆f ′ k′ ; δf, sf !ckk′ (22) Marginals p(ckk′) ∝µ↓(ckk′)µ↓′(ckk′) N x′ k′ −xk ∆xk + ∆x′ k′ ; δt, st N f ′ k′ −fk ∆fk + ∆f ′ k′ ; δf, sf !ckk′ (23) Table 1: Inference algorithm. and 30Hz. The subjects comprised two study groups: the first consisted of a group of 22 patients diagnosed as suffering from MCI, who subsequently developed mild AD. The other group was a control set of 38 age-matched, healthy subjects who had no memory or other cognitive impairments. Pre-selection was conducted to ensure that the data were of a high quality, as determined by the presence of at least 20s of artifact free data. We computed a large variety of synchrony measures for both data sets; the results are summarized in Table 2. We report results for global synchrony, obtained by averaging the synchrony measures over 5 brain regions (frontal, temporal left and right, central, occipital). For SES, the bump models were clustered by means of the aggregation algorithm described in [17]. The strongest observed effect is a significantly higher degree of background noise (ρspur) in MCI patients, more specifically, a high number of spurious, non-synchronous oscillatory events (p = 0.00021). We verified that the SES measures are not correlated (Pearson r) with other synchrony measures (p > 0.10); in contrast to the other measures, SES quantifies the synchrony of oscillatory events (instead of more conventional amplitude or phase synchrony). Combining ρspur with ffDTF yields good classification of MCI vs. Control patients (see Fig.5(a)). Interestingly, we did not observe a significant effect on the timing jitter st of the non-spurious events (p = 0.91). In other words, AD seems to be associated with a significant increase of spurious background activity, while the non-spurious activity remains well synchronized. Moreover, only the non-spurious activity slows down (p = 0.0012; see Fig.5(c)), the average frequency of the spurious activity is not affected in MCI patients (see Fig.5(c)). In future work, we will verify those observations by means of additional data sets. Measure Cross-correlation Coherence Phase Coherence Corr-entropy Wave-entropy p-value 0.028∗ 0.060 0.72 0.27 0.012∗ References [16] [18] [20] Measure Granger coherence Partial Coherence PDC DTF ffDTF dDTF p-value 0.15 0.16 0.60 0.34 0.0012∗∗ 0.030∗ References [13] Measure Kullback-Leibler R´enyi Jensen-Shannon Jensen-R´enyi IW I p-value 0.072 0.076 0.084 0.12 0.060 0.080 References [23] [22] Measure N k Sk Hk S-estimator p-value 0.032∗ 0.29 0.090 0.33 References [15] [21] Measure Hilbert Phase Wavelet Phase Evolution Map Instantaneous Period p-value 0.15 0.082 0.072 0.020∗ References [24] [19] Measure st ρspur p-value 0.91 0.00021∗∗ Table 2: Sensitivity of synchrony measures for early prediction of AD (p-values for Mann-Whitney test; * and ** indicate p < 0.05 and p < 0.005 respectively). N k, Sk, and Hk are three measures of nonlinear interdependence [15]. 7 0.045 0.05 0.055 0.06 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 MCI CTR ρspur F 2 ij (a) ρspur vs. ffDTF 12 13 14 15 16 17 18 19 fspur CTR MCI (b) Av. frequency of the spurious activity (p = 0.87) 12 13 14 15 16 17 18 19 fnon-spur CTR MCI (c) Av. frequency of the nonspurious activity (p = 0.0019) Figure 5: Results. References [1] F. Varela, J. P. Lachaux, E. Rodriguez, and J. Martinerie, “The Brainweb: Phase Synchronization and Large-Scale Integration”, Nature Reviews Neuroscience, 2(4):229–39, 2001. [2] W. Singer, “Consciousness and the Binding Problem,” Annals of the New York Academy of Sciences, 929:123–146, April 2001. [3] M. Abeles, H. Bergman, E. Margalit, and E. Vaadia, “Spatiotemporal Firing Patterns in the Frontal Cortex of Behaving Monkeys,” J. Neurophysiol, 70(4):1629–1638. 1993. [4] S. Amari, H. Nakahara, S. Wu, and Y. 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Robinson, “The Biophysical Basis of Firing Variability in Cortical Neurons,” Chapter 6 in Computational Neuroscience: A Comprehensive Approach, Mathematical Biology & Medicine Series, Edited By Jianfeng Feng, Chapman & Hall/CRC, 2003. [11] E. Pereda, R. Q. Quiroga, and J. Bhattacharya, “Nonlinear Multivariate Analysis of Neurophsyiological Signals,” Progress in Neurobiology, 77 (2005) 1–37. [12] M. Breakspear, “Dynamic Connectivity in Neural Systems: Theoretical and Empirical Considerations,” Neuroinformatics, vol. 2, no. 2, 2004. [13] M. Kami´nski and Hualou Liang, “Causal Influence: Advances in Neurosignal Analysis,” Critical Review in Biomedical Engineering, 33(4):347–430 (2005). [14] C. J. Stam, “Nonlinear Dynamical Analysis of EEG and MEG: Review of an Emerging Field,” Clinical Neurophysiology 116:2266–2301 (2005). [15] R. Q. Quiroga, A. Kraskov, T. Kreuz, and P. Grassberger, “Performance of Different Synchronization Measures in Real Data: A Case Study on EEG Signals,” Physical Review E, vol. 65, 2002. [16] P. Nunez and R. Srinivasan, Electric Fields of the Brain: The Neurophysics of EEG, Oxford University Press, 2006. [17] F. Vialatte, C. Martin, R. Dubois, J. Haddad, B. Quenet, R. Gervais, and G. Dreyfus, “A Machine Learning Approach to the Analysis of Time-Frequency Maps, and Its Application to Neural Dynamics,” Neural Networks, 2007, 20:194–209. [18] Jian-Wu Xu, H. Bakardjian, A. Cichocki, and J. C. Principe, “EEG Synchronization Measure: a Reproducing Kernel Hilbert Space Approach,” submitted to IEEE Transactions on Biomedical Engineering Letters, Sept. 2006. [19] M. G. Rosenblum, L. Cimponeriu, A. Bezerianos, A. Patzak, and R. Mrowka, “Identification of Coupling Direction: Application to Cardiorespiratory Interaction,” Physical Review E, 65 041909, 2002. [20] C. S. Herrmann, M. Grigutsch, and N. A. Busch, “EEG Oscillations and Wavelet Analysis,” in Todd Handy (ed.) Event-Related Potentials: a Methods Handbook, pp. 229-259, Cambridge, MIT Press, 2005. [21] C. Carmeli, M. G. Knyazeva, G. M. Innocenti, and O. De Feo, “Assessment of EEG Synchronization Based on State-Space Analysis,” Neuroimage, 25:339–354 (2005). [22] A. Kraskov, H. St¨ogbauer, and P. Grassberger, “Estimating Mutual Information,” Phys. Rev. E 69 (6) 066138, 2004. [23] S. Aviyente, “A Measure of Mutual Information on the Time-Frequency Plane,” Proc. of ICASSP 2005, vol. 4, pp. 481–484, March 18–23, 2005, Philadelphia, PA, USA. [24] J.-P. Lachaux, E. Rodriguez, J. Martinerie, and F. J. Varela, “Measuring Phase Synchrony in Brain Signals,” Human Brain Mapping 8:194208 (1999). 8
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Discriminative Batch Mode Active Learning Yuhong Guo and Dale Schuurmans Department of Computing Science University of Alberta {yuhong, dale}@cs.ualberta.ca Abstract Active learning sequentially selects unlabeled instances to label with the goal of reducing the effort needed to learn a good classifier. Most previous studies in active learning have focused on selecting one unlabeled instance to label at a time while retraining in each iteration. Recently a few batch mode active learning approaches have been proposed that select a set of most informative unlabeled instances in each iteration under the guidance of heuristic scores. In this paper, we propose a discriminative batch mode active learning approach that formulates the instance selection task as a continuous optimization problem over auxiliary instance selection variables. The optimization is formulated to maximize the discriminative classification performance of the target classifier, while also taking the unlabeled data into account. Although the objective is not convex, we can manipulate a quasi-Newton method to obtain a good local solution. Our empirical studies on UCI datasets show that the proposed active learning is more effective than current state-of-the art batch mode active learning algorithms. 1 Introduction Learning a good classifier requires a sufficient number of labeled training instances. In many circumstances, unlabeled instances are easy to obtain, while labeling is expensive or time consuming. For example, it is easy to download a large number of webpages, however, it typically requires manual effort to produce classification labels for these pages. Randomly selecting unlabeled instances for labeling is inefficient in many situations, since non-informative or redundant instances might be selected. Hence, active learning (i.e., selective sampling) methods have been adopted to control the labeling process in many areas of machine learning, with the goal of reducing the overall labeling effort. Given a large pool of unlabeled instances, active learning provides a way to iteratively select the most informative unlabeled instances—the queries—to label. This is the typical setting of poolbased active learning. Most active learning approaches, however, have focused on selecting only one unlabeled instance at one time, while retraining the classifier on each iteration. When the training process is hard or time consuming, this repeated retraining is inefficient. Furthermore, if a parallel labeling system is available, a single instance selection system can make wasteful use of the resource. Thus, a batch mode active learning strategy that selects multiple instances each time is more appropriate under these circumstances. Note that simply using a single instance selection strategy to select more than one unlabeled instance in each iteration does not work well, since it fails to take the information overlap between the multiple instances into account. Principles for batch mode active learning need to be developed to address the multi-instance selection specifically. In fact, a few batch mode active learning approaches have been proposed recently [2, 8, 9, 17, 19]. However, most extend existing single instance selection strategies into multi-instance selection simply by using a heuristic score or greedy procedure to ensure both the instance diversity and informativeness. In this paper, we propose a new discriminative batch mode active learning strategy that exploits information from an unlabeled set to attempt to learn a good classifier directly. We define a good classifier to be one that obtains high likelihood on the labeled training instances and low uncertainty on labels of the unlabeled instances. We therefore formulate the instance selection problem as an optimization problem with respect to auxiliary instance selection variables, taking a combination of discriminative classification performance and label uncertainty as the objective function. Unfortunately, this optimization problem is NP-hard, thus seeking the optimal solution is intractable. However, we can approximate it locally using a second order Taylor expansion and obtain a suboptimal solution using a quasi-Newton local optimization technique. The instance selection variables we introduce can be interpreted as indicating self-supervised, optimistic guesses for the labels of the selected unlabeled instances. A concern about the instance selection process, therefore, is that some information in the unlabeled data that is inconsistent with the true classification partition might mislead instance selection. Fortunately, the active learning method can immediately tell whether it has been misled, by comparing the true labels with its optimized guesses. Therefore, one can then adjust the active selection strategy to avoid such over-fitting in the next iteration, whenever a mismatch between the labeled and unlabeled data has been detected. An empirical study on UCI datasets shows that the proposed batch mode active learning method is more effective than some current state-of-the-art batch mode active learning algorithms. 2 Related Work Many researchers have addressed the active learning problem in a variety of ways. Most have focused on selecting a single most informative unlabeled instance to label at a time. Many such approaches therefore make myopic decisions based solely on the current learned classifier, and select the unlabeled instance for which there is the greatest uncertainty. [10] chooses the unlabeled instance with conditional probability closest to 0.5 as the most uncertain instance. [5] takes the instance on which a committee of classifiers disagree the most. [3, 18] suggest choosing the instance closest to the classification boundary, where [18] analyzes this active learning strategy as a version space reduction process. Approaches that exploit unlabeled data to provide complementary information for active learning have also been proposed. [4, 20] exploit unlabeled data by using the prior density p(x) as uncertainty weights. [16] selects the instance that optimizes the expected generalization error over the unlabeled data. [11] uses an EM approach to integrate information from unlabeled data. [13, 22] consider combining active learning with semi-supervised learning. [14] presents a mathematical model that explicitly combines clustering and active learning. [7] presents a discriminative approach that implicitly exploits the clustering information contained in the unlabeled data by considering optimistic labelings. Since single instance selection strategies require tedious retraining with each instance labeled (and, moreover, since they cannot take advantage of parallel labeling systems), many batch mode active learning methods have recently been proposed. [2, 17, 19] extend single instance selection strategies that use support vector machines. [2] takes the diversity of the selected instances into account, in addition to individual informativeness. [19] proposes a representative sampling approach that selects the cluster centers of the instances lying within the margin of a support vector machine. [8, 9] choose multiple instances that efficiently reduce the Fisher information. Overall, these approaches use a variety of heuristics to guide the instance selection process, where the selected batch should be informative about the classification model while being diverse enough so that their information overlap is minimized. Instead of using heuristic measures, in this paper, we formulate batch mode active learning as an optimization problem that aims to learn a good classifier directly. Our optimization selects the best set of unlabeled instances and their labels to produce a classifier that attains maximum likelihood on labels of the labeled instances while attaining minimum uncertainty on labels of the unlabeled instances. It is intractable to conduct an exhaustive search for the optimal solution; our optimization problem is NP-hard. Nevertheless we can exploit a second-order Taylor approximation and use a quasi-Newton optimization method to quickly reach a local solution. Our proposed approach provides an example of exploiting optimization techniques in batch model active learning research, much like other areas of machine learning where optimization techniques have been widely applied [1]. 3 Logistic Regression In this paper, we use binary logistic regression as the base classification algorithm. Logistic regression is a well-known and mature statistical model for probabilistic classification that has been actively studied and applied in machine learning. Given a test instance x, binary logistic regression models the conditional probability of the class label y ∈{+1, −1} by p(y|x, w) = 1 1 + exp(−yw⊤x) where w is the model parameter. Here the bias term is omitted for simplicity of notation. The model parameters can be trained by maximizing the likelihood of the labeled training data, i.e., minimizing the logloss of the training instances min w X i∈L log(1 + exp(−yiw⊤xi)) + λ 2 w⊤w (1) where L indexes the training instances, and λ 2 w⊤w is a regularization term introduced to avoid over-fitting problems. Logistic regression is a robust classifier that can be trained efficiently using various convex optimization techniques [12]. Although it is a linear classifier, it is easy to obtain nonlinear classifications by simply introducing kernels [21]. 4 Discriminative Batch Mode Active Learning For active learning, one typically encounters a small number of labeled instances and a large number of unlabeled instances. Instance selection strategies based only on the labeled data therefore ignore potentially useful information embodied in the unlabeled instances. In this section, we present a new discriminative batch mode active learning algorithm for binary classification that exploits information in the unlabeled instances. The proposed approach is discriminative in the sense that (1) it selects a batch of instances by optimizing a discriminative classification model; and (2) it selects instances by considering the best discriminative configuration of their labels leading to the best classifier. Unlike other batch mode active learning methods, which identify the most informative batch of instances using heuristic measures, our approach aims to identify the batch of instances that directly optimizes classification performance. 4.1 Optimization Problem An optimal active learning strategy selects a set of instances to label that leads to learning the best classifier. We assume the learner selects a set of a fixed size m, which is chosen as a parameter. Supervised learning methods typically maximize the likelihood of training instances. With unlabeled data being available, semi-supervised learning methods have been proposed that train by simultaneously maximizing the likelihood of labeled instances and minimizing the uncertainty of the labels for unlabeled instances [6]. That is, to achieve a classifier with better generalization performance, one can maximizing the expected log likelihood of the labeled data and minimize the entropy of the missing labels on the unlabeled data, according to X i∈L log P(yi|xi, w) + α X j∈U X y=±1 P(y|xj, w) log P(y|xj, w) (2) where α is a tradeoff parameter used to adjust the relative influence of the labeled and unlabeled data, w specifies the conditional model, L indexes the labeled instances, and U indexes the unlabeled instances. The new active learning approach we propose is motivated by this semi-supervised learning principle. We propose to select a batch of m unlabeled instances, S, to label in each iteration from the total unlabeled set U , with the goal of maximizing the objective (2). Specifically, we define the score function for a set of selected instances S in iteration t + 1 as follows f(S) = X i∈Lt∪S log P(yi|xi, wt+1) −α X j∈U t\S H(y|xj, wt+1) (3) where wt+1 is the parameter set for the conditional classification model trained on the new labeled set Lt+1 = Lt ∪S, and H(y|xj, wt+1) denotes the entropy of the conditional distribution P(y|xj, wt+1), such that H(y|xj, wt+1) = − X y=±1 P(y|xj, wt+1) log P(y|xj, wt+1) The proposed active learning strategy is to select the batch of instances that has the highest score. In practice, however it is problematic to use the f(S) score directly to guide instance selection: the labels for instances S are not known when the selection is conducted. One typical solution for this problem is to use the expected f(S) score computed under the current conditional model specified by wt E[f(S)] = X yS P(yS|xS, wt)f(S) However, using P(yS|xS, wt) as weights, this expectation might aggravate any ambiguity that already exists in the current classification model wt, since it has been trained on a very small labeled set Lt. Instead, we propose an optimistic strategy: use the best f(S) score that the batch of unlabeled instances S can achieve over all possible label configurations. This optimistic scoring function can be written as f(S) = max yS X i∈Lt∪S log P(yi|xi, wt+1) −α X j∈U t\S H(y|xj, wt+1) (4) Thus the problem becomes how to select a set of instances S that achieves the best optimistic f(S) score defined in (4). Although this problem can be solved using an exhaustive search on all size m subsets, S, of the unlabeled set U, it is intractable to do so in practice since the search space is exponentially large. Explicit heuristic search approaches seeking a local optima do not exist either, since it is hard to define an efficient set of operators that can transfer from one position to another one within the search space while guaranteeing improvements to the optimistic score. Instead, in this paper we propose to approach the problem by formulating optimistic batch mode active learning as an explicit mathematical optimization. Given the labeled set Lt and unlabeled set U t after iteration t, the task in iteration t + 1 is to select a size m subset S from U t that achieves the best score defined in (4). To do so, we first introduce a set of {0, 1}-valued instance selection variables µ. In particular, µ is a |U t| × 2 sized indicator matrix, where each row vector µj corresponds to the two possible labels {+1, −1} of the jth instance in U t. Then the optimistic instance selection for iteration t + 1 can be formulated as the following optimization problem max µ X i∈Lt log P(yi|xi, wt+1) + β X j∈U t vt+1 j µ⊤ j −α X j∈U t (1 −µje)H(y|xj, wt+1) (5) s.t. µ ∈{0, 1}|U t|×2 (6) µ • E = m (7) µje ≤1, ∀j (8) 1⊤µ ≤ 1 2 + ϵ me⊤ (9) where vt+1 j is a row vector [log P(y = 1|xj, wt+1), log P(y = −1|xj, wt+1)]; e is a 2-entry column vector with all 1s; 1 is a |U t|-entry column vector with all 1s; E is a U t × 2 sized matrix with all 1s; • is matrix inner product; ϵ is a user-provided parameter that controls class balance during instance selection; and β is a parameter that we will use later to adjust our belief in the guessed labels. Note that, the selection variables µ not only choose instances from U t, but also select labels for the selected instances. Solving this optimization yields the optimal µ for instance selection in iteration t + 1. The optimization problem (5) is an integer programming problem that produces equivalent results to using exhaustive search to optimize (4), except that we have additional class balance constraints (9). Integer programming is an NP-hard problem. Thus, the first step toward solving this problem in practice is to relax it into a continuous optimization by replacing the integer constraints (6) with continuous constraints 0 ≤µ ≤1, yielding the relaxed formulation max µ X i∈Lt log P(yi|xi, wt+1) + β X j∈U t vt+1 j µ⊤ j −α X j∈U t (1 −µje)H(y|xj, wt+1) (10) s.t. 0 ≤µ ≤1 (11) µ • E = m (12) µje ≤1, ∀j (13) 1⊤µ ≤ 1 2 + ϵ me⊤ (14) If we can solve this continuous optimization problem, a greedy strategy can then be used to recover the integer solution by iteratively setting the largest non-integer µ value to 1 with respect to the constraints. However, this relaxed optimization problem is still very complex: the objective function (10) is not a concave function of µ.1 Nevertheless, standard continuous optimization techniques can be used to solve for a local maxima. 4.2 Quasi-Newton Method To derive a local optimization technique, consider the objective function (10) as a function of the instance selection variables µ f(µ) = X i∈Lt log P(yi|xi, wt+1) + β X j∈U t vt+1 j µ⊤ j −α X j∈U t (1 −µje)H(y|xj, wt+1) (15) As noted, this function is non-concave, therefore convenient convex optimization techniques that achieve global optimal solutions cannot be applied. Nevertheless, a local optimization approach exploiting quasi-Newton methods can quickly determine a local optimal solution µ∗. Such a local optimization approach iteratively updates µ to improve the objective (15), and stops when a local maximum is reached. At each iteration, it makes a local move that allows it to achieve the largest improvement in the objective function along the direction decided by cumulative information obtained from the sequence of local gradients. Suppose ¯µ(k) is the starting point for iteration k. We first derive a second-order Taylor approximation ˜f(µ) for the objective function f(µ) at ¯µ(k) ˜f(µ) = f(¯µ(k)) + ∇f ⊤ k vec(µ −¯µ(k)) + 1 2vec(µ −¯µ(k))⊤Hk vec(µ −¯µ(k)) (16) where vec(·) is a function that transforms a matrix into a column vector, and ∇fk = ∇f(¯µ(k)) and Hk denote the gradient vector and Hessian matrix of f(µ) at point ¯µ(k), respectively. Since our original optimization function f(µ) is smooth, the quadratic function ˜f(µ) can reasonably approximate it in a small neighborhood of ¯µ(k). Thus we can determine our update direction by solving a quadratic programming with the objective (16) and linear constraints (11), (12), (13) and (14). Suppose the optimal solution for this quadratic program is ˜µ∗ (k). Then a reasonable update direction dk = ˜µ∗ (k)−¯µ(k) can be obtained for iteration k. Given this direction, a backtrack line search can be used to guarantee improvement over the original objective (15). Note that for each different value of µ, wt+1 has to be retrained on Lt ∪S to evaluate the new objective value, since S is determined by µ. In order to reduce the computational cost, we approximate the training of wt+1 in our empirical study, by limiting it to a few Newton-steps with a starting point given by wt trained only on Lt. The remaining issue is to compute the local gradient ∇f(¯µ(k)) and the Hessian matrix Hk. We assume wt+1 remains constant with small local updates on ¯µ. Thus the local gradient can be approximated as ∇f(¯µj(k)) = β vt+1 j + α [H(y|xj, wt+1), H(y|xj, wt+1)] and therefore ∇f(¯µ(k)) can be constructed from the individual ∇f(¯µj(k)). We then use BFGS (Broyden-Fletcher-Goldfarb-Shanno) to compute the Hessian matrix, which starts as an identity matrix for the first iteration, and is updated in each iteration as follows [15] Hk+1 = Hk −Hksks⊤ k Hk s⊤ k Hksk + yky⊤ k y⊤ k sk 1Note that wt+1 is the classification model parameter set trained on Lt+1 = Lt ∪S, where S indexes the unlabeled instances selected by µ. Therefore wt+1 is a function of µ. where yk = ∇fk+1 −∇fk, and sk = ¯µ(k+1) −¯µ(k). This Hessian matrix accumulates information from the sequences of local gradients to help determine better update directions. 4.3 Adjustment Strategy In the discriminative optimization problem formulated in Section 4.1, the µ variables are used to optimistically select both instances and their labels, with the goal of achieving the best classification model according to the objective (5). However, when the labeled set is small and the discriminative partition (clustering) information contained in the large unlabeled set is inconsistent with the true classification, the labels optimistically guessed for the selected instances through µ might not match the underlying true labels. When this occurs, the instance selected will not be very useful for identifying the true classification model. Furthermore, the unlabeled data might continue to mislead the next instance selection iteration. Fortunately, we can immediately identify when the process has been misled once the true labels for the selected instances have been obtained. If the true labels are different from the labels guessed by the optimization, we need to make an adjustment for the next instance selection iteration. We have tried a few adjustment strategies in our study, but report the most effective one in this paper. Note that the being-misled problem is caused by the unlabeled data, which affects the target classification model through the term β P j∈U t vt+1 j µ⊤ j . Therefore, a simple way to fix the problem is to adjust the parameter β. Specifically, at the end of each iteration t, we obtain the true labels yS for the selected instances S, and compare them with our guessed labels ˆyS indicated by µ∗. If they are consistent, we will set β = 1, which means we trust the partition information from the unlabeled data as same as the label information in the labeled data for building the classification model. If yS ̸= ˆyS, apparently we should reduce the β value, that is, reducing the influence of the unlabeled data for the next selection iteration t + 1. We use a simple heuristic procedure to determine the β value in this case. Starting from β = 1, we then multiplicatively reduce its value by a small factor, 0.5, until a better objective value for (15) can be obtained when replacing the guessed indicator variables µ∗with the true label indicators. Note that, if we reduce β to zero, our optimization problem will be exactly equivalent to picking the most uncertain instance (when m = 1). 5 Experiments To investigate the empirical performance of the proposed discriminative batch mode active learning algorithm (Discriminative), we conducted a set of experiments on nine two-class UCI datasets, comparing with a baseline random instance selection algorithm (Random), a non-batch myopic active learning method that selects the most uncertain instance each time (MostUncertain), and two batch mode active learning methods proposed in the literature: svmD, an approach that incorporates diversity in active learning with SVM [2]; and Fisher, an approach that uses Fisher information matrix for instance selection [9]. The UCI datasets we used include (we show the name, followed by the number of instances and the number of attributes): Australian(690;14), Cleve(303;13), Corral(128;6), Crx(690;15), Flare(1066;10), Glass2(163;9), Heart(270;13), Hepatitis(155;20) and Vote(435;15). We consider a hard case of active learning, where only a few labeled instances are given at the start. In each experiment, we start with four randomly selected labeled instances, two in each class. We then randomly select 2/3 of the remaining instances as the unlabeled set, using the remaining instances for testing. All the algorithms start with the same initial labeled set, unlabeled set and testing set. For a fixed batch size m, each algorithm repeatedly select m instances to label each time. In this section, we report the experimental results with m = 5, averaged over 20 times repetitions. Figure 1 shows the comparison results on the nine UCI datasets. These results suggest that although the baseline random sampling method, Random, works surprisingly well in our experiments, the proposed algorithm, Discriminative, always performs better or at least achieves a comparable performance. Moreover, Discriminative also apparently outperforms the other two batch mode algorithms, svmD and Fisher, on five datasets—Australian, Cleve, Flare, Heart and Hepatitis, and reaches a tie on two datasets—Crx and Vote. The myopic most uncertain selection method, MostUncertain, shows an overall inferior performance to Discriminative on Australian, Cleve, Crx, Heart and Hepatitis, and achieves a tie on Flare and Vote. However, Discriminative demonstrates weak perfor0 20 40 60 80 100 0.55 0.6 0.65 0.7 0.75 0.8 0.85 Number of Labeled Instances Accuracy australian Random MostUncertain svmD Fisher Discriminative 0 20 40 60 80 100 0.55 0.6 0.65 0.7 0.75 0.8 0.85 Number of Labeled Instances Accuracy cleve Random MostUncertain svmD Fisher Discriminative 0 10 20 30 40 50 60 70 80 0.7 0.75 0.8 0.85 0.9 Number of Labeled Instances Accuracy corral Random MostUncertain svmD Fisher Discriminative 0 20 40 60 80 100 0.55 0.6 0.65 0.7 0.75 0.8 0.85 Number of Labeled Instances Accuracy crx Random MostUncertain svmD Fisher Discriminative 0 20 40 60 80 100 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 Number of Labeled Instances Accuracy flare Random MostUncertain svmD Fisher Discriminative 0 20 40 60 80 100 0.5 0.55 0.6 0.65 0.7 0.75 Number of Labeled Instances Accuracy glass2 Random MostUncertain svmD Fisher Discriminative 0 20 40 60 80 100 0.55 0.6 0.65 0.7 0.75 0.8 0.85 Number of Labeled Instances Accuracy heart Random MostUncertain svmD Fisher Discriminative 0 5 10 15 20 25 30 35 40 45 0.6 0.65 0.7 0.75 0.8 0.85 Number of Labeled Instances Accuracy hepatitis Random MostUncertain svmD Fisher Discriminative 0 20 40 60 80 100 0.8 0.85 0.9 Number of Labeled Instances Accuracy vote Random MostUncertain svmD Fisher Discriminative Figure 1: Results on UCI Datasets mance on two datasets—Corral and Glass2, where the evaluation lines for most algorithms in the figures are strangely very bumpy. The reason behind this remains to be investigated. These empirical results suggest that selecting unlabeled instances through optimizing the classification model directly would obtain more relevant and informative instances, comparing with using heuristic scores to guide the selection. Although the original optimization problem formulated is NP-hard, a relaxed local optimization method that leads to a local optimal solution still works effectively. 6 Conclusion In this paper, we proposed a discriminative batch mode active learning approach that exploits information in unlabeled data and selects a batch of instances by optimizing the target classification model. Although the proposed technique could be overly optimistic about the information presented by the unlabeled set, and consequently be misled, this problem can be identified immediately after obtaining the true labels. A simple adjustment strategy can then be used to rectify the problem in the following iteration. Experimental results on UCI datasets show that this approach is generally more effective comparing with other batch mode active learning methods, a random sampling method, and a myopic non-batch mode active learning method. Our current work is focused on 2-class classification problems, however, it is easy to be extended to multiclass classification problems. References [1] K. Bennett and E. Parrado-Hernandez. The interplay of optimization and machine learning research. Journal of Machine Learning Research, 7, 2006. [2] K. Brinker. Incorporating diversity in active learning with support vector machines. In Proceedings of the 20th International Conference on Machine learning, 2003. [3] C. Campbell, N. Cristianini, and A. Smola. Query learning with large margin classifiers. In Proceedings of the 17th International Conference on Machine Learning, 2000. [4] D. Cohn, Z. Ghahramani, and M. Jordan. Active learning with statistical models. Journal of Artificial Intelligence Research, 4, 1996. 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In Proceedings of the 18th International Conference on Machine Learning, 2001. [17] G. Schohn and D. Cohn. Less is more: Active learning with support vector machines. In Proceedings of the 17th International Conference on Machine Learning, 2000. [18] S. Tong and D. Koller. Support vector machine active learning with applications to text classification. In Proceedings of the 17th International Conference on Machine Learning, 2000. [19] Z. Xu, K. Yu, V. Tresp, X. Xu, and J. Wang. Representative sampling for text classification using support vector machines. In Proceedings of the 25th European Conference on Information Retrieval Research, 2003. [20] C. Zhang and T. Chen. An active learning framework for content-based information retrieval. IEEE Trans on Multimedia, 4:260–258, 2002. [21] J. Zhu and T. Hastie. Kernel logistic regression and the import vector machine. Journal of Computational and Graphical Statistics, 14, 2005. [22] X. Zhu, J. Lafferty, and Z. Ghahramani. 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Comparing Bayesian models for multisensory cue combination without mandatory integration Ulrik R. Beierholm Computation and Neural Systems California Institute of Technology Pasadena, CA 91025 beierh@caltech.edu Konrad P. K¨ording Rehabilitation Institute of Chicago Northwestern University, Dept. PM&R Chicago, IL 60611 konrad@koerding.com Ladan Shams Department of Psychology University of California, Los Angeles Los Angeles, CA 90095 ladan@psych.ucla.edu Wei Ji Ma Department of Brain and Cognitive Sciences University of Rochester Rochester, NY 14620 weijima@gmail.com Abstract Bayesian models of multisensory perception traditionally address the problem of estimating an underlying variable that is assumed to be the cause of the two sensory signals. The brain, however, has to solve a more general problem: it also has to establish which signals come from the same source and should be integrated, and which ones do not and should be segregated. In the last couple of years, a few models have been proposed to solve this problem in a Bayesian fashion. One of these has the strength that it formalizes the causal structure of sensory signals. We first compare these models on a formal level. Furthermore, we conduct a psychophysics experiment to test human performance in an auditory-visual spatial localization task in which integration is not mandatory. We find that the causal Bayesian inference model accounts for the data better than other models. Keywords: causal inference, Bayesian methods, visual perception. 1 Multisensory perception In the ventriloquist illusion, a performer speaks without moving his/her mouth while moving a puppet’s mouth in synchrony with his/her speech. This makes the puppet appear to be speaking. This illusion was first conceptualized as ”visual capture”, occurring when visual and auditory stimuli exhibit a small conflict ([1, 2]). Only recently has it been demonstrated that the phenomenon may be seen as a byproduct of a much more flexible and nearly Bayes-optimal strategy ([3]), and therefore is part of a large collection of cue combination experiments showing such statistical near-optimality [4, 5]. In fact, cue combination has become the poster child for Bayesian inference in the nervous system. In previous studies of multisensory integration, two sensory stimuli are presented which act as cues about a single underlying source. For instance, in the auditory-visual localization experiment by Alais and Burr [3], observers were asked to envisage each presentation of a light blob and a sound click as a single event, like a ball hitting the screen. In many cases, however, the brain is not only posed with the problem of identifying the position of a common source, but also of determining whether there was a common source at all. In the on-stage ventriloquist illusion, it is indeed primarily the causal inference process that is being fooled, because veridical perception would attribute independent causes to the auditory and the visual stimulus. 1 To extend our understanding of multisensory perception to this more general problem, it is necessary to manipulate the degree of belief assigned to there being a common cause within a multisensory task. Intuitively, we expect that when two signals are very different, they are less likely to be perceived as having a common source. It is well-known that increasing the discrepancy or inconsistency between stimuli reduces the influence that they have on each other [6, 7, 8, 9, 10, 11]. In auditoryvisual spatial localization, one variable that controls stimulus similarity is spatial disparity (another would be temporal disparity). Indeed, it has been reported that increasing spatial disparity leads to a decrease in auditory localization bias [1, 12, 13, 14, 15, 16, 17, 2, 18, 19, 20, 21]. This decrease also correlates with a decrease in the reports of unity [19, 21]. Despite the abundance of experimental data on this issue, no general theory exists that can explain multisensory perception across a wide range of cue conflicts. 2 Models The success of Bayesian models for cue integration has motivated attempts to extend them to situations of large sensory conflict and a consequent low degree of integration. In one of recent studies taking this approach, subjects were presented with concurrent visual flashes and auditory beeps and asked to count both the number of flashes and the number of beeps [11]. The advantage of the experimental paradigm adopted here was that it probed the joint response distribution by requiring a dual report. Human data were accounted for well by a Bayesian model in which the joint prior distribution over visual and auditory number was approximated from the data. In a similar study, subjects were presented with concurrent flashes and taps and asked to count either the flashes or the taps [9, 22]. The Bayesian model proposed by these authors assumed a joint prior distribution with a near-diagonal form. The corresponding generative model assumes that the sensory sources somehow interact with one another. A third experiment modulated the rates of flashes and beeps. The task was to judge either the visual or the auditory modulation rate relative to a standard [23]. The data from this experiment were modeled using a joint prior distribution which is the sum of a near-diagonal prior and a flat background. While all these models are Bayesian in a formal sense, their underlying generative model does not formalize the model selection process that underlies the combination of cues. This makes it necessary to either estimate an empirical prior [11] by fitting it to human behavior or to assume an ad hoc form [22, 23]. However, we believe that such assumptions are not needed. It was shown recently that human judgments of spatial unity in an auditory-visual spatial localization task can be described using a Bayesian inference model that infers causal structure [24, 25]. In this model, the brain does not only estimate a stimulus variable, but also infers the probability that the two stimuli have a common cause. In this paper we compare these different models on a large data set of human position estimates in an auditory-visual task. In this section we first describe the traditional cue integration model, then the recent models based on joint stimulus priors, and finally the causal inference model. To relate to the experiment in the next section, we will use the terminology of auditory-visual spatial localization, but the formalism is very general. 2.1 Traditional cue integration The traditional generative model of cue integration [26] has a single source location s which produces on each trial an internal representation (cue) of visual location, xV and one of auditory location, xA. We assume that the noise processes by which these internal representations are generated are conditionally independent from each other and follow Gaussian distributions. That is, p (xV |s) ∼N (xV ; s, σV )and p (xA|s) ∼N (xA; s, σA), where N (x; µ, σ) stands for the normal distribution over x with mean µ and standard deviation σ. If on a given trial the internal representations are xV and xA, the probability that their source was s is given by Bayes’ rule, p (s|xV , xA) ∝p (xV |s) p (xA|s) . If a subject performs maximum-likelihood estimation, then the estimate will be ˆs = wV xV +wAxA wV +wA , where wV = 1 σ2 V and wA = 1 σ2 A . It is important to keep in mind that this is the estimate on a single trial. A psychophysical experimenter can never have access to xV and xA, which 2 are the noisy internal representations. Instead, an experimenter will want to collect estimates over many trials and is interested in the distribution of ˆs given sV and sA, which are the sources generated by the experimenter. In a typical cue combination experiment, xV and xA are not actually generated by the same source, but by different sources, a visual one sV and an auditory one sA. These sources are chosen close to each other so that the subject can imagine that the resulting cues originate from a single source and thus implicitly have a common cause. The experimentally observed distribution is then p (ˆs|sV , sA) = Z Z p (ˆs|xV , xA) p (xV |sV ) p (xA|sA) dxV dxA Given that ˆs is a linear combination of two normally distributed variables, it will itself follow a normal distribution, with mean⟨ˆs⟩= wV sV +wAsA wV +wA and variance σ2 ˆs = 1 wV +wA . The reason that we emphasize this point is because many authors identify the estimate distribution p (ˆs|sV , sA) with the posterior distribution p (s|xV , xA). This is justified in this case because all distributions are Gaussian and the estimate is a linear combination of cues. However, in the case of causal inference, these conditions are violated and the estimate distribution will in general not be the same as the posterior distribution. 2.2 Models with bisensory stimulus priors Models with bisensory stimulus priors propose the posterior over source positions to be proportional to the product of unimodal likelihoods and a two-dimensional prior: p (sV , sA|xV , xA) = p (sV , sA) p (xV |sV ) p (xA|sA) The traditional cue combination model has p (sV , sA) = p (sV ) δ (sV −sA), usually (as above) even with p (sV ) uniform. The question arises what bisensory stimulus prior is appropriate. In [11], the prior is estimated from data, has a large number of parameters, and is therefore limited in its predictive power. In [23], it has the form p (sV , sA) ∝ω + e −(sV −sA)2 2σ2 coupling while in [22] the additional assumption ω = 0 is made1. XA XV S C XV XA S SV A C=1 C=2 Figure 1: Generative model of causal inference. In all three models, the response distribution p (ˆsV , ˆsA|sV , sA) is obtained by identifying it with the posterior distribution p (sV , sA|xV , xA). This procedure thus implicitly assumes that marginalizing over the latent variables xV and xA is not necessary, which leads to a significant error for non-Gaussian priors. In this paper we correctly deal with these issues and in all cases marginalize over the latent variables. The parametric models used for the coupling between the cues lead to an elegant low-dimensional model of cue integration that allows for estimates of single cues that differ from one another. 2.3 Causal inference model In the causal inference model [24, 25], we start from the traditional cue integration model but remove the assumption that two signals are caused by the same source. Instead, the number of sources can be one or two and is itself a variable that needs to be inferred from the cues. 1This family of Bayesian posterior distributions also includes one used to successfully model cue combination in depth perception [27, 28]. In depth perception, however, there is no notion of segregation as always a single surface is assumed. 3 If there are two sources, they are assumed to be independent. Thus, we use the graphical model depicted in Fig. 1. We denote the number of sources by C. The probability distribution over C given internal representations xV and xA is given by Bayes’ rule: p (C|xV , xA) ∝p (xV , xA|C) p (C) . In this equation, p (C) is the a priori probability of C. We will denote the probability of a common cause by pcommon, so that p (C = 1) = pcommon and p (C = 2) = 1 −pcommon. The probability of generating xV and xA given C is obtained by inserting a summation over the sources: p (xV , xA|C = 1) = Z p (xV , xA|s)p (s) ds = Z p (xV |s) p (xA|s)p (s) ds Here p (s) is a prior for spatial location, which we assume to be distributed as N (s; 0, σP ). Then all three factors in this integral are Gaussians, allowing for an analytic solution: p (xV , xA|C = 1) = 1 2π√ σ2 V σ2 A+σ2 V σ2 P +σ2 Aσ2 P exp h −1 2 (xV −xA)2σ2 P +x2 V σ2 A+x2 Aσ2 V σ2 V σ2 A+σ2 V σ2 P +σ2 Aσ2 P i . For p (xV , xA|C = 2) we realize that xV and xA are independent of each other and thus obtain p (xV , xA|C = 2) = Z p (xV |sV )p (sV ) dsV Z p (xA|sA)p (sA) dsA Again, as all these distributions are assumed to be Gaussian, we obtain an analytic solution, p (xV , xA|C = 2) = 1 2πp (σ2 V +σ2 p)(σ2 A+σ2 p) exp h −1 2 x2 V σ2 V +σ2 p + x2 A σ2 A+σ2 p i . Now that we have computed p (C|xV , xA), the posterior distribution over sources is given by p (si|xV , xA) = X C=1,2 p (si|xV , xA, C) p (C|xV , xA) where i can be V or A and the posteriors conditioned on C are well-known: p (si|xA, xV , C = 1) = p (xA|si) p (xV |si) p (si) R p (xA|s) p (xV |s) p (s) ds, p (si|xA, xV , C = 2) = p (xi|si) p (si) R p (xi|si) p (si) dsi The former is the same as in the case of mandatory integration with a prior, the latter is simply the unimodal posterior in the presence of a prior. Based on the posterior distribution on a given trial, p (si|xV , xA), an estimate has to be created. For this, we use a sum-squared-error cost function, Cost = D p (C = 1|xV , xA) (ˆs −s)2E + D p (C = 2|xV , xA) (ˆs −sV or A)2E . Then the best estimate is the mean of the posterior distribution, for instance for the visual estimation: ˆsV = p (C = 1|xA, xV ) ˆsV,C=1 + p (C = 2|xA, xV ) ˆsV,C=2 where ˆsV,C=1 = xV σ−2 V +xAσ−2 A +xP σ−2 P σ−2 V +σ−2 A +σ−2 P and ˆsV,C=2 = xV σ−2 V +xP σ−2 P σ−2 V +σ−2 P . If pcommonequals 0 or 1, this estimate reduces to one of the conditioned estimates and is linear in xV and xA. If 0 < pcommon < 1, the estimate is a nonlinear combination of xV and xA, because of the functional form of p (C|xV , xA). The response distributions, that is the distributions of ˆsV and ˆsA given sV and sA over many trials, now cannot be identified with the posterior distribution on a single trial and cannot be computed analytically either. The correct way to obtain the response distribution is to simulate an experiment numerically. Note that the causal inference model above can also be cast in the form of a bisensory stimulus prior by integrating out the latent variable C, with: p (sA, sV ) = p (C = 1) δ (sA −sV ) p (sA) + p (sA) p (sV ) p (C = 2) However, in addition to justifying the form of the interaction between the cues, the causal inference model has the advantage of being based on a generative model that well formalizes salient properties of the world, and it thereby also allows to predict judgments of unity. 4 3 Model performance and comparison To examine the performance of the causal inference model and to compare it to previous models, we performed a human psychophysics experiment in which we adopted the same dual-report paradigm as was used in [11]. Observers were simultaneously presented with a brief visual and also an auditory stimulus, each of which could originate from one of five locations on an imaginary horizontal line (-10◦, -5◦, 0◦, 5◦, or 10◦with respect to the fixation point). Auditory stimuli were 32 ms of white noise filtered through an individually calibrated head related transfer function (HRTF) and presented through a pair of headphones, whereas the visual stimuli were high contrast Gabors on a noisy background presented on a 21-inch CRT monitor. Observers had to report by means of a key press (1-5) the perceived positions of both the visual and the auditory stimulus. Each combination of locations was presented with the same frequency over the course of the experiment. In this way, for each condition, visual and auditory response histograms were obtained. We obtained response distributions for each the three models described above by numeral simulation. On each trial, estimation is followed by a step in which, the key is selected which corresponds to the position closed to the best estimate. The simulated histograms obtained in this way were compared to the measured response frequencies of all subjects by computing the R2 statistic. The parameters in the causal inference model were optimized using fminsearch in MATLAB to maximize R2. The best combination of parameters yielded an R2 of 0.97. The response frequencies are depicted in Fig. 2. The bisensory prior models also explain most of the variance, with R2 = 0.96 for the Roach model and R2 = 0.91 for the Bresciani model. This shows that it is possible to model cue combination for large disparities well using such models. 1 0 no audio no vision Auditory response Auditory model Visual response Visual model Figure 2: A comparison between subjects’ performance and the causal inference model. The blue line indicates the frequency of subjects responses to visual stimuli, red line is the responses to auditory stimuli. Each set of lines is one set of audio-visual stimulus conditions. Rows of conditions indicate constant visual stimulus, columns is constant audio stimulus. Model predictions is indicated by the red and blue dotted line. 5 3.1 Model comparison To facilitate quantitative comparison with other models, we now fit the parameters of each model2 to individual subject data, maximizing the likelihood of the model, i.e., the probability of the response frequencies under the model. The causal inference model fits human data better than the other models. Compared to the best fit of the causal inference model, the Bresciani model has a maximal log likelihood ratio (base e) of the data of −22 ± 6 (mean ± s.e.m. over subjects), and the Roach model has a maximal log likelihood ratio of the data of −18 ± 6. A causal inference model that maximizes the probability of being correct instead of minimizing the mean squared error has a maximal log likelihood ratio of −18 ± 3. These values are considered decisive evidence in favor of the causal inference model that minimizes the mean squared error (for details, see [25]). The parameter values found in the likelihood optimization of the causal model are as follows: pcommon = 0.28 ± 0.05, σV = 2.14 ± 0.22◦, σA = 9.2 ± 1.1◦, σP = 12.3 ± 1.1◦(mean ± s.e.m. over subjects). We see that there is a relatively low prior probability of a common cause. In this paradigm, auditory localization is considerably less precise than visual localization. Also, there is a weak prior for central locations. 3.2 Localization bias A useful quantity to gain more insight into the structure of multisensory data is the cross-modal bias. In our experiment, relative auditory bias is defined as the difference between the mean auditory estimate in a given condition and the real auditory position, divided by the difference between the real visual position and the real auditory position in this condition. If the influence of vision on the auditory estimate is strong, then the relative auditory bias will be high (close to one). It is well-known that bias decreases with spatial disparity and our experiment is no exception (solid line in Fig. 3; data were combined between positive and negative disparities). 5 10 15 20 20 25 30 35 40 45 50 Spatial Disparity (deg.) % Auditory Bias 5 10 15 20 20 25 30 35 40 45 50 Figure 3: Auditory bias as a function of spatial disparity. Solid blue line: data. Red: Causal inference model. Green: Model by Roach et al. [23]. Purple: Model by Bresciani et al. [22]. Models were optimized on response frequencies (as in Fig. 2), not on the bias data. It can easily be shown that a traditional cue integration model would predict a bias equal to 1 + σ2 V σ2 A −1 , which would be close to 1 and independent of disparity, unlike the data. This shows that a mandatory integration model is an insufficient model of multisensory interactions. We used the individual subject fittings from above and and averaged the auditory bias values obtained from those fits (i.e. we did not fit the bias data themselves). Fits are shown in Fig. 3 (dashed lines). We applied a paired t-test to the differences between the 5◦and 20◦disparity conditions (model-subject comparison). Using a double-sided test, the null hypothesis that the difference between the bias in the 5◦and 20◦conditions is correctly predicted by each model is rejected for the Bresciani model (p < 0.002) and the Roach model (p < 0.042) and accepted for the causal inference model (p > 0.17). Alternatively, with a single-sided test, the hypothesis is rejected for the Bresciani model (p < 0.001) and the Roach model (p < 0.021) and accepted for the causal inference model (> 0.9). The reason that the Bresciani model fares worst is that its prior distribution does not include a component that corresponds to independent causes. On 2The Roach et al. model has four free parameters (ω,σV , σA, σcoupling), the Bresciani et al. model has three (σV , σA, σcoupling), and the causal inference model has four (pcommon,σV , σA, σP ). We do not consider the Shams et al. model here, since it has many more parameters and it is not immediately clear how in this model the erroneous identification of posterior with response distribution can be corrected. 6 the contrary, the prior used in the Roach model contains two terms, one term that is independent of the disparity and one term that decreases with increasing disparity. It is thus functionally somewhat similar to the causal inference model. 4 Discussion We have argued that any model of multisensory perception should account not only for situations of small, but also of large conflict. In these situations, segregation is more likely, in which the two stimuli are not perceived to have the same cause. Even when segregation occurs, the two stimuli can still influence each other. We compared three Bayesian models designed to account for situations of large conflict by applying them to auditory-visual spatial localization data. We pointed out a common mistake: for nonGaussian bisensory priors without mandatory integration, the response distribution can no longer be identified with the posterior distribution. After correct implementation of the three models, we found that the causal inference model is superior to the models with ad hoc bisensory priors. This is expected, as the nervous system actually needs to solve the problem of deciding which stimuli have a common cause and which stimuli are unrelated. We have seen that multisensory perception is a suitable tool for studying causal inference. However, the causal inference model also has the potential to quantitatively explain a number of other perceptual phenomena, including perceptual grouping and binding, as well as within-modality cue combination [27, 28]. Causal inference is a universal problem: whenever the brain has multiple pieces of information it must decide if they relate to one another or are independent. As the causal inference model describes how the brain processes probabilistic sensory information, the question arises about the neural basis of these processes. Neural populations encode probability distributions over stimuli through Bayes’ rule, a type of coding known as probabilistic population coding. Recent work has shown how the optimal cue combination assuming a common cause can be implemented in probabilistic population codes through simple linear operations on neural activities [29]. 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What Makes Some POMDP Problems Easy to Approximate? David Hsu∗ Wee Sun Lee∗ Nan Rong† ∗Department of Computer Science National University of Singapore Singapore, 117590, Singapore †Department of Computer Science Cornell University Ithaca, NY 14853, USA Abstract Point-based algorithms have been surprisingly successful in computing approximately optimal solutions for partially observable Markov decision processes (POMDPs) in high dimensional belief spaces. In this work, we seek to understand the belief-space properties that allow some POMDP problems to be approximated efficiently and thus help to explain the point-based algorithms’ success often observed in the experiments. We show that an approximately optimal POMDP solution can be computed in time polynomial in the covering number of a reachable belief space, which is the subset of the belief space reachable from a given belief point. We also show that under the weaker condition of having a small covering number for an optimal reachable space, which is the subset of the belief space reachable under an optimal policy, computing an approximately optimal solution is NP-hard. However, given a suitable set of points that “cover” an optimal reachable space well, an approximate solution can be computed in polynomial time. The covering number highlights several interesting properties that reduce the complexity of POMDP planning in practice, e.g., fully observed state variables, beliefs with sparse support, smooth beliefs, and circulant state-transition matrices. 1 Introduction Computing an optimal policy for a partially observable Markov decision process (POMDP) is an intractable problem [10, 9]. Intuitively, the intractability is due to the “curse of dimensionality”: the belief space B used in solving a POMDP typically has dimensionality equal to |S|, the number of states in the POMDP, and therefore the size of B grows exponentially with |S|. As a result, the number of states is often used in practice as an important measure of the complexity of POMDP planning. However, in recent years, point-based POMDP algorithms have made impressive progress in computing approximate solutions by sampling the belief space: POMDPs with hundreds of states have been solved in a matter of seconds [14, 4]. It seems surprising that even an approximate solution can be obtained in seconds in a space of hundreds of dimensions. Thus, we would like to investigate why these point-based algorithms work well, whether there are sub-classes of POMDPs that are computationally easier, and whether there are alternative measures that better capture the complexity of POMDP planning for point-based algorithms. Our work is motivated by a benchmark problem called Tag [11], in which a robot needs to search and tag a moving target that tends to move away from it. The environment is modeled as a grid. The robot’s position is fully observable. The target’s position is not observable, i.e., unknown to the robot, unless the target is in the same grid position as the robot. The joint state of the robot and target positions is thus only partially observable. The problem has 870 states in total, resulting in a belief space of 870 dimensions. Tag was introduced in the work on Point-Based Value Iteration (PBVI) [11], one of the first point-based POMDP algorithms. At the time, it was among the largest POMDP problems ever attempted and was considered a challenge for fast, scalable POMDP algorithms [11]. Surprisingly, only two years later, another point-based algorithm [14] computed an approximate solution to Tag, a problem with an 870-dimensional belief space, in less than a minute! One important feature that underlies the success of many point-based algorithms is that they only explore a subset R(b0) ⊆B, usually called the reachable space from b0. The reachable space R(b0) contains all points reachable from a given initial belief point b0 ∈B under arbitrary sequences of actions and observations. One may then speculate that the reason for point-based algorithms’ good performance on Tag is that its reachable space R(b0) has much lower dimensionality than B. This is, however, not true. By checking the dimensionality of a large set of points sampled from R(b0), we have found that the dimensionality of R(b0) is at least 860 and thus almost as large as B. In this paper, we propose to use the covering number as an alternative measure of the complexity of POMDP planning ( Section 4). Intuitively, the covering number of a space is the minimum number of given size balls that needed to cover the space fully. We show that an approximately optimal POMDP solution can be computed in time polynomial in the covering number of R(b0). The covering number also reveals that the belief space for Tag behaves more like the union of some 29-dimensional spaces rather than an 870-dimensional space, as the robot’s position is fully observed. Therefore, Tag is probably not as hard as it was thought to be, and the covering number captures the complexity of the Tag problem better than the dimensionality of the belief space (the number of states) or the dimensionality of the reachable space. We further ask whether it is possible to compute an approximate solution efficiently under the weaker condition of having a small covering number for an optimal reachable R∗(b0), which contains only points in B reachable from b0 under an optimal policy. Unfortunately, we can show that this problem is NP-hard. The problem remains NP-hard, even if the optimal policies have a compact piecewiselinear representation using α-vectors. However, we can also show that given a suitable set of points that “cover” R∗(b0) well, a good approximate solution can be computed in polynomial time. Together, the negative and the positive results indicate that using sampling to approximate an optimal reachable space, and not just the reachable space, may be a promising approach in practice. We have already obtained initial experimental evidence that supports this idea. Through careful sampling and pruning, our new point-based algorithm solved the Tag problem in less than 5 seconds [4]. The covering number highlights several properties that reduce the complexity of POMDP planning in practice, and it helps to quantify their effects (Section 5). Highly informative observations usually result in beliefs with sparse support and substantially reduce the covering number. For example, fully observed state variables reduce the covering number by a doubly exponential factor. Interestingly, smooth beliefs, usually a result of imperfect actions and uninformative observations, also reduce the covering number. In addition, state-transition matrices with special structures, such as circulant matrices [1], restrict the space of reachable beliefs and reduce the covering number correspondingly. 2 Related Works POMDPs provide a principled mathematical framework for planning and decision-making under uncertainty [13, 5], but they are notoriously hard to solve [10, 7, 9, 8]. It has been shown that finding an optimal policy over the entire belief space for a finite-horizon POMDP is PSPACE-complete [10] and that finding an optimal policy over an infinite horizon is undecidable [9]. As a result, there has been a lot of work on computing approximate POMDP solutions [2], including a number of point-based POMDP algorithms [16, 11, 15, 14, 3]. Some point-based algorithms were able to compute reasonably good policies for very large POMDPs with hundreds of thousands states. The success of these algorithms motivated us to try to understand why and when they work well. The approximation errors of some point-based algorithms have been analyzed [11, 14], but these analyses do not address the general question of when an approximately optimal policy can be computed efficiently in polynomial time. We provide both positive and negative results showing the difficulty of computing approximate POMDP solutions. The proof techniques used for Theorems 1 and 2 are similar to those used for analyzing an approximation algorithm for large (fully observable) MDPs [6]. While the algorithm in [6] handles large state spaces well, it does not run in polynomial time: it appears that additional assumptions such as those made in this paper are required for polynomial time results. Our hardness result is closely related to that for finite-horizon POMDPs [8], but we give a direct reduction from the Hamiltonian cycle problem. 3 Preliminaries A POMDP models an agent taking a sequence of actions under uncertainty to maximize its total reward. Formally it is specified as a tuple (S, A, O, T, Z, R, γ), where S is a set of discrete states, A is a finite set of actions, and O is a set of discrete observations. At each time step, the agent takes some action a ∈A and moves from a start state s to an end state s′. The end state s′ is given by a state-transition function T(s, a, s′) = p(s′|s, a), which gives the probability that the agent lies in s′, after taking action a in state s. The agent then makes an observation to gather information on its current state. The outcome of observing o ∈O is given by an observation function Z(s, a, o) = p(o|s, a) for s ∈S and a ∈A. The reward function R gives the agent a real-valued reward R(s, a) if it takes action a in state s, and the goal of the agent is to maximize its expected total reward by choosing a suitable sequence of actions. In this paper, we consider only infinite-horizon POMDPs with discounted reward. Thus, the expected total reward is given by E[P∞ t=0 γtR(st, at)], where γ ∈(0, 1) is a discount factor, and st and at denote the agent’s state and the action at time t. Since the agent’s state is only partially observable, we rely on the concept of a belief, which is simply a probability distribution over S, represented disretely as a vector. A POMDP solution is a policy π that specifies the action π(b) for every belief b. Our goal is to find an optimal policy π∗that maximizes the expected total reward. A policy π induces a value function V π that specifies the value V π(b) of every belief b under π. It is known that V ∗, the value function associated the optimal policy π∗, can be approximated arbitrarily closely by a convex, piecewiselinear function V (b) = maxα∈Γ(α · b), where Γ is a finite set of vectors called α-vectors. The optimal value function V ∗satisfies the following Lipschitz condition: Lemma 1 For any two belief points b and b′, if ||b −b′|| ≤δ, then |V ∗(b) −V ∗(b′)| ≤Rmax 1−γ δ.1 a1 a2 o1 o2 b0 Figure 1: The belief tree rooted at b0. Throughout this paper, we always use the l1 metric to measure the distance between belief points: for b, b′ ∈Rd, ||b −b′|| = Pd i=1 |bi −b′ i|. The Lipschitz condition bounds the change of a value function using the distance between belief points. It provides the basis for approximating the value at a belief point by the values of other belief points nearby. To find an approximately optimal policy, pointbased algorithms explore only the reachable belief space R(b0) from a given initial belief point b0. Strictly speaking, these algorithms compute only a policy over R(b0), rather than the entire belief space B. We can view the exploration of R(b0) as searching a belief tree TR rooted at b0 (Figure 1). The nodes of TR correspond to beliefs in R(b0). The edges correspond to action-observation pairs. Suppose that a child node b′ is connected to its parent b by an edge (a, o). We can compute b′ using the formula b′(s′) = τ(b, a, o) = ηZ(s′, a, o) P s T(s, a, s′)b(s), where η is a normalizing constant. After obtaining enough belief points from R(b0), point-based algorithms perform backup operations over them to compute an approximately optimal value function. 4 The Covering Number and the Complexity of POMDP Planning Our first goal is to show that if the covering number of a reachable space R(b0) is small, then an approximately optimal policy in R(b0) can be computed efficiently. We start with the definition of the covering number: Definition 1 Given a metric space X, a δ-cover of a set B ⊆X is a set of point C ⊆X such that for every point b ∈B, there is a point c ∈C with ||b −c|| < δ. If all the points in C also lie in B, then we say that C is a proper cover of B. The δ-covering number of B, denoted by C(δ), is the size of the smallest δ-cover of B. Intuitively, the covering number is equal to the minimum number of balls of radius δ needed to cover the set B. A closely related notion is that of the packing number: Definition 2 Given a metric space X, a δ-packing of a set B ⊆X is a set of points P ⊆B such that for any two points p1, p2 ∈P, ||p1 −p2|| ≥δ. The δ-packing number of a set B, denoted by P(δ), is the size of the largest δ-packing of B. 1The proofs of this and other results are available as an appendix at http://motion.comp.nus.edu. sg/papers/nips07.pdf. For any set B, the following relationship holds between packing and covering numbers. Lemma 2 C(δ) ≤P(δ) ≤C(δ/2). We are now ready to state our first main result. It shows that for any point b0 ∈B, if the covering number of R(b0) grows polynomially with the parameters of interest, then a good approximation of the value at b0 can be computed in polynomial time. Theorem 1 For any b0 ∈B, let C(δ) be the δ-covering number of R(b0). Given any constant ϵ > 0, an approximation V (b0) of V ∗(b0), with error |V ∗(b0) −V (b0)| ≤ϵ, can be found in time O C (1 −γ)2ϵ 4γRmax 2 logγ (1 −γ)ϵ 2Rmax ! . Proof. To prove the result, we give an algorithm that computes the required approximation. It performs a depth-first search on a depth-bounded belief tree and uses approximate memorization to avoid unnecessarily computing the values of very similar beliefs. Intuitively, to achieve a polynomial time algorithm, we bound the height of the tree by exploiting the discount factor and bound the width of the tree by exploiting the covering number. We perform the depth-first search recursively on a belief tree TR that has root b0 and height h, while maintaining a δ-packing of R(b0) at every level of TR. Suppose that the search encounters a new belief node b at level i of TR. If b is within a distance δ of a point b′ in the current packing at level i, we set V (b) = V (b′), abort the recursion at b, and backtrack. Otherwise, we recursively search the children of b. When the search returns, we perform a backup operation to compute V (b) and add b to the packing at level i. If b is a leaf node of TR, we set V (b) = 0. We build a separate packing at each level of TR, as each level has a different approximation error. We now calculate the values for h and δ required to achieve the given approximation bound ϵ at b0. Let ϵi = |V ∗(b)−V (b)| denote the approximation error for a node b at level i of TR, if the recursive search continues in the children of b. By convention, the leaf nodes are at level 0. Similarly, let ϵ′ i denote the error for b, if the search aborts at b and V (b) = V (b′) for some b′ in the packing at level i. Hence, ϵ′ i = |V ∗(b) −V (b′)| ≤ |V ∗(b) −V ∗(b′)| + |V ∗(b′) −V (b′)| ≤ Rmax 1 −γ δ + ϵi, where the last inequality uses Lemma 1 and the definition of ϵi. Clearly, ϵ0 ≤Rmax/(1 −γ). To calculate ϵi for a node b at level i, we establish a recurrence. The children of b, which are at level i −1, have error at most ϵ′ i−1. Since a backup operation is performed at b, we have ϵi ≤γϵ′ i−1 and thus the recurrence ϵi ≤γ(ϵi−1 + Rmax 1−γ δ). Expanding the recurrence, we find that the error ϵh at the root b0 is given by |V ∗(b0) −V (b0)| ≤ γRmax(1 −γh) (1 −γ)2 δ + γh Rmax 1 −γ ≤ γRmax (1 −γ)2 δ + γh Rmax 1 −γ . By setting δ = (1−γ)2ϵ 2γRmax and h = logγ (1−γ)ϵ 2Rmax , we can guarantee |V ∗(b0) −V (b0)| ≤ϵ. We now work out the running time of the algorithm. For each node b in the packings, the algorithm expands it by calculating the beliefs and the corresponding values for all its children and performing a backup operation at b to compute V (b). It takes O(|S|2) time to calculate the belief at a child node. We then perform a nearest neighbor search in O(P(δ)|S|) time to check whether the child node lies within a distance δ of any point in the packing at that level. Since b has |A||O| children, the expansion operation takes O(|A||O||S|(|S| + P(δ)) time. The backup operation then computes V (b) as an average of its children’s values, weighted by the probabilities specified by the observation function, and takes only O(|A||O|) time. Since there are h packings of size P(δ) each and by Lemma 2, P(δ) ≤C(δ/2), the total running time of our algorithm is given by O (hC(δ/2)|A||O||S|(|S| + C(δ/2))) . We assume that |S|, |A|, and |O| are constant to focus on the dependency on the covering number, and the above expression then becomes O(hC(δ/2)2). Substituting in the values for h and δ, we get the final result. 2 The algorithm in the above proof can be used on-line to choose an approximately optimal action at b0. We first estimate the values for all the child nodes of b0 and then select the action resulting in the highest value. Suppose that at each belief point reachable from b0, we perform such an on-line search for action selection. Using the technique in [12], one can show that if the value function approximations at all the child nodes have error at most ϵ, then the policy π implicitly defined by the on-line search has approximation error |V ∗(b) −V π(b)| ≤2γϵ/(1 −γ) for all b in R(b0). Instead of performing the on-line search, one may want to precompute an approximately optimal value function over R(b0) and perform one-step look-ahead on it at runtime for action selection. The algorithm in Theorem 1 is not sufficient for this purpose, as it samples only enough points from R(b0) to give a good value estimate at b0, but the sampled points do not form a cover of R(b0). One possibility would be to find a cover of R(b0) first and then apply PBVI [11] over the points in the cover. Unfortunately, we do not know how to find a cover of R(b0) efficiently. Instead, we give a randomized algorithm that computes an approximately optimal value function with high probability. Roughly, this algorithm incrementally builds a packing of R(b0) at each level of TR. It first runs the algorithm in Theorem 1 to obtain an initial packing Pi for each level i and estimate the values of belief points in Pi. Then, to test whether the current packing Pi covers R(b0) well, it runs a set of simulations of a fixed size. If the simulations encounter new points not covered by Pi, we estimate their values and insert them into Pi. The process repeats until no more new belief points are discovered within a set of simulation. We show that if the set of simulations is sufficiently large, then the probability that in any future run of the policy, we encounter new belief points not covered by the final set of packings can be made arbitrarily small. Theorem 2 For any b0 ∈B, let C(δ) be the δ-covering number of R(b0). Given constants β ∈(0, 1) and ϵ > 0, a randomized algorithm can compute, with probability at least 1 −β, an approximately optimal value function in time O Rmax (1 −γ)ϵ C (1 −γ)3ϵ 16γRmax logγ (1 −γ)ϵ 4Rmax 2 log 1 β C (1 −γ)3ϵ 16γRmax logγ (1 −γ)ϵ 4Rmax ! . such that the one-step look-ahead policy π induced by this value function has error |V ∗(b0) − V π(b0)| ≤ϵ. It takes O C (1−γ)3ϵ 16γRmax time to use this value function to select an action at runtime. Both theorems above assume tha a small covering number of R(b0) for efficient computation. To relax this assumption, we may require only that the covering number for an optimal reachable space R∗(b0) is small, as R∗(b0) contains only points reachable under an optimal policy and can be much smaller than R(b0). Unfortunately, under the relaxed condition, approximating the value at b0 is NP-hard. We prove this by reduction from the Hamiltonian cycle problem. The main idea is to show that a Hamiltonian cycle exists in a given graph if and only an approximation to V ∗(b0), with a suitably chosen error, can be computed for a POMDP whose optimal reachable space R∗(b0) has a small covering number. The result is closely related to one for finite-horizon POMDPs [8]. Theorem 3 Given constant ϵ > 0, computing an approximation V (b0) of V ∗(b0), with error |V (b0) −V ∗(b0)| ≤ϵ|V ∗(b0)|, is NP-hard, even if the covering number of R∗(b0) is polynomialsized. The result above assumes the standard encoding of POMDP input with state-transition functions, observation functions, and reward functions all represented discretely by matrices of suitable sizes. By slightly extending the proof of Theorem 3, we can also show a related hardness result, which assumes that the optimal policy has a compact representation. Theorem 4 Given constant ϵ > 0, computing an approximation V (b0) of V ∗(b0), with error |V (b0)−V ∗(b0)| ≤ϵ|V ∗(b0)|, is NP-hard, even if the number of α-vectors required to represent an optimal policy is polynomial-sized. On the other hand, if an oracle provides us a proper cover of an optimal reachable space R∗(b0), then a good approximation of V ∗(b0) can be found efficiently. Theorem 5 For any b0 ∈B, given constant ϵ > 0 and a proper δ-cover C of R∗(b0) with δ = (1−γ)2ϵ 2γRmax , an approximation V (b0) of V ∗(b0), with error |V ∗(b0) −V (b0)| ≤ϵ, can be found in time O |C|2 + |C| logγ (1 −γ)ϵ 2RMax . Together, the negative and the positive results (Theorems 3 to 5) indicate that a key difficulty for point-based algorithms lies in finding a cover for R∗(b0). In practice, to overcome the difficulty, one may use problem-specific knowledge or heuristics to approximate R∗(b0) through sampling. Most point-based POMDP algorithms [11, 15, 14] interpolate the value function using α-vectors. Although we use the nearest neighbor approximation to simplify the proofs of Theorems 1, 2, and 5, we want to point out that very similar results can be obtained using the α-vector representation if we slightly modify the analysis of the approximation errors in the proofs. 5 Bounding the Covering Number The covering number highlights several properties that reduce the complexity of POMDP planning in practice. We describe them below and show how they affect the covering number. 5.1 Fully Observed State Variables Suppose that there are d state variables, each of which has at most k possible values. If d′ of these variables are fully observed, then for every such belief point, its vector representation contains at most m = kd−d′ non-zero elements out of kd elements in total. For a given initial belief b0, the belief vectors with the same non-zero pattern form a subspace in R(b0), and R(b0) is a union of these subspaces. We can compute a δ-cover for each subspace by discretizing each non-zero element of the belief vectors to an accuracy of δ/m, and the size of the resulting δ-cover is at most ( m δ )m. There are kd′ such subspaces. So the δ-covering number of R(b0) is at most kd′( m δ )m = kd′( kd−d′ δ )kd−d′ . The fully observed variables thus give a doubly exponential reduction in the covering number: it reduces the exponent by a factor of kd′ at the cost of a multiplicative factor of kd′. Proposition 1 Suppose that a POMDP has d state variables, each of which has at most k possible values. If d′ state variables are fully observed, then for any belief point b0, the δ-covering number of the reachable belief space R(b0) is at most kd′( kd−d′ δ )kd−d′ . Consider again the Tag problem described in Section 1. The state consists of both the robot’s and the target’s positions, as well as the status indicating whether the target is tagged. The robot and the target can occupy any position in an environment modeled as a grid of 29 cells. If the robot has the target tagged, they must be in the same position. So, there are 29 × 29 + 29 = 870 states in total, and the belief space B is 870-dimensional. However, the robot’s position is fully observed. By Proposition 1, the δ-covering number is at most 30 · (30/δ)30. Indeed, for Tag, any reachable belief space R(b0) is effectively a union of two sets. One set corresponds to the case when the target is not tagged and consists of the union of 29 sub-spaces of 29 dimensions. The other set corresponds to the case when the target is tagged and consists of exactly 29 points. Clearly, the covering number captures the underlying complexity of R(b0) more accurately than the dimensionality of R(b0). 5.2 Sparse Beliefs Highly informative observations often result in sparse beliefs, i.e., beliefs whose vector representation is sparse. For example, in the Tag problem, the state is known exactly if the robot and the target are in the same position, leaving only a single non-zero element in the belief vector. Fully observed state variables usually result in very sparse beliefs and can be considered a special case. If the beliefs are always sparse, we can exploit the sparsity to bound the covering number. Otherwise, sparsity may still give a hint that the covering number is smaller than what would be suggested by the dimensionality of the belief space. By exploiting the non-zeros patterns of belief vectors in a way similar to that in Section 5.1, we can derive the following result: Proposition 2 Let B be a set in an n-dimensional belief space. If every belief in B can be represented as a vector with at most m non-zero elements, then the δ-covering number of B is O(nm m δ m). 5.3 Smooth Beliefs Sparse beliefs are often peaky. Interestingly, when the beliefs are sufficiently smooth, e.g., when their Fourier representations are sparse, the covering number is also small. Below we give a more general result, assuming that the beliefs can be represented as a linear combination of a small number of basis vectors. Proposition 3 Let B be a set in an n-dimensional belief space. Assume that every belief b ∈B can be represented as a linear combination of m basis vectors such that the magnitudes of both the elements of the basis vectors and the coefficients representing b are bounded by a constant C. The δcovering number of B is O(( 2C2mn δ )m) when the basis vectors are real-valued, and O(( 4C2mn δ )2m) when they are complex-valued. Smooth beliefs are usually a result of actions with high uncertainty and uninformative observations. 5.4 Circulant State-Transition Matrices Let us now shift our attention from observations to actions, in particular, actions that can be represented by state-transition matrices with special structures. We start with an example. A mobile robot scout needs to navigate from a known start position to a goal position in a large environment modeled as a grid. It must not enter certain danger zones to avoid detection by enemies. The robot can take four actions to move in the {N, S, E, W} directions, but have imperfect control. Since the environment is large, we assume that the robot always operates far away from the boundary and the boundary effect can be ignored. At each grid cell, the robot moves to the intended cell with probability 1−p and moves diagonally to the two cells adjacent to the intended one with probability 0.5p. The robot can use its sensors to make highly accurate observations on its current position, but by doing so, it runs the risk of being detected. Under our assumptions, the state-transition functions representing robot actions are invariant over the grid cells and can thus be represented by circulant matrices [1]. Circulant matrices are widely used in signal processing and control theory, as they can represent all discrete-time linear translationinvariant systems. In the context of POMDPs, if applying a state-transition matrix to a belief b corresponds to convolution with a suitable distribution, then the state-transition matrix is circulant. One of the key properties of circulant matrices is that they all share the same eigenvectors. Therefore, we can multiply them in any arbitrary order and obtain the same result. In our example, this means that given a set of robot moves, we can apply them in any order and the resulting belief on the robot’s position is the same. This greatly reduces the number of possible beliefs and correspondingly the covering number in open-loop POMDPs, where there are no observations involved. Proposition 4 Suppose that all ℓstate-transition matrices representing actions are circulant and that each matrix has at most m eigenvalues whose magnitudes are greater than ζ, with 0 < ζ < 1. In an open-loop POMDP, for any point b0 in an n-dimensional belief space, the δ-covering number of the reachable belief space R(b0) is O 8ℓmn δ 2ℓm + hℓ , where h = logζ(δ/2n). In our example, suppose that the robot scout makes a sequences of moves and needs to decide when to take occasional observations along the way to localize itself. To bound the covering number, we divide the sequence of moves into subsequences such that each subsequence starts with an observation and ends right before the next observation. In each subsequence, the robot starts at a specific belief and moves without additional observations. So, within a subsequence, the beliefs encountered have a δ-cover of size O((8ℓmn/δ)2ℓm + hℓ) by Proposition 4. Furthermore, since all the observations are highly informative, we assume that the initial beliefs of all subsequences can be represented as vectors with at most m′ non-zero elements. The set of all initial beliefs then has a δ-cover of size O(nm′(m′/δ)m′) by Proposition 2. From Lemma 3 below, we know that in an open-loop POMDP, two belief trajectories can only get closer to each other, as they progress. Lemma 3 Let M be a Markov matrix and ||b1 −b2|| ≤δ. Then ||Mb1 −Mb2|| ≤δ. Therefore, to get a δ-cover of the space R(b0) that the robot scout can reach from a given b0, it suffices to first compute a δ/2-cover C of the initial belief points for all possible subsequences of moves and then take the union of the δ/2-covers of the belief points traversed by the subsequences whose initial belief points lie in C. The δ-cover of R(b0) then has its size bounded by O(nm′(2m′/δ)m′(16ℓmn/δ)2ℓm + hℓ), where h = logζ(δ/4n). The requirement of translation invariance means that circulant matrices have some limitations in modeling certain phenomena well. In mobile robot navigation, obstacles or boundaries in the environment often cause difficulties. However, if the environment is sufficiently large and the obstacles are sparse, the behaviors of some systems can be approximated by circulant matrices. 6 Conclusion We propose the covering number as a measure of the complexity of POMDP planning. We believe that for point-based algorithms, the covering number captures the difficulty of computing approximate solutions to POMDPs better than other commonly used measures, such as the number of states. The covering number highlights several interesting properties that reduce the complexity of POMDP planning, and quantifies their effects. Using the covering number, we have shown several results that help to identify the main difficulty of POMDP planning using point-based algorithms. These results indicate that a promising approach in practice is to approximate an optimal reachable space through sampling. We are currently exploring this idea and have already obtained promising initial results [4]. On a set of standard test problems, our new point-based algorithm outperformed the fastest existing point-based algorithm by 5 to 10 times on some problems, while remaining competitive on others. Acknowledgements. We thank Leslie Kaelbling and Tom´as Lozano-P´erez for many insightful discussions on POMDPs. This work is supported in part by NUS ARF grants R-252-000-240-112 and R-252-000-243-112. References [1] R.M. Gray. Toeplitz and Circulant Matrices: A Review. Now Publishers Inc, 2006. [2] M. Hauskrecht. Value-function approximations for partially observable Markov decision processes. J. Artificial Intelligence Research, 13:33–94, 2000. [3] J. Hoey, A. von Bertoldi, P. Poupart, and A. Mihailidis. Assisting persons with dementia during handwashing using a partially observable Markov decision process. In Proc. Int. Conf. on Vision Systems, 2007. [4] D. Hsu, W.S. Lee, and N. Rong. Accelerating point-based POMDP algorithms through successive approximations of the optimal reachable space. Technical Report TRA4/07, National University of Singapore. School of Computing, April 2007. [5] L.P. Kaelbling, M.L. Littman, and A.R. Cassandra. Planning and acting in partially observable stochastic domains. Artificial Intelligence, 101(1–2):99–134, 1998. [6] M. Kearns, Y. Mansour, and A.Y. Ng. A sparse sampling algorithm for near optimal planning in large Markov decision processes. Machine Learning, 49(2-3):193–208, 2002. [7] M.L. Littman. Algorithms for sequential decision making. PhD thesis, Dept. of Computer Science, Brown University, 1996. [8] C. Lusena, J. Goldsmith, and M. Mundhenk. Nonapproximability results for partially observable Markov decision processes. J. Artificial Intelligence Research, 14:83–103, 2002. [9] O. Madani, S. Hanks, and A. Condon. On the undecidability of probabilistic planning and infinite-horizon partially observable Markov decision problems. In Proc. Nat. Conf. on Artificial Intelligence, pages 541– 548, 1999. [10] C. Papadimitriou and J.N. Tsisiklis. The complexity of Markov decision processes. Mathematics of Operations Research, 12(3):441–450, 1987. [11] J. Pineau, G. Gordon, and S. Thrun. Point-based value iteration: An anytime algorithm for POMDPs. In Proc. Int. Jnt. Conf. on Artificial Intelligence, pages 477–484, 2003. [12] S.P. Singh and R.C. Yee. An upper bound on the loss from approximate optimal-value functions. Machine Learning, 16(3):227–233, 1994. [13] R.D. Smallwood and E.J. Sondik. The optimal control of partially observable Markov processes over a finite horizon. Operations Research, 21:1071–1088, 1973. [14] T. Smith and R. Simmons. Point-based POMDP algorithms: Improved analysis and implementation. In Proc. Uncertainty in Artificial Intelligence, 2005. [15] M.T.J. Spaan and N. Vlassis. A point-based POMDP algorithm for robot planning. In Proc. IEEE Int. Conf. on Robotics & Automation, 2004. [16] N.L. Zhang and W. Zhang. Speeding up the convergence of value iteration in partially observable Markov decision processes. Journal of Artificial Intelligence Research, 14:29–51, 2002.
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Boosting Algorithms for Maximizing the Soft Margin Manfred K. Warmuth∗ Dept. of Engineering University of California Santa Cruz, CA, U.S.A. Karen Glocer Dept. of Engineering University of California Santa Cruz, CA, U.S.A. Gunnar R¨atsch Friedrich Miescher Laboratory Max Planck Society T¨ubingen, Germany Abstract We present a novel boosting algorithm, called SoftBoost, designed for sets of binary labeled examples that are not necessarily separable by convex combinations of base hypotheses. Our algorithm achieves robustness by capping the distributions on the examples. Our update of the distribution is motivated by minimizing a relative entropy subject to the capping constraints and constraints on the edges of the obtained base hypotheses. The capping constraints imply a soft margin in the dual optimization problem. Our algorithm produces a convex combination of hypotheses whose soft margin is within δ of its maximum. We employ relative entropy projection methods to prove an O( ln N δ2 ) iteration bound for our algorithm, where N is number of examples. We compare our algorithm with other approaches including LPBoost, BrownBoost, and SmoothBoost. We show that there exist cases where the number of iterations required by LPBoost grows linearly in N instead of the logarithmic growth for SoftBoost. In simulation studies we show that our algorithm converges about as fast as LPBoost, faster than BrownBoost, and much faster than SmoothBoost. In a benchmark comparison we illustrate the competitiveness of our approach. 1 Introduction Boosting methods have been used with great success in many applications like OCR, text classification, natural language processing, drug discovery, and computational biology [13]. For AdaBoost [7] it was frequently observed that the generalization error of the combined hypotheses kept decreasing after the training error had already reached zero [19]. This sparked a series of theoretical studies trying to understand the underlying principles that govern the behavior of ensemble methods [19, 1]. It became apparent that some of the power of ensemble methods lies in the fact that they tend to increase the margin of the training examples. This was consistent with the observation that AdaBoost works well on low-noise problems, such as digit recognition tasks, but not as well on tasks with high noise. On such tasks, better generalizaton can be achieved by not enforcing a large margin on all training points. This experimental observation was supported by the study of [19], where the generalization error of ensemble methods was bounded by the sum of two terms: the fraction of training points which have a margin smaller than some value ρ plus a complexity term that depends on the base hypothesis class and ρ. While this worst-case bound can only capture part of what is going on in practice, it nevertheless suggests that in some cases it pays to allow some points to have small margin or be misclassified if this leads to a larger overall margin on the remaining points. To cope with this problem, it was necessary to construct variants of AdaBoost which trade off the fraction of examples with margin at least ρ with the size of the margin ρ. This was typically done by preventing the distribution maintained by the algorithm from concentrating too much on the most difficult examples. This idea is implemented in many algorithms including AdaBoost with soft margins [15], MadaBoost [5], ν-Arc [16, 14], SmoothBoost [21], LPBoost [4], and several others (see references in [13]). For some of these algorithms, significant improvements were shown compared to the original AdaBoost algorithm on high noise data. ∗Supported by NSF grant CCR 9821087. 1 In parallel, there has been a significant interest in how the linear combination of hypotheses generated by AdaBoost is related to the maximum margin solution [1, 19, 4, 18, 17]. It was shown that AdaBoost generates a combined hypothesis with a large margin, but not necessarily the maximum hard margin [15, 18]. This observation motivated the development of many Boosting algorithms that aim to maximize the margin [1, 8, 4, 17, 22, 18]. AdaBoost∗[17] and TotalBoost [22] provable converge to the maximum hard margin within precision δ in 2 ln(N/δ2) iterations. The other algorithms have worse or no known convergence rates. However, such margin-maximizing algorithms are of limited interest for a practitioner working with noisy real-world data sets, as overfitting is even more problematic for such algorithms than for the original AdaBoost algorithm [1, 8]. In this work we combine these two lines of research into a single algorithm, called SoftBoost, that for the first time implements the soft margin idea in a practical boosting algorithm. SoftBoost finds in O(ln(N)/δ2) iterations a linear combination of base hypotheses whose soft margin is at least the optimum soft margin minus δ. BrownBoost [6] does not always optimize the soft margin. SmoothBoost and MadaBoost can be related to maximizing the soft margin, but while they have known iterations bounds in terms of other criteria, it is unknown how quickly they converge to the maximum soft margin. From a theoretical point of view the optimization problems underlying SoftBoost as well as LPBoost are appealing, since they directly maximize the margin of a (typically large) subset of the training data [16]. This quantity plays a crucial role in the generalization error bounds [19]. Our new algorithm is most similar to LPBoost because its goal is also to optimize the soft margin. The most important difference is that we use slightly relaxed constraints and a relative entropy to the uniform distribution as the objective function. This leads to a distribution on the examples that is closer to the uniform distribution. An important result of our work is to show that this strategy may help to increase the convergence speed: We will give examples where LPBoost converges much more slowly than our algorithm—linear versus logarithmic growth in N. The paper is organized as follows: in Section 2 we introduce the notation and the basic optimization problem. In Section 3 we discuss LPBoost and give a separable setting where N/2 iterations are needed by LPBoost to achieve a hard margin within precision .99. In Section 4 we present our new SoftBoost algorithm and prove its iteration bound. We provide an experimental comparison of the algorithms on real and synthetic data in Section 5, and conclude with a discussion in Section 6. 2 Preliminaries In the boosting setting, we are given a set of N labeled training examples (xn, yn), n = 1 . . . N, where the instances xn are in some domain X and the labels yn ∈±1. Boosting algorithms maintain a distribution d on the N examples, i.e. d lies in the N dimensional probability simplex PN. Intuitively, the hard to classify examples receive more weight. In each iteration, the algorithm gives the current distribution to an oracle (a.k.a. base learning algorithm), which returns a new base hypothesis h : X →[−1, 1]N with a certain guarantee of performance. This guarantee will be discussed at the end of this section. One measure of the performance of a base hypothesis h with respect to distribution d is its edge, γh = PN n=1 dnynh(xn). When the range of h is ±1 instead of the interval [-1,1], then the edge is just an affine transformation of the weighted error ǫh of hypothesis h: i.e. ǫh(d) = 1 2 −1 2γh. A hypothesis that predicts perfectly has edge γ = 1, a hypothesis that always predicts incorrectly has edge γ = −1, and a random hypothesis has edge γ ≈0. The higher the edge, the more useful is the hypothesis for classifying the training examples. The edge of a set of hypotheses is defined as the maximum edge of the set. After a hypothesis is received, the algorithm must update its distribution d on the examples. Boosting algorithms (for the separable case) commonly update their distribution by placing a constraint on the edge of most recent hypothesis. Such algorithms are called corrective [17]. In totally corrective updates, one constrains the distribution to have small edge with respect to all of the previous hypotheses [11, 22]. The update developed in this paper is an adaptation of the totally corrective update of [22] that handles the inseparable case. The final output of the boosting algorithm is always a convex combination of base hypotheses fw(xn) = PT t=1 wtht(xn), where ht is the hypothesis added at iteration t and wt is its coefficient. The margin of a labeled example (xn, yn) is defined as 2 ρn = ynfw(xn). The (hard) margin of a set of examples is taken to be the minimum margin of the set. It is convenient to define an N-dimensional vector um that combines the base hypothesis hm with the labels yn of the N examples: um n := ynhm(xn). With this notation, the edge of the t-th hypothesis becomes d · ut and the margin of the n-th example w.r.t. a convex combination w of the first t −1 hypotheses is Pt−1 m=1 um n wm. For a given set of hypotheses {h1, . . . , ht}, the following linear programming problem (1) optimizes the minimum soft margin. The term “soft” here refers to a relaxation of the margin constraint. We now allow examples to lie below the margin but penalize them linearly via slack variables ψn. The dual problem (2) minimizes the maximum edge when the distribution is capped with 1/ν, where ν ∈{1, . . ., N}: ρ∗ t (ν) = max w,ρ,ψ ρ −1 ν XN n=1 ψn (1) s.t. Xt m=1 um n wm ≥ρ −ψn, for 1 ≤n ≤N, w ∈Pt, ψ ≥0. γ∗ t (ν) = min d,γ γ (2) s.t. d · um ≤γ, for 1 ≤m ≤t, d ∈PN, d ≤1 ν 1. By duality, ρ∗ t (ν) = γ∗ t (ν). Note that the relationship between capping and the hinge loss has long been exploited by the SVM community [3, 20] and has also been used before for Boosting in [16, 14]. In particular, it is known that ρ in (1) is chosen such that N −ν examples have margin at least ρ. This corresponds to ν active constraints in (2). The case ν = 1 is degenerate: there are no capping constraints in (2) and this is equivalent to the hard margin case.1 Assumption on the weak learner We assume that for any distribution d ≤1 ν 1 on the examples, the oracle returns a hypothesis h with edge at least g, for some fixed g. This means that for the corresponding u vector, d · u ≥g. For binary valued features, this is equivalent to the assumption that the base learner always returns a hypothesis with error at most 1 2 −1 2g. Adding a new constraint can only increase the value γ∗ t (ν) of the minimization problem (2) and therefore γ∗ t (ν) is non-decreasing in t. It is natural to define γ∗(ν) as the value of (2) w.r.t. the entire hypothesis set from which the oracle can choose. Clearly γ∗ t (ν) approaches γ∗(ν) from below. Also, the guarantee g of the oracle can be at most γ∗(ν) because for the optimal distribution d∗that realizes γ∗(ν), all hypotheses have edge at most γ∗(ν). For computational reasons, g might however be lower than γ∗(ν) and in that case the optimum soft margin we can achieve is g. 3 LPBoost In iteration t, the LPBoost algorithm [4] sends its current distribution dt−1 to the oracle and receives a hypothesis ht that satisfies dt−1 ·ut ≥g. It then updates its distribution to dt by solving the linear programming problem (1) based on the t hypotheses received so far. The goal of the boosting algorithms is to produce a convex combination of T hypotheses such that γT (ν) ≥g −δ. The simplest way to achieve this is to break when this condition is satisfied. Although the guarantee g is typically not known, it is upper bounded by bγt = min1≤m≤t dt−1 · ut and therefore LPBoost uses the more stringent stopping criterion γt(ν) ≥bγt −δ. To our knowledge, there is no known iteration bound for LPBoost even though it provably converges to the δ-optimal solution of the optimization problem after it has seen all hypotheses [4, 10]. Empirically, the convergence speed depends on the linear programming optimizer, e.g. simplex or interior point solver [22]. For the first time, we are able to establish a lower bound showing that, independent of the optimizer, LPBoost can require Ω(N) iterations: Theorem 1 There exists a case where LPBoost requires N/2 iterations to achieve a hard margin that is within δ = .99 of the optimum hard margin. Proof. Assume we are in the hard margin case (ν = 1). The counterexample has N examples and N 2 + 1 base hypothesis. After N 2 iterations, the optimal value γ∗ t (1) for the chosen hypotheses will 1Please note that [20] have previously used the parameter ν with a slightly different meaning, namely ν/N in our notation. We use an unnormalized version of ν denoting a number of examples instead of a fraction. 3 Algorithm 1 LPBoost with accuracy param. δ and capping parameter ν 1. Input: S = ⟨(x1, y1), . . . , (xN, yN)⟩, accuracy δ, capping parameter ν ∈[1, N]. 2. Initialize: d0 to the uniform distribution and bγ0 to 1. 3. Do for t = 1, . . . (a) Send dt−1 to oracle and obtain hypothesis ht. Set ut n = ht(xn)yn and bγt = min{bγt−1, dt−1 · ut}. (Assume dt−1 · ut ≥g, where edge guarantee g is unknown.) (b) Update the distribution to any dt that solves the LP problem (dt, γ∗ t ) = argmin d,γ γ s.t. d · um ≤γ, for 1 ≤m ≤t; d ∈PN, d ≤1 ν 1. (c) If γ∗ t ≥bγt −δ then set T = t and break.2 4. Output: fw(x) = PT m=1 wmhm(x), where the coefficients wm maximize the soft margin over the hypothesis set {h1, . . . , hT } using the LP problem (1). 2When g is known, then one can break already when γ∗ t (ν) ≥g −δ. still be close to −1, whereas after the last hypothesis is added, this value is at least ǫ/2. Here ǫ is a precision parameter that is an arbitrary small number. Figure 1 shows the case where N = 8 and T = 5, but it is trivial to generalize this example to any even N. There are 8 examples/rows and the five columns are the ut’s of the five available base hypotheses. The examples are separable because if we put half of the weight on the first and last hypothesis, then n \ t 1 2 3 4 5 1 +1 −1 + 5ǫ −1 + 7ǫ −1 + 9ǫ −1 + ǫ 2 +1 −1 + 5ǫ −1 + 7ǫ −1 + 9ǫ −1 + ǫ 3 +1 −1 + 5ǫ −1 + 7ǫ −1 + 9ǫ −1 + ǫ 4 +1 −1 + 5ǫ −1 + 7ǫ −1 + 9ǫ −1 + ǫ 5 −1 + 2ǫ +1 −1 + 7ǫ −1 + 9ǫ +1 −ǫ 6 −1 + 3ǫ −1 + 4ǫ +1 −1 + 9ǫ +1 −ǫ 7 −1 + 3ǫ −1 + 5ǫ −1 + 6ǫ +1 +1 −ǫ 8 −1 + 3ǫ −1 + 5ǫ −1 + 7ǫ −1 + 8ǫ +1 −ǫ γ∗ t (1) −1 + 2ǫ −1 + 4ǫ −1 + 6ǫ −1 + 8ǫ ≥ǫ/2 Figure 1: The ut vectors that are hard for LPBoost (for ν = 1). the margins of all examples are at least ǫ/2. We assume that in each iteration the oracle will return the remaining hypothesis with maximum edge. This will result in LPBoost choosing the hypotheses in order, and there will never be any ties. The initial distribution d0 is uniform. At the end of iteration t (1 ≤t ≤N/2), the distribution dt will focus all its weight on example N/2 + t, and the optimum mixture of the columns will put all of its weight on the tth hypothesis that was just received. In other words the value will be the bolded entries in Figure 1: −1 + 2ǫt at the end of iteration t = 1, . . . , N/2. After N/2 iterations the value γ∗ t (1) of the underlying LP problem will still be close to −1, because ǫ can be made arbitrary small. We reasoned already that the value for all N/2 + 1 hypotheses will be positive. So if ǫ is small enough, then after N/2 iterations LPBoost is still at least .99 away from the optimal solution. □ Although the example set used in the above proof is linearly separable, we can modify it explicitly to argue that capping the distribution on examples will not help in the sense that “soft” LPBoost with ν > 1 can still have linear iteration bounds. To negate the effect of capping, simply pad out the problem by duplicating all of the rows ν times. There will now be ˜N = Nν examples, and after N 2 = ˜ N 2ν iterations, the value of the game is still close to −1. This is not a claim that capping has no value. It remains an important technique for making an algorithm more robust to noise. However, it is not sufficient to improve the iteration bound of LPBoost from linear growth in N to logarithmic. Another attempt might be to modify LPBoost so that at each iteration a base hypothesis is chosen that increases the value of the optimization problem the most. Unfortunately we found similar Ω(N) counter examples to this heuristic (not shown). It is also easy to see that the algorithms related to the below SoftBoost algorithm choose the last hypothesis after first and finish in just two iterations. 4 Algorithm 2 SoftBoost with accuracy param. δ and capping parameter ν 1. Input: S = ⟨(x1, y1), . . . , (xN, yN)⟩, desired accuracy δ, and capping parameter ν ∈[1, N]. 2. Initialize: d0 to the uniform distribution and bγ0 to 1. 3. Do for t = 1, . . . (a) Send dt−1 to the oracle and obtain hypothesis ht. Set ut n = ht(xn)yn and bγt = min{bγt−1, dt−1 · ut}. (Assume dt−1 · ut ≥g, where edge guarantee g is unknown.) (b) Update3 dt = argmin d ∆(d, d0), s.t. d·um ≤bγt−δ, for 1 ≤m ≤t, X n dn = 1, d ≤1 ν 1. (c) If above infeasible or dt contains a zero then T = t and break. 4. Output: fw(x) = PT m=1 wmhm(x), where the coefficients wm maximize the soft margin over the hypothesis set {h1, . . . , ht} using the LP problem (1). 3 When g is known, replace the upper bound bγt −δ by g −δ. 4 SoftBoost In this section, we present the SoftBoost algorithm, which adds capping to the TotalBoost algorithm of [22]. SoftBoost takes as input a sequence of examples S = ⟨(x1, y1), . . . , (xN, yN)⟩, an accuracy parameter δ, and a capping parameter ν. The algorithm has an oracle available with unknown guarantee g. Its initial distribution d0 is uniform. In each iteration t, the algorithm prompts the oracle for a new base hypothesis, incorporates it into the constraint set, and updates its distribution dt−1 to dt by minimizing the relative entropy ∆(d, d0) := P n dn ln dn d0n subject to linear constraints: dt+1 = argmind ∆(d, d0) s.t. d · um ≤bγt −δ, for 1 ≤m ≤t (where bγt = min1≤m≤t dm−1 · um), P n dn = 1, d ≤1 ν 1. It is easy to solve this optimization problem with vanilla sequential quadratic programming methods (see [22] for details). Observe that removing the relative entropy term from the objective, results in a feasibility problem for linear programming where the edges are upper bounded by bγt −δ. If we remove the relative entropy and minimize the upper bound on the edges, then we arrive at the optimization problem of LPBoost, and logarithmic growth in the number of examples is no longer possible. The relative entropy in the objective assures that the probabilities of the examples are always proportional to their exponentiated negative soft margins (not shown). That is, more weight is put on the examples with low soft margin, which are the examples that are hard to classify. 4.1 Iteration bounds for SoftBoost Our iteration bound for SoftBoost is very similar to the bound proven for TotalBoost [22], differing only in the additional details related to capping. Theorem 2 SoftBoost terminates after at most ⌈2 δ2 ln(N/ν)⌉iterations with a convex combination that is at most δ below the optimum value g. Proof. We begin by observing that if the optimization problem at iteration t is infeasible, then γ∗ t (ν) > bγt −δ ≥g −δ. Also if dt contains a zero, then since the objective function ∆(d, d0) is strictly convex in d and minimized at the interior point d0, there is no optimal solution in the interior of the simplex. Hence, γ∗ t (ν) = bγt −δ ≥g −δ. Let Ct be the convex subset of probability vectors d ∈PN satisfying d ≤1 ν 1 and maxt m=1 d·ut ≤ bγt −δ. Notice that C0 is the N dimensional probability simplex where the components are capped to 1 ν . The distribution dt−1 at iteration t −1 is the projection of d0 onto the closed convex set Ct−1. Because adding a new hypothesis in iteration t results in an additional constraint and bγt ≤bγt−1, 5 we have Ct ⊆Ct−1. If t ≤T −1, then our termination condition assures that at iteration t −1 the set Ct−1 has a feasible solution in the interior of the simplex. Also, d0 lies in the interior and dt ∈Ct ⊆Ct−1. These preconditions assure that at iteration t −1, the projection dt−1 of d0 onto Ct−1, exists and the Generalized Pythagorean Theorem for Bregman divergences [2, 9] is applicable: ∆(dt, d0) −∆(dt−1, d0) ≥∆(dt, dt−1). (3) By Pinsker’s inequality, ∆(dt, dt−1) ≥(||dt−dt−1||1)2 2 , and by H¨older’s inequality, ||dt−1−dt||1 ≥ ||dt−1 −dt||1||ut||∞≥dt−1 · ut −dt · ut. Also dt−1 · ut ≥bγt by the definition of bγt, and the constraints on the optimization problem assure that dt · ut ≤bγt −δ and thus dt−1 · ut −dt · ut ≥ bγt−(bγt−δ) = δ. We conclude that ∆(dt, dt−1) ≥δ2 2 at iterations 1 through T −1. By summing (3) over the first T −1 iterations, we obtain ∆(dT , d0) −∆(d0, d0) ≥(T −1)δ2 2 . Since the left side is at most ln(N/ν), the bound of the theorem follows. □ When ν = 1, then capping is vacuous and the algorithm and its iteration bound coincides with the bound for TotalBoost. Note that the upper bound ln(N/ν) on the relative entropy decreases with ν. When ν = N, then the distribution stays at d0 and the iteration bound is zero. 5 Experiments In a first study, we use experiments on synthetic data to illustrate the general behavior of the considered algorithms.2 We generated a synthetic data set by starting with a random matrix of 2000 rows and 100 columns, where each entry was chosen uniformly in [0, 1]. For the first 1000 rows, we added 1/2 to the first 10 columns and rescaled such that the entries in those columns were again in [0, 1]. The rows of this matrix are our examples and the columns and their negation are the base hypotheses, giving us a total of 200 of them. The first 1000 examples were labeled +1 and the rest −1. This results in a well separable dataset. To illustrate how the algorithms deal with the inseparable case, we flipped the sign of a random 10% of the data set. We then chose a random 500 examples as our training set and the rest as our test set. In every boosting iteration we chose the base hypothesis which has the largest edge with respect to the current distribution on the examples. We have trained LPBoost and SoftBoost for different values of ν and recorded the generalization error (cf. Figure 2; δ = 10−3). We should expect that for small ν (e.g. ν/N < 10%) the data is not easily separable, even when allowing ν wrong predictions. Hence the algorithm may mistakenly concentrate on the random directions for discrimination. If ν is large enough, most incorrectly labeled examples are likely to be identified as margin errors (ψi > 0) and the performance should stabilize. In Figure 2 we observe this expected behavior and also that for large ν the classification performance decays again. The generalization performances of LPBoost and SoftBoost are very similar, which is expected as they both attempt to maximize the soft-margin. Using the same data set, we analysed the convergence speed of several algorithms: LPBoost, SoftBoost, BrownBoost, and SmoothBoost. We chose δ = 10−2 and ν = 200.3 For every iteration we record all margins and compute the soft margin objective (1) for optimally chosen ρ and ψ’s. Figure 3 plots this value against the number of iterations for the four algorithms. SmoothBoost takes dramatically longer to converge to the maximum soft margin than the other other three algorithms. In our experiments it nearly converges to the maximum soft margin objective, even though no theoretical evidence is known for this observed convergence. Among the three remaining algorithms, LPBoost and SoftBoost converge in roughly the same number of iterations, but SoftBoost has a slower start. BrownBoost terminates in fewer iterations than the other algorithms but does not maximize the soft margin.4 This is not surprising as there is no theoretical reason to expect such a result. 2Our code is available at https://sourceforge.net/projects/nboost 3Smaller choices of ν lead to an even slower convergence of SmoothBoost. 4SmoothBoost has two parameters: a guarantee g on the edge of the base learner and the target margin θ. We chose g = γ∗(ν) (computed with LPBoost) and θ = g/2 2+g/2 as proposed in [21]. Brownboost’s one parameter, c = 0.35, was chosen via cross-validation. 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.06 0.08 0.1 0.12 0.14 0.16 0.18 ← LPBoost SoftBoost → ν/N classification error Figure 2: Generalization performance of SoftBoost (solid) and LPBoost (dotted) on a synthetic data set with 10% label-noise for different values of ν. 10 0 10 1 10 2 10 3 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 LPBoost → SmoothBoost → ← BrownBoost ← SoftBoost number of iterations soft margin objective Figure 3: Soft margin objective vs. the number of iterations for LPBoost, SoftBoost, BrownBoost and SmoothBoost. Finally, we present a small comparison on ten benchmark data sets derived from the UCI benchmark repository as previously used in [15]. We analyze the performance of AdaBoost, LPBoost, SoftBoost, BrownBoost [6] and AdaBoostReg [15] using RBF networks as base learning algorithm.5 The data comes in 100 predefined splits into training and test sets. For each of the splits we use 5-fold cross-validation to select the optimal regularization parameter for each of the algorithms. This leads to 100 estimates of the generalization error for each method and data set. The means and standard deviations are given in Table 1.6 As before, the generalization performances of SoftBoost and LPBoost are very similar. However, the soft margin algorithms outperform AdaBoost on most data sets. The genaralization error of BrownBoost lies between that of AdaBoost and SoftBoost. AdaBoostReg performs as well as SoftBoost, but there are no iteration bounds known for this algorithm. Even though SoftBoost and LPBoost often have similar generalization error on natural datasets, the number of iterations needed by both algorithms can be radically different (see Theorem 1). Also, in [22] there are some artificial data sets where TotalBoost (i.e. SoftBoost with ν = 1) outperformed LPBoost i.t.o. generalization error. AdaBoost LPBoost SoftBoost BrownBoost AdaBoost reg Banana 13.3 ± 0.7 11.1 ± 0.6 11.1 ± 0.5 12.9 ± 0.7 11.3 ± 0.6 B.Cancer 32.1 ± 3.8 27.8 ± 4.3 28.0 ± 4.5 30.2 ± 3.9 27.3 ± 4.3 Diabetes 27.9 ± 1.5 24.4 ± 1.7 24.4 ± 1.7 27.2 ± 1.6 24.5 ± 1.7 German 26.9 ± 1.9 24.6 ± 2.1 24.7 ± 2.1 24.8 ± 1.9 25.0 ± 2.2 Heart 20.1 ± 2.7 18.4 ± 3.0 18.2 ± 2.7 20.0 ± 2.8 17.6 ± 3.0 Ringnorm 1.9 ± 0.3∗ 1.9 ± 0.2 1.8 ± 0.2 1.9 ± 0.2 1.7 ± 0.2 F.Solar 36.1 ± 1.5 35.7 ± 1.6 35.5 ± 1.4 36.1 ± 1.4 34.4 ± 1.7 Thyroid 4.4 ± 1.9∗ 4.9 ± 1.9 4.9 ± 1.9 4.6 ± 2.1 4.9 ± 2.0 Titanic 22.8 ± 1.0 22.8 ± 1.0 23.0 ± 0.8 22.8 ± 0.8 22.7 ± 1.0 Waveform 10.5 ± 0.4 10.1 ± 0.5 9.8 ± 0.5 10.4 ± 0.4 10.4 ± 0.7 Table 1: Generalization error estimates and standard deviations for ten UCI benchmark data sets. SoftBoost and LPBoost outperform AdaBoost and BrownBoost on most data sets. 6 Conclusion We prove by counterexample that LPBoost cannot have an O(ln N) iteration bound. This counterexample may seem similar to the proof that the Simplex algorithm for LP can take exponentially more steps than interior point methods. However this similarity is only superficial. First, our iteration bound does not depend on the LP solver used within LPBoost. This is because in the construction, the interim solutions are always unique and thus all LP solvers will produce the same solution. Second, the iteration bound essentially says that column generation methods (of which LPBoost is a canonical example) should not solve the current subproblem at iteration t optimally. Instead a good algorithm should loosen the constraints and spread the weight via a regularization such as the relative entropy. These two tricks used by the SoftBoost algorithm make it possible to obtain iteration 5The data is from http://theoval.cmp.uea.ac.uk/∼gcc/matlab/index.shtml. The RBF networks were obtained from the authors of [15], including the hyper-parameter settings for each data set. 6Note that [15] contains a similar benchmark comparison. It is based on a different model selection setup leading to underestimates of the generalization error. Presumably due to slight differences in the RBF hyperparameters settings, our results for AdaBoost often deviate by 1-2%. 7 bounds that grow logarithmic in N. The iteration bound for our algorithm is a straightforward extension of a bound given in [22] that is based on Bregman projection methods. By using a different divergence in SoftBoost, such as the sum of binary relative entropies, the algorithm morphs into a “soft” version of LogitBoost (see discussion in [22]) which has essentially the same iteration bound as SoftBoost. We think that the use of Bregman projections illustrates the generality of the methods. 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Semi-infinite programming: Theory, methods and applications. SIAM Review, 3:380–429, September 1993. [11] J. Kivinen and M. K. Warmuth. Boosting as entropy projection. In Proc. 12th Annu. Conference on Comput. Learning Theory, pages 134–144. ACM Press, New York, NY, 1999. [12] J. Liao. Totally Corrective Boosting Algorithms that Maximize the Margin. PhD thesis, University of California at Santa Cruz, December 2006. [13] R. Meir and G. R¨atsch. An introduction to boosting and leveraging. In S. Mendelson and A. Smola, editors, Proc. 1st Machine Learning Summer School, Canberra, LNCS, pages 119–184. Springer, 2003. [14] G. R¨atsch. Robust Boosting via Convex Optimization: Theory and Applications. PhD thesis, University of Potsdam, Germany, December 2001. [15] G. R¨atsch, T. Onoda, and K.-R. M¨uller. Soft margins for AdaBoost. Machine Learning, 42(3):287–320, 2001. [16] G. R¨atsch, B. Sch¨olkopf, A.J. Smola, S. Mika, T. Onoda, and K.-R. M¨uller. Robust ensemble learning. In A.J. 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Ensemble Clustering using Semidefinite Programming Vikas Singh Biostatistics and Medical Informatics University of Wisconsin – Madison vsingh @ biostat.wisc.edu Lopamudra Mukherjee Computer Science and Engineering State University of New York at Buffalo lm37 @ cse.buffalo.edu Jiming Peng Industrial and Enterprise System Engineering University of Illinois at Urbana-Champaign pengj @ uiuc.edu Jinhui Xu Computer Science and Engineering State University of New York at Buffalo jinhui @ cse.buffalo.edu Abstract We consider the ensemble clustering problem where the task is to ‘aggregate’ multiple clustering solutions into a single consolidated clustering that maximizes the shared information among given clustering solutions. We obtain several new results for this problem. First, we note that the notion of agreement under such circumstances can be better captured using an agreement measure based on a 2D string encoding rather than voting strategy based methods proposed in literature. Using this generalization, we first derive a nonlinear optimization model to maximize the new agreement measure. We then show that our optimization problem can be transformed into a strict 0-1 Semidefinite Program (SDP) via novel convexification techniques which can subsequently be relaxed to a polynomial time solvable SDP. Our experiments indicate improvements not only in terms of the proposed agreement measure but also the existing agreement measures based on voting strategies. We discuss evaluations on clustering and image segmentation databases. 1 Introduction In the so-called Ensemble Clustering problem, the target is to ‘combine’ multiple clustering solutions or partitions of a set into a single consolidated clustering that maximizes the information shared (or ‘agreement’) among all available clustering solutions. The need for this form of clustering arises in many applications, especially real world scenarios with a high degree of uncertainty such as image segmentation with poor signal to noise ratio and computer assisted disease diagnosis. It is quite common that a single clustering algorithm may not yield satisfactory results, while multiple algorithms may individually make imperfect choices, assigning some elements to wrong clusters. Usually, by considering the results of several different clustering algorithms together, one may be able to mitigate degeneracies in individual solutions and consequently obtain better solutions. The idea has been employed successfully for microarray data classification analysis [1], computer assisted diagnosis of diseases [2] and in a number of other applications [3]. Formally, given a data set D = {d1, d2, . . . , dn}, a set of clustering solutions C = {C1, C2, . . . , Cm} obtained from m different clustering algorithms is called a cluster ensemble. Each solution, Ci, is the partition of the data into at most k different clusters. The Ensemble Clustering problem requires one to use the individual solutions in C to partition D into k clusters such that information shared (‘agreement’) among the solutions of C is maximized. 1.1 Previous works The Ensemble Clustering problem was recently introduced by Strehl and Ghosh [3], in [4] a related notion of correlation clustering was independently proposed by Bansal, Blum, and Chawla. The problem has attracted a fair amount of attention and a number of interesting techniques have been proposed [3, 2, 5, 6], also see [7, 4]. Formulations primarily differ in how the objective of shared information maximization or agreement is chosen, we review some of the popular techniques next. The Instance Based Graph Formulation (IBGF) [2, 5] first constructs a fully connected graph G = (V, W) for the ensemble C = (C1, . . . , Cm), each node represents an element of D = {d1, . . . , dn}. The edge weight wij for the pair (di, dj) is defined as the number of algorithms in C that assign the nodes di and dj to the same cluster (i.e., wij measures the togetherness frequency of di and dj). Then, standard graph partitioning techniques are used to obtain a final clustering solution. In Cluster Based Graph Formulation (CBGF), a given cluster ensemble is represented as C = {C11, . . . , Cmk} = { ¯C1, . . . , ¯Cm·k} where Cij denotes the ith cluster of the jth algorithm in C. Like IBGF, this approach also constructs a graph, G = (V, W), to model the correspondence (or ‘similarity’) relationship among the mk clusters, where the similarity matrix W reflects the Jaccard’s similarity measure between clusters ¯Ci and ¯Cj. The graph is then partitioned so that the clusters of the same group are similar to one another. Variants of the problem have also received considerable attention in the theoretical computer science and machine learning communities. A recent paper by Ailon, Charikar, and Newman [7] demonstrated connections to other well known problems such as Rank Aggregation, their algorithm is simple and obtains an expected constant approximation guarantee (via linear programming duality). In addition to [7], other results include [4, 8]. A commonality of existing algorithms for Ensemble Clustering [3, 2, 9] is that they employ a graph construction, as a first step. Element pairs (cluster pairs or item pairs) are then evaluated and their edges are assigned a weight that reflects their similarity. A natural question relates to whether we can find a better representation of the available information. This will be the focus of the next section. 2 Key Observations: Two is a company, is three a crowd? Consider an example where one is ‘aggregating’ recommendations made by a group of family and friends for dinner table seating assignments at a wedding. The hosts would like each ‘table’ to be able to find a common topic of dinner conversation. Now, consider three persons, Tom, Dick, and Harry invited to this reception. Tom and Dick share a common interest in Shakespeare, Dick and Harry are both surfboard enthusiasts, and Harry and Tom attended college together. Because they had strong pairwise similarities, they were seated together but had a rather dull evening. A simple analysis shows that the three guests had strong common interests when considered two at a time, but there was weak communion as a group. The connection of this example to the ensemble clustering problem is clear. Existing algorithms represent the similarity between elements in D as a scalar value assigned to the edge joining their corresponding nodes in the graph. This weight is essentially a ‘vote’ reflecting the number of algorithms that assigned those two elements to the same cluster. The mechanism seems perfect until we ask if strong pairwise coupling necessarily implies coupling for a larger group as well. The weight metric considering two elements does not retain information about which algorithms assigned them together. When expanding the group to include more elements, one is not sure if a common feature exists under which the larger group is similar. It seems natural to assign a higher priority to triples or larger groups of people that were recommended to be seated together (must be similar under at least one feature) compared to groups that were never assigned to the same table by any person in the recommendation group (clustering algorithm), notwithstanding pairwise evaluations, for an illustrative example see [10]. While this problem seems to be a distinctive disadvantage for only the IBGF approach; it also affects the CBGF approach. This can be seen by looking at clusters as items and the Jaccard’s similarity measure as the vote (weight) on the edges. 3 Main Ideas To model the intuition above, we generalize the similarity metric to maximize similarity or ‘agreement’ by an appropriate encoding of the solutions obtained from individual clustering algorithms. More precisely, in our generalization the similarity is no longer just a scalar value but a 2D string. The ensemble clustering problem thus reduces to a form of string clustering problem where our objective is to assign similar strings to the same cluster. The encoding into a string is done as follows. The data item set is given as D with |D| = n. Let m be the number of clustering algorithms with each solution having no more than k clusters. We represent all input information (ensemble) as a single 3D matrix, A ∈ℜn×m×k. For every data element dl ∈D, Al ∈ℜm×k is a matrix whose elements are defined by Al(i, j) = 1 if dl is assigned to cluster i by Cj; 0 otherwise (1) It is easy to see that the summation of every row of Al equals 1. We call each Al an A-string. Our goal is to cluster the elements D = {d1, d2, . . . , dn} based on the similarity of their A-strings. We now consider how to compute the clusters based on the similarity (or dissimilarity) of strings. We note that the paper [11] by Gasieniec et al., discussed the so-called Hamming radius p-clustering and Hamming diameter p-clustering problems on strings. Though their results shed considerable light on the hardness of string clustering with the selected distance measures, those techniques cannot be directly applied to the problem at hand because the objective here is fairly different from the one in [11]. Fortunately, our analysis reveals that a simpler objective is sufficient to capture the essence of similarity maximization in clusters using certain special properties of the A-strings. Our approach is partly inspired by the classical k-means clustering where all data points are assigned to the cluster based on the shortest distance to the cluster center. Imagine an ideal input instance for the ensemble clustering problem (all clustering algorithms behave similarly) – one with only k unique members among n A-strings. The partitioning simply assigns similar strings to the same partition. The representative for each cluster will then be exactly like its members, is a valid Astring, and can be viewed as a center in a geometric sense. General input instances will obviously be non-ideal and are likely to contain far more than k unique members. Naturally, the centers of the clusters will vary from its members. This variation can be thought of as noise or disagreement within the clusters, our objective is to find a set of clusters (and centers) such that the noise is minimized and we move very close to the ideal case. To model this, we consider the centers to be in the same high dimensional space as the A-strings in D (though it may not belong to D). Consider an example where a cluster i in this optimal solution contains items (d1, d2, . . . , d7). A certain algorithm Cj in the input ensemble clusters items (d1, d2, d3, d4) in cluster s and (d5, d6, d7) in cluster p. How would Cj behave if evaluating the center of cluster i as a data item? The probability it assigns the center to cluster s is 4/7 and the probability it assigns the center to cluster p is 3/7. If we emulate this logic – we must pick the choice with the higher probability and assign the center to such a cluster. It can be verified that this choice minimizes the dissent of all items in cluster i to the center. The Astring for the center of cluster i will have a “1” at position (j, s). The assignment of A-string (items) to clusters is unknown; however, if it were somehow known, we could find the centers for all other clusters i ∈[1, k] by computing the average value at every cell of the A matrices corresponding to the members of the cluster and rounding the largest value in every row to 1 (rest to 0) and assigning this as the cluster center. Hence, the dissent within a cluster can be quantified simply by averaging the matrices of elements that belong to the cluster and computing the difference to the center. Our goal is to find such an assignment and group the A-strings so that the sum of the absolute differences of the averages of clusters to their centers (dissent) is minimized. In the subsequent sections, we will introduce our optimization framework for ensemble clustering based on these ideas. 4 Integer Program for Model 1 We start with a discussion of an Integer Program (IP, for short) formulation for ensemble clustering. For convenience, we denote the final clustering solution by C∗= {C∗ 1, . . . , C∗ k} and Cij denotes the cluster i by the algorithm j. The variables that constitute the IP are as follows. Xli′ = 1 if dl ∈C∗ i′; 0 otherwise (2) siji′ = ( 1 if C∗ i′ = arg max i=1,...,k{|C∗ i′ T Cij|} 0 otherwise (3) We mention that the above definition implies that for a fixed index i′, its center, siji′ also provides an indicator to the cluster most similar to C∗ i′ in the set of clusters produced by the clustering algorithm Cj. We are now ready to introduce the following IP. min k X i′=1 k X i=1 m X j=1 siji′ − Pn l=1 AlijXli′ Pn l=1 Xli′ (4) s.t. k X i′=1 Xli′ = 1 ∀l ∈[1, n], n X l=1 Xli′ ≥1 ∀i′ ∈[1, k], (5) k X i=1 siji′ = 1 ∀j ∈[1, m], i′ ∈[1, k], Xli′ ∈{0, 1}, siji′ ∈{0, 1}. (6) (4) minimizes the sum of the difference between siji′ (the center for cluster C∗ i′) and the average of all Alij bits of the data elements dl assigned to cluster C∗ i′. Recall that siji′ will be 1 if Cij is the most similar cluster to C∗ i′ among all the clusters produced by algorithm Cj. Hence, if siji′ = 0 and Pn l=1 AlijXli′ Pn l=1 Xli′ ̸= 0, the value siji′ − Pn l=1 AlijXli′ Pn l=1 Xli′ represents the percentage of data elements in C∗ i′ that do not consent with the majority of the other elements in the group w.r.t. the clustering solution provided by Cj. In other words, we are trying to minimize the dissent and maximize the consent simultaneously. The remaining constraints are relatively simple – (5) enforces the condition that a data element should belong to precisely one cluster in the final solution and that every cluster must have size at least 1; (6) ensures that siji′ is an appropriate A-string for every cluster center. 5 0-1 Semidefinite Program for Model 1 The formulation given by (4)-(6) is a mixed integer program (MIP, for short) with a nonlinear objective function in (4). Solving this model optimally, however, is extremely challenging – (a) the constraints in (5)-(6) are discrete; (b) the objective is nonlinear and nonconvex. One possible way of attacking the problem is to ‘relax’ it to some polynomially solvable problems such as SDP (the problem of minimizing a linear function over the intersection of a polyhedron and the cone of symmetric and positive semidefinite matrices, see [12] for an introduction). Our effort would be to convert the nonlinear form in (4) into a 0-1 SDP form. By introducing artificial variables, we rewrite (4) as min k X i=1 m X j=1 k X i′=1 tiji′ (7) siji′ −ciji′ ≤tiji′, ciji′ −siji′ ≤tiji′ ∀i, i′, j, (8) where the term ciji′ represents the second term in (4) defined by ciji′ = Pn l=1 AlijXli′ Pn l=1 Xli′ ∀i, i′, j. (9) Since both Alij and Xli′ are binary, (9) can be rewritten as ciji′ = Pn l=1 A2 lijX2 li′ Pn l=1 X2 li′ ∀i, i′, j. (10) Let us introduce a matrix variable yi′ ∈ℜn whose lth column is defined by y(l) i′ = Xli′ pPn l=1 X2 li′ = Xli′ ∥Xi′∥2 . (11) Let Aij ∈ℜn be a vector whose lth element has value Al(i, j). This allows us to represent (10) as ciji′ = tr(BijZi′), Z2 i′ = Zi′, Zi′ ⪰0, (12) where Bij = diag(Aij) is a diagonal matrix with (Bij)ll = Al(i, j), the second and third properties follow from Zi′ = yi′yT i′ being a positive semidefinite matrix. Now, we rewrite the constraints for X in terms of Z. (5) is automatically satisfied by the following constraints on the elements of Zi′. n X l=1 Z(ll) i′ = 1 ∀i′ ∈[1, k], n X l′=1 Z(ll′) i′ ≤1 ∀i′ ∈[1, k], ∀l ∈[1, n]. (13) where Z(uv) i′ refers to the (u, v) entry of matrix Zi′. Since Z′ i is a symmetric projection matrix by construction, (7)-(13) constitute a precisely defined 0-1 SDP that can be expressed in trace form as min k X i′=1 tr(diag(Ti′ek)) (14) s.t. (Si′ −Ti′ −Qi′) ≤0, (Qi′ −Si′ −Ti′) ≤0 ∀i′ ∈[1, k], (15) ( k X i′=1 Zi′)en = en ∀i′ ∈[1, k], tr(Zi′) = 1 ∀i′ ∈[1, k], tr( k X i′=1 Zi′) = k, (16) Si′ek = em ∀i′ ∈[1, k], Z ≥0; Z2 i′ = Zi′; Zi′ = ZT i′ ; Si′ ∈{0, 1}. (17) where Qi′(i, j) = ciji′ = tr(BijZi′), and en ∈ℜn is a vector of all 1s. The experimental results for this model indicate that it performs very well in practice (see [10]). However, because we must solve the model while maintaining the requirement that Si′ be binary (otherwise, the problem becomes ill-posed), a branch and bound type method is needed. Such approaches are widely used in many application areas, but its worst case complexity is exponential in the input data size. In the subsequent sections, we will make several changes to this framework based on additional observations in order to obtain a polynomial algorithm for the problem. 6 Integer Program and 0-1 Semidefinite Program for Model 2 Recall the definition of the variables ciji′, which can be interpreted as the size of the overlap between the cluster C∗ i′ in the final solution and Cij, and is proportional to the cardinality of C∗ i′. Let us define ci∗ji′ = max i=1,...,k ciji′. Let us also define vector variables qji′ whose ith element is siji′ −ciji′. In the IP model 1, we try to minimize the sum of all the L1-norms of qji′. The main difficulty in the previous formulation stems from the fact that ciji′ is a fractional function w.r.t the assignment matrix X. Fortunately, we note that since entries of ciji′ are fractional satisfying Pk i=1 ciji′ = 1 for any fixed j, i′, their sum of squares is maximized when its largest entry is as high as possible. Thus, minimizing the function 1 −Pk i=1(ciji′)2 is a reasonable substitute to minimizing the sum of the L1-norms in the IP model 1. The primary advantage of this observation is that we do not need to know the ‘index’ (i∗) of the maximal element ci∗ji′. As before, X denotes the assignment matrix. We no longer need the variable s, as it can be easily determined from the solution. This yields the following IP. min k X i′=1 m X j=1 ( n X l=1 Xli′) 1 − k X i=1 (ciji′)2 ! (18) s.t. k X i′=1 Xli′ = 1 ∀l ∈[1, n], n X l=1 Xli′ ≥1 ∀i′ ∈[1, k], Xli′ ∈{0, 1}. (19) We next discuss how to transform the above problem to a 0-1 SDP. For this, we first note that the objective function (18) can be expressed as follows. min k X i′=1 m X j=1 ( n X l=1 Xli′) − k X i=1 (Pn l=1 AlijXli′)2 Pn l=1 Xli′ ! , (20) which can be equivalently stated as min nm − k X i′=1 m X j=1 k X i=1 (Pn l=1 AlijXli′)2 Pn l=1 Xli′ , (21) The numerator of the second term above can be rewritten as ( n X l=1 AlijXli′)2 = (A1ijX1i′ + . . . + AnijXni′)2 = (AT ijXi′)2 = XT i′ AijAT ijXi′, (22) where X′ i is the i′th column vector of X. Therefore, the second term of (21) can be written as = tr( k X i′=1 m X j=1 k X i=1 XT i′ AijAT ijXi′(XT i′ Xi′)−1) = tr( k X i′=1 m X j=1 k X i=1 AijAT ijZi′) = tr( m X j=1 k X i=1 AijAT ijZ) = tr( m X j=1 BjZ) = tr(BZ). (23) In (23), Zi′ = Xi′(XT i′ Xi′)−1XT i′ (same as in IP model 1) and Z = Pk i′=1 Z′ i and B = Pm j=1 Bj. Since each matrix Zi′ is a symmetric projection matrix and Xi′ 1 and Xi′ 2 are orthogonal to each other when i′ 1 ̸= i′ 2, Z is a projection matrix of the form X(XT X)−1X. The last fact also used in [13] is originally attributed to an anonymous referee in [14]. Finally, we derive the 0-1 SDP formulation for the problem (18)-(19) as follows. min (nm −tr(BZ)) (24) s.t. Zen = en ∀i′ ∈[1, k], (25) tr(Z) = k, Z ≥0; Z2 = Z; Z = ZT . (26) Relaxing and Solving the 0-1 SDP: The relaxation to (24)-(26) exploits the fact that Z is a projection matrix satisfying Z2 = Z. This allows replacing the last three constraints in (26) as I ⪰Z ⪰0. By establishing the result that any feasible solution to the second formulation of 0-1 SDP, Zfeas is a rank k matrix, we first solve the relaxed SDP using SeDuMi [15], take the rank k projection of Z∗ and then adopt a rounding based on a variant of the winner-takes-all approach to obtain a solution in polynomial time. For the technical details and their proofs, please refer to [10]. 7 Experimental Results Our experiments included evaluations on several classification datasets, segmentation databases and simulations. Due to space limitations, we provide a brief summary here. Our first set of experiments illustrates an application to several datasets from the UCI Machine Learning Repository: (1) Iris dataset, (2) Soybean dataset and (3) Wine dataset; these include ground truth data, see http://www.ics.uci.edu/ mlearn/MLRepository.html. To create the ensemble, we used a set of [4, 10] clustering schemes (by varying the clustering criterion and/or algorithm) from the CLUTO clustering toolkit. The multiple solutions comprised the input ensemble, our model was then used to determine a agreement maximizing solution. The ground-truth data was used at this stage to evaluate accuracy of the ensemble (and individual schemes). The results are shown in Figure 1(a)-(c). For each case, we can see that the ensemble clustering solution is at least as good as the best clustering algorithm. Observe, however, that while such results are expected for this and many other datasets (see [3]), the consensus solution may not always be superior to the ‘best’ clustering solution. For instance, in Fig. 1(c) (for m = 7) the best solution has a marginally lower error rate than the ensemble. An ensemble solution is useful because we do not know a priori that which algorithm will perform the best (especially if ground truth is unavailable). 3 4 5 6 7 8 9 10 11 Number of clustering algorithms in ensemble (m) 0.1 0.2 0.3 0.4 0.5 Mislabelled cases in each algorithm cluster ensemble individual algorithms 3 4 5 6 7 8 9 10 11 Number of clustering algorithms in ensemble (m) 0.1 0.2 0.3 0.4 Mislabelled cases in each algorithm cluster ensemble individual algorithms 3 4 5 6 7 8 9 10 11 Number of clustering algorithms in ensemble (m) 0.1 0.2 0.3 0.4 0.5 0.6 Mislabelled cases in each algorithm cluster ensemble individual algorithms (a) (b) (c) Figure 1: Synergy. The fraction of mislabeled cases ([0, 1]) in a consensus solution (∗) is compared to the number of mislabelled cases (∆) in individual clustering algorithms. We illustrate the ensemble effect for the Iris dataset in (a), the Soybean dataset in (b), and the Wine dataset in (c). Our second set of experiments focuses on a novel application of ensembles to the problem of image segmentation. Even sophisticated segmentation algorithms may yield ‘different’ results on the same image, when multiple segmentations are available, it seems reasonable to ‘combine’ segmentations to reduce degeneracies. Our experimental results indicate that in many cases, we can obtain a better overall segmentation that captures (more) details in the images more accurately with fewer outlying clusters. In Fig. 2, we illustrate the results on an image from the Berkeley dataset. The segmentations were generated using several powerful algorithms including (a) Normalized Cuts, (b) Energy Minimization by Graph Cuts and (c)–(d) Curve Evolution. Notice that each algorithm performs well but misses out on some details. For instance, (a) and (d) do not segment the eyes; (b) does well in segmenting the shirt collar region but can only recognize one of the eyes and (c) creates an additional cut across the forehead. The ensemble (extreme right) is able to segment these details (eyes, shirt collar and cap) nicely by combining (a)–(d). For implementation details of the algorithm including settings, preprocessing and additional evaluations, please refer to [10]. (a) (b) (c) (d) ensemble Figure 2: A segmentation ensemble on an image from the Berkeley Segmentation dataset. (a)–(d) show the individual segmentations overlaid on the input image, the right-most image shows the segmentation generated from ensemble clustering. The final set of our experiments were performed on 500 runs of artificially generated cluster ensembles. We first constructed an initial set segmentation, this was then repeatedly permuted (up to 15%) yielding a set of clustering solutions. The solutions from our model and [3] were compared w.r.t. our objective functions and Normalized Mutual Information used in [3]. In Figure 3(a), we see that our algorithm (Model 1) outperforms [3] on all instances. In the average case, the ratio is slightly more than 1.5. We must note the time-quality trade-off because solving Model 1 requires a branchand-bound approach. In Fig. 3(b), we compare the results of [3] with solutions from the relaxed SDP Model 2 on (24). We can see that our model performs better in ∼95% cases. Finally, Figure 1(b) shows a comparison of relaxed SDP Model 2 with [3] on the objective function optimized in [3] (using best among two techniques). We observed that our solutions achieve superior results in 80% of the cases. The results show that even empirically our objective functions model similarity rather well, and that Normalized Mutual Information may be implicitly optimized within this framework. Remarks. We note that the graph partitioning methods used in [3] are typically much faster than the time needed by SDP solvers (e.g., SeDuMi [15] and SDPT3) for comparable problem sizes. However, given the increasing interest in SDP in the last few years, we may expect the development of new algorithms, and faster/more efficient software tools. 8 Conclusions We have proposed a new algorithm for ensemble clustering based on a SDP formulation. Among the important contributions of this paper is, we believe, the observation that the notion of agreement in an ensemble can be captured better using string encoding rather than a voting strategy. While a partition problem defined directly on such strings yields a non-linear optimization problem, we illustrate a transformation into a strict 0-1 SDP via novel convexification techniques. The last result of this paper is the design of a modified model of the SDP based on additional observations on the structure of the underlying matrices. We discuss extensive experimental evaluations on simulations and real datasets, in addition, we illustrate application of the algorithm for segmentation ensembles. We feel that the latter application is of independent research interest; to the best of our knowledge, this is the first algorithmic treatment of generating segmentation ensembles for improving accuracy. 0 1 2 3 4 5 Ratios of solutions of objective functions of the two algorithms 0 50 100 150 200 Number of instances Better Worse 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Ratios of solutions of objective functions of the two algorithms 0 100 200 300 400 500 Number of instances Better Worse -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 Normalized difference 0 50 100 150 200 Number of instances Worse Better (a) (b) (c) Figure 3: A comparison of [3] with SDP Model 1 in (a), and with SDP Model 2 on (24) in (b). The solution from [3] was used as the numerator. In (c), comparisons (difference in normalized values) between our solution and the best among IBGF and CBGF based on the Normalized Mutual Information (NMI) objective function used in [3]. Acknowledgments. This work was supported in part by NSF grants CCF-0546509 and IIS0713489. The first author was also supported by start-up funds from the Dept. of Biostatistics and Medical Informatics, UW – Madison. We thank D. Sivakumar for useful discussions, Johan L¨ofberg for a thorough explanation of the salient features of Yalmip [16], and the reviewers for suggestions regarding the presentation of the paper. One of the reviewers also pointed out a typo in the derivations in §6. References [1] V. Filkov and S. Skiena. Integrating microarray data by consensus clustering. In Proc. of International Conference on Tools with Artificial Intelligence, page 418, 2003. [2] X. Z. Fern and C. E. Brodley. Solving cluster ensemble problems by bipartite graph partitioning. In Proc. of International Conference on Machine Learning, page 36, 2004. [3] A. Strehl and J. Ghosh. Cluster Ensembles – A Knowledge Reuse Framework for Combining Partitionings. In Proc. of AAAI 2002, pages 93–98, 2002. [4] N. Bansal, A. Blum, and S. Chawla. Correlation clustering. In Proc. Symposium on Foundations of Computer Science, page 238, 2002. [5] S. Monti, P. Tamayo, J. Mesirov, and T. Golub. Consensus clustering: A resampling-based method for class discovery and visualization of gene expression microarray data. Mach. Learn., 52(1-2):91–118, 2003. [6] A. Gionis, H. Mannila, and P. Tsaparas. Clustering aggregation. In Proc. of International Conference on Data Engineering, pages 341–352, 2005. [7] N. Ailon, M. Charikar, and A. Newman. Aggregating inconsistent information: ranking and clustering. In Proc. of Symposium on Theory of Computing, pages 684–693, 2005. [8] M. Charikar, V. Guruswami, and A. Wirth. Clustering with qualitative information. J. Comput. Syst. Sci., 71(3):360–383, 2005. [9] X. Z. Fern and C. E. Brodley. Random projection for high dimensional data clustering: A cluster ensemble approach. In Proceedings of International Conference on Machine Learning, 2003. [10] V. Singh. On Several Geometric Optimization Problems in Biomedical Computation. PhD thesis, State University of New York at Buffalo, 2007. [11] L. Gasieniec, J. Jansson, and A. Lingas. Approximation algorithms for hamming clustering problems. In Proc. of Symposium on Combinatorial Pattern Matching, pages 108–118, 2000. [12] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, New York, 2004. [13] J. Peng and Y. Wei. Approximating k-means-type clustering via semidefinite programming. SIAM Journal on Optimization, 18(1):186–205, 2007. [14] A. D. Gordon and J. T. Henderson. An algorithm for euclidean sum of squares classification. Biometrics, 33:355–362, 1977. [15] J. F. Sturm. Using SeDuMi 1.02, A Matlab Toolbox for Optimization over Symmetric Cones. Optimization Methods and Software, 11-12:625–653, 1999. [16] J. L¨ofberg. YALMIP : A toolbox for modeling and optimization in MATLAB. In CCA/ISIC/CACSD, September 2004.
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Gaussian Process Models for Link Analysis and Transfer Learning Kai Yu NEC Laboratories America Cupertino, CA 95014 Wei Chu Columbia University, CCLS New York, NY 10115 Abstract This paper aims to model relational data on edges of networks. We describe appropriate Gaussian Processes (GPs) for directed, undirected, and bipartite networks. The inter-dependencies of edges can be effectively modeled by adapting the GP hyper-parameters. The framework suggests an intimate connection between link prediction and transfer learning, which were traditionally two separate research topics. We develop an efficient learning algorithm that can handle a large number of observations. The experimental results on several real-world data sets verify superior learning capacity. 1 Introduction In many scenarios the data of interest consist of relational observations on the edges of networks. Typically, a given finite collection of such relational data can be represented as an M × N matrix Y = {yi,j}, which is often partially observed because many elements are missing. Sometimes accompanying Y are attributes of nodes or edges. As an important nature of networks, {yi,j} are highly inter-dependent even conditioned on known node or edge attributes. The phenomenon is extremely common in real-world data, for example, • Bipartite Graphs. The data represent relations between two different sets of objects or measurements under a pair of heterogeneous conditions. One notable example is transfer learning, also known as multi-task learning, which jointly learns multiple related but different predictive functions based on the M × N observed labels Y, namely, the results of N functions acting on a set of M data examples. Collaborative filtering is an important application of transfer learning that learns many users’ interests on a large set of items. • Undirected and Directed Graphs. The data are measurements of existences, strengths, and types of links between a set of nodes in a graph, where a given collection of observations are an M ×M (in this case N = M) matrix Y, which can be symmetric or asymmetric, depending on whether the links are undirected or directed. Examples include protein-protein interactions, social networks, citation networks, and hyperlinks on the WEB. Link prediction aims to recover those missing measurements in Y, for example, predicting unknown protein-protein interactions based on known interactions. The goal of this paper is to design a Gaussian process (GP) [13] framework to model the dependence structure of networks, and to contribute an efficient algorithm to learn and predict large-scale relational data. We explicitly construct a series of parametric models indexed by their dimensionality, and show that in the limit we obtain nonparametric GP priors consistent with the dependence of edge-wise measurements. Since the kernel matrix is on a quadratic number of edges and the computation cost is even cubic of the kernel size, we develop an efficient algorithm to reduce the computational complexity. We also demonstrate that transfer learning has an intimate connection to link prediction. Our method generalizes several recent transfer learning algorithms by additionally learning a task-specific kernel that directly expresses the dependence between tasks. 1 The application of GPs to learning on networks or graphs has been fairly recent. Most of the work in this direction has focused on GPs over nodes of graphs and targeted at the classification of nodes [20, 6, 10]. In this paper, we regard the edges as the first-class citizen and develop a general GP framework for modeling the dependence of edge-wise observations on bipartite, undirected and directed graphs. This work extends [19], which built GPs for only bipartite graphs and proposed an algorithm scaling cubically to the number of nodes. In contrast, the work here is more general and the algorithm scales linearly to the number of edges. Our study promises a careful treatment to model the nature of edge-wise observations and offers a promising tool for link prediction. 2 Gaussian Processes for Network Data 2.1 Modeling Bipartite Graphs We first review the edge-wise GP for bipartite graphs [19], where each observation is a measurement on a pair of objects of different types, or under a pair of heterogenous conditions. Formally, let U and V be two index sets, then yi,j denotes a measurement on edge (i, j) with i ∈U and j ∈V. In the context of transfer learning, the pair involves a data instance i and a task j, and yi,j denotes the label of data i within task j. The probabilistic model assumes that yi,j are noisy outcomes of a real-valued function f : U × V →R, which follows a Gaussian process GP(b, K), characterized by mean function b and covariance (kernel) function between edges K ((i, j), (i′, j′)) = Σ(i, i′)Ω(j, j′) (1) where Σ and Ωare kernel functions on U and V, respectively. As a result, the realizations of f on a finite set i = 1, . . . , M and j = 1, . . . , N form a matrix F, following a matrix-variate normal distribution NM×N(B, Σ, Ω), or equivalently a normal distribution N(b, K) with mean b = vec(B) and covariance K = Ω⊗Σ, where ⊗means Kronecker product. The dependence structure of edges is decomposed into the dependence of nodes. Since a kernel is a notion of similarity, the model expresses a prior belief – if node i is similar to node i′ and node j is similar node j′, then so are f(i, j) and f(i′, j′). It is essential to learn the kernels Σ and Ωbased on the partially observed Y, in order to capture the dependence structure of the network. For transfer learning, this means to learn the kernel Σ between data instances and the kernel Ωbetween tasks. Having Σ and Ωis it then possible to predict those missing yi,j based on known observations by using GP inference. Theorem 2.1 ([19]). Let f(i, j) = D−1/2 PD k=1 gk(i)hk(j) + b(i, j), where gk iid∼GP(0, Σ) and hk iid∼GP(0, Ω), then f ∼GP(b, K) in the limit D →∞, and the covariance between pairs is K ((i, j), (i′, j′)) = Σ(i, i′)Ω(j, j′). Theorem (2.1) offers an alternative view to understand the model. The edge-wise function f can be decomposed into a product of two sets of intermediate node-wise functions, {gk}∞ k=1 and {hk}∞ k=1, which are i.i.d. samples from two GP priors GP(0, Σ) and GP(0, Ω). The theorem suggests that the GP model for bipartite relational data is a generalization of a Bayesian low-rank matrix factorization F = HG⊤+ B, under the prior H ∼NM×D(0, Σ, I) and G ∼NN×D(0, Ω, I). When D is finite, the elements of F are not Gaussian random variables. 2.2 Modeling Directed and Undirected Graphs In this section we model observations on pairs of nodes of the same set U. This case includes both directed and undirected graphs. It turns out that the directed graph is relatively easy to handle while deriving a GP prior for undirected graphs is slightly non-trivial. For the case of directed graphs, we let the function f : U × U →R follow GP(b, K), where the covariance function between edges is K ((i, j), (i′, j′)) = C(i, i′)C(j, j′) (2) and C : U × U →R is a kernel function between nodes. Since a random function f drawn from the GP is generally asymmetric (even if b is symmetric), namely f(i, j) ̸= f(j, i), the direction of edges can be modeled. The covariance function Eq. (2) can be derived from Theorem (2.1) by setting that {gk} and {hk} are two independent sets of functions i.i.d. sampled from the same GP prior 2 GP(0, C), modeling the situation that each node’s behavior as a sender is different but statistically related to it’s behavior as a receiver. This is a reasonable modeling assumption. For example, if two papers cite a common set of papers, their are also likely to be cited by a common set of other papers. For the case of undirected graphs, we need to design a GP that ensures any sampled function to be symmetric. Following the construction of GP in Theorem (2.1), it seems that f is symmetric if gk ≡hk for k = 1, . . . , D. However a calculation reveals that f is not bounded in the limit D →∞. Theorem (2.2) shows that the problem can be solved by subtracting a growing quantity D1/2C(i, j) as D →∞, and suggests the covariance function K ((i, j), (i′, j′)) = C(i, i′)C(j, j′) + C(i, j′)C(j, i′). (3) With such covariance function , f is ensured to be symmetric because the covariance between f(i, j) and f(j, i) equals the variance of either. Theorem 2.2. Let f(i, j) = D−1/2 PD k=1 tk(i)tk(j)+b(i, j)−D1/2C(i, j), where tk iid∼GP(0, C), then f ∼GP(b, K) in the limit D →∞, and the covariance between pairs is K ((i, j), (i′, j′)) = C(i, i′)C(j, j′) + C(i, j′)C(j, i′). If b(i, j) = b(j, i), then f(i, j) = f(j, i). Proof. Without loss of generality, let b(i, j) ≡0. Based on the central limit theorem, for every (i, j), f(i, j) converges to a zero-mean Gaussian random variable as D →∞, because {tk(i)tk(j)}D k=1 is a collection of random variables independently following the same distribution, and has the mean C(i, j). The covariance function is Cov(f(i, j), f(i′, j′)) = 1 D PD k=1{E[tk(i)tk(j)tk(i′)tk(j′)] − C(i, j)E[tk(i′)tk(j′)] −C(i′, j′)E[tk(i)tk(j)] + C(i, j)C(i′, j′)} = C(i, i′)C(j, j′) + C(i, j′)C(j, i′) + C(i, j)C(i′, j′) −C(i, j)C(i′, j′) = C(i, i′)C(j, j′) + C(i, j′)C(j, i′). Interestingly, Theorem (2.2) recovers Theorem (2.1) and is thus more general. To see the connection, let hk ∼GP(0, Σ) and gk ∼GP(0, Ω) be concatenated to form a function tk, then we have tk ∼GP(0, C) and the covariance is C(i, j) = Σ(i, j), if i, j ∈U, Ω(i, j), if i, j ∈V, 0, if i, j are in different sets. (4) For i, i′ ∈U and j, j′ ∈V, applying Theorem (2.2) leads to f(i, j) = D−1/2 D X k=1 tk(i)tk(j) + b(i, j) −D1/2C(i, j) = D−1/2 D X k=1 hk(i)gk(j) + b(i, j), (5) K ((i, j), (i′, j′)) = C(i, i′)C(j, j′) + C(i, j′)C(j, i′) = Σ(i, i′)Ω(j, j′). (6) Theorems (2.1) and (2.2) suggest a general GP framework to model directed or undirected relationships connecting heterogeneous types of nodes. Basically, we learn node-wise covariance functions, like Σ, Ω, and C, such that edge-wise covariances composed by Eq. (1), (2), or (3) can explain the happening of observations yi,j on edges. The proposed framework can be extended to cope with more complex network data, for example, networks containing both undirected links and directed links. We will briefly discuss some extensions in Sec. 6. 3 An Efficient Learning Algorithm We consider the regression case under a Gaussian noise model, and later briefly discuss extensions to the classification case. Let y = [yi,j](i,j)∈O be the observational vector of length |O|, f be the corresponding quantities of the latent function f, and K be the |O|×|O| matrix of K between edges having observations, computed by Eq. (1)-(3). Then observations on edges are generated by yi,j = f(i, j) + bi,j + ϵi,j (7) where f ∼N(0, K), ϵi,j iid∼N(0, β−1), and the mean has a parametric form bi,j = µi + νj. In the directed/undirected graph case we let µi = νi for any i ∈U. f can be analytically marginalized out, the marginal distribution of observations is then p(y|θ) = N(y; b, K + β−1I), (8) 3 where θ = {β, b, K}. The parameters can be estimated by minimizing the penalized negative loglikelihood L(θ) = −ln p(y|θ) + ℓ(θ) under a suitable regularization ℓ(θ). The objective function has the form: L(θ) = |O| 2 log 2π + 1 2 ln |C| + 1 2tr £ C−1mm⊤¤ + ℓ(θ), (9) where C = K + β−1I, m = y −b and b = [bi,j], (i, j) ∈O. ℓ(θ) will be configured in Sec. 3.1. Gradient-based optimization packages can be applied to find a local optimum of θ. However the computation can be prohibitively high when the size |O| of measured edges is very big, because the memory cost is O(|O|2), and the computational cost is O(|O|3). In our experiments |O| is about tens of thousands or even millions. A slightly improved algorithm was introduced in [19], with a complexity O(M 3 + N 3) cubic to the size of nodes. The algorithm employed a non-Gaussian approximation based on Theorem (2.1) and is applicable to only bipartite graphs. We reduce the memory and computational cost by exploring the special structure of K as discussed in Sec. 2 and assume K to be composed by node-wise linear kernels Σ(i, i′) = ⟨xi, xi′⟩, Ω(i, i′) = ⟨zj, zj′⟩, and C(i, j) = ⟨xi, xj⟩, with x ∈RL1 and z ∈RL2. The edge-wise covariance is then • Bipartite Graphs: K ((i, j), (i′, j′)) = ⟨xi ⊗zj, xi′ ⊗zj′⟩. • Directed Graphs: K ((i, j), (i′, j′)) = ⟨xi ⊗xj, xi′ ⊗xj′⟩. • Undirected Graphs: K ((i, j), (i′, j′)) = ⟨xi ⊗xj, xi′ ⊗xj′⟩+ ⟨xi ⊗xj, xj′ ⊗xi′⟩ We turn the problem of optimizing K into the problem of optimizing X = [x1, . . . , xM]⊤and Z = [z1, . . . , zN]⊤. It is important to note that in all the cases the kernel matrix has the form K = UU⊤, where U is an |O| × L matrix, L ≪|O|, therefore applying the Woodbury identity C−1 = β[I−U(U⊤U+β−1I)−1U⊤] can dramatically reduce the computational cost. For example, in the bipartite graph case and the directed graph case, respectively there are U⊤= £ xi ⊗zj ¤ (i,j)∈O, and U⊤= £ xi ⊗xj ¤ (i,j)∈O, (10) where the rows of U are indexed by (i, j) ∈O. For the undirected graph case, we first rewrite the kernel function K ((i, j), (i′, j′)) = ⟨xi ⊗xj, xi′ ⊗xj′⟩+ ⟨xi ⊗xj, xj′ ⊗xi′⟩ = 1 2 h ⟨xi ⊗xj, xi′ ⊗xj′⟩+ ⟨xj ⊗xi, xj′ ⊗xi′⟩+ ⟨xi ⊗xj, xj′ ⊗xi′⟩+ ⟨xj ⊗xi, xi′ ⊗xj′⟩ i = 1 2 h (xi ⊗xj + xj ⊗xi), (xi′ ⊗xj′ + xj′ ⊗xi′) ®i , (11) and then obtain a simple form for the undirected graph case U⊤= 1 √ 2 h xi ⊗xj + xj ⊗xi i (i,j)∈O (12) The overall computational cost is at O(L3 + |O|L2). Empirically we found that the algorithm is efficient to handle L = 500 when |O| is about millions. The gradients with respect to U can be found in [12]. Further calculation of gradients with respect to X and Z can be easily derived. Here we omit the details for saving the space. Finally, in order to predict the missing measurements, we only need to estimate a simple linear model f(i, j) = w⊤ui,j + bi,j. 3.1 Incorporating Additional Attributes and Learning from Discrete Observations There are different ways to incorporate node or edge attributes into our model. A common practice is to let the kernel K, Σ, or Ωbe some parametric function of attributes. One such choice is the RBF function. However, node or edge attributes are typically local information while the network itself is rather a global dependence structure, thus the network data often has a large part of patterns that are independent of those known predictors. In the following, via the example of placing a Bayesian prior on Σ : U ×U →R, we describe a flexible solution to incorporate additional knowledge. Let Σ0 be the covariance that we wish Σ to be apriori close to. We apply the prior p(Σ) = 1 Z exp(−τE(Σ)) and use its negative log-likelihood as a regularization for Σ: ℓ(Σ) = τE(Σ) = τ 2 h log |Σ + γ−1I| + tr ¡ (Σ + γ−1I)−1Σ0 ¢ i (13) 4 where τ is a hyperparameter predetermined on validation data, and γ−1 is a small number to be optimized. The energy function E(Σ) is related to the KL divergence DKL(GP(0, Σ0)||GP(0, Σ + γ−1δ)), where δ(·, ·) is the dirac kernel. If we let Σ0 be the linear kernel of attributes, normalized by the dimensionality, then E(Σ) can be derived from a likelihood of Σ as if each dimension of the attributes is a random sample from GP(0, Σ + γ−1δ). If the attributes are nonlinear predictors we can conveniently set Σ0 by a nonlinear kernel. We set Σ0 = I if the corresponding attributes are absent. ℓ(Ω), ℓ(C) and ℓ(K) can be set in the same way. The observations can be discrete variables rather than real values. In this case, an appropriate likelihood function can be devised accordingly. For example, the probit function could be employed as the likelihood function for binary labels, which relates f(i, j) to the target yi,j ∈{−1, +1}, by a cumulative normal Φ (yi,j(f(i, j) + bi,j)). To preserve computationally tractability, a family of inference techniques, e.g. Laplace approximation, can be applied to finding a Gaussian distribution that approximates the true likelihood. Then, the marginal likelihood (8) can be written as an explicit expression and the gradient can be derived analytically as well. 4 Discussions on Related Work Transfer Learning: As we have suggested before, the link prediction for bipartite graphs has a tight connection to transfer learning. To make it clear, let fj(·) = f(·, j), then the edge-wise function f : U × V →R consists of N node-wise functions fj : U →R for j = 1, . . . , N. If we fix Ω(j, j′) ≡δ(j, j′), namely a Dirac delta function, then fj are assumed to be i.i.d. GP functions from GP(0, Σ), where each function corresponds to one learning task. This is the hierarchical Baysian model that assumes multiple tasks sharing the same GP prior [18]. In particular, the negative logarithm of p ¡ {yi,j}, {fj}|Σ ¢ is L ³ {fj}, Σ ´ = N X j=1 X i∈Oj l ¡ yi,j, fj(i) ¢ + 1 2f jΣ−1f j + N 2 log |Σ|, (14) where l(yi,j, fj(i)) = −log p(yi,j|fj(i)). The form is close to the recent convex multi-task learning in a regularization framework [3], if the log-determinant term is replaced by a trace regularization term λtr(Σ). It was proven in [3] that if l(·, ·) is convex with fj, then the minimization of (14) is convex with jointly {fj} and Σ. The GP approach differs from the regularization approach in two aspects: (1) fj are treated as random variables which are marginalized out, thus we only need to estimate Σ; (2) The regularization for Σ is a non-convex log-determinant term. Interestingly, because log |Σ| ≤tr(Σ)−M, the trace norm is the convex envelope for the log-determinant, and thus the two minimization problems are somehow doing similar things. However, the framework introduced in this paper goes beyond the two methods by introducing an informative kernel Ωbetween tasks. From a probabilistic modeling point of view, the independence of {fj} conditioned on Σ is a restrictive assumption and even incorrect when some task-specific attributes are given (which means that {fj} are not exchangeable anymore). The task-specific kernel for transfer learning has been recently introduced in [4], which however increased the computational complexity by a factor of N 2. One contribution of this paper on transfer learning is an algorithm that can efficiently solve the learning problem with both data kernel Σ and task kernel Ω. Gaussian Process Latent-Variable Model (GPLVM): Our learning algorithm is also a generalization of GPLVM. If we enforce Ω(j, j′) = δ(j, j′) in the model of bipartite graphs, then the evidence Eq. (9) is equivalent to the form of GPLVM, L(Σ, β) = MN 2 log 2π + N 2 ln |(Σ + β−1I)| + 1 2tr h (Σ + β−1I)−1YY⊤i , (15) where Y is a fully observed M × N matrix, the mean B = 0, and there is no further regularization on Σ. GPLVM assumes that columns of Y are conditionally independent given Σ. In this paper we consider a situation with complex dependence of edges in network graphs. Other Related Work: Getoor et al. [7] introduced link uncertainty in the framework of probabilistic relational models. Latent-class relational models [17, 11, 1] have been popular, aiming to find the block structure of links. Link prediction was casted as structured-output prediction in [15, 2]. Statistical models based on matrix factorization was studied by [8]. Our work is similar to [8] in the 5 Figure 1: The left-hand side: the subset of the UMist Faces data that contains 10 people at 10 different views. The blank blocks indicate the ten knocked-off images as test cases; The right-hand side: the ten knocked-off images (the first row) along with predictive images. The second row is of our results, the third row is of the MMMF results, and the fourth row is of the bilinear results. sense that relations are modeled by multiplications of node-wise factors. Very recently, Hoff showed in [9] that the multiplicative model generalizes the latent-class models [11, 1] and can encode the transitivity of relations. 5 Numerical Experiments We set the dimensionality of the model via validation on 10% of training data. In cases that the additional attributes on nodes or edges are either unavailable or very weak, we compare our method with max-margin matrix factorization (MMMF) [14] using a square loss, which is similar to singular value decomposition (SVD) but can handle missing measurements. 5.1 A Demonstration on Face Reconstruction A subset of the UMist Faces images of size 112 × 92 was selected to illustrate our algorithm, which consists of 10 people at 10 different views. We manually knocked 10 images off as test cases, as presented in Figure 1, and treated each image as a vector that leads to a 103040 × 10 matrix with 103040 missing values, where each column corresponds a view of faces. GP was trained by setting L1 = L2 = 4 on this matrix to learn from the appearance relationships between person identity and pose. The images recovered by GP for the test cases are presented as the second row of Figure 1-right (RMSE=0.2881). The results of MMMF are presented as the third row (RMSE=0.4351). We also employed the bilinear models introduced by [16], which however does not handle missing data of a matrix, and put the results at the bottom row for comparison. Quantitatively and perceptually our model offers a better generalization to unseen views of known persons. 5.2 Collaborative Filtering Collaborative filtering is a typical case of bipartite graphs, where ratings are measurements on edges of user-item pairs. We carried out a serial of experiments on the whole EachMovie data, which includes 61265 users’ 2811718 distinct numeric ratings on 1623 movies. We randomly selected 80% of each user’s ratings for training and used the remaining 20% as test cases. The random selection was carried out 20 times independently. For comparison purpose, we also evaluated the predictive performance of four other approaches: 1) Movie Mean: the empirical mean of ratings per movie was used as the predictive value of all users’ rating on the movie; 2) User Mean: the empirical mean of ratings per user was used as the predictive value of the users’ rating on all movies; 3) Pearson Score: the Pearson correlation coefficient corresponds to a dot product between normalized rating vectors. We computed the Gram matrices of the Pearson score with mean imputation for movies and users respectively, and took principal components as their individual attributes. We tried 20 or 50 principal components as attributes in this experiment and carried out least square regression on observed entries. 4) MMMF. The optimal rank was decided by validation. 6 Table 1: Test results on the EachMovie data. The number in bracket indicates the rank we applied. The results are averaged over 20 trials, along with the standard deviation. To evaluate accuracy, we utilize root mean squared error (RMSE), mean absolute error (MAE), and normalized mean squared error, i.e. ,the RMSE normalized by the standard deviation of observations. METHODS RMSE MAE NMSE MOVIE MEAN 1.3866±0.0013 1.1026±0.0010 0.7844±0.0012 USER MEAN 1.4251±0.0011 1.1405±0.0009 0.8285±0.0008 PEARSON(20) 1.3097±0.0012 1.0325±0.0013 0.6999±0.0011 PEARSON(50) 1.3034±0.0018 1.0277±0.0015 0.6931±0.0019 MMMF(3) 1.2245±0.0503 0.9392±0.0246 0.6127±0.0516 MMMF(15) 1.1696±0.0283 0.8918±0.0146 0.5585±0.0286 GP(3) 1.1557±0.0010 0.8781±0.0009 0.5449±0.0011 Table 2: Test results on the Cora data. The classification accuracy rate is averaged over 5 trials, each with 4 folds for training and one fold for test. METHODS DS HA ML PL CONTENT 53.70±0.50 67.50±1.70 68.30±1.60 56.40±0.70 LINK 48.90±1.70 65.80±1.40 60.70±1.10 58.20±0.70 PCA(50) 61.61±1.42 69.36±1.36 70.06±0.90 60.26±1.16 GP(50) 62.10±0.84 75.40±0.80 78.30±0.78 63.25±0.60 The results of these approaches are reported in Table 1. The per-movie average yields much better results than the per-user average, which is consistent with the findings previously reported by [5]. The improvement is noticeable by using more components of the Pearson score, but not significant. The generalization performance of our algorithm is better than that of others. T-test showed a significant difference with p-value 0.0387 of GP over MMMF (with 15 dimensions) in terms of RMSE. It is well worth highlighting another attractiveness of our algorithm – the compact representation of factors. On the EachMovie data, there are only three factors that well represent thousands of items individually. We also trained MMMF with 3 factors as well. Although the three-factor solution GP found is also accessible to other models, MMMF failed to achieve comparable performance on this case (i.e., see results of MMMF(3)). In each trial, the number of training samples is around 2.25 million. Our program took about 865 seconds to accomplish 500 L-BFGS updates on all 251572 parameters using an AMD Opteron 2.6GHz processor. 5.3 Text Categorization based on Contents and Links We used a part of Cora corpus including 751 papers on data structure (DS), 400 papers on hardware and architecture (HA), 1617 on machine learning (ML) and 1575 on programming language (PL). We treated the citation network as a directed graph and modeled the link existence as binary labels. Our model applied the probit likelihood and learned a node-wise covariance function C, L = 50 × 50, which composes an edge-wise covariance K by Eq. (2). We set the prior covariance C0 by the linear kernel computed by bag-of-word content attributes. Thus the learned linear features encode both link and content information, which were then used for document classification. We compare several other methods that provide linear features for one-against-all categorization using SVM: 1) CONTENT: bag-of-words features; 2) LINK: each paper’s citation list; 3) PCA: 50 components by PCA on the concatenation of bag-of-word features and citation list for each paper. We chose the dimensionality 50 for both GP and PCA, because their performances both saturated when the dimensionality exceeds 50. We reported results based on 5-fold cross validation in Table 2. GP clearly outperformed other methods in 3 out of 4 categories. The main reason we believe is that our approach models the in-bound and out-bound behaviors simultaneously for each paper . 6 Conclusion and Extensions In this paper we proposed GPs for modeling data living on links of networks. We described solutions to handle directed and undirected links, as well as links connecting heterogenous nodes. This work paves a way for future extensions for learning more complex relational data. For example, we can model a network containing both directed and undirected links. Let (i, j) be directed and (i′, j′) be undirected. Based on the feature representations, Eq.(10)-right for directed links and Eq.(12) for undirected links, the covaraince is K((i, j), (i′, j′)) = 1/ √ 2[C(i, i′)C(j, j′)+C(i, j′)C(j, i′)], 7 which indicates that dependence between a directed link and an undirected link is penalized compared to dependence between two undirected links. Moreover, GPs can be employed to model multiple networks involving multiple different types of nodes. For each type, we use one node-wise covariance. Letting covariance between two different types of nodes be zero, we obtain a huge block-diagonal node-wise covariance matrix, where each block corresponds to one type of nodes. This big covariance matrix will induce the edge-wise covariance for links connecting nodes of the same or different types. In the near future it is promising to apply the model to various link prediction or network completion problems. References [1] E. M. Airoldi, D. M. Blei, S. E. Fienberg, and E. P. Xing, Mixed membership stochastic block models for relational data with application to protein-protein interactions. Biometrics Society Annual Meeting, 2006. [2] S. Andrews and T. Jebara, Structured Network Learning. NIPS Workshop on Learning to Compare Examples, 2006. [3] A. Argyriou, T. Evgeniou, and M. Pontil. Convex multi-task feature learning. Machine Learning, 2007. [4] E. V. Bonilla, F. V. Agakov, and C. K. I. Williams. Kernel multi-task learning using taskspecific features. International Conferences on Artificial Intelligence and Statistics, 2007. [5] J. Canny. Collaborative filtering with privacy via factor analysis. International ACM SIGIR Conference , 2002. [6] W. Chu, V. Sindhwani, Z. Ghahramani, and S. S. Keerthi. Relational learning with gaussian processes. Neural Informaiton Processing Systems 19, 2007. [7] L. Getoor, E. Segal, B. Taskar, and D. Koller. Probabilistic models of text and link structure for hypertext classification. ICJAI Workshop, 2001. [8] P. Hoff. Multiplicative latent factor models for description and prediction of social networks. to appear in Computational and Mathematical Organization Theory, 2007. [9] P. Hoff. Modeling homophily and stochastic equivalence in symmetric relational data. to appear in Neural Informaiton Processing Systems 20, 2007. [10] A. Kapoor, Y. Qi, H. Ahn, and R. W. Picard. Hyperparameter and kernel learning for graph based semi-supervised classification. Neural Informaiton Processing Systems 18, 2006. [11] C. Kemp, J. B. Tenenbaum, T. L. Griffiths, T. Yamada, and N. Ueda. Learning systems of concepts with an infinite relational model. AAAI Conference on Artificial Intelligence, 2006. [12] N. Lawrence. 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Near-Maximum Entropy Models for Binary Neural Representations of Natural Images Matthias Bethge and Philipp Berens Max Planck Institute for Biological Cybernetics Spemannstrasse 41, 72076, T¨ubingen, Germany mbethge,berens@tuebingen.mpg.de Abstract Maximum entropy analysis of binary variables provides an elegant way for studying the role of pairwise correlations in neural populations. Unfortunately, these approaches suffer from their poor scalability to high dimensions. In sensory coding, however, high-dimensional data is ubiquitous. Here, we introduce a new approach using a near-maximum entropy model, that makes this type of analysis feasible for very high-dimensional data—the model parameters can be derived in closed form and sampling is easy. Therefore, our NearMaxEnt approach can serve as a tool for testing predictions from a pairwise maximum entropy model not only for low-dimensional marginals, but also for high dimensional measurements of more than thousand units. We demonstrate its usefulness by studying natural images with dichotomized pixel intensities. Our results indicate that the statistics of such higher-dimensional measurements exhibit additional structure that are not predicted by pairwise correlations, despite the fact that pairwise correlations explain the lower-dimensional marginal statistics surprisingly well up to the limit of dimensionality where estimation of the full joint distribution is feasible. 1 Introduction A core issue in sensory coding is to seek out and model statistical regularities in high-dimensional data. In particular, motivated by developments in information theory, it has been hypothesized that modeling these regularities by means of redundancy reduction constitutes an important goal of early visual processing [2]. Recent studies conjectured that the binary spike responses of retinal ganglion cells may be characterized completely in terms of second-order correlations when using a maximum entropy approach [13, 12]. In light of what we know about the statistics of the visual input, however, this would be very surprising: Natural images are known to exhibit complex higherorder correlations which are extremely difficult to model yet being perceptually relevant. Thus, if we assume that retinal ganglion cells do not discard the information underlying these higher-order correlations altogether, it would be a very difficult signal processing task to remove all of those already within the retinal network. Oftentimes, neurons involved in early visual processing are modeled as rather simple computational units akin to generalized linear models, where a linear filter is followed by a point-wise nonlinearity. For such simple neuron models, the possibility of removing higher-order correlations present in the input is very limited [3]. Here, we study the role of second-order correlations in the multivariate binary output statistics of such linear-nonlinear model neurons with a threshold nonlinearity responding to natural images. That is, each unit can be described by an affine transformation zk = wT k x + ϑ followed by a point-wise signum function sk = sgn(zk). Our interest in this model is twofold: (A) It can be regarded a parsimonious model for the analysis of population codes of natural images for which the 1 10 12 14 16 18 20 10 −2 10 −1 10 0 2 4 6 8 10 0 3 x 10 −3 Dimension ∆H (%) 2 4 6 8 10 0 1 2 3 4 5 6 x 10 −5 Dimension JS−Divergence (bits) 12 14 16 18 20 0 0.5 1 Dimension ∆H (%) A B C D log 2(Number of Samples) log∆H (%) Figure 1: Similarity between the Ising and the DG model. A+C: Entropy difference ∆H between the Ising model and the Dichotomized Gaussian distribution as a function of dimensionality. A: Up to 10 dimensions we can compute HDG directly by evaluating Eq. 6. Gray dots correspond to different sets of parameters. For m ≥4, the relatively large scatter and the existence of negative values is due to the limited numerical precision of the Monte-Carlo integration. Errorbars show standard error of the mean. B. JS-divergence DJS between PI and PDG. C. ∆H as above, for higher dimensions. Up to 20 dimensions ∆H remains very small. The increase for m →20 is most likely due to undersampling of the distributions. D. ∆H as function of sample size used to estimate HDG, at seven (black) and ten (grey) dimensions (note log scale on both axes). ∆H decreases with a power law with increasing sample sizes. computational power and the bandwidth of each unit is limited. (B) The same model can also be used more generally to fit multivariate binary data with given pairwise correlations, if x is drawn from a Gaussian distribution. In particular, we will show that the resulting distribution closely resembles the binary maximum entropy models known as Ising models or Boltzmann machines which have recently become popular for the analysis of spike train recordings from retinal ganglion cell responses [13, 12]. Motivated by the analysis in [12, 13] and the discussion in [10] we are interested at a more general level in the following questions: are pairwise interactions enough for understanding the statistical regularities in high-dimensional natural data (given that they provide a good fit in the lowdimensional case)? If we suppose that pairwise interactions are enough, what can we say about the amount of redundancies in high-dimensional data? In comparison with neural spike data, natural images provide two advantages for studying these questions: 1) It is much easier to obtain large amounts of data with millions of samples which are less prone to nonstationarities. 2) Often differences in the higher-order statistics such as between pink noise and natural images can be recognized by eye. 2 Second order models for binary variables In order to study whether pairwise interactions are enough to determine the statistical regularities in high-dimensional data, it is necessary to be able to compute the maximum entropy distribution for large number of dimensions N. Given a set of measured statistics, maximum entropy models yield a full probability distribution that is consistent with these constraints but does not impose any 2 0 1 2 0 0.05 α 0 1 2 0 1 2 x 10 −4 α Figure 2: Examples of covariance matrices (A+B.) and their learned approximations (C+D) at m = 10 for clarity. α is the parameter controlling the steepness of correlation decrease. E+F. Eigenvalue spectra of both matrices. G. Entropy difference ∆H and H. JS-divergence between the distribution of samples obtained from the two models at m = 7. additional structure on the distribution [7]. For binary data with given mean activations µi = ⟨si⟩ and correlations between neurons Σij = ⟨sisj⟩−⟨si⟩⟨sj⟩, one obtains a quadratic exponential probability mass function known as the Ising model in physics or as the Boltzmann machine in machine learning. Currently all methods used to determine the parameters of such binary maximum entropy models suffer from the same drawback: since the parameters do not correspond directly to any of the measured statistics, they have to be inferred (or ‘learned’) from data. In high dimensions though, this poses a difficult computational problem. Therefore the characterization of complete neural circuits with possibly hundreds of neurons is still out of reach, even though analysis was recently extended to up to forty neurons [14]. To make the maximum entropy approach feasible in high dimensions, we propose a new strategy: Sampling from a ‘near-maximum’ entropy model that does not require any complicated learning of parameters. In order to justify this approach, we verify empirically that the entropy of the full probability distributions obtained with the near-maximum entropy model are indistinguishable from those obtained with classical methods such as Gibbs sampling for up to 20 dimensions. 2.1 Boltzmann machine learning For a binary vector of neural activities s ∈{−1, 1}m and specified µi and Σij the Ising model takes the form PI(s) = 1 Z exp m X i=1 hisi + 1 2 X i̸=j Jijsisj , (1) where the local fields hi and the couplings Jij have to be chosen such that ⟨si⟩= µi and ⟨sisj⟩− ⟨si⟩⟨sj⟩= Σij. Unfortunately, finding the correct parameters turns out to be a difficult problem which cannot be solved in closed form. Therefore, one has to resort to an optimization approach to learn the model parameters hi and Jij from data. This problem is called Boltzmann machine learning and is based on maximization of the log-likelihood L = ln PI({si}N i=1|h, J) [1] where N is the number of samples. The gradient of the likelihood can be computed in terms of the empirical covariance and the covariance of si and sj as produced by the current model: ∂L ∂Jij = ⟨sisj⟩Data −⟨sisj⟩Model (2) The second term on the right hand side is difficult to compute, as it requires sampling from the model. Since the partition function Z in Eq. (1) is not available in closed form, Monte-Carlo methods such 3 Figure 3: Random samples of dichotomized 4x4 patches from the van Hateren image data base (left) and from the corresponding dichotomized Gaussian distribution with equal covariance matrix (middle). It is not possible to see any systematic difference between the samples from the two distributions. For comparison, this is not so for the sample from the independent model (right). as Gibbs sampling are employed [9] in order to approximate the required model average. This is computationally demanding as sampling is necessary for each individual update. While efficient sampling algorithms exist for special cases [6], it still remains a hard and time consuming problem in the general case. Additionally, most sampling algorithms do not come with guarantees for the quality of the approximation of the required average. In conclusion, parameter fitting of the Ising model is slow and oftentimes painstaking, especially in high dimensions. 2.2 Modeling with the dichotomized Gaussian Here we explore an intriguing alternative to the Monte-Carlo approach: We replace the Ising model by a ’near-maximum’ entropy model, for which both parameter computation and sampling is easy. A very convenient, but in this context rarely recognized, candidate model is the dichotomized Gaussian distribution (DG) [11, 5, 4]. It is obtained by supposing that the observed binary vector s is generated from a hidden Gaussian variable z ∼N(γ, Λ) , si = sgn(zi). (3) Without loss of generality, we can assume unit variances for the Gaussian, i.e. Λii = 1, the mean µ and the covariance matrix Σ of s are given by µi = 2Φ(γi) −1 , Σii = 4Φ(γi)Φ(−γi) , Σij = 4Ψ(γi, γj, Λij) for i ̸= j (4) where Ψ(x, y, λ) = Φ2(x, y, λ) −Φ(x)Φ(y) . Here Φ is the univariate standardized cumulative Gaussian distribution and Φ2 its bivariate counterpart. While the computation of the model parameters was hard for the Ising model, these equations can be easily inverted to find the parameters of the hidden Gaussian distribution: γi = Φ−1 µi + 1 2 (5) Determining Λij generally requires to find a suitable value such that Σij −4Ψ(γi, γj, Λij) = 0. This can be efficently solved by numerical computations, since the function is monotonic in Λij and has a unique zero crossing. We obtain an especially easy case, when γi = γj = 0, as then Λij = sin π 2 Σij . It is also possible to evaluate the probability mass function of the DG model by numerical integration, PDG(s) = 1 (2π)N/2|Λ|1/2 Z b1 a1 . . . Z bm am exp −(s −γ)T Λ−1(s −γ) , (6) where the integration limits are chosen as ai = 0 and bi = ∞, if si = 1, and ai = −∞and bi = 0, otherwise. In summary, the proposed model has two advantages over the traditional Ising model: (1) Sampling is easy, and (2) finding the model parameters is easy too. 4 3 Near-maximum entropy behavior of the dichotomized Gaussian distribution In the previous section we introduced the dichotomized Gaussian distribution. Our conjecture is that in many cases it can serve as a convenient approximation to the Ising model. Now, we investigate how good this approximation is. For a wide range of interaction terms and mean activations we verify that the DG model closely resembles the Ising model. In particular we show that the entropy of the DG distribution is not smaller than the entropy of the Ising model even at rather high dimensions. 3.1 Random Connectivity We created randomly connected networks of varying size m, where mean activations hi and interactions terms Jij were drawn from N(0, 0.4). First, we compared the entropy HI = −P s PI(s) log2 PI(s) of the thus specified Ising model obtained by evaluating Eq. 1 with the entropy of the DG distribution HDG computed by numerical integration1 from Eq. 6 (twenty parameter sets). The entropy difference ∆H = HI −HDG was smaller than 0.002 percent of HI (Fig. 1 A, note scale) and probably within the range of the numerical integration accuracy. In addition, we computed the Jensen-Shannon divergence DJS[PI∥PDG] = 1 2 (DKL[PI∥M] + DKL[PDG∥M]), where M = 1 2(PI + PDG) [8]. We find that DJS[PI∥PDG] is extremly small up to 10 dimensions (Fig. 1 B). Therefore, the distributions seem to be not only close in their respective entropy, but also to have a very similar structure. Next, we extended this analysis to networks of larger size and repeated the same analysis for up to twenty dimensions. Since the integration in Eq. 6 becomes too time-consuming for m →20 due to the large number of states, we used a histogram based estimate of PDG (using 3 · 106 samples for m < 15 and 15 · 106 samples for m ≥15). The estimate of ∆H is still very small at high dimensions (Fig. 1 C, below 0.5%). We also computed DJS, which scaled similarly to ∆H (data not shown). In Fig. 1 C, ∆H seems to increase with dimensionality. Therefore, we investigated how the estimate of ∆H is influenced by the number of samples used. We computed both quantities for varying numbers of samples from the DG distribution (for m = 7, 10). As ∆H decreases according to a power law with increasing m, the rise of ∆H observed in Fig. 1 C is most likely due to undersampling of the distribution. 3.2 Specified covariance structure To explore the relationship between the two techniques more systematically, we generated covariance matrices with varying eigenvalue spectra. We used a parametric Toeplitz form, where the nth diagonal is set to a constant value exp(−α · n) (Fig. 2A and B, m = 7, 10). We varied the decay parameter α, which led to a widely varying covariance structure (For eigenvalue spectra, see Fig. 2E and F). We fit the Ising models using the Boltzmann machine gradient descent procedure. The covariance matrix of the samples drawn from the Ising model resembles the original very closely (Fig. 2C and D). We also computed the entropy of the DG model using the desired covariance structure. We estimated ∆H and DJS[PG∥PDG] averaged over 10 trials with 105 samples obtained by Gibbs sampling from the Ising model. ∆H is very close to zero (Fig. 2G, m = 7) except for small αs and never exceeded 0.05%. Moreover, the structure of both distributions seems to be very similar as well (Fig. 2H, m = 7). At m = 10, both quantities scaled qualitatively similair (data not shown). We also repeated this analysis using equations 1 and 6 as before, which lead to similar results (data not shown). Our experiments demonstrate clearly that the dichotomized Gaussian distribution constitutes a good approximation to the quadratic exponential distribution for a large parameter range. In the following section, we will exploit the similarity between the two models to study how the role of second-order correlations may change between low-dimensional and high-dimensional statistics in case of natural images. 1For integration, we used the mvncdf function of Matlab. For m ≥4 this function employs Monte-Carlo integration. 5 Figure 4: A: Negative log probabilities of the DG model are plotted against ground truth (red dots). Identical distributions fall on the diagonal. Data points outside the area enclosed by the dashed lines indicate significant differences between the model and ground truth. The DG model matches the true distribution very well. For comparison the independent model is shown as well (blue crosses). B: The multi-information of the true distribution (blue dots) accurately agrees with the multi-information of the DG model (red line). Similar to the analysis in [12], we observe a power law behavior of the entropy of the independent model (black solid line) and the mutli-information. Linear extrapolation (in the log-log plot) to higher dimensions is indicated by dashed lines. C: Different way of presentation of the same data as in B: the joint entropy H = Hindep −I (blue dots) is plotted instead of I and the axis are in linear scale. The dashed red line represents the same extrapolation as in B. 4 Natural images: Second order and beyond We now investigate to which extent the statistics of natural images with dichotomized pixel intensities can be characterized by pairwise correlations only. In particular, we would like to know how the role of pairwise correlations opposed to higher-order correlations changes depending on the dimensionality. Thanks to the DG model introduced above, we are in the position to study the effect of pairwise correlations for high-dimensional binary random variables (N ≈1000 or even larger). We use the van Hateren image database in log-intensity scale, from which we sample small image patches at random positions. The threshold for the dichotomization is set to the median of pixel intensities. That is, each binary variable encodes whether the corresponding pixel intensity is above or below the median over the ensemble. Up to patch sizes of 4 × 4 pixel, the true joint statistics can be assessed using nonparametric histogram methods. Before we present quantitative comparisons, it is instructive to look at random samples from the true distribution (Fig. 3, left), from the DG model with same mean and covariance (Fig. 3, middle), and from the corresponding independent model (Fig. 3, right). By visual inspection, it seems that the DG model fits the true distribution well. In order to quantify how well the DG model matches the true distribution, we draw two independent sets of samples from each (N = 2 · 106 for each set) and generate a scatter plot as shown in Fig. 4 A for 4×4 image patches. Each dot corresponds to one of the 216 = 65536 possible different binary patterns. The relative frequencies of these patterns according to the DG model (red dots) and according to the independent model (blue dots) are plotted against the relative frequencies obtained from the natural image patches. The solid diagonal line corresponds to a perfect match between model and ground truth. The dashed lines enclose the regions within which deviations are to be expected due to the finite sampling size. Since most of the red dots fall within this region, the DG model fits the data distribution very well. We also systematically evaluated the JS-divergence and the multi-information I[S] = P k H[Sk] − H[S] as a function of dimensionality. That is, we started with the bivariate marginal distribution of two randomly selected pixels. Then we incrementally added more pixels of random location until the random vector contains all the 16 pixels of the 4 × 4 image patches. Independent of the dimension, the JS-divergence between the DG model and the true distribution is smaller than 0.015 bits. For comparison, the JS-divergence between the independent model and the true distribution increases with dimensionality from roughly 0.2 bits in the case of two pixels up to 0.839 bits in the case of 16 pixels. For two independent sets of samples both drawn from natural image data the JS-divergence ranges between 0.006 and 0.007 bits for 4 × 4 patches setting the gold standard for the minimal possible JS-divergence one could achieve with any model due to finite sampling size. Carrying out the same type of analysis as in [12], we make qualitatively the same observations as it was reported there: as shown above, we find a quite accurate match between the two distributions. 6 Figure 5: Random samples of dichotomized 32x32 patches from the van Hateren image data base (left) and from the corresponding dichotomized Gaussian distribution with equal covariance matrix (right). For the latter, the percept of typical objects is missing due to the ignorance of higher-order correlations. This striking difference is not obvious, however, at the level of 4x4 patches, for which we found an excellent match of the dichotomized Gaussian to the ensemble of natural images. Furthermore, the multi-information of the DG model (red solid line) and of the true distribution (blue dots) increases linearly on a loglog-scale with the number of dimensions (Fig. 4 B). Both findings can be verified only up to a rather limited number of dimensions (less than 20). Nevertheless, in [12], two claims about the higher-dimensional statistics have been based on these two observations: First, that pairwise correlations may be sufficient to determine the full statistics of binary responses, and secondly, that the convergent scaling behavior in the log-log plot may indicate a transition towards strong order. Using natural images instead of retinal ganglion cell data, we would like to verify to what extent the low-dimensional observations can be used to support these claims about the high-dimensional statistics [10]. To this end we study the same kind of extrapolation (Fig. 4 B) to higher dimensions (dashed lines) as in [12]. The difference between the entropy of the independent model and the multi-information yields the joint entropy of the respective distribution. If the extrapolation is taken seriously, this difference seems to vanish at the order of 50 dimensions suggesting that the joint entropy of the neural responses approaches zero at this size—say for 7×7 image patches (Fig. 4 C). Though it was not taken literally, this point of ‘freezing’ has been pointed out in [12] as a critical network size at which a transition to strong order is to be expected. The meaning of this assertion, however, is not clear. First of all, the joint entropy of a distribution can never be smaller than the joint entropy of any of its marginals. Therefore, the joint entropy cannot decrease with increasing number of dimensions as the extrapolation would suggest (Fig. 4 C). Instead it would be necessary to ask more precisely how the growth rate of the joint entropy can be characterized and whether there is a critical number of dimensions at which the growth rate suddenly drops. In our study with natural images, visual inspection does not indicate anything special to happen at the ‘critical patch size’ of 7 × 7 pixels. Rather, for all patch sizes, the DG model yields dichotomized pink noise. In Fig. 5 (right) we show a sample from the DG model for 32×32 image patches (i.e. 1024 dimensions) which provides no indication for a particularly interesting change in the statistics towards strong order. The exact law according to which the multi-information grows with the number of dimensions for large m, however, is not easily assessed and remains to be explored. Finally, we point out that the sufficiency of pairwise correlations at the level of m = 16 dimensions does not hold any more in the case of large m: the samples from the true distribution at the left hand side of Fig. 5 clearly show much more structure than the samples from the DG model (Fig. 5, right), indicating that pairwise correlations do not suffice to determine the full statistics of large image patches. Even if the match between the DG model and the Ising model may turn out to be less accurate in high dimensions, this would not affect our conclusion. Any mismatch would only introduce more order in the DG model than justified by pairwise correlations only. 5 Conclusion and Outlook We proposed a new approach to maximum entropy modeling of binary variables, extending maximum entropy analysis to previously infeasible high dimensions: As both sampling and finding pa7 rameters is easy for the dichotomized Gaussian model, it overcomes the computational drawbacks of Monte-Carlo methods. We verified numerically that the empirical entropy of the DG model is comparable to that obtained with Gibbs sampling at least up to 20 dimensions. For practical purposes, the DG distribution can even be superior to the Gibbs sampler in terms of entropy maximization due to the lack of independence between consecutive samples in the Gibbs sampler. Although the Ising model and the DG model are in principle different, the match between the two turns out to be surprisingly good for a large region of the parameter space. Currently, we are trying to determine where the close similarity between the Ising model and the DG model breaks down. In addition, we explore the possibility to use the dichotomized Gaussian distribution as a proposal density for Monte-Carlo methods such as importance sampling. As it is a very close approximation to the Ising model, we expect this combination to yield highly efficient sampling behaviour. In summary, by linking the DG model to the Ising model, we believe that maximum entropy modeling of multivariate binary random variables will become much more practical in the future. We used the DG model to investigate the role of second-order correlations in the context of sensory coding of natural images. While for small image patches the DG model provided an excellent fit to the true distribution, we were able to show that this agreement breakes down in the case of larger image patches. Thus caution is required when extrapolating from low-dimensional measurements to higher-dimensional distributions because higher-order correlations may be invisible in low-dimensional marginal distributions. Nevertheless, the maximum entropy approach seems to be a promising tool for the analysis of correlated neural activities, and the DG model can facilitate its use significantly in practice. Acknowledgments We thank Jakob Macke, Pierre Garrigues, and Greg Stephens for helpful comments and stimulating discussions, as well as Alexander Ecker and Andreas Hoenselaar for last minute advice. An implementation of the DG model in Matlab and R will be avaible at our website http://www.kyb.tuebingen.mpg.de/bethgegroup/code/DGsampling. References [1] D.H. Ackley, G.E. Hinton, and T.J. Sejnowski. A learning algorithm for boltzmann machines. Cognitive Science, 9:147–169, 1985. [2] H.B. Barlow. Sensory mechanisms, the reduction of redundancy, and intelligence. In The Mechanisation of Thought Processes, pages 535–539, London: Her Majesty’s Stationery Office, 1959. [3] M. Bethge. Factorial coding of natural images: How effective are linear model in removing higher-order dependencies? J. Opt. Soc. Am. A, 23(6):1253–1268, June 2006. [4] D.R. Cox and N. Wermuth. On some models for multivariate binary variables parallel in complexity with the multivariate gaussian distribution. Biometrika, 89:462–469, 2002. [5] L.J. Emrich and M.R. Piedmonte. A method for generating high-dimensional multivariate binary variates. The American Statistician, 45(4):302–304, 1991. [6] M. Huber. A bounding chain for swendsen-wang. Random Structures & Algorithms, 22:53–59, 2002. [7] E.T. Jaynes. Where do we stand on maximum entropy inference. In R.D. Levine and M. Tribus, editors, The Maximum Entropy Formalism. MIT Press, Cambridge, MA, 1978. [8] J. Linn. Divergence measures based on the shannon entropy. IEEE Trans Inf Theory, 37:145–151, 1991. [9] D. J. C. MacKay. Information Theory, Inference and Learning Algorithms. Cambridge University Press, 2003. [10] Sheila H Nirenberg and Jonathan D Victor. Analyzing the activity of large populations of neurons: how tractable is the problem? Current Opinion in Neurobiology, 17:397–400, August 2007. [11] Karl Pearson. On a new method of determining correlation between a measured character a, and a character b, of which only the percentage of cases wherein b exceeds (or falls short of) a given intensity is recorded for each grade of a. Biometrika, 7:96–105, 1909. [12] Elad Schneidman, Michael J Berry, Ronen Segev, and William Bialek. Weak pairwise correlations imply strongly correlated network states in a neural population. Nature, 440(7087):1007–1012, Apr 2006. [13] J Shlens, JD Field, JL Gauthier, MI Grivich, D Petrusca, A Sher, AM Litke, and EJ Chichilnisky. The structure of multi-neuron firing patterns in primate retina. J Neurosci, 26(32):8254–8266, Aug 2006. [14] G. Tkacik, E. Schneidman, M.J. Berry, and W. Bialek. Ising models for networks of real neurons. arXiv:qbio.NC/0611072, 1:1–4, 2006. 8
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Privacy-Preserving Belief Propagation and Sampling Michael Kearns, Jinsong Tan, and Jennifer Wortman Department of Computer and Information Science University of Pennsylvania, Philadelphia, PA 19104 Abstract We provide provably privacy-preserving versions of belief propagation, Gibbs sampling, and other local algorithms — distributed multiparty protocols in which each party or vertex learns only its final local value, and absolutely nothing else. 1 Introduction In this paper we provide provably privacy-preserving versions of belief propagation, Gibbs sampling, and other local message-passing algorithms on large distributed networks. Consider a network of human social contacts, and imagine that each party would like to compute or estimate their probability of having contracted a contagious disease, which depends on the exposures to the disease of their immediate neighbors in the network. If network participants (or their proxy algorithms) engage in standard belief propagation, each party would learn their probability of exposure conditioned on any evidence — and a great deal more, including information about the exposure probabilities of their neighbors. Obviously such leakage of non-local information is highly undesirable in settings where we regard each party in the network as a self-interested agent, and privacy is paramount. Other examples include inference problems in distributed military sensor networks (where we would like the “capture” of one sensor to reveal as little non-local state information as possible), settings where networks of financial organizations would like to share limited information, and so on. By a privacy-preserving version of inference (for example), we informally mean a protocol in which each party learns their conditional probability of exposure to the disease and absolutely nothing else. More precisely, anything a party can efficiently compute after having participated in the protocol, they could have efficiently computed alone given only the value of their conditional probability — thus, the protocol leaked no additional information beyond its desired outputs. Classical and powerful tools from cryptography [6] provide solutions to this problem, but with the significant drawback of entirely centralizing the privacy-preserving computation. Put another way, the straightforward solution from cryptography requires every party in the network to have the ability to broadcast to all others, and the resulting algorithm may bear little resemblance to standard belief propagation. Clearly this is infeasible in settings where the network is very large and entirely distributed, where individuals may not even know the size of the overall network, much less its structure and the identity of its constituents. While there has been work on minimizing the number of messages exchanged in centralized privacy-preserving protocols [9], ours are the first results that preserve the local communication structure of distributed algorithms like belief propagation. Our protocols are faithful to the network topology, requiring only the passing of messages between parties separated by one or two hops in the network. Furthermore, our protocols largely preserve the algebraic structure of the original algorithms (for instance, the sum-product structure of belief propagation) and enjoy all the correctness guarantees of the originals (such as exact inference in trees for belief prop or convergence of Gibbs sampling to the joint distribution). Our technical methods show how to blend tools from cryptography (secure multiparty computation and blindable encryption) with local message-passing algorithms in a way that preserves the original computations, but in which all messages appear to be randomly distributed from the viewpoint of any individual. 1 All results in this paper apply to the “semi-honest” or “honest but curious” model in the cryptography literature, in which participants obediently execute the protocol but may attempt to infer non-private information from it. We expect that via the use of zero-knowledge proof techniques, our protocols may be strengthened to models in which participants who deviate from the protocol are detected. 2 Background and Tools from Cryptography 2.1 Secure Multiparty Function Computation Let f(x1, . . . , xk) be any function on k inputs. Imagine a scenario in which there are k distinct parties, each in possession of exactly one of these inputs (that is, party i initially knows xi) and the k parties would like to jointly compute the value of f(x1, . . . , xk). Perhaps the simplest protocol would have all parties share their private inputs and then individually compute the value of f. However, in many natural settings, we would like the parties to be able to perform this joint computation in a privacy-preserving fashion, with each party revealing as little as possible about their private input. Simple examples include voting — we would all like to learn the results of the election without having to broadcast our private votes — and the so-called “Millionaire’s Problem” in which two individuals would like to learn who is wealthier, without revealing their precise wealth to each other. If a trusted “third party” is available, one solution would be to provide the private inputs to them, and have them perform the computation in secrecy, only announcing the final result. The purpose of the theory of secure multiparty function computation [6] is to show that under extremely general circumstances, a third party is surprisingly unnecessary. Note that it is typically inevitable that some information is revealed just by the result of the computation of f itself. For example, in the Millionaire’s Problem, there is no avoiding the poorer party learning a lower bound on the richer’s wealth (namely, the poorer party’s wealth). The goal is thus to reveal nothing beyond what it implied by the value of f. To formalize this notion in a complexity-theoretic framework, let us assume without loss of generality that each input xi is n bits in length. We make the natural and common assumptions that the function f can be computed in time polynomial in kn, and that each party’s computational resources are bounded by a polynomial in n. We (informally) define a protocol Π for the k parties to compute f to be a specific mechanism by which the parties exchange messages and perform computations, ending with every party learning the value y = f(x1, . . . , xk). One (uninteresting) protocol is the one in which each party sends their private inputs to all others, and every party computes y alone. Definition 1 1 Let Π be any protocol for the k parties to jointly compute the value y = f(x1, . . . , xk) from their n-bit private inputs. We say that Π is privacy-preserving if for every 1 ≤i ≤k, anything that party i can compute in time polynomial in n following the execution of Π, they could also compute in polynomial time given only their private input xi and the value y. In other words, whatever information party i is able to obtain from their view of the execution of protocol Π, it does not let them efficiently compute anything they couldn’t efficiently compute just from being told the final output y of Π (and their private input xi). This captures the notion that while y itself may “leak” some information about the other private inputs xj, the protocol Π yields nothing further.2 Further, for the following theorem we can consider more general vector outputs and randomized functionalities, which we need for our technical results. Theorem 1 (See e.g. [6]) Let f(x1, . . . , xk) = (y1, . . . , yk) be any (possibly randomized) k-input, k-output functionality that can be computed in polynomial time. Then under standard cryptographic assumptions, 3 there exists a polynomial time privacy-preserving protocol Π for f (that is, a protocol in which party i learns nothing not already implied by their private input xi and private output yi). 1We state this definition informally, as the complete technical definition is somewhat lengthy and adds little intuition. It involves both formalizing the notion of a multiparty computation protocol, as well as defining the “view” of an individual party of a specific execution of the protocol. The definition involves computational indistinguishability of probability distributions since the protocols may often use randomization. 2Our definition of privacy does not imply that coalitions of parties cannot together compute additional information. In the extended version of this paper, we discuss the difficulty of achieving this stronger notion of privacy with any protocol that uses a truly distributed method of computation. 3An example would be the existence of trapdoor permutations [6]. 2 This remarkable and important theorem essentially says that whatever a population can jointly compute, it can jointly compute with arbitrary restrictions on who learns what. A powerful use of vector outputs is to enforce knowledge asymmetries on the parties. For instance, in the Millionaire’s Problem, by defining one player’s output to always be nil, we can ensure that this player learns absolutely nothing from the protocol, while the other learns which player is wealthier. The proof of Theorem 1 is constructive, providing an algorithm to transform any polynomial circuit into a polynomial-time privacy-preserving protocol for k parties. As discussed in the introduction, this theorem can be immediately applied to (say) belief propagation to yield centralized privacy-preserving protocols for inference; our contribution is preserving the highly distributed, local message-passing structure of belief propagation and similar algorithms. 2.2 Public-Key Encryption with Blinding The second cryptographic primitive that we shall require is standard public-key encryption with an additional property known as blinding. A standard public-key cryptosystem allows any party to generate a pair of keys (P, S), which we can think of as k-bit strings; k is often called the security parameter. Associated with the public key P there is a (possibly probabilistic) encryption function EP and associated with the secret or private key S there is a (deterministic) decryption function DS. Informally, the system should have the following security properties: • For any n-bit x, the value of the function EP (x) can be computed in polynomial time from inputs x and P. Similarly, DS(y) can be computed efficiently given y and S. • For any n-bit input x, DS(EP (x)) = x. Thus, decryption is the inverse of encryption. • For any n-bit x, it is hard for a party knowing only the public key P and the encryption EP (x) to compute x. 4 Thus, in such a scheme, anyone knowing the public key of Alice can efficiently compute and send encrypted messages to Alice, but only Alice, who is the sole party knowing her private key, can decrypt those messages. Such cryptosystems are widely believed to exist and numerous concrete proposals have been examined for decades. As one specific example that allows probabilistic encryption of individual bits, let the public key consist of an integer N = p · q that is the product of two k/2-bit randomly generated prime numbers p and q, as well as a number z that has the property that z is not equal to x2 mod N for any x. It is easy to generate such (N, z) pairs. In order to encrypt a 0, one simply chooses x at random and lets the encryption be y = x2 mod N, known as a quadratic residue. In order to encrypt a 1, one instead sends y = x2 · z mod N, which is guaranteed to not be a quadratic residue. It is not difficult to show that given the factors p and q (which constitute the secret key), one can efficiently compute whether y is a quadratic residue and thus learn the decrypted bit. Furthermore, it is widely believed that decryption is actually equivalent to factoring N, and thus intractable without the secret key. This simple public-key cryptosystem also has the additional blinding property that we will require. Given only the public key (N, z) and an encrypted bit y as above, it is the case that for any value w, w2y mod N is a quadratic residue if and only if y is a quadratic residue, and that furthermore w2y mod N is uniformly distributed among all (non-)quadratic residues if y is a (non-)quadratic residue. Thus, a party knowing only Alice’s public key can nevertheless take any bit encrypted for Alice and generate random re-encryptions of that bit without needing to actually know the decryption. We refer to this operation as blinding an encrypted bit. 3 Privacy-Preserving Belief Propagation In this section we briefly review the standard algorithm for belief propagation on trees [10] and outline how to run this algorithm in a privacy-preserving manner such that each variable learns only its final marginals and no additional new information that is not implied by these marginals. In standard belief propagation, we are given an undirected graphical model with vertex set X for which the underlying network is a tree. We denote by V(Xi) the set of possible values of Xi ∈X, 4This is often formalized by asserting that the distribution of the encryption is computationally indistinguishable from true randomness in time polynomial in n and k. 3 and by N(Xi) the set of Xi’s neighbors. For each Xi ∈X, we are given a non-negative potential function ψi over possible values xi ∈V(Xi). Similarly, for each pair of adjacent vertices Xi and Xj, we are given a non-negative potential function ψi,j over joint assignments to Xi and Xj. The main inductive phase of the belief propagation algorithm is the message-passing phase. At each step, a node Xi computes a message µi→j to send to some Xj ∈N(Xi). This message is indexed by all possible assignments xj ∈V(Xj), and is defined inductively by µi→j(xj) = X xi∈V(Xi) ψi(xi)ψi,j(xi, xj) Y Xk∈N(Xi)\Xj µk→i(xi). (1) Belief propagation follows the so-called message-passing protocol, in which any vertex of degree d that has received the incoming messages from any d−1 of its neighbors can perform the computation above in order to send an outgoing message to its remaining neighbor. Eventually, the vertex will receive a message back from this last neighbor, at which point it will be able to calculate messages to send to its remaining d −1 neighbors. The protocol begins at the leaves of the tree, and it follows from standard arguments that until all nodes have received incoming messages from all of their neighbors, there must be some vertex that is ready to compute and send a new message. The message-passing phase ends when all vertices have received messages from all of their neighbors. Once vertex Xi has received all of its incoming messages, the marginal distribution P is proportional to their product. That is, if xi is any setting to Xi, then P[Xi = xi] ∝ψi(xi) Y Xj∈N(Xi) µj→i(xi). (2) When there is evidence in the network, represented as a partial assignment ⃗e to some subset E of the variables, we can simply hard-wire this evidence into the potential functions ψj for each Xj ∈E. In this case it is well-known that the algorithm computes the conditional marginals P[Xi = xi|E = ⃗e]. For a more in-depth review of belief propagation, see Yedidia et al. [13] or Chapter 8 of Bishop [1]. 3.1 Mask Propagation and the Privacy-Preserving Protocol We assume that at the beginning of the privacy-preserving protocol, each node Xi knows its own individual potential function ψi, as well as the joint potential functions ψi,j for all Xj ∈N(Xi). Recall that our fundamental privacy goal is to allow each vertex Xi to compute its own marginal distribution P[Xi = xi] (or P[Xi = xi|E = ⃗e] if there is evidence), but absolutely nothing else. In particular, Xi should not be able to compute the values of any of the incoming messages from its neighbors. Knowledge of µj→i(xi), for example, along with µi→j and ψi,j, may give Xi information about the marginals over Xj, a clear privacy violation. We thus must somehow prevent Xi from being able to “read” any of its incoming messages — nor even its own outgoing messages — yet still allow each variable to learn its own set of marginals at the end. To accomplish this we combine tools from secure multiparty function computation with a method we call “mask propagation”, in which messages remain “masked” (that is, provably unreadable) to the vertices at all times. The keys required to unmask the messages are generated locally as the computation propagates through the tree, thus preserving the original communication pattern of the standard (non-private) algorithm. Before diving into the secure protocol, we first must fix conventions regarding the encoding of numerical values. We will assume throughout that all potential function values, all message values and all the required products computed by the algorithm can be represented as n-bit natural numbers and thus fall in ZN = {0, . . . , N −1} where N = 2n. As expressed by Equation (2), marginal probabilities are obtained by taking products of such n-bit numbers and then normalizing to obtain finite-precision real-valued numbers in the range [0, 1]. It will be convenient to think of values in ZN as elements of the cyclic group of order N with addition and subtraction modulo N. In particular, we will make frequent use of the following simple fact: for any fixed x ∈ZN, if r ∈ZN is chosen randomly among all n-bit numbers, then x+r mod N is also distributed randomly among all n-bit numbers. We can think of the random value r as “masking” or hiding the value of x to a party that does not know r, while leaving it readable to a party that does. Let us now return to the message-passing phase of the algorithm described by Equation (1), and let us focus on the computation of µi→j for a fixed setting xj of Xj. For the secure version of the algorithm, we make the following inductive message and knowledge assumptions: 4 • For each Xℓ∈N(Xi)\Xj, and for each setting xi of Xi, Xi has already obtained a masked version of µℓ→i(xi): µℓ→i(xi) + βj,ℓ(xi) mod N (3) where βj,ℓ(xi) is uniformly distributed in ZN. • Xi knows only the sum in Equation (3) (which again is uniformly distributed in ZN and thus meaningless by itself), and does not know the masking values βj,ℓ(xi). • Vertex Xj knows only the masking values βj,ℓ(xi), and not the sum in Equation (3). For all leaf nodes, these assumptions hold trivially at the start of the protocol, providing the base case for the induction. Now under these informational assumptions, vertex Xi knows the set Ii = {µℓ→i(xi) + βj,ℓ(xi) mod N : Xℓ∈N(Xi)\Xj, xi ∈V(Xi)} while vertex Xj knows the set Ij = {βj,ℓ(xi) mod N : Xℓ∈N(Xi)\Xj, xi ∈V(Xi)}. Let us first consider the case in which Xj is not a leaf node and thus has neighbors other than Xi itself. In order to complete the inductive step, it will be necessary for each Xk ∈N(Xj)\Xi to provide a set of masking values βk,i(xj) so that Xj can obtain a set of masked messages of the form µi→j(xj) + βk,i(xj). Here we focus on a single neighbor Xk of Xj. Vertex Xk privately generates a masking value βk,i(xj) that is uniformly distributed in Zn. It is clear that, ignoring privacy concerns, Xi and Xj together could compute ψi(xi)ψi,j(xi, xj) Q Xℓ∈N(Xi)\Xj µℓ→i(xi) for each fixed pair xi and xj. Thus from their joint inputs Ii, Ij, and βk,i(xj), ignoring privacy, Xi, Xj, and Xk could compute: X xi∈V(Xi) ψi(xi)ψi,j(xi, xj) Y Xℓ∈N(Xi)\Xj µℓ→i(xi) + βk,i(xj) mod N = µi→j(xj) + βk,i(xj) mod N (4) Since this expression can be computed jointly by Xi, Xj and Xk without privacy considerations, Theorem 1 establishes that we can construct an efficient protocol for them to compute it securely, allowing Xj to learn only the value of the expression in Equation (4), while Xi and Xk learn no new information at all (i.e. nil output). Note that this expression, due to the presence of the unknown masking value βk,i(xj), is a uniform randomly distributed number in Zn from Xj’s point of view. After this masking process has been completed for all Xk ∈N(Xj)\Xi, we will have begun to satisfy the inductive informational assumptions a step further in the propagation: for each neighbor Xk of Xj excluding Xi, Xj will know a masked version of µi→j(xj) in which the masking value βk,i(xj) is known only to Xk. Xj will obtain masked messages in a similar manner from all but one of its other neighbors in turn, and for all of its other values, until the inductive assumptions are fully satisfied at Xj. Every value received by Xi, Xj, and Xk during the above protocol is distributed uniformly at random in Zn from the perspective of its recipient, and thus conveys no information. It remains to consider the case in which Xj is a leaf node. In this case, there is no need to satisfy the inductive assumptions at the next level, as the propagation ends at the leaves. Furthermore, it is acceptable for Xj to learn its incoming messages directly, since these messages will be implied by its final marginal. From their joint input Ii and Ij, it is clear that Xi and Xj together could compute µi→j(xj) as given in Equation (1). Thus by Theorem 1, we can construct a protocol for them to efficiently compute this value in such a way that Xj learns only µi→j(xj) and Xi learns nothing. At the end of the message-passing phase, each internal (non-leaf) node Xi will know a set of masked messages from each of its neighbors. In particular, for each pair Xj, Xℓ∈N(Xi), for each xi ∈ V(Xi), Xi will know the values of µj→i(xi) + βℓ,j(xi). Ignoring privacy concerns, it is the case that Xi and any pair of its neighbors could compute the marginal of Xi in Equation (2). Invoking Theorem 1 again, we can construct an efficient protocol for Xi and this pair of neighbors to together compute the marginals such that Xi learns only the marginals and the neighbors learn nothing. Each leaf vertex Xi will be in possession of its unmasked messages µj→i(xi) for every xi ∈V(Xi) from its neighbor Xj, and can easily compute its marginals as given in Equation (2) without having learned anything not already implied by its initial potential functions and the marginals themselves. We use PrivateBeliefProp(T) to denote the algorithm above when applied to a particular tree T. The full proof of the following is omitted, but follows the logic sketched in the preceding sections. 5 Theorem 2 Under standard cryptographic assumptions, PrivateBeliefProp(T) allows every variable Xi to compute its own marginal distribution P[Xi] and nothing else (that is, nothing not already computable in polynomial time from only P[Xi] and the initial potential functions). Direct communication occurs only between variables who are immediate neighbors or two steps away in T, and secure function computation is never invoked on sets of more than three variables. 5 We briefly note a number of extensions to Theorem 2 and the methods described above. Loopy Belief Propagation: Theorem 2 can be extended to privacy-preserving loopy belief propagation on graphs that contain cycles. Because of the protocol’s faithfulness to the original algorithm, the same convergence and correctness claims hold as in standard loopy belief propagation [7]. Computing Only Partial Information: Allowing a variable to learn its exact numerical marginal distribution may actually convey a great deal of information. We might instead only want each variable to learn, for instance, whether its probability of taking on a given value is greater than 0.1 or not. Theorem 2 can easily be generalized to allow each variable to learn only any partial information about its own marginal. Privacy-Preserving Junction Tree: The protocol can also be modified to perform privacypreserving belief propagation on a junction tree [11]. Here it is necessary to take intra-clique privacy into account in order to enforce that variables can learn only their own marginals and not, for example, the marginals of other nodes within the same clique. NashProp and Other Message-Passing Algorithms: The methods described here can also be applied to provide privacy-preserving versions of the NashProp algorithm [8], allowing players in a multiparty game to jointly compute and draw actions from a Nash equilibrium, with each player learning only his own action and nothing else.6 We are investigating more general applications of our methods to a broad class of message-passing algorithms that would include many others. 4 Privacy-Preserving Gibbs Sampling We now move on to the problem of secure Gibbs sampling on an undirected graphical model G. The local potential functions accompanying G can be preprocessed to obtain conditional distributions for each variable given a setting of all its neighbors (Markov blanket). Thus we henceforth assume that each variable has access to its local conditional distribution, which it will be convenient to represent in a particular tabular form. To simplify presentation, we will assume each variable is binary, taking on values in {0, 1}, but this assumption is easy to relax. If a node Xi is of degree d, the conditional distribution of Xi given a particular assignment to its neighbors will be represented by a table Ti with 2d rows and d + 1 columns. The first d columns range over all 2d possible assignments ⃗x to N(Xi), while the final column contains the numerical value P[Xi = 1|N(Xi) = ⃗x]. We will use Ti(⃗x) to denote the value P[Xi = 1|N(Xi) = ⃗x] stored in the d + 1st column in the row corresponding to the assignment ⃗x. With this notation, the standard (non-private) Gibbs sampling algorithm [4, 2] can be easily described. After choosing an initial assignment to all of the variables in G (for instance, uniformly at random), the algorithm repeatedly resamples values for individual variables conditioned on the current values of their neighbors. More precisely, at each step, a variable Xi is chosen for resampling. Its current value is replaced by randomly drawing value 1 with probability Ti(⃗x) and value 0 with probability 1 −Ti(⃗x) where ⃗x is the current set of assignments to N(Xi). To implement a privacy-preserving variant of Gibbs sampling, we must solve the following cryptographic problem: how can a set of vertices communicate with their neighbors in order to repeatedly resample their values from their conditional distributions given their neighbors’ current assignments, without learning any information except their own final values at the end of the process and anything that is implied by these values? Again, we would like to accomplish this with limited communication so that no vertex is required to communicate with a vertex more than two hops away. 5Since the application of standard secure function computation requires broadcast among all participants, it is a feature of the algorithm that it limits such invocations to three parties at a time. 6See work by Dodis et al. [3] and Teague [12] for more on privacy-preserving computation in game theory. 6 In order for each variable to learn only its final sampled value after some number of iterations, and not its intermediate resampled values (which may be enough to provide a good approximation of the marginal distribution on the variable), we first provide a way of distributing the current value of a vertex so that it cannot be learned by any vertex in isolation. One way of accomplishing this is by assigning each vertex Xi a “distinguished neighbor” N ∗(Xi). Xi will hold one bit bi while N ∗(Xi) will hold a second bit b′ i such that the current value of Xi is bi ⊕b′ i. Using such an encoding, there is a simple but relatively inefficient construction for privacypreserving Gibbs sampling that uses only secure multiparty function computation, but that invokes Theorem 1 on entire neighborhoods of the graph. In graphs with high degree, this requires broadcast communication between a large number of parties, which we would like to avoid. Here we describe a much more communication-efficient protocol using blinded encryption. For concreteness the reader may imagine below that we are using the blindable cryptosystem based on quadratic residues described in Section 2.2, though other choices are possible. We begin by describing a sub-protocol for preprocessing the table Ti before resampling begins. Let S be the 2d indices of the rows of the table Ti. For ease of notation, we will refer to the d neighbors of Xi as V1, . . . , Vd. The purpose of the sub-protocol is for Xi and its neighbors to compute a random permutation π of S (which can be thought of as a random permutation of the rows of Ti) in such a way that during the protocol, each Vj ∈N(Xi) learns only the sets {π(⃗x) : Vj = 0} and {π(⃗x) : Vj = 1} and Xi learns nothing. The sub-protocol is quite simple. First each neighbor Vj of Xi encrypts column j of Ti using its own public key and passes the encrypted column to Xi. Next Xi encrypts column d + 1 using its own public key. Xi then concatenates the d + 1 encrypted columns together to form an encrypted version of Ti in which column j is encrypted using the public key of Vj for 1 ≤j ≤d and column d + 1 is encrypted using the public key of Xi. Xi then takes the resulting table, randomly permutes the rows, and blinds (randomly re-encrypts) each entry using the appropriate public keys (i.e. the key of Vj for column j where 1 ≤j ≤d and its own public key for column d + 1). At this point, Xi sends the resulting table to its distinguished neighbor N ∗(Xi). The purpose of the blinding steps here is to prevent parties from tracking correspondences between cleartext and encrypted table entries. For instance, without blinding above, N ∗(Xi) could reconstruct the permutation chosen by Xi by seeing how its own encrypted values have been rearranged. Now from the perspective of N ∗(Xi), d columns of the table will look like uniformly distributed random bits. N ∗(Xi) will still be able to decrypt the column of the table that corresponds to its own values, but it will become clear that decrypting this column alone cannot yield useful information. In the next step in the protocol, N ∗(Xi) re-encrypts column d + 1 of the table with its own public key. It then randomly permutes the rows of the table, blinds each entry using the appropriate public keys (those of Vj for columns 1 ≤j ≤d and its own for column d + 1), and sends the updated table back to Xi. At this point, every entry in the table will look random bits to Xi. Each column j will be encrypted by the public key of Vj, with the exception of the final column, which will be encrypted by both Xi and N ∗(Xi). Call this new table T ′ i. Once these encrypted tables have been computed for each node, we begin the main Gibbs sampling protocol. We inductively assume that at the start of each step, for each Xj ∈X, the current value of Xj is distributed between Xj and N ∗(Xj). At the end of the step, the only information that has been learned is the new value of a particular node Xi, but distributed between Xi and N ∗(Xi). Consider a neighbor Vj of Xi. Vj can decrypt column j of T ′ i in order to learn which rows correspond to its value being 0 and which rows correspond to its values being 1. While Vj alone does not know what its current value is, Vj and N ∗(Vj) could compute it together, and thus could together figure out which rows of the permutation correspond to Vj’s current value. By Theorem 1, since there is a way for them to compute this information ignoring privacy, we can construct an efficient protocol for Vj, N ∗(Vj), and Xi to perform this computation such that Xi learns only the rows that correspond to Vj’s value (and in particular does not learn what this value is), while Vj and N ∗(Vj) learn nothing. After this secure computation of partitions has been completed for all neighbors of Xi, Xi will be able to compute the intersection of the subsets of rows it has received from each neighbor. This intersection will be a single row corresponding to the current values of all nodes in N(Xi). Initially, Xi will not be able to decrypt any of the entries in this row. However, Xi and N ∗(Xi) could together 7 decrypt the value in column d + 1, use this value in order to sample Xi’s new value according to the appropriate distribution, and distribute the new value between themselves. Calling upon Theorem 1 once again, this means that we can construct an efficient protocol for Xi and N ∗(Xi) to together complete these computations in such a way that they only learn the new bits bi and b′ i respectively. Each time the value of a node Xi is resampled, Xi and N ∗(Xi) repeat the process of blinding and permuting the rows of T ′ i. This prevents Xi and its neighbors from learning how frequently they take on different values throughout the sampling process. After the value of each node has been privately resampled sufficiently many times, we can use one final application of secure multi-party computation between each node Xi and its distinguished neighbor N ∗(Xi) to allow Xi to learn its final value. As with standard Gibbs sampling, we also need to specify a schedule by which vertices in the Markov network will have their values updated, as well as the number of iterations of this schedule, which will in turn determine how close the sampled distribution is to the true joint (stationary) distribution. Since our interests are in privacy considerations only, let us use PrivateGibbs to refer to the protocol described above when applied to any fixed Markov network, combined with some fixed updating schedule (such as random or a fixed ordering) and some number r of iterations. Theorem 3 Under standard cryptographic assumptions7, PrivateGibbs computes a sample from the joint distribution after r iterations, with every variable learning its own value and nothing else. Direct communication occurs only between variables who are immediate neighbors or two steps away, and secure function computation is never invoked on sets of more than three variables. The full proof is again omitted, but largely follows the sketch above. We note that PrivateGibbs enjoys an even stronger privacy property — even if any subset of parties collude by combining their post-protocol views, they can learn nothing not implied by their combined sampled values. Furthermore, any convergence guarantees that hold for standard Gibbs sampling [4, 5] with the same updating schedule will also hold for the secure version. References [1] C. Bishop. Pattern Recognition and Machine Learning. Springer, 2006. [2] G. Casella and E. George. Explaining the Gibbs sampler. The American Statistician, 46:167–174, 1992. [3] Y. Dodis, S. Halevi, and T. Rabin. A cryptographic solution to a game theoretic problem. In CRYPTO, pages 112–130, 2000. [4] S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6:721–741, 1984. [5] A. Gibbs. Bounding convergence time of the Gibbs sampler in Bayesian image restoration. Biometrika, 87:749–766, 2000. [6] O. Goldreich. Foundations of Cryptography, Volume 2. Cambridge University Press, 2004. [7] A. Ihler, J. Fisher III, and A. Willsky. Loopy belief propagation: Convergence and effects of message errors. Journal of Machine Learning Research, 6:905–936, 2005. [8] M. Kearns, M. Littman, and S. Singh. Graphical models for game theory. In Uncertainty in Artificial Intelligence, 2001. [9] M. Naor and K. Nissim. Communication preserving protocols for secure function evaluation. In ACM Symposium on Theory of Computing, pages 590–599, 2001. [10] J. Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, 1988. [11] P. Shenoy and G. Shafer. Axioms for probability and belief-function propagation. In Uncertainty in Artificial Intelligence, pages 169–198, 1990. [12] V. Teague. Selecting correlated random actions. In Financial Cryptography, pages 181–195, 2004. [13] J. Yedidia, W. Freeman, and Y. Weiss. Understanding belief propagation and its generalizations. In Exploring Artificial Intelligence in the New Millennium. Morgan Kaufmann, 2003. 7An example would be intractability of recognizing quadratic residues. 8
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Bayesian Co-Training Shipeng Yu, Balaji Krishnapuram, Romer Rosales, Harald Steck, R. Bharat Rao CAD & Knowledge Solutions, Siemens Medical Solutions USA, Inc. firstname.lastname@siemens.com Abstract We propose a Bayesian undirected graphical model for co-training, or more generally for semi-supervised multi-view learning. This makes explicit the previously unstated assumptions of a large class of co-training type algorithms, and also clarifies the circumstances under which these assumptions fail. Building upon new insights from this model, we propose an improved method for co-training, which is a novel co-training kernel for Gaussian process classifiers. The resulting approach is convex and avoids local-maxima problems, unlike some previous multi-view learning methods. Furthermore, it can automatically estimate how much each view should be trusted, and thus accommodate noisy or unreliable views. Experiments on toy data and real world data sets illustrate the benefits of this approach. 1 Introduction Data samples may sometimes be characterized in multiple ways, e.g., web-pages can be described both in terms of the textual content in each page and the hyperlink structure between them. [1] have shown that the error rate on unseen test samples can be upper bounded by the disagreement between the classification-decisions obtained from independent characterizations (i.e., views) of the data. Thus, in the web-page example, misclassification rate can be indirectly minimized by reducing the rate of disagreement between hyperlink-based and content-based classifiers, provided these characterizations are independent conditional on the class. In many application domains class labels can be expensive to obtain and hence scarce, whereas unlabeled data are often cheap and abundantly available. Moreover, the disagreement between the class labels suggested by different views can be computed even when using unlabeled data. Therefore, a natural strategy for using unlabeled data to minimize the misclassification rate is to enforce consistency between the classification decisions based on several independent characterizations of the unlabeled samples. For brevity, unless otherwise specified, we shall use the term co-training to describe the entire genre of methods that rely upon this intuition, although strictly it should only refer to the original algorithm of [2]. Some co-training algorithms jointly optimize an objective function which includes misclassification penalties (loss terms) for classifiers from each view and a regularization term that penalizes lack of agreement between the classification decisions of the different views. In recent times, this coregularization approach has become the dominant strategy for exploiting the intuition behind multiview consensus learning, rendering obsolete earlier alternating-optimization strategies. We survey in Section 2 the major approaches to co-training, the theoretical guarantees that have spurred interest in the topic, and the previously published concerns about the applicability to certain domains. We analyze the precise assumptions that have been made and the optimization criteria to better understand why these approaches succeed (or fail) in certain situations. Then in Section 3 we propose a principled undirected graphical model for co-training which we call the Bayesian cotraining, and show that co-regularization algorithms provide one way for maximum-likelihood (ML) learning under this probabilistic model. By explicitly highlighting previously unstated assumptions, 1 Bayesian co-training provides a deeper understanding of the co-regularization framework, and we are also able to discuss certain fundamental limitations of multi-view consensus learning. In Section 4, we show that even simple and visually illustrated 2-D problems are sometimes not amenable to a co-training/co-regularization solution (no matter which specific model/algorithm is used – including ours). Empirical studies on two real world data sets are also illustrated. Summarizing our algorithmic contributions, co-regularization is exactly equivalent to the use of a novel co-training kernel for support vector machines (SVMs) and Gaussian processes (GP), thus allowing one to leverage the large body of available literature for these algorithms. The kernel is intrinsically non-stationary, i.e., the level of similarity between any pair of samples depends on all the available samples, whether labeled or unlabeled, thus promoting semi-supervised learning. Therefore, this approach is significantly simpler and more efficient than the alternating-optimization that is used in previous co-regularization implementations. Furthermore, we can automatically estimate how much each view should be trusted, and thus accommodate noisy or unreliable views. 2 Related Work Co-Training and Theoretical Guarantees: The iterative, alternating co-training method originally introduced in [2] works in a bootstrap mode, by repeatedly adding pseudo-labeled unlabeled samples into the pool of labeled samples, retraining the classifiers for each view, and pseudo-labeling additional unlabeled samples where at least one view is confident about its decision. The paper provided PAC-style guarantees that if (a) there exist weakly useful classifiers on each view of the data, and (b) these characterizations of the sample are conditionally independent given the class label, then the co-training algorithm can utilize the unlabeled data to learn arbitrarily strong classifiers. [1] proved PAC-style guarantees that if (a) sample sizes are large, (b) the different views are conditionally independent given the class label, and (c) the classification decisions based on multiple views largely agree with each other, then with high probability the misclassification rate is upper bounded by the rate of disagreement between the classifiers based on each view. [3] tried to reduce the strong theoretical requirements. They showed that co-training would be useful if (a) there exist low error rate classifiers on each view, (b) these classifiers never make mistakes in classification when they are confident about their decisions, and (c) the two views are not too highly correlated, in the sense that there would be at least some cases where one view makes confident classification decisions while the classifier on the other view does not have much confidence in its own decision. While each of these theoretical guarantees is intriguing and theoretically interesting, they are also rather unrealistic in many application domains. The assumption that classifiers do not make mistakes when they are confident and that of class conditional independence are rarely satisfied in practice. Nevertheless empirical success has been reported. Co-EM and Related Algorithms: The Co-EM algorithm of [4] extended the original bootstrap approach of the co-training algorithm to operate simultaneously on all unlabeled samples in an iterative batch mode. [5] used this idea with SVMs as base classifiers and subsequently in unsupervised learning by [6]. However, co-EM also suffers from local maxima problems, and while each iteration’s optimization step is clear, the co-EM is not really an expectation maximization algorithm (i.e., it lacks a clearly defined overall log-likelihood that monotonically improves across iterations). Co-Regularization: [7] proposed an approach for two-view consensus learning based on simultaneously learning multiple classifiers by maximizing an objective function which penalized misclassifications by any individual classifier, and included a regularization term that penalized a high level of disagreement between different views. This co-regularization framework improves upon the cotraining and co-EM algorithms by maximizing a convex objective function; however the algorithm still depends on an alternating optimization that optimizes one view at a time. This approach was later adapted to two-view spectral clustering [8]. Relationship to Current Work: The present work provides a probabilistic graphical model for multi-view consensus learning; alternating optimization based co-regularization is shown to be just one algorithm that accomplishes ML learning in this model. A more efficient, alternative strategy is proposed here for fully Bayesian classification under the same model. In practice, this strategy offers several advantages: it is easily extended to multiple views, it accommodates noisy views which are less predictive of class labels, and reduces run-time and memory requirements. 2 (a) (b) y1 fc(x1) fc(x2) fc(xn) f1(x1 (1)) … f1(x2 (1)) f1(xn (1)) … f2(x1 (2)) f2(x2 (2)) f2(xn (2)) f(x1) … f(x2) f(xn) yn y1 y2 y2 yn Figure 1: Factor graph for (a) one-view and (b) two-view models. 3 Bayesian Co-Training 3.1 Single-View Learning with Gaussian Processes A Gaussian Process (GP) defines a nonparametric prior over functions in Bayesian statistics [9]. A random real-valued function f : Rd →R follows a GP, denoted by GP(h, κ), if for every finite number of data points x1, . . . , xn ∈Rd, f = {f(xi)}n i=1 follows a multivariate Gaussian N(h, K) with mean h = {h(xi)}n i=1 and covariance K = {κ(xi, xj)}n i,j=1. Normally we fix the mean function h ≡0, and take a parametric (and usually stationary) form for the kernel function κ (e.g., the Gaussian kernel κ(xk, xℓ) = exp(−ρ∥xk −xℓ∥2) with ρ > 0 a free parameter). In a single-view, supervised learning scenario, an output or target yi is given for each observation xi (e.g., for regression yi ∈R and for classification yi ∈{−1, +1}). In the GP model we assume there is a latent function f underlying the output, p(yi|xi) = R p(yi|f, xi)p(f) df, with the GP prior p(f) = GP(h, κ). Given the latent function f, p(yi|f, xi) = p(yi|f(xi)) takes a Gaussian noise model N(f(xi), σ2) for regression, and a sigmoid function λ(yif(xi)) for classification. The dependency structure of the single-view GP model can be shown as an undirected graph as in Fig. 1(a). The maximal cliques of the graphical model are the fully connected nodes (f(x1), . . . , f(xn)) and the pairs (yi, f(xi)), i = 1, . . . , n. Therefore, the joint probability of random variables f = {f(xi)} and y = {yi} is defined as p(f, y) = 1 Z ψ(f) Q i ψ(yi, f(xi)), with potential functions1 ψ(f) = exp(−1 2f ⊤K−1f), ψ(yi, f(xi)) = ½exp(−1 2σ2 ∥yi −f(xi)∥2) for regression λ(yif(xi)) for classification (1) and normalization factor Z (hereafter Z is defined such that the joint probability sums to 1). 3.2 Undirected Graphical Model for Multi-View Learning In multi-view learning, suppose we have m different views of a same set of n data samples. Let x(j) i ∈Rdj be the features for the i-th sample obtained using the j-th view, where dj is the dimensionality of the input space for view j. Then the vector xi ≜(x(1) i , . . . , x(m) i ) is the complete representation of the i-th data sample, and x(j) ≜(x(j) 1 , . . . , x(j) n ) represents all sample observations for the j-th view. As in the single-view learning, let y = (y1, . . . , yn) where yi is the single output assigned to the i-th data point. One can clearly concatenate the multiple views into a single view and apply a single-view GP model, but the basic idea of multi-view learning is to introduce one function per view which only uses the features from that view, and then jointly optimize these functions such that they come to a consensus. Looking at this problem from a GP perspective, let fj denote the latent function for the j-th view (i.e., using features only from view j), and let fj ∼GP(0, κj) be its GP prior in view j. Since one data sample i has only one single label yi even though it has multiple features from the multiple 1The definition of ψ in this paper has been overloaded to simplify notation, but its meaning should be clear from the function arguments. 3 views (i.e., latent function value fj(x(j) i ) for view j), the label yi should depend on all of these latent function values for data sample i. The challenge here is to make this dependency explicit in a graphical model. We tackle this problem by introducing a new latent function, the consensus function fc, to ensure conditional independence between the output y and the m latent functions {fj} for the m views (see Fig. 1(b) for the undirected graphical model). At the functional level, the output y depends only on fc, and latent functions {fj} depend on each other only via the consensus function fc. That is, we have the joint probability: p(y, fc, f1, . . . , fm) = 1 Z ψ(y, fc) m Y j=1 ψ(fj, fc), with some potential functions ψ. In the ground network with n data samples, let f c = {fc(xi)}n i=1 and f j = {fj(x(j) i )}n i=1. The graphical model leads to the following factorization: p (y, f c, f 1, . . . , f m) = 1 Z Y i ψ(yi, fc(xi)) m Y j=1 ψ(f j)ψ(f j, f c). (2) Here the within-view potential ψ(f j) specifies the dependency structure within each view j, and the consensus potential ψ(f j, f c) describes how the latent function in each view is related with the consensus function fc. With a GP prior for each of the views, we can define the following potentials: ψ(f j) = exp µ −1 2f ⊤ j K−1 j f j ¶ , ψ(f j, f c) = exp µ −∥f j −f c∥2 2σ2 j ¶ , (3) where Kj is the covariance matrix of view j, i.e., Kj(xk, xℓ) = κj(x(j) k , x(j) ℓ), and σj > 0 a scalar which quantifies how far away the latent function f j is from f c. The output potential ψ(yi, fc(xi)) is defined the same as that in (1) for regression or classification. Some more insight may be gained by taking a careful look at these definitions: 1) The within-view potentials only rely on the intrinsic structure of each view, i.e., through the covariance Kj in a GP setting; 2) Each consensus potential actually defines a Gaussian over the difference of f j and f c, i.e., f j −f c ∼N(0, σ2 j I), and it can also be interpreted as assuming a conditional Gaussian for f j with the consensus f c being the mean. Alternatively if we focus on f c, the joint consensus potentials effectively define a conditional Gaussian prior for f c, f c|f 1, . . . , f m, as N(µc, σ2 cI) where µc = σ2 c X j f j σ2 j , σ2 c = µ X j 1 σ2 j ¶−1 . (4) This can be easily verified as a product of Gaussians. This indicates that the prior mean of the consensus function f c is a weighted combination of the latent functions from all the views, and the weight is given by the inverse variance of each consensus potential. The higher the variance, the smaller the contribution to the consensus function. More insights of this undirected graphical model can be seen from the marginals, which we discuss in detail in the following subsections. One advantage of this representation is that is allows us to see that many existing multi-view learning models are actually a special case of the proposed framework. In addition, this Bayesian interpretation also helps us understand both the benefits and the limitations of co-training. 3.3 Marginal 1: Co-Regularized Multi-View Learning By taking the integral of (2) over f c (and ignoring the output potential for the moment), we obtain the joint marginal distribution of the m latent functions: p(f 1, . . . , f m) = 1 Z exp −1 2 m X j=1 f jK−1 j f j −1 2 X j<k ∥f j −f k∥2 σ2 j + σ2 k . (5) It is clearly seen that the negation of the logarithm of this marginal exactly recovers the regularization terms in co-regularized multi-view learning: The first part regularizes the functional space of each 4 view, and the second part constrains that all the functions need to agree on their outputs (inversely weighted by the sum of the corresponding variances). From the GP perspective, (5) actually defines a joint multi-view prior for the m latent functions, (f 1, . . . , f m) ∼N(0, Λ−1), where Λ is a mn × mn matrix with block-wise definition Λ(j, j) = K−1 j + X k̸=j 1 σ2 j + σ2 k I, Λ(j, j′) = − 1 σ2 j + σ2 j′ I, j = 1, . . . , m, j′ ̸= j. (6) Jointly with the target variable y, the marginal is (for instance for regression): p(y, f 1, . . . , f m) = 1 Z exp −1 2 X j ∥f j −y∥2 σ2 j + σ2 −1 2 m X j=1 f jK−1 j f j −1 2 X j<k ∥f j −f k∥2 σ2 j + σ2 k . (7) This recovers the co-regularization with least square loss in its log-marginal form. 3.4 Marginal 2: The Co-Training Kernel The joint multi-view kernel defined in (6) is interesting, but it has a large dimension and is difficult to work with. A more interesting kernel can be obtained if we instead integrate out all the m latent functions in (2). This leads to a Gaussian prior p(f c) = N(0, Kc) for the consensus function fc, where Kc = X j (Kj + σ2 j I)−1 −1 . (8) In the following we call Kc the co-training kernel for multi-view learning. This marginalization is very important, because it reveals the previously unclear insight of how the kernels from different views are combined together in a multi-view learning framework. This allows us to transform a multi-view learning problem into a single-view problem, and simply use the co-training kernel Kc to solve GP classification or regression. Since this marginalization is equivalent to (5), we will end up with solutions that are largely similar to any other co-regularization algorithm, but however a key difference is the Bayesian treatement contrasting previous ML-optimization methods. Additional benefits of the co-training kernel include the following: 1. The co-training kernel avoids repeated alternating optimizations over the different views f j, and directly works with a single consensus view f c. This reduces both time complexity and space complexity (only maintains Kc in memory) of multi-view learning. 2. While other alternating optimization algorithms might converge to local minima (because they optimize, not integrate), the single consensus view guarantees the global optimal solution for multiview learning. 3. Even if all the individual kernels are stationary, Kc is in general non-stationary. This is because the inverse-covariances are added and then inverted again. In a transductive setting where the data are partially labeled, the co-training kernel between labeled data is also dependent on the unlabeled data. Hence the proposed co-training kernel can be used for semi-supervised GP learning [10]. 3.5 Benefits of Bayesian Co-Training The proposed undirected graphical model provides better understandings of multi-view learning algorithms. The co-training kernel in (8) indicates that the Bayesian co-training is equivalent to single-view learning with a special (non-stationary) kernel. This is also the preferable way of working with multi-view learning since it avoids alternating optimizations. Here are some other benefits which are not mentioned before: Trust-worthiness of each view: The graphical model allows each view j to have its own levels of uncertainty (or trust-worthiness) σ2 j . In particular, a larger value of σ2 j implies less confidence on the observation of evidence provided by the j-th view. Thus when some views of the data are better at predicting the output than the others, they are weighted more while forming consensus opinions. 5 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 x(1) x(2) −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 x(1) x(2) −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 x(1) x(2) −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 x(1) x(2) −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 x(1) x(2) −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 −0.5 −0.5 −0.5 −0.5 0 0 0 0 0 0 0.5 0.5 0.5 0.5 0.5 x(1) x(2) Figure 2: Toy examples for co-training. Big red/blue markers denote +1/−1 labeled points; remaining points are unlabeled. TOP left: co-training result on two-Gaussian data with mean (2, −2) and (−2, 2); center and right: canonical and Bayesian co-training on two-Gaussian data with mean (2, 0) and (−2, 0); BOTTOM left: XOR data with four Gaussians; center and right: Bayesian co-training and pure GP supervised learning result (with RBF kernel). Co-training is much worse than GP supervised learning in this case. All Gaussians have unit variance. RBF kernel uses width 1 for supervised learning and 1/ √ 2 for each feature in two-view learning. These uncertainties can be easily optimized in the GP framework by maximizing the marginal of output y (omitted in this paper due to space limit). Unsupervised and semi-supervised multi-view learning: The proposed graphical model also motivates new methods for unsupervised multi-view learning such as spectral clustering. While the similarity matrix of each view j is encoded in Kj, the co-training kernel Kc encodes the similarity of two data samples with multiple views, and thus can be used directly in spectral clustering. The extension to semi-supervised learning is also straightforward since Kc by definition depends on unlabeled data as well. Alternative interaction potential functions: Previous discussions about multi-view learning rely on potential definitions in (3) (which we call the consensus-based potentials), but other definitions are also possible and will lead to different co-training models. Actually, the definition in (3) has fundamental limitations and leads only to consensus-based learning, as seen from the next subsection. 3.6 Limitations of Consensus-based Potentials As mentioned before, the consensus-based potentials in (3) can be interpreted as defining a Gaussian prior (4) to f c, where the mean is a weighted average of the m individual views. This averaging indicates that the value of f c is never higher (or lower) than that of any single view. While the consensus-based potentials are intuitive and useful for many applications, they are limited for some real world problems where the evidence from different views should be additive (or enhanced) rather than averaging. For instance, when a radiologist is making a diagnostic decision about a lung cancer patient, he might look at both the CT image and the MRI image. If either of the two images gives a strong evidence of cancer, he can make decision based on a single view; if both images give an evidence of 0.6 (in a [0,1] scale), the final evidence of cancer should be higher (say, 0.8) than either of them. It’s clear that the multi-view learning in this scenario is not consensus-based. While all the previously proposed co-training and co-regularization algorithms have thus far been based on enforcing consensus between the views, in principle our graphical model allows other forms of 6 Table 1: Results for Citeseer with different numbers of training data (pos/neg). Bold face indicates best performance. Bayesian co-training is significantly better than the others (p-value 0.01 in Wilcoxon rank sum test) except in AUC with “Train +2/-10”. # TRAIN +2/-10 # TRAIN +4/-20 MODEL AUC F1 AUC F1 TEXT 0.5725 ± 0.0180 0.1359 ± 0.0565 0.5770 ± 0.0209 0.1443 ± 0.0705 INBOUND LINK 0.5451 ± 0.0025 0.3510 ± 0.0011 0.5479 ± 0.0035 0.3521 ± 0.0017 OUTBOUND LINK 0.5550 ± 0.0119 0.3552 ± 0.0053 0.5662 ± 0.0124 0.3600 ± 0.0059 TEXT+LINK 0.5730 ± 0.0177 0.1386 ± 0.0561 0.5782 ± 0.0218 0.1474 ± 0.0721 CO-TRAINED GPLR 0.6459 ± 0.1034 0.4001 ± 0.2186 0.6519 ± 0.1091 0.4042 ± 0.2321 BAYESIAN CO-TRAINING 0.6536 ± 0.0419 0.4210 ± 0.0401 0.6880 ± 0.0300 0.4530 ± 0.0293 relationships between the views. In particular, potentials other than those in (3) should be of great interest for future research. 4 Experimental Study Toy Examples: We show some 2D toy classification problems to visualize the co-training result (in Fig. 2). Our first example is a two-Gaussian case where either feature x(1) or x(2) can fully solve the problem (top left). This is an ideal case for co-training since: 1) each single view is sufficient to train a classifier, and 2) both views are conditionally independent given the class labels. The second toy data is a bit harder since the two Gaussians are aligned to the x(1)-axis. In this case the feature x(2) is totally irrelevant to the classification problem. The canonical co-training fails here (top center) since when we add labels using the x(2) feature , noisy labels will be introduced and expanded to future training. The proposed model can handle this situation since we can adapt the weight of each view and penalize the feature x(2) (top right). Our third toy data follows an XOR shape where four Gaussians form a binary classification problem that is not linearly separable (bottom left). In this case both assumptions mentioned above are violated, and co-training failed completely (bottom center). A supervised learning model can however easily recover the non-linear underlying structure (bottom right). This indicates that the co-training kernel Kc is not suitable for this problem. Web Data: We use two sets of linked documents for our experiment. The Citeseer data set contains 3,312 entries that belong to six classes. There are three natural views: the text view consists of title and abstract of a paper; the two link views are inbound and outbound references. We pick up the largest class which contains 701 documents and test the one-vs-rest classification performance. The WebKB data set is a collection of 4,502 academic web pages manually grouped into six classes (student, faculty, staff, department, course, project). There are two views containing the text on the page and the anchor text of all inbound links, respectively. We consider the binary classification problem “student” against “faculty”, for which there are 1,641 and 1,119 documents, respectively. We compare the single-view learning methods (TEXT, INBOUND LINK, etc), concatenated-view method (TEXT+LINK), and co-training methods CO-TRAINED GPLR (Co-Trained Gaussian Process Logistic Regression) and BAYESIAN CO-TRAINING. Linear kernels are used for all the competing methods. For the canonical co-training method we repeat 50 times and in each iteration add the most predictable 1 positive sample and r negative samples into the training set where r depends on the number of negative/positive ratio of each data set. Performance is evaluated using AUC score and F1 measure. We vary the number of training documents (with ratio proportional to the true positive/negative ratio), and all the co-training algorithms use all the unlabeled data in the training process. The experiments are repeated 20 times and the prediction means and standard deviations are shown in Table 1 and 2. It can be seen that for Citeseer the co-training methods are better than the supervised methods. In this cases Bayesian co-training is better than canonical co-training and achieves the best performance. For WebDB, however, canonical co-trained GPLR is not as good as supervised algorithms, and thus Bayesian co-training is also worse than supervised methods though a little better than co-trained GPLR. This is maybe because the TEXT and LINK features are not independent given the class labels (especially when two classes “faculty” and “staff” might share features). Canonical co-training has higher deviations than other methods due to the possibility of adding noisy labels. We have also tried other number of iterations but 50 seems to give an overall best performance. 7 Table 2: Results for WebKB with different numbers of training data (pos/neg). Bold face indicates best performance. No results are significantly better than all the others (p-value 0.01 in Wilcoxon rank sum test). MODEL # TRAIN +2/-2 # TRAIN +4/-4 AUC F1 AUC F1 TEXT 0.5767 ± 0.0430 0.4449 ± 0.1614 0.6150 ± 0.0594 0.5338 ± 0.1267 INBOUND LINK 0.5211 ± 0.0017 0.5761 ± 0.0013 0.5210 ± 0.0019 0.5758 ± 0.0015 TEXT+LINK 0.5766 ± 0.0429 0.4443 ± 0.1610 0.6150 ± 0.0594 0.5336 ± 0.1267 CO-TRAINED GPLR 0.5624 ± 0.1058 0.5437 ± 0.1225 0.5959 ± 0.0927 0.5737 ± 0.1203 BAYESIAN CO-TRAINING 0.5794 ± 0.0491 0.5562 ± 0.1598 0.6140 ± 0.0675 0.5742 ± 0.1298 Note that the single-view learning with TEXT almost achieves the same performance as concatenated-view method. This is because the number of text features are much more than the link features (e.g., for WebKB there are 24,480 text features and only 901 link features). So these multiple views are very unbalanced and should be taken into account in co-training with different weights. Bayesian co-training provides a natural way of doing it. 5 Conclusions This paper has two principal contributions. We have proposed a graphical model for combining multi-view data, and shown that previously derived co-regularization based training algorithms maximize the likelihood of this model. In the process, we showed that these algorithms have been making an intrinsic assumption of the form p(fc, f1, f2, . . . , fm) ∝ψ(fc, f1)ψ(fc, f2) . . . ψ(fc, fm), even though it was not explicitly realized earlier. We also studied circumstances when this assumption proves unreasonable. Thus, our first contribution was to clarify the implicit assumptions and limitations in multi-view consensus learning in general, and co-regularization in particular. Motivated by the insights from the graphical model, our second contribution was the development of alternative algorithms for co-regularization; in particular the development of a non-stationary cotraining kernel, and the development of methods for using side-information in classification. Unlike previously published co-regularization algorithms, our approach: (a) handles naturally more than 2 views; (b) automatically learns which views of the data should be trusted more while predicting class labels; (c) shows how to leverages previously developed methods for efficiently training GP/SVM; (d) clearly explains our assumptions, what is being optimized overall, etc; (e) does not suffer from local maxima problems; (f) is less computationally demanding in terms of both speed and memory requirements. References [1] S. Dasgupta, M. Littman, and D. McAllester. PAC generalization bounds for co-training. In NIPS, 2001. [2] A. Blum and T. Mitchell. Combining labeled and unlabeled data with co-training. In COLT, 1998. [3] N. Balcan, A. Blum, and K. Yang. Co-training and expansion: Towards bridging theory and practice. In NIPS, 2004. [4] K. Nigam and R. Ghani. Analyzing the effectiveness and applicability of co-training. In Workshop on information and knowledge management, 2000. [5] U. Brefeld and T. Scheffer. Co-em support vector learning. In ICML, 2004. [6] Steffen Bickel and Tobias Scheffer. Estimation of mixture models using co-em. In ECML, 2005. [7] B. Krishnapuram, D. Williams, Y. Xue, A. Hartemink, L. Carin, and M. Figueiredo. On semi-supervised classification. In NIPS, 2004. [8] Virginia de Sa. Spectral clustering with two views. In ICML Workshop on Learning With Multiple Views, 2005. [9] C. E. Rasmussen and C. K. I. Williams. Gaussian Processes for Machine Learning. MIT Press, 2006. [10] Xiaojin Zhu, John Lafferty, and Zoubin Ghahramani. Semi-supervised learning: From Gaussian fields to gaussian processes. 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Supervised topic models David M. Blei Department of Computer Science Princeton University Princeton, NJ blei@cs.princeton.edu Jon D. McAuliffe Department of Statistics University of Pennsylvania, Wharton School Philadelphia, PA mcjon@wharton.upenn.edu Abstract We introduce supervised latent Dirichlet allocation (sLDA), a statistical model of labelled documents. The model accommodates a variety of response types. We derive a maximum-likelihood procedure for parameter estimation, which relies on variational approximations to handle intractable posterior expectations. Prediction problems motivate this research: we use the fitted model to predict response values for new documents. We test sLDA on two real-world problems: movie ratings predicted from reviews, and web page popularity predicted from text descriptions. We illustrate the benefits of sLDA versus modern regularized regression, as well as versus an unsupervised LDA analysis followed by a separate regression. 1 Introduction There is a growing need to analyze large collections of electronic text. The complexity of document corpora has led to considerable interest in applying hierarchical statistical models based on what are called topics. Formally, a topic is a probability distribution over terms in a vocabulary. Informally, a topic represents an underlying semantic theme; a document consisting of a large number of words might be concisely modelled as deriving from a smaller number of topics. Such topic models provide useful descriptive statistics for a collection, which facilitates tasks like browsing, searching, and assessing document similarity. Most topic models, such as latent Dirichlet allocation (LDA) [4], are unsupervised: only the words in the documents are modelled. The goal is to infer topics that maximize the likelihood (or the posterior probability) of the collection. In this work, we develop supervised topic models, where each document is paired with a response. The goal is to infer latent topics predictive of the response. Given an unlabeled document, we infer its topic structure using a fitted model, then form its prediction. Note that the response is not limited to text categories. Other kinds of document-response corpora include essays with their grades, movie reviews with their numerical ratings, and web pages with counts of how many online community members liked them. Unsupervised LDA has previously been used to construct features for classification. The hope was that LDA topics would turn out to be useful for categorization, since they act to reduce data dimension [4]. However, when the goal is prediction, fitting unsupervised topics may not be a good choice. Consider predicting a movie rating from the words in its review. Intuitively, good predictive topics will differentiate words like “excellent”, “terrible”, and “average,” without regard to genre. But topics estimated from an unsupervised model may correspond to genres, if that is the dominant structure in the corpus. The distinction between unsupervised and supervised topic models is mirrored in existing dimension-reduction techniques. For example, consider regression on unsupervised principal components versus partial least squares and projection pursuit [7], which both search for covariate linear combinations most predictive of a response variable. These linear supervised methods have non1 parametric analogs, such as an approach based on kernel ICA [6]. In text analysis, McCallum et al. developed a joint topic model for words and categories [8], and Blei and Jordan developed an LDA model to predict caption words from images [2]. In chemogenomic profiling, Flaherty et al. [5] proposed “labelled LDA,” which is also a joint topic model, but for genes and protein function categories. It differs fundamentally from the model proposed here. This paper is organized as follows. We first develop the supervised latent Dirichlet allocation model (sLDA) for document-response pairs. We derive parameter estimation and prediction algorithms for the real-valued response case. Then we extend these techniques to handle diverse response types, using generalized linear models. We demonstrate our approach on two real-world problems. First, we use sLDA to predict movie ratings based on the text of the reviews. Second, we use sLDA to predict the number of “diggs” that a web page will receive in the www.digg.com community, a forum for sharing web content of mutual interest. The digg count prediction for a page is based on the page’s description in the forum. In both settings, we find that sLDA provides much more predictive power than regression on unsupervised LDA features. The sLDA approach also improves on the lasso, a modern regularized regression technique. 2 Supervised latent Dirichlet allocation In topic models, we treat the words of a document as arising from a set of latent topics, that is, a set of unknown distributions over the vocabulary. Documents in a corpus share the same set of K topics, but each document uses a mix of topics unique to itself. Thus, topic models are a relaxation of classical document mixture models, which associate each document with a single unknown topic. Here we build on latent Dirichlet allocation (LDA) [4], a topic model that serves as the basis for many others. In LDA, we treat the topic proportions for a document as a draw from a Dirichlet distribution. We obtain the words in the document by repeatedly choosing a topic assignment from those proportions, then drawing a word from the corresponding topic. In supervised latent Dirichlet allocation (sLDA), we add to LDA a response variable associated with each document. As mentioned, this variable might be the number of stars given to a movie, a count of the users in an on-line community who marked an article interesting, or the category of a document. We jointly model the documents and the responses, in order to find latent topics that will best predict the response variables for future unlabeled documents. We emphasize that sLDA accommodates various types of response: unconstrained real values, real values constrained to be positive (e.g., failure times), ordered or unordered class labels, nonnegative integers (e.g., count data), and other types. However, the machinery used to achieve this generality complicates the presentation. So we first give a complete derivation of sLDA for the special case of an unconstrained real-valued response. Then, in Section 2.3, we present the general version of sLDA, and explain how it handles diverse response types. Focus now on the case y ∈R. Fix for a moment the model parameters: the K topics β1:K (each βk a vector of term probabilities), the Dirichlet parameter α, and the response parameters η and σ 2. Under the sLDA model, each document and response arises from the following generative process: 1. Draw topic proportions θ | α ∼Dir(α). 2. For each word (a) Draw topic assignment zn | θ ∼Mult(θ). (b) Draw word wn | zn, β1:K ∼Mult(βzn). 3. Draw response variable y | z1:N, η, σ 2 ∼N η⊤¯z, σ 2 . Here we define ¯z := (1/N) PN n=1 zn. The family of probability distributions corresponding to this generative process is depicted as a graphical model in Figure 1. Notice the response comes from a normal linear model. The covariates in this model are the (unobserved) empirical frequencies of the topics in the document. The regression coefficients on those frequencies constitute η. Note that a linear model usually includes an intercept term, which amounts to adding a covariate that always equals one. Here, such a term is redundant, because the components of ¯z always sum to one. 2 Figure 1: (Left) A graphical model representation of Supervised Latent Dirichlet allocation. (Bottom) The topics of a 10-topic sLDA model fit to the movie review data of Section 3. both motion simple perfect fascinating power complex however cinematography screenplay performances pictures effective picture his their character many while performance between !30 !20 !10 0 10 20 ! ! ! ! !! ! ! ! ! more has than films director will characters one from there which who much what awful featuring routine dry offered charlie paris not about movie all would they its have like you was just some out bad guys watchable its not one movie least problem unfortunately supposed worse flat dull θd Zd,n Wd,n N D K βk α Yd η, σ2 By regressing the response on the empirical topic frequencies, we treat the response as nonexchangeable with the words. The document (i.e., words and their topic assignments) is generated first, under full word exchangeability; then, based on the document, the response variable is generated. In contrast, one could formulate a model in which y is regressed on the topic proportions θ. This treats the response and all the words as jointly exchangeable. But as a practical matter, our chosen formulation seems more sensible: the response depends on the topic frequencies which actually occurred in the document, rather than on the mean of the distribution generating the topics. Moreover, estimating a fully exchangeable model with enough topics allows some topics to be used entirely to explain the response variables, and others to be used to explain the word occurrences. This degrades predictive performance, as demonstrated in [2]. We treat α, β1:K , η, and σ 2 as unknown constants to be estimated, rather than random variables. We carry out approximate maximum-likelihood estimation using a variational expectation-maximization (EM) procedure, which is the approach taken in unsupervised LDA as well [4]. 2.1 Variational E-step Given a document and response, the posterior distribution of the latent variables is p(θ, z1:N | w1:N, y, α, β1:K , η, σ 2) = p(θ | α) QN n=1 p(zn | θ)p(wn | zn, β1:K ) p(y | z1:N, η, σ 2) R dθ p(θ | α) P z1:N QN n=1 p(zn | θ)p(wn | zn, β1:K ) p(y | z1:N, η, σ 2) . (1) The normalizing value is the marginal probability of the observed data, i.e., the document w1:N and response y. This normalizer is also known as the likelihood, or the evidence. As with LDA, it is not efficiently computable. Thus, we appeal to variational methods to approximate the posterior. Variational objective function. We maximize the evidence lower bound (ELBO) L(·), which for a single document has the form log p w1:N, y | α, β1:K , η, σ 2 ≥L(γ, φ1:N; α, β1:K , η, σ 2) = E[log p(θ | α)] + N X n=1 E[log p(Zn | θ)] + N X n=1 E[log p(wn | Zn, β1:K )] + E[log p(y | Z1:N, η, σ 2)] + H(q) . (2) Here the expectation is taken with respect to a variational distribution q. We choose the fully factorized distribution, q(θ, z1:N | γ, φ1:N) = q(θ | γ ) QN n=1 q(zn | φn), (3) 3 where γ is a K-dimensional Dirichlet parameter vector and each φn parametrizes a categorical distribution over K elements. Notice E[Zn] = φn. The first three terms and the entropy of the variational distribution are identical to the corresponding terms in the ELBO for unsupervised LDA [4]. The fourth term is the expected log probability of the response variable given the latent topic assignments, E[log p(y | Z1:N, η, σ 2)] = = −1 2 log 2πσ 2 − y2 −2yη⊤E ¯Z + η⊤E ¯Z ¯Z⊤ η . 2σ 2 . (4) The first expectation is E ¯Z = ¯φ := (1/N) PN n=1 φn, and the second expectation is E ¯Z ¯Z⊤ = (1/N 2) PN n=1 P m̸=n φnφ⊤ m + PN n=1 diag{φn} . (5) To see (5), notice that for m ̸= n, E[Zn Z⊤ m] = E[Zn]E[Zm]⊤= φnφ⊤ m because the variational distribution is fully factorized. On the other hand, E[Zn Z⊤ n ] = diag(E[Zn]) = diag(φn) because Zn is an indicator vector. For a single document-response pair, we maximize (2) with respect to φ1:N and γ to obtain an estimate of the posterior. We use block coordinate-ascent variational inference, maximizing with respect to each variational parameter vector in turn. Optimization with respect to γ . The terms that involve the variational Dirichlet γ are identical to those in unsupervised LDA, i.e., they do not involve the response variable y. Thus, the coordinate ascent update is as in [4], γ new ←α + PN n=1 φn. (6) Optimization with respect to φ j. Define φ−j := P n̸= j φn. Given j ∈{1, . . . , N}. In [3], we maximize the Lagrangian of the ELBO, which incorporates the constraint that the components of φ j sum to one, and obtain the coordinate update φnew j ∝exp E[log θ | γ ] + E[log p(w j | β1:K )] + y Nσ 2 η − 2 η⊤φ−j η + (η ◦η) 2N 2σ 2 . (7) Exponentiating a vector means forming the vector of exponentials. The proportionality symbol means the components of φnew j are computed according to (7), then normalized to sum to one. Note that E[log θi | γ ] = 9(γi) −9(P γ j), where 9(·) is the digamma function. The central difference between LDA and sLDA lies in this update. As in LDA, the jth word’s variational distribution over topics depends on the word’s topic probabilities under the actual model (determined by β1:K ). But w j’s variational distribution, and those of all other words, affect the probability of the response, through the expected residual sum of squares (RSS), which is the second term in (4). The end result is that the update (7) also encourages φ j to decrease this expected RSS. The update (7) depends on the variational parameters φ−j of all other words. Thus, unlike LDA, the φ j cannot be updated in parallel. Distinct occurrences of the same term are treated separately. 2.2 M-step and prediction The corpus-level ELBO lower bounds the joint log likelihood across documents, which is the sum of the per-document log-likelihoods. In the E-step, we estimate the approximate posterior distribution for each document-response pair using the variational inference algorithm described above. In the M-step, we maximize the corpus-level ELBO with respect to the model parameters β1:K , η, and σ 2. For our purposes, it suffices simply to fix α to 1/K times the ones vector. In this section, we add document indexes to the previous section’s quantities, so y becomes yd and ¯Z becomes ¯Zd. Estimating the topics. The M-step updates of the topics β1:K are the same as for unsupervised LDA, where the probability of a word under a topic is proportional to the expected number of times that it was assigned to that topic [4], ˆβnew k,w ∝ D X d=1 N X n=1 1(wd,n = w)φk d,n. (8) 4 Here again, proportionality means that each ˆβnew k is normalized to sum to one. Estimating the regression parameters. The only terms of the corpus-level ELBO involving η and σ 2 come from the corpus-level analog of (4). Define y = y1:D as the vector of response values across documents. Let A be the D × (K + 1) matrix whose rows are the vectors ¯Z⊤ d . Then the corpus-level version of (4) is E[log p(y | A, η, σ 2)] = −D 2 log(2πσ 2) − 1 2σ 2 E h (y −Aη)⊤(y −Aη) i . (9) Here the expectation is over the matrix A, using the variational distribution parameters chosen in the previous E-step. Expanding the inner product, using linearity of expectation, and applying the first-order condition for η, we arrive at an expected-value version of the normal equations: E A⊤A η = E[A]⊤y ⇒ ˆηnew ← E A⊤A −1 E[A]⊤y . (10) Note that the dth row of E[A] is just ¯φd, and all these average vectors were fixed in the previous Estep. Also, E A⊤A = P d E ¯Zd ¯Z⊤ d , with each term having a fixed value from the previous E-step as well, given by (5). We caution again: formulas in the previous section, such as (5), suppress the document indexes which appear here. We now apply the first-order condition for σ 2 to (9) and evaluate the solution at ˆηnew, obtaining: ˆσ 2 new ←(1/D){y⊤y −y⊤E[A] E A⊤A −1 E[A]⊤y} . (11) Prediction. Our focus in applying sLDA is prediction. Specifically, we wish to compute the expected response value, given a new document w1:N and a fitted model {α, β1:K , η, σ 2}: E[Y | w1:N, α, β1:K , η, σ 2] = η⊤E[ ¯Z | w1:N, α, β1:K ]. (12) The identity follows easily from iterated expectation. We approximate the posterior mean of ¯Z using the variational inference procedure of the previous section. But here, the terms depending on y are removed from the φ j update in (7). Notice this is the same as variational inference for unsupervised LDA: since we averaged the response variable out of the right-hand side in (12), what remains is the standard unsupervised LDA model for Z1:N and θ. Thus, given a new document, we first compute Eq[Z1:N], the variational posterior distribution of the latent variables Zn. Then, we estimate the response with E[Y | w1:N, α, β1:K , η, σ 2] ≈η⊤Eq[ ¯Z] = η⊤¯φ. (13) 2.3 Diverse response types via generalized linear models Up to this point, we have confined our attention to an unconstrained real-valued response variable. In many applications, however, we need to predict a categorical label, or a non-negative integral count, or a response with other kinds of constraints. Sometimes it is reasonable to apply a normal linear model to a suitably transformed version of such a response. When no transformation results in approximate normality, statisticians often make use of a generalized linear model, or GLM [9]. In this section, we describe sLDA in full generality, replacing the normal linear model of the earlier exposition with a GLM formulation. As we shall see, the result is a generic framework which can be specialized in a straightforward way to supervised topic models having a variety of response types. There are two main ingredients in a GLM: the “random component” and the “systematic component.” For the random component, one takes the distribution of the response to be an exponential dispersion family with natural parameter ζ and dispersion parameter δ: p(y | ζ, δ) = h(y, δ) exp ζ y −A(ζ) δ . (14) For each fixed δ, (14) is an exponential family, with base measure h(y, δ), sufficient statistic y, and log-normalizer A(ζ). The dispersion parameter provides additional flexibility in modeling the variance of y. Note that (14) need not be an exponential family jointly in (ζ, δ). 5 In the systematic component of the GLM, we relate the exponential-family parameter ζ of the random component to a linear combination of covariates – the so-called linear predictor. For sLDA, the linear predictor is η⊤¯z. In fact, we simply set ζ = η⊤¯z. Thus, in the general version of sLDA, the previous specification in step 3 of the generative process is replaced with y | z1:N, η, δ ∼GLM(¯z, η, δ) , (15) so that p(y | z1:N, η, δ) = h(y, δ) exp η⊤(¯zy) −A(η⊤¯z) δ . (16) The reader familiar with GLMs will recognize that our choice of systematic component means sLDA uses only canonical link functions. In future work, we will relax this constraint. We now have the flexibility to model any type of response variable whose distribution can be written in exponential dispersion form (14). As is well known, this includes many commonly used distributions: the normal; the binomial (for binary response); the Poisson and negative binomial (for count data); the gamma, Weibull, and inverse Gaussian (for failure time data); and others. Each of these distributions corresponds to a particular choice of h(y, δ) and A(ζ). For example, it is easy to show that the normal distribution corresponds to h(y, δ) = (1/ √ 2πδ) exp{−y2/(2δ)} and A(ζ) = ζ 2/2. In this case, the usual parameters µ and σ 2 just equal ζ and δ, respectively. Variational E-step. The distribution of y appears only in the cross-entropy term (4). Its form under the GLM is E[log p(y | Z1:N, η, δ)] = log h(y, δ) + 1 δ h η⊤ E ¯Z y −E A(η⊤¯Z) i . (17) This changes the coordinate ascent step for each φ j, but the variational optimization is otherwise unaffected. In particular, the gradient of the ELBO with respect to φ j becomes ∂L ∂φ j = E[log θ | γ ]+E[log p(w j | β1:K )]−log φ j +1+ y Nδ η− 1 δ ∂ ∂φ j n E A(η⊤¯Z) o . (18) Thus, the key to variational inference in sLDA is obtaining the gradient of the expected GLM lognormalizer. Sometimes there is an exact expression, such as the normal case of Section 2. As another example, the Poisson GLM leads to an exact gradient, which we omit for brevity. Other times, no exact gradient is available. In a longer paper [3], we study two methods for this situation. First, we can replace −E[A(η⊤¯Z)] with an adjustable lower bound whose gradient is known exactly; then we maximize over the original variational parameters plus the parameter controlling the bound. Alternatively, an application of the multivariate delta method for moments [1], plus standard exponential family theory, shows E A(η⊤¯Z) ≈A(η⊤¯φ) + VarGLM(Y | ζ = η⊤¯φ) · η⊤Varq( ¯Z)η . (19) Here, VarGLM denotes the response variance under the GLM, given a specified value of the natural parameter—in all standard cases, this variance is a closed-form function of φ j. The variancecovariance matrix of ¯Z under q is already known in closed from from E[ ¯Z] and (5). Thus, computing ∂/∂φ j of (19) exactly is mechanical. However, using this approximation gives up the usual guarantee that the ELBO lower bounds the marginal likelihood. We forgo details and further examples due to space constraints. The GLM contribution to the gradient determines whether the φ j coordinate update itself has a closed form, as it does in the normal case (7) and the Poisson case (omitted). If the update is not closed-form, we use numerical optimization, supplying a gradient obtained from one of the methods described in the previous paragraph. Parameter estimation (M-step). The topic parameter estimates are given by (8), as before. For the corpus-level ELBO, the gradient with respect to η becomes ∂ ∂η 1 δ D X d=1 n η⊤¯φd yd −E A(η⊤¯Zd) o = 1 δ ( D X d=1 ¯φd yd − D X d=1 Eq µ(η⊤¯Zd) ¯Zd ) . (20) The appearance of µ(·) = EGLM[Y | ζ = ·] follows from exponential family properties. This GLM mean response is a known function of η⊤¯Zd in all standard cases. However, Eq[µ(η⊤¯Zd) ¯Zd] has 6 ● ●● ● ● ● ● ● ● ●● ● ● ● 2 4 10 20 30 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Number of topics Predictive R2 ●●●● ● ● ● ● ● ●● ● ● ● 2 4 10 20 30 −8.6 −8.5 −8.4 −8.3 −8.2 −8.1 −8.0 Number of topics Per−word held out log likelihood ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 5 10 15 20 25 30 35 40 45 50 −6.42 −6.41 −6.40 −6.39 −6.38 −6.37 Number of topics Per−word held out log likelihood ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 5 10 15 20 25 30 35 40 45 50 0.0 0.1 0.2 0.3 0.4 0.5 Number of topics Predictive R2 sLDA LDA Movie corpus Digg corpus Figure 2: Predictive R2 and per-word likelihood for the movie and Digg data (see Section 3). an exact solution only in some cases (e.g. normal, Poisson). In other cases, we approximate the expectation with methods similar to those applied for the φ j coordinate update. Reference [3] has details, including estimation of δ and prediction, where we encounter the same issues. The derivative with respect to δ, evaluated at ˆηnew, is ( D X d=1 ∂h(yd, δ)/∂δ h(yd, δ) ) − 1 δ2 ( D X d=1 ¯φd yd − D X d=1 Eq µ(ˆη⊤ new ¯Zd) ¯Zd ) . (21) Given that the rightmost summation has been evaluated, exactly or approximately, during the η optimization, (21) has a closed form. Depending on h(y, δ) and its partial with respect to δ, we obtain ˆδnew either in closed form or via one-dimensional numerical optimization. Prediction. We form predictions just as in Section 2.2. The difference is that we now approximate the expected response value of a test document as E[Y | w1:N, α, β1:K , η, δ] ≈Eq[µ(η⊤¯Z)]. (22) Again, this follows from iterated expectation plus the variational approximation. When the variational expectation cannot be computed exactly, we apply the approximation methods we relied on for the GLM E-step and M-step. We defer specifics to [3]. 3 Empirical results We evaluated sLDA on two prediction problems. First, we consider “sentiment analysis” of newspaper movie reviews. We use the publicly available data introduced in [10], which contains movie reviews paired with the number of stars given. While Pang and Lee treat this as a classification problem, we treat it as a regression problem. With a 5000-term vocabulary chosen by tf-idf, the corpus contains 5006 documents and comprises 1.6M words. Second, we introduce the problem of predicting web page popularity on Digg.com. Digg is a community of users who share links to pages by submitting them to the Digg homepage, with a short description. Once submitted, other users “digg” the links they like. Links are sorted on the Digg homepage by the number of diggs they have received. Our Digg data set contains a year of link descriptions, paired with the number of diggs each received during its first week on the homepage. (This corpus will be made publicly available at publication.) We restrict our attention to links in the technology category. After trimming the top ten outliers, and using a 4145-term vocabulary chosen by tf-idf, the Digg corpus contains 4078 documents and comprises 94K words. For both sets of response variables, we transformed to approximate normality by taking logs. This makes the data amenable to the continuous-response model of Section 2; for these two problems, generalized linear modeling turned out to be unnecessary. We initialized β1:K to uniform topics, σ 2 to the sample variance of the response, and η to a grid on [−1, 1] in increments of 2/K. We ran EM until the relative change in the corpus-level likelihood bound was less than 0.01%. In the E-step, we ran coordinate-ascent variational inference for each document until the relative change in the 7 per-document ELBO was less than 0.01%. For the movie review data set, we illustrate in Figure 1 a matching of the top words from each topic to the corresponding coefficient ηk. We assessed the quality of the predictions with “predictive R2.” In our 5-fold cross-validation (CV), we defined this quantity as the fraction of variability in the out-of-fold response values which is captured by the out-of-fold predictions: pR2 := 1 −(P(y −ˆy)2)/(P(y −¯y)2). We compared sLDA to linear regression on the ¯φd from unsupervised LDA. This is the regression equivalent of using LDA topics as classification features [4].Figure 2 (L) illustrates that sLDA provides improved predictions on both data sets. Moreover, this improvement does not come at the cost of document model quality. The per-word hold-out likelihood comparison in Figure 2 (R) shows that sLDA fits the document data as well or better than LDA. Note that Digg prediction is significantly harder than the movie review sentiment prediction, and that the homogeneity of Digg technology content leads the model to favor a small number of topics. Finally, we compared sLDA to the lasso, which is L1-regularized least-squares regression. The lasso is a widely used prediction method for high-dimensional problems. We used each document’s empirical distribution over words as its lasso covariates, setting the lasso complexity parameter with 5-fold CV. On Digg data, the lasso’s optimal model complexity yielded a CV pR2 of 0.088. The best sLDA pR2 was 0.095, an 8.0% relative improvement. On movie data, the best Lasso pR2 was 0.457 versus 0.500 for sLDA, a 9.4% relative improvement. Note moreover that the Lasso provides only a prediction rule, whereas sLDA models latent structure useful for other purposes. 4 Discussion We have developed sLDA, a statistical model of labelled documents. The model accommodates the different types of response variable commonly encountered in practice. We presented a variational procedure for approximate posterior inference, which we then incorporated in an EM algorithm for maximum-likelihood parameter estimation. We studied the model’s predictive performance on two real-world problems. In both cases, we found that sLDA moderately improved on the lasso, a state-of-the-art regularized regression method. Moreover, the topic structure recovered by sLDA had higher hold-out likelihood than LDA on one problem, and equivalent hold-out likelihood on the other. These results illustrate the benefits of supervised dimension reduction when prediction is the ultimate goal. Acknowledgments David M. Blei is supported by grants from Google and the Microsoft Corporation. References [1] P. Bickel and K. Doksum. Mathematical Statistics. Prentice Hall, 2000. [2] D. Blei and M. Jordan. Modeling annotated data. In SIGIR, pages 127–134. ACM Press, 2003. [3] D. Blei and J. McAuliffe. Supervised topic models. In preparation, 2007. [4] D. Blei, A. Ng, and M. Jordan. Latent Dirichlet allocation. JMLR, 3:993–1022, 2003. [5] P. Flaherty, G. Giaever, J. Kumm, M. Jordan, and A. Arkin. A latent variable model for chemogenomic profiling. Bioinformatics, 21(15):3286–3293, 2005. [6] K. Fukumizu, F. Bach, and M. Jordan. Dimensionality reduction for supervised learning with reproducing kernel Hilbert spaces. Journal of Machine Learning Research, 5:73–99, 2004. [7] T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning. 2001. [8] A. McCallum, C. Pal, G. Druck, and X. Wang. Multi-conditional learning: Generative/discriminative training for clustering and classification. In AAAI, 2006. [9] P. McCullagh and J. A. Nelder. Generalized Linear Models. Chapman & Hall, 1989. [10] B. Pang and L. Lee. Seeing stars: Exploiting class relationships for sentiment categorization with respect to rating scales. In Proceedings of the ACL, 2005. 8
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A Kernel Statistical Test of Independence Arthur Gretton MPI for Biological Cybernetics T¨ubingen, Germany arthur@tuebingen.mpg.de Kenji Fukumizu Inst. of Statistical Mathematics Tokyo Japan fukumizu@ism.ac.jp Choon Hui Teo NICTA, ANU Canberra, Australia choonhui.teo@gmail.com Le Song NICTA, ANU and University of Sydney lesong@it.usyd.edu.au Bernhard Sch¨olkopf MPI for Biological Cybernetics T¨ubingen, Germany bs@tuebingen.mpg.de Alexander J. Smola NICTA, ANU Canberra, Australia alex.smola@gmail.com Abstract Although kernel measures of independence have been widely applied in machine learning (notably in kernel ICA), there is as yet no method to determine whether they have detected statistically significant dependence. We provide a novel test of the independence hypothesis for one particular kernel independence measure, the Hilbert-Schmidt independence criterion (HSIC). The resulting test costs O(m2), where m is the sample size. We demonstrate that this test outperforms established contingency table and functional correlation-based tests, and that this advantage is greater for multivariate data. Finally, we show the HSIC test also applies to text (and to structured data more generally), for which no other independence test presently exists. 1 Introduction Kernel independence measures have been widely applied in recent machine learning literature, most commonly in independent component analysis (ICA) [2, 11], but also in fitting graphical models [1] and in feature selection [22]. One reason for their success is that these criteria have a zero expected value if and only if the associated random variables are independent, when the kernels are universal (in the sense of [23]). There is presently no way to tell whether the empirical estimates of these dependence measures indicate a statistically significant dependence, however. In other words, we are interested in the threshold an empirical kernel dependence estimate must exceed, before we can dismiss with high probability the hypothesis that the underlying variables are independent. Statistical tests of independence have been associated with a broad variety of dependence measures. Classical tests such as Spearman’s ρ and Kendall’s τ are widely applied, however they are not guaranteed to detect all modes of dependence between the random variables. Contingency tablebased methods, and in particular the power-divergence family of test statistics [17], are the best known general purpose tests of independence, but are limited to relatively low dimensions, since they require a partitioning of the space in which each random variable resides. Characteristic functionbased tests [6, 13] have also been proposed, which are more general than kernel density-based tests [19], although to our knowledge they have been used only to compare univariate random variables. In this paper we present three main results: first, and most importantly, we show how to test whether statistically significant dependence is detected by a particular kernel independence measure, the Hilbert Schmidt independence criterion (HSIC, from [9]). That is, we provide a fast (O(m2) for sample size m) and accurate means of obtaining a threshold which HSIC will only exceed with small probability, when the underlying variables are independent. Second, we show the distribution 1 of our empirical test statistic in the large sample limit can be straightforwardly parameterised in terms of kernels on the data. Third, we apply our test to structured data (in this case, by establishing the statistical dependence between a text and its translation). To our knowledge, ours is the first independence test for structured data. We begin our presentation in Section 2, with a short overview of cross-covariance operators between RKHSs and their Hilbert-Schmidt norms: the latter are used to define the Hilbert Schmidt Independence Criterion (HSIC). In Section 3, we describe how to determine whether the dependence returned via HSIC is statistically significant, by proposing a hypothesis test with HSIC as its statistic. In particular, we show that this test can be parameterised using a combination of covariance operator norms and norms of mean elements of the random variables in feature space. Finally, in Section 4, we give our experimental results, both for testing dependence between random vectors (which could be used for instance to verify convergence in independent subspace analysis [25]), and for testing dependence between text and its translation. Software to implement the test may be downloaded from http : //www.kyb.mpg.de/bs/people/arthur/indep.htm 2 Definitions and description of HSIC Our problem setting is as follows: Problem 1 Let Pxy be a Borel probability measure defined on a domain X × Y, and let Px and Py be the respective marginal distributions on X and Y. Given an i.i.d sample Z := (X, Y ) = {(x1, y1), . . . , (xm, ym)} of size m drawn independently and identically distributed according to Pxy, does Pxy factorise as PxPy (equivalently, we may write x ⊥⊥y)? We begin with a description of our kernel dependence criterion, leaving to the following section the question of whether this dependence is significant. This presentation is largely a review of material from [9, 11, 22], the main difference being that we establish links to the characteristic function-based independence criteria in [6, 13]. Let F be an RKHS, with the continuous feature mapping φ(x) ∈F from each x ∈X, such that the inner product between the features is given by the kernel function k(x, x′) := ⟨φ(x), φ(x′)⟩. Likewise, let G be a second RKHS on Y with kernel l(·, ·) and feature map ψ(y). Following [7], the cross-covariance operator Cxy : G →F is defined such that for all f ∈F and g ∈G, ⟨f, Cxyg⟩F = Exy ([f(x) −Ex(f(x))] [g(y) −Ey(g(y))]) . The cross-covariance operator itself can then be written Cxy := Exy[(φ(x) −µx) ⊗(ψ(y) −µy)], (1) where µx := Exφ(x), µy := Eyφ(y), and ⊗is the tensor product [9, Eq. 6]: this is a generalisation of the cross-covariance matrix between random vectors. When F and G are universal reproducing kernel Hilbert spaces (that is, dense in the space of bounded continuous functions [23]) on the compact domains X and Y, then the largest singular value of this operator, ∥Cxy∥, is zero if and only if x ⊥⊥y [11, Theorem 6]: the operator therefore induces an independence criterion, and can be used to solve Problem 1. The maximum singular value gives a criterion similar to that originally proposed in [18], but with more restrictive function classes (rather than functions of bounded variance). Rather than the maximum singular value, we may use the squared Hilbert-Schmidt norm (the sum of the squared singular values), which has a population expression HSIC(Pxy, F, G) = Exx′yy′[k(x, x′)l(y, y′)] + Exx′[k(x, x′)]Eyy′[l(y, y′)] −2Exy [Ex′[k(x, x′)]Ey′[l(y, y′)]] (2) (assuming the expectations exist), where x′ denotes an independent copy of x [9, Lemma 1]: we call this the Hilbert-Schmidt independence criterion (HSIC). We now address the problem of estimating HSIC(Pxy, F, G) on the basis of the sample Z. An unbiased estimator of (2) is a sum of three U-statistics [21, 22], HSIC(Z) = 1 (m)2 X (i,j)∈im 2 kijlij + 1 (m)4 X (i,j,q,r)∈im 4 kijlqr −2 1 (m)3 X (i,j,q)∈im 3 kijliq, (3) 2 where (m)n := m! (m−n)!, the index set im r denotes the set all r-tuples drawn without replacement from the set {1, . . . , m}, kij := k(xi, xj), and lij := l(yi, yj). For the purpose of testing independence, however, we will find it easier to use an alternative, biased empirical estimate [9, Definition 2], obtained by replacing the U-statistics with V-statistics1 HSICb(Z) = 1 m2 m X i,j kijlij + 1 m4 m X i,j,q,r kijlqr −2 1 m3 m X i,j,q kijliq = 1 m2 trace(KHLH), (4) where the summation indices now denote all r-tuples drawn with replacement from {1, . . . , m} (r being the number of indices below the sum), K is the m×m matrix with entries kij, H = I−1 m11⊤, and 1 is an m × 1 vector of ones (the cost of computing this statistic is O(m2)). When a Gaussian kernel kij := exp −σ−2 ∥xi −xj∥2 is used (or a kernel deriving from [6, Eq. 4.10]), the latter statistic is equivalent to the characteristic function-based statistic [6, Eq. 4.11] and the T 2n statistic of [13, p. 54]: details are reproduced in [10] for comparison. Our setting allows for more general kernels, however, such as kernels on strings (as in our experiments in Section 4) and graphs (see [20] for further details of kernels on structures): this is not possible under the characteristic function framework, which is restricted to Euclidean spaces (Rd in the case of [6, 13]). As pointed out in [6, Section 5], the statistic in (4) can also be linked to the original quadratic test of Rosenblatt [19] given an appropriate kernel choice; the main differences being that characteristic function-based tests (and RKHS-based tests) are not restricted to using kernel densities, nor should they reduce their kernel width with increasing sample size. Another related test described in [4] is based on the functional canonical correlation between F and G, rather than the covariance: in this sense the test statistic resembles those in [2]. The approach in [4] differs with both the present work and [2], however, in that the function spaces F and G are represented by finite sets of basis functions (specifically B-spline kernels) when computing the empirical test statistic. 3 Test description We now describe a statistical test of independence for two random variables, based on the test statistic HSICb(Z). We begin with a more formal introduction to the framework and terminology of statistical hypothesis testing. Given the i.i.d. sample Z defined earlier, the statistical test, T(Z) : (X × Y)m 7→{0, 1} is used to distinguish between the null hypothesis H0 : Pxy = PxPy and the alternative hypothesis H1 : Pxy ̸= PxPy. This is achieved by comparing the test statistic, in our case HSICb(Z), with a particular threshold: if the threshold is exceeded, then the test rejects the null hypothesis (bearing in mind that a zero population HSIC indicates Pxy = PxPy). The acceptance region of the test is thus defined as any real number below the threshold. Since the test is based on a finite sample, it is possible that an incorrect answer will be returned: the Type I error is defined as the probability of rejecting H0 based on the observed sample, despite x and y being independent. Conversely, the Type II error is the probability of accepting Pxy = PxPy when the underlying variables are dependent. The level α of a test is an upper bound on the Type I error, and is a design parameter of the test, used to set the test threshold. A consistent test achieves a level α, and a Type II error of zero, in the large sample limit. How, then, do we set the threshold of the test given α? The approach we adopt here is to derive the asymptotic distribution of the empirical estimate HSICb(Z) of HSIC(Pxy, F, G) under H0. We then use the 1 −α quantile of this distribution as the test threshold.2 Our presentation in this section is therefore divided into two parts. First, we obtain the distribution of HSICb(Z) under both H0 and H1; the latter distribution is also needed to ensure consistency of the test. We shall see, however, that the null distribution has a complex form, and cannot be evaluated directly. Thus, in the second part of this section, we describe ways to accurately approximate the 1 −α quantile of this distribution. Asymptotic distribution of HSICb(Z) We now describe the distribution of the test statistic in (4) The first theorem holds under H1. 1The U- and V-statistics differ in that the latter allow indices of different sums to be equal. 2An alternative would be to use a large deviation bound, as provided for instance by [9] based on Hoeffding’s inequality. It has been reported in [8], however, that such bounds are generally too loose for hypothesis testing. 3 Theorem 1 Let hijqr = 1 4! (i,j,q,r) X (t,u,v,w) ktultu + ktulvw −2ktultv, (5) where the sum represents all ordered quadruples (t, u, v, w) drawn without replacement from (i, j, q, r), and assume E h2 < ∞. Under H1, HSICb(Z) converges in distribution as m →∞ to a Gaussian according to m 1 2 (HSICb(Z) −HSIC(Pxy, F, G)) D →N 0, σ2 u . (6) The variance is σ2 u = 16 Ei Ej,q,rhijqr 2 −HSIC(Pxy, F, G) , where Ej,q,r := Ezj,zq,zr. Proof We first rewrite (4) as a single V-statistic, HSICb(Z) = 1 m4 m X i,j,q,r hijqr, (7) where we note that hijqr defined in (5) does not change with permutation of its indices. The associated U-statistic HSICs(Z) converges in distribution as (6) with variance σ2 u [21, Theorem 5.5.1(A)]: see [22]. Since the difference between HSICb(Z) and HSICs(Z) drops as 1/m (see [9], or Theorem 3 below), HSICb(Z) converges asymptotically to the same distribution. The second theorem applies under H0 Theorem 2 Under H0, the U-statistic HSICs(Z) corresponding to the V-statistic in (7) is degenerate, meaning Eihijqr = 0. In this case, HSICb(Z) converges in distribution according to [21, Section 5.5.2] mHSICb(Z) D → ∞ X l=1 λlz2 l , (8) where zl ∼N(0, 1) i.i.d., and λl are the solutions to the eigenvalue problem λlψl(zj) = Z hijqrψl(zi)dFi,q,r, where the integral is over the distribution of variables zi, zq, and zr. Proof This follows from the discussion of [21, Section 5.5.2], making appropriate allowance for the fact that we are dealing with a V-statistic (which is why the terms in (8) are not centred: in the case of a U-statistic, the sum would be over terms λl(z2 l −1)). Approximating the 1 −α quantile of the null distribution A hypothesis test using HSICb(Z) could be derived from Theorem 2 above by computing the (1 −α)th quantile of the distribution (8), where consistency of the test (that is, the convergence to zero of the Type II error for m →∞) is guaranteed by the decay as m−1 of the variance of HSICb(Z) under H1. The distribution under H0 is complex, however: the question then becomes how to accurately approximate its quantiles. One approach, taken by [6], is to use a Monte Carlo resampling technique: the ordering of the Y sample is permuted repeatedly while that of X is kept fixed, and the 1 −α quantile is obtained from the resulting distribution of HSICb values. This can be very expensive, however. A second approach, suggested in [13, p. 34], is to approximate the null distribution as a two-parameter Gamma distribution [12, p. 343, p. 359]: this is one of the more straightforward approximations of an infinite sum of χ2 variables (see [12, Chapter 18.8] for further ways to approximate such distributions; in particular, we wish to avoid using moments of order greater than two, since these can become expensive to compute). Specifically, we make the approximation mHSICb(Z) ∼xα−1e−x/β βαΓ(α) where α = (E(HSICb(Z)))2 var(HSICb(Z)) , β = mvar(HSICb(Z)) E(HSICb(Z)) . (9) 4 Figure 1: mHSICb cumulative distribution function (Emp) under H0 for m = 200, obtained empirically using 5000 independent draws of mHSICb. The two-parameter Gamma distribution (Gamma) is fit using α = 1.17 and β = 8.3 × 10−4 in (9), with mean and variance computed via Theorems 3 and 4. 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 mHSICb P(mHSICb(Z) < mHSICb) Emp Gamma An illustration of the cumulative distribution function (CDF) obtained via the Gamma approximation is given in Figure 1, along with an empirical CDF obtained by repeated draws of HSICb. We note the Gamma approximation is quite accurate, especially in areas of high probability (which we use to compute the test quantile). The accuracy of this approximation will be further evaluated experimentally in Section 4. To obtain the Gamma distribution from our observations, we need empirical estimates for E(HSICb(Z)) and var(HSICb(Z)) under the null hypothesis. Expressions for these quantities are given in [13, pp. 26-27], however these are in terms of the joint and marginal characteristic functions, and not in our more general kernel setting (see also [14, p. 313]). In the following two theorems, we provide much simpler expressions for both quantities, in terms of norms of mean elements µx and µy, and the covariance operators Cxx := Ex[(φ(x) −µx) ⊗(φ(x) −µx)] and Cyy, in feature space. The main advantage of our new expressions is that they are computed entirely in terms of kernels, which makes possible the application of the test to any domains on which kernels can be defined, and not only Rd. Theorem 3 Under H0, E(HSICb(Z)) = 1 mTrCxxTrCyy = 1 m 1 + ∥µx∥2 ∥µy∥2 −∥µx∥2 −∥µy∥2 , (10) where the second equality assumes kii = lii = 1. An empirical estimate of this statistic is obtained by replacing the norms above with \ ∥µx∥2 = (m)−1 2 P (i,j)∈im 2 kij, bearing in mind that this results in a (generally negligible) bias of O(m−1) in the estimate of ∥µx∥2 ∥µy∥2. Theorem 4 Under H0, var(HSICb(Z)) = 2(m −4)(m −5) (m)4 ∥Cxx∥2 HS ∥Cyy∥2 HS + O(m−3). Denoting by ⊙the entrywise matrix product, A·2 the entrywise matrix power, and B = ((HKH) ⊙(HLH))·2, an empirical estimate with negligible bias may be found by replacing the product of covariance operator norms with 1⊤(B −diag(B)) 1: this is slightly more efficient than taking the product of the empirical operator norms (although the scaling with m is unchanged). Proofs of both theorems may be found in [10], where we also compare with the original characteristic function-based expressions in [13]. We remark that these parameters, like the original test statistic in (4), may be computed in O(m2). 4 Experiments General tests of statistical independence are most useful for data having complex interactions that simple correlation does not detect. We investigate two cases where this situation arises: first, we test vectors in Rd which have a dependence relation but no correlation, as occurs in independent subspace analysis; and second, we study the statistical dependence between a text and its translation. Independence of subspaces One area where independence tests have been applied is in determining the convergence of algorithms for independent component analysis (ICA), which involves separating random variables that have been linearly mixed, using only their mutual independence. ICA generally entails optimisation over a non-convex function (including when HSIC is itself the optimisation criterion [9]), and is susceptible to local minima, hence the need for these tests (in fact, for classical approaches to ICA, the global minimum of the optimisation might not correspond to independence for certain source distributions). Contingency table-based tests have been applied [15] 5 in this context, while the test of [13] has been used in [14] for verifying ICA outcomes when the data are stationary random processes (through using a subset of samples with a sufficiently large delay between them). Contingency table-based tests may be less useful in the case of independent subspace analysis (ISA, see e.g. [25] and its bibliography), where higher dimensional independent random vectors are to be separated. Thus, characteristic function-based tests [6, 13] and kernel independence measures might work better for this problem. In our experiments, we tested the independence of random vectors, as a way of verifying the solutions of independent subspace analysis. We assumed for ease of presentation that our subspaces have respective dimension dx = dy = d, but this is not required. The data were constructed as follows. First, we generated m samples of two univariate random variables, each drawn at random from the ICA benchmark densities in [11, Table 3]: these include super-Gaussian, sub-Gaussian, multimodal, and unimodal distributions. Second, we mixed these random variables using a rotation matrix parameterised by an angle θ, varying from 0 to π/4 (a zero angle means the data are independent, while dependence becomes easier to detect as the angle increases to π/4: see the two plots in Figure 2, top left). Third, we appended d −1 dimensional Gaussian noise of zero mean and unit standard deviation to each of the mixtures. Finally, we multiplied each resulting vector by an independent random d-dimensional orthogonal matrix, to obtain vectors dependent across all observed dimensions. We emphasise that classical approaches (such as Spearman’s ρ or Kendall’s τ) are completely unable to find this dependence, since the variables are uncorrelated; nor can we recover the subspace in which the variables are dependent using PCA, since this subspace has the same second order properties as the noise. We investigated sample sizes m = 128, 512, 1024, 2048, and d = 1, 2, 4. We compared two different methods for computing the 1 −α quantile of the HSIC null distribution: repeated random permutation of the Y sample ordering as in [6] (HSICp), where we used 200 permutations; and Gamma approximation (HSICg) as in [13], based on (9). We used a Gaussian kernel, with kernel size set to the median distance between points in input space. We also compared with two alternative tests, the first based on a discretisation of the variables, and the second on functional canonical correlation. The discretisation based test was a power-divergence contingency table test from [17] (PD), which consisted in partitioning the space, counting the number of samples falling in each partition, and comparing this with the number of samples that would be expected under the null hypothesis (the test we used, described in [15], is more refined than this short description would suggest). Rather than a uniform space partitioning, we divided our space into roughly equiprobable bins as in [15], using a Gessaman partition for higher dimensions [5, Figure 21.4] (Ku and Fine did not specify a space partitioning strategy for higher dimensions, since they dealt only with univariate random variables). All remaining parameters were set according to [15]. The functional correlationbased test (fCorr) is described in [4]: the main differences with respect to our test are that it uses the spectrum of the functional correlation operator, rather than the covariance operator; and that it approximates the RKHSs F and G by finite sets of basis functions. Parameter settings were as in [4, Table 1], with the second order B-spline kernel and a twofold dyadic partitioning. Note that fCorr applies only in the univariate case. Results are plotted in Figure 2 (average over 500 repetitions). The y-intercept on these plots corresponds to the acceptance rate of H0 at independence, or 1 −(Type I error), and should be close to the design parameter of 1 −α = 0.95. Elsewhere, the plots indicate acceptance of H0 where the underlying variables are dependent, i.e. the Type II error. As expected, we observe that dependence becomes easier to detect as θ increases from 0 to π/4, when m increases, and when d decreases. The PD and fCorr tests perform poorly at m = 128, but approach the performance of HSIC-based tests for increasing m (although PD remains slightly worse than HSIC at m = 512 and d = 1, while fCorr becomes slightly worse again than PD). PD also scales very badly with d, and never rejects the null hypothesis when d = 4, even for m = 2048. Although HSIC-based tests are unreliable for small θ, they generally do well as θ approaches π/4 (besides m = 128, d = 2). We also emphasise that HSICp and HSICg perform identically, although HSICp is far more costly (by a factor of around 100, given the number of permutations used). Dependence and independence between text In this section, we demonstrate independence testing on text. Our data are taken from the Canadian Hansard corpus (http : //www.isi.edu/natural−language/download/hansard/). These consist of the official records of the 36th Canadian parliament, in English and French. We used debate transcripts on the three topics of Agriculture, Fisheries, and Immigration, due to the relatively large volume of data in these categories. Our goal was to test whether there exists a statistical dependence between 6 −2 0 2 −3 −2 −1 0 1 2 3 Rotation θ = π/8 X Y −2 0 2 −3 −2 −1 0 1 2 3 Rotation θ = π/4 X Y 0 0.5 1 0 0.2 0.4 0.6 0.8 1 Angle (×π/4) % acceptance of H0 Samp:128, Dim:1 PD fCorr HSICp HSICg 0 0.5 1 0 0.2 0.4 0.6 0.8 1 Angle (×π/4) % acceptance of H0 Samp:128, Dim:2 0 0.5 1 0 0.2 0.4 0.6 0.8 1 Angle (×π/4) % acceptance of H0 Samp:512, Dim:1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 Angle (×π/4) % acceptance of H0 Samp:512, Dim:2 0 0.5 1 0 0.2 0.4 0.6 0.8 1 Angle (×π/4) % acceptance of H0 Samp:1024, Dim:4 0 0.5 1 0 0.2 0.4 0.6 0.8 1 Angle (×π/4) % acceptance of H0 Samp:2048, Dim:4 Figure 2: Top left plots: Example dataset for d = 1, m = 200, and rotation angles θ = π/8 (left) and θ = π/4 (right). In this case, both sources are mixtures of two Gaussians (source (g) in [11, Table 3]). We remark that the random variables appear “more dependent” as the angle θ increases, although their correlation is always zero. Remaining plots: Rate of acceptance of H0 for the PD, fCorr, HSICp, and HSICg tests. “Samp” is the number m of samples, and “dim” is the dimension d of x and y. English text and its French translation. Our dependent data consisted of a set of paragraph-long (5 line) English extracts and their French translations. For our independent data, the English paragraphs were matched to random French paragraphs on the same topic: for instance, an English paragraph on fisheries would always be matched with a French paragraph on fisheries. This was designed to prevent a simple vocabulary check from being used to tell when text was mismatched. We also ignored lines shorter than five words long, since these were not always part of the text (e.g. identification of the person speaking). We used the k-spectrum kernel of [16], computed according to the method of [24]. We set k = 10 for both languages, where this was chosen by cross validating on an SVM classifier for Fisheries vs National Defense, separately for each language (performance was not especially sensitive to choice of k; k = 5 also worked well). We compared this kernel with a simple kernel between bags of words [3, pp. 186–189]. Results are in Table 1. Our results demonstrate the excellent performance of the HSICp test on this task: even for small sample sizes, HSICp with a spectral kernel always achieves zero Type II error, and a Type I error close to the design value (0.95). We further observe for m = 10 that HSICp with the spectral kernel always has better Type II error than the bag-of words kernel. This suggests that a kernel with a more sophisticated encoding of text structure induces a more sensitive test, although for larger sample sizes, the advantage vanishes. The HSICg test does less well on this data, always accepting H0 for m = 10, and returning a Type I error of zero, rather than the design value of 5%, when m = 50. It appears that this is due to a very low variance estimate returned by the Theorem 4 expression, which could be caused by the high diagonal dominance of kernels on strings. Thus, while the test threshold for HSICg at m = 50 still fell between the dependent and independent values of HSICb, this was not the result of an accurate modelling of the null distribution. We would therefore recommend the permutation approach for this problem. Finally, we also tried testing with 2-line extracts and 10-line extracts, which yielded similar results. 5 Conclusion We have introduced a test of whether significant statistical dependence is obtained by a kernel dependence measure, the Hilbert-Schmidt independence criterion (HSIC). Our test costs O(m2) for sample size m. In our experiments, HSIC-based tests always outperformed the contingency table [17] and functional correlation [4] approaches, for both univariate random variables and higher dimensional vectors which were dependent but uncorrelated. We would therefore recommend HSIC-based tests for checking the convergence of independent component analysis and independent subspace analysis. Finally, our test also applies on structured domains, being able to detect the dependence 7 Table 1: Independence tests for cross-language dependence detection. Topics are in the first column, where the total number of 5-line extracts for each dataset is in parentheses. BOW(10) denotes a bag of words kernel and m = 10 sample size, Spec(50) is a k-spectrum kernel with m = 50. The first entry in each cell is the null acceptance rate of the test under H0 (i.e. 1−(Type I error); should be near 0.95); the second entry is the null acceptance rate under H1 (the Type II error, small is better). Each entry is an average over 300 repetitions. Topic BOW(10) Spec(10) BOW(50) Spec(50) HSICg HSICp HSICg HSICp HSICg HSICp HSICg HSICp Agriculture (555) 1.00, 0.99 0.94, 0.18 1.00, 1.00 0.95, 0.00 1.00, 0.00 0.93, 0.00 1.00, 0.00 0.95, 0.00 Fisheries (408) 1.00, 1.00 0.94, 0.20 1.00, 1.00 0.94, 0.00 1.00, 0.00 0.93, 0.00 1.00, 0.00 0.95, 0.00 Immigration (289) 1.00, 1.00 0.96, 0.09 1.00, 1.00 0.91, 0.00 0.99, 0.00 0.94, 0.00 1.00, 0.00 0.95, 0.00 of passages of text and their translation.Another application along these lines might be in testing dependence between data of completely different types, such as images and captions. Acknowledgements: NICTA is funded through the Australian Government’s Backing Australia’s Ability initiative, in part through the ARC. This work was supported in part by the IST Programme of the European Community, under the PASCAL Network of Excellence, IST-2002-506778. References [1] F. Bach and M. Jordan. Tree-dependent component analysis. In UAI 18, 2002. 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Discriminative Keyword Selection Using Support Vector Machines W. M. Campbell, F. S. Richardson MIT Lincoln Laboratory Lexington, MA 02420 wcampbell,frichard@ll.mit.edu Abstract Many tasks in speech processing involve classification of long term characteristics of a speech segment such as language, speaker, dialect, or topic. A natural technique for determining these characteristics is to first convert the input speech into a sequence of tokens such as words, phones, etc. From these tokens, we can then look for distinctive sequences, keywords, that characterize the speech. In many applications, a set of distinctive keywords may not be known a priori. In this case, an automatic method of building up keywords from short context units such as phones is desirable. We propose a method for the construction of keywords based upon Support Vector Machines. We cast the problem of keyword selection as a feature selection problem for n-grams of phones. We propose an alternating filter-wrapper method that builds successively longer keywords. Application of this method to language recognition and topic recognition tasks shows that the technique produces interesting and significant qualitative and quantitative results. 1 Introduction A common problem in speech processing is to identify properties of a speech segment such as the language, speaker, topic, or dialect. A typical solution to this problem is to apply a detection paradigm. A set of classifiers is applied to a speech segment to produce a decision. For instance, for language recognition, we might construct detectors for English, French, and Spanish. The maximum scoring detector on a speech segment would be the predicted language. Two basic categories of systems have been applied to the detection problem. A first approach uses short-term spectral characteristics of the speech and models these with Gaussian mixture models (GMMs) or support vector machines (SVMs) directly producinga decision. Although quite accurate, this type of system produces only a classification decision with no qualitative interpretation. A second approach uses high level features of the speech such as phones and words to detect the properties. An advantage of this approach is that, in some instances, we can explain why we made a decision. For example, a particular phone or word sequence might indicate the topic. We adopt this latter approach for our paper. SVMs have become a common method of extracting high-level properties of sequences of speech tokens [1, 2, 3, 4]. Sequence kernels are constructed by viewing a speech segment as a document of tokens. The SVM feature space in this case is a scaling of co-occurrence probabilities of tokens in an utterance. This technique is analogous to methods for applying SVMs to text classification [5]. SVMs have been applied at many linguistic levels of tokens as detectors. Our focus in this paper is at the acoustic phone level. Our goal is to automatically derive long sequences of phones which ∗This work was sponsored by the Department of Homeland Security under Air Force Contract FA872105-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the United States Government. 1 we call keywords which are characteristic of a given class. Prior work, for example, in language recognition [6], has shown that certain words are a significant predictor of a language. For instance, the presence of the phrase “you know” in a conversational speech segment is a strong indicator of English. A difficulty in using words as the indicator of the language is that we may not have available a speech-to-text (STT) system in all languages of interest. In this case, we’d like to automatically construct keywords that are indicative of the language. Note that a similar problem can occur in other property extraction problems. For instance, in topic recognition, proper names not in our STT system dictionary may be a strong indicator of topic. Our basic approach is to view keyword construction as a feature selection problem. Keywords are composed of sequences of phones of length n, i.e. n-grams. We would like to find the set of n-grams that best discriminates between classes. Unfortunately, this problem is difficult to solve directly, since the number of unique n-grams grows exponentially with increasing n. To alleviate this difficultly, we propose a method that starts with lower order n-grams and successively builds higher order n-grams. The outline of the paper is as follows. In Section 2.1, we review the basic architecture that we use for phone recognition and how it is applied to the problem. In Section 2.2, we review the application of SVMs to determining properties. Section 3.1 describes a feature selection method for SVMs. Section 3.2 presents our method for constructing long context units of phones to automatically create keywords. We use a novel feature selection approach that attempts to find longer strings that discriminate well between classes. Finally, in Section 4, we show the application of our method to language and topic recognition problems. We show qualitatively that the method produces interesting keywords. Quantitatively, we show that the method produces keywords which are good discriminators between classes. 2 Phonotactic Classification 2.1 Phone Recognition The high-level token extraction component of our system is a phone recognizer based upon the Brno University (BUT) design [7]. The basic architecture of this system is a monophone HMM system with a null grammar. Monophones are modeled by three states. This system uses two powerful components to achieve high accuracy. First, split temporal context (STC) features provide contextual cues for modeling monophones. Second, the BUT recognizer extensively uses discriminatively trained feedforward artificial neural networks (ANNs) to model HMM state posterior probabilities. We developed a phone recognizer for English units using the BUT architecture and automatically generated STT transcripts on the Switchboard 2 Cell corpora [8]. Training data consisted of approximately 10 hours of speech. ANN training was accomplished using the ICSI Quicknet package [9]. The resulting system has 49 monophones including silence. The BUT recognizer is used along with the HTK HMM toolkit [10] to produce lattices. Lattices encode multiple hypotheses with acoustic likelihoods. From a lattice, a 1-best (Viterbi) output can be produced. Alternatively, we use the lattice to produce expected counts of tokens and n-grams of tokens. Expected counts of n-grams can be easily understood as an extension of standard counts. Suppose we have a hypothesized string of tokens, W = w1, · · · , wn. Then bigrams are created by grouping two tokens at a time to form, W2 = w1_w2, w2_w3, · · · , wn−1_wn. Higher order n-grams are formed from longer juxtapositions of tokens. The count function for a given bigram, di, is count(di|W2) is the number of occurrences of di in the sequence W2. To extend counts to a lattice, L, we find the expected count over all all possible hypotheses in the lattice, count(di|L) = EW [count(di|W)] = X W∈L p(W|L) count(di|W). (1) The expected counts can be computed efficiently by a forward-backward algorithm; more details can be found in Section 3.3 and [11]. 2 A useful application of expected counts is to find the probability of an n-gram in a lattice. For a lattice, L, the joint probability of an n-gram, di, is p(di|L) = count(di|L) P j count(dj|L) (2) where the sum in (2) is performed over all unique n-grams in the utterance. 2.2 Discriminative Language Modeling: SVMs We focus on token-based language recognition with SVMs using the approach from [1, 4]. Similar to [1], a lattice of tokens, L, is modeled using a bag-of-n-grams approach. Joint probabilities of the unique n-grams, dj, on a per conversation basis are calculated, p(dj|L), see (2). Then, the probabilities are mapped to a sparse vector with entries Djp(dj|W). (3) The selection of the weighting, Dj, in (3) is critical for good performance. A typical choice is of the form Dj = min Cj, gj 1 p(dj|all) (4) where gj(·) is a function which squashes the dynamic range, and Cj is a constant. The probability p(dj|all) in (4) is calculated from the observed probability across all classes. The squashing function should monotonically map the interval [1, ∞) to itself to suppress large inverse probabilities. Typical choices for gj are gj(x) = √x and gj(x) = log(x) + 1. In both cases, the squashing function gj normalizes out the typicality of a feature across all classes. The constant Cj limits the effect of any one feature on the kernel inner product. If we set Cj = 1, then this makes Dj = 1 for all j. For the experiments in this paper, we use gj(x) = √x, which is suited to high frequency token streams. The general weighting of probabilities is then combined to form a kernel between two lattices, see [1] for more details. For two lattices, L1 and L2, the kernel is K(L1, L2) = X j D2 jp(dj|L1)p(dj|L2). (5) Intuitively, the kernel in (5) says that if the same n-grams are present in two sequences and the normalized frequencies are similar there will be a high degree of similarity (a large inner product). If n-grams are not present, then this will reduce similarity since one of the probabilities in (5) will be zero. The normalization Dj insures that n-grams with large probabilities do not dominate the kernel function. The kernel can alternatively be viewed as a linearization of the log-likelihood ratio [1]. Incorporating the kernel (5) into an SVM system is straightforward. SVM training and scoring require only a method of kernel evaluation between two objects that produces positive definite kernel matrices (the Mercer condition). We use the package SVMTorch [12]. Training is performed with a one-versus-all strategy. For each target class, we group all remaining class data and then train with these two classes. 3 Discriminative Keyword Selection 3.1 SVM Feature Selection A first step towards an algorithm for automatic keyword generation using phones is to examine feature selection methods. Ideally, we would like to select over all possible n-grams, where n is varying, the most discriminative sequences for determining a property of a speech segment. The number of features in this case is prohibitive, since it grows exponentially with n. Therefore, we have to consider alternate methods. As a first step, we examine feature selection for fixed n and look for keywords with n or less phones. Suppose that we have a set of candidate keywords. Since we are already using an SVM, a natural algorithm for discriminative feature selection in this case is to use a wrapper method [13]. 3 Suppose that the optimized SVM solution is f(X) = X i αiK(X, Xi) + c (6) and w = X i αib(Xi) (7) where b(Xi) is the vector of weighted n-gram probabilities in (3). We note that the kernel presented in (5) is linear. Also, the n-gram probabilities have been normalized in (3) by their probability across the entire data set. Intuitively, because of this normalization and since f(X) = wtb(X) + c, large magnitude entries in w correspond to significant features. A confirmation of this intuitive idea is the algorithm of Guyon, et. al. [14]. Guyon proposes an iterative wrapper method for feature selection for SVMs which has these basic steps: • For a set of features, S, find the SVM solution with model w. • Rank the features by their corresponding model entries w2 i . Here, wi is the ith entry of w in (7). • Eliminate low ranking features using a threshold. The algorithm may be iterated multiple times. Guyon’s algorithm for feature selection can be used for picking significant n-grams as keywords. We can create a kernel which is the sum of kernels as in (5) up to the desired n. We then train an SVM and rank n-grams according to the magnitude of the entries in the SVM model vector, w. As an example, we have looked at this feature selection method for a language recognition task with trigrams (to be described in Section 4). Figure 1 provides a motivation for the applicability of Guyon’s feature selection method. The figure shows two functions. First, the cumulative density function (CDF) of the SVM model values, |wi|, is shown. The CDF has an S-curve shape; i.e., only a small set of models weights has large magnitudes. The second curve shows the equal error rate (EER) of the task as a function of applying one iteration of the Guyon algorithm and retraining the SVM. EER is defined as the value where the miss and false alarm rates are equal. All features with |wi| below the value on the x-axis are discarded in the first iteration. From the figure, we see that only a small fraction (< 5%) of the features are needed to obtain good error rates. This interesting result provides motivation that a small subset of keywords are significant to the task. 10 −4 10 −3 10 −2 10 −1 10 0 0 0.25 0.5 0.75 1 Threshold CDF |wi| 10 −4 10 −3 10 −2 10 −1 10 00 0.05 0.1 0.15 0.2 Equal Error Rate CDF EER Figure 1: Feature selection for a trigram language recognition task using Guyon’s method 4 3.2 Keywords via an alternating wrapper/filter method The algorithm in Section 3.1 gives a method for n-gram selection for fixed n. Now, suppose we want to find keywords for arbitrary n. One possible hypothesis for keyword selection is that since higher order n-grams are discriminative, lower order n-grams in the keywords will also be discriminative. Therefore, it makes sense to finding distinguishing lower order n-grams and then construct longer units from these. On the basis of this idea, we propose the following algorithm for keyword construction: Keyword Building Algorithm • Start with an initial value of n = ns. Initialize the set, S′ n, to all possible n-grams of phones including lower order grams. By default, let S1 be the set of all phones. • (Wrapper Step) General n. Apply the feature selection algorithm in Section 3.1 to produce a subset of distinguishing n-grams, Sn ⊂S′ n. • (Filter Step) Construct a new set of (n + 1)-grams by juxtaposing elements from Sn with phones. Nominally, we take this step to be juxtaposition on the right and left, S′ n+1 = {dp, qd|d ∈Sn, p ∈S1, q ∈S1}. • Iterate to the wrapper step. • Output: Sn at some stopping n. A few items should be noted about the proposed keyword building algorithm. First, we call the second feature selection process a filter step, since induction has not been applied to the (n + 1)-gram features. Second, note that the purpose of the filter step is to provide a candidate set of possible (n + 1)-grams which can then be more systematically reduced. Third, several potential algorithms exist for the filter step. In our experiments and in the algorithm description, we nominally append one phone to the beginning and end of an n-gram. Another possibility is to try to combine overlapping n-grams. For instance, suppose the keyword is some_people which has phone transcript s_ah_m_p_iy_p_l. Then, if we are looking at 4-grams, we might see as top features s_ah_m_p and p_iy_p_l and combine these to produce a new keyword. 3.3 Keyword Implementation The expected n-gram counts were computed from lattices using the forward-backward algorithm. Equation (8) gives the posterior probability of a connected sequence of arcs in the lattice where src_nd(a) and dst_nd(a) are the source and destination node of arc a, ℓ(a)is the likelihood associated with arc a, α(n) and β(n) are the forward and backward probabilities of reaching node n from the beginning or end of the lattice L respectively, and ℓ(L) is the total likelihood of the lattice (the α(·) of the final node or β(·) of the initial node of the lattice). p(aj, ..., aj+n) = α(src_nd(aj))ℓ(aj) . . . ℓ(aj+n)β(dst_nd(aj+n)) ℓ(L) (8) Now if we define the posterior probability of a node p(n) as p(n) = (α(n)β(n))/ℓ(L). Then equation (8) becomes: p(aj, ..., aj+n) = p(aj) . . . p(aj+n) p(src_nd(aj+1)) . . . p(src_nd(aj+n)). (9) Equation (9) is attractive because it provides a way of computing the path posteriors locally using only the individual arc and node posteriors along the path. We use this computation along with a trie structure [15] to compute the posteriors of our keywords. 4 Experiments 4.1 Language Recognition Experimental Setup The phone recognizer described in Section 2.1 was used to generate lattices across a train and an evaluation data set. The training data set consists of more than 360 hours of telephone speech 5 spanning 13 different languages and coming from a variety of different sources including Callhome, Callfriend and Fisher. The evaluation data set is the NIST 2005 Language Recognition Evaluation data consisting of roughly 20,000 utterances (with duration of 30, 10 or 3 seconds depending on the task) coming from three collection sources including Callfriend, Mixer and OHSU. We evaluated our system for the 30 and 10 second task under the the NIST 2005 closed condition which limits the evaluation data to 7 languages (English, Hindi, Japanese, Korean, Mandarin, Spanish and Tamil) coming only from the OHSU data source. The training and evaluation data was segmented using an automatic speech activity detector and segments smaller than 0.5 seconds were thrown out. We also sub-segmented long audio files in the training data to keep the duration of each utterance to around 5 minutes (a shorter duration would have created too many training instances). Lattice arcs with posterior probabilities lower than 10−6 were removed and lattice expected counts smaller than 10−3 were ignored. The top and bottom 600 ranking keywords for each language were selected after each training iteration. The support vector machine was trained using a kernel formulation which requires pre-computing all of the kernel distances between the data points and using an alternate kernel which simply indexes into the resulting distance matrix (this approach becomes difficult when the number of data points is too large). 4.2 Language Recognition Results (Qualitative and Quantitative) To get a sense of how well our keyword building algorithm was working, we looked at the top ranking keywords from the English model only (since our phone recognizer is trained using the English phone set). Table 1 summarizes a few of the more compelling phone 5-grams, and a possible keyword that corresponds to each one. Not suprisingly, we noticed that in the list of top-ranking n-grams there were many variations or partial n-gram matches to the same keyword, as well as n-grams that didn’t correspond to any apparent keyword. The equal error rates for our system on the NIST 2005 language recognition evaluation are summarized in Table 2. The 4-gram system gave a relative improvement of 12% on the 10 second task and 9% on the 30 second task, but despite the compelling keywords produced by the 5-gram system, the performance actually degraded significantly compared to the 3-gram and 4-gram systems. Table 1: Top ranking keywords for 5-gram SVM for English language recognition model phones Rank keyword SIL_Y_UW_N_OW 1 you know !NULL_SIL_Y_EH_AX 3 <s> yeah !NULL_SIL_IY_M_TH 4 <s> ??? P_IY_P_AX_L 6 people R_IY_L_IY_SIL 7 really Y_UW_N_OW_OW 8 you know (var) T_L_AY_K_SIL 17 ? like L_AY_K_K_SIL 23 like (var) R_AY_T_SIL_!NULL 27 right </s> HH_AE_V_AX_N 29 have an !NULL_SIL_W_EH_L 37 <s> well Table 2: %EER for 10 and 30 second NIST language recognition tasks N 1 2 3 4 5 10sec 25.3 16.5 11.3 10.0 13.6 30sec 18.3 07.4 04.3 03.9 05.6 6 4.3 Topic Recognition Experimental Setup Topic recognition was performed using a subset of the phase I Fisher corpus (English) from LDC. This corpus consists of 5, 851 telephone conversations. Participants were given instructions to discuss a topic for 10 minutes from 40 different possible topics. Topics included “Education”, “Hobbies,” “Foreign Relations”, etc. Prompts were used to elicit discussion on the topics. An example prompt is: Movies: Do each of you enjoy going to the movies in a theater, or would you rather rent a movie and stay home? What was the last movie that you saw? Was it good or bad and why? For our experiments, we used 2750 conversation sides for training. We also constructed development and test sets of 1372 conversation sides each. The training set was used to find keywords and models for topic detection. 4.4 Topic Recognition Results We first looked at top ranking keywords for several topics; some results are shown in Table 3. We can see that many keywords show a strong correspondence with the topic. Also, there are partial keywords which correspond to what appears to be longer keywords, e.g. “eh_t_s_ih_k” corresponds to get sick. As in the language recognition task, we used EER as the performance measure. Results in Table 4 show the performance for several n-gram orders. Performance improves going from 3-grams to 4grams. But, as with the language recognition task, we see a degradation in performance for 5-grams. 5 Conclusions and future work We presented a method for automatic construction of keywords given a discriminative speech classification task. Our method was based upon successively building longer span keywords from shorter span keywords using phones as a fundamental unit. The problem was cast as a feature selection problem, and an alternating filter and wrapper algorithm was proposed. Results showed that reasonable keywords and improved performance could be achieved using this methodology. Table 3: Top keyword for 5-gram SVM in Topic Recognition Topic Phones Keyword Professional Sports on TV S_P_AO_R_T sport Hypothetical: Time Travel G_OW_B_AE_K go back Affirmative Action AX_V_AE_K_CH [affirmat]ive act[ion] US Public Schools S_K_UW_L_Z schools Movies IY_V_IY_D_IY DVD Hobbies HH_OH_B_IY_Z hobbies September 11 HH_AE_P_AX_N happen Issues in the Middle East IH_Z_R_IY_L Israel Illness EH_T_S_IH_K [g]et sick Hypothetical: One Million Dollars to leave the US Y_UW_M_AY_Y you may Table 4: Performance of Topic Detection for Different n-gram orders n-gram order 3 4 5 EER (%) 10.22 8.95 9.40 7 Numerous possibilities exist for future work on this task. First, extension and experimentation on other tasks such as dialect and speaker recognition would be interesting. The method has the potential for discovery of new interesting characteristics. Second, comparison of this method with other feature selection methods may be appropriate [16]. A third area for extension is various technical improvements. For instance, we might want to consider more general keyword models where skips are allowed (or more general finite state transducers [17]). Also, alternate methods for the filter for constructing higher order n-grams is a good area for exploration. References [1] W. M. Campbell, J. P. Campbell, D. A. Reynolds, D. A. Jones, and T. R. Leek, “Phonetic speaker recognition with support vector machines,” in Advances in Neural Information Processing Systems 16, Sebastian Thrun, Lawrence Saul, and Bernhard Schölkopf, Eds. MIT Press, Cambridge, MA, 2003. [2] W. M. Campbell, T. Gleason, J. Navratil, D. Reynolds, W. Shen, E. Singer, and P. TorresCarrasquillo, “Advanced language recognition using cepstra and phonotactics: MITLL system performance on the NIST 2005 language recognition evaluation,” in Proc. IEEE Odyssey, 2006. [3] Bin Ma and Haizhou Li, “A phonotactic-semantic paradigm for automatic spoken document classification,” in The 28th Annual International ACM SIGIR Conference, Brazil, 2005. [4] Lu-Feng Zhai, Man hung Siu, Xi Yang, and Herbert Gish, “Discriminatively trained language models using support vector machines for language identification,” in Proc. IEEE Odyssey: The Speaker and Language Recognition Workshop, 2006. [5] T. Joachims, Learning to Classify Text Using Support Vector Machines, Kluwer Academic Publishers, 2002. [6] W. M. Campbell, F. Richardson, and D. A. Reynolds, “Language recognition with word lattices and support vector machines,” in Proceedings of ICASSP, 2007, pp. IV–989 – IV–992. [7] Petr Schwarz, Matejka Pavel, and Jan Cernocky, “Hierarchical structures of neural networks for phoneme recognition,” in Proceedings of ICASSP, 2006, pp. 325–328. [8] Linguistic Data Consortium, “Switchboard-2 corpora,” http://www.ldc.upenn.edu. [9] “ICSI QuickNet,” http://www.icsi.berkeley.edu/Speech/qn.html. [10] S. Young, Gunnar Evermann, Thomas Hain, D. Kershaw, Gareth Moore, J. Odell, D. Ollason, V. Valtchev, and P. Woodland, The HTK book, Entropic, Ltd., Cambridge, UK, 2002. [11] L. Rabiner and B.-H. Juang, Fundamentals of Speech Recognition, Prentice-Hall, 1993. [12] Ronan Collobert and Samy Bengio, “SVMTorch: Support vector machines for large-scale regression problems,” Journal of Machine Learning Research, vol. 1, pp. 143–160, 2001. [13] Avrim L. Blum and Pat Langley, “Selection of relevant features and examples in machine learning,” Artificial Intelligence, vol. 97, no. 1-2, pp. 245–271, Dec. 1997. [14] I. Guyon, J. Weston, S. Barnhill, and V. Vapnik, “Gene selection for cancer classification using support vector machines,” Machine Learning, vol. 46, no. 1-3, pp. 389–422, 2002. [15] Konrad Rieck and Pavel Laskov, “Language models for detection of unknown attacks in network traffic,” Journal of Computer Virology, vol. 2, no. 4, pp. 243–256, 2007. [16] Takaaki Hori, I. Lee Hetherington, Timothy J. Hazen, and James R. Glass, “Open-vocabulary spoken utterance retrieval using confusion neworks,” in Proceedings of ICASSP, 2007. [17] C. Cortes, P. Haffner, and M. Mohri, “Rational kernels,” in Advances in Neural Information Processing Systems 15, S. Thrun S. Becker and K. Obermayer, Eds., Cambridge, MA, 2003, pp. 601–608, MIT Press. 8
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Probabilistic Matrix Factorization Ruslan Salakhutdinov and Andriy Mnih Department of Computer Science, University of Toronto 6 King’s College Rd, M5S 3G4, Canada {rsalakhu,amnih}@cs.toronto.edu Abstract Many existing approaches to collaborative filtering can neither handle very large datasets nor easily deal with users who have very few ratings. In this paper we present the Probabilistic Matrix Factorization (PMF) model which scales linearly with the number of observations and, more importantly, performs well on the large, sparse, and very imbalanced Netflix dataset. We further extend the PMF model to include an adaptive prior on the model parameters and show how the model capacity can be controlled automatically. Finally, we introduce a constrained version of the PMF model that is based on the assumption that users who have rated similar sets of movies are likely to have similar preferences. The resulting model is able to generalize considerably better for users with very few ratings. When the predictions of multiple PMF models are linearly combined with the predictions of Restricted Boltzmann Machines models, we achieve an error rate of 0.8861, that is nearly 7% better than the score of Netflix’s own system. 1 Introduction One of the most popular approaches to collaborative filtering is based on low-dimensional factor models. The idea behind such models is that attitudes or preferences of a user are determined by a small number of unobserved factors. In a linear factor model, a user’s preferences are modeled by linearly combining item factor vectors using user-specific coefficients. For example, for N users and M movies, the N ×M preference matrix R is given by the product of an N ×D user coefficient matrix U T and a D × M factor matrix V [7]. Training such a model amounts to finding the best rank-D approximation to the observed N × M target matrix R under the given loss function. A variety of probabilistic factor-based models has been proposed recently [2, 3, 4]. All these models can be viewed as graphical models in which hidden factor variables have directed connections to variables that represent user ratings. The major drawback of such models is that exact inference is intractable [12], which means that potentially slow or inaccurate approximations are required for computing the posterior distribution over hidden factors in such models. Low-rank approximations based on minimizing the sum-squared distance can be found using Singular Value Decomposition (SVD). SVD finds the matrix ˆR = U T V of the given rank which minimizes the sum-squared distance to the target matrix R. Since most real-world datasets are sparse, most entries in R will be missing. In those cases, the sum-squared distance is computed only for the observed entries of the target matrix R. As shown by [9], this seemingly minor modification results in a difficult non-convex optimization problem which cannot be solved using standard SVD implementations. Instead of constraining the rank of the approximation matrix ˆR = U T V , i.e. the number of factors, [10] proposed penalizing the norms of U and V . Learning in this model, however, requires solving a sparse semi-definite program (SDP), making this approach infeasible for datasets containing millions of observations. 1 U V j i Rij j=1,...,M i=1,...,N V σ U σ σ i Y V j Rij j=1,...,M U i iI i=1,...,N V σ U σ W k=1,...,M k W σ σ Figure 1: The left panel shows the graphical model for Probabilistic Matrix Factorization (PMF). The right panel shows the graphical model for constrained PMF. Many of the collaborative filtering algorithms mentioned above have been applied to modelling user ratings on the Netflix Prize dataset that contains 480,189 users, 17,770 movies, and over 100 million observations (user/movie/rating triples). However, none of these methods have proved to be particularly successful for two reasons. First, none of the above-mentioned approaches, except for the matrix-factorization-based ones, scale well to large datasets. Second, most of the existing algorithms have trouble making accurate predictions for users who have very few ratings. A common practice in the collaborative filtering community is to remove all users with fewer than some minimal number of ratings. Consequently, the results reported on the standard datasets, such as MovieLens and EachMovie, then seem impressive because the most difficult cases have been removed. For example, the Netflix dataset is very imbalanced, with “infrequent” users rating less than 5 movies, while “frequent” users rating over 10,000 movies. However, since the standardized test set includes the complete range of users, the Netflix dataset provides a much more realistic and useful benchmark for collaborative filtering algorithms. The goal of this paper is to present probabilistic algorithms that scale linearly with the number of observations and perform well on very sparse and imbalanced datasets, such as the Netflix dataset. In Section 2 we present the Probabilistic Matrix Factorization (PMF) model that models the user preference matrix as a product of two lower-rank user and movie matrices. In Section 3, we extend the PMF model to include adaptive priors over the movie and user feature vectors and show how these priors can be used to control model complexity automatically. In Section 4 we introduce a constrained version of the PMF model that is based on the assumption that users who rate similar sets of movies have similar preferences. In Section 5 we report the experimental results that show that PMF considerably outperforms standard SVD models. We also show that constrained PMF and PMF with learnable priors improve model performance significantly. Our results demonstrate that constrained PMF is especially effective at making better predictions for users with few ratings. 2 Probabilistic Matrix Factorization (PMF) Suppose we have M movies, N users, and integer rating values from 1 to K1. Let Rij represent the rating of user i for movie j, U ∈RD×N and V ∈RD×M be latent user and movie feature matrices, with column vectors Ui and Vj representing user-specific and movie-specific latent feature vectors respectively. Since model performance is measured by computing the root mean squared error (RMSE) on the test set we first adopt a probabilistic linear model with Gaussian observation noise (see fig. 1, left panel). We define the conditional distribution over the observed ratings as p(R|U, V, σ2) = N Y i=1 M Y j=1 N(Rij|U T i Vj, σ2) Iij , (1) where N(x|µ, σ2) is the probability density function of the Gaussian distribution with mean µ and variance σ2, and Iij is the indicator function that is equal to 1 if user i rated movie j and equal to 1Real-valued ratings can be handled just as easily by the models described in this paper. 2 0 otherwise. We also place zero-mean spherical Gaussian priors [1, 11] on user and movie feature vectors: p(U|σ2 U) = N Y i=1 N (Ui|0, σ2 UI), p(V |σ2 V ) = M Y j=1 N(Vj|0, σ2 V I). (2) The log of the posterior distribution over the user and movie features is given by ln p(U, V |R, σ2, σ2 V , σ2 U) = − 1 2σ2 N X i=1 M X j=1 Iij(Rij −U T i Vj)2 − 1 2σ2 U N X i=1 U T i Ui − 1 2σ2 V M X j=1 V T j Vj −1 2 N X i=1 M X j=1 Iij ln σ2 + ND ln σ2 U + MD ln σ2 V + C, (3) where C is a constant that does not depend on the parameters. Maximizing the log-posterior over movie and user features with hyperparameters (i.e. the observation noise variance and prior variances) kept fixed is equivalent to minimizing the sum-of-squared-errors objective function with quadratic regularization terms: E = 1 2 N X i=1 M X j=1 Iij Rij −U T i Vj 2 + λU 2 N X i=1 ∥Ui ∥2 F ro +λV 2 M X j=1 ∥Vj ∥2 F ro, (4) where λU = σ2/σ2 U, λV = σ2/σ2 V , and ∥· ∥2 F ro denotes the Frobenius norm. A local minimum of the objective function given by Eq. 4 can be found by performing gradient descent in U and V . Note that this model can be viewed as a probabilistic extension of the SVD model, since if all ratings have been observed, the objective given by Eq. 4 reduces to the SVD objective in the limit of prior variances going to infinity. In our experiments, instead of using a simple linear-Gaussian model, which can make predictions outside of the range of valid rating values, the dot product between user- and movie-specific feature vectors is passed through the logistic function g(x) = 1/(1 + exp(−x)), which bounds the range of predictions: p(R|U, V, σ2) = N Y i=1 M Y j=1 N (Rij|g(U T i Vj), σ2) Iij . (5) We map the ratings 1, ..., K to the interval [0, 1] using the function t(x) = (x −1)/(K −1), so that the range of valid rating values matches the range of predictions our model makes. Minimizing the objective function given above using steepest descent takes time linear in the number of observations. A simple implementation of this algorithm in Matlab allows us to make one sweep through the entire Netflix dataset in less than an hour when the model being trained has 30 factors. 3 Automatic Complexity Control for PMF Models Capacity control is essential to making PMF models generalize well. Given sufficiently many factors, a PMF model can approximate any given matrix arbitrarily well. The simplest way to control the capacity of a PMF model is by changing the dimensionality of feature vectors. However, when the dataset is unbalanced, i.e. the number of observations differs significantly among different rows or columns, this approach fails, since any single number of feature dimensions will be too high for some feature vectors and too low for others. Regularization parameters such as λU and λV defined above provide a more flexible approach to regularization. Perhaps the simplest way to find suitable values for these parameters is to consider a set of reasonable parameter values, train a model for each setting of the parameters in the set, and choose the model that performs best on the validation set. The main drawback of this approach is that it is computationally expensive, since instead of training a single model we have to train a multitude of models. We will show that the method proposed by [6], originally applied to neural networks, can be used to determine suitable values for the regularization parameters of a PMF model automatically without significantly affecting the time needed to train the model. 3 As shown above, the problem of approximating a matrix in the L2 sense by a product of two low-rank matrices that are regularized by penalizing their Frobenius norm can be viewed as MAP estimation in a probabilistic model with spherical Gaussian priors on the rows of the low-rank matrices. The complexity of the model is controlled by the hyperparameters: the noise variance σ2 and the the parameters of the priors (σ2 U and σ2 V above). Introducing priors for the hyperparameters and maximizing the log-posterior of the model over both parameters and hyperparameters as suggested in [6] allows model complexity to be controlled automatically based on the training data. Using spherical priors for user and movie feature vectors in this framework leads to the standard form of PMF with λU and λV chosen automatically. This approach to regularization allows us to use methods that are more sophisticated than the simple penalization of the Frobenius norm of the feature matrices. For example, we can use priors with diagonal or even full covariance matrices as well as adjustable means for the feature vectors. Mixture of Gaussians priors can also be handled quite easily. In summary, we find a point estimate of parameters and hyperparameters by maximizing the logposterior given by ln p(U, V, σ2, ΘU, ΘV |R) = ln p(R|U, V, σ2) + ln p(U|ΘU) + ln p(V |ΘV )+ ln p(ΘU) + ln p(ΘV ) + C, (6) where ΘU and ΘV are the hyperparameters for the priors over user and movie feature vectors respectively and C is a constant that does not depend on the parameters or hyperparameters. When the prior is Gaussian, the optimal hyperparameters can be found in closed form if the movie and user feature vectors are kept fixed. Thus to simplify learning we alternate between optimizing the hyperparameters and updating the feature vectors using steepest ascent with the values of hyperparameters fixed. When the prior is a mixture of Gaussians, the hyperparameters can be updated by performing a single step of EM. In all of our experiments we used improper priors for the hyperparameters, but it is easy to extend the closed form updates to handle conjugate priors for the hyperparameters. 4 Constrained PMF Once a PMF model has been fitted, users with very few ratings will have feature vectors that are close to the prior mean, or the average user, so the predicted ratings for those users will be close to the movie average ratings. In this section we introduce an additional way of constraining user-specific feature vectors that has a strong effect on infrequent users. Let W ∈RD×M be a latent similarity constraint matrix. We define the feature vector for user i as: Ui = Yi + PM k=1 IikWk PM k=1 Iik . (7) where I is the observed indicator matrix with Iij taking on value 1 if user i rated movie j and 0 otherwise2. Intuitively, the ith column of the W matrix captures the effect of a user having rated a particular movie has on the prior mean of the user’s feature vector. As a result, users that have seen the same (or similar) movies will have similar prior distributions for their feature vectors. Note that Yi can be seen as the offset added to the mean of the prior distribution to get the feature vector Ui for the user i. In the unconstrained PMF model Ui and Yi are equal because the prior mean is fixed at zero (see fig. 1). We now define the conditional distribution over the observed ratings as p(R|Y, V, W, σ2) = N Y i=1 M Y j=1 N(Rij|g Yi + PM k=1 IikWk PM k=1 Iik T Vj , σ2) Iij . (8) We regularize the latent similarity constraint matrix W by placing a zero-mean spherical Gaussian prior on it: p(W|σW ) = M Y k=1 N(Wk|0, σ2 W I). (9) 2If no rating information is available about some user i, i.e. all entries of Ii vector are zero, the value of the ratio in Eq. 7 is set to zero. 4 0 10 20 30 40 50 60 70 80 90 100 0.91 0.92 0.93 0.94 0.95 0.96 0.97 Epochs RMSE PMF1 PMF2 Netflix Baseline Score SVD PMFA1 0 5 10 15 20 25 30 35 40 45 50 55 60 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 Epochs RMSE PMF Constrained PMF Netflix Baseline Score SVD The Netflix Dataset 10D 30D Figure 2: Left panel: Performance of SVD, PMF and PMF with adaptive priors, using 10D feature vectors, on the full Netflix validation data. Right panel: Performance of SVD, Probabilistic Matrix Factorization (PMF) and constrained PMF, using 30D feature vectors, on the validation data. The y-axis displays RMSE (root mean squared error), and the x-axis shows the number of epochs, or passes, through the entire training dataset. As with the PMF model, maximizing the log-posterior is equivalent to minimizing the sum-ofsquared errors function with quadratic regularization terms: E = 1 2 N X i=1 M X j=1 Iij Rij −g Yi + PM k=1 IikWk PM k=1 Iik T Vj 2 (10) +λY 2 N X i=1 ∥Yi ∥2 F ro +λV 2 M X j=1 ∥Vj ∥2 F ro +λW 2 M X k=1 ∥Wk ∥2 F ro, with λY = σ2/σ2 Y , λV = σ2/σ2 V , and λW = σ2/σ2 W . We can then perform gradient descent in Y , V , and W to minimize the objective function given by Eq. 10. The training time for the constrained PMF model scales linearly with the number of observations, which allows for a fast and simple implementation. As we show in our experimental results section, this model performs considerably better than a simple unconstrained PMF model, especially on infrequent users. 5 Experimental Results 5.1 Description of the Netflix Data According to Netflix, the data were collected between October 1998 and December 2005 and represent the distribution of all ratings Netflix obtained during this period. The training dataset consists of 100,480,507 ratings from 480,189 randomly-chosen, anonymous users on 17,770 movie titles. As part of the training data, Netflix also provides validation data, containing 1,408,395 ratings. In addition to the training and validation data, Netflix also provides a test set containing 2,817,131 user/movie pairs with the ratings withheld. The pairs were selected from the most recent ratings for a subset of the users in the training dataset. To reduce the unintentional overfitting to the test set that plagues many empirical comparisons in the machine learning literature, performance is assessed by submitting predicted ratings to Netflix who then post the root mean squared error (RMSE) on an unknown half of the test set. As a baseline, Netflix provided the test score of its own system trained on the same data, which is 0.9514. To provide additional insight into the performance of different algorithms we created a smaller and much more difficult dataset from the Netflix data by randomly selecting 50,000 users and 1850 movies. The toy dataset contains 1,082,982 training and 2,462 validation user/movie pairs. Over 50% of the users in the training dataset have less than 10 ratings. 5.2 Details of Training To speed-up the training, instead of performing batch learning we subdivided the Netflix data into mini-batches of size 100,000 (user/movie/rating triples), and updated the feature vectors after each 5 mini-batch. After trying various values for the learning rate and momentum and experimenting with various values of D, we chose to use a learning rate of 0.005, and a momentum of 0.9, as this setting of parameters worked well for all values of D we have tried. 5.3 Results for PMF with Adaptive Priors To evaluate the performance of PMF models with adaptive priors we used models with 10D features. This dimensionality was chosen in order to demonstrate that even when the dimensionality of features is relatively low, SVD-like models can still overfit and that there are some performance gains to be had by regularizing such models automatically. We compared an SVD model, two fixed-prior PMF models, and two PMF models with adaptive priors. The SVD model was trained to minimize the sum-squared distance only to the observed entries of the target matrix. The feature vectors of the SVD model were not regularized in any way. The two fixed-prior PMF models differed in their regularization parameters: one (PMF1) had λU = 0.01 and λV = 0.001, while the other (PMF2) had λU = 0.001 and λV = 0.0001. The first PMF model with adaptive priors (PMFA1) had Gaussian priors with spherical covariance matrices on user and movie feature vectors, while the second model (PMFA2) had diagonal covariance matrices. In both cases, the adaptive priors had adjustable means. Prior parameters and noise covariances were updated after every 10 and 100 feature matrix updates respectively. The models were compared based on the RMSE on the validation set. The results of the comparison are shown on Figure 2 (left panel). Note that the curve for the PMF model with spherical covariances is not shown since it is virtually identical to the curve for the model with diagonal covariances. Comparing models based on the lowest RMSE achieved over the time of training, we see that the SVD model does almost as well as the moderately regularized PMF model (PMF2) (0.9258 vs. 0.9253) before overfitting badly towards the end of training. While PMF1 does not overfit, it clearly underfits since it reaches the RMSE of only 0.9430. The models with adaptive priors clearly outperform the competing models, achieving the RMSE of 0.9197 (diagonal covariances) and 0.9204 (spherical covariances). These results suggest that automatic regularization through adaptive priors works well in practice. Moreover, our preliminary results for models with higher-dimensional feature vectors suggest that the gap in performance due to the use of adaptive priors is likely to grow as the dimensionality of feature vectors increases. While the use of diagonal covariance matrices did not lead to a significant improvement over the spherical covariance matrices, diagonal covariances might be well-suited for automatically regularizing the greedy version of the PMF training algorithm, where feature vectors are learned one dimension at a time. 5.4 Results for Constrained PMF For experiments involving constrained PMF models, we used 30D features (D = 30), since this choice resulted in the best model performance on the validation set. Values of D in the range of [20, 60] produce similar results. Performance results of SVD, PMF, and constrained PMF on the toy dataset are shown on Figure 3. The feature vectors were initialized to the same values in all three models. For both PMF and constrained PMF models the regularization parameters were set to λU = λY = λV = λW = 0.002. It is clear that the simple SVD model overfits heavily. The constrained PMF model performs much better and converges considerably faster than the unconstrained PMF model. Figure 3 (right panel) shows the effect of constraining user-specific features on the predictions for infrequent users. Performance of the PMF model for a group of users that have fewer than 5 ratings in the training datasets is virtually identical to that of the movie average algorithm that always predicts the average rating of each movie. The constrained PMF model, however, performs considerably better on users with few ratings. As the number of ratings increases, both PMF and constrained PMF exhibit similar performance. One other interesting aspect of the constrained PMF model is that even if we know only what movies the user has rated, but do not know the values of the ratings, the model can make better predictions than the movie average model. For the toy dataset, we randomly sampled an additional 50,000 users, and for each of the users compiled a list of movies the user has rated and then discarded the actual ratings. The constrained PMF model achieved a RMSE of 1.0510 on the validation set compared to a RMSE of 1.0726 for the simple movie average model. This experiment strongly suggests that knowing only which movies a user rated, but not the actual ratings, can still help us to model that user’s preferences better. 6 0 20 40 60 80 100 120 140 160 180 200 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 1.22 1.24 1.26 1.28 1.3 Epochs RMSE PMF Constrained PMF SVD 1−5 6−10 11−20 21−40 41−80 81−160 >161 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Number of Observed Ratings RMSE PMF Constrained PMF Movie Average Toy Dataset Figure 3: Left panel: Performance of SVD, Probabilistic Matrix Factorization (PMF) and constrained PMF on the validation data. The y-axis displays RMSE (root mean squared error), and the x-axis shows the number of epochs, or passes, through the entire training dataset. Right panel: Performance of constrained PMF, PMF, and the movie average algorithm that always predicts the average rating of each movie. The users were grouped by the number of observed ratings in the training data. 1−5 6−10 11−20 21−40 41−80 81−160 161−320 321−640 >641 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Number of Observed Ratings RMSE PMF Constrained PMF Movie Average 1−5 6−10 11−20 21−40 41−80 81−160 161−320 321−640 >641 0 2 4 6 8 10 12 14 16 18 20 Number of Observed Ratings Users (%) 0 5 10 15 20 25 30 35 40 45 50 55 60 0.9 0.902 0.904 0.906 0.908 0.91 0.912 0.914 0.916 0.918 0.92 Epochs RMSE Constrained PMF (using Test rated/unrated id) Constrained PMF Figure 4: Left panel: Performance of constrained PMF, PMF, and the movie average algorithm that always predicts the average rating of each movie. The users were grouped by the number of observed rating in the training data, with the x-axis showing those groups, and the y-axis displaying RMSE on the full Netflix validation data for each such group. Middle panel: Distribution of users in the training dataset. Right panel: Performance of constrained PMF and constrained PMF that makes use of an additional rated/unrated information obtained from the test dataset. Performance results on the full Netflix dataset are similar to the results on the toy dataset. For both the PMF and constrained PMF models the regularization parameters were set to λU = λY = λV = λW = 0.001. Figure 2 (right panel) shows that constrained PMF significantly outperforms the unconstrained PMF model, achieving a RMSE of 0.9016. A simple SVD achieves a RMSE of about 0.9280 and after about 10 epochs begins to overfit. Figure 4 (left panel) shows that the constrained PMF model is able to generalize considerably better for users with very few ratings. Note that over 10% of users in the training dataset have fewer than 20 ratings. As the number of ratings increases, the effect from the offset in Eq. 7 diminishes, and both PMF and constrained PMF achieve similar performance. There is a more subtle source of information in the Netflix dataset. Netflix tells us in advance which user/movie pairs occur in the test set, so we have an additional category: movies that were viewed but for which the rating is unknown. This is a valuable source of information about users who occur several times in the test set, especially if they have only a small number of ratings in the training set. The constrained PMF model can easily take this information into account. Figure 4 (right panel) shows that this additional source of information further improves model performance. When we linearly combine the predictions of PMF, PMF with a learnable prior, and constrained PMF, we achieve an error rate of 0.8970 on the test set. When the predictions of multiple PMF models are linearly combined with the predictions of multiple RBM models, recently introduced by [8], we achieve an error rate of 0.8861, that is nearly 7% better than the score of Netflix’s own system. 7 6 Summary and Discussion In this paper we presented Probabilistic Matrix Factorization (PMF) and its two derivatives: PMF with a learnable prior and constrained PMF. We also demonstrated that these models can be efficiently trained and successfully applied to a large dataset containing over 100 million movie ratings. Efficiency in training PMF models comes from finding only point estimates of model parameters and hyperparameters, instead of inferring the full posterior distribution over them. If we were to take a fully Bayesian approach, we would put hyperpriors over the hyperparameters and resort to MCMC methods [5] to perform inference. While this approach is computationally more expensive, preliminary results strongly suggest that a fully Bayesian treatment of the presented PMF models would lead to a significant increase in predictive accuracy. Acknowledgments We thank Vinod Nair and Geoffrey Hinton for many helpful discussions. This research was supported by NSERC. References [1] Delbert Dueck and Brendan Frey. Probabilistic sparse matrix factorization. Technical Report PSI TR 2004-023, Dept. of Computer Science, University of Toronto, 2004. [2] Thomas Hofmann. Probabilistic latent semantic analysis. In Proceedings of the 15th Conference on Uncertainty in AI, pages 289–296, San Fransisco, California, 1999. Morgan Kaufmann. [3] Benjamin Marlin. Modeling user rating profiles for collaborative filtering. In Sebastian Thrun, Lawrence K. Saul, and Bernhard Sch¨olkopf, editors, NIPS. MIT Press, 2003. [4] Benjamin Marlin and Richard S. Zemel. The multiple multiplicative factor model for collaborative filtering. In Machine Learning, Proceedings of the Twenty-first International Conference (ICML 2004), Banff, Alberta, Canada, July 4-8, 2004. ACM, 2004. [5] Radford M. Neal. Probabilistic inference using Markov chain Monte Carlo methods. Technical Report CRG-TR-93-1, Department of Computer Science, University of Toronto, September 1993. [6] S. J. Nowlan and G. E. Hinton. Simplifying neural networks by soft weight-sharing. Neural Computation, 4:473–493, 1992. [7] Jason D. M. Rennie and Nathan Srebro. Fast maximum margin matrix factorization for collaborative prediction. In Luc De Raedt and Stefan Wrobel, editors, Machine Learning, Proceedings of the TwentySecond International Conference (ICML 2005), Bonn, Germany, August 7-11, 2005, pages 713–719. ACM, 2005. [8] Ruslan Salakhutdinov, Andriy Mnih, and Geoffrey Hinton. Restricted Boltzmann machines for collaborative filtering. In Machine Learning, Proceedings of the Twenty-fourth International Conference (ICML 2004). ACM, 2007. [9] Nathan Srebro and Tommi Jaakkola. Weighted low-rank approximations. In Tom Fawcett and Nina Mishra, editors, Machine Learning, Proceedings of the Twentieth International Conference (ICML 2003), August 21-24, 2003, Washington, DC, USA, pages 720–727. AAAI Press, 2003. [10] Nathan Srebro, Jason D. M. Rennie, and Tommi Jaakkola. Maximum-margin matrix factorization. In Advances in Neural Information Processing Systems, 2004. [11] Michael E. Tipping and Christopher M. Bishop. Probabilistic principal component analysis. Technical Report NCRG/97/010, Neural Computing Research Group, Aston University, September 1997. [12] Max Welling, Michal Rosen-Zvi, and Geoffrey Hinton. Exponential family harmoniums with an application to information retrieval. In NIPS 17, pages 1481–1488, Cambridge, MA, 2005. MIT Press. 8
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Density Estimation under Independent Similarly Distributed Sampling Assumptions Tony Jebara, Yingbo Song and Kapil Thadani Department of Computer Science Columbia University New York, NY 10027 { jebara,yingbo,kapil }@cs.columbia.edu Abstract A method is proposed for semiparametric estimation where parametric and nonparametric criteria are exploited in density estimation and unsupervised learning. This is accomplished by making sampling assumptions on a dataset that smoothly interpolate between the extreme of independently distributed (or id) sample data (as in nonparametric kernel density estimators) to the extreme of independent identically distributed (or iid) sample data. This article makes independent similarly distributed (or isd) sampling assumptions and interpolates between these two using a scalar parameter. The parameter controls a Bhattacharyya affinity penalty between pairs of distributions on samples. Surprisingly, the isd method maintains certain consistency and unimodality properties akin to maximum likelihood estimation. The proposed isd scheme is an alternative for handling nonstationarity in data without making drastic hidden variable assumptions which often make estimation difficult and laden with local optima. Experiments in density estimation on a variety of datasets confirm the value of isd over iid estimation, id estimation and mixture modeling. 1 Introduction Density estimation is a popular unsupervised learning technique for recovering distributions from data. Most approaches can be split into two categories: parametric methods where the functional form of the distribution is known a priori (often from the exponential family (Collins et al., 2002; Efron & Tibshirani, 1996)) and non-parametric approaches which explore a wider range of distributions with less constrained forms (Devroye & Gyorfi, 1985). Parametric approaches can underfit or may be mismatched to real-world data if they are built on incorrect a priori assumptions. A popular non-parametric approach is kernel density estimation or the Parzen windows method (Silverman, 1986). However, these may over-fit thus requiring smoothing, bandwidth estimation and adaptation (Wand & Jones, 1995; Devroye & Gyorfi, 1985; Bengio et al., 2005). Semiparametric efforts (Olking & Spiegelman, 1987) combine the complementary advantages of both schools. For instance, mixture models in their infinite-component setting (Rasmussen, 1999) as well as statistical processes (Teh et al., 2004) make only partial parametric assumptions. Alternatively, one may seed non-parametric distributions with parametric assumptions (Hjort & Glad, 1995) or augment parametric models with nonparametric factors (Naito, 2004). This article instead proposes a continuous interpolation between iid parametric density estimation and id kernel density estimation. It makes independent similarly distributed (isd) sampling assumptions on the data. In isd, a scalar parameter λ trades off parametric and non-parametric properties to produce an overall better density estimate. The method avoids sampling or approximate inference computations and only recycles well known parametric update rules for estimation. It remains computationally efficient, unimodal and consistent for a wide range of models. This paper is organized as follows. Section 2 shows how id and iid sampling setups can be smoothly interpolated using a novel isd posterior which maintains log-concavity for many popular models. Section 3 gives analytic formulae for the exponential family case as well as slight modifications to familiar maximum likelihood updates for recovering parameters under isd assumptions. Some consistency properties of the isd posterior are provided. Section 4 then extends the method to hidden variable models or mixtures and provides simple update rules. Section 5 provides experiments comparing isd with id and iid as well as mixture modeling. We conclude with a brief discussion. 2 A Continuum between id and iid Assume we are given a dataset of N −1 inputs x1, . . . , xN−1 from some sample space Ω. Given a new query input xN also in the same sample space, density estimation aims at recovering a density function p(x1, . . . , xN−1, xN) or p(xN|x1, . . . , xN−1) using a Bayesian or frequentist approach. Therefore, a general density estimation task is, given a dataset X = x1, . . . , xN, recover p(x1, . . . , xN). A common subsequent assumption is that the data points are id or independently sampled which leads to the following simplification: pid(X) = N Y n=1 pn(xn). The joint likelihood factorizes into a product of independent singleton marginals pn(xn) each of which can be different. A stricter assumption is that all samples share the same singleton marginal: piid(X) = N Y n=1 p(xn). which is the popular iid sampling situation. In maximum likelihood estimation, either of the above likelihood scores (pid or piid) is maximized by exploring different settings of the marginals. The id setup gives rise to what is commonly referred to as kernel density or Parzen estimation. Meanwhile, the iid setup gives rise to traditional iid parametric maximum likelihood (ML) or maximum a posteriori (MAP) estimation. Both methods have complementary advantages and disadvantages. The iid assumption may be too aggressive for many real world problems. For instance, data may be generated by some slowly time-varying nonstationary distribution or (more distressingly) from a distribution that does not match our parametric assumptions. Similarly, the id setup may be too flexible and might over-fit when the marginal pn(x) is myopically recovered from a single xn. Consider the parametric ML and MAP setting where parameters Θ = {θ1, . . . , θN} are used to define the marginals. We will use p(x|θn) = pn(x) interchangeably. The MAP id parametric setting involves maximizing the following posterior (likelihood times a prior) over the models: pid(X, Θ) = N Y n=1 p(xn|θn)p(θn). To mimic ML, simply set p(θn) to uniform. For simplicity assume that these singleton priors are always kept uniform. Parameters Θ are then estimated by maximizing pid. To obtain the iid setup, we can maximize pid subject to constraints that force all marginals to be equal, in other words θm = θn for all m, n ∈{1, . . . , N}. Instead of applying N(N −1)/2 hard pairwise constraints in an iid setup, consider imposing penalty functions across pairs of marginals. These penalty functions reduce the posterior score when marginals disagree and encourage some stickiness between models (Teh et al., 2004). We measure the level of agreement between two marginals pm(x) and pn(x) using the following Bhattacharyya affinity metric (Bhattacharyya, 1943) between two distributions: B(pm, pn) = B(p(x|θm), p(x|θn)) = Z pβ(x|θm)pβ(x|θn)dx. This is a symmetric non-negative quantity in both distributions pm and pn. The natural choice for the setting of β is 1/2 and in this case, it is easy to verify the affinity is maximal and equals one if and only if pm(x) = pn(x). A large family of alternative information divergences exist to relate pairs of distributions (Topsoe, 1999) and are discussed in the Appendix. In this article, the Bhattacharyya affinity is preferred since it has some useful computational, analytic, and logconcavity properties. In addition, it leads to straightforward variants of the estimation algorithms as in the id and iid situations for many choices of parametric densities. Furthermore, (unlike Kullback Leibler divergence) it is possible to compute the Bhattacharyya affinity analytically and efficiently for a wide range of probability models including hidden Markov models (Jebara et al., 2004). We next define (up to a constant scaling) the posterior score for independent similarly distributed (isd) data: pλ(X, Θ) ∝ Y n p(xn|θn)p(θn) Y m̸=n Bλ/N(p(x|θm), p(x|θn)). (1) Here, a scalar power λ/N is applied to each affinity. The parameter λ adjusts the importance of the similarity between pairs of marginals. Clearly, if λ →0, then the affinity is always unity and the marginals are completely unconstrained as in the id setup. Meanwhile, as λ →∞, the affinity is zero unless the marginals are exactly identical. This produces the iid setup. We will refer to the isd posterior as Equation 1 and when p(θn) is set to uniform, we will call it the isd likelihood. One can also view the additional term in isd as id estimation with a modified prior ˜p(Θ) as follows: ˜p(Θ) ∝ Y n p(θn) Y m̸=n Bλ/N(p(x|θm), p(x|θn)). This prior is a Markov random field tying all parameters in a pairwise manner in addition to the standard singleton potentials in the id scenario. However, this perspective is less appealing since it disguises the fact that the samples are not quite id or iid. One of the appealing properties of iid and id maximum likelihood estimation is its unimodality for log-concave distributions. The isd posterior also benefits from a unique optimum and log-concavity. However, the conditional distributions p(x|θn) are required to be jointly log-concave in both parameters θn and data x. This set of distributions includes the Gaussian distribution (with fixed variance) and many exponential family distributions such as the Poisson, multinomial and exponential distribution. We next show that the isd posterior score for log-concave distributions is log-concave in Θ. This produces a unique estimate for the parameters as was the case for id and iid setups. Theorem 1 The isd posterior is log-concave for jointly log-concave density distributions and for log-concave prior distributions. Proof 1 The isd log-posterior is the sum of the id log-likelihoods, the singleton log-priors and pairwise log-Bhattacharyya affinities: log pλ(X, Θ) = const + X n log p(xn|θn) + X n log p(θn) + λ N X n X m̸=n log B(pm, pn). The id log-likelihood is the sum of the log-probabilities of distributions that are log-concave in the parameters and is therefore concave. Adding the log-priors maintains concavity since these are logconcave in the parameters. The Bhattacharyya affinities are log-concave by the following key result (Prekopa, 1973). The Bhattacharyya affinity for log-concave distributions is given by the integral over the sample space of the product of two distributions. Since the term in the integral is a product of jointly log-concave distributions (by assumption), the integrand is a jointly log-concave function. Integrating a log-concave function over some of its arguments produces a log-concave function in the remaining arguments (Prekopa, 1973). Therefore, the Bhattacharyya affinity is log-concave in the parameters of jointly log-concave distributions. Finally, since the isd log-posterior is the sum of concave terms and concave log-Bhattacharyya affinities, it must be concave. This log-concavity permits iterative and greedy maximization methods to reliably converge in practice. Furthermore, the isd setup will produce convenient update rules that build upon iid estimation algorithms. There are additional properties of isd which are detailed in the following sections. We first explore the β = 1/2 setting and subsequently discuss the β = 1 setting. 3 Exponential Family Distributions and β = 1/2 We first specialize the above derivations to the case where the singleton marginals obey the exponential family form as follows: p(x|θn) = exp H(x) + θT nT (x) −A(θn) . An exponential family distribution is specified by providing H, the Lebesgue-Stieltjes integrator, θn the vector of natural parameters, T , the sufficient statistic, and A the normalization factor (which is also known as the cumulant-generating function or the log-partition function). Tables of these values are shown in (Jebara et al., 2004). The function A is obtained by normalization (a Legendre transform) and is convex by construction. Therefore, exponential family distributions are always log-concave in the parameters θn. For the exponential family, the Bhattacharyya affinity is computable in closed form as follows: B(pm, pn) = exp (A(θm/2 + θn/2) −A(θm)/2 −A(θn)/2) . Assuming uniform priors on the exponential family parameters, it is now straightforward to write an iterative algorithm to maximize the isd posterior. We find settings of θ1, . . . , θN that maximize the isd posterior or log pλ(X, Θ) using a simple greedy method. Assume a current set of parameters is available ˜θ1, . . . , ˜θN. We then update a single θn to increase the posterior while all other parameters (denoted ˜Θ/n) remain fixed at their previous settings. It suffices to consider only terms in log pλ(X, Θ) that are variable with θn: log pλ(X, θn, ˜Θ/n) = const + θT nT (xn) −N + λ(N −1) N A(θn) + 2λ N X m̸=n A(˜θm/2 + θn/2). If the exponential family is jointly log-concave in parameters and data (as is the case for Gaussians), this term is log-concave in θn. Therefore, we can take a partial derivative of it with respect to θn and set to zero to maximize: A′(θn) = N N + λ(N −1) T (xn) + λ N X m̸=n A′(˜θm/2 + θn/2) . (2) For the Gaussian mean case (i.e. a white Gaussian with covariance locked at identity), we have A(θ) = θT θ. Then a closed-form formula is easy to recover from the above1. However, a simpler iterative update rule for θn is also possible as follows. Since A(θ) is a convex function, we can compute a linear variational lower bound on each A(θm/2 + θn/2) term for the current setting of θn: log pλ(X, θn, ˜Θ/n) ≥ const + θT nT (xn) −N + λ(N −1) N A(θn) + λ N X m̸=n 2A(˜θm/2 + ˜θn/2) + A′(˜θm/2 + ˜θn/2) T(θn −˜θn). This gives an iterative update rule of the form of Equation 2 where the θn on the right hand side is kept fixed at its previous setting (i.e. replace the right hand side θn with ˜θn) while the equation is iterated multiple times until the value of θn converges. Since we have a variational lower bound, each iterative update of θn monotonically increases the isd posterior. We can also work with a robust (yet not log-concave) version of the isd score which has the form: log ˆpλ(X, Θ) = const + X n log p(xn|θn) + X n log p(θn) + λ N X n log X m̸=n B(pm, pn) . and leads to the general update rule (where α = 0 reproduces isd and larger α increases robustness): A′(θn) = N N + λ(N −1) T (xn) + λ N X m̸=n (N −1)Bα(p(x|˜θm), p(x|˜θn)) P l̸=n Bα(p(x|˜θl), p(x|˜θn)) A′(˜θm/2 + ˜θn/2) . We next examine marginal consistency, another important property of the isd posterior. 1The update for the Gaussian mean with covariance=I is: θn = 1 N+λ(N−1)/2(Nxn + λ/2 P m̸=n ˜θm). 3.1 Marginal Consistency in the Gaussian Mean Case For marginal consistency, if a datum and model parameter are hidden and integrated over, this should not change our estimate. It is possible to show that the isd posterior is marginally consistent at least in the Gaussian mean case (one element of the exponential family). In other words, marginalizing over an observation and its associated marginal’s parameter (which can be taken to be xN and θN without loss of generality) still produces a similar isd posterior on the remaining observations X/N and parameters Θ/N. Thus, we need: Z Z pλ(X, Θ)dxNdθN ∝ pλ(X/N, Θ/N). We then would recover the posterior formed using the formula in Equation 1 with only N −1 observations and N −1 models. Theorem 2 The isd posterior with β = 1/2 is marginally consistent for Gaussian distributions. Proof 2 Start by integrating over xN: Z pλ(X, Θ)dxN ∝ N−1 Y i=1 p(xi|θi) N Y n=1 p(θn) N Y m=n+1 B2λ/N(pm, pn) Assume the singleton prior p(θN) is uniform and integrate over θN to obtain: Z Z pλ(X, Θ)dxNdθN ∝ N−1 Y i=1 p(xi|θi) N−1 Y n=1 N−1 Y m=n+1 B2λ/N(pm, pn) Z N−1 Y m=1 B2λ/N(pm, pN)dθN Consider only the right hand integral and impute the formula for the Bhattacharyya affinity: Z N−1 Y m=1 B2λ/N(pm, pN)dθN = Z exp 2λ N N−1 X m=1 A θm 2 + θN 2 −A(θm) 2 −A(θN) 2 ! dθN In the (white) Gaussian case A(θ) = θT θ which simplifies the above into: Z N−1 Y m=1 B2λ/N(pm, pN)dθN = Z exp −2λ N N−1 X m=1 A θm 2 −θN 2 ! dθN ∝ exp 2λ N(N −1) N−1 X n=1 N−1 X m=n+1 A θm 2 + θn 2 −A(θm) 2 −A(θn) 2 ! ∝ N−1 Y n=1 N−1 Y m=n+1 B 2λ N(N−1) (pm, pn) Reinserting the integral changes the exponent of the pairs of Bhattacharyya affinities between the (N −1) models raising it to the appropriate power λ/(N −1): Z Z pλ(X, Θ)dxNdθN ∝ N−1 Y i=1 p(xi|θi) N−1 Y n=1 N−1 Y m=n+1 B2λ/(N−1)(pm, pn) = pλ(X/N, Θ/N). Therefore, we get the same isd score that we would have obtained had we started with only (N −1) data points. We conjecture that it is possible to generalize the marginal consistency argument to other distributions beyond the Gaussian. The isd estimator thus has useful properties and still agrees with id when λ = 0 and iid when λ = ∞. Next, the estimator is generalized to handle distributions beyond the exponential family where latent variables are implicated (as is the case for mixtures of Gaussians, hidden Markov models, latent graphical models and so on). 4 Hidden Variable Models and β = 1 One important limitation of most divergences between distributions is that they become awkward when dealing with hidden variables or mixture models. This is because they may involve intractable integrals. The Bhattacharyya affinity with the setting β = 1, also known as the probability product kernel, is an exception to this since it only involves integrating the product of two distributions. In fact, it is known that this affinity is efficient to compute for mixtures of Gaussians, multinomials and even hidden Markov models (Jebara et al., 2004). This permits the affinity metric to efficiently pull together parameters θm and θn. However, for mixture models, there is the presence of hidden variables h in addition to observed variables. Therefore, we replace all the marginals p(x|θn) = P h p(x, h|θn). The affinity is still straightforward to compute for any pair of latent variable models (mixture models, hidden Markov models and so on). Thus, evaluating the isd posterior is straightforward for such models when β = 1. We next provide a variational method that makes it possible to maximize a lower bound on the isd posterior in these cases. Assume a current set of parameters is available ˜Θ = ˜θ1, . . . , ˜θN. We will find a new setting for θn that increases the posterior while all other parameters (denoted ˜Θ/n) remain fixed at their previous settings. It suffices to consider only terms in log pλ(X, Θ) that depend on θn. This yields: log pλ(X, θn, ˜Θ/n) = const + log p(xn|θn)p(θn) + 2λ N X m̸=n log Z p(x|˜θm)p(x|θn)dx ≥ const + log p(xn|θn)p(θn) + 2λ N X m̸=n Z p(x|˜θm) log p(x|θn)dx The application of Jensen’s inequality above produces an auxiliary function Q(θn|˜Θ/n) which is a lower-bound on the log-posterior. Note that each density function has hidden variables, p(xn|θn) = P h p(xn, h|θn). Applying Jensen’s inequality again (as in the Expectation-Maximization or EM algorithm) replaces the log-incomplete likelihoods over h with expectations over the complete posteriors given the previous parameters ˜θn. This gives isd the following auxiliary function Q(θn|˜Θ) = X h p(h|xn, ˜θn) log p(xn, h|θn) + log p(θn) + 2λ N X m̸=n Z p(x|˜θm) X h p(h|x, ˜θn) log p(x, h|θn)dx. This is a variational lower bound which can be iteratively maximized instead of the original isd posterior. While it is possible to directly solve for the maximum of Q(θn|˜Θ) in some mixture models, in practice, a further simplification is to replace the integral over x with synthesized samples drawn from p(x|˜θm). This leads to the following approximate auxiliary function (based on the law of large numbers) which is merely the update rule for EM for θn with s = 1, . . . , S virtual samples xm,s obtained from the m’th model p(x|˜θm) for each of the other N −1 models, ˜Q(θn|˜Θ) = X h p(h|xn, ˜θn) log p(xn, h|θn) + log p(θn) + 2λ SN X m̸=n X s X h p(h|xm,s, ˜θn) log p(xm,s, h|θn). We now have an efficient update rule for latent variable models (mixtures, hidden Markov models, etc.) which maximizes a lower bound on pλ(X, Θ). Unfortunately, as with most EM implementations, the arguments for log-concavity no longer hold. 5 Experiments A preliminary way to evaluate the usefulness of the isd framework is to explore density estimation over real-world datasets under varying λ. If we set λ large, we have the standard iid setup and only fit a single parametric model to the dataset. For small λ, we obtain the kernel density or Parzen estimator. In between, an iterative algorithm is available to maximize the isd posterior to obtain potentially superior models θ∗ 1, . . . , θ∗ N. Figure 1 shows the isd estimator with Gaussian models on a ring-shaped 2D dataset. The new estimator recovers the shape of the distribution more accurately. To evaluate performance on real data, we aggregate the isd learned models into a single density estimate as is done with Parzen estimators and compute the iid likelihood of held out test −2 −1 0 1 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −1.5 −1 −0.5 0 0.5 1 1.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −1.5 −1 −0.5 0 0.5 1 1.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −1.5 −1 −0.5 0 0.5 1 1.5 λ = 0, α = 0 λ = 1, α = 0 λ = 2, α = 0 λ = ∞, α = 0 −2 −1 0 1 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −1.5 −1 −0.5 0 0.5 1 1.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −1.5 −1 −0.5 0 0.5 1 1.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −1.5 −1 −0.5 0 0.5 1 1.5 λ = 0, α = 1 2 λ = 1, α = 1 2 λ = 2, α = 1 2 λ = ∞, α = 1 2 Figure 1: Estimation with isd for Gaussian models (mean and covariance) on synthetic data. Dataset id iid-1 iid-2 iid-3 iid-4 iid-5 iid-∞ isd α = 0 isd α = 1 2 SPIRAL -5.61e3 -1.36e3 -1.36e3 -1.19e3 -7.98e2 -6.48e2 -4.86e2 -2.26e2 -1.19e2 MIT-CBCL -9.82e2 -1.39e3 -1.19e3 -1.00e3 -1.01e3 -1.10e3 -3.14e3 -9.79e2 -9.79e2 HEART -1.94e3 -2.02e4 -3.23e4 -2.50e4 -1.68e4 -3.15e4 -4.02e2 -4.51e2 -4.47e2 DIABETES -6.25e3 -2.12e5 -2.85e5 -4.48e5 -2.03e5 -3.40e5 -8.22e2 -8.28e2 -8.09e2 CANCER -5.80e3 -7.22e6 -2.94e6 -3.92e6 -4.08e6 -3.96e6 -1.22e2 -5.54e2 -5.54e2 LIVER -3.41e3 -2.53e4 -1.88e4 -2.79e4 -2.62e4 -3.23e4 -4.56e2 -4.74e2 -4.69e2 Table 1: Gaussian test log-likelihoods using id, iid, EM, ∞GMM and isd estimation. data via P τ log 1 N P n p(xτ|θ∗ n) . A larger score implies a better p(x) density estimate. Table 1 summarizes experiments with the Gaussian (mean and covariance) models. On 6 standard datasets, we show the average test log-likelihood of Gaussian estimation while varying the settings of λ compared to a single iid Gaussian, an id Parzen RBF estimator and a mixture of 2 to 5 Gaussians using EM. Comparisons with (Rasmussen, 1999) are also shown. Cross-validation was used to choose the σ, λ or EM local minimum (from ten initializations), for the id, isd and EM algorithms respectively. Train, cross-validation and test split sizes where 80%, 10% and 10% respectively. The test log-likelihoods show that isd outperformed iid, id and EM estimation and was comparable to infinite Gaussian mixture (iid−∞) models (Rasmussen, 1999) (which is a far more computationally demanding method). In another synthetic experiment with hidden Markov models, 40 sequences of 8 binary symbols were generated using 2 state HMMs with 2 discrete emissions. However, the parameters generating the HMMs were allowed to slowly drift during sampling (i.e. not iid). The data was split into 20 training and 20 testing examples. Table 2 shows that the isd estimator for certain values of λ produced higher test log-likelihoods than id and iid. 6 Discussion This article has provided an isd scheme to smoothly interpolate between id and iid assumptions in density estimation. This is done by penalizing divergence between pairs of models using a Bhattacharyya affinity. The method maintains simple update rules for recoveringparameters for exponential families as well as mixture models. In addition, the isd posterior maintains useful log-concavity and marginal consistency properties. Experiments show its advantages in real-world datasets where id or iid assumptions may be too extreme. Future work involves extending the approach into other aspects of unsupervised learning such as clustering. We are also considering computing the isd posλ = 0 λ = 1 λ = 2 λ = 3 λ = 4 λ = 5 λ = 10 λ = 20 λ = 30 λ = ∞ -5.7153 -5.5875 -5.5692 -5.5648 -5.5757 -5.5825 -5.5849 -5.5856 -5.6152 -5.5721 Table 2: HMM test log-likelihoods using id, iid and isd estimation. terior with a normalizing constant which depends on λ and thus permits a direct estimate of λ by maximization instead of cross-validation2. 7 Appendix: Alternative Information Divergences There is a large family of information divergences (Topsoe, 1999) between pairs of distributions (Renyi measure, variational distance, χ2 divergence, etc.) that can be used to pull models pm and pn towards each other. The Bhattacharya, though, is computationally easier to evaluate and minimize over a wide range of probability models (exponential families, mixtures and hidden Markov models). An alternative is the Kullback-Leibler divergence D(pm∥pn) = R pm(x)(log pm(x)−log pn(x))dx and its symmetrized variant D(pm∥pn)/2 + D(pn∥pm)/2. The Bhattacharyya affinity is related to the symmetrized variant of KL. Consider a variational distribution q that lies between the input pm and pn. The log Bhattacharyya affinity with β = 1/2 can be written as follows: log B(pm, pn) = log Z q(x) p pm(x)pn(x) q(x) dx ≥−D(q∥pm)/2 −D(q∥pn)/2. Thus, B(pm, pn) ≥exp(−D(q∥pm)/2 −D(q∥pn)/2). The choice of q that maximizes the lower bound on the Bhattacharyya is q(x) = 1 Z p pm(x)pn(x). Here, Z = B(pm, pn) normalizes q(x) and is therefore equal to the Bhattacharyya affinity. Thus we have the following property: −2 log B(pm, pn) = min q D(q∥pm) + D(q∥pn). It is interesting to note that the Jensen-Shannon divergence (another symmetrized variant of KL) emerges by placing the variational q distribution as the second argument in the divergences: 2JS(pm, pn) = D(pm∥pm/2 + pn/2) + D(pn∥pm/2 + pn/2) = min q D(pm∥q) + D(pn∥q). Simple manipulations then show 2JS(pm, pn) ≤min(D(pm∥pn), D(pn∥pm)). Thus, there are close ties between Bhattacharyya, Jensen-Shannon and symmetrized KL divergences. References Bengio, Y., Larochelle, H., & Vincent, P. (2005). Non-local manifold Parzen windows. Neural Information Processing Systems. Bhattacharyya, A. (1943). On a measure of divergence between two statistical populations defined by their probability distributions. Bull. Calcutta Math Soc. Collins, M., Dasgupta, S., & Schapire, R. (2002). A generalization of principal components analysis to the exponential family. NIPS. Devroye, L., & Gyorfi, L. (1985). Nonparametric density estimation: The l1 view. John Wiley. Efron, B., & Tibshirani, R. (1996). Using specially designed exponential families for density estimation. The Annals of Statistics, 24, 2431–2461. Hjort, N., & Glad, I. (1995). Nonparametric density estimation with a parametric start. The Annals of Statistics, 23, 882–904. Jebara, T., Kondor, R., & Howard, A. (2004). Probability product kernels. Journal of Machine Learning Research, 5, 819–844. Naito, K. (2004). Semiparametric density estimation by local l2-fitting. The Annals of Statistics, 32, 1162– 1192. Olking, I., & Spiegelman, C. (1987). A semiparametric approach to density estimation. Journal of the American Statistcal Association, 82, 858–865. Prekopa, A. (1973). On logarithmic concave measures and functions. Acta. Sci. Math., 34, 335–343. Rasmussen, C. (1999). The infinite Gaussian mixture model. NIPS. Silverman, B. (1986). Density estimation for statistics and data analysis. Chapman and Hall: London. Teh, Y., Jordan, M., Beal, M., & Blei, D. (2004). Hierarchical Dirichlet processes. NIPS. Topsoe, F. (1999). Some inequalities for information divergence and related measures of discrimination. Journal of Inequalities in Pure and Applied Mathematics, 2. Wand, M., & Jones, M. (1995). Kernel smoothing. CRC Press. 2Work supported in part by NSF Award IIS-0347499 and ONR Award N000140710507.
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Efficient Inference for Distributions on Permutations Jonathan Huang Carnegie Mellon University jch1@cs.cmu.edu Carlos Guestrin Carnegie Mellon University guestrin@cs.cmu.edu Leonidas Guibas Stanford University guibas@cs.stanford.edu Abstract Permutations are ubiquitous in many real world problems, such as voting, rankings and data association. Representing uncertainty over permutations is challenging, since there are n! possibilities, and typical compact representations such as graphical models cannot efficiently capture the mutual exclusivity constraints associated with permutations. In this paper, we use the “low-frequency” terms of a Fourier decomposition to represent such distributions compactly. We present Kronecker conditioning, a general and efficient approach for maintaining these distributions directly in the Fourier domain. Low order Fourier-based approximations can lead to functions that do not correspond to valid distributions. To address this problem, we present an efficient quadratic program defined directly in the Fourier domain to project the approximation onto a relaxed form of the marginal polytope. We demonstrate the effectiveness of our approach on a real camera-based multi-people tracking setting. 1 Introduction Permutations arise naturally in a variety of real situations such as card games, data association problems, ranking analysis, etc. As an example, consider a sensor network that tracks the positions of n people, but can only gather identity information when they walk near certain sensors. Such mixed-modality sensor networks are an attractive alternative to exclusively using sensors which can measure identity because they are potentially cheaper, easier to deploy, and less intrusive. See [1] for a real deployment. A typical tracking system maintains tracks of n people and the identity of the person corresponding to each track. What makes the problem difficult is that identities can be confused when tracks cross in what we call mixing events. Maintaining accurate track-to-identity assignments in the face of these ambiguities based on identity measurements is known as the Identity Management Problem [2], and is known to be NP-hard. Permutations pose a challenge for probabilistic inference, because distributions on the group of permutations on n elements require storing at least n! −1 numbers, which quickly becomes infeasible as n increases. Furthermore, typical compact representations, such as graphical models, cannot capture the mutual exclusivity constraints associated with permutations. Diaconis [3] proposes maintaining a small subset of Fourier coefficients of the actual distribution allowing for a principled tradeoff between accuracy and complexity. Schumitsch et al. [4] use similar ideas to maintain a particular subset of Fourier coefficients of the log probability distribution. Kondor et al. [5] allow for general sets of coefficients, but assume a restrictive form of the observation model in order to exploit an efficient FFT factorization. The main contributions of this paper are: • A new, simple and general algorithm, Kronecker Conditioning, which performs all probabilistic inference operations completely in the Fourier domain. Our approach is general, in the sense that it can address any transition model or likelihood function that can be represented in the Fourier domain, such as those used in previous work, and can represent the probability distribution with any desired set of Fourier coefficients. • We show that approximate conditioning can sometimes yield Fourier coefficients which do not correspond to any valid distribution, and present a method for projecting the result back onto a relaxation of the marginal polytope. • We demonstrate the effectiveness of our approach on a real camera-based multi-people tracking setting. 1 2 Filtering over permutations In identity management, a permutation σ represents a joint assignment of identities to internal tracks, with σ(i) being the track belonging to the ith identity. When people walk too closely together, their identities can be confused, leading to uncertainty over σ. To model this uncertainty, we use a Hidden Markov Model on permutations, which is a joint distribution over P(σ(1), . . . , σ(T ), z(1), . . . , z(T )) which factors as: P(σ(1), . . . , σ(T ), z(1), . . . , z(T )) = P(z(1)|σ(1)) Y t P(zt|σ(t)) · P(σ(t)|σ(t−1)), where the σ(t) are latent permutations and the z(t) denote observed variables. The conditional probability distribution P(σ(t)|σ(t−1)) is called the transition model, and might reflect for example, that the identities belonging to two tracks were swapped with some probability. The distribution P(z(t)|σ(t)) is called the observation model, which might capture a distribution over the color of clothing for each individual. We focus on filtering, in which one queries the HMM for the posterior at some timestep, conditioned on all past observations. Given the distribution P(σ(t)|z(1), . . . , z(t)), we recursively compute P(σ(t+1)|z(1), . . . , z(t+1)) in two steps: a prediction/rollup step and a conditioning step. The first updates the distribution by multiplying by the transition model and marginalizing out the previous timestep: P(σ(t+1)|z(1), . . . , z(t)) = P σ(t) P(σ(t+1)|σ(t))P(σ(t)|z(1), . . . , z(t)). The second conditions the distribution on an observation z(t+1) using Bayes rule: P(σ(t+1)|z(1), . . . , z(t+1)) ∝ P(z(t+1)|σ(t+1))P(σ(t+1)|z(1), . . . , z(t)). Since there are n! permutations, a single update requires O((n!)2) flops and is consequently intractable for all but very small n. The approach that we advocate is to maintain a compact approximation to the true distribution based on the Fourier transform. As we discuss later, the Fourier based approximation is equivalent to maintaining a set of low-order marginals, rather than the full joint, which we regard as being analagous to an Assumed Density Filter [6]. 3 Fourier projections of functions on the Symmetric Group Over the last 50 years, the Fourier Transform has been ubiquitously applied to everything digital, particularly with the invention of the Fast Fourier Transform. On the real line, the Fourier Transform is a well-studied method for decomposing a function into a sum of sine and cosine terms over a spectrum of frequencies. Perhaps less familiar, is its group theoretic generalization, which we review in this section with an eye towards approximating functions on the group of permutations, the Symmetric Group. For permutations on n objects, the Symmetric Group will be abbreviated by Sn. The formal definition of the Fourier Transform relies on the theory of group representations, which we briefly discuss first. Our goal in this section is to motivate the idea that the Fourier transform of a distribution P is related to certain marginals of P. For references on this subject, see [3]. Definition 1. A representation of a group G is a map ρ from G to a set of invertible dρ × dρ matrix operators which preserves algebraic structure in the sense that for all σ1, σ2 ∈ G, ρ(σ1σ2) = ρ(σ1) · ρ(σ2). The matrices which lie in the image of this map are called the representation matrices, and we will refer to dρ as the degree of the representation. Representations play the role of basis functions, similar to that of sinusoids, in Fourier theory. The simplest basis functions are constant functions — and our first example of a representation is the trivial representation ρ0 : G →R which maps every element of G to 1. As a more pertinent example, we define the 1st order permutation representation of Sn to be the degree n representation, τ1, which maps a permutation σ to its corresponding permutation matrix given by: [τ1(σ)]ij = 1 {σ(j) = i}. For example, the permutation in S3 which swaps the second and third elements maps to: τ1(1 7→1, 2 7→3, 3 7→2) = 0 @ 1 0 0 0 0 1 0 1 0 1 A . The τ1 representation can be thought of as a collection of n2 functions at once, one for each matrix entry, [τ1(σ)]ij. There are other possible permutation representations - for example the 2nd order unordered permutation representation, τ2, is defined by the action of a permutation on unordered pairs of objects, ([ρ(σ)]{i,j},{ℓ,k} = 1 {σ({ℓ, k}) = {i, j}}), and is a degree n(n−1) 2 representation. And the list goes on to include many more complicated representations. 2 It is useful to think of two representations as being the same if the representation matrices are equal up to some consistent change of basis. This idea is formalized by declaring two representations ρ and τ to be equivalent if there exists an invertible matrix C such that C−1 · ρ(σ) · C = τ(σ) for all σ ∈G. We write this as ρ ≡τ. Most representations can be seen as having been built up by smaller representations. We say that a representation ρ is reducible if there exist smaller representations ρ1, ρ2 such that ρ ≡ρ1 ⊕ρ2 where ⊕is defined to be the direct sum representation: ρ1 ⊕ρ2(g) ≜ „ ρ1(g) 0 0 ρ2(g) « . (1) In general, there are infinitely many inequivalent representations. However, for any finite group, there is always a finite collection of atomic representations which can be used to build up any other representation using direct sums. These representations are referred to as the irreducibles of a group, and they are simply the collection of representations which are not reducible. We will refer to the set of irreducibles by R. It can be shown that any representation of a finite group G is equivalent to a direct sum of irreducibles [3], and hence, for any representation τ, there exists a matrices C for which C−1 · τ · C = ⊕ρi∈R ⊕ρi, where the inner ⊕refers to some finite number of copies of the irreducible ρi. Describing the irreducibles of Sn up to equivalence is a subject unto itself; We will simply say that there is a natural way to order the irreducibles of Sn that corresponds to ‘simplicity’ in the same way that low frequency sinusoids are simpler than higher frequency ones. We will refer to the irreducibles in this order as ρ0, ρ1, . . . . For example, the first two irreducibles form the first order permutation representation (τ1 ≡ρ0 ⊕ρ1), and the second order permutation representation can be formed by the first 3 irreducibles. Irreducible representation matrices are not always orthogonal, but they can always be chosen to be so (up to equivalence). For notational convenience, the irreducible representations in this paper will always be assumed to be orthogonal. 3.1 The Fourier transform On the real line, the Fourier Transform corresponds to computing inner products of a function with sines and cosines at varying frequencies. The analogous definition for finite groups replaces the sinusoids by group representations. Definition 2. Let f : G →R be any function on a group G and let ρ be any representation on G. The Fourier Transform of f at the representation ρ is defined to be: ˆfρ = P σ f(σ)ρ(σ). There are two important points which distinguish this Fourier Transform from the familiar version on the real line — it is matrix-valued, and instead of real numbers, the inputs to ˆf are representations of G. The collection of Fourier Transforms of f at all irreducibles form the Fourier Transform of f. As in the familiar case, there is an inverse transform given by: f(σ) = 1 |G| X k dρkTr h ˆf T ρk · ρk(σ) i , (2) where k indexes over the collection of irreducibles of G. We provide two examples for intuition. For functions on the real line, the Fourier Transform at zero gives the DC component of a signal. This is also true for functions on a group; If f : G →R is any function, then the Fourier Transform of f at the trivial representation is constant with ˆfρ0 = P σ f(σ). Thus, for any probability distribution P, we have ˆPρ0 = 1. If P were the uniform distribution, then ˆPρ = 0 at all irreducibles except at the trivial representation. The Fourier Transform at τ1 also has a simple interpretation: [ ˆfτ1]ij = X σ∈Sn f(σ)[τ1(σ)]ij = X σ∈Sn f(σ)1 {σ(j) = i} = X σ:σ(j)=i f(σ). Thus, if P is a distribution, then ˆPτ1 is a matrix of marginal probabilties, where the ij-th element is the marginal probability that a random permutation drawn from P maps element j to i. Similarly, the Fourier transform of P at the second order permutation representation is a matrix of marginal probabilities of the form P(σ({i, j}) = {k, ℓ}). 3 In Section 5, we will discuss function approximation by bandlimiting the Fourier coefficients, but this example should illustrate the fact that maintaining Fourier coefficients at low-order irreducibles is the same as maintaining low-order marginal probabilities, while higher order irreducibles correspond to more complicated marginals. 4 Inference in the Fourier domain Bandlimiting allows for compactly storing a distribution over permutations, but the idea is rather moot if it becomes necessary to transform back to the primal domain each time an inference operation is called. Naively, the Fourier Transform on Sn scales as O((n!)2), and even the fastest Fast Fourier Transforms for functions on Sn are no faster than O(n! log(n!)) (see [7] for example). To resolve this issue, we present a formulation of inference which operates solely in the Fourier domain, allowing us to avoid a costly transform. We begin by discussing exact inference in the Fourier domain, which is no more tractable than the original problem because there are n! Fourier coefficients, but it will allow us to discuss the bandlimiting approximation in the next section. There are two operations to consider: prediction/rollup, and conditioning. The assumption for the rest of this section is that the Fourier Transforms of the transition and observation models are known. We discuss methods for obtaining the models in Section 7. 4.1 Fourier prediction/rollup We will consider one particular type of transition model — that of a random walk over a group. This model assumes that σ(t+1) is generated from σ(t) by drawing a random permutation τ (t) from some distribution Q(t) and setting σ(t+1) = τ (t)σ(t). In our identity management example, τ (t) represents a random identity permutation that might occur among tracks when they get close to each other (a mixing event), but the random walk model appears in other applications such as modeling card shuffles [3]. The Fourier domain Prediction/Rollup step is easily formulated using the convolution theorem (see also [3]): Proposition 3. Let Q and P be probability distributions on Sn. Define the convolution of Q and P to be the function [Q ∗P] (σ1) = P σ2 Q(σ1 ·σ−1 2 )P(σ2). Then for any representation ρ, h \ Q ∗P i ρ = bQρ · bPρ, where the operation on the right side is matrix multiplication. The Prediction/Rollup step for the random walk transition model can be written as a convolution: P(σ(t+1)) = X {(σ(t),τ(t)) : σ(t+1)=τ(t)·σ(t)} Q(t)(τ (t))·P(σ(t)) = X σ(t) Q(t)(σ(t+1)·(σ(t))−1)P(σ(t)) = h Q(t) ∗P i (σ(t+1)). Then assuming that bP (t) ρ and bQ(t) ρ are given, the prediction/rollup update rule is simply: bP (t+1) ρ ←bQ(t) ρ · bP (t) ρ . Note that the update requires only knowledge of ˆP and does not require P. Furthermore, the update is pointwise in the Fourier domain in the sense that the coefficients at the representation ρ affect bP (t+1) ρ only at ρ. 4.2 Fourier conditioning An application of Bayes rule to find a posterior distribution P(σ|z) after observing some evidence z requires a pointwise product of likelihood L(z|σ) and prior P(σ), followed by a normalization step. We showed earlier that the normalization constant P σ L(z|σ) · P(σ) is given by the Fourier transform of \ L(t)P (t) at the trivial representation — and therefore the normalization step of conditioning can be implemented by simply dividing each Fourier coefficient by the scalar h \ L(t)P (t) i ρ0 . The pointwise product of two functions f and g, however, is trickier to formulate in the Fourier domain. For functions on the real line, the pointwise product of functions can be implemented by convolving the Fourier coefficients of ˆf and ˆg, and so a natural question is: can we apply a similar operation for functions over other groups? Our answer to this is that there is an analogous (but more complicated) notion of convolution in the Fourier domain of a general finite group. We present a convolution-based conditioning algorithm which we call Kronecker Conditioning, which, in contrast to the pointwise nature of the Fourier Domain prediction/rollup step, and much like convolution, smears the information at an irreducible ρk to other irreducibles. 4 Fourier transforming the pointwise product Our approach to Fourier Transforming the pointwise product in terms of ˆf and ˆg is to manipulate the function f(σ)g(σ) so that it can be seen as the result of an inverse Fourier Transform. Hence, the goal will be to find matrices Ak (as a function of ˆf, ˆg) such that for any σ ∈G, f(σ) · g(σ) = 1 |G| X k dρkTr “ AT k · ρk(σ) ” , (3) where Ak = h c fg i ρk . For any σ ∈G we can write the pointwise product in terms ˆf and ˆg using the inverse Fourier Transform (Equation 2): f(σ) · g(σ) = " 1 |G| X i dρiTr “ ˆf T ρi · ρi(σ) ”# · " 1 |G| X j dρjTr “ ˆgT ρj · ρj(σ) ”# = „ 1 |G| «2 X i,j dρidρj h Tr “ ˆf T ρi · ρi(σ) ” · Tr “ ˆgT ρj · ρj(σ) ”i . (4) Now we want to manipulate this product of traces in the last line to be just one trace (as in Equation 3), by appealing to some properties of the matrix Kronecker product. The connection to the pointwise product (first observed in [8]), lies in the property that for any matrices U, V , Tr (U ⊗V ) = (Tr U) · (Tr V ). Applying this to Equation 4, we have: Tr “ ˆf T ρi · ρi(σ) ” · Tr “ ˆgT ρj · ρj(σ) ” = Tr ““ ˆf T ρi · ρi(σ) ” ⊗ “ ˆgT ρj · ρj(σ) ”” = Tr „“ ˆfρi ⊗ˆgρj ”T · (ρi(σ) ⊗ρj(σ)) « , (5) where the last line follows by standard matrix properties. The term on the right, ρi(σ) ⊗ρj(σ), itself happens to be a representation, called the Kronecker Product Representation. In general, the Kronecker Product representation is reducible, and so it can decomposed into a direct sum of irreducibles. This means that if ρi and ρj are any two irreducibles of G, there exists a similarity transform Cij such that for any σ ∈G, C−1 ij · [ρi ⊗ρj] (σ) · Cij = M k zijk M ℓ=1 ρk(σ). The ⊕symbols here refer to a matrix direct sum as in Equation 1, k indexes over all irreducible representations of Sn, while ℓindexes over a number of copies of ρk which appear in the decomposition. We index blocks on the right side of this equation by pairs of indices (k, ℓ). The number of copies of each ρk is denoted by the integer zijk, the collection of which, taken over all triples (i, j, k), are commonly referred to as the Clebsch-Gordan series. Note that we allow the zijk to be zero, in which case ρk does not contribute to the direct sum. The matrices Cij are known as the Clebsch-Gordan coefficients. The Kronecker Product Decomposition problem is that of finding the irreducible components of the Kronecker product representation, and thus to find the Clebsch-Gordan series/coefficients for each pair of representations (ρi, ρj). Decomposing the Kronecker product inside Equation 5 using the Clebsch-Gordan series/coefficients yields the desired Fourier Transform, which we summarize here: Proposition 4. Let ˆf, ˆg be the Fourier Transforms of functions f and g respectively, and for each ordered pair of irreducibles (ρi, ρj), define the matrix: Aij ≜C−1 ij · ˆfρi ⊗ˆgρj · Cij. Then the Fourier tranform of the pointwise product fg is: h c fg i ρk = 1 dρk|G| X ij dρidρj zijk X ℓ=1 Akℓ ij , (6) where Akℓ ij is the block of Aij corresponding to the (k, ℓ) block in ⊕k ⊕zijk ℓ ρk. See the Appendix for a full proof of Proposition 4. The Clebsch-Gordan series, zijk, plays an important role in Equation 6, which says that the (ρi, ρj) crossterm contributes to the pointwise product at ρk only when zijk > 0. For example, ρ1 ⊗ρ1 ≡ρ0 ⊕ρ1 ⊕ρ2 ⊕ρ3. (7) So z1,1,k = 1 for k ≤3 and is zero otherwise. 5 Unfortunately, there are no analytical formulas for finding the Clebsch-Gordan series or coefficients, and in practice, these computations can take a long time. We emphasize however, that as fundamental quantities, like the digits of π, they need only be computed once and stored in a table for future reference. Due to space limitations, we will not provide complete details on computing these numbers. We refer the reader to Murnaghan [9], who provides general formulas for computing ClebschGordan series for pairs of low-order irreducibles, and to Appendix 1 for details about computing Clebsch-Gordan coefficients. We will also make precomputed coefficients available on the web. 5 Approximate inference by bandlimiting We approximate the probability distribution P(σ) by fixing a bandlimit B and maintaining the Fourier transform of P only at irreducibles ρ0, . . . ρB. We refer to this set of irreducibles as B. As on the real line, smooth functions are generally well approximated by only a few Fourier coefficients, while “wigglier” functions require more. For example, when B = 3, B is the set ρ0, ρ1, ρ2, and ρ3, which corresponds to maintaining marginal probabilities of the form P(σ((i, j)) = (k, ℓ)). During inference, we follow the procedure outlined in the previous section but ignore the higher order terms which are not maintained. Pseudocode for bandlimited prediction/rollup and Kronecker conditioning is given in Figures 1 and 2. Since the Prediction/Rollup step is pointwise in the Fourier domain, the update is exact for the maintained irreducibles because higher order irreducibles cannot affect those below the bandlimit. As in [5], we find that the error from bandlimiting creeps in through the conditioning step. For example, Equation 7 shows that if B = 1 (so that we maintain first-order marginals), then the pointwise product spreads information to second-order marginals. Conversely, pairs of higher-order irreducibles may propagate information to lower-order irreducibles. If a distribution is diffuse, then most of the energy is stored in low-order Fourier coefficients anyway, and so this is not a big problem. However, it is when the distribution is sharply concentrated at a small subset of permutations, that the low-order Fourier projection is unable to faithfully approximate the distribution, in many circumstances, resulting in a bandlimited Fourier Transform with negative “marginal probabilities”! To combat this problem, we present a method for enforcing nonnnegativity. Projecting to a relaxed marginal polytope The marginal polytope, M, is the set of marginals which are consistent with some joint distribution over permutations. We project our approximation onto a relaxation of the marginal polytope, M′, defined by linear inequality constraints that marginals be nonnegative, and linear equality constraints that they correspond to some legal Fourier transform. Intuitively, our relaxation produces matrices of marginals which are doubly stochastic (rows and columns sum to one and all entries are nonnegative), and satisfy lower-order marginal consistency (different high-order marginals are consistent at lower orders). After each conditioning step, we apply a ‘correction’ to the approximate posterior P (t) by finding the bandlimited function in M′ which is closest to P (t) in an L2 sense. To perform the projection, we employ the Plancherel Theorem [3] which relates the L2 distance between functions on Sn to a distance metric in the Fourier domain. Proposition 5. X σ (f(σ) −g(σ))2 = 1 |G| X k dρkTr ˆfρk −ˆgρk T · ˆfρk −ˆgρk . (8) We formulate the optimization as a quadratic program where the objective is to minimize the right side of Equation 8 — the sum is taken only over the set of maintained irreducibles, B, and subject to the linear constraints which define M′. We remark that even though the projection will always produce a Fourier transform corresponding to nonnegative marginals, there might not necessarily exist a joint probability distribution on Sn consistent with those marginals. In the case of first-order marginals, however, the existence of a consistent joint distribution is guaranteed by the Birkhoff-von Neumann theorem [10], which states that a matrix is doubly stochastic if and only if it can be written as a convex combination of permutation matrices. And so for the case of first-order marginals, our relaxation is in fact, exact. 6 Related Work The Identity Management problem was first introduced in [2] which maintains a doubly stochastic first order belief matrix to reason over data associations. Schumitsch et al. [4] exploits a similar idea, but formulated the problem in log-space. 6 Figure 1: Pseudocode for the Fourier Prediction/Rollup Algorithm. PREDICTIONROLLUP foreach ρk ∈B do ˆP (t+1) ρk ←ˆQ(t) ρk · ˆP (t) ρk ; Figure 2: Pseudocode for the Kronecker Conditioning Algorithm. KRONECKERCONDITIONING foreach ρk ∈B do h \ L(t)P (t) i ρk ←0 //Initialize Posterior //Pointwise Product foreach ρi ∈B do foreach ρj ∈B do z ←CGseries(ρi, ρj) ; Cij ←CGcoefficients(ρi, ρj) ; Aij ←CT ij · “ ˆfρi ⊗ˆgρj ” · Cij ; for ρk ∈B such that zijk ̸= 0 do for ℓ= 1 to zk do h \ L(t)P (t) i ρk ← h \ L(t)P (t) i ρk + dρi dρj dρk n! Akℓ ij //Akℓ ij is the (k, ℓ) block of Aij Z ← h \ L(t)P (t) i ρ0 ; foreach ρk ∈B do h \ L(t)P (t) i ρk ←1 Z h \ L(t)P (t) i ρk //Normalization Kondor et al. [5] were the first to show that the data association problem could be approximately handled via the Fourier Transform. For conditioning, they exploit a modified FFT factorization which works on certain simplified observation models. Our approach generalizes the type of observations that can be handled in [5] and is equivalent in the simplified model that they present. We require O(D3n2) time in their setting. Their FFT method saves a factor of D due to the fact that certain representation matrices can be shown to be sparse. Though we do not prove it, we observe that the Clebsch-Gordan coefficients, Cij are typically similarly sparse, which yields an equivalent running time in practice. In addition, Kondor et al. do not address the issue of projecting onto valid marginals, which, as we show in our experimental results, is fundamental in practice. Willsky [8] was the first to formulate a nonabelian version of the FFT algorithm (for Metacyclic groups) as well as to note the connection between pointwise products and Kronecker product decompositions for general finite groups. In this paper, we address approximate inference, which is necessary given the n! complexity of inference for the Symmetric group. 7 Experimental results For small n, we compared our algorithm to exact inference on synthetic datasets in which tracks are drawn at random to be observed or swapped. For validation we measure the L1 distance between true and approximate marginal distributions. In (Fig. 3(a)), we call several mixings followed by a single observation, after which we measured error. As expected, the Fourier approximation is better when there are either more mixing events, or when more Fourier coefficients are maintained. In (Fig. 3(b)) we allow for consecutive conditioning steps and we see that that the projection step is fundamental, especially when mixing events are rare, reducing the error dramatically. Comparing running times, it is clear that our algorithm scales gracefully compared to the exact solution (Fig. 3(c)). We also evaluated our algorithm on data taken from a real network of 8 cameras (Fig. 3(d)). In the data, there are n = 11 people walking around a room in fairly close proximity. To handle the fact that people can freely leave and enter the room, we maintain a list of the tracks which are external to the room. Each time a new track leaves the room, it is added to the list and a mixing event is called to allow for m2 pairwise swaps amongst the m external tracks. The number of mixing events is approximately the same as the number of observations. For each observation, the network returns a color histogram of the blob associated with one track. The task after conditioning on each observation is to predict identities for all tracks inside the room, and the evaluation metric is the fraction of accurate predictions. We compared against a baseline approach of predicting the identity of a track based on the most recently observed histogram at that track. This approach is expected to be accurate when there are many observations and discriminative appearance models, neither of which our problem afforded. As (Fig. 3(e)) shows, 7 0 5 10 15 0 0.02 0.04 0.06 0.08 0.1 Error of Kronecker Conditioning, n=8 # Mixing Events L1 error at 1st order marginals b=1 b=2 b=3 (a) Error of Kronecker Conditioning 0.2 0.4 0.6 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 Projection versus No Projection (n=6) Fraction of Observation events L1 error at 1st order Marginals Averaged over 250 timesteps b=1, w/o Projection b=2, w/o Projection b=3, w/o Projection b=1, w/Projection b=2, w/Projection b=3, w/Projection b=0 (Uniform distribution) (b) Projection vs. No Projection 4 5 6 7 8 0 1 2 3 4 5 Running time of 10 forward algorithm iterations n Running time in seconds b=1 b=2 b=3 exact (c) n versus Running Time (d) Sample Image 0 10 20 30 40 50 60 % Tracks correctly Identified Baseline w/o Projection w/Projection Omniscient (e) Accuracy for Camera Data Figure 3: Evaluation on synthetic ((a)-(c)) and real camera network ((d),(e)) data. both the baseline and first order model(without projection) fared poorly, while the projection step dramatically boosted the accuracy. To illustrate the difficulty of predicting based on appearance alone, the rightmost bar reflects the performance of an omniscient tracker who knows the result of each mixing event and is therefore left only with the task of distinguishing between appearances. 8 Conclusions We presented a formulation of hidden Markov model inference in the Fourier domain. In particular, we developed the Kronecker Conditioning algorithm which performs a convolution-like operation on Fourier coefficients to find the Fourier transform of the posterior distribution. We argued that bandlimited conditioning can result in Fourier coefficients which correspond to no distribution, but that the problem can be remedied by projecting to a relaxation of the marginal polytope. Our evaluation on data from a camera network shows that our methods outperform well when compared to the optimal solution in small problems, or to an omniscient tracker in large problems. Furthermore, we demonstrated that our projection step is fundamental to obtaining these high-quality results. We conclude by remarking that the mathematical framework developed in this paper is quite general. In fact, both the prediction/rollup and conditioning formulations hold over any finite group, providing a principled method for approximate inference for problems with underlying group structure. Acknowledgments This work is supported in part by the ONR under MURI N000140710747, the ARO under grant W911NF-06-1-0275, the NSF under grants DGE-0333420, EEEC-540865, Nets-NOSS 0626151 and TF 0634803, and by the Pennsylvania Infrastructure Technology Alliance (PITA). Carlos Guestrin was also supported in part by an Alfred P. Sloan Fellowship. We thank Kyle Heath for helping with the camera data and Emre Oto, and Robert Hough for valuable discussions. References [1] Y. Ivanov, A. Sorokin, C. Wren, and I. Kaur. Tracking people in mixed modality systems. Technical Report TR2007-11, MERL, 2007. [2] J. Shin, L. Guibas, and F. Zhao. A distributed algorithm for managing multi-target identities in wireless ad-hoc sensor networks. In IPSN, 2003. [3] P. Diaconis. Group Representations in Probability and Statistics. IMS Lecture Notes, 1988. [4] B. Schumitsch, S. Thrun, G. Bradski, and K. Olukotun. The information-form data association filter. In NIPS. 2006. [5] R. Kondor, A. Howard, and T. Jebara. Multi-object tracking with representations of the symmetric group. In AISTATS, 2007. [6] X. Boyen and D. Koller. Tractable inference for complex stochastic processes. In UAI, 1998. [7] R. Kondor. Snob: a C++ library for fast Fourier transforms on the symmetric group, 2006. Available at http://www.cs.columbia.edu/˜risi/Snob/. [8] A. Willsky. On the algebraic structure of certain partially observable finite-state markov processes. Information and Control, 38:179–212, 1978. [9] F.D. Murnaghan. The analysis of the kronecker product of irreducible representations of the symmetric group. American Journal of Mathematics, 60(3):761–784, 1938. [10] J. van Lint and R.M. Wilson. A Course in Combinatorics. Cambridge University Press, 2001. 8
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Theoretical Analysis of Heuristic Search Methods for Online POMDPs St´ephane Ross McGill University Montr´eal, Qc, Canada sross12@cs.mcgill.ca Joelle Pineau McGill University Montr´eal, Qc, Canada jpineau@cs.mcgill.ca Brahim Chaib-draa Laval University Qu´ebec, Qc, Canada chaib@ift.ulaval.ca Abstract Planning in partially observable environments remains a challenging problem, despite significant recent advances in offline approximation techniques. A few online methods have also been proposed recently, and proven to be remarkably scalable, but without the theoretical guarantees of their offline counterparts. Thus it seems natural to try to unify offline and online techniques, preserving the theoretical properties of the former, and exploiting the scalability of the latter. In this paper, we provide theoretical guarantees on an anytime algorithm for POMDPs which aims to reduce the error made by approximate offline value iteration algorithms through the use of an efficient online searching procedure. The algorithm uses search heuristics based on an error analysis of lookahead search, to guide the online search towards reachable beliefs with the most potential to reduce error. We provide a general theorem showing that these search heuristics are admissible, and lead to complete and ǫ-optimal algorithms. This is, to the best of our knowledge, the strongest theoretical result available for online POMDP solution methods. We also provide empirical evidence showing that our approach is also practical, and can find (provably) near-optimal solutions in reasonable time. 1 Introduction Partially Observable Markov Decision Processes (POMDPs) provide a powerful model for sequential decision making under state uncertainty. However exact solutions are intractable in most domains featuring more than a few dozen actions and observations. Significant efforts have been devoted to developing approximate offline algorithms for larger POMDPs [1, 2, 3, 4]. Most of these methods compute a policy over the entire belief space. This is both an advantage and a liability. On the one hand, it allows good generalization to unseen beliefs, and this has been key to solving relatively large domains. Yet it makes these methods impractical for problems where the state space is too large to enumerate. A number of compression techniques have been proposed, which handle large state spaces by projecting into a sub-dimensional representation [5, 6]. Alternately online methods are also available [7, 8, 9, 10, 11]. These achieve scalability by planning only at execution time, thus allowing the agent to only consider belief states that can be reached over some (small) finite planning horizon. However despite good empirical performance, both classes of approaches lack theoretical guarantees on the approximation. So it would seem we are constrained to either solving small to mid-size problems (near-)optimally, or solving large problems possibly badly. This paper suggests otherwise, arguing that by combining offline and online techniques, we can preserve the theoretical properties of the former, while exploiting the scalability of the latter. In previous work [11], we introduced an anytime algorithm for POMDPs which aims to reduce the error made by approximate offline value iteration algorithms through the use of an efficient online searching procedure. The algorithm uses search heuristics based on an error analysis of lookahead search, to guide the online search towards reachable beliefs with the most potential to reduce error. In this paper, we derive formally the heuristics from our error minimization point of view and provide theoretical results showing that these search heuristics are admissible, and lead to complete and ǫoptimal algorithms. This is, to the best of our knowledge, the strongest theoretical result available for online POMDP solution methods. Furthermore the approach works well with factored state representations, thus further enhancing scalability, as suggested by earlier work [2]. We also provide empirical evidence showing that our approach is computationally practical, and can find (provably) near-optimal solutions within a smaller overall time than previous online methods. 2 Background: POMDP A POMDP is defined by a tuple (S, A, Ω, T, R, O, γ) where S is the state space, A is the action set, Ωis the observation set, T : S × A × S →[0, 1] is the state-to-state transition function, R : S × A →R is the reward function, O : Ω× A × S →[0, 1] is the observation function, and γ is the discount factor. In a POMDP, the agent often does not know the current state with full certainty, since observations provide only a partial indicator of state. To deal with this uncertainty, the agent maintains a belief state b(s), which expresses the probability that the agent is in each state at a given time step. After each step, the belief state b is updated using Bayes rule. We denote the belief update function b′ = τ(b, a, o), defined as b′(s′) = ηO(o, a, s′) P s∈S T(s, a, s′)b(s), where η is a normalization constant ensuring P s∈S b′(s) = 1. Solving a POMDP consists in finding an optimal policy, π∗: ∆S →A, which specifies the best action a to do in every belief state b, that maximizes the expected return (i.e., expected sum of discounted rewards over the planning horizon) of the agent. We can find the optimal policy by computing the optimal value of a belief state over the planning horizon. For the infinite horizon, the optimal value function is defined as V ∗(b) = maxa∈A[R(b, a) + γ P o∈ΩP(o|b, a)V ∗(τ(b, a, o))], where R(b, a) represents the expected immediate reward of doing action a in belief state b and P(o|b, a) is the probability of observing o after doing action a in belief state b. This probability can be computed according to P(o|b, a) = P s′∈S O(o, a, s′) P s∈S T(s, a, s′)b(s). We also denote the value Q∗(b, a) of a particular action a in belief state b, as the return we will obtain if we perform a in b and then follow the optimal policy Q∗(b, a) = R(b, a) + γ P o∈ΩP(o|b, a)V ∗(τ(b, a, o)). Using this, we can define the optimal policy π∗(b) = argmaxa∈A Q∗(b, a). While any POMDP problem has infinitely many belief states, it has been shown that the optimal value function of a finite-horizon POMDP is piecewise linear and convex. Thus we can define the optimal value function and policy of a finite-horizon POMDP using a finite set of |S|-dimensional hyper plans, called α-vectors, over the belief state space. As a result, exact offline value iteration algorithms are able to compute V ∗in a finite amount of time, but the complexity can be very high. Most approximate offline value iteration algorithms achieve computational tractability by selecting a small subset of belief states, and keeping only those α-vectors which are maximal at the selected belief states [1, 3, 4]. The precision of these algorithms depend on the number of belief points and their location in the space of beliefs. 3 Online Search in POMDPs Contrary to offline approaches, which compute a complete policy determining an action for every belief state, an online algorithm takes as input the current belief state and returns the single action which is the best for this particular belief state. The advantage of such an approach is that it only needs to consider belief states that are reachable from the current belief state. This naturally provides a small set of beliefs, which could be exploited as in offline methods. But in addition, since online planning is done at every step (and thus generalization between beliefs is not required), it is sufficient to calculate only the maximal value for the current belief state, not the full optimal α-vector. A lookahead search algorithm can compute this value in two simple steps. First we build a tree of reachable belief states from the current belief state. The current belief is the top node in the tree. Subsequent belief states (as calculated by the τ(b, a, o) function) are represented using OR-nodes (at which we must choose an action) and actions are included in between each layer of belief nodes using AND-nodes (at which we must consider all possible observations). Note that in general the belief MDP could have a graph structure with cycles. Our algorithm simply handle such structure by unrolling the graph into a tree. Hence, if we reach a belief that is already elsewhere in the tree, it will be duplicated.1 Second, we estimate the value of the current belief state by propagating value estimates up from the fringe nodes, to their ancestors, all the way to the root. An approximate value function is generally used at the fringe of the tree to approximate the infinite-horizon value. We are particularly interested in the case where a lower bound and an upper bound on the value of the fringe belief states is available, as this allows us to get a bound on the error at any specific node. The lower and upper bounds can be propagated to parent nodes according to: UT (b) = U(b) if b is a leaf in T, maxa∈A UT (b, a) otherwise; (1) UT (b, a) = RB(b, a) + γ X o∈Ω P(o|b, a)UT (τ(b, a, o)); (2) LT (b) = L(b) if b is a leaf in T, maxa∈A LT (b, a) otherwise; (3) LT (b, a) = RB(b, a) + γ X o∈Ω P(o|b, a)LT (τ(b, a, o)); (4) where UT (b) and LT (b) represent the upper and lower bounds on V ∗(b) associated to belief state b in the tree T, UT (b, a) and LT (b, a) represent corresponding bounds on Q∗(b, a), and L(b) and U(b) are the bounds on fringe nodes, typically computed offline. Performing a complete k-step lookahead search multiplies the error bound on the approximate value function used at the fringe by γk ([13]), and thus ensures better value estimates. However, it has complexity exponential in k, and may explore belief states that have very small probabilities of occurring (and an equally small impact on the value function) as well as exploring suboptimal actions (which have no impact on the value function). We would evidently prefer to have a more efficient online algorithm, which can guarantee equivalent or better error bounds. In particular, we believe that the best way to achieve this is to have a search algorithm which uses estimates of error reduction as a criteria to guide the search over the reachable beliefs. 4 Anytime Error Minimization Search In this section, we review the Anytime Error Minimization Search (AEMS) algorithm we had first introduced in [11] and present a novel mathematical derivation of the heuristics that we had suggested. We also provide new theoretical results describing sufficient conditions under which the heuristics are guaranteed to yield ǫ-optimal solutions. Our approach uses a best-first search of the belief reachability tree, where error minimization (at the root node) is used as the search criteria to select which fringe nodes to expand next. Thus we need a way to express the error on the current belief (i.e. root node) as a function of the error at the fringe nodes. This is provided in Theorem 1. Let us denote (i) F(T), the set of fringe nodes of a tree T; (ii) eT (b) = V ∗(b) −LT (b), the error function for node b in the tree T; (iii) e(b) = V ∗(b) −L(b), the error at a fringe node b ∈F(T); (iv) hb0,b T , the unique action/observation sequence that leads from the root b0 to belief b in tree T; (v) d(h), the depth of an action/observation sequence h (number of actions); and (vi) P(h|b0, π∗) = Qd(h) i=1 P(hi o|bhi−1 0 , hi a)π∗(bhi−1, hi a), the probability of executing the action/observation sequence h if we follow the optimal policy π∗from the root node b0 (where hi a and hi o refers to the ith action and observation in the sequence h, and bhi is the belief obtained after taking the i first actions and observations from belief b. π∗(b, a) is the probability that the optimal policy chooses action a in belief b). By abuse of notation, we will use b to represent both a belief node in the tree and its associated belief2. 1We are considering using a technique proposed in the LAO* algorithm [12] to handle cycle, but we have not investigated this fully, especially in terms of how it affects the heuristic value presented below. 2e.g. P b∈F(T ) should be interpreted as a sum over all fringe nodes in the tree, while e(b) to be the error associated to the belief in fringe node b. Theorem 1. In any tree T, eT (b0) ≤P b∈F(T ) γd(hb0,b T )P(hb0,b T |b0, π∗)e(b). Proof. Consider an arbitrary parent node b in tree T and let’s denote ˆaT b = argmaxa∈A LT (b, a). We have eT (b) = V ∗(b) −LT (b). If ˆaT b = π∗(b), then eT (b) = γ P o∈ΩP(o|b, π∗(b))e(τ(b, π∗(b), o)). On the other hand, when ˆaT b ̸= π∗(b), then we know that LT (b, π∗(b)) ≤LT (b, ˆaT b ) and therefore eT (b) ≤γ P o∈ΩP(o|b, π∗(b))e(τ(b, π∗(b), o)). Consequently, we have the following: eT (b) ≤ ( e(b) if b ∈F(T) γ P o∈Ω P(o|b, π∗(b))eT (τ(b, π∗(b), o)) otherwise Then eT (b0) ≤P b∈F(T ) γd(hb0,b T )P(hb0,b T |b0, π∗)e(b) can be easily shown by induction. 4.1 Search Heuristics From Theorem 1, we see that the contribution of each fringe node to the error in b0 is simply the term γd(hb0,b T )P(hb0,b T |b0, π∗)e(b). Consequently, if we want to minimize eT (b0) as quickly as possible, we should expand fringe nodes reached by the optimal policy π∗that maximize the term γd(hb0,b T )P(hb0,b T |b0, π∗)e(b) as they offer the greatest potential to reduce eT (b0). This suggests us a sound heuristic to explore the tree in a best-first-search way. Unfortunately we do not know V ∗ nor π∗, which are required to compute the terms e(b) and P(hb0,b T |b0, π∗); nevertheless, we can approximate them. First, the term e(b) can be estimated by the difference between the lower and upper bound. We define ˆe(b) = U(b) −L(b) as an estimate of the error introduced by our bounds at fringe node b. Clearly, ˆe(b) ≥e(b) since U(b) ≥V ∗(b). To approximate P(hb0,b T |b0, π∗), we can view the term π∗(b, a) as the probability that action a is optimal in belief b. Thus, we consider an approximate policy ˆπT that represents the probability that action a is optimal in belief state b given the bounds LT (b, a) and UT (b, a) that we have on Q∗(b, a) in tree T. More precisely, to compute ˆπT (b, a), we consider Q∗(b, a) as a random variable and make some assumptions about its underlying probability distribution. Once cumulative distribution functions F b,a T , s.t. F b,a T (x) = P(Q∗(b, a) ≤x), and their associated density functions f b,a T are determined for each (b, a) in tree T, we can compute the probability ˆπT (b, a) = P(Q∗(b, a′) ≤Q∗(b, a)∀a′ ̸= a) = R ∞ −∞f b,a T (x) Q a′̸=a F b,a′ T (x)dx. Computing this integral may not be computationally efficient depending on how we define the functions f b,a T . We consider two approximations. One possible approximation is to simply compute the probability that the Q-value of a given action is higher than its parent belief state value (instead of all actions’ Q-value). In this case, we get ˆπT (b, a) = R ∞ −∞f b,a T (x)F b T (x)dx, where F b T is the cumulative distribution function for V ∗(b), given bounds LT (b) and UT (b) in tree T. Hence by considering both Q∗(b, a) and V ∗(b) as random variables with uniform distributions between their respective lower and upper bounds, we get: ˆπT (b, a) = ( η (UT (b,a)−LT (b))2 UT (b,a)−LT (b,a) if UT (b, a) > LT (b), 0 otherwise. (5) where η is a normalization constant such that P a∈A ˆπT (b, a) = 1. Notice that if the density function is 0 outside the interval between the lower and upper bound, then ˆπT (b, a) = 0 for dominated actions, thus they are implicitly pruned from the search tree by this method. A second practical approximation is: ˆπT (b, a) = 1 if a = argmaxa′∈A UT (b, a′), 0 otherwise. (6) which simply selects the action that maximizes the upper bound. This restricts exploration of the search tree to those fringe nodes that are reached by sequence of actions that maximize the upper bound of their parent belief state, as done in the AO∗algorithm [14]. The nice property of this approximation is that these fringe nodes are the only nodes that can potentially reduce the upper bound in b0. Using either of these two approximations for ˆπT , we can estimate the error contribution ˆeT (b0, b) of a fringe node b on the value of root belief b0 in tree T, as: ˆeT (b0, b) = γd(hb0,b T )P(hb0,b T |b0, ˆπT )ˆe(b). Using this as a heuristic, the next fringe node eb(T) to expand in tree T is defined as eb(T) = argmaxb∈F(T ) γd(hb0,b T )P(hb0,b T |b0, ˆπT )ˆe(b). We use AEMS13 to denote the heuristic that uses ˆπT as defined in Equation 5, and AEMS24 to denote the heuristic that uses ˆπT as defined in Equation 6. 4.2 Algorithm Algorithm 1 presents the anytime error minimization search. Since the objective is to provide a near-optimal action within a finite allowed online planning time, the algorithm accepts two input parameters: t, the online search time allowed per action, and ǫ, the desired precision on the value function. Algorithm 1 AEMS: Anytime Error Minimization Search Function SEARCH(t, ǫ) Static : T: an AND-OR tree representing the current search tree. t0 ←TIME() while TIME() −t0 ≤t and not SOLVED(ROOT(T), ǫ) do b∗←eb(T) EXPAND(b∗) UPDATEANCESTORS(b∗) end while return argmaxa∈A LT (ROOT(T), a) The EXPAND function expands the tree one level under the node b∗by adding the next action and belief nodes to the tree T and computing their lower and upper bounds according to Equations 14. After a node is expanded, the UPDATEANCESTORS function simply recomputes the bounds of its ancestors according to Equations determining b′(s′), V ∗(b), P(o|b, a) and Q∗(b, a), as outlined in Section 2. It also recomputes the probabilities ˆπT (b, a) and the best actions for each ancestor node. To find quickly the node that maximizes the heuristic in the whole tree, each node in the tree contains a reference to the best node to expand in their subtree. These references are updated by the UPDATEANCESTORS function without adding more complexity, such that when this function terminates, we always know immediatly which node to expand next, as its reference is stored in the root node. The search terminates whenever there is no more time available, or we have found an ǫoptimal solution (verified by the SOLVED function). After an action is executed in the environment, the tree T is updated such that our new current belief state becomes the root of T; all nodes under this new root can be reused at the next time step. 4.3 Completeness and Optimality We now provide some sufficient conditions under which our heuristic search is guaranteed to converge to an ǫ-optimal policy after a finite number of expansions. We show that the heuristics proposed in Section 4.1 satisfy those conditions, and therefore are admissible. Before we present the main theorems, we provide some useful preliminary lemmas. Lemma 1. In any tree T, the approximate error contribution ˆeT (b0, bd) of a belief node bd at depth d is bounded by ˆeT (b0, bd) ≤γd supb ˆe(b). Proof. P(hb0,b T |b0, ˆπT ) ≤1 and ˆe(b) ≤supb′ ˆe(b′) for all b. Thus ˆeT (b0, bd) ≤γd supb ˆe(b). For the following lemma and theorem, we will denote P(ho|b0, ha) = Qd(h) i=1 P(hi o|bhi−1 0 , hi a) the probability of observing the sequence of observations ho in some action/observation sequence h, given that the sequence of actions ha in h is performed from current belief b0, and bF(T) ⊆F(T) the set of all fringe nodes in T such that P(hb0,b T |b0, ˆπT ) > 0, for ˆπT defined as in Equation 6 (i.e. 3This heuristic is slightly different from the AEMS1 heuristic we had introduced in [11]. 4This is the same as the AEMS2 heuristic we had introduced in [11]. the set of fringe nodes reached by a sequence of actions in which each action maximizes UT (b, a) in its respective belief state.) Lemma 2. For any tree T, ǫ > 0, and D such that γD supb ˆe(b) ≤ǫ, if for all b ∈bF(T), either d(hb0,b T ) ≥D or there exists an ancestor b′ of b such that ˆeT (b′) ≤ǫ, then ˆeT (b0) ≤ǫ. Proof. Let’s denote ˆaT b = argmaxa∈A UT (b, a). Notice that for any tree T, and parent belief b ∈T, ˆeT (b) = UT (b)−LT (b) ≤UT (b, ˆaT b )−LT (b, ˆaT b ) = γ P o∈ΩP(o|b, ˆaT b )ˆeT (τ(b, ˆaT b , o)). Consequently, the following recurrence is an upper bound on ˆeT (b): ˆeT (b) ≤ 8 > < > : ˆe(b) if b ∈F(T) ǫ if ˆeT (b) ≤ǫ γ P o∈Ω P(o|b, ˆaT b )ˆeT (τ(b, ˆaT b , o)) otherwise By unfolding the recurrence for b0, we get ˆeT (b0) ≤ P b∈A(T ) γd(hb0,b T )P(hb0,b T,o |b0, hb0,b T,a )ˆe(b) + ǫ P b∈B(T ) γd(hb0,b T )P(hb0,b T,o |b0, hb0,b T,a ), where B(T) is the set of parent nodes b′ having a descendant in b F(T) such that ˆeT (b′) ≤ǫ and A(T) is the set of fringe nodes b′′ in b F(T) not having an ancestor in B(T). Hence if for all b ∈b F(T), d(hb0,b T ) ≥D or there exists an ancestor b′ of b such that ˆeT (b′) ≤ǫ, then this means that for all b in A(T), d(hb0,b T ) ≥D, and therefore, ˆeT (b0) ≤γD supb ˆe(b) P b′∈A(T ) P(hb0,b′ T,o |b0, hb0,b′ T,a ) + ǫ P b′∈B(T ) P(hb0,b′ T,o |b0, hb0,b′ T,a ) ≤ǫ P b′∈A(T )∪B(T ) P(hb0,b′ T,o |b0, hb0,b′ T,a ) = ǫ. Theorem 2. For any tree T and ǫ > 0, if ˆπT is defined such that infb,T |ˆeT (b)>ǫ ˆπT (b, ˆaT b ) > 0 for ˆaT b = argmaxa∈A UT (b, a), then Algorithm 1 using eb(T) is complete and ǫ-optimal. Proof. If γ = 0, then the proof is immediate. Consider now the case where γ ∈(0, 1). Clearly, since U is bounded above and L is bounded below, then ˆe is bounded above. Now using γ ∈(0, 1), we can find a positive integer D such that γD supb ˆe(b) ≤ǫ. Let’s denote AT b the set of ancestor belief states of b in the tree T, and given a finite set A of belief nodes, let’s define ˆemin T (A) = minb∈A ˆeT (b). Now let’s define Tb = {T|Tfinite, b ∈b F(T), ˆemin T (AT b ) > ǫ} and B = {b|ˆe(b) infT ∈Tb P(hb0,b T |b0, ˆπT ) > 0, d(hb0,b T ) ≤D}. Clearly, by the assumption that infb,T |ˆeT (b)>ǫ ˆπT (b, ˆaT b ) > 0, then B contains all belief states b within depth D such that ˆe(b) > 0, P(hb0,b T,o |b0, hb0,b T,a ) > 0 and there exists a finite tree T where b ∈b F(T) and all ancestors b′ of b have ˆeT (b′) > ǫ. Furthermore, B is finite since there are only finitely many belief states within depth D. Hence there exist a Emin = minb∈B γd(hb0,b T )ˆe(b) infT ∈Tb P(hb0,b T |b0, ˆπT ). Clearly, Emin > 0 and we know that for any tree T, all beliefs b in B ∩b F(T) have an approximate error contribution ˆeT (b0, b) ≥Emin. Since Emin > 0 and γ ∈(0, 1), there exist a positive integer D′ such that γD′ supb ˆe(b) < Emin. Hence by Lemma 1, this means that Algorithm 1 cannot expand any node at depth D′ or beyond before expanding a tree T where B ∩b F(T) = ∅. Because there are only finitely many nodes within depth D′, then it is clear that Algorithm 1 will reach such tree T after a finite number of expansions. Furthermore, for this tree T, since B ∩b F(T) = ∅, we have that for all beliefs b ∈b F(T), either d(hb0,b T ) ≥D or ˆemin T (AT b ) ≤ǫ. Hence by Lemma 2, this implies that ˆeT (b0) ≤ǫ, and consequently Algorithm 1 will terminate after a finite number of expansions (SOLVED(b0, ǫ) will evaluate to true) with an ǫ-optimal solution (since eT (b0) ≤ˆeT (b0)). From this last theorem, we notice that we can potentially develop many different admissible heuristics for Algorithm 1; the main sufficient condition being that ˆπT (b, a) > 0 for a = argmaxa′∈A UT (b, a′). It also follows from this theorem that the two heuristics described above, AEMS1 and AEMS2, are admissible. The following corollaries prove this: Corollary 1. Algorithm 1, using eb(T), with ˆπT as defined in Equation 6 is complete and ǫ-optimal. Proof. Immediate by Theorem 2 and the fact that ˆπT (b, ˆaT b ) = 1 for all b, T. Corollary 2. Algorithm 1, using eb(T), with ˆπT as defined in Equation 5 is complete and ǫ-optimal. Proof. We first notice that (UT (b, a) −LT (b))2/(UT (b, a) −LT (b, a)) ≤ˆeT (b, a), since LT (b) ≥ LT (b, a) for all a. Furthermore, ˆeT (b, a) ≤supb′ ˆe(b′). Therefore the normalization constant η ≥(|A| supb ˆe(b))−1. For ˆaT b = argmaxa∈A UT (b, a), we have UT (b, ˆaT b ) = UT (b), and therefore UT (b, ˆaT b ) −LT (b) = ˆeT (b). Hence this means that ˆπT (b, ˆaT b ) = η(ˆeT (b))2/ˆeT (b, ˆaT b ) ≥ (|A|(supb′ ˆe(b′))2)−1(ˆeT (b))2 for all T, b. Hence, for any ǫ > 0, infb,T |ˆeT (b)>ǫ ˆπT (b, ˆaT b ) ≥ (|A|(supb ˆe(b))2)−1ǫ2 > 0. Hence, corrolary follows from Theorem 2. 5 Experiments In this section we present a brief experimental evaluation of Algorithm 1, showing that in addition to its useful theoretical properties, the empirical performance matches, and in some cases exceeds, that of other online approaches. The algorithm is evaluated in three large POMDP environments: Tag [1], RockSample [3] and FieldVisionRockSample (FVRS) [11]; all are implemented using a factored state representation. In each environments we compute the Blind policy5 to get a lower bound and the FIB algorithm [15] to get an upper bound. We then compare performance of Algorithm 1 with both heuristics (AEMS1 and AEMS2) to the performance achieved by other online approaches (Satia [7], BI-POMDP [8], RTBSS [10]). For all approaches we impose a real-time constraint of 1 sec/action, and measure the following metrics: average return, average error bound reduction6 (EBR), average lower bound improvement7 (LBI), number of belief nodes explored at each time step, percentage of belief nodes reused in the next time step, and the average online time per action (< 1s means the algorithm found an ǫ-optimal action)8. Satia, BI-POMDP, AEMS1 and AEMS2 were all implemented using the same algorithm since they differ only in their choice of search heuristic used to guide the search. RTBSS served as a base line for a complete k-step lookahead search using branch & bound pruning. All results were obtained on a Xeon 2.4 Ghz with 4Gb of RAM; but the processes were limited to use a max of 1Gb of RAM. Table 1 shows the average value (over 1000+ runs) of the different statistics. As we can see from these results, AEMS2 provides the best average return, average error bound reduction and average lower bound improvement in all considered environments. The higher error bound reduction and lower bound improvement obtained by AEMS2 indicates that it can guarantee performance closer to the optimal. We can also observe that AEMS2 has the best average reuse percentage, which indicates that AEMS2 is able to guide the search toward the most probable nodes and allows it to generally maintain a higher number of belief nodes in the tree. Notice that AEMS1 did not perform very well, except in FVRS[5,7]. This could be explained by the fact that our assumption that the values of the actions are uniformly distributed between the lower and upper bounds is not valid in the considered environments. Finally, we also examined how fast the lower and upper bounds converge if we let the algorithm run up to 1000 seconds on the initial belief state. This gives an indication of which heuristic would be the best if we extended online planning time past 1sec. Results for RockSample[7,8] are presented in Figure 2, showing that the bounds converge much more quickly for the AEMS2 heuristic. 6 Conclusion In this paper we examined theoretical properties of online heuristic search algorithms for POMDPs. To this end, we described a general online search framework, and examined two admissible heuristics to guide the search. The first assumes that Q∗(b, a) is distributed uniformly at random between the bounds (Heuristic AEMS1), the second favors an optimistic point of view, and assume the Q∗(b, a) is equal to the upper bound (Heuristic AEMS2). We provided a general theorem that shows that AEMS1 and AEMS2 are admissible and lead to complete and ǫ-optimal algorithms. Our experimental work supports the theoretical analysis, showing that AEMS2 is able to outperform online approaches. Yet it is equally interesting to note that AEMS1 did not perform nearly as well. This highlights the fact that not all admissible heuristics are equally useful. Thus it will be interesting in the future to develop further guidelines and theoretical results describing which subclasses of heuristics are most appropriate. 5The policy obtained by taking the combination of the |A| α-vectors that each represents the value of a policy performing the same action in every belief state. 6The error bound reduction is defined as 1 −UT (b0)−LT (b0) U(b0)−L(b0) , when the search process terminates on b0 7The lower bound improvement is defined as LT (b0) −L(b0), when the search process terminates on b0 8For RTBSS, the maximum search depth under the 1sec time constraint is show in parenthesis. Figure 1: Comparison of different online search algorithm in different environments. Heuristic / Belief Reuse Time Algorithm Return EBR (%) LBI Nodes (%) (ms) ± 0.01 ± 0.1 ± 0.01 ±0.1 ±1 Tag (|S| = 870, |A| = 5, |Ω| = 30) RTBSS(5) -10.30 22.3 3.03 45067 0 580 Satia & Lave -8.35 22.9 2.47 36908 10.0 856 AEMS1 -6.73 49.0 3.92 43693 25.1 814 BI-POMDP -6.22 76.2 7.81 79508 54.6 622 AEMS2 -6.19 76.3 7.81 80250 54.8 623 RockSample[7,8] (|S| = 12545, |A| = 13, |Ω| = 2) Satia & Lave 7.35 3.6 0 509 8.9 900 AEMS1 10.30 9.5 0.90 579 5.3 916 RTBSS(2) 10.30 9.7 1.00 439 0 896 BI-POMDP 18.43 33.3 4.33 2152 29.9 953 AEMS2 20.75 52.4 5.30 3145 36.4 859 FVRS[5,7] (|S| = 3201, |A| = 5, |Ω| = 128) RTBSS(1) 20.57 7.7 2.07 516 0 254 BI-POMDP 22.75 11.1 2.08 4457 0.4 923 Satia & Lave 22.79 11.1 2.05 3683 0.4 947 AEMS1 23.31 12.4 2.24 3856 1.4 942 AEMS2 23.39 13.3 2.35 4070 1.6 944 10 −2 10 −1 10 0 10 1 10 2 10 3 5 10 15 20 25 30 Time (s) V(b0) AEMS2 AEMS1 BI−POMDP Satia Figure 2: Evolution of the upper / lower bounds on the initial belief state in RockSample[7,8]. Acknowledgments This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds Qu´eb´ecois de la Recherche sur la Nature et les Technologies (FQRNT). References [1] J. Pineau. Tractable planning under uncertainty: exploiting structure. PhD thesis, Carnegie Mellon University, Pittsburgh, PA, 2004. [2] P. Poupart. Exploiting structure to efficiently solve large scale partially observable Markov decision processes. PhD thesis, University of Toronto, 2005. [3] T. Smith and R. Simmons. Point-based POMDP algorithms: improved analysis and implementation. In UAI, 2005. [4] M. T. J. Spaan and N. Vlassis. Perseus: randomized point-based value iteration for POMDPs. JAIR, 24:195–220, 2005. [5] N. Roy and G. Gordon. Exponential family PCA for belief compression in POMDPs. In NIPS, 2003. [6] P. Poupart and C. Boutilier. Value-directed compression of POMDPs. In NIPS, 2003. [7] J. K. Satia and R. E. Lave. Markovian decision processes with probabilistic observation of states. Management Science, 20(1):1–13, 1973. [8] R. Washington. BI-POMDP: bounded, incremental partially observable Markov model planning. In 4th Eur. Conf. on Planning, pages 440–451, 1997. [9] D. McAllester and S. Singh. Approximate Planning for Factored POMDPs using Belief State Simplification. In UAI, 1999. [10] S. Paquet, L. Tobin, and B. Chaib-draa. An online POMDP algorithm for complex multiagent environments. In AAMAS, 2005. [11] S. Ross and B. Chaib-draa. AEMS: an anytime online search algorithm for approximate policy refinement in large POMDPs. In IJCAI, 2007. [12] E. A. Hansen and S. Zilberstein. LAO * : A heuristic search algorithm that finds solutions with loops. Artificial Intelligence, 129(1-2):35–62, 2001. [13] M. L. Puterman. Markov Decision Processes: Discrete Stochastic Dynamic Programming. John Wiley & Sons, Inc., New York, NY, USA, 1994. [14] N.J. Nilsson. Principles of Artificial Intelligence. Tioga Publishing, 1980. [15] M. Hauskrecht. Value-function approximations for POMDPs. JAIR, 13:33–94, 2000.
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Fitted Q-iteration in continuous action-space MDPs Andr´as Antos Computer and Automation Research Inst. of the Hungarian Academy of Sciences Kende u. 13-17, Budapest 1111, Hungary antos@sztaki.hu R´emi Munos SequeL project-team, INRIA Lille 59650 Villeneuve d’Ascq, France remi.munos@inria.fr Csaba Szepesv´ari∗ Department of Computing Science University of Alberta Edmonton T6G 2E8, Canada szepesva@cs.ualberta.ca Abstract We consider continuous state, continuous action batch reinforcement learning where the goal is to learn a good policy from a sufficiently rich trajectory generated by some policy. We study a variant of fitted Q-iteration, where the greedy action selection is replaced by searching for a policy in a restricted set of candidate policies by maximizing the average action values. We provide a rigorous analysis of this algorithm, proving what we believe is the first finite-time bound for value-function based algorithms for continuous state and action problems. 1 Preliminaries We will build on the results from [1, 2, 3] and for this reason we use the same notation as these papers. The unattributed results cited in this section can be found in the book [4]. A discounted MDP is defined by a quintuple (X, A, P, S, γ), where X is the (possible infinite) state space, A is the set of actions, P : X × A →M(X) is the transition probability kernel with P(·|x, a) defining the next-state distribution upon taking action a from state x, S(·|x, a) gives the corresponding distribution of immediate rewards, and γ ∈(0, 1) is the discount factor. Here X is a measurable space and M(X) denotes the set of all probability measures over X. The Lebesguemeasure shall be denoted by λ. We start with the following mild assumption on the MDP: Assumption A1 (MDP Regularity) X is a compact subset of the dX -dimensional Euclidean space, A is a compact subset of [−A∞, A∞]dA. The random immediate rewards are bounded by ˆRmax and that the expected immediate reward function, r(x, a) = R rS(dr|x, a), is uniformly bounded by Rmax: ∥r∥∞≤Rmax. A policy determines the next action given the past observations. Here we shall deal with stationary (Markovian) policies which choose an action in a stochastic way based on the last observation only. The value of a policy π when it is started from a state x is defined as the total expected discounted reward that is encountered while the policy is executed: V π(x) = Eπ [P∞ t=0 γtRt|X0 = x]. Here Rt ∼S(·|Xt, At) is the reward received at time step t, the state, Xt, evolves according to Xt+1 ∼ ∗Also with: Computer and Automation Research Inst. of the Hungarian Academy of Sciences Kende u. 13-17, Budapest 1111, Hungary. 1 P(·|Xt, At), where At is sampled from the distribution determined by π. We use Qπ : X × A →R to denote the action-value function of policy π: Qπ(x, a) = Eπ [P∞ t=0 γtRt|X0 = x, A0 = a]. The goal is to find a policy that attains the best possible values, V ∗(x) = supπ V π(x), at all states x ∈X. Here V ∗is called the optimal value function and a policy π∗that satisfies V π∗(x) = V ∗(x) for all x ∈X is called optimal. The optimal action-value function Q∗(x, a) is Q∗(x, a) = supπ Qπ(x, a). We say that a (deterministic stationary) policy π is greedy w.r.t. an action-value function Q ∈B(X × A), and we write π = ˆπ(·; Q), if, for all x ∈X, π(x) ∈argmaxa∈A Q(x, a). Under mild technical assumptions, such a greedy policy always exists. Any greedy policy w.r.t. Q∗ is optimal. For π : X →A we define its evaluation operator, T π : B(X × A) →B(X × A), by (T πQ)(x, a) = r(x, a) + γ R X Q(y, π(y)) P(dy|x, a). It is known that Qπ = T πQπ. Further, if we let the Bellman operator, T : B(X × A) →B(X × A), defined by (TQ)(x, a) = r(x, a) + γ R X supb∈A Q(y, b) P(dy|x, a) then Q∗= TQ∗. It is known that V π and Qπ are bounded by Rmax/(1 −γ), just like Q∗and V ∗. For π : X →A, the operator Eπ : B(X × A) →B(X) is defined by (EπQ)(x) = Q(x, π(x)), while E : B(X × A) →B(X) is defined by (EQ)(x) = supa∈A Q(x, a). Throughout the paper F ⊂{f : X × A →R} will denote a subset of real-valued functions over the state-action space X × A and Π ⊂AX will be a set of policies. For ν ∈M(X) and f : X →R measurable, we let (for p ≥1) ∥f∥p p,ν = R X |f(x)|pν(dx). We simply write ∥f∥ν for ∥f∥2,ν. Further, we extend ∥·∥ν to F by ∥f∥2 ν = R A R X |f|2(x, a) dν(x) dλA(a), where λA is the uniform distribution over A. We shall use the shorthand notation νf to denote the integral R f(x)ν(dx). We denote the space of bounded measurable functions with domain X by B(X). Further, the space of measurable functions bounded by 0 < K < ∞shall be denoted by B(X; K). We let ∥·∥∞denote the supremum norm. 2 Fitted Q-iteration with approximate policy maximization We assume that we are given a finite trajectory, {(Xt, At, Rt)}1≤t≤N, generated by some stochastic stationary policy πb, called the behavior policy: At ∼πb(·|Xt), Xt+1 ∼P(·|Xt, At), Rt ∼ S(·|Xt, At), where πb(·|x) is a density with π0 def= inf(x,a)∈X×A πb(a|x) > 0. The generic recipe for fitted Q-iteration (FQI) [5] is Qk+1 = Regress(Dk(Qk)), (1) where Regress is an appropriate regression procedure and Dk(Qk) is a dataset defining a regression problem in the form of a list of data-point pairs: Dk(Qk) = ½h (Xt, At), Rt + γ max b∈A Qk(Xt+1, b) i 1≤t≤N ¾ .1 Fitted Q-iteration can be viewed as approximate value iteration applied to action-value functions. To see this note that value iteration would assign the value (TQk)(x, a) = r(x, a) + γ R maxb∈A Qk(y, b) P(dy|x, a) to Qk+1(x, a) [6]. Now, remember that the regression function for the jointly distributed random variables (Z, Y ) is defined by the conditional expectation of Y given Z: m(Z) = E [Y |Z]. Since for any fixed function Q, E [Rt + γ maxb∈A Q(Xt+1, b)|Xt, At] = (TQ)(Xt, At), the regression function corresponding to the data Dk(Q) is indeed TQ and hence if FQI solved the regression problem defined by Qk exactly, it would simulate value iteration exactly. However, this argument itself does not directly lead to a rigorous analysis of FQI: Since Qk is obtained based on the data, it is itself a random function. Hence, after the first iteration, the “target” function in FQI becomes random. Furthermore, this function depends on the same data that is used to define the regression problem. Will FQI still work despite these issues? To illustrate the potential difficulties consider a dataset where X1, . . . , XN is a sequence of independent random variables, which are all distributed uniformly at random in [0, 1]. Further, let M be a random integer greater than N which is independent of the dataset (Xt)N t=1. Let U be another random variable, uniformly distributed in [0, 1]. Now define the regression problem by Yt = fM,U(Xt), where fM,U(x) = sgn(sin(2M2π(x + U))). Then it is not hard to see that no matter how big N is, no procedure can 1Since the designer controls Qk, we may assume that it is continuous, hence the maximum exists. 2 estimate the regression function fM,U with a small error (in expectation, or with high probability), even if the procedure could exploit the knowledge of the specific form of fM,U. On the other hand, if we restricted M to a finite range then the estimation problem could be solved successfully. The example shows that if the complexity of the random functions defining the regression problem is uncontrolled then successful estimation might be impossible. Amongst the many regression methods in this paper we have chosen to work with least-squares methods. In this case Equation (1) takes the form Qk+1 = argmin Q∈F N X t=1 1 πb(At|Xt) µ Q(Xt, At) − · Rt + γ max b∈A Qk(Xt+1, b) ¸¶2 . (2) We call this method the least-squares fitted Q-iteration (LSFQI) method. Here we introduced the weighting 1/πb(At|Xt) since we do not want to give more weight to those actions that are preferred by the behavior policy. Besides this weighting, the only parameter of the method is the function set F. This function set should be chosen carefully, to keep a balance between the representation power and the number of samples. As a specific example for F consider neural networks with some fixed architecture. In this case the function set is generated by assigning weights in all possible ways to the neural net. Then the above minimization becomes the problem of tuning the weights. Another example is to use linearly parameterized function approximation methods with appropriately selected basis functions. In this case the weight tuning problem would be less demanding. Yet another possibility is to let F be an appropriate restriction of a Reproducing Kernel Hilbert Space (e.g., in a ball). In this case the training procedure becomes similar to LS-SVM training [7]. As indicated above, the analysis of this algorithm is complicated by the fact that the new dataset is defined in terms of the previous iterate, which is already a function of the dataset. Another complication is that the samples in a trajectory are in general correlated and that the bias introduced by the imperfections of the approximation architecture may yield to an explosion of the error of the procedure, as documented in a number of cases in, e.g., [8]. Nevertheless, at least for finite action sets, the tools developed in [1, 3, 2] look suitable to show that under appropriate conditions these problems can be overcome if the function set is chosen in a judicious way. However, the results of these works would become essentially useless in the case of an infinite number of actions since these previous bounds grow to infinity with the number of actions. Actually, we believe that this is not an artifact of the proof techniques of these works, as suggested by the counterexample that involved random targets. The following result elaborates this point further: Proposition 2.1. Let F ⊂B(X × A). Then even if the pseudo-dimension of F is finite, the fatshattering function of F∨ max = ½ VQ : VQ(·) = max a∈A Q(·, a), Q ∈F ¾ can be infinite over (0, 1/2).2 Without going into further details, let us just note that the finiteness of the fat-shattering function is a sufficient and necessary condition for learnability and the finiteness of the fat-shattering function is implied by the finiteness of the pseudo-dimension [9].The above proposition thus shows that without imposing further special conditions on F, the learning problem may become infeasible. One possibility is of course to discretize the action space, e.g., by using a uniform grid. However, if the action space has a really high dimensionality, this approach becomes unfeasible (even enumerating 2dA points could be impossible when dA is large). Therefore we prefer alternate solutions. Another possibility is to make the functions in F, e.g., uniformly Lipschitz in their state coordinates. Then the same property will hold for functions in F∨ max and hence by a classical result we can bound the capacity of this set (cf. pp. 353–357 of [10]). One potential problem with this approach is that this way it might be difficult to get a fine control of the capacity of the resulting set. 2The proof of this and the other results are given in the appendix, available in the extended version of this paper, downloadable from http://hal.inria.fr/inria-00185311/en/. 3 In the approach explored here we modify the fitted Q-iteration algorithm by introducing a policy set Π and a search over this set for an approximately greedy policy in a sense that will be made precise in a minute. Our algorithm thus has four parameters: F, Π, K, Q0. Here F is as before, Π is a user-chosen set of policies (mappings from X to A), K is the number of iterations and Q0 is an initial value function (a typical choice is Q0 ≡0). The algorithm computes a sequence of iterates (Qk, ˆπk), k = 0, . . . , K, defined by the following equations: ˆπ0 = argmax π∈Π N X t=1 Q0(Xt, π(Xt)), Qk+1 = argmin Q∈F N X t=1 1 πb(At|Xt) ³ Q(Xt, At) − £ Rt + γQk(Xt+1, ˆπk(Xt+1)) ¤´2 , (3) ˆπk+1 = argmax π∈Π N X t=1 Qk+1(Xt, π(Xt)). (4) Thus, (3) is similar to (2), while (4) defines the policy search problem. The policy search will generally be solved by a gradient procedure or some other appropriate method. The cost of this step will be primarily determined by how well-behaving the iterates Qk+1 are in their action arguments. For example, if they were quadratic and if π was linear then the problem would be a quadratic optimization problem. However, except for special cases3 the action value functions will be more complicated, in which case this step can be expensive. Still, this cost could be similar to that of searching for the maximizing actions for each t = 1, . . . , N if the approximately maximizing actions are similar across similar states. This algorithm, which we could also call a fitted actor-critic algorithm, will be shown to overcome the above mentioned complexity control problem provided that the complexity of Π is controlled appropriately. Indeed, in this case the set of possible regression problems is determined by the set F∨ Π = { V : V (·) = Q(·, π(·)), Q ∈F, π ∈Π } , and the proof will rely on controlling the complexity of F∨ Π by selecting F and Π appropriately. 3 The main theoretical result 3.1 Outline of the analysis In order to gain some insight into the behavior of the algorithm, we provide a brief summary of its error analysis. The main result will be presented subsequently. For f,Q ∈F and a policy π, we define the tth TD-error as follows: dt(f; Q, π) = Rt + γQ(Xt+1, π(Xt+1)) −f(Xt, At). Further, we define the empirical loss function by ˆLN(f; Q, π) = 1 N N X t=1 d2 t(f; Q, π) λ(A)πb(At|Xt), where the normalization with λ(A) is introduced for mathematical convenience. Then (3) can be written compactly as Qk+1 = argminf∈F ˆLN(f; Qk, ˆπk). The algorithm can then be motivated by the observation that for any f,Q, and π, ˆLN(f; Q, π) is an unbiased estimate of L(f; Q, π) def= ∥f −T πQ∥2 ν + L∗(Q, π), (5) where the first term is the error we are interested in and the second term captures the variance of the random samples: L∗(Q, π) = Z A E [Var [R1 + γQ(X2, π(X2))|X1, A1 = a]] dλA(a). 3Linear quadratic regulation is such a nice case. It is interesting to note that in this special case the obvious choices for F and Π yield zero error in the limit, as can be proven based on the main result of this paper. 4 This result is stated formally by E h ˆLN(f; Q, π) i = L(f; Q, π). Since the variance term in (5) is independent of f, argminf∈F L(f; Q, π) = argminf∈F ∥f −T πQ∥2 ν. Thus, if ˆπk were greedy w.r.t. Qk then argminf∈F L(f; Qk, ˆπk) = argminf∈F ∥f −TQk∥2 ν. Hence we can still think of the procedure as approximate value iteration over the space of action-value functions, projecting TQk using empirical risk minimization on the space F w.r.t. ∥·∥ν distances in an approximate manner. Since ˆπk is only approximately greedy, we will have to deal with both the error coming from the approximate projection and the error coming from the choice of ˆπk. To make this clear, we write the iteration in the form Qk+1 = T ˆπkQk + ε′ k = TQk + ε′ k + (T ˆπkQk −TQk) = TQk + εk, where ε′ k is the error committed while computing T ˆπkQk, ε′′ k def= T ˆπkQk −TQk is the error committed because the greedy policy is computed approximately and εk = ε′ k + ε′′ k is the total error of step k. Hence, in order to show that the procedure is well behaved, one needs to show that both errors are controlled and that when the errors are propagated through these equations, the resulting error stays controlled, too. Since we are ultimately interested in the performance of the policy obtained, we will also need to show that small action-value approximation errors yield small performance losses. For these we need a number of assumptions that concern either the training data, the MDP, or the function sets used for learning. 3.2 Assumptions 3.2.1 Assumptions on the training data We shall assume that the data is rich, is in a steady state, and is fast-mixing, where, informally, mixing means that future depends weakly on the past. Assumption A2 (Sample Path Properties) Assume that {(Xt, At, Rt)}t=1,...,N is the sample path of πb, a stochastic stationary policy. Further, assume that {Xt} is strictly stationary (Xt ∼ν ∈ M(X)) and exponentially β-mixing with the actual rate given by the parameters (β, b, κ).4 We further assume that the sampling policy πb satisfies π0 = inf(x,a)∈X×A πb(a|x) > 0. The β-mixing property will be used to establish tail inequalities for certain empirical processes.5 Note that the mixing coefficients do not need to be known. In the case when no mixing condition is satisfied, learning might be impossible. To see this just consider the case when X1 = X2 = . . . = XN. Thus, in this case the learner has many copies of the same random variable and successful generalization is thus impossible. We believe that the assumption that the process is in a steady state is not essential for our result, as when the process reaches its steady state quickly then (at the price of a more involved proof) the result would still hold. 3.2.2 Assumptions on the MDP In order to prevent the uncontrolled growth of the errors as they are propagated through the updates, we shall need some assumptions on the MDP. A convenient assumption is the following one [11]: Assumption A3 (Uniformly stochastic transitions) For all x ∈X and a ∈A, assume that P(·|x, a) is absolutely continuous w.r.t. ν and the Radon-Nikodym derivative of P w.r.t. ν is bounded uniformly with bound Cν: Cν def= supx∈X,a∈A °°° dP (·|x,a) dν °°° ∞< +∞. Note that by the definition of measure differentiation, Assumption A3 means that P(·|x, a) ≤ Cνν(·). This assumption essentially requires the transitions to be noisy. We will also prove (weaker) results under the following, weaker assumption: 4For the definition of β-mixing, see e.g. [2]. 5We say “empirical process” and “empirical measure”, but note that in this work these are based on dependent (mixing) samples. 5 Assumption A4 (Discounted-average concentrability of future-state distributions) Given ρ, ν, m ≥1 and an arbitrary sequence of stationary policies {πm}m≥1, assume that the futurestate distribution ρP π1P π2 . . . P πm is absolutely continuous w.r.t. ν. Assume that c(m) def= supπ1,...,πm °°° d(ρP π1P π2...P πm) dν °°° ∞satisfies P m≥1 mγm−1c(m) < +∞. We shall call Cρ,ν def= max © (1 −γ)2 P m≥1 mγm−1c(m), (1 −γ) P m≥1 γmc(m) ª the discounted-average concentrability coefficient of the future-state distributions. The number c(m) measures how much ρ can get amplified in m steps as compared to the reference distribution ν. Hence, in general we expect c(m) to grow with m. In fact, the condition that Cρ,µ is finite is a growth rate condition on c(m). Thanks to discounting, Cρ,µ is finite for a reasonably large class of systems (see the discussion in [11]). A related assumption is needed in the error analysis of the approximate greedy step of the algorithm: Assumption A5 (The random policy “makes no peak-states”) Consider the distribution µ = (ν × λA)P which is the distribution of a state that results from sampling an initial state according to ν and then executing an action which is selected uniformly at random.6 Then Γν = ∥dµ/dν∥∞< +∞. Note that under Assumption A3 we have Γν ≤Cν. This (very mild) assumption means that after one step, starting from ν and executing this random policy, the probability of the next state being in a set is upper bounded by Γν-times the probability of the starting state being in the same set. Besides, we assume that A has the following regularity property: Let Py(a, h, ρ) def= © (a′, v) ∈RdA+1 : ∥a −a′∥1 ≤ρ, 0 ≤v/h ≤1 −∥a −a′∥1 /ρ ª denote the pyramid with hight h and base given by the ℓ1-ball B(a, ρ) def= © a′ ∈RdA : ∥a −a′∥1 ≤ρ ª centered at a. Assumption A6 (Regularity of the action space) We assume that there exists α > 0, such that for all a ∈A, for all ρ > 0, λ(Py(a, 1, ρ) ∩(A × R)) λ(Py(a, 1, ρ)) ≥min µ α, λ(A) λ(B(a, ρ)) ¶ . For example, if A is an ℓ1-ball itself, then this assumption will be satisfied with α = 2−dA. Without assuming any smoothness of the MDP, learning in infinite MDPs looks hard (see, e.g., [12, 13]). Here we employ the following extra condition: Assumption A7 (Lipschitzness of the MDP in the actions) Assume that the transition probabilities and rewards are Lipschitz w.r.t. their action variable, i.e., there exists LP , Lr > 0 such that for all (x, a, a′) ∈X × A × A and measurable set B of X, |P(B|x, a) −P(B|x, a′)| ≤LP ∥a −a′∥1 , |r(x, a) −r(x, a′)| ≤Lr ∥a −a′∥1 . Note that previously Lipschitzness w.r.t. the state variables was used, e.g., in [11] to construct consistent planning algorithms. 3.2.3 Assumptions on the function sets used by the algorithm These assumptions are less demanding since they are under the control of the user of the algorithm. However, the choice of these function sets will greatly influence the performance of the algorithm, as we shall see it from the bounds. The first assumption concerns the class F: Assumption A8 (Lipschitzness of candidate action-value functions) Assume F ⊂B(X × A) and that any elements of F is uniformly Lipschitz in its action-argument in the sense that |Q(x, a)− Q(x, a′)| ≤LA ∥a −a′∥1 holds for any x ∈X, a,a′ ∈A, and Q ∈F. 6Remember that λA denotes the uniform distribution over the action set A. 6 We shall also need to control the capacity of our function sets. We assume that the reader is familiar with the concept of VC-dimension.7 Here we use the pseudo-dimension of function sets that builds upon the concept of VC-dimension: Definition 3.1 (Pseudo-dimension). The pseudo-dimension VF+ of F is defined as the VCdimension of the subgraphs of functions in F (hence it is also called the VC-subgraph dimension of F). Since A is multidimensional, we define VΠ+ to be the sum of the pseudo-dimensions of the coordinate projection spaces, Πk of Π: VΠ+ = dA X k=1 VΠ+ k , Πk = { πk : X →R : π = (π1, . . . , πk, . . . , πdA) ∈Π } . Now we are ready to state our assumptions on our function sets: Assumption A9 (Capacity of the function and policy sets) Assume that F ⊂B(X × A; Qmax) for Qmax > 0 and VF+ < +∞. Also, A ⊂[−A∞, A∞]dA and VΠ+ < +∞. Besides their capacity, one shall also control the approximation power of the function sets involved. Let us first consider the policy set Π. Introduce e∗(F, Π) = sup Q∈F inf π∈Π ν(EQ −EπQ). Note that infπ∈Π ν(EQ −EπQ) measures the quality of approximating νEQ by νEπQ. Hence, e∗(F, Π) measures the worst-case approximation error of νEQ as Q is changed within F. This can be made small by choosing Π large. Another related quantity is the one-step Bellman-error of F w.r.t. Π. This is defined as follows: For a fixed policy π, the one-step Bellman-error of F w.r.t. T π is defined as E1(F; π) = sup Q∈F inf Q′∈F ∥Q′ −T πQ∥ν . Taking again a pessimistic approach, the one-step Bellman-error of F is defined as E1(F, Π) = sup π∈Π E1(F; π). Typically by increasing F, E1(F, Π) can be made smaller (this is discussed at some length in [3]). However, it also holds for both Π and F that making them bigger will increase their capacity (pseudo-dimensions) which leads to an increase of the estimation errors. Hence, F and Π must be selected to balance the approximation and estimation errors, just like in supervised learning. 3.3 The main result Theorem 3.2. Let πK be a greedy policy w.r.t. QK, i.e. πK(x) ∈argmaxa∈A QK(x, a). Then under Assumptions A1, A2, and A5–A9, for all δ > 0 we have with probability at least 1 −δ: given Assumption A3 (respectively A4), ∥V ∗−V πK∥∞(resp. ∥V ∗−V πK∥1,ρ), is bounded by C E1(F, Π) + e∗(F, Π) + (log N + log(K/δ)) κ+1 4κ N 1/4 1 dA+1 + γK , where C depends on dA, VF+, (VΠ+ k )dA k=1, γ, κ, b, β, Cν (resp. Cρ,ν), Γν, LA, LP ,Lr, α, λ(A), π0, Qmax, Rmax, ˆRmax, and A∞. In particular, C scales with V κ+1 4κ(dA+1) , where V = 2VF+ + VΠ+ plays the role of the “combined effective” dimension of F and Π. 7Readers not familiar with VC-dimension are suggested to consult a book, such as the one by Anthony and Bartlett [14]. 7 4 Discussion We have presented what we believe is the first finite-time bounds for continuous-state and actionspace RL that uses value functions. Further, this is the first analysis of fitted Q-iteration, an algorithm that has proved to be useful in a number of cases, even when used with non-averagers for which no previous theoretical analysis existed (e.g., [15, 16]). In fact, our main motivation was to show that there is a systematic way of making these algorithms work and to point at possible problem sources the same time. We discussed why it can be difficult to make these algorithms work in practice. We suggested that either the set of action-value candidates has to be carefully controlled (e.g., assuming uniform Lipschitzness w.r.t. the state variables), or a policy search step is needed, just like in actorcritic algorithms. The bound in this paper is similar in many respects to a previous bound of a Bellman-residual minimization algorithm [2]. It looks that the techniques developed here can be used to obtain results for that algorithm when it is applied to continuous action spaces. Finally, although we have not explored them here, consistency results for FQI can be obtained from our results using standard methods, like the methods of sieves. We believe that the methods developed here will eventually lead to algorithms where the function approximation methods are chosen based on the data (similar to adaptive regression methods) so as to optimize performance, which in our opinion is one of the biggest open questions in RL. Currently we are exploring this possibility. Acknowledgments Andr´as Antos would like to acknowledge support for this project from the Hungarian Academy of Sciences (Bolyai Fellowship). Csaba Szepesv´ari greatly acknowledges the support received from the Alberta Ingenuity Fund, NSERC, the Computer and Automation Research Institute of the Hungarian Academy of Sciences. References [1] A. Antos, Cs. Szepesv´ari, and R. Munos. Learning near-optimal policies with Bellman-residual minimization based fitted policy iteration and a single sample path. In COLT-19, pages 574–588, 2006. [2] A. Antos, Cs. Szepesv´ari, and R. Munos. Learning near-optimal policies with Bellman-residual minimization based fitted policy iteration and a single sample path. Machine Learning, 2007. (accepted). [3] A. Antos, Cs. Szepesv´ari, and R. Munos. Value-iteration based fitted policy iteration: learning with a single trajectory. In IEEE ADPRL, pages 330–337, 2007. [4] D. P. Bertsekas and S.E. Shreve. Stochastic Optimal Control (The Discrete Time Case). Academic Press, New York, 1978. [5] D. Ernst, P. Geurts, and L. Wehenkel. Tree-based batch mode reinforcement learning. Journal of Machine Learning Research, 6:503–556, 2005. [6] R.S. Sutton and A.G. Barto. Reinforcement Learning: An Introduction. Bradford Book. MIT Press, 1998. [7] N. Cristianini and J. Shawe-Taylor. An introduction to support vector machines (and other kernel-based learning methods). Cambridge University Press, 2000. [8] J.A. Boyan and A.W. Moore. Generalization in reinforcement learning: Safely approximating the value function. In NIPS-7, pages 369–376, 1995. [9] P.L. Bartlett, P.M. Long, and R.C. Williamson. Fat-shattering and the learnability of real-valued functions. Journal of Computer and System Sciences, 52:434–452, 1996. [10] A.N. Kolmogorov and V.M. Tihomirov. ϵ-entropy and ϵ-capacity of sets in functional space. American Mathematical Society Translations, 17(2):277–364, 1961. [11] R. Munos and Cs. Szepesv´ari. Finite time bounds for sampling based fitted value iteration. Technical report, Computer and Automation Research Institute of the Hungarian Academy of Sciences, Kende u. 13-17, Budapest 1111, Hungary, 2006. [12] A.Y. Ng and M. Jordan. PEGASUS: A policy search method for large MDPs and POMDPs. In Proceedings of the 16th Conference in Uncertainty in Artificial Intelligence, pages 406–415, 2000. [13] P.L. Bartlett and A. Tewari. Sample complexity of policy search with known dynamics. In NIPS-19. MIT Press, 2007. [14] M. Anthony and P. L. Bartlett. Neural Network Learning: Theoretical Foundations. Cambridge University Press, 1999. [15] M. Riedmiller. Neural fitted Q iteration – first experiences with a data efficient neural reinforcement learning method. In 16th European Conference on Machine Learning, pages 317–328, 2005. [16] S. Kalyanakrishnan and P. Stone. Batch reinforcement learning in a complex domain. In AAMAS-07, 2007. 8
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Blind channel identification for speech dereverberation using l1-norm sparse learning Yuanqing Lin†, Jingdong Chen‡, Youngmoo Kim♯, Daniel D. Lee† †GRASP Laboratory, Department of Electrical and Systems Engineering, University of Pennsylvania ‡Bell Laboratories, Alcatel-Lucent ♯Department of Electrical and Computer Engineering, Drexel University Abstract Speech dereverberation remains an open problem after more than three decades of research. The most challenging step in speech dereverberation is blind channel identification (BCI). Although many BCI approaches have been developed, their performance is still far from satisfactory for practical applications. The main difficulty in BCI lies in finding an appropriate acoustic model, which not only can effectively resolve solution degeneracies due to the lack of knowledge of the source, but also robustly models real acoustic environments. This paper proposes a sparse acoustic room impulse response (RIR) model for BCI, that is, an acoustic RIR can be modeled by a sparse FIR filter. Under this model, we show how to formulate the BCI of a single-input multiple-output (SIMO) system into a l1norm regularized least squares (LS) problem, which is convex and can be solved efficiently with guaranteed global convergence. The sparseness of solutions is controlled by l1-norm regularization parameters. We propose a sparse learning scheme that infers the optimal l1-norm regularization parameters directly from microphone observations under a Bayesian framework. Our results show that the proposed approach is effective and robust, and it yields source estimates in real acoustic environments with high fidelity to anechoic chamber measurements. 1 Introduction Speech dereverberation, which may be viewed as a denoising technique, is crucial for many speech related applications, such as hands-free teleconferencing and automatic speech recognition. It is a challenging signal processing task and remains an open problem after more than three decades of research. Although many approaches [1] have been developed for speech dereverberation, blind channel identification (BCI) is believed to be the key to thoroughly solving the dereverberation problem. Most BCI approaches rely on source statistics (higher order statistics [2] or statistics of LPC coefficients [3]), or spatial difference among multiple channels [4] for resolving solution degeneracies due to the lack of knowledge of the source. The performance of these approaches depends on how well they model real acoustic systems (mainly sources and channels). The BCI approaches using source statistics need a long sequence of data to build up the statistics, and their performance often degrades significantly in real acoustic environments where acoustic systems are time-varying and only approximately time-invariant during a short time window. Besides the data efficiency issue, there are some other difficulties in the BCI approaches using source statistics, for example, non-stationarity of a speech source, whitening side effect, and non-minimum phase of a filter [2]. In contrast, the BCI approaches exploiting channel spatial difference are blind to the source, and thus they avoid those difficulties arising in assuming source statistics. Unfortunately, these approaches are often too ill-conditioned to tolerate even a very small amount of ambient noise. In general, BCI for speech dereverberation is an active research area, and the main challenge is how to build an effective acoustic model that not only can resolve solution degeneracies due to the lack of knowledge of the source, but also robustly models real acoustic environments. 1 To address the challenge, this paper proposes a sparse acoustic room impulse response (RIR) model for BCI, that is, an acoustic RIR can be modeled by a sparse FIR filter. The sparse RIR model is theoretically sound [5], and it has been shown to be useful for estimating RIRs in real acoustic environments when the source is given a priori [6]. In this paper, the sparse RIR model is incorporated with channel spatial difference, resulting a blind sparse channel identification (BSCI) approach for a single-input multiple-output (SIMO) acoustic system. The BSCI approach aims to resolve some of the difficulties in conventional BCI approaches. It is blind to the source and therefore avoids the difficulties arising in assuming source statistics. Meanwhile, the BSCI approach is expected to be robust to ambient noise. It has been shown that, when the source is given a priori [7], the prior knowledge about sparse RIRs plays an important role in robustly estimating RIRs in noisy acoustic environments. Furthermore, the statistics describing the sparseness of RIRs are governed by acoustic room characteristics, and thus they are close to be stationary with respect to a specific room. This is advantageous in terms of both learning the statistics and applying them in channel identification. Based on the cross relation formulation [4] of BCI, this paper develops a BSCI algorithm that incorporates the sparse RIR model. Our choice for enforcing sparsity is l1-norm regularization [8], which has been the driving force for many emerging fields in signal processing, such as sparse coding and compressive sensing. In the context of BCI, two important issues need to be addressed when using l1-norm regularization. First, the existing cross relation formulation for BCI is nonconvex, and directly enforcing l1-norm regularization will result in an intractable optimization. Second, l1-norm regularization parameters are critical for deriving correct solutions, and their improper setting may lead to totally irrelevant solutions. To address these two issues, this paper shows how to formulate the BCI of a SIMO system into a convex optimization, indeed an unconstrained least squares (LS) problem, which provides a flexible platform for incorporating l1-norm regularization; it also shows how to infer the optimal l1-norm regularization parameters directly from microphone observations under a Bayesian framework. We evaluate the proposed BSCI approach using both simulations and experiments in real acoustic environments. Simulation results illustrate the effectiveness of the proposed sparse RIR model in resolving solution degeneracies, and they show that the BSCI approach is able to robustly and accurately identify filters from noisy microphone observations. When applied to speech dereverberation in real acoustic environments, the BSCI approach yields source estimates with high fidelity to anechoic chamber measurements. All of these demonstrate that the BSCI approach has the potential for solving the difficult speech dereverberation problem. 2 Blind sparse channel identification (BSCI) 2.1 Previous work Our BSCI approach is based on the cross relation formulation for blind SIMO channel identification [4]. In a one-speaker two-microphone system, the microphone signals at time k can be written as: xi(k) = s(k) ∗hi + ni(k), i = 1, 2, (1) where ∗denotes linear convolution, s(k) is a source signal, hi represents the channel impulse response between the source and the ith microphone, and ni(k) is ambient noise. The cross relation formulation is based on a clever observation, x2(k) ∗h1 = x1(k) ∗h2 = s(k) ∗h1 ∗h2, if the microphone signals are noiseless [4]. Then, without requiring any knowledge from the source signal, the channel filters can be identified by minimizing the squared cross relation error. In matrix-vector form, the optimization can be written as h∗ 1, h∗ 2 = argmin ∥h1∥2+∥h2∥2=1 1 2∥X2h1 −X1h2∥2 (2) where Xi is the (N + L −1) × L convolution Toeplitz matrix whose first row and first column are [xi(k −N + 1), 0, . . . , 0] and [xi(k −N + 1), xi(k −N + 2), ..., xi(k), 0, . . . , 0]T, respectively, N is the microphone signal length, L is the filter length, hi(i = 1, 2) are L×1 vectors representing the filters, ∥· ∥denotes l2-norm, and the constraint is to avoid the trivial zero solution. It is easy to see that the above optimization is a minimum eigenvalue problem, and it can be solved by eigenvalue decomposition. As shown in [4], the eigenvalue decomposition approach finds the true solution within a constant time delay and a constant scalar factor when 1) the system is noiseless; 2) the two 2 filters are co-prime (namely, no common zeros); and 3) the system is sufficiently excited (i.e., the source needs to have enough frequency bands). Unfortunately, the eigenvalue decomposition approach has not been demonstrated to be useful for speech dereverberation in real acoustic environments. This is because the conditions for finding true solutions are difficult to sustain. First, microphone signals in real acoustic environments are always immersed in excessive ambient noise (such as air-conditioning noise), and thus the noiseless assumption is never true. Second, it requires precise information about filter order for the filters to be co-prime, however, the filter order itself is hard to compute accurately since the filters modeling RIRs are often thousands of taps long. As a result, eigenvalue decomposition approach is often ill-conditioned and very sensitive to even a very small amount of ambient noise. Our proposed sparse RIR model aims to alleviate those difficulties. Under the sparse RIR model, sparsity regularization automatically determines filter order since surplus filter coefficients are forced to be zero. Furthermore, previous work [7] has demonstrated that, when the source is given a priori, sparsity regularization plays an important role in robustly estimating RIRs in noisy acoustic environments. In order to exploit the sparse RIR model, we first formulate the BCI using cross relation into a convex optimization, which will provide a flexible platform for enforcing l1-norm sparsity regularization. 2.2 Convex formulation The optimization in Eq. 2 is nonconvex because its domain, ∥h1∥2 + ∥h2∥2 = 1, is nonconvex. We propose to replace it with a convex singleton linear constraint, and the optimization becomes h∗ 1, h∗ 2 = argmin h1(l)=1 1 2∥X2h1 −X1h2∥2 (3) where h1(l) is the lth element of filter h1. It is easy to see that, when microphone signals are noiseless, the optimizations in Eqs. 2 and 3 yield equivalent solutions within a constant time delay and a constant scalar factor. Because the optimization is a minimization, h1(l) tends to align with the largest coefficient in filter h1, which normally is the coefficient corresponding to the direct path. Consequently, the singleton linear constraint removes two degrees of freedom in filter estimates: a constant time delay (by fixing l) and a constant scalar factor [by fixing h1(l) = 1]. The choice of l (0 ≤l ≤L −1) is arbitrary as long as the direct path in filter h2 is no more than l samples earlier than the one in filter h1. The new formulation in Eq. 3 has many advantages. It is convex and indeed an unconstrained LS problem since the singleton linear constraint can be easily substituted into the objective function. Furthermore, the new LS formulation is more robust to ambient noise than the eigenvalue decomposition approach in Eq. 2. This can be better viewed in the frequency domain. Because the squared cross relation error (the objective function in Eqs. 2 and 3) is weighted in the frequency domain by the power spectrum density of a common source, the total filter energy constraint in Eq. 2 may be filled with less significant frequency bands which contribute little to the source and are weighted less in the objective function. As a result, the eigenvalue decomposition approach is very sensitive to noise. In contrast, the singleton linear constraint in Eq. 3 has much less coupling in filter energy allocation, and the new LS approach is more robust to ambient noise. Then, the BSCI approach is to incorporate the LS formulation with l1-norm sparsity regularization, and the optimization becomes h∗ 1, h∗ 2 = argmin h1(l)=1 1 2∥X2h1 −X1h2∥2 + λ′ L−1 X j=0 [|h1(j)| + |h2(j)|] (4) where λ′ is a nonnegative scalar regularization parameter that balances the preference between the squared cross relation error and the sparseness of solutions described by their l1-norm. The setting of λ′ is critical for deriving appropriate solutions, and we will show how to compute its optimal setting in a Bayesian framework in Section 2.3. Given a λ′, the optimization in Eq. 4 is convex and can be solved by various methods with guaranteed global convergence. We implemented the Mehrotra predictor-corrector primal-dual interior point method [9], which is known to yield better search directions than the Newton’s method. Our implementation usually solves the optimization in Eq. 4 with extreme accuracy (relative duality gap less than 10−14) in less than 20 iterations. 3 2.3 Bayesian l1-norm sparse learning for blind channel identification The l1-norm regularization parameter λ′ in Eq. 4 is critical for deriving appropriately sparse solutions. How to determine its optimal setting is still an open research topic. A recent development is to solve the optimization in Eq. 4 with respect to all possible values of λ′ [10], and cross-validation is then employed to find an appropriate solution. However, it is not easy to obtain extra data for crossvalidation in BCI since real acoustic environments are often time-varying. In this study, we develop a Bayesian framework for inferring the optimal regularization parameters for the BSCI formulation in Eq. 4. A similar Bayesian framework can be found in [7], where the source was assumed to be known a priori. The optimization in Eq. 4 is a maximum-a-posteriori estimation under the following probabilistic assumptions P X2h1 −X1h2|σ2, h1, h2 = 1 (2πσ2)(N+L−1)/2 exp −1 2σ2 ∥X2h1 −X1h2∥2 , (5) P (h1, h2|λ) = λ 2 2L exp −λ L−1 X j=0 [|h1(j)| + |h2(j)|] (6) where the cross relation error is an I.I.D. zero-mean Gaussian with variance σ2, and the filter coefficients are governed by a Laplacian sparse prior with the scalar parameter λ. Then, the regularization parameter λ′ in Eq. 4 can be written as λ′ = σ2λ. (7) When the ambient noise [n1(k) and n2(k) in Eq. 1] is an I.I.D. zero-mean Gaussian with variance σ2 0, the parameter σ2 can be approximately written as σ2 = σ2 0(∥h1∥2 + ∥h2∥2), (8) because x2(k) ∗h1 −x1(k) ∗h2 = n2(k) ∗h1 −n1(k) ∗h2. The above form of σ2 is only an approximation because the cross relation error is temporally correlated through the convolution. Nevertheless, since the cross relation error is the result of the convolutive mixing, its distribution will be close to the Gaussian with its variance described by Eq. 8 according to the central limit theorem. We choose to estimate the ambient noise level (σ2 0) directly from microphone observations via restricted maximum likelihood [11]: σ2 0 = min s,h1,h2 1 N −L −1 2 X i=1 N−1 X k=0 ∥xi(k) −s(k) ∗hi∥2 (9) where the denominator N −L −1 (but not 2N) accounts for the loss of the degrees of freedom during the optimization. The above minimization is solved by coordinate descent alternatively with respect to the source and the filters. It is initialized with the LS solution by Eq. 3 and often able to yield a good σ2 0 estimate in a few iterations. Note that each iteration can be computed efficiently in the frequency domain. Meanwhile, the parameter λ can be computed by λ = 2L PL−1 j=0 [|h1(j)| + |h2(j)|] , (10) as a result of finding the optimal Laplacian distribution given its sufficient statistics. With the Eqs. 8 and 10, finding the optimal regularization parameters becomes computing the statistics of filters, ∥h1∥2 + ∥h2∥2 and PL−1 j=0 [|h1(j)| + |h2(j)|]. These statistics are closely related to acoustic room characteristics and may be computed from them if they are known a priori. For example, the reverberation time of a room defines how fast echoes decay −60 dB, and it can be used to compute the filter statistics. More generally, we choose to compute the statistics directly from microphone observations in the Baysian framework by maximizing the marginal likelihood, P(X2h1 −X1h2|σ2, λ) = R h1(l)=1 P(X2h1 −X1h2, h1, h2|σ2, λ)dh1dh2. The optimization is through Expectation-Maximization (EM) updates [7]: σ2 ←− σ2 0 Z h(l)=1 (∥h1∥2 + ∥h2∥2)Q(h1, h2)dh1dh2 (11) λ ←− 2L R h(l)=1(PL−1 j=0 |h1(j)| + |h2(j)|)Q(h1, h2)dh1dh2 (12) 4 where h1 and h2 are treated as hidden variables, σ2 and λ are parameters, and Q(h1, h2) ∝ exp{−1 2σ2 ∥X2h1 −X1h2∥2 −λ[PL−1 j=0 |h1(j)| + |h2(j)|]} is the probability distribution of h1 and h2 given the current estimate of σ2 and λ. The integrals in Eqs. 11 and 12 can be computed using the variational scheme described in [7]. The EM updates often converge to a good estimate of σ2 and λ in a few iterations. Moreover, since the filter statistics are relatively stationary for a specified room, the Bayesian inference may be carried out off-line and only once if the room conditions stay the same. After the filters are identified by BCI approaches, the source can be computed by various methods [12]. We choose to estimate the source by the following optimization s∗= argmin s 2 X i=1 N−1 X k=0 ∥xi(k) −s(k) ∗hi∥2, (13) which will yield maximum-likelihood (ML) estimation if the filter estimates are accurate. 3 Simulations and Experiments 3.1 Simulations 3.1.1 Simulations with artificial RIRs We first employ a simulated example to illustrate the effectiveness of the proposed sparse RIR model for BCI. In the simulation, we used a speech sequence of 1024 samples (with 16 kHz sampling rate) as the source (s) and simulated two 16-sample FIR filters (h1 and h2). The filter h1 had nonzero elements only at indices 0, 2, and 12 with amplitudes of 1, -0.7, and 0.5, respectively; the filter h2 had nonzero elements only at indices 2, 6, 8, and 10 with amplitudes of 1, -0.6, 0.6, and 0.4, respectively. Notice that both h1 and h2 are sparse. Then the simulated microphone observations (x1 and x2) were computed by Eq. 1 with the ambient noise being real noise recorded in a classroom. The noise was scaled so that the signal-to-noise ratio (SNR) of the microphone signals was approximately 20 dB. Because a big portion of the noise (mainly air-conditioning noise) was at low frequency, the microphone observations were high-passed with a cut-off frequency of 100 Hz before they were fed to BCI algorithms. In the BSCI algorithm, the l1-norm regularization parameters, σ2 and λ, were estimated in the Bayesian framework using the update rules given in Eqs. 11 and 12. Figure 1 shows the filters identified by different BCI approaches. Compared to the conventional eigenvalue decomposition method (Eq. 2), the new convex LS approach (Eq. 3) is more robust to ambient noise and yielded better filter estimates even though the estimates still seem to be convolved by a common filter. The proposed BSCI approach (Eq. 4) yielded filter estimates that are almost identical to the true ones. It is evident that the proposed sparse RIR model played a crucial role in robustly and accurately identifying filters in blind manners. The robustness and accuracy gained by the BSCI approach will become essential when the filters are thousands of taps long in real acoustic environments. 3.1.2 Simulations with measured RIRs Here we employ simulations using RIRs measured in real rooms to demonstrate the effectiveness of the proposed BSCI approach for speech dereverberation. Its performance is compared to the beamforming, the eigenvalue decomposition (Eq. 2), and the LS (Eq. 3) approaches. In the simulation, the source sequence (s) was a sentence of speech (approximately 1.5 seconds), and the filters (h1 and h2) were two measured RIRs from York MARDY database (http://www.commsp.ee.ic.ac.uk/ sap/mardy.htm) but down-sampled to 16 kHz (from originally 48 kHz). The original filters in the database were not sparse, but they had many tiny coefficients which were in the range of measurement uncertainty. To make the simulated filters sparse, we simply zeroed out those coefficients whose amplitudes were less than 2% of the maximum. Finally, we truncated the filters to have length of 2048 since there were very few nonzero coefficients after that. With the simulated source and filters, we then computed microphone observations using Eq. 1 with ambient noise being real noise recorded in a classroom. For testing the robustness of different BCI algorithms, the ambient noise was scaled to different levels so that the SNRs varied from 60 dB to 10 dB. Similar to the previous simulations, the simulated observations were high-passed with a cutoff 5 0 5 10 15 −0.5 0 0.5 1 h1 0 5 10 15 −1 0 1 h2 0 5 10 15 −1 0 1 0 5 10 15 −1 0 1 0 5 10 15 −1 0 1 Time (sample) 0 5 10 15 −1 0 1 Time (sample) Estimated True Eig− decomp LS BSCI Figure 1: Identified filters by three different BCI approaches in a simulated example: the eigenvalue decomposition approach (denoted as eig-decomp) in Eq. 2, the LS approach in Eq. 3, and the blind sparse channel identification (BSCI) approach in Eq. 4. The solid-dot lines represent the estimated filters, and the dot-square lines indicate the true filters within a constant time delay and a constant scalar factor. −60 −50 −40 −30 −20 −10 0 20 40 60 80 100 Noise level (dB) Normalized correlation (%) Filter estimates −60 −50 −40 −30 −20 −10 0 20 40 60 80 100 Noise level (dB) Normalized correlation (%) Source estimates eigen−decomp beamforming LS BSCI eigen−decomp LS BSCI Figure 2: The simulation results using measured real RIRs. The normalized correlation (defined in Eq. 14) of the estimates were computed with respect to their true values. The filters were identified by three different approaches: the eigenvalue decomposition approach (denoted as eigen-decomp) in Eq. 2 , the LS approach in Eq. 3, and the blind sparse channel identification (BSCI) approach in Eq. 4. After the filters were identified, the source was estimated by Eq. 13. The source estimated by beamforming is also presented as a baseline reference. frequency of 100 Hz before they were fed to different BCI algorithms. In the BSCI approach, the l1-norm regularization parameters were iteratively computed using the updates in Eqs. 11 and 12. After filters were identified, the source was estimated using Eq. 13. Because both filter and source estimates by BCI algorithms are within a constant time delay and a constant scalar factor, we use normalized correlation for evaluating the estimates. Let ˆs and s0 denote an estimated source and the true source, respectively, then the normalized correlation C(ˆs, s0) is defined as C(ˆs, s0) = max m P k ˆs(k −m)s0(k) ∥ˆs∥∥s0∥ (14) where m and k are sample indices, and ∥· ∥denotes l2-norm. It is easy to see that, the normalized correlation is between 0% and 100%: it is equal to 0% when the two signals are uncorrelated, and it is equal to 100% only when the two signal are identical within a constant time delay and a constant scalar factor. The definition in Eq. 14 is also applicable to the evaluation of filter estimates. The simulation results are shown in Fig. 2. Similar to what we observed in the previous example, the convex LS approach (Eq. 3) shows significant improvement in both filter and source estimation compared to the eigenvalue decomposition approach (Eq. 2). In fact, the eigenvalue decomposition 6 1 2 3 4 5 6 7 8 9 10 30 40 50 60 70 80 90 100 Experiments Normalized correlation (%) Beamforming Eig−decomp LS BSCI Figure 3: The source estimates of 10 experiments in real acoustic environments. The normalized correlation was with respect to their anechoic chamber measurement. The filters were identified by three different BCI approaches: the eigenvalue decomposition approach (denoted as eig-decomp) in Eq. 2, the LS approach in Eq. 3, and the blind sparse channel identification (BSCI) approach in Eq. 4. The beamforming results serve as the baseline performance for comparison. 0 500 1000 1500 -1 -0.5 0 0.5 1 Amplitude h1 0 500 1000 1500 -0.5 0 0.5 1 Time (samples) Amplitude h2 0 0 0 100 200 300 400 500 600 700 800 900 −10 0 10 Amplitude Anechoic chamber measurement 0 100 200 300 400 500 600 700 800 900 −5 0 5 Amplitude Real room recording (left microphone) 0 100 200 300 400 500 600 700 800 900 −5 0 5 Amplitude Source estimate using the filters identified by BSCI Time (samples) A B C Figure 4: Results of Experiment 6 in Fig. 3. Left: the filters estimated by the proposed blind sparse channel identification (BSCI) approach. They are sparse as indicated by the enlarged segments. Right: a segment of source estimate (shown in C) using the BSCI approach. It is compared with its anechoic measurement (shown in A) and its microphone recording (shown in B). approach did not yield relevant results because it was too ill-conditioned due to the long filters. The remarkable performance came from the BSCI approach, which incorporates the convex LS formulation with the sparse RIR model. In particular, the BSCI approach yielded higher than 90% normalized correlation in source estimates when SNR was better than 20 dB, and it yielded higher than 99% normalized correlation in the low noise limit. The performance of the canonical delayand-sum beamforming is also presented as the baseline for all BCI algorithms. 3.2 Experiments We also evaluated the proposed BSCI approach using signals recorded in real acoustic environments. We carried out 10 experiments in total in a reverberant room. In each experiment, a sentence of speech (approximately 1.5 seconds, and the same for all experiments) was played through a loudspeaker (NSW2-326-8A, Aura Sound) and recorded by a matched omnidirectional microphone pair (M30MP, Earthworks). The speaker-microphone positions (and thus RIRs) were different in different experiments. Because the recordings had a large amount of low-frequency noise, they were high-passed with a cutoff frequency of 100 Hz before they were fed to BCI algorithms. In the BSCI approach, the l1-norm regularization parameters, σ2 and λ, were iteratively computed using the updates in Eq. 11 and 12. After the filters were identified, the sources were computed using Eq. 13. We also had recordings in the anechoic chamber at Bell Labs using the same instruments and settings, and the anechoic measurement served as the approximated ground truth for evaluating the performance of different BCI approaches. 7 Figure 3 shows the source estimates in the 10 experiments in terms of their normalized correlation to the anechoic measurement. The performance of the proposed BSCI is compared with the beamforming, the eigenvalue decomposition (Eq. 2), and the convex LS (Eq. 3) approaches. The results of the 10 experiments unanimously support our previous findings in simulations. First, the convex LS approach yielded significantly better source estimates than the eigenvalue decomposition method. Second, the proposed BSCI approach, which incorporates the convex LS formulation with the sparse RIR model, yielded the most dramatic results, achieving 85% or higher of normalized correlation in source estimates in most experiments while the LS approach only obtained approximately 70% of normalized correlation. Figure 4 shows one instance of filter and source estimates. The estimated filters have about 2000 zeros out of totally 3072 coefficients, and thus they are sparse. This observation experimentally validates our hypothesis of the sparse RIR models, namely, an acoustic RIR can be modeled by a sparse FIR filter. The source estimate shown in Fig. 4 vividly illustrates the convolution and dereverberation process. It only plots a small segment to reveal greater details. As we see, the anechoic measurement was clean and had clear harmonic structure; the signal recorded in the reverberant room was smeared by echoes during the convolution process; and then, the dereverberation using our BSCI approach deblurred the signal and recovered the underlying harmonic structure. 4 Discussion We propose a blind sparse channel identification (BSCI) approach for speech dereverberation. It consists of three important components. The first is the sparse RIR model, which effectively resolves solution degeneracies and robustly models real acoustic environments. The second is the convex formulation, which guarantees global convergence of the proposed BSCI algorithm. And the third is the Bayesian l1-norm sparse learning scheme that infers the optimal regularization parameters for deriving optimally sparse solutions. The results demonstrate that the proposed BSCI approach holds the potential to solve the speech dereverberation problem in real acoustic environments, which has been recognized as a very difficult problem in signal processing. The acoustic data used in this paper are available at http://www.seas.upenn.edu/∼linyuanq/Research.html. Our future work includes side-by-side comparison between our BSCI approach and existing source statistics based BCI approaches. Our goal is to build a uniform framework that combines various prior knowledge about acoustic systems for best solving the speech dereverberation problem. References [1] T. Nakatani, M. Miyoshi, and K. Kinoshita, “One microphone blind dereverberation based on quasiperiodicity of speech signals,” in NIPS 16. 2004. [2] A. Hyvarinen, J. Karhunen, and E. Oja, Independent Component Analysis, New York, NY: John Wiley and Sons, 2001. [3] H. Attias, J. C. Platt, A. Acero, and L. Deng, “Speech denoising and dereverberation using probabilistic models,” in NIPS 13, 2000. [4] L. Tong, G. Xu, and T. Kailath, “Blind identification and equalization based on second-order statistics: A time domain approach,” IEEE Trans. Information Theory, vol. 40, no. 2, pp. 340–349, 1994. [5] J. B. Allen and D. A. Berkley, “Image method for efficiently simulating small-room acoustics,” J. Acoustical Society America, vol. 65, pp. 943–950, 1979. [6] D. L. Duttweiler, “Proportionate normalized least-mean-squares adaptation in echo cancelers,” IEEE Trans. Speech Audio Processing, vol. 8, pp. 508–518, 2000. [7] Y. Lin and D. D. Lee, “Bayesian L1-norm sparse learning,” in Proc. ICASSP, 2006. [8] S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Scientific Computing, vol. 20, no. 1, pp. 33–61, 1998. [9] S. J. Wright, Primal-Dual Interior Point Methods, Philadelphia, PA: SIAM, 1997. [10] D. M. Malioutov, M. Cetin, and A. S. Willsky, “Homotopy continuation for sparse signal representation,” in Proc. ICASSP, 2005. [11] D.A. Harville, “Maximum likelihood approaches to variance component estimation and to related problems,” J. American Statistical Association, vol. 72, pp. 320–338, 1977. [12] M. Miyoshi and Y. Kaneda, “Inverse filtering of room acoustics,” IEEE Trans. Acoustics, Speech, and Signal Processing, vol. 36, no. 2, pp. 145–152, 1988. 8
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Predicting Brain States from fMRI Data: Incremental Functional Principal Component Regression S. Ghebreab ISLA/HCS lab, Informatics Institute University of Amsterdam, The Netherlands ghebreab@science.uva.nl A.W.M. Smeulders ISLA lab, Informatics Institute University of Amsterdam, The Netherlands smeulders@science.uva.nl P. Adriaans HCS lab, Informatics Institute University of Amsterdam, The Netherlands pietera@science.uva.nl Abstract We propose a method for reconstruction of human brain states directly from functional neuroimaging data. The method extends the traditional multivariate regression analysis of discretized fMRI data to the domain of stochastic functional measurements, facilitating evaluation of brain responses to complex stimuli and boosting the power of functional imaging. The method searches for sets of voxel time courses that optimize a multivariate functional linear model in terms of R2statistic. Population based incremental learning is used to identify spatially distributed brain responses to complex stimuli without attempting to localize function first. Variation in hemodynamic lag across brain areas and among subjects is taken into account by voxel-wise non-linear registration of stimulus pattern to fMRI data. Application of the method on an international test benchmark for prediction of naturalistic stimuli from new and unknown fMRI data shows that the method successfully uncovers spatially distributed parts of the brain that are highly predictive of a given stimulus. 1 Introduction To arrive at a better understanding of human brain function, functional neuroimaging traditionally studies the brain’s responses to controlled stimuli. Controlled stimuli have the benefit of leading to clear and often localized response signals in fMRI as they are specifically designed to affect only certain brain functions. The drawback of controlled stimuli is that they are a reduction of reality: one cannot be certain whether the response is due to the reduction or due to the stimulus. Naturalistic stimuli open the possibility to avoid the question whether the response is due to the reduction or the signal. Naturalistic stimuli, however, carry a high information content in their spatio-temporal structure that is likely to instigate complex brain states. The immediate consequence hereof is that one faces the task of isolating relevant responses amids complex patterns. To reveal brain responses to naturalistic stimuli, advanced signal processing methods are required that go beyond conventional mass univariate data analysis. Univariate techniques generally lack sufficient power to capture the spatially distributed response of the brain to naturalistic stimuli. Multivariate pattern techniques, on the other hand, have the capacity to identify patterns of information when they are present across the full spatial extent of the brain without attempting to localize function. Here, we propose a multivariate pattern analysis approach for predicting naturalistic stimuli on the basis of fMRI data. Inverting the task from correlating stimuli with fMRI data to predicting stimuli from fMRI data makes it easier to evaluate brain responses to naturalistic stimuli and may extend the power of functional imaging substantially [1]. Various multivariate approaches for reconstruction of brain states directly from fMRI measurements have recently been proposed. In most of these approaches, a classifier is trained directly on the fMRI data to discriminate between known different brain states. This classifier is then used to predict brain states on the basis of new and unknown fMRI data alone. Such approaches have been used to predict what percept is dominant in a binocular rivalry protocol [2], what the orientation is of structures subjects are viewing [3] and what the semantic category is of objects [4] and words [5] subjects see on a screen. In one competition [6], participants trained pattern analyzers on fMRI of subjects viewing two short movies as well as on the subject’s movie feature ratings. Then participants employed the analyzers to predict the experience of subjects watching a third movie based purely on fMRI data. Very accurate predictions were reported for identifying the presence of specific time varying movie features (e.g. faces, motion) and the observers who coded the movies [7]. We propose an incremental multivariate linear modeling approach for functional covariates, i.e. where both the fMRI data and external stimuli are continuous. This approach differs fundamentally from existing multivariate linear approaches (e.g. [8]) that instantly fit a given model to the data within the linear framework under the assumption that both the data and the model are discrete. Contemporary neuroimaging studies increasingly use high-resolution fMRI to accurately capture continuous brain processes, frequently instigated by continuous stimulations. Hence, we propose the use of functional data analysis [9], which treats data, or the processes giving rise to them, as functions. This not only allows to overcome limitations in neuroimaing studies due to the large number of data points compared to the number of samples, but also allows to exploit the fact that functions defined on a specific domain form an inner product vector space, and in most circumstances can be treated algebraically like vectors [10]. We extend classical multivariate regression analysis of fMRI data [11] to stochastic functional measurements. We show that, cast into an incremental pattern searching framework, functional multivariate regression provides a powerful technique for fMRI-based prediction of naturalistic stimuli. 2 Method In the remainder, we consider stimuli data and data produced by fMRI scanners as continuous functions of time, sampled at the scan interval and subject to observational noise. We treat the data within a functional linear model where both the predictant and predictor are functional, but where the design matrix that takes care of the linear mapping between the two is vectorial. 2.1 The Predictor The predictor data are derived directly from the four-dimensional fMRI data I(x, t), where x ∈ℜ3 denotes the spatial position of a voxel and t denotes its temporal position. We represent each of the S voxel time courses in functional form by fs(t), with t denoting the continuous path parameter and s = 1, ..., S . Rather than directly using voxel time courses for prediction, we use their principal components to eliminate collinearity in the predictor set. Following [10], we use functional principal component analysis. Viviani et al. [10] showed that functional principal components analysis is more effective than is its ordinary counterpart in recovering the signal of interest in fMRI data, even if limited or no prior knowledge of the hemodynamic function or experimental design is specified. In contrast to [10], however, our approach incrementally zooms in on stimuli-related voxel time courses for dimension reduction (see section 2.5). Given the set of S voxel time courses represented by the vector of functionals f(t) = [ f1(t), ..., fS (t)]T, functional principal components analysis extracts main modes of variation in f(t). The number of modes to retain is determined from the proportion of the variance that needs to be explained. Assuming this is Q, the central concept is that of taking the linear combination fsq = Z t fs(t)αq(t)dt (1) where fsq is the principal component score value of voxel time course fs(t) in dimension q. Principal components αq(t), q = 1, .., Q are sought for one-by-one by optimizing αq(t) = max α∗q(t) 1 S SX s=1 f 2 sq (2) where αq(t) is subject to the following orthonormal constraints Z t αq(t)2dt = 1 Z t αk(t)αq(t)dt = 0, k ≤q. (3) The mapping of fs(t) onto the subspace spanned by the first Q principal component curves results in the vector of scalars fs = [ fs1, ..., fsQ]. We define the S × Q matrix F = [f1, ..., fS ]T of principal components scores as our predictor data in linear regression. That is, we perform principal component regression with F as model, allowing to naturally deal with temporal correlations, multicollinearity and systematic signal variation. 2.2 The Predictand We represent the stimulus pattern by the functional 1(t), t being the continuous time parameter. We register 1(t) to each voxel time course fs(t) in order to be able to compare equivalent time points on stimulus and brain activity data. Alignment reduces to finding the warping function ωs(t) that produces the warped stimulus function gs(t) = 1(ωs(t)). (4) The time warping function ωs(t) is strictly monotonic, differentiable up to a certain order and takes care of a small shift and nonlinear transformation. A global alignment criteria and least squares estimation is used: ωs(t) = min ω∗s Z t (1(ω∗ s(t)) −fs(t))2dt. (5) Registration of 1(t) to all voxel time courses S results in predictand data g(t) = [g1(t), ..., gS (t)]T, where g(t) is 1(t) registered onto voxel times-course f(t). Our motivation for using voxel-wise registration over standard convolution of stimulus 1(t) with the hemodynamic reponse function, is the large variability in hemodynamic delays across brain regions and subjects. A non -linear warp of 1(t) does not guarantee an outcome that is associated with brain physiology, however it allows to capture unknown subtle localized variations in hemodynamic delays across brain regions and subjects. 2.3 The Model We employ the predictor data to explain the predictand data within a linear modeling approach, i.e. our multivariate linear model is defined as g(t) = Fβ(t) + ǫ(t) (6) with β(t) = [β1(t), ..., βQ(t)]T being the Q×1 vector of regression functions. The regression functions are estimated by least squares minimization such that ˆβ(t) = min β ∗(t) Z t (g(t) −Fβ∗(t))2dt, (7) under the assumption that the residual functions ǫ(t) = [ǫ1(t), ...., ǫS (t)]T are independent and normally distributed with zero mean. The estimated regression functions provide the best estimate of g(t) in least squares sense: ˆg(t) = Fˆβ(t). (8) Given a new (sub)set of voxel time courses, prediction of a stimulus pattern now reduces to computing the matrix of principal component scores from this new set and weighting these scores by the estimated regression functions ˆβ(t). 2.4 The Objective The overall fit of the model to the data is expressed in terms of adjusted R2 statistic. The functional counterpart of the traditional R2 is computed on the basis of g(t), its mean ¯g(t) and its estimation ˆg(t). For the voxel set S , ˙gS (t) = SX s=1 (gs(t) −¯g(t))2 (9) ¨gS (t) = SX s=1 (gs(t) −ˆgs(t))2 (10) are derived, where the first term is the variation of the response about its mean and the second the error sum of squares function. The adjusted R-square function is then defined as RS (t) = 1 −¨gS (t)/S −Q −1 ˙gS (t)/S −1 (11) where degrees of freedom S −Q −1 and S −1 adjust the R-square. Our objective is to find the set of voxel time courses S defined as S = max S ∗⊂S Z t RS ∗(t)dt (12) where S ∗denotes a subset of the entire collection of voxels time courses S extracted from a single fMRI scan. That is, we aim at finding spatially distributed voxel responses S that best explain the naturalistic stimuli, without making any prior assumptions about location and size of voxel subsets. 2.5 The Search In order to efficiently find the subset of voxels that maximizes Equation (12), we use PopulationBased Incremental Learning (PBIL) [12], which combines Genetic Algorithms with Competitive Learning. The PBIL algorithm uses a probability vector to explore the space of solutions. It incrementally generates solutions by sampling from that probability vector, evaluates these solutions and selects promising ones to update the probability vector. Here, at increment i, the probability vector pi = [pi 1, ..., pi S ] is used to generate a population of N solutions Mi = [mi 1, ..., mi N], where each member is an S-vector of binary values: mi n = [mi n1, ..., mi nS ]. A value of 1 for mns means that for solution n the corresponding voxel time course fs(t) is included in the predictor set, while a value 0 indicates exclusion. Each member mi n is evaluated in terms of its adjusted R2 value, and the members with highest values form the joint probability vector p∗. A new probability vector is subsequently constructed for the next generation via competitive learning: pi+1 = γpi + (1 −γ)p∗. (13) The learning parameter γ controls the search: a low value enables to focus entirely on the most recent voxel subset while a low value ensures that previously selected voxel subsets are exploited. In order to ensure spatial coherence and limit computation load, we employ the PBIl algorithm not on single time courses, but on averages of spatial clusters of voxel time courses. That is, we first spatially cluster voxel locations as shown in Figure 1, then compute average time course for each cluster and then explore the averages via PBIL for model building. 2.6 The Prediction The subset of voxel time courses that results from population based incremental learning defines the most predictive voxel locations and associated regression functions. Given new and spatially normalized fMRI data, represented by ˜f(t) = [ ˜f1(t), ..., ˜fS (t)]T, prediction of a stimulus then reduces to computing ˜g(t) = ˜Fˆβ(t). (14) In here, ˜g(t) is the vector of predicted stimuli of which the mean is considered to be the sought stimulus. The matrix ˜F is the principal component scores matrix obtained from performing functional principal components analysis on subset ˜fS(t), with S referring to the set of most predictive voxels as determined by training. Figure 1: Examples of K-means clustering of voxel locations using Euclidean distance. Left: 1024means clustering output. Right: 512-means clustering output. Different gray values indicate different clusters in a spatially normalized brain atlas. 3 Experiments and Results 3.1 Experiment Evaluation of our method is done on a data subset from the 2006 Pittsburgh brain activity interpretation competition (PBAIC) [6, 7], involving fMRI scans of three different subjects and two movie sessions. In each session, a subject viewed a new Home Improvement sitcom movie for approximately 20 minutes. The 20-minute movie contained 5 interruptions where no video was present, only a white fixation cross on a black background. All three subjects watched the same two movies. The scans produced volumes with approximately 35,000 brain voxels, each approximately 3.28mm by 3.28mm by 3.5mm, with one volume produced every 1.75 seconds. These scans were preprocessed (motion correction, slice time correction, linear trend removal) and spatially normalized (non-linear registration to the Montreal Neurological Institute brain atlas). After fMRI scanning, the three subjects watched the movie again to rate 30 movie features at time intervals corresponding to the fMRI scan rate. In our experiments, we focus on the 13 core movie features: amusement, attention, arousal, body parts, environmental sounds, faces, food, language, laughter, motion, music, sadness and tools. The real-valued ratings were convolved with a hemodynamic response function (HRF) modeled by two gamma functions, then subjected to voxel-wise non-linear registration as described in 2.2. For training and testing our model, we removed parts corresponding with video presentations of a white fixation cross on a black background. Taking into account the hemodynamic lag, we divided each fMRI scan and each subject rating into 6 parts corresponding with the movie on parts. On average each movie part contained 105 discrete measurements. We then functionalized these parts by fitting a 30 coefficient B-spline to each voxel’s discrete time course. This resulted in 18 data sets for training (3 subjects × 6 movie parts) and another 18 for testing. We used movie 1 data for training and movie 2 data for prediction, and vice versa. We performed data analysis at two levels. For each feature, first the individual brain scans were analyzed with our method, resulting in a first sifting of voxels. First-level analysis results for a given feature were then subjected to second level analysis to identify across subject predictive voxels. Pearson product-moment correlation coefficient between manual feature rating functions and the automatically predicted feature functions was used as an evaluation measure. 3.2 Results All results were obtained with Q = 4 principal component dimensions, learning parameter value γ = 0.6 and K-means clustering with 1024 clusters for all movie features. These values for Q and γ produced overall highest average cross correlation value in a small parameter optimization experiment (data not shown here). Little performance differences were seen for various numbers of dimensions, indicating that the essential information can be captured with as little as 4 dimension. Significant performance differences across features, however, were observed for different learning parameter values, indicating considerable variation in brain response to distinct stimuli. 0 0.2 0.4 0.6 0.8 1 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 arguments functions Manual versus Predicted Feature Ratings Manual Prediction Figure 2: Left: normalized cross correlation values from cross-validation for 13 core movie features. Right: functionalized subject3 (solid red) and predicted (dotted blue) rating for the language feature of part 5 of movie 1. Figure 2 (left) shows the average of 2 × 18 cross correlation coefficients from cross validation for all 13 movie features. For features faces, language and motion cross correlation values above 0.5 were obtained, meaning that there is a significant degree of match between the subject ratings and the predicted ratings. Reasonable predictions were also obtained for features arousal and body parts. Our results are consistent with top 3 rank entries of 2006 PBAIC in that features faces and language are reliably predicted. These entries used recurrent neural networks, ridge regression and a dynamic Gaussian Markov Random Field modeling on the entire test data benchmark, yielding across feature average cross correlations of: 0.49, 0.49 and 0.47 respectively. Here, the feature average cross correlation value based on the reduced training data set is 0.36. Note, that in the 2006 competition our method ranked first in the actor category [6]. We were able to accurately predict which actor the subjects were seeing purely based on fMRI scans [7]. The best single result, with highest cross correlation value of 0.76, was obtained for feature language of subject 3 watching part 5 of movie 1. For this feature, first level analysis of each of the 18 training data sets associated with movie 2 produced a total number of 1738 predictive voxels. In the second level analysis, these voxels were analyzed again to arrive at a reduced data set of 680 voxels for building the multivariate functional linear model and determining regression functions β(t). For prediction of feature language, corresponding voxel time courses were extracted from the fMRI data of subject 3 watching movie 1 part 5, and weighted by β(t). The manual rating of feature language of movie 1 part 5 by subject 3 and the average of the automatically predicted feature functions are shown in Figure 2 (right). Figure 3: Glass view, gray level image with color overlay and surface rendering of 1738 voxels from first level analysis. Color denotes predictive power and cross hair shows most predictive location. Figure 3 shows glass view, gray level image with color overlay and surface rendering of the 1738 voxels (approximately 40 clusters) from first level analysis. The cross hair shows the voxel location in Brodman area 47 that was found to be predictive across most subjects and movie parts: it was selected in 6 out of 18 training items (see color bar). The predictive locations correspond with the left and right inferior frontal gyrus, which are known to be involved in language processing. The distributed nature of these clusters is consistent with earlier findings that processing involved in language occurs in diffuse brain regions, including primary auditory and visual cortex, frontal regions in the left and right hemisphere, in homologues regions [13]. As we are dealing with curves, the possibility exists to explore additional data characteristics such as curvature. We performed an experiment with 1st order derivative functions, rather than the original functions to exploit potentially available higher order structure. Figure 4 (left) shows the cross correlation for 1st order derivative functions. The cross correlation values are similar to the ones shown in Figure 2. The average cross correlation value is slightly better than for the original data: 0.38. This may indicate that higher order structures may contain more predictive power. In order to get insight in the effect of non-linear warping on prediction performance, we conducted an experiment in which we used convolutions of the stimulus 1(t) with different forms of a HRF function modeled by two gamma functions. Various HRF functions were obtained by varing the delay of response (relative to onset), delay of undershoot (relative to onset), dispersion of response, dispersion of undershoot, ratio of response to undershoot. To determine gs(t), we convolved 1(t) with 16 different HRF functions, and selected the convolved one with highest cross correlation with fs(t) to be gs(t). Hence, we parametrically modeled the HRF and learned its parameters from the data. Figure 4 (right) shows the results of the experiments with convolution of stimuli data with HRF models learned from the data. As can be seen, the cross correlation values are much lower compared to the values in Figure 2 (left). The average cross correlation value is 0.31. Hence, non-linear warping of stimulus onto voxel time course significantly enhances the predictive power of our model. This suggests that non-linear warping is a potential alternative for determining the best possible HRF estimate to overcome potential negative consequences of assuming HRF consistency across subjects or brain regions [14]. Figure 4: Left: normalized cross correlation values from cross-validation for 13 core movie features, using 1st order derivative data. Right: cross correlation values from cross-validation for 13 core movie features, using HRF convoluted rather than warped stimuli data. 4 Conclusion Functional data analysis provides the possibility to fully exploit structure in inherently continuous data such as fMRI. The advantage of functional data analysis for principal component analysis of fMRI data was recently demonstrated in [10]. Here, we proposed a functional linear model that treats fMRI and stimuli as stochastic functional measurements. Cast into an incremental pattern searching framework, the method provides the ability to identify important covariance structure of spatially distributed brain responses and stimuli, i.e. it directly couples activation across brain regions rather than first localizing and then integrating function. The method is suited for unbiased probing of functional characteristics of brain areas as well as for exposing meaningful relations between complex stimuli and distributed brain responses. This finding is supported by the good prediction performance of our method in the 2006 PBAIC international competition for brain activity interpretation. We are currently extending the method with new objective functions, dimension reduction techniques and multi-target search techniques to cope with multiple (interacting) stimuli. Also, in this work we made use of spatial clusters at a single hierarchical level. Preliminary results with hierarchical clustering to arrive at ”supervoxels” at different spatial resolutions, seem to further improve prediction power. References [1] J. Haynes and G. Rees. Decoding mental states from brain activity in humans. Nature Neuroscience, 7(8):523–534, 2006. [2] J. Haynes and G. Rees. Predicting the orientation of invisible stimuli from activity in human primary visual cortex. Nature Neuroscience, 7(5):686–691, 2005. [3] Y. Kamitani and F. Tong. Decoding the visual and subjective contents of the human brain. Nature Neuroscience, 8(5):679–685, 2005. [4] S.M. Polyn, V.S. Natu, J.D. Cohen, and K.A. Norman. Category-specific cortical activity precedes retrieval during memory search. Science, 310(5756):1963–1966, 2005. [5] T.M. Mitchell, R. Hutchinson, R.S. Niculescu, F. Pereira, X. Wang, M. Just, and S. Newman. Learning to decode cognitive states from brain images. Machine Learning, 57(1-2), 2004. [6] W. Schneider, A. Bartels, E. Formisano, J. Haxby, R. Goebel, T. Mitchell, T. Nichols, and G. Siegle. Competition: Inferring experience based cognition from fmri. In Proceedings Organization of Human Brain Mapping Florence Italy June 15, 2006. [7] Editorial. What’s on your mind. Nature Neuroscience, 6(8):981, 2006. [8] K.J. Worsley, J.B. Poline, K.J. Friston, and A.C. Evans. Characterizing the response of pet and fmri data using multivariate linear models. Neuroimage, 6, 1997. [9] J. Ramsay and B. Silverman. Functional Data Analysis. Springer-Verlag, 1997. [10] R. Viviani, G. Grohn, and M. Spitzer. Functional principal component analysis of fmri data. Human Brain Mapping, 24:109–129, 2005. [11] D.B. Rowe and R.G. Hoffmann. Multivariate statistical analysis in fmri. IEEE Engineering in Medicine and Biology, 25:60–64, 2006. [12] Shumeet Baluja. Population-based incremental learning: A method for integrating genetic search based function optimization and competitive learning. Technical Report CMU-CS-94163, Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, 1994. [13] M.A. Gernsbacher and M.P. Kaschak. Neuroimaging studies of language production and comprehension. Annual Review of Psychology, 54:91–114, 2003. [14] D.A. Handwerker, J.M. Ollinger, and M. D’Esposito. Variation of bold hemodynamic response function across subjects and brain regions and their effects on statistical analysis. NeuroImage, 8(21):1639–1651, 2004.
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Agreement-Based Learning Percy Liang Computer Science Division University of California Berkeley, CA 94720 pliang@cs.berkeley.edu Dan Klein Computer Science Division University of California Berkeley, CA 94720 klein@cs.berkeley.edu Michael I. Jordan Computer Science Division University of California Berkeley, CA 94720 jordan@cs.berkeley.edu Abstract The learning of probabilistic models with many hidden variables and nondecomposable dependencies is an important and challenging problem. In contrast to traditional approaches based on approximate inference in a single intractable model, our approach is to train a set of tractable submodels by encouraging them to agree on the hidden variables. This allows us to capture non-decomposable aspects of the data while still maintaining tractability. We propose an objective function for our approach, derive EM-style algorithms for parameter estimation, and demonstrate their effectiveness on three challenging real-world learning tasks. 1 Introduction Many problems in natural language, vision, and computational biology require the joint modeling of many dependent variables. Such models often include hidden variables, which play an important role in unsupervised learning and general missing data problems. The focus of this paper is on models in which the hidden variables have natural problem domain interpretations and are the object of inference. Standard approaches for learning hidden-variable models involve integrating out the hidden variables and working with the resulting marginal likelihood. However, this marginalization can be intractable. An alternative is to develop procedures that merge the inference results of several tractable submodels. An early example of such an approach is the use of pseudolikelihood [1], which deals with many conditional models of single variables rather than a single joint model. More generally, composite likelihood permits a combination of the likelihoods of subsets of variables [7]. Another approach is piecewise training [10, 11], which has been applied successfully to several large-scale learning problems. All of the above methods, however, focus on fully-observed models. In the current paper, we develop techniques in this spirit that work for hidden-variable models. The basic idea of our approach is to create several tractable submodels and train them jointly to agree on their hidden variables. We present an intuitive objective function and efficient EM-style algorithms for training a collection of submodels. We refer to this general approach as agreement-based learning. Sections 2 and 3 presents the general theory for agreement-based learning. In some applications, it is infeasible computationally to optimize the objective function; Section 4 provides two alternative objectives that lead to tractable algorithms. Section 5 demonstrates that our methods can be applied successfully to large datasets in three real world problem domains—grammar induction, word alignment, and phylogenetic hidden Markov modeling. 1 2 Agreement-based learning of multiple submodels Assume we have M (sub)models pm(x, z; θm), m = 1, . . . , M, where each submodel specifies a distribution over the observed data x ∈X and some hidden state z ∈Z. The submodels could be parameterized in completely different ways as long as they are defined on the common event space X × Z. Intuitively, each submodel should capture a different aspect of the data in a tractable way. To learn these submodels, the simplest approach is to train them independently by maximizing the sum of their log-likelihoods: Oindep(θ) def = log Y m X z pm(x, z; θm) = X m log pm(x; θm), (1) where θ = (θ1, . . . , θM) is the collective set of parameters and pm(x; θm) = P z pm(x, z; θm) is the likelihood under submodel pm.1 Given an input x, we can then produce an output z by combining the posteriors pm(z | x; θm) of the trained submodels. If we view each submodel as trying to solve the same task of producing the desired posterior over z, then it seems advantageous to train the submodels jointly to encourage “agreement on z.” We propose the following objective which realizes this insight: Oagree(θ) def = log X z Y m pm(x, z; θm) = X m log pm(x; θm) + log X z Y m pm(z | x; θm). (2) The last term rewards parameter values θ for which the submodels assign probability mass to the same z (conditioned on x); the summation over z reflects the fact that we do not know what z is. Oagree has a natural probabilistic interpretation. Imagine defining a joint distribution over M independent copies over the data and hidden state, (x1, z1), . . . , (xM, zM), which are each generated by a different submodel: p((x1, z1), . . . , (xM, zM); θ) = Q m p(xm, zm; θm). Then Oagree is the probability that the submodels all generate the same observed data x and the same hidden state: p(x1 = · · · = xM = x, z1 = · · · = zM; θ). Oagree is also related to the likelihood of a proper probabilistic model pnorm, obtained by normalizing the product of the submodels, as is done in [3]. Our objective Oagree is then a lower bound on the likelihood under pnorm: pnorm(x; θ) def = P z Q m pm(x, z; θm) P x,z Q m pm(x, z; θm) ≥ P z Q m pm(x, z; θm) Q m P x,z pm(x, z; θm) = Oagree(θ). (3) The inequality holds because the denominator of the lower bound contains additional cross terms. The bound is generally loose, but becomes tighter as each pm becomes more deterministic. Note that pnorm is distinct from the product-of-experts model [3], in which each “expert” model pm has its own set of (nuisance) hidden variables: ppoe(x) ∝Q m P z pm(x, z; θm). In contrast, pnorm has one set of hidden variables z common to all submodels, which is what provides the mechanism for agreement-based learning. 2.1 The product EM algorithm We now derive the product EM algorithm to maximize Oagree. Product EM bears many striking similarities to EM: both are coordinate-wise ascent algorithms on an auxiliary function and both increase the original objective monotonically. By introducing an auxiliary distribution q(z) and applying Jensen’s inequality, we can lower bound Oagree with an auxiliary function L: Oagree(θ) = log X z q(z) Q m pm(x, z; θm) q(z) ≥Eq(z) log Q m pm(x, z; θm) q(z) def = L(θ, q) (4) The product EM algorithm performs coordinate-wise ascent on L(θ, q). In the (product) E-step, we optimize L with respect to q. Simple algebra reveals that this optimization is equivalent to minimizing a KL-divergence: L(θ, q) = −KL(q(z)|| Q m pm(x, z; θm)) + constant, where the constant 1To simplify notation, we consider one data point x. Extending to a set of i.i.d. points is straightforward. 2 does not depend on q. This quantity is minimized by setting q(z) ∝Q m pm(x, z; θm). In the (product) M-step, we optimize L with respect to θ, which decomposes into M independent objectives: L(θ, q) = P m Eq log pm(x, z; θm) + constant, where this constant does not depend on θ. Each term corresponds to an independent M-step, just as in EM for maximizing Oindep. Thus, our product EM algorithm differs from independent EM only in the E-step, in which the submodels are multiplied together to produce one posterior over z rather than M separate ones. Assuming that there is an efficient EM algorithm for each submodel pm, there is no difficulty in performing the product M-step. In our applications (Section 5), each pm is composed of multinomial distributions, so the M-step simply involves computing ratios of expected counts. On the other hand, the product E-step can become intractable and we must develop approximations (Section 4). 3 Exponential family formulation Thus far, we have placed no restrictions on the form of the submodels. To develop a richer understanding and provide a framework for making approximations, we now assume that each submodel pm is an exponential family distribution: pm(x, z; θm) = exp{θT mφm(x, z) −Am(θm)} for x ∈X, z ∈Zm and 0 otherwise, (5) where φm are sufficient statistics (features) and Am(θm) = log P x∈X,z∈Zm exp{θT mφm(x, z)} is the log-partition function,2 defined on θm ∈Θm ⊂RJ. We can think of all the submodels pm as being defined on a common space Z∪= ∪mZm, but the support of q(z) as computed in the E-step is only the intersection Z∩= ∩mZm. Controlling this support will be essential in developing tractable approximations (Section 4.1). In the general formulation, we required only that the submodels share the same event space X × Z. Now we make explicit the possibility of the submodels sharing features, which give us more structure for deriving approximations. In particular, suppose each feature j of submodel pm can be decomposed into a part that depends on x (which is specific to that particular submodel) and a part that depends on z (which is the same for all submodels): φmj(x, z) = I X i=1 φX mji(x)φZ i (z), or in matrix notation, φm(x, z) = φX m(x)φZ(z), (6) where φX m(x) is a J × I matrix and φZ(z) is a I × 1 vector. When z is discrete, such a decomposition always exists by defining φZ(z) to be an |Z∪|-dimensional indicator vector which is 1 on the component corresponding to z. Fortunately, we can usually obtain more compact representations of φZ(z). We can now express our objective L(θ, q) (4) using (5) and (6): L(θ, q) = X m θT mφX m(x) (Eq(z)φZ(z)) + H(q) − X m Am(θm) for q ∈Q(Z∩), (7) where Q(Z′) def = {q : q(z) = 0 for z ̸∈Z′} is the set of distributions with support Z′. For convenience, define bT m = θT mφX m(x) and b = P m bm, which summarize the parameters θ for the E-step. Note that for any θ, the q maximizing L always has the following exponential family form: q(z; β) = exp{βT φZ(z) −AZ∩(β)} for z ∈Z∩and 0 otherwise, (8) where AZ∩(β) = log P z∈Z∩exp{βT φZ(z)} is the log-partition function. In a minor abuse of notation, we write L(θ, β) = L(θ, q(·; β)). Specifically, L(θ, β) is maximized by setting β = b. It will be useful to express (7) using convex duality [12]. The key idea of convex duality is the existence of a mapping between the canonical exponential parameters β ∈RI of an exponential family distribution q(z; β) and the mean parameters defined by µ = Eq(z;β)φZ(z) ∈M(Z∩) ⊂RI, where M(Z′) = {µ : ∃q ∈Q(Z′) : EqφZ(z) = µ} is the set of realizable mean parameters. The Fenchel-Legendre conjugate of the log-partition function AZ∩(β) is A∗ Z∩(µ) def = sup β∈RI{βT µ −AZ∩(β)} for µ ∈M(Z∩), (9) 2Our applications use directed graphical models, which correspond to curved exponential families where each Θm is defined by local normalization constraints and Am(θm) = 0. 3 which is also equal to −H(q(z; β)), the negative entropy of any distribution q(z; β) corresponding to µ. Substituting µ and A∗ Z∩(µ) into (7), we obtain an objective in terms of the dual variables µ: L∗(θ, µ) def = X m θT mφX m(x) µ −A∗ Z∩(µ) − X m Am(θm) for µ ∈M(Z∩). (10) Note that the two objectives are equivalent: supβ∈RI L(θ, β) = supµ∈M(Z∩) L∗(θ, µ) for each θ. The mean parameters µ are exactly the z-specific expected sufficient statistics computed in the product E-step. The dual is an attractive representation because it allows us to form convex combinations of different µ, an operation does not have a direct correlate in the primal formulation. The product EM algorithm is summarized below: Product EM E-step: µ = argmaxµ′∈M(Z∩){bT µ′ −A∗ Z∩(µ′)} M-step: θm = argmaxθ′ m∈Θm{θ′T m φX (x)µ −Am(θ′ m)} 4 Approximations The product M-step is tractable provided that the M-step for each submodel is tractable, which is generally the case. The corresponding statement is not true for the E-step, which in general requires explicitly summing over all possible z ∈Z∩, often an exponentially large set. We will thus consider alternative E-steps, so it will be convenient to succinctly characterize an E-step. An E-step is specified by a vector b′ (which depends on θ and x) and a set Z′ (which we sum z over): E(b′, Z′) computes µ = argmax µ′∈M(Z′) {b′T µ′ −A∗ Z′(µ′)}. (11) Using this notation, E(bm, Zm) is the E-step for training the m-th submodel independently using EM and E(b, Z∩) is the E-step of product EM. Though we write E-steps in the dual formulation, in practice, we compute µ as an expectation over all z ∈Z′, perhaps leveraging dynamic programming. If E(bm, Zm) is tractable and all submodels have the same dynamic programming structure (e.g., if z is a tree and all features are local with respect to that tree), then E(b, Z∩) is also tractable: we can incorporate all the features into the same dynamic program and simply run product EM (see Section 5.1 for an example). However, E(b, Z∩) is intractable in general, owing to two complications: (1) we can sum over each Zm efficiently but not the intersection Z∩; and (2) each bm corresponds to a decomposable graphical model, but the combined b = P m bm corresponds to a loopy graph. In the sequel, we describe two approximate objective functions addressing each complication, whose maximization can be carried out by performing M independent tractable E-steps. 4.1 Domain-approximate product EM Assume that for each submodel pm, E(b, Zm) is tractable (see Section 5.2 for an example). We propose maximizing the following objective: L∗ dom(θ, µ1, . . . , µm) def = 1 M X m h X m′ θT m′φX m′(x) µm −A∗ Zm(µm) i − X m Am(θm), (12) with each µm ∈M(Zm). This objective can be maximized via coordinate-wise ascent: Domain-approximate product EM E-step: µm = argmaxµ′ m∈M(Zm){bT µ′ m −A∗ Zm(µ′ m)} [E(b, Zm)] M-step: θm = argmaxθ′ m∈Θm{θ′T m φX (x) 1 M P m′ µm′ −Am(θ′ m)} The product E-step consists of M separate E-steps, which are each tractable because each involves the respective Zm instead of Z∩. The resulting expected sufficient statistics are averaged and used in the product M-step, which breaks down into M separate M-steps. 4 While we have not yet established any relationship between our approximation L∗ dom and the original objective L∗, we can, however, relate L∗ dom to L∗ ∪, which is defined as an analogue of L∗by replacing Z∩with Z∪in (10). Proposition 1. L∗ dom(θ, µ1, . . . , µM) ≤L∗ ∪(θ, ¯µ) for all θ and µm ∈M(Zm) and ¯µ = 1 M P m µm. Proof. First, since M(Zm) ⊂M(Z∪) and M(Z∪) is a convex set, ¯µ ∈M(Z∪), so L∗ ∪(θ, ¯µ) is well-defined. Subtracting the L∪version of (10) from (12), we obtain L∗ dom(θ, µ1, . . . , µM) − L∗ ∪(θ, ¯µ) = A∗ Z∪(¯µ) − 1 M P m A∗ Zm(µm). It suffices to show A∗ Z∪(¯µ) ≤ 1 M P m A∗ Z∪(µm) ≤ 1 M P m A∗ Zm(µm). The first inequality follows from convexity of A∗ Z∪(·). For the second inequality: since Zm ⊃Z∪, AZ∪(µm) ≥AZm(µm); by inspecting (9), it follows that A∗ Z∪(µm) ≤A∗ Zm(µm). 4.2 Parameter-approximate product EM Now suppose that for each submodel pm, E(bm, Z∩) is tractable (see Section 5.3 for an example). We propose maximizing the following objective: L∗ par(θ, µ1, . . . , µm) def = 1 M X m h (MθT mφX m(x))µm −A∗ Z∩(µm) i − X m Am(θm), (13) with each µm ∈M(Z∩). This objective can be maximized via coordinate-wise ascent, which again consists of M separate E-steps E(Mbm, Z∩) and the same M-step as before: Parameter-approximate product EM E-step: µm = argmaxµ′ m∈M(Zm){(Mbm)T µ′ m −A∗ Z∩(µ′ m)} [E(Mbm, Z∩)] M-step: θm = argmaxθ′ m∈Θm{θ′T m φX (x) 1 M P m′ µm′ −Am(θ′ m)} We can show that the maximum value of L∗ par is at least that of L∗, which leaves us maximizing an upper bound of L∗. Although less logical than maximizing a lower bound, in Section 5.3, we show that our approach is nonetheless a reasonable approximation which importantly is tractable. Proposition 2. maxµ1∈M(Z∩),...,µM∈M(Z∩) L∗ par(θ, µ1, . . . , µM) ≥maxµ∈M(Z∩) L∗(θ, µ). Proof. From the definitions of L∗ par (13) and L∗(10), it is easy to see that L∗ par(θ, µ, . . . , µ) = L∗(θ, µ) for all µ ∈M(Z∩). If we maximize L∗ par with M distinct arguments, we cannot end up with a smaller value. The product E-step could also be approximated by mean-field or loopy belief propagation variants. These methods and the two we propose all fall under the general variational framework for approximate inference [12]. The two approximations we developed have the advantage of permitting exact tractable solutions without resorting to expensive iterative methods which are only guaranteed to converge to a local optima. While we still lack a complete theory relating our approximations L∗ dom and L∗ par to the original objective L∗, we can give some intuitions. Since we are operating in the space of expected sufficient statistics µm, most of the information about the full posterior pm(z | x) must be captured in these statistics alone. Therefore, we expect our approximations to be accurate when each submodel has enough capacity to represent the posterior pm(z | x; θm) as a low-variance unimodal distribution. 5 Applications We now empirically validate our algorithms on three concrete applications: grammar induction using product EM (Section 5.1), unsupervised word alignment using domain-approximate product EM (Section 5.2), and prediction of missing nucleotides in DNA sequences using parameter-approximate product EM (Section 5.3). 5 e1 e2 e3 a1 a2 a3 a4 f1 f2 f3 f4 e1 e2 e3 a1 a2 a3 f1 f2 f3 f4 (a) Submodel p1 (b) Submodel p2 0.07 0.08 0.09 0.1 0.11 0.12 1 2 3 4 5 6 7 8 9 10 alignment error rate iteration HMM model Independent EM Domain-approximate product EM Figure 1: The two instances of IBM model 1 for word alignment are shown in (a) and (b). The graph shows gains from agreement-based learning. 5.1 Grammar induction Grammar induction is the problem of inducing latent syntactic structures given a set of observed sentences. There are two common types of syntactic structure (one based on word dependencies and the other based on constituent phrases), which can each be represented as a submodel. [5] proposed an algorithm to train these two submodels. Their algorithm is a special case of our product EM algorithm, although they did not state an objective function. Since the shared hidden state is a tree structure, product EM is tractable. They show that training the two submodels to agree significantly improves accuracy over independent training. See [5] for more details. 5.2 Unsupervised word alignment Word alignment is an important component of machine translation systems. Suppose we have a set of sentence pairs. Each pair consists of two sentences, one in a source language (say, English) and its translation in a target language (say, French). The goal of unsupervised word alignment is to match the words in a source sentence to the words in the corresponding target sentence. Formally, let x = (e, f) be an observed pair of sentences, where e = (e1, . . . , e|e|) and f = (f1, . . . , f|f|); z is a set of alignment edges between positions in the English and positions in the French. Classical models for word alignment include IBM models 1 and 2 [2] and the HMM model [8]. These are asymmetric models, which means that they assign non-zero probability only to alignments in which each French word is aligned to at most one English word; we denote this set Z1. An element z ∈Z1 can be parameterized by a vector a = (a1, . . . , a|f|), with aj ∈{NULL, 1, . . . , |e|}, corresponding to the English word (if any) that French word fj is aligned to. We define the first submodel on X × Z1 as follows (specializing to IBM model 1 for simplicity): p1(x, z; θ1) = p1(e, f, a; θ1) = p1(e) |f| Y j=1 p1(aj)p1(fj | eaj; θ1), (14) where p1(e) and p1(aj) are constant and the canonical exponential parameters θ1 are the transition log-probabilities {log t1;ef} for each English word e (including NULL) and French word f. Written in exponential family form, φZ(z) is an (|e| + 1)(|f| + 1)-dimensional vector whose components are {φZ ij(z) ∈{0, 1} : i = NULL, 1, . . . , |e|, j = NULL, 1, . . . , |f|}. We have φZ ij(z) = 1 if and only if English word ei is aligned to French word fj and zNULLj = 1 if and only if fj is not aligned to any English word. Also, φX ef;ij(x) = 1 if and only if ei = e and fj = f. The mean parameters associated with an E-step are {µ1;ij}, the posterior probabilities of ei aligning to fj; these can be computed independently for each j. We can define a second submodel p2(x, z; θ2) on X × Z2 by reversing the roles of English and French. Figure 1(a)–(b) shows the two models. We cannot use product EM algorithm to train p1 and p2 because summing over all alignments in Z∩= Z1 ∩Z2 is NP-hard. However, we can use domain-approximate product EM because E(b1 + b2, Zm) is tractable—the tractability here does not depend on decomposability of b but the asymmetric alignment structure of Zm. The concrete change from independent EM is slight: we need to only change the E-step of each pm to use the product of translation probabilities t1;eft2;fe and change the M-step to use the average of the edge posteriors obtained from the two E-steps. 6 dA1 dB1 dC1 dD1 dE1 dA2 dB2 dC2 dD2 dE2 dA3 dB3 dC3 dD3 dE3 dA4 dB4 dC4 dD4 dE4 dA1 dB1 dC1 dD1 dE1 dA2 dB2 dC2 dD2 dE2 dA3 dB3 dC3 dD3 dE3 dA4 dB4 dC4 dD4 dE4 (a) Submodel p1 (b) Submodel p2 Figure 2: The two phylogenetic HMM models, one for the even slices, the other for the odd ones. [6] proposed an alternative method to train two models to agree. Their E-step computes µ1 = E(b1, Z1) and µ2 = E(b2, Z2), whereas our E-steps incorporate the parameters of both models in b1 + b2. Their M-step uses the elementwise product of µ1 and µ2, whereas we use the average 1 2(µ1 + µ2). Finally, while their algorithm appears to be very stable and is observed to converge empirically, no objective function has been developed; in contrast, our algorithm maximizes (12). In practice, both algorithms perform comparably. We conducted our experiments according to the setup of [6]. We used 100K unaligned sentences for training and 137 for testing from the English-French Hansards data of the NAACL 2003 Shared Task. Alignments are evaluated using alignment error rate (AER); see [6] for more details. We trained two instances of the HMM model [8] (English-to-French and French-to-English) using 10 iterations of domain-approximate product EM, initializing with independently trained IBM model 1 parameters. For prediction, we output alignment edges with sufficient posterior probability: {(i, j) : 1 2(µ1;ij + µ2;ij) ≥δ}. Figure 1 shows how agreement-based training improves the error rate over independent training for the HMM models. 5.3 Phylogenetic HMM models Suppose we have a set of species s ∈S arranged in a fixed phylogeny (i.e., S are the nodes of a directed tree). Each species s is associated with a length L sequence of nucleotides ds = (ds1, . . . , dsL). Let d = {ds : s ∈S} denote all the nucleotides, which consist of some observed ones x and unobserved ones z. A good phylogenetic model should take into consideration both the relationship between nucleotides of the different species at the same site and the relationship between adjacent nucleotides in the same species. However, such a model would have high tree-width and be intractable to train. Past work has focused on traditional variational inference in a single intractable model [9, 4]. Our approach is to instead create two tractable submodels and train them to agree. Define one submodel to be p1(x, z; θ1) = p1(d; θ1) = Y j odd Y s∈S Y s′∈CH(s) p1(ds′j | dsj; θ1)p1(ds′j+1 | ds′j, ds(j+1); θ1), (15) where CH(s) is the set of children of s in the tree. The second submodel p2 is defined similarly, only with the product taken over j even. The parameters θm consist of first-order mutation logprobabilities and second-order mutation log-probabilities. Both submodels permit the same set of assignments of hidden nucleotides (Z∩= Z1 = Z2). Figure 2(a)–(b) shows the two submodels. Exact product EM is not tractable since b = b1 + b2 corresponds to a graph with high tree-width. We can apply parameter-approximate product EM, in which the E-step only involves computing µm = E(2bm, Z∩). This can be done via dynamic programming along the tree for each twonucleotide slice of the sequence. In the M-step, the average 1 2(µ1 + µ2) is used for each model, which has a closed form solution. Our experiments used a multiple alignment consisting of L = 20, 000 consecutive sites belonging to the L1 transposons in the Cystic Fibrosis Transmembrane Conductance Regulator (CFTR) gene (chromosome 7). Eight eutherian species were arranged in the phylogeny shown in Figure 3. The data we used is the same as that of [9]. Some nucleotides in the sequences were already missing. In addition, we held out some fraction of the observed ones for evaluation. We trained two models using 30 iterations of parameter-approximate product EM.3 For prediction, the posteriors over heldout 3We initialized with a small amount of noise around uniform parameters plus a small bias towards identity mutations. 7 (hidden) (hidden) baboon (hidden) chimp human (hidden) (hidden) cow pig (hidden) cat dog (hidden) mouse rat 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0 5 10 15 20 25 accuracy iteration 20% heldout Independent EM Parameter-approximate product EM 0.4 0.5 0.6 0.7 0.8 0 5 10 15 20 25 accuracy iteration 50% heldout Independent EM Parameter-approximate product EM Figure 3: The tree is the phylogeny topology used in experiments. The graphs show the prediction accuracy of independent versus agreement-based training (parameter-approximate product EM) when 20% and 50% of the observed nodes are held out. nucleotides under each model are averaged and the one with the highest posterior is chosen. Figure 3 shows the prediction accuracy. Though independent and agreement-based training eventually obtain the same accuracy, agreement-based training converges much faster. This gap grows as the amount of heldout data increases. 6 Conclusion We have developed a general framework for agreement-based learning of multiple submodels. Viewing these submodels as components of an overall model, our framework permits the submodels to be trained jointly without paying the computational cost associated with an actual jointly-normalized probability model. We have presented an objective function for agreement-based learning and three EM-style algorithms that maximize this objective or approximations to this objective. We have also demonstrated the applicability of our approach to three important real-world tasks. For grammar induction, our approach yields the existing algorithm of [5], providing an objective for that algorithm. For word alignment and phylogenetic HMMs, our approach provides entirely new algorithms. Acknowledgments We would like to thank Adam Siepel for providing the phylogenetic data and acknowledge the support of the Defense Advanced Research Projects Agency under contract NBCHD030010. References [1] J. Besag. The analysis of non-lattice data. The Statistician, 24:179–195, 1975. [2] P. F. Brown, S. A. D. Pietra, V. J. D. Pietra, and R. L. Mercer. The mathematics of statistical machine translation: Parameter estimation. Computational Linguistics, 19:263–311, 1993. [3] G. Hinton. Products of experts. In International Conference on Artificial Neural Networks, 1999. [4] V. Jojic, N. Jojic, C. Meek, D. Geiger, A. Siepel, D. Haussler, and D. Heckerman. Efficient approximations for learning phylogenetic HMM models from data. Bioinformatics, 20:161–168, 2004. [5] D. Klein and C. D. Manning. Corpus-based induction of syntactic structure: Models of dependency and constituency. In Association for Computational Linguistics (ACL), 2004. [6] P. Liang, B. Taskar, and D. Klein. Alignment by agreement. In Human Language Technology and North American Association for Computational Linguistics (HLT/NAACL), 2006. [7] B. Lindsay. Composite likelihood methods. Contemporary Mathematics, 80:221–239, 1988. [8] H. Ney and S. Vogel. HMM-based word alignment in statistical translation. In International Conference on Computational Linguistics (COLING), 1996. [9] A. Siepel and D. Haussler. Combining phylogenetic and hidden Markov models in biosequence analysis. Journal of Computational Biology, 11:413–428, 2004. [10] C. Sutton and A. McCallum. Piecewise training of undirected models. In Uncertainty in Artificial Intelligence (UAI), 2005. [11] C. Sutton and A. McCallum. Piecewise pseudolikelihood for efficient CRF training. In International Conference on Machine Learning (ICML), 2007. [12] M. Wainwright and M. I. Jordan. Graphical models, exponential families, and variational inference. 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Extending position/phase-shift tuning to motion energy neurons improves velocity discrimination Stanley Yiu Man Lam and Bertram E. Shi Department of Electronic and Computer Engineering Hong Kong Univeristy of Science and Technology Clear Water Bay, Kowloon, Hong Kong {eelym,eebert}@ee.ust.hk Abstract We extend position and phase-shift tuning, concepts already well established in the disparity energy neuron literature, to motion energy neurons. We show that Reichardt-like detectors can be considered examples of position tuning, and that motion energy filters whose complex valued spatio-temporal receptive fields are space-time separable can be considered examples of phase tuning. By combining these two types of detectors, we obtain an architecture for constructing motion energy neurons whose center frequencies can be adjusted by both phase and position shifts. Similar to recently described neurons in the primary visual cortex, these new motion energy neurons exhibit tuning that is between purely spacetime separable and purely speed tuned. We propose a functional role for this intermediate level of tuning by demonstrating that comparisons between pairs of these motion energy neurons can reliably discriminate between inputs whose velocities lie above or below a given reference velocity. 1 Introduction Image motion is an important cue used by both biological and artificial visual systems to extract information about the environment. Although image motion is commonly used, there are different models for image motion processing in different systems. The Reichardt model is a dominant model for motion detection in insects, where image motion analysis occurs at a very early stage [1]. For mammals, the bulk of visual processing for motion is thought to occur in the cortex, and the motion energy model is one of the dominant models [2][3]. However, despite the differences in complexity between these two models, they are mathematically equivalent given appropriate choices of the spatial and temporal filters [4]. The motion energy model is very closely related to the disparity energy model, which has been used to model the outputs of disparity selective neurons in the visual cortex [5]. The disparity tuning of neurons in this model can be adjusted via two mechanisms: a position shift between the center locations of the receptive fields in the left and right eyes or a phase shift between the receptive field organization in the left and right eyes [6][7]. It appears that biological systems use a combination of these two mechanisms. In Section 2, we extend the concepts of position and phase tuning to the construction of motion energy neurons. We combine the Reichardt model and the motion energy model to obtain an architecture for constructing motion energy neurons whose tuning can be adjusted by the analogs of position and phase shifts. In Section 3, we investigate the functional advantages of position and phase shifts, inspired by a similar comparison from the disparity energy literature. We show that by simply comparing the outputs of pair of motion energy cells with combined position/phase shift tuning enables us to discriminate reliably between stimuli moving above and below a reference velocity. Finally, in Section 4, we place this work in the context of recent results on speed tuning in V1 neurons. 2 Extending Position/Phase Tuning to Motion Energy Models Figure 1(a) shows a 1D array of three Reichardt detectors[1] tuned to motion from left to right. Each detector computes the correlation between its photosensor input and the delayed input from the photosensor to the left. The delay could be implemented by a low pass filter. Usually, the correlation is assumed to be computed by a multiplication between the current and delayed signals. For consistency with the following discussion, we show the output as a summation followed by a squaring. Squaring the sum is essentially equivalent to the product, since the product could be recovered by subtracting the sum of the squared inputs from the squared sum (e.g. ). Delbruck proposed a modification of the Reichardt detector (Figure 1(b)), which switches the order of the delay and the sum, resulting in a delay-line architecture [8]. The output of a detector is the sum of its photosensor input and the delayed output of the detector to the left. This recurrent connection extends the spatio-temporal receptive field of the detector, since the input from the secondnearest-neighboring photosensor to the left is now connected to the detector through two delays, whereas the Reichardt detector never sees the output of its second-nearest-neighboring photosensor. The velocity tuning of these detectors is determined by the combination of the temporal delay and the position shift between the neighboring detectors. As the delay increases, the tuned velocity decreases. As the position shift increases, the tuned velocity also increases. This position-tuning of velocity is reminiscent of the position-tuning of disparity energy neurons, where the larger the position shift between the spatial receptive fields being combined from the left and right eyes, the larger the disparity tuning [9]. Figure 1(c) shows a 1D array of three motion energy detectors[2][3]. At each spatial location, the outputs of the photosensors in a neighborhood around each spatial location are combined through even and odd symmetric linear spatial receptive fields, which are here modelled by spatial Gabor functions. In 1D, the even and odd symmetric Gabor receptive field profiles are the real and imaginary parts of the function (1) where determines the preferred spatial frequency of the receptive field, and determines its spatial extent. The even and odd spatial filter outputs are then combined through temporal filters to produce two outputs which are then squared and summed to produce the motion energy. In many cases, the temporal receptive field profiles are also Gabor functions. The combined spatial and temporal receptive fields of the two neurons are separable when considered as a single complex valued function: (2) where and determine the preferred temporal frequency and temporal extent of the temporal receptive fields. Strictly speaking, these spatio-temporal filters are not velocity tuned, since the velocity at which a moving sine-wave grating stimulus produces maximum response varies with the spatial frequency of the sine-wave grating. However, since spatial frequencies of lead to the largest responses, the filter is sometimes thought of as having a preferred velocity . a b + ( )2 a2 b2 + ( ) – 2ab = gs x ( ) 1 2πσx ----------------x2 2σx 2 --------– ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ jxΩx ( ) exp exp 1 2πσx ----------------x2 2σx 2 --------– ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ Ωxx ( ) cos j Ωxx ( ) sin + ( ) exp = = Ωx σx g x t , ( ) 1 2πσx ----------------x2 2σx 2 --------– ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ jΩxx ( ) exp exp 1 2πσt ----------------t2 2σt 2 --------– ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ jΩtt ( ) exp exp ⋅ = Ωt σt Ωx vpref Ωt Ωx ⁄ – = One problem with using spatio-temporal Gabor functions is that they are non-causal in time. In this work, we consider the use of a causal recurrently implemented temporal filter. If we let the real and imaginary parts of denote the even and odd spatial filter outputs, then the two temporal fil(a) (b) (c) (d) Figure 1. (a) 1D array of three Reichardt detectors tuned to motion from left to right. The block represents a temporal delay. The semi-circles represent photosensors. (b) Delbruck delay-line detector. (c) 1D array of three motion energy detectors. The bottom blocks represent even and odd symmetric spatial receptive fields modelled by Gabor functions. (d) The proposed motion detector by combining the position and phase tuning mechanisms of (b) and (c). τ 2 2 τ 2 τ 2 2 τ 2 τ ae jΩ τ 2 2 τ ae jΩ τ 2 2 τ ae jΩ τ 2 2 ae jΩ im re im re a cosΩ -a sinΩ a sinΩ a cosΩ im re im re τ τ 2 2 τ τ 2 2 τ τ 2 2 ae jΩ ae jΩ τ u x t , ( ) ter outputs of the temporal filter are given by the real and imaginary parts of , which satisfies (3) where and are real valued constants. We derive this equation from Fig. 1(c) by considering the time delay as a unit sample discrete time delay. We consider discrete time operation here for consistency with our experimental results, however, a corresponding continuous time temporal filter can be obtained by replacing the time delay by a first order continuous-time recurrent filter with time constant . The frequency response of this complex-valued filter is (4) where and are spatial and temporal frequency variables. This function achieves unity maximum value at , independently of . Assuming the same Gabor spatial receptive field, the combined spatio-temporal receptive field can be approximated by the continuous function: (5) where is the unit step function, and . Again, strictly speaking, the filter is not velocity tuned, but for input sine-wave gratings with a spatial frequency near , the composite spatio-temporal filter has a preferred velocity near . The velocity tuning of this filter is determined by the combination of the time delay and a phase shift between the input and the output . The longer the time delay, the slower the preferred velocity. However, the larger the phase-shift, the higher the preferred velocity. This phase-tuning of velocity is reminiscent of the phase-tuning of disparity tuned neurons, where the larger the phase shift between the left and right receptive fields, the larger the preferred disparity. The possibility to adjust velocity tuning using two complementary mechanisms, suggests that it should be possible to combine these two methods, as observed in disparity neurons. Figure 1(d) shows how the position and phase tuning mechanisms of Figures 1(b) and 1(c) can be combined. The preferred velocity for spatial frequencies will be determined by the sum of the preferred velocities determined by the position and phase-shift mechanisms, i.e. , assuming a unit spatial displacement between adjacent photosensors. 3 Motion energy pairs for velocity discrimination Given the possibility of combining the position and phase tuning mechanisms, an interesting question is how these two mechanisms might be exploited when constructing populations of motion energy neurons. Velocity can be estimated using a population of neurons tuned to different spatiotemporal frequencies [10][11]. However, the output of a single motion energy neuron is an ambiguous indicator of velocity, since its output depends upon other stimulus dimensions in addition to motion, (e.g. orientation, contrast). Given the long history of position/phase shifts in disparity tuning, it is natural to start with an inspiration taken from the context of binocular vision. It has been shown that the responses from a population of phase-tuned disparity energy are more comparable than the responses from a population of position-tuned disparity energy neurons [12]. In particular, the preferred disparity of the neuron with maximum response in a population of phase tuned neurons is a more reliable indicator of the stimulus disparity than the preferred disparity of the neuron with maximum response in a population of position tuned neurons, especially for neurons with small phase shifts. The disadvantage of purely phase tuned neurons is that their preferred disparities can be tuned only over a limited range due to phase-wraparound in the sinusoidal modulation of the spatial Gabor. However, there is no v x t , ( ) v x t , ( ) a jΩt ( ) exp v x t 1 – , ( ) 1 a – ( ) u x t , ( ) ⋅ + ⋅ = a 1 < Ωt τ τ V ωx ωt , ( ) U ωx ωt , ( ) ---------------------1 a – 1 a j ωt Ωt – ( ) – ( ) exp ⋅ – -------------------------------------------------------------= ωx ωt ωt Ωt – = ωx g x t , ( ) 1 2πσx ----------------x2 2σx 2 --------– ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ jΩxx ( ) exp exp τ 1 – t τ ⁄ – ( ) jΩtt ( )h t ( ) exp exp ⋅ = h t ( ) τ 1 – 1 a – ( ) ≈ Ωx vpref Ωt Ωx ⁄ – = Ωt u x t , ( ) v x t 1 – , ( ) Ωx vpref 1 Ωt Ωx ⁄ – = restriction on the range of preferred disparities when using position shifts. Thus, it has been suggested that position shifts can be used to “bias” the preferred disparity of a population around a rough estimate of the stimulus disparity, and then use a population of neurons tuned by phase shifts to obtain a more accurate estimation of the actual disparity. In this section, we demonstrate that a similar phenomenon holds for motion energy neurons. In particular, we show that we can use position shifts to place the tuned velocity (for a spatial frequency of ) in a population of two neurons around a desired bias velocity, , and then use phase shifts with equal magnitude but opposite sign to place the preferred velocities symmetrically around this bias velocity. We then show that by comparing the outputs of these two neurons, we can accurately discriminate between velocities above and below . The equation describing the complex valued output of the spatio-temporal filtering stage for the detector shown in Figure 1(d) is (6) The frequency response is (7) and achieves its maximum along the line , as seen in the contour plot of the spatiotemporal frequency response magnitude of the cascade of (1) and (7) in Fig. 2(a). In comparison, the spatio-temporal frequency response of the cascade of (1) and (4) shown in Fig. 2(e), achieves its maximum at independently of . For a moving sine wave grating input with spatial and temporal frequencies and , the steady state motion energy outputs will be proportional to the squared magnitudes of the spatio-temporal frequency response evaluated at . Assume that we have two such motion cells with the same preferred spatial frequency but opposite temporal frequencies . The motion energy cell with positive is tuned to fast velocities, while the motion energy cell with negative is tuned to slow velocities. If we compare the frequency response magnitudes at frequency , the boundary between the regions in the plane where the magnitude of one is larger than the other is a line passing thorough the origin with slope equal to 1, as shown in Fig. 2(c). This suggests that we can determine whether the velocity of the grating is faster or slower than 1 pixel per frame by checking the relative magnitude of the motion energy outputs, at least for sine-wave gratings. Although the sine-wave grating is a particularly simple input, this property is not shared by other pairs of motion energy neurons. For example, Fig. 2(f) shows the spatio-temporal frequency responses two motion energy neurons that have the same spatio-temporal center frequencies as considered above, but are constructed by phase tuning (the cascade of (1) and (4)). In this case, the boundary is a horizontal line. Thus, the velocity boundary depends upon the spatial frequency. For lower spatial frequencies, the relative magnitudes will switch at higher velocities. Another commonly considered arrangement of Gabor-filters is to place the center frequencies around a circle. For two neurons, this corresponds to displacing the two center frequencies by an equal amount perpendicularly to the line (Fig. 2(k)). For motion energy filters built from non-causal Gabor filters, the spatio-temporal frequency responses exhibit perfect circular symmetry, and the decision boundary also coincides with the diagonal line (see Figure 9 in [13]). However, non-causal filters are not physically realizable. If we consider motion energy neurons constructed from temporally causal functions (e.g. the cascade (1) and (4)), the boundary only matches the diagonal line in a small neighborhood of , as shown in Fig. 2(i). We have characterized the performance of the three motion pairs on the fast/slow velocity discrimination task for a variety of inputs, including sine-wave gratings, square wave gratings, and drifting random dot stimuli with varying coherence. We first consider drifting sinusoidal gratings with spatial frequencies and velocities . For each spatial frequency and velocity, we compare the two motion energy Ωx vbias vbias w x t , ( ) w x t , ( ) a jΩt ( ) exp w x 1 – t 1 – , ( ) 1 a – ( ) u x t , ( ) ⋅ + ⋅ = W ωx ωt , ( ) U ωx ωt , ( ) -----------------------1 a – 1 a j ωt ωx Ωt – + ( ) – ( ) exp ⋅ – --------------------------------------------------------------------------= ωt ωx Ωt + = Ωt ωx ωx ωt ωx ωt , ( ) Ωx 2π 20 ⁄ = Ωt 2π 20 ⁄ ± = Ωt Ωt ωx ωt , ( ) ωx ωt – ωx ωt = ωx ωt = ωx Ωx = ωx 0 2π 10 ⁄ , [ ] ∈ vinput 0 2 , [ ] ∈ outputs at different phase shifts of the input grating, and calculate the percentage where the response of the fast cell is larger than that of the slow cell. Fig. 3(a)-(c) show the percentages as the grey scale value for each combination of input spatial frequency and velocity. Ideally, the top half should be white (i.e. the fast cell’s response is larger for all inputs whose velocity is greater than one), and the bottom half should be black. For the phase-shifted motion cells with unit positiontuned velocity bias, the responses are correct over a wide range of spatial frequencies. On the other hand, for the motion pairs with the same center frequencies but tuned by pure phase shifts (Fig. 3(c)), the velocity at which the relative responses switch decreases with spatial frequency. This is consistent with the horizontal decision boundary computed by comparing the frequency response magnitudes. For the phase-tuned motion-energy cells with orthogonally displaced center frequencies, the boundary rapidly diverges from the horizontal as the spatial frequency moves away from . Fig. 3(d) shows the overall accuracy by combining the responses over all velocities. The detector utilizing the phase-tuned cells with position bias have the highest accuracy over the widest range of spatial frequencies. Fig. 3(e)-(h) show the responses of the motion pairs to square wave gratings. The results are similar to the case of sinusoidal gratings, except that the performance at low spatial frequencies is worse. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Figure 2. Frequency response amplitudes of the motion pairs formed by types of motion cells. First row: Phase and position tuned motion cells. The center frequencies of the fast (a) and slow (b) cells are and respectively. Second row: Vertically displaced phase-tuned motion energy cells. The center frequencies of the fast (e) and slow (f) cells are and respectively. Third row: Orthogonally displaced phase-tuned motion energy cells. The center frequencies of the fast (i) and slow (j) cells are and respectively. The third column shows the contour plot of difference between the frequency response amplitudes of the fast cell from the slow cell. The dashed line shows the decision boundary at zero. The fourth column shows the cross sections of the frequency response amplitudes along the line connecting the two center frequencies (fast = solid, slow = dashed). Zero denotes the point on the line that crosses . fast cell slow cell motion pair tuning curve phase/position-tuned -1 0 1 -1.5 -1 -0.5 0 0.5 1 1.5 temporal frequency spatial frequency -1 0 1 -1.5 -1 -0.5 0 0.5 1 1.5 temporal frequency spatial frequency -1 0 1 -1.5 -1 -0.5 0 0.5 1 1.5 spatial frequency temporal frequency -0.4 -0.2 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 1 amplitude distance phase-tuned (vertical) -1 0 1 -1.5 -1 -0.5 0 0.5 1 1.5 temporal frequency spatial frequency -1 0 1 -1.5 -1 -0.5 0 0.5 1 1.5 temporal frequency spatial frequency -1 0 1 -1.5 -1 -0.5 0 0.5 1 1.5 temporal frequency spatial frequency -0.4 -0.2 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 1 amplitude distance phase-tuned (orthogonal) -1 0 1 -1.5 -1 -0.5 0 0.5 1 1.5 temporal frequency spatial frequency -1 0 1 -1.5 -1 -0.5 0 0.5 1 1.5 temporal frequency spatial frequency -1 0 1 -1.5 -1 -0.5 0 0.5 1 1.5 temporal frequency spatial frequency -0.4 -0.2 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 1 amplitude distance ωx ωt , ( ) 0.314 0.628 , ( ) = 0.314 0 , ( ) 0.314 0.628 , ( ) 0.314 0 , ( ) 0.092 0.536 , ( ) 0.536 0.092 , ( ) ωt ωx = Ωx this is expected, since for low spatial frequencies, the square wave gratings have large constant intensity areas that convey no motion information. Fig. 3(i)-(l) show the responses for drifting random dot stimuli at different velocities and coherence levels. The dots were one pixel wide. The motion pair using the phase-shifted cells with position tuned bias velocity maintain a consistently higher accuracy over all coherence levels tested. 4 Discussion We described a new architecture for motion energy filters obtained by combining the position tuning mechanism of the Reichardt-like detectors and the phase tuning mechanism of motion energy detectors based on complex-valued spatio-temporal separable filters. Motivated by results with disparity energy neurons indicating that the responses of phase-tuned neurons with small phase shifts are more comparable, we have examined the ability of the proposed velocity detectors to discriminate between input stimuli above and below a fixed velocity. Our experimental and analytical results confirm that comparisons between pairs constructed by using a position shift to center the tuned velocities around the border and using phase shifts to offset the tuned velocity of the pair to opposite sides of the boundary is consistently better than previously proposed architectures that were based on pure phase tuning. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Figure 3. Performance on the velocity discrimination task for different stimuli. First row: sine wave gratings; second row: square wave gratings; third row: drifting random dots. The first three columns show the percentage of stimuli where the fast motion energy cell’s response is larger than the slow cell’s response. First column: motion cells with position-tuned velocity bias; second column: phase tuned motion cells with the same center frequencies; third column: phase-tuned motion cells with orthogonal offset. The fourth column shows the average accuracy over all input velocities. Solid line: motion cells with position-tuned velocity bias; dashed line: phase tuned motion cells with the same center frequencies; dash-dot line: phase-tuned motion cells with orthogonal offset. sine wave gratings position/phase tuned phase-tuned (vertical) phase-tuned (orthogonal) average accuracy 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.5 1 1.5 2 input spatial frequency velocity 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.5 1 1.5 2 input spatial frequency velocity 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.5 1 1.5 2 input spatial frequency velocity 0 0.1 0.2 0.3 0.4 0.5 0.6 0.5 0.6 0.7 0.8 0.9 1 input spatial frequency square wave gratings 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.5 1 1.5 2 input spatial frequency velocity 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.5 1 1.5 2 velocity input spatial frequency 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.5 1 1.5 2 velocity input spatial frequency 0 0.1 0.2 0.3 0.4 0.5 0.6 0.5 0.6 0.7 0.8 0.9 1 accuracy input spatial frequency drifting random dots 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 coherence level velocity 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 velocity coherence level 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 velocity coherence level 0.5 0.6 0.7 0.8 0.9 1 0.5 0.6 0.7 0.8 0.9 1 coherence level accuracy Recent experimental evidence has cast doubt upon the belief that the motion neurons in V1 and MT have very distinct properties. Traditionally, the tuning of V1 motion sensitive neurons is thought to be separable along the spatial and temporal frequency dimensions, while the frequency tuning MT neurons is inseparable, consistent with constant speed tuning. However, it now seems that both V1 and MT neurons actually show a continuum in the degree to which preferred velocity changes with spatial frequency [14][15][16]. Our proposed neurons constructed by position and phase shifts also show an intermediate behavior between speed tuning and space-time separable tuning. With pure phase shifts, the tuning is space-time separable. With position shifts, the neurons become speed tuned. An intermediate tuning is obtained by combining position and phase tuning. Our results on a simple velocity discrimination task suggest a functional role for this intermediate level of tuning in creating motion energy pairs whose relative responses truly indicate changes in velocity around a reference level for stimuli with a broad band of spatial frequency content. Pair-wise comparisons have been previously proposed as a potential method for coding image speed [17][18]. Here, we have demonstrated a systematic way of constructing reliably comparable pairs of neurons using simple neurally plausible circuits. Acknowledgements This work was supported in part by the Hong Kong Research Grants Council under Grant HKUST6300/04E. References [1] W. Reichardt, “Autocorrelation, a principle for the evaluation of sensory information by the central nervous system,” in Sensory Communication, W. A. Rosenblith, ed. (Wiley, New York, 1961). [2] E. Adelson and J. Bergen, “Spatiotemporal energy models for the perception of motion,” Optical Society of America, Journal, A: Optics and Image Science, vol. 2, pp. 284-299, 1985. [3] A. B. Watson and J. A. J. Ahumada, “Model of human visual-motion sensing,” Journal Optical Society of America A, vol. 2, pp. 322-342, 1985. [4] J. P. H. van Santen and G. Sperling, “Elaborated Reichardt detectors,” Journal of the Optical Society of America A, vol. 2, pp. 300-321, 1985. [5] I. Ohzawa, G. C. DeAngelis, and R. D. Freeman, “Stereoscopic depth discrimination in the visual cortex: Neurons ideally suited as disparity detectors,” Science, vol. 249, pp. 1037-1041, 1990. [6] N. Qian, “Computing stereo disparity and motion with known binocular cell properties,” Neural Computation, vol. 6, pp. 390-404, 1994. [7] D. Fleet, H. Wagner, and D. Heeger, “Neural encoding of binocular disparity: Energy models, position shifts and phase shifts,” Vision Research, vol. 36, pp. 1839-1857, 1996. [8] T. Delbruck, “Silicon Retina with Correlation-Based, Velocity-Tuned Pixels,” IEEE Transactions on Neural Networks, vol. 4, pp. 529-541, 1993. [9] A. Anzai, I. Ohzawa and R. D. Freeman, “Neural mechanisms for encoding binocular disparity: Position vs. phase,” J. Neurophysiology, vol. 82, pp. 874-890, 1999. [10] D. J. Heeger, “Model for the extraction of image flow,” Journal Optical Society of America A, vol. 4, pp. 1455-1471, 1987. [11] E. Simoncelli and D. Heeger, “A model of neuronal responses in visual area MT,” Vision Research, vol. 38, pp. 743-61, 1998. [12] Y. Chen and N. Qian, “A course-to-fine disparity energy model with both phase-shift and positionshift receptive field mechanisms,” Neural Computation, vol. 16, pp. 1545-1578, 2004. [13] M. V. Srinivasan, M. Poteser and K. Kral, “Motion detection in insect orientation and navigation,” Vision Research, vol. 39, pp. 2749-2766, 1999. [14] N. Priebe, C. Cassanello, and S. Lisberger, “The Neural Representation of Speed in Macaque Area MT/V5,” Journal of Neuroscience, vol. 23, pp. 5650, 2003. [15] N. Priebe, S. Lisberger, and J. Movshon, “Tuning for Spatiotemporal Frequency and Speed in Directionally Selective Neurons of Macaque Striate Cortex,” Journal of Neuroscience, vol. 26, pp. 29412950, 2006. [16] J. Perrone, “A Single Mechanism Can Explain the Speed Tuning Properties of MT and V1 Complex Neurons,” Journal of Neuroscience, vol. 26, pp. 11987-11991, 2006. [17] P. Thompson, “Discrimination of moving gratings at and above detection threshold,” Vision Research, vol. 23, pp. 1533-1538, 1983. [18] J. A. Perrone, “Simulating the speed and direction tuning of MT neurons using spatiotemporal tuned V1-neuron inputs”. Investigative Opthalmology and Visual Science (Supplement), vol. 35, pp. 2158, 1994.
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A Bayesian Framework for Cross-Situational Word-Learning Michael C. Frank, Noah D. Goodman, and Joshua B. Tenenbaum Department of Brain and Cognitive Science Massachusetts Institute of Technology {mcfrank, ndg, jbt}@mit.edu Abstract For infants, early word learning is a chicken-and-egg problem. One way to learn a word is to observe that it co-occurs with a particular referent across different situations. Another way is to use the social context of an utterance to infer the intended referent of a word. Here we present a Bayesian model of cross-situational word learning, and an extension of this model that also learns which social cues are relevant to determining reference. We test our model on a small corpus of mother-infant interaction and find it performs better than competing models. Finally, we show that our model accounts for experimental phenomena including mutual exclusivity, fast-mapping, and generalization from social cues. To understand the difficulty of an infant word-learner, imagine walking down the street with a friend who suddenly says “dax blicket philbin na fivy!” while at the same time wagging her elbow. If you knew any of these words you might infer from the syntax of her sentence that blicket is a novel noun, and hence the name of a novel object. At the same time, if you knew that this friend indicated her attention by wagging her elbow at objects, you might infer that she intends to refer to an object in a nearby show window. On the other hand if you already knew that “blicket” meant the object in the window, you might be able to infer these elements of syntax and social cues. Thus, the problem of early word-learning is a classic chicken-and-egg puzzle: in order to learn word meanings, learners must use their knowledge of the rest of language (including rules of syntax, parts of speech, and other word meanings) as well as their knowledge of social situations. But in order to learn about the facts of their language they must first learn some words, and in order to determine which cues matter for establishing reference (for instance, pointing and looking at an object but normally not waggling your elbow) they must first have a way to know the intended referent in some situations. For theories of language acquisition, there are two common ways out of this dilemma. The first involves positing a wide range of innate structures which determine the syntax and categories of a language and which social cues are informative. (Though even when all of these elements are innately determined using them to learn a language from evidence may not be trivial [1].) The other alternative involves bootstrapping: learning some words, then using those words to learn how to learn more. This paper gives a proposal for the second alternative. We first present a Bayesian model of how learners could use a statistical strategy—cross-situational word-learning—to learn how words map to objects, independent of syntactic and social cues. We then extend this model to a true bootstrapping situation: using social cues to learn words while using words to learn social cues. Finally, we examine several important phenomena in word learning: mutual exclusivity (the tendency to assign novel words to novel referents), fast-mapping (the ability to assign a novel word in a linguistic context to a novel referent after only a single use), and social generalization (the ability to use social context to learn the referent of a novel word). Without adding additional specialized machinery, we show how these can be explained within our model as the result of domain-general probabilistic inference mechanisms operating over the linguistic domain. 1 ℓ Os ∀s Is r, b Ws Ss Figure 1: Graphical model describing the generation of words (Ws) from an intention (Is) and lexicon (ℓ), and intention from the objects present in a situation (Os). The plate indicates multiple copies of the model for different situation/utterance pairs (s). Dotted portions indicate additions to include the generation of social cues Ss from intentions. 1 The Model Behind each linguistic utterance is a meaning that the speaker intends to communicate. Our model operates by attempting to infer this intended meaning (which we call the intent) on the basis of the utterance itself and observations of the physical and social context. For the purpose of modeling early word learning—which consists primarily of learning words for simple object categories—in our model, we assume that intents are simply groups of objects. To state the model formally, we assume the non-linguistic situation consists of a set Os of objects and that utterances are unordered sets of words Ws1. The lexicon ℓis a (many-to-many) map from words to objects, which captures the meaning of those words. (Syntax enters our model only obliquely by different treatment of words depending on whether they are in the lexicon or not—that is, whether they are common nouns or other types of words.) In this setting the speaker’s intention will be captured by a set of objects in the situation to which she intends to refer: Is ⊆Os. This setup is indicated in the graphical model of Fig. 1. Different situation-utterance pairs Ws, Os are independent given the lexicon ℓ, giving: P(W|ℓ, O) = Y s X Is P(Ws|Is, ℓ) · P(Is|Os). (1) We further simplify by assuming that P(Is|Os) ∝1 (which could be refined by adding a more detailed model of the communicative intentions a person is likely to form in different situations). We will assume that words in the utterance are generated independently given the intention and the lexicon and that the length of the utterance is observed. Each word is then generated from the intention set and lexicon by first choosing whether the word is a referential word or a non-referential word (from a binomial distribution of weight γ), then, for referential words, choosing which object in the intent it refers to (uniformly). This process gives: P(Ws|Is, ℓ) = Y w∈Ws " (1 −γ)PNR(w|ℓ) + γ X x∈Is 1 |Is|PR(w|x, ℓ) # . (2) The probability of word w referring to object x is PR(w|x, ℓ) ∝δx∈ℓ(w), and the probability of word w occurring as a non-referring word is PNR(w|ℓ) ∝ 1 if ℓ(w) = ∅, κ otherwise. (3) (this probability is a distribution over all words in the vocabulary, not just those in lexicon ℓ). The constant κ is a penalty for using a word in the lexicon as a non-referring word—this penalty indirectly enforces a light-weight difference between two different groups of words (parts-of-speech): words that refer and words that do not refer. Because the generative structure of this model exposes the role of speaker’s intentions, it is straightforward to add non-linguistic social cues. We assume that social cues such as pointing are generated 1Note that, since we ignore word order, the distribution of words in a sentence should be exchangeable given the lexicon and situation. This implies, by de Finetti’s theorem, that they are independent conditioned on a latent state—we assume that the latent state giving rise to words is the intention of the speaker. 2 from the speaker’s intent independently of the linguistic aspects (as shown in the dotted arrows of Fig. 1). With the addition of social cues Ss, Eq. 1 becomes: P(W|ℓ, O) = Y s X Is P(Ws|Is, ℓ) · P(Ss|Is) · P(Is|Os). (4) We assume that the social cues are a set Si(x) of independent binary (cue present or not) feature values for each object x ∈Os, which are generated through a noisy-or process: P(Si(x)=1|Is, ri, bi) = 1 −(1 −bi)(1 −ri)δx∈Is . (5) Here ri is the relevance of cue i, while bi is its base rate. For the model without social cues the posterior probability of a lexicon given a set of situated utterances is: P(ℓ|W, O) ∝P(W|ℓ, O)P(ℓ). (6) And for the model with social cues the joint posterior over lexicon and cue parameters is: P(ℓ, r, b|W, O) ∝P(W|ℓ, r, b, O)P(ℓ)P(r, b). (7) We take the prior probability of a lexicon to be exponential in its size: P(ℓ) ∝e−α|ℓ|, and the prior probability of social cue parameters to be uniform. Given the model above and the corpus described below, we found the best lexicon (or lexicon and cue parameters) according to Eq. 6 and 7 by MAP inference using stochastic search2. 2 Previous work While cross-situational word-learning has been widely discussed in the empirical literature, e.g., [2], there have been relatively few attempts to model this process computationally. Siskind [3] created an ambitious model which used deductive rules to make hypotheses about propositional word meanings their use across situations. This model achieved surprising success in learning word meanings in artificial corpora, but was extremely complex and relied on the availability of fully coded representations of the meaning of each sentence, making it difficult to extend to empirical corpus data. More recently, Yu and Ballard [4] have used a machine translation model (similar to IBM Translation Model I) to learn word-object association probabilities. In their study, they used a pre-existing corpus of mother-infant interactions and coded the objects present during each utterance (an example from this corpus—illustrated with our own coding scheme—is shown in Fig. 2). They applied their translation model to estimate the probability of an object given a word, creating a table of associations between words and objects. Using this table, they extracted a lexicon (a group of word-object mappings) which was relatively accurate in its guesses about the names of objects that were being talked about. They further extended their model to incorporate prosodic emphasis on words (a useful cue which we will not discuss here) and joint attention on objects. Joint attention was coded by hand, isolating a subset of objects which were attended to by both mother and infant. Their results reflected a sizable increase in recall with the use of social cues. 3 Materials and Assessment Methods To test the performance of our model on natural data, we used the Rollins section of the CHILDES corpus[5]. For comparison with the model by Yu and Ballard [4], we chose the files me03 and di06, each of which consisted of approximately ten minutes of interaction between a mother and a preverbal infant playing with objects found in a box of toys. Because we were not able to obtain the exact corpus Yu and Ballard used, we recoded the objects in the videos and added a coding of social cues co-occurring with each utterance. We annotated each utterance with the set of objects visible to the infant and with a social coding scheme (for an illustrated example, see Figure 2). Our social code included seven features: infants eyes, infants hands, infants mouth, infant touching, mothers hands, mothers eyes, mother touching. For each utterance, this coding created an object by social feature matrix. 2In order to speed convergence we used a simulated tempering scheme with three temperature chains and a range of data-driven proposals. 3 Figure 2: A still frame from our corpus showing the coding of objects and social cues. We coded all mid-sized objects visible to the infant as well as social information including what both mother and infant were touching and looking at. We evaluated all models based on their coverage of a gold-standard lexicon, computing precision (how many of the word-object mappings in a lexicon were correct relative to the gold-standard), recall (how many of the total correct mappings were found), and their geometric mean, F-score. However, the gold-standard lexicon for word-learning is not obvious. For instance, should it include the mapping between the plural “pigs” or the sound “oink” and the object PIG? Should a goldstandard lexicon include word-object pairings that are correct but were not present in the learning situation? In the results we report, we included those pairings which would be useful for a child to learn (e.g., “oink” →PIG) but not including those pairings which were not observed to co-occur in the corpus (however, modifying these decisions did not affect the qualitative pattern of results). 4 Results For the purpose of comparison, we give scores for several other models on the same corpus. We implemented a range of simple associative models based on co-occurrence frequency, conditional probability (both word given object and object given word), and point-wise mutual information. In each of these models, we computed the relevant statistic across the entire corpus and then created a lexicon by including all word-object pairings for which the association statistic met a threshold value. We additionally implemented a translation model (based on Yu and Ballard [4]). Because Yu and Ballard did not include details on how they evaluated their model, we scored it in the same way as the other associative models, by creating an association matrix based on the scores P(O|W) (as given in equation (3) in their paper) and then creating a lexicon based on a threshold value. In order to simulate this type of threshold value for our model, we searched for the MAP lexicon over a range of parameters α in our prior (the larger the prior value, the less probable a larger lexicon, thus this manipulation served to create more or less selective lexicons) . Base model. In Figure 3, we plot the precision and the recall for lexicons across a range of prior parameter values for our model and the full range of threshold values for the translation model and two of the simple association models (since results for the conditional probability models were very similar but slightly inferior to the performance of mutual information, we did not include them). For our model, we averaged performance at each threshold value across three runs of 5000 search iterations each. Our model performed better than any of the other models on a number of dimensions (best lexicon shown in Table 1), both achieving the highest F-score and showing a better tradeoff between precision and recall at sub-optimal threshold values. The translation model also performed well, increasing precision as the threshold of association was raised. Surprisingly, standard cooccurrence statistics proved to be relatively ineffective at extracting high-scoring lexicons: at any given threshold value, these models included a very large number of incorrect pairs. Table 1: The best lexicon found by the Bayesian model (α=11, γ=0.2, κ=0.01). baby →book bigbird →bird bird →rattle birdie →duck book →book hand →hand hat →hat meow →kitty moocow →cow oink →pig on →ring ring →ring sheep →sheep 4 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 precision recall F=0.12 F=0.12 F=0.21 F=0.44 F=0.54 Co!occurrence frequency Mutual information Translation model Bayesian model Figure 3: Comparison of models on corpus data: we plot model precision vs. recall across a range of threshold values for each model (see text). Unlike standard ROC curves for classification tasks, the precision and recall of a lexicon depends on the entire lexicon, and irregularities in the curves reflect the small size of the lexicons). One additional virtue of our model over other associative models is its ability to determine which objects the speaker intended to refer to. In Table 2, we give some examples of situations in which the model correctly inferred the objects that the speaker was talking about. Social model. While the addition of social cues did not increase corpus performance above that found in the base model, the lexicons which were found by the social model did have several properties that were not present in the base model. First, the model effectively and quickly converged on the social cues that we found subjectively important in viewing the corpus videos. The two cues which were consistently found relevant across the model were (1) the target of the infant’s gaze and (2) the caregiver’s hand. These data are especially interesting in light of the speculation that infants initially believe their own point of gaze is a good cue to reference, and must learn over the second year that the true cue is the caregiver’s point of gaze, not their own [6]. Second, while the social model did not outperform the base model on the full corpus (where many words were paired with their referents several times), on a smaller corpus (taking every other utterance), the social cue model did slightly outperform a model without social cues (max F-score=0.43 vs. 0.37). Third, the addition of social cues allowed the model to infer the intent of a speaker even in the absence of a word being used. In the right-hand column of Table 2, we give an example of a situation in which the caregiver simply says ”see that?” but from the direction of the infant’s eyes and the location of her hand, the model correctly infers that she is talking about the COW, not either of the other possible referents. This kind of inference might lead the way in allowing infants to learn words like pronouns, which serve pick out an unambiguous focus of attention (one that is so obvious based on social and contextual cues that it does not need to be named). Finally, in the next section we show that the addition of social cues to the model allows correct performance in experimental tests of social generalization which only children older than 18 months can pass, suggesting perhaps that the social model is closer to the strategy used by more mature word learners. Table 2: Intentions inferred by the Bayesian model after having learned a lexicon from the corpus. (IE=Infant’s eyes, CH=Caregiver’s hands). Words “look at the moocow” “see the bear by the rattle?” “see that?” Objects COW GIRL BEAR BEAR RATTLE COW BEAR RATTLE COW Social Cues IE & CH→COW Inferred intention COW BEAR RATTLE COW 5 "dax" BALL DAX situation: !7.3, corpus: !631.1, total: !638.4 "dax" BALL DAX situation: !3.4, corpus: !638.9, total: !642.3 "dax" BALL DAX situation: !2.7, corpus: !635.1, total: !637.8 "dax" BALL DAX situation: !2.3, corpus: !642.9, total: !645.2 "this" "is" "a" "koba" KOBA situation: !14.1, corpus: !1582.0, total: !1596.2 "this" "is" "a" "koba" KOBA situation: !11.8, corpus: !1570.2, total: !1582.0 "this" "is" "a" "koba" KOBA situation: !11.8, corpus: !1570.2, total: !1582.0 Figure 4: Possible outcomes in (right) a mutual-exclusivity situation and (left) a fast-mapping situation. Situation score is the log probability of the situation (blue dots represent words and objects) under a lexicon (mappings are red lines). Corpus score is the posterior log likelihood of the entire old corpus, including both prior and likelihood terms. 5 Coverage of experimental phenomena Mutual exclusivity. When children as young as sixteen months hear a request for a novel word (e.g. where is the dax?) they make a surprising inference: they conclude that the novel word applies to a novel object[7, 8]. This inference is surprising because there seems to be no prima facie reason why children should make it—after all, why shouldnt dax simply be another name for a ball? The experimental phenomenon of “mutual exclusivity” has become a touchstone for theories of wordlearning: while some authors argue that children use a piece of language-specific knowledge, a principle of mutual exclusivity (that objects do not have two labels), to make this inference [7], others have argued that childrens mapping of the novel noun is a consequence of more general social-pragmatic principles [9]. We test whether, instead of following from language-specific knowledge or pragmatic principles, the same inference could simply be a result of the probabilistic structure of our model. We use the model to infer the best lexicon for a simple artificial corpus (similar to that used in [10]). We then present the model with a single new situation, analogous to the mutual exclusivity experiments (left side of Figure 4). This new situation consists of hearing a novel word (“dax”) and seeing both a familiar object and a novel object (BALL and DAX). We then compare four different lexicons on their coverage of both this situation and the previous corpus: (1) one that learns nothing new from the new situation, (2) one that maps dax to BALL, (3) one that maps “dax” to DAX, and (4) one that maps dax to both. We evaluate the scores of these lexicons on both the new situation and the old corpus. While learning both words produces the best score on the new situation, explaining with high probability why the word “dax” was produced, it performs worst on the rest of the corpus. In particular, it gives a low probability to the coincidence that, while “dax” meant BALL the entire time, the model happened never to hear dax when there was a ball around. In contrast, a lexicon learning no new words scores best on the corpus (because of the prior on smaller lexicons) but has no explanation for why it heard the word “dax” in the new situation. The lexicon which learns “dax”→BALL scores well on neither the corpus nor the new situation: it has no explanation for why it never heard “dax” before, but it also must take into account the fact that “dax” is only half as likely to be spoken when a BALL is present because the word “ball” also could have been produced. Thus, the correct lexicon, which learns that dax→DAX, performs best when taking into account both the current situation and the model’s prior experience. The success of this lexicon (robust across a variety of simulations and parameter settings) suggests that explaining the phenomenon of mutual exclusivity may not require appeals to special principles, either pragmatic or languagespecific. Instead, the mutual exclusivity phenomenon may come from a general goal: to learn the lexicon which best explains the utterances the learner hears, given their context. 6 "modi" MODI DAX situation: −10.2, corpus: −772.2, total: −782.4 "modi" MODI DAX situation: −6.2, corpus: −774.2, total: −780.3 "modi" MODI DAX situation: −9.1, corpus: −774.2, total: −783.2 "modi" MODI DAX situation: −6.1, corpus: −776.2, total: −782.3 Figure 5: Possible outcomes in a social generalization experiment. The eye-gaze of the speaker (pointing to the MODI) is the only cue which determines that the word “modi” should be mapped to the MODI object; despite this, our model finds the correct mapping. Fast-mapping. A second phenomenon which has been much discussed in the psychological literature is fast-mapping [11]. This label refers to the ability of older children to learn a novel label for a novel object in a well-understood sentence frame after only one or a few exposures and retain it over a significant delay. There are two surprising components to this task: first, the ability of children to learn from a single exposure, and second, the retention of the word for a long period. Although our model cannot speak to the retention interval, our non-social model predicts that a single, ambiguous situation can give enough evidence to learn a new word. Our scenario is similar to the experimental setup used by Markson and Bloom [12]. We learn a lexicon for a small artificial corpus that contains some number of function words, which do not co-occur regularly with any object. We then present the model with a new situation in which there is a novel referent, three words that had been “function words” in our corpus, and one new word (analogous to seeing a novel object, a KOBA, and hearing the utterance “this is a Koba!”; see Figure 4 right side for details). In this scenario, the model strongly favors learning “koba”→KOBA. If it learns nothing, it is penalized on its inability to explain the new situation; if it learns a mapping to a function word, it must explain why this function word was not used referentially in the rest of its experience. Thus, when the other words in the utterance are familiar, our model will learn an appropriate lexical mapping from even a single situation. Social generalization. By adding the ability to learn social cues, our model gains the ability to learn words even in fully ambiguous situations. An experimental demonstration of this phenomena with children is given by Hollich, Hirsh-Pasek, and Golinkoff [6]. In one study, they showed children two novel objects while an experimenter said “Look at the modi!” and looked directly at one of the objects. While 12-month-olds were not able to learn that the word “modi” mapped onto the object that the experimenter looked at, both 18- and 24-month-olds correctly made this inference. As pictured in Figure 5, our model shows this same pattern of inference. While the best explanation of this situation was given by assuming that the word “modi” mapped to both novel objects (bottom right), this alternative was not preferred because it added two mappings to the lexicon rather than one. On the other hand, the most parsimonious option according to the prior was not to learn any new words, but this did not account for the new evidence. Of the two remaining options, the mapping of “modi” to the correct object was preferred exclusively on account of the distribution of social cues. Much like the older children in Hollich and colleagues’ experiment, our model was first able to learn the relevance of particular social cues over the course of its experience (e.g., by processing the corpus) and then apply this knowledge in a novel, ambiguous situation (Figure 5). 7 6 Conclusions We have presented a Bayesian model of cross-situational world-learning which outperforms both baseline associative models and a more sophisticated translation model on learning from noisy corpus data. However, the strength of our model is not just its performance on the corpus, but also a more natural formulation which may contribute to the clarity of our understanding of word learning. By organizing our model around determining the speaker’s referential intent, we find that several puzzling empirical phenomena in word-learning can be explained as consequences of the structure of the model. The first is mutual exclusivity, the tendency to avoid mapping a novel word to a familiar object when a novel object is available. Researchers in the psychological literature have attempted to explain this type of phenomenon in terms of both language-specific constraints and more general social principles. We suggest, however, that mutual exclusivity may be explained as one of a variety of rational inferences that word-learners can make when presented with an ambiguous situation. The same principle applies to the phenomenon of fast-mapping: given the evidence against other mappings, a rational word-learner would do best to learn the novel mapping. In both of these cases, the relevant phenomena come from the basic model design and domain general principles of inference; as do, for instance, the taxonomic inferences observed by Xu & Tenenbaum [13]. Because it is based on a well-posed generative process, the model can be easily extended to account for joint learning with other domains. We have illustrated this by giving an extension to our basic model of social intention, in which social cues independently contribute to establishing the focus of referential intention in a particular situation. A strength of this extension is that the model does not need to know before learning which cues are relevant for establishing referential intention (and indeed, these cues may vary across cultures where pointing is accomplished in different ways). While Yu and Ballard [4] modify their model to incorporate the focus of intention, their social model assumes that the socially-salient objects are externally indicated—it cannot learn what cues signal that focus or their relative weights. Using these learned social cues our model is able to succeed in learning words even when there is no consistent pattern of co-occurrece (either because of a lack of data or because of a truly ambiguous situation). This brings us to the question of the psychological status of our model. Our model does not embody a theory about the process or algorithm that children follow to learn words. Instead, our model can been seen as a proposal about the representations and principles underlying word-learning. According to this proposal, it is not necessary to represent association probabilities for all word-concept pairs in order to learn words statistically. Instead, learners can learn a lexicon consisting only of guesses about the meanings of words. And by applying principles of probabilistic inference to this lexicon, it may be possible to bootstrap into the broader social, communicative system. References [1] S. Pinker. Learnability and cognition: the acquisition of argument structure. MIT Press, 1989. [2] L. Gleitman. The structural sources of verb meanings. Language acquisition, 1:3–55, 1990. [3] J.M. Siskind. A computational study of cross-situational techniques for learning word-to-meaning mappings. Cognition, 61(1):39–91, 1996. [4] C. Yu and D. Ballard. A unified model of word learning: Integrating statistical and social cues. Neurocomputing, in press. [5] B. MacWhinney. The CHILDES Project: Tools for Analyzing Talk. Lawrence Erlbaum, 2000. [6] G. Hollich, K. Hirsh-Pasek, and R.M. Golinkoff. II. The Emergentist Coalition Model. Monographs of the Society for Research in Child Development, 65(3):17–29, 2000. [7] E.M. Markman. Categorization and Naming in Childern: problems of induction. Bradford Book, 1989. [8] C.B. Mervis and J. Bertrand. Acquisition of the Novel Name-Nameless Category (N3C) Principle. Child Development, 65(6):1646–1662, 1994. [9] E. V. Clark. On the logic of contrast. Journal of Child Language, 15:317–335, 1988. [10] C Yu and L Smith. Rapid word learning under uncertainty via cross-situational statistics. Psychological Science, in press. [11] S. Carey. The child as word learner. In Linguistic theory and psychological reality. MA: MIT Press, 1978. [12] L. Markson and P. Bloom. Evidence against a dedicated system for word learning in children. Nature, 385(6619):813–815, 1997. [13] F. Xu and J. B. Tenenbaum. Word learning as bayesian inference. Psychological Review, 2007. 8
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PSVM: Parallelizing Support Vector Machines on Distributed Computers Edward Y. Chang∗, Kaihua Zhu, Hao Wang, Hongjie Bai, Jian Li, Zhihuan Qiu, & Hang Cui Google Research, Beijing, China Abstract Support Vector Machines (SVMs) suffer from a widely recognized scalability problem in both memory use and computational time. To improve scalability, we have developed a parallel SVM algorithm (PSVM), which reduces memory use through performing a row-based, approximate matrix factorization, and which loads only essential data to each machine to perform parallel computation. Let n denote the number of training instances, p the reduced matrix dimension after factorization (p is significantly smaller than n), and m the number of machines. PSVM reduces the memory requirement from O(n2) to O(np/m), and improves computation time to O(np2/m). Empirical study shows PSVM to be effective. PSVM Open Source is available for download at http://code.google.com/p/psvm/. 1 Introduction Let us examine the resource bottlenecks of SVMs in a binary classification setting to explain our proposed solution. Given a set of training data X = {(xi, yi)|xi ∈Rd}n i=1, where xi is an observation vector, yi ∈{−1, 1} is the class label of xi, and n is the size of X, we apply SVMs on X to train a binary classifier. SVMs aim to search a hyperplane in the Reproducing Kernel Hilbert Space (RKHS) that maximizes the margin between the two classes of data in X with the smallest training error (Vapnik, 1995). This problem can be formulated as the following quadratic optimization problem: min P(w, b, ξ) = 1 2∥w∥2 2 + C n X i=1 ξi (1) s.t. 1 −yi(wT φ(xi) + b) ≤ξi, ξi > 0, where w is a weighting vector, b is a threshold, C a regularization hyperparameter, and φ(·) a basis function which maps xi to an RKHS space. The decision function of SVMs is f(x) = wT φ(x)+b, where the w and b are attained by solving P in (1). The optimization problem in (1) is the primal formulation of SVMs. It is hard to solve P directly, partly because the explicit mapping via φ(·) can make the problem intractable and partly because the mapping function φ(·) is often unknown. The method of Lagrangian multipliers is thus introduced to transform the primal formulation into the dual one min D(α) = 1 2αT Qα −αT 1 (2) s.t. 0 ≤α ≤C, yT α = 0, where [Q]ij = yiyjφT (xi)φ(xj), and α ∈Rn is the Lagrangian multiplier variable (or dual variable). The weighting vector w is related with α in w = Pn i=1 αiφ(xi). ∗This work was initiated in 2005 when the author was a professor at UCSB. 1 The dual formulation D(α) requires an inner product of φ(xi) and φ(xj). SVMs utilize the kernel trick by specifying a kernel function to define the inner-product K(xi, xj) = φT (xi)φ(xj). We thus can rewrite [Q]ij as yiyjK(xi, xj). When the given kernel function K is psd (positive semidefinite), the dual problem D(α) is a convex Quadratic Programming (QP) problem with linear constraints, which can be solved via the Interior-Point method (IPM) (Mehrotra, 1992). Both the computational and memory bottlenecks of the SVM training are the IPM solver to the dual formulation of SVMs in (2). Currently, the most effective IPM algorithm is the primal-dual IPM (Mehrotra, 1992). The principal idea of the primal-dual IPM is to remove inequality constraints using a barrier function and then resort to the iterative Newton’s method to solve the KKT linear system related to the Hessian matrix Q in D(α). The computational cost is O(n3) and the memory usage O(n2). In this work, we propose a parallel SVM algorithm (PSVM) to reduce memory use and to parallelize both data loading and computation. Given n training instances each with d dimensions, PSVM first loads the training data in a round-robin fashion onto m machines. The memory requirement per machine is O(nd/m). Next, PSVM performs a parallel row-based Incomplete Cholesky Factorization (ICF) on the loaded data. At the end of parallel ICF, each machine stores only a fraction of the factorized matrix, which takes up space of O(np/m), where p is the column dimension of the factorized matrix. (Typically, p can be set to be about √n without noticeably degrading training accuracy.) PSVM reduces memory use of IPM from O(n2) to O(np/m), where p/m is much smaller than n. PSVM then performs parallel IPM to solve the quadratic optimization problem in (2). The computation time is improved from about O(n2) of a decomposition-based algorithm (e.g., SVMLight (Joachims, 1998), LIBSVM (Chang & Lin, 2001), SMO (Platt, 1998), and SimpleSVM (Vishwanathan et al., 2003)) to O(np2/m). This work’s main contributions are: (1) PSVM achieves memory reduction and computation speedup via a parallel ICF algorithm and parallel IPM. (2) PSVM handles kernels (in contrast to other algorithmic approaches (Joachims, 2006; Chu et al., 2006)). (3) We have implemented PSVM on our parallel computing infrastructures. PSVM effectively speeds up training time for large-scale tasks while maintaining high training accuracy. PSVM is a practical, parallel approximate implementation to speed up SVM training on today’s distributed computing infrastructures for dealing with Web-scale problems. What we do not claim are as follows: (1) We make no claim that PSVM is the sole solution to speed up SVMs. Algorithmic approaches such as (Lee & Mangasarian, 2001; Tsang et al., 2005; Joachims, 2006; Chu et al., 2006) can be more effective when memory is not a constraint or kernels are not used. (2) We do not claim that the algorithmic approach is the only avenue to speed up SVM training. Data-processing approaches such as (Graf et al., 2005) can divide a serial algorithm (e.g., LIBSVM) into subtasks on subsets of training data to achieve good speedup. (Data-processing and algorithmic approaches complement each other, and can be used together to handle large-scale training.) 2 PSVM Algorithm The key step of PSVM is parallel ICF (PICF). Traditional column-based ICF (Fine & Scheinberg, 2001; Bach & Jordan, 2005) can reduce computational cost, but the initial memory requirement is O(np), and hence not practical for very large data set. PSVM devises parallel row-based ICF (PICF) as its initial step, which loads training instances onto parallel machines and performs factorization simultaneously on these machines. Once PICF has loaded n training data distributedly on m machines, and reduced the size of the kernel matrix through factorization, IPM can be solved on parallel machines simultaneously. We present PICF first, and then describe how IPM takes advantage of PICF. 2.1 Parallel ICF ICF can approximate Q (Q ∈Rn×n) by a smaller matrix H (H ∈Rn×p, p ≪n), i.e., Q ≈ HHT . ICF, together with SMW (the Sherman-Morrison-Woodbury formula), can greatly reduce the computational complexity in solving an n × n linear system. The work of (Fine & Scheinberg, 2001) provides a theoretical analysis of how ICF influences the optimization problem in Eq.(2). The authors proved that the error of the optimal objective value introduced by ICF is bounded by C2lϵ/2, where C is the hyperparameter of SVM, l is the number of support vectors, and ϵ is the bound of 2 Algorithm 1 Row-based PICF Input: n training instances; p: rank of ICF matrix H; m: number of machines Output: H distributed on m machines Variables: v: fraction of the diagonal vector of Q that resides in local machine k: iteration number; xi: the ith training instance M: machine index set, M = {0, 1, . . . , m −1} Ic: row-index set on machine c (c ∈M), Ic = {c, c + m, c + 2m, . . .} 1: for i = 0 to n −1 do 2: Load xi into machine imodulom. 3: end for 4: k ←0; H ←0; v ←the fraction of the diagonal vector of Q that resides in local machine. (v(i)(i ∈Im) can be obtained from xi) 5: Initialize master to be machine 0. 6: while k < p do 7: Each machine c ∈M selects its local pivot value, which is the largest element in v: lpvk,c = max i∈Ic v(i). and records the local pivot index, the row index corresponds to lpvk,c: lpik,c = arg max i∈Ic v(i). 8: Gather lpvk,c’s and lpik,c’s (c ∈M) to master. 9: The master selects the largest local pivot value as global pivot value gpvk and records in ik, row index corresponding to the global pivot value. gpvk = max c∈M lpvk,c. 10: The master broadcasts gpvk and ik. 11: Change master to machine ik%m. 12: Calculate H(ik, k) according to (3) on master. 13: The master broadcasts the pivot instance xik and the pivot row H(ik, :). (Only the first k + 1 values of the pivot row need to be broadcast, since the remainder are zeros.) 14: Each machine c ∈M calculates its part of the kth column of H according to (4). 15: Each machine c ∈M updates v according to (5). 16: k ←k + 1 17: end while ICF approximation (i.e. tr(Q −HHT ) < ϵ). Experimental results in Section 3 show that when p is set to √n, the error can be negligible. Our row-based parallel ICF (PICF) works as follows: Let vector v be the diagonal of Q and suppose the pivots (the largest diagonal values) are {i1, i2, . . . , ik}, the kth iteration of ICF computes three equations: H(ik, k) = p v(ik) (3) H(Jk, k) = (Q(Jk, k) − k−1 X j=1 H(Jk, j)H(ik, j))/H(ik, k) (4) v(Jk) = v(Jk) −H(Jk, k)2, (5) where Jk denotes the complement of {i1, i2, . . . , ik}. The algorithm iterates until the approximation of Q by HkHT k (measured by trace(Q −HkHT k )) is satisfactory, or the predefined maximum iterations (or say, the desired rank of the ICF matrix) p is reached. As suggested by G. Golub, a parallelized ICF algorithm can be obtained by constraining the parallelized Cholesky Factorization algorithm, iterating at most p times. However, in the proposed algorithm (Golub & Loan, 1996), matrix H is distributed by columns in a round-robin way on m machines (hence we call it column-based parallelized ICF). Such column-based approach is optimal for the single-machine setting, but cannot gain full benefit from parallelization for two major reasons: 3 1. Large memory requirement. All training data are needed for each machine to calculate Q(Jk, k). Therefore, each machine must be able to store a local copy of the training data. 2. Limited parallelizable computation. Only the inner product calculation (Pk−1 j=1 H(Jk, j)H(ik, j)) in (4) can be parallelized. The calculation of pivot selection, the summation of local inner product result, column calculation in (4), and the vector update in (5) must be performed on one single machine. To remedy these shortcomings of the column-based approach, we propose a row-based approach to parallelize ICF, which we summarize in Algorithm 1. Our row-based approach starts by initializing variables and loading training data onto m machines in a round-robin fashion (Steps 1 to 5). The algorithm then performs the ICF main loop until the termination criteria are satisfied (e.g., the rank of matrix H reaches p). In the main loop, PICF performs five tasks in each iteration k: • Distributedly find a pivot, which is the largest value in the diagonal v of matrix Q (steps 7 to 10). Notice that PICF computes only needed elements in Q from training data, and it does not store Q. • Set the machine where the pivot resides as the master (step 11). • On the master, PICF calculates H(ik, k) according to (3) (step 12). • The master then broadcasts the pivot instance xik and the pivot row H(ik, :) (step 13). • Distributedly compute (4) and (5) (steps 14 and 15). At the end of the algorithm, H is stored distributedly on m machines, ready for parallel IPM (presented in the next section). PICF enjoys three advantages: parallel memory use (O(np/m)), parallel computation (O(p2n/m)), and low communication overhead (O(p2 log(m))). Particularly on the communication overhead, its fraction of the entire computation time shrinks as the problem size grows. We will verify this in the experimental section. This pattern permits a larger problem to be solved on more machines to take advantage of parallel memory use and computation. 2.2 Parallel IPM As mentioned in Section 1, the most effective algorithm to solve a constrained QP problem is the primal-dual IPM. For detailed description and notations of IPM, please consult (Boyd, 2004; Mehrotra, 1992). For the purpose of SVM training, IPM boils down to solving the following equations in the Newton step iteratively. △λ = −λ + vec µ 1 t(C −αi) ¶ + diag( λi C −αi )△x (6) △ξ = −ξ + vec µ 1 tαi ¶ −diag( ξi αi )△x (7) △ν = yT Σ−1z + yT α yT Σ−1y (8) D = diag( ξi αi + λi C −αi ) (9) △x = Σ−1(z −y△ν), (10) where Σ and z depend only on [α, λ, ξ, ν] from the last iteration as follows: Σ = Q + diag( ξi αi + λi C −αi ) (11) z = −Qα + 1n −νy + 1 t vec( 1 αi − 1 C −αi ). (12) The computation bottleneck is on matrix inverse, which takes place on Σ for solving △ν in (8) and △x in (10). Equation (11) shows that Σ depends on Q, and we have shown that Q can be approximated through PICF by HHT . Therefore, the bottleneck of the Newton step can be sped up from O(n3) to O(p2n), and be parallelized to O(p2n/m). Distributed Data Loading To minimize both storage and communication cost, PIPM stores data distributedly as follows: 4 • Distribute matrix data. H is distributedly stored at the end of PICF. • Distribute n × 1 vector data. All n × 1 vectors are distributed in a round-robin fashion on m machines. These vectors are z, α, ξ, λ, ∆z, ∆α, ∆ξ, and ∆λ. • Replicate global scalar data. Every machine caches a copy of global data including ν, t, n, and ∆ν. Whenever a scalar is changed, a broadcast is required to maintain global consistency. Parallel Computation of △ν Rather than walking through all equations, we describe how PIPM solves (8), where Σ−1 appears twice. An interesting observation is that parallelizing Σ−1z (or Σ−1y) is simpler than parallelizing Σ−1. Let us explain how parallelizing Σ−1z works, and parallelizing Σ−1y can follow suit. According to SMW (the Sherman-Morrison-Woodbury formula), we can write Σ−1z as Σ−1z = (D + Q)−1z ≈(D + HHT )−1z = D−1z −D−1H(I + HT D−1H)−1HT D−1z = D−1z −D−1H(GGT )−1HT D−1z. Σ−1z can be computed in four steps: 1. Compute D−1z. D can be derived from locally stored vectors, following (9). D−1z is a n × 1 vector, and can be computed locally on each of the m machines. 2. Compute t1 = HT D−1z. Every machine stores some rows of H and their corresponding part of D−1z. This step can be computed locally on each machine. The results are sent to the master (which can be a randomly picked machine for all PIPM iterations) to aggregate into t1 for the next step. 3. Compute (GGT )−1t1. This step is completed on the master, since it has all the required data. G can be obtained from H in a straightforward manner as shown in SMW. Computing t2 = (GGT )−1t1 is equivalent to solving the linear equation system t1 = (GGT )t2. PIPM first solves t1 = Gy0, then y0 = GT t2. Once it has obtained y0, PIPM can solve GT t2 = y0 to obtain t2. The master then broadcasts t2 to all machines. 4. Compute D−1Ht2 All machines have a copy of t2, and can compute D−1Ht2 locally to solve for Σ−1z. Similarly, Σ−1y can be computed at the same time. Once we have obtained both, we can solve ∆ν according to (8). 2.3 Computing b and Writing Back When the IPM iteration stops, we have the value of α and hence the classification function f(x) = Ns X i=1 αiyik(si, x) + b Here Ns is the number of support vectors and si are support vectors. In order to complete this classification function, b must be computed. According to the SVM model, given a support vector s, we obtain one of the two results for f(s): f(s) = +1, if ys = +1, or f(s) = −1, if ys = −1. In practice, we can select M, say 1, 000, support vectors and compute the average of the bs in parallel using MapReduce (Dean & Ghemawat, 2004). 3 Experiments We conducted experiments on PSVM to evaluate its 1) class-prediction accuracy, 2) scalability on large datasets, and 3) overheads. The experiments were conducted on up to 500 machines in our data center. Not all machines are identically configured; however, each machine is configured with a CPU faster than 2GHz and memory larger than 4GBytes. 5 Table 1: Class-prediction Accuracy with Different p Settings. dataset samples (train/test) LIBSVM p = n0.1 p = n0.2 p = n0.3 p = n0.4 p = n0.5 svmguide1 3, 089/4, 000 0.9608 0.6563 0.9 0.917 0.9495 0.9593 mushrooms 7, 500/624 1 0.9904 0.9920 1 1 1 news20 18, 000/1, 996 0.7835 0.6949 0.6949 0.6969 0.7806 0.7811 Image 199, 957/84, 507 0.849 0.7293 0.7210 0.8041 0.8121 0.8258 CoverType 522, 910/58, 102 0.9769 0.9764 0.9762 0.9766 0.9761 0.9766 RCV 781, 265/23, 149 0.9575 0.8527 0.8586 0.8616 0.9065 0.9264 3.1 Class-prediction Accuracy PSVM employs PICF to approximate an n × n kernel matrix Q with an n × p matrix H. This experiment evaluated how the choice of p affects class-prediction accuracy. We set p of PSVM to nt, where t ranges from 0.1 to 0.5 incremented by 0.1, and compared its class-prediction accuracy with that achieved by LIBSVM. The first two columns of Table 1 enumerate the datasets and their sizes with which we experimented. We use Gaussian kernel, and select the best C and σ for LIBSVM and PSVM, respectively. For CoverType and RCV, we loosed the terminate condition (set -e 1, default 0.001) and used shrink heuristics (set -h 0) to make LIBSVM terminate within several days. The table shows that when t is set to 0.5 (or p = √n), the class-prediction accuracy of PSVM approaches that of LIBSVM. We compared only with LIBSVM because it is arguably the best open-source SVM implementation in both accuracy and speed. Another possible candidate is CVM (Tsang et al., 2005). Our experimental result on the CoverType dataset outperforms the result reported by CVM on the same dataset in both accuracy and speed. Moreover, CVM’s training time has been shown unpredictable by (Loosli & Canu, 2006), since the training time is sensitive to the selection of stop criteria and hyper-parameters. For how we position PSVM with respect to other related work, please refer to our disclaimer in the end of Section 1. 3.2 Scalability For scalability experiments, we used three large datasets. Table 2 reports the speedup of PSVM on up to m = 500 machines. Since when a dataset size is large, a single machine cannot store the factorized matrix H in its local memory, we cannot obtain the running time of PSVM on one machine. We thus used 10 machines as the baseline to measure the speedup of using more than 10 machines. To quantify speedup, we made an assumption that the speedup of using 10 machines is 10, compared to using one machine. This assumption is reasonable for our experiments, since PSVM does enjoy linear speedup when the number of machines is up to 30. Table 2: Speedup (p is set to √n); LIBSVM training time is reported on the last row for reference. Image (200k) CoverType (500k) RCV (800k) Machines Time (s) Speedup Time (s) Speedup Time (s) Speedup 10 1, 958 (9) 10∗ 16, 818 (442) 10∗ 45, 135 (1373) 10∗ 30 572 (8) 34.2 5, 591 (10) 30.1 12, 289 (98) 36.7 50 473 (14) 41.4 3, 598 (60) 46.8 7, 695 (92) 58.7 100 330 (47) 59.4 2, 082 (29) 80.8 4, 992 (34) 90.4 150 274 (40) 71.4 1, 865 (93) 90.2 3, 313 (59) 136.3 200 294 (41) 66.7 1, 416 (24) 118.7 3, 163 (69) 142.7 250 397 (78) 49.4 1, 405 (115) 119.7 2, 719 (203) 166.0 500 814 (123) 24.1 1, 655 (34) 101.6 2, 671 (193) 169.0 LIBSVM 4, 334 NA NA 28, 149 NA NA 184, 199 NA NA We trained PSVM three times for each dataset-m combination. The speedup reported in the table is the average of three runs with standard deviation provided in brackets. The observed variance in speedup was caused by the variance of machine loads, as all machines were shared with other tasks 6 running on our data centers. We can observe in Table 2 that the larger is the dataset, the better is the speedup. Figures 1(a), (b) and (c) plot the speedup of Image, CoverType, and RCV, respectively. All datasets enjoy a linear speedup when the number of machines is moderate. For instance, PSVM achieves linear speedup on RCV when running on up to around 100 machines. PSVM scales well till around 250 machines. After that, adding more machines receives diminishing returns. This result led to our examination on the overheads of PSVM, presented next. (a) Image (200k) speedup (b) Covertype (500k) speedup (c) RCV (800k) speedup (d) Image (200k) overhead (e) Covertype (500k) overhead (f) RCV (800k) overhead (g) Image (200k) fraction (h) Covertype (500k) fraction (i) RCV (800k) fraction Figure 1: Speedup and Overheads of Three Datasets. 3.3 Overheads PSVM cannot achieve linear speedup when the number of machines continues to increase beyond a data-size-dependent threshold. This is expected due to communication and synchronization overheads. Communication time is incurred when message passing takes place between machines. Synchronization overhead is incurred when the master machine waits for task completion on the slowest machine. (The master could wait forever if a child machine fails. We have implemented a checkpoint scheme to deal with this issue.) The running time consists of three parts: computation (Comp), communication (Comm), and synchronization (Sync). Figures 1(d), (e) and (f) show how Comm and Sync overheads influence the speedup curves. In the figures, we draw on the top the computation only line (Comp), which approaches the linear speedup line. Computation speedup can become sublinear when adding machines beyond a threshold. This is because the computation bottleneck of the unparallelizable step 12 in Algorithm 1 (which computation time is O(p2)). When m is small, this bottleneck is insignificant in the total computation time. According to the Amdahl’s law; however, even a small fraction of unparallelizable computation can cap speedup. Fortunately, the larger the dataset is, the smaller is this unparallelizable fraction, which is O(m/n). Therefore, more machines (larger m) can be employed for larger datasets (larger n) to gain speedup. 7 When communication overhead or synchronization overhead is accounted for (the Comp + Comm line and the Comp + Comm + Sync line), the speedup deteriorates. Between the two overheads, the synchronization overhead does not impact speedup as much as the communication overhead does. Figures 1(g), (h), and (i) present the percentage of Comp, Comm, and Sync in total running time. The synchronization overhead maintains about the same percentage when m increases, whereas the percentage of communication overhead grows with m. As mentioned in Section 2.1, the communication overhead is O(p2 log(m)), growing sub-linearly with m. But since the computation time per node decreases as m increases, the fraction of the communication overhead grows with m. Therefore, PSVM must select a proper m for a training task to maximize the benefit of parallelization. 4 Conclusion In this paper, we have shown how SVMs can be parallelized to achieve scalable performance. PSVM distributedly loads training data on parallel machines, reducing memory requirement through approximate factorization on the kernel matrix. PSVM solves IPM in parallel by cleverly arranging computation order. We have made PSVM open source at http://code.google.com/p/psvm/. Acknowledgement The first author is partially supported by NSF under Grant Number IIS-0535085. References Bach, F. R., & Jordan, M. I. (2005). Predictive low-rank decomposition for kernel methods. Proceedings of the 22nd International Conference on Machine Learning. Boyd, S. (2004). Convex optimization. Cambridge University Press. Chang, C.-C., & Lin, C.-J. (2001). LIBSVM: a library for support vector machines. Software available at http://www.csie.ntu.edu.tw/ cjlin/libsvm. Chu, C.-T., Kim, S. K., Lin, Y.-A., Yu, Y., Bradski, G., Ng, A. Y., & Olukotun, K. (2006). Map reduce for machine learning on multicore. NIPS. Dean, J., & Ghemawat, S. (2004). Mapreduce: Simplified data processing on large clusters. OSDI’04: Symposium on Operating System Design and Implementation. Fine, S., & Scheinberg, K. (2001). Efficient svm training using low-rank kernel representations. Journal of Machine Learning Research, 2, 243–264. Ghemawat, S., Gobioff, H., & Leung, S.-T. (2003). The google file system. 19th ACM Symposium on Operating Systems Principles. Golub, G. H., & Loan, C. F. V. (1996). Matrix computations. Johns Hopkins University Press. Graf, H. P., Cosatto, E., Bottou, L., Dourdanovic, I., & Vapnik, V. (2005). Parallel support vector machines: The cascade svm. In Advances in neural information processing systems 17, 521–528. Joachims, T. (1998). Making large-scale svm learning practical. Advances in Kernel Methods Support Vector Learning. Joachims, T. (2006). Training linear svms in linear time. ACM KDD, 217–226. Lee, Y.-J., & Mangasarian, O. L. (2001). Rsvm: Reduced support vector machines. First SIAM International Conference on Data Mining. Chicago. Loosli, G., & Canu, S. (2006). Comments on the core vector machines: Fast svm training on very large data sets (Technical Report). Mehrotra, S. (1992). On the implementation of a primal-dual interior point method. SIAM J. Optimization, 2. Platt, J. (1998). Sequential minimal optimization: A fast algorithm for training support vector machines (Technical Report MSR-TR-98-14). Microsoft Research. Tsang, I. W., Kwok, J. T., & Cheung, P.-M. (2005). Core vector machines: Fast svm training on very large data sets. Journal of Machine Learning Research, 6, 363–392. Vapnik, V. (1995). The nature of statistical learning theory. New York: Springer. Vishwanathan, S., Smola, A. J., & Murty, M. N. (2003). Simplesvm. ICML. 8
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Object Recognition by Scene Alignment Bryan C. Russell Antonio Torralba Ce Liu Rob Fergus William T. Freeman Computer Science and Artificial Intelligence Laboratory Massachusetts Institute of Technology Cambrige, MA 02139 USA {brussell,torralba,celiu,fergus,billf}@csail.mit.edu Abstract Current object recognition systems can only recognize a limited number of object categories; scaling up to many categories is the next challenge. We seek to build a system to recognize and localize many different object categories in complex scenes. We achieve this through a simple approach: by matching the input image, in an appropriate representation, to images in a large training set of labeled images. Due to regularities in object identities across similar scenes, the retrieved matches provide hypotheses for object identities and locations. We build a probabilistic model to transfer the labels from the retrieval set to the input image. We demonstrate the effectiveness of this approach and study algorithm component contributions using held-out test sets from the LabelMe database. 1 Introduction The recognition of objects in a scene often consists of matching representations of image regions to an object model while rejecting background regions. Recent examples of this approach include aligning pictorial cues [4], shape correspondence [1], and modeling the constellation of parts [5]. Other models, exploiting knowledge of the scene context in which the objects reside, have proven successful in boosting object recognition performance [18, 20, 15, 7, 13]. These methods model the relationship between scenes and objects and allow information transfer across the two. Here, we exploit scene context using a different approach: we formulate the object detection problem as one of aligning elements of the entire scene to a large database of labeled images. The background, instead of being treated as a set of outliers, is used to guide the detection process. Our approach relies on the observation that when we have a large enough database of labeled images, we can find with high probability some images in the database that are very close to the query image in appearance, scene contents, and spatial arrangement [6, 19]. Since the images in the database are partially labeled, we can transfer the knowledge of the labeling to the query image. Figure 1 illustrates this idea. With these assumptions, the problem of object detection in scenes becomes a problem of aligning scenes. The main issues are: (1) Can we find a big enough dataset to span the required large number of scene configurations? (2) Given an input image, how do we find a set of images that aligns well with the query image? (3) How do we transfer the knowledge about objects contained in the labels? The LabelMe dataset [14] is well-suited for this task, having a large number of images and labels spanning hundreds of object categories. Recent studies using non-parametric methods for computer vision and graphics [19, 6] show that when a large number of images are available, simple indexing techniques can be used to retrieve images with object arrangements similar to those of a query image. The core part of our system is the transfer of labels from the images that best match the query image. We assume that there are commonalities amongst the labeled objects in the retrieved images and we cluster them to form candidate scenes. These scene clusters give hints as to what objects are depicted 1 desk 3 mouse 1 keyboard 2 mousepad 2 (a) Input image (b) Images with similar scene configuration (c) Output image with object labels transferred screen 2 Figure 1: Overview of our system. Given an input image, we search for images having a similar scene configuration in a large labeled database. The knowledge contained in the object labels for the best matching images is then transfered onto the input image to detect objects. Additional information, such as depth-ordering relationships between the objects, can also be transferred. Figure 2: Retrieval set images. Each of the two rows depicts an input image (on the left) and 30 images from the LabelMe dataset [14] that best match the input image using the gist feature [12] and L1 distance (the images are sorted by their distances in raster order). Notice that the retrieved images generally belong to similar scene categories. Also the images contain mostly the same object categories, with the larger objects often matching in spatial location within the image. Many of the retrieved images share similar geometric perspective. in the query image and their likely location. We describe a relatively simple generative model for determining which scene cluster best matches the query image and use this to detect objects. The remaining sections are organized as follows: In Section 2, we describe our representation for scenes and objects. We formulate a model that integrates the information in the object labels with object detectors in Section 3. In Section 4, we extend this model to allow clustering of the retrieved images based on the object labels. We show experimental results of our system output in Section 5, and conclude in Section 6. 2 Matching Scenes and Objects with the Gist Feature We describe the gist feature [12], which is a low dimensional representation of an image region and has been shown to achieve good performance for the scene recognition task when applied to an entire image. To construct the gist feature, an image region is passed through a Gabor filter bank comprising 4 scales and 8 orientations. The image region is divided into a 4x4 non-overlapping grid and the output energy of each filter is averaged within each grid cell. The resulting representation is a 4 × 8 × 16 = 512 dimensional vector. Note that the gist feature preserves spatial structure information and is similar to applying the SIFT descriptor [9] to the image region. We consider the task of retrieving a set of images (which we refer to as the retrieval set) that closely matches the scene contents and geometrical layout of an input image. Figure 2 shows retrieval sets for two typical input images using the gist feature. We show the top 30 closest matching images from the LabelMe database based on the L1-norm distance, which is robust to outliers. Notice that the gist feature retrieves images that match the scene type of the input image. Furthermore, many of the objects depicted in the input image appear in the retrieval set, with the larger objects residing in approximately the same spatial location relative to the image. Also, the retrieval set has many 2 images that share a similar geometric perspective. Of course, not every retrieved image matches well and we account for outliers in Section 4. 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Retrieval set SVM (local appearance) tree car window person sky road sidewalk plant door chair table screen cabinet lamp pole book keyboard bookshelf bottle mouse head cup blind mug paper phone motorbike mousepad speaker streetlight Figure 3: Evaluation of the goodness of the retrieval set by how well it predicts which objects are present in the input image. We build a simple classifier based on object counts in the retrieval set as provided by their associated LabelMe object labels. We compare this to detection based on local appearance alone using an SVM applied to bounding boxes in the input image (the maximal score is used). The area under the ROC curve is computed for many object categories for the two classifiers. Performance is shown as a scatter plot where each point represents an object category. Notice that the retrieval set predicts well object presence and in a majority cases outperforms the SVM output, which is based only on local appearance. We evaluate the ability of the retrieval set to predict the presence of objects in the input image. For this, we found a retrieval set of 200 images and formed a normalized histogram (the histogram entries sum to one) of the object categories that were labeled. We compute performance for object categories with at least 200 training examples and that appear in at least 15 test images. We compute the area under the ROC curve for each object category. As a comparison, we evaluate the performance of an SVM applied to gist features by using the maximal score over a set of bounding boxes extracted from the image. The area under ROC performance of the retrieval set versus the SVM is shown in Figure 3 as a scatter plot, with each point corresponding to a tested object category. As a guide, a diagonal line is displayed; those points that reside above the diagonal indicate better SVM performance (and vice versa). Notice that the retrieval set predicts well the objects present in the input image and outperforms the detectors based on local appearance information (the SVM) for most object classes. 3 Utilizing Retrieval Set Images for Object Detection In Section 2, we observed that the set of labels corresponding to images that best match an input image predict well the contents of the input image. In this section, we will describe a model that integrates local appearance with object presence and spatial likelihood information given by the object labels belonging to the retrieval set. We wish to model the relationship between object categories o, their spatial location x within an image, and their appearance g. For a set of N images, each having Mi object proposals over L object categories, we assume a joint model that factorizes as follows: p(o, x, g|θ, φ, η) = N Y i=1 Mi Y j=1 1 X hi,j=0 p(oi,j|hi,j, θ) p(xi,j|oi,j, hi,j, φ) p(gi,j|oi,j, hi,j, η) (1) We assume that the joint model factorizes as a product of three terms: (i) p(oi,j|hi,j = m, θm), the likelihood of which object categories will appear in the image, (ii) p(xi,j|oi,j = l, hi,j = m, φm,l), the likely spatial locations of observing object category l in the image, and (iii) p(gi,j|oi,j = l, hi,j = m, ηm,l), the appearance likelihood of object category l. We let hi,j = 1 indicate whether object category oi,j is actually present in location xi,j (hi,j = 0 indicates absence). Figure 4 depicts the above as a graphical model. We use plate notation, where the variable nodes inside a plate are duplicated based on the counts depicted in the top-left corner of the plate. We instantiate the model as follows. The spatial location of objects are parameterized as bounding boxes xi,j = (cx i,j, cy i,j, cw i,j, ch i,j) where (cx i,j, cy i,j) is the centroid and (cw i,j,cw i,j) is the width and 3 height (bounding boxes are extracted from object labels by tightly cropping the polygonal annotation). Each component of xi,j is normalized with respect to the image to lie in [0, 1]. We assume θm are multinomial parameters and φm,l = (µm,l, Λm,l) are Gaussian means and covariances over the bounding box parameters. Finally, we assume gi,j is the output of a trained SVM applied to a gist feature ˜gi,j. We let ηm,l parameterize the logistic function (1 + exp(−ηm,l [1 gi,j]T ))−1. gi,j hi,j xi,j Mi N µm Ám,l ´m,l f0,1g f0,1g L ° ¯ » oi,j L Figure 4: Graphical model that integrates information about which objects are likely to be present in the image o, their appearance g, and their likely spatial location x. The parameters for object appearance η are learned offline using positive and negative examples for each object class. The parameters for object presence likelihood θ and spatial location φ are learned online from the retrieval set. For all possible bounding boxes in the input image, we wish to infer h, which indicates whether an object is present or absent. The parameters ηm,l are learned offline by first training SVMs for each object class on the set of all labeled examples of object class l and a set of distractors. We then fit logistic functions to the positive and negative examples of each class. We learn the parameters θm and φm,l online using the object labels corresponding to the retrieval set. These are learned by simply counting the object class occurrences and fitting Gaussians to the bounding boxes corresponding to the object labels. For the input image, we wish to infer the latent variables hi,j corresponding to a dense sampling of all possible bounding box locations xi,j and object classes oi,j using the learned parameters θm, φm,l, and ηm,l. For this, we compute the postierior distribution p(hi,j = m|oi,j = l, xi,j, gi,j, θm, φm,l, ηm,l), which is proportional to the product of the three learned distributions, for m = {0, 1}. The procedure outlined here allows for significant computational savings over naive application of an object detector. Without finding similar images that match the input scene configuration, we would need to apply an object detector densely across the entire image for all object categories. In contrast, our model can constrain which object categories to look for and where. More precisely, we only need to consider object categories with relatively high probability in the scene model and bounding boxes within the range of the likely search locations. These can be decided based on thresholds. Also note that the conditional independences implied by the graphical model allows us to fit the parameters from the retrieval set and train the object detectors separately. Note that for tractability, we assume Dirichlet and Normal-Inverse-Wishart conjugate prior distributions over θm and φm,l with hyperparemters β and γ = (κ, ϑ, ν, ∆) (expected mean ϑ, κ pseudocounts on the scale of the spatial observations, ν degrees of freedom, and sample covariance ∆). Furthermore, we assume a Bernoulli prior distribution over hi,j parameterized by ξ = 0.5. We hand-tuned the remaining parameters in the model. For hi,j = 0, we assume the noninformative distributions oi,j ∼Uniform(1/L) and each component of xi,j ∼Uniform(1). 4 Clustering Retrieval Set Images for Robustness to Mismatches While many images in the retrieval set match the input image scene configuration and contents, there are also outliers. Typically, most of the labeled objects in the outlier images are not present in the input image or in the set of correctly matched retrieval images. In this section, we describe a process to organize the retrieval set images into consistent clusters based on the co-occurrence of the object labels within the images. The clusters will typically correspond to different scene types and/or viewpoints. The task is to then automatically choose the cluster of retrieval set images that will best assist us in detecting objects in the input image. We augment the model of Section 3 by assigning each image to a latent cluster si. The cluster assignments are distributed according to the mixing weights π. We depict the model in Figure 5(a). Intuitively, the model finds clusters using the object labels oi,j and their spatial location xi,j within the retrieved set of images. To automatically infer the number of clusters, we use a Dirichlet Process prior on the mixing weights π ∼Stick(α), where Stick(α) is the stick-breaking process of Grif4 gi,j hi,j xi,j Mi N µk,m Ák,m,l ´m,l f0,1g f0,1g L ° ¯ » si ¼ ® 1 oi,j L (a) Input image (b) Cluster 1 Cluster 2 Cluster 3 Cluster 4 Cluster 5 (d) 1 2 3 4 5 0 20 40 60 Clusters Counts Cluster counts (c) (e) 0 0.05 0.1 0.15 0.2 screen desk keyboard mouse chair bookshelf floor person poster 0 0.05 0.1 wall chair window floor picture table cabinet lamp book 0 0.05 0.1 0.15 0.2 car window sidewalk building roadsky tree pedestrian doorway 0 0.05 0.1 0.15 0.2 tree plant sky clock flower greenery land berries brush 0 0.2 0.4 0.6 0.8 person bag bedside dish foot furniture garden glass head (f) (g) Figure 5: (a) Graphical model for clustering retrieval set images using their object labels. We extend the model of Figure 4 to allow each image to be assigned to a latent cluster si, which is drawn from mixing weights π. We use a Dirichlet process prior to automatically infer the number of clusters. We illustrate the clustering process for the retrieval set corresponding to the input image in (b). (c) Histogram of the number of images assigned to the five clusters with highest likelihood. (d) Montages of retrieval set images assigned to each cluster, along with their object labels (colors show spatial extent), shown in (e). (f) The likelihood of an object category being present in a given cluster (the top nine most likely objects are listed). (g) Spatial likelihoods for the objects listed in (f). Note that the montage cells are sorted in raster order. fiths, Engen, and McCloskey [8, 11, 16] with concentration parameter α. In the Chinese restaurant analogy, the different clusters correspond to tables and the parameters for object presence θk and spatial location φk are the dishes served at a given table. An image (along with its object labels) corresponds to a single customer that is seated at a table. We illustrate the clustering process for a retrieval set belonging to the input image in Figure 5(b). The five clusters with highest likelihood are visualized in the columns of Figure 5(d)-(g). Figure 5(d) shows montages of retrieval images with highest likelihood that were assigned to each cluster. The total number of retrieval images that were assigned to each cluster are shown as a histogram in Figure 5(c). The number of images assigned to each cluster is proportional to the cluster mixing weights, π. Figure 5(e) depicts the object labels that were provided for the images in Figure 5(d), with the colors showing the spatial extent of the object labels. Notice that the images and labels belonging to each cluster share approximately the same object categories and geometrical configuration. Also, the cluster that best matches the input image tends to have the highest number of retrieval images assigned to it. Figure 5(f) shows the likelihood of objects that appear in the cluster 5 (the nine objects with highest likelihood are shown). This corresponds to θ in the model. Figure 5(g) depicts the spatial distribution of the object centroid within the cluster. The montage of nine cells correspond to the nine objects listed in Figure 5(f), sorted in raster order. The spatial distributions illustrate φ. Notice that typically at least one cluster predicts well the objects contained in the input image, in addition to their location, via the object likelihoods and spatial distributions. To learn θk and φk, we use a Rao-Blackwellized Gibbs sampler to draw samples from the posterior distribution over si given the object labels belonging to the set of retrieved images. We ran the Gibbs sampler for 100 iterations. Empirically, we observed relatively fast convergence to a stable solution. Note that improved performance may be achieved with variational inference for Dirichlet Processes [10, 17]. We manually tuned all hyperparameters using a validation set of images, with concentration parameter α = 100 and spatial location parameters κ = 0.1, ϑ = 0.5, ν = 3, and ∆= 0.01 across all bounding box parameters (with the exception of ∆= 0.1 for the horizontal centroid location, which reflects less certainty a priori about the horizontal location of objects). We used a symmetric Dirichlet hyperparameter with βl = 0.1 across all object categories l. For final object detection, we use the learned parameters π, θ, and φ to infer hi,j. Since si and hi,j are latent random variables for the input image, we perform hard EM by marginalizing over hi,j to infer the best cluster s∗ i . We then in turn fix s∗ i and infer hi,j, as outlined in Section 3. 5 Experimental Results In this section we show qualitative and quantitative results for our model. We use a subset of the LabelMe dataset for our experiments, discarding spurrious and nonlabeled images. The dataset is split into training and test sets. The training set has 15691 images and 105034 annotations. The test set has 560 images and 3571 annotations. The test set comprises images of street scenes and indoor office scenes. To avoid overfitting, we used street scene images that were photographed in a different city from the images in the training set. To overcome the diverse object labels provided by users of LabelMe, we used WordNet [3] to resolve synonyms. For object detection, we extracted 3809 bounding boxes per image. For the final detection results, we used non-maximal suppression. Example object detections from our system are shown in Figure 6(b),(d),(e). Notice that our system can find many different objects embedded in different scene type configurations. When mistakes are made, the proposed object location typically makes sense within the scene. In Figure 6(c), we compare against a baseline object detector using only appearance information and trained with a linear kernel SVM. Thresholds for both detectors were set to yield a 0.5 false positive rate per image for each object category (∼1.3e-4 false positives per window). Notice that our system produces more detections and rejects objects that do not belong to the scene. In Figure 6(e), we show typical failures of the system, which usually occurs when the retrieval set is not correct or an input image is outside of the training set. In Figure 7, we show quantitative results for object detection for a number of object categories. We show ROC curves (plotted on log-log axes) for the local appearance detector, the detector from Section 3 (without clustering), and the full system with clustering. We scored detections using the PASCAL VOC 2006 criteria [2], where the outputs are sorted from most confident to least and the ratio of intersection area to union area is computed between an output bounding box and groundtruth bounding box. If the ratio exceeds 0.5, then the output is deemed correct and the ground-truth label is removed. While this scoring criteria is good for some objects, other objects are not well represented by bounding boxes (e.g. buildings and sky). Notice that the detectors that take into account context typically outperforms the detector using local appearance only. Also, clustering does as well and in some cases outperforms no clustering. Finally, the overall system sometimes performs worse for indoor scenes. This is due to poor retrieval set matching, which causes a poor context model to be learned. 6 Conclusion We presented a framework for object detection in scenes based on transferring knowledge about objects from a large labeled image database. We have shown that a relatively simple parametric 6 (d) (e) sidewalk sky sky sky building sky car road tree building road car car car sky sky road table chair keyboard table car road tree sky road car car car sidewalk sky building building keyboard tabletable road car car person windowwindow person window road window keyboard screen screen keyboard building wall wall table screen wall wall keyboard screen keyboard car car chair (c) (b) (a) keyboard keyboard chair keyboard car road road person building Figure 6: (a) Input images. (b) Object detections from our system combining scene alignment with local detection. (c) Object detections using appearance information only with an SVM. Notice that our system detects more objects and rejects out-of-context objects. (d) More outputs from our system. Notice that many different object categories are detected across different scenes. (e) Failure cases for our system. These often occur when the retrieval set is incorrect. model, trained on images loosely matching the spatial configuration of the input image, is capable of accurately inferring which objects are depicted in the input image along with their location. We showed that we can successfully detect a wide range of objects depicted in a variety of scene types. 7 Acknowledgments This work was supported by the National Science Foundation Grant No. 0413232, the National Geospatial-Intelligence Agency NEGI-1582-04-0004, and the Office of Naval Research MURI Grant N00014-06-1-0734. References [1] A. Berg, T. Berg, and J. Malik. Shape matching and object recognition using low distortion correspondence. In CVPR, volume 1, pages 26–33, June 2005. [2] M. Everingham, A. Zisserman, C.K.I. Williams, and L. Van Gool. The pascal visual object classes challenge 2006 (voc 2006) results. Technical report, September 2006. The PASCAL2006 dataset can be downloaded at http : //www.pascal −network.org/challenges/VOC/voc2006/. 7 10 −4 10 −3 10 −2 10 −1 10 −1 10 0 tree (531) 10 −4 10 −3 10 −2 10 −1 10 −1 10 0 building (547) 10 −4 10 −3 10 −2 10 −1 10 −1 10 0 person (113) 10 −4 10 −3 10 −2 10 −1 10 −1 10 0 screen (268) 10 −4 10 −3 10 −2 10 −1 10 −1 10 0 sidewalk (196) 10 −4 10 −3 10 −2 10 −1 10 −1 10 0 car (138) 10 −4 10 −3 10 −2 10 −1 10 −1 10 0 road (232) 10 −4 10 −3 10 −2 10 −1 10 −1 10 0 keyboard (154) 10 −4 10 −3 10 −2 10 −1 10 −1 10 0 sky (144) 10 −4 10 −3 10 −2 10 −1 10 −1 10 0 motorbike (40) 10 −4 10 −3 10 −2 10 −1 10 −1 10 0 wall (69) 10 −4 10 −3 10 −2 10 −1 10 −1 10 0 bookshelf (47) SVM No clustering Clustering Figure 7: Comparison of full system against local appearance only detector (SVM). Detection rate for a number of object categories tested at a fixed false positive per window rate of 2e-04 (0.8 false positives per image per object class). The number of test examples appear in parenthesis next to the category name. We plot performance for a number of classes for the baseline SVM object detector (blue), the detector of Section 3 using no clustering (red), and the full system (green). Notice that detectors taking into account context performs better in most cases than using local appearance alone. Also, clustering does as well, and sometimes exceeds no clustering. Notable exceptions are for some indoor object categories. This is due to poor retrieval set matching, which causes a poor context model to be learned. [3] C. Fellbaum. Wordnet: An Electronic Lexical Database. Bradford Books, 1998. [4] P. Felzenszwalb and D. Huttenlocher. Pictorial structures for object recognition. Intl. J. Computer Vision, 61(1), 2005. [5] R. Fergus, P. Perona, and A. Zisserman. Object class recognition by unsupervised scale-invariant learning. In CVPR, 2003. [6] James Hays and Alexei Efros. Scene completion using millions of photographs. In ”SIGGRAPH”, 2007. [7] D. Hoiem, A. Efros, and M. Hebert. Putting objects in perspective. In CVPR, 2006. [8] H. Ishwaran and M. Zarepour. Exact and approximate sum-representations for the dirichlet process. Canadian Journal of Statistics, 30:269–283, 2002. [9] David G. Lowe. Distinctive image features from scale-invariant keypoints. Intl. J. Computer Vision, 60(2):91–110, 2004. [10] J. McAuliffe, D. Blei, and M. Jordan. Nonparametric empirical bayes for the Dirichlet process mixture model. Statistics and Computing, 16:5–14, 2006. [11] R. M. Neal. Density modeling and clustering using Dirichlet diffusion trees. In Bayesian Statistics, 7:619–629, 2003. [12] A. Oliva and A. Torralba. Modeling the shape of the scene: a holistic representation of the spatial envelope. Intl. J. Computer Vision, 42(3):145–175, 2001. [13] A. Rabinovich, A. Vedaldi, C. Galleguillos, E. Wiewiora, and S. Belongie. Objects in context. In IEEE Intl. Conf. on Computer Vision, 2007. [14] B. C. Russell, A. Torralba, K. P. Murphy, and W. T. Freeman. Labelme: a database and web-based tool for image annotation. Technical Report AIM-2005-025, MIT AI Lab Memo, September, 2005. [15] E. Sudderth, A. Torralba, W. T. Freeman, and W. Willsky. Learning hierarchical models of scenes, objects, and parts. In IEEE Intl. Conf. on Computer Vision, 2005. [16] Y. W. Teh, M. I. Jordan, M. J. Beal, and D. M. Blei. Hierarchical Dirichlet processes. Journal of the American Statistical Association, 2006. [17] Y. W. Teh, D. Newman, and Welling M. A collapsed variational bayesian inference algorithm for latent dirichlet allocation. In Advances in Neural Info. Proc. Systems, 2006. [18] A. Torralba. Contextual priming for object detection. Intl. J. Computer Vision, 53(2):153–167, 2003. [19] A. Torralba, R. Fergus, and W.T. Freeman. Tiny images. Technical Report AIM-2005-025, MIT AI Lab Memo, September, 2005. [20] A. Torralba, K. Murphy, W. Freeman, and M. Rubin. Context-based vision system for place and object recognition. In Intl. Conf. Computer Vision, 2003. 8
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A neural network implementing optimal state estimation based on dynamic spike train decoding Omer Bobrowski1, Ron Meir1, Shy Shoham2 and Yonina C. Eldar1 Department of Electrical Engineering1 and Biomedical Engineering2 Technion, Haifa 32000, Israel {bober@tx},{rmeir@ee},{sshoham@bm},{yonina@ee}.technion.ac.il Abstract It is becoming increasingly evident that organisms acting in uncertain dynamical environments often employ exact or approximate Bayesian statistical calculations in order to continuously estimate the environmental state, integrate information from multiple sensory modalities, form predictions and choose actions. What is less clear is how these putative computations are implemented by cortical neural networks. An additional level of complexity is introduced because these networks observe the world through spike trains received from primary sensory afferents, rather than directly. A recent line of research has described mechanisms by which such computations can be implemented using a network of neurons whose activity directly represents a probability distribution across the possible “world states”. Much of this work, however, uses various approximations, which severely restrict the domain of applicability of these implementations. Here we make use of rigorous mathematical results from the theory of continuous time point process filtering, and show how optimal real-time state estimation and prediction may be implemented in a general setting using linear neural networks. We demonstrate the applicability of the approach with several examples, and relate the required network properties to the statistical nature of the environment, thereby quantifying the compatibility of a given network with its environment. 1 Introduction A key requirement of biological or artificial agents acting in a random dynamical environment is estimating the state of the environment based on noisy observations. While it is becoming clear that organisms employ some form of Bayesian inference, it is not yet clear how the required computations may be implemented in networks of biological neurons. We consider the problem of a system, receiving multiple state-dependent observations (possibly arising from different sensory modalities) in the form of spike trains, and construct a neural network which, based on these noisy observations, is able to optimally estimate the probability distribution of the hidden world state. The present work continues a line of research attempting to provide a probabilistic Bayesian framework for optimal dynamical state estimation by biological neural networks. In this framework, first formulated by Rao (e.g., [8, 9]), the time-varying probability distributions are represented in the neurons’ activity patterns, while the network’s connectivity structure and intrinsic dynamics are responsible for performing the required computation. Rao’s networks use linear dynamics and discrete time to approximately compute the log-posterior distributions from noisy continuous inputs (rather than actual spike trains). More recently, Beck and Pouget [1] introduced networks in which the neurons directly represent and compute the posterior probabilities (rather than their logarithms) from discrete-time approximate firing rate inputs, using non-linear mechanisms such as multiplicative interactions and divisive normalization. Another relevant line of work, is that of Brown and colleagues as well as others (e.g., [4, 11, 13]) where approximations of optimal dynamical estima1 tors from spike-train based inputs are calculated, however, without addressing the question of neural implementation. Our approach is formulated within a continuous time point process framework, circumventing many of the difficulties encountered in previous work based on discrete time approximations and input smoothing. Moreover, using tools from the theory of continuous time point process filtering (e.g., [3]), we are able to show that a linear system suffices to yield the exact posterior distribution for the state. The key element in the approach is switching from posterior distributions to a new set of functions which are simply non-normalized forms of the posterior distribution. While posterior distributions generally obey non-linear differential equations, these non-normalized functions obey a linear set of equations, known as the Zakai equations [15]. Intriguingly, these linear equations contain the full information required to reconstruct the optimal posterior distribution! The linearity of the exact solution provides many advantages of interpretation and analysis, not least of which is an exact solution, which illustrates the clear distinction between observation-dependent and independent contributions. Such a separation leads to a characterization of the system performance in terms of prior knowledge and real-time observations. Since the input observations appear directly as spike trains, no temporal information is lost. The present formulation allows us to consider inputs arising from several sensory modalities, and to determine the contribution of each modality to the posterior estimate, thereby extending to the temporal domain previous work on optimal multimodal integration, which was mostly restricted to the static case. Inherent differences between the modalities, related to temporal delays and different shapes of tuning curves can be incorporated and quantified within the formalism. In a historical context we note that a mathematically rigorous approach to point process based filtering was developed during the early 1970s following the seminal work of Wonham [14] for finite state Markov processes observed in Gaussian noise, and of Kushner [7] and Zakai [15] for diffusion processes. One of the first papers presenting a mathematically rigorous approach to nonlinear filtering in continuous time based on point process observations was [12], where the exact nonlinear differential equations for the posterior distributions are derived. The presentation in Section 4 summarizes the main mathematical results initiated by the latter line of research, adapted mainly from [3], and serves as a convenient starting point for many possible extensions. 2 A neural network as an optimal filter Consider a dynamic environment characterized at time t by a state Xt, belonging to a set of N states, namely Xt ∈{s1, s2, . . . , sN}. We assume the state dynamics is Markovian with generator matrix Q. The matrix Q, [Q]ij = qij, is defined [5] by requiring that for small values of h, Pr[Xt+h = si|Xt = si] = 1 + qiih + o(h) and Pr[Xt+h = sj|Xt = si] = qijh + o(h) for j ̸= i. The normalization requirement is that P j qij = 0. This matrix controls the process’ infinitesimal progress according to ˙π(t) = π(t)Q, where πi(t) = Pr[Xt = si]. The state Xt is not directly observable, but is only sensed through a set of M random state-dependent observation point processes {N (k) t }M k=1. We take each point process N (k) t to represent the spiking activity of the k-th sensory cell, and assume these processes to be doubly stochastic Poisson counting processes1 with state-dependent rates λk(Xt). These processes are assumed to be independent, given the current state Xt. The objective of state estimation (a.k.a. nonlinear filtering) is to obtain a differential equation for the posterior probabilities pi(t) △= Pr h Xt = si N (1) [0,t], . . . , N (M) [0,t] i , (1) where N (k) [0,t] = {N (k) s }0≤s≤t. In the sequel we denote Y t 0 △= n N (1) [0,t], . . . , N (M) [0,t] o , and refer the reader to Section 4 for precise mathematical definitions. We interpret the rate λk as providing the tuning curve for the k-th sensory input. In other words, the k-th sensory cell responds with strength λk(si) when the input state is Xt = si. The required differential equations for pi(t) are considerably simplified, with no loss of information [3], by considering a set of non-normalized ‘probability functions’ ρi(t), such that pi(t) = ρi(t)/ PN j=1 ρj(t). 1Implying that the rate function itself is a random process. 2 Based on the theory presented in Section 4 we obtain ˙ρi(t) = N X j=1 Qjiρj(t) + M X k=1 (λk(si) −1) "X n δ(t −tk n) −1 # ρi(t), (2) where {tk n} denote the spiking times of the k-th sensory cell. This equation can be written in vector form by defining Λk = diag(λk(s1) −1, λk(s2) −1, . . . λk(sN) −1) ; Λ = M X k=1 Λk, (3) and ρ = (ρ1, . . . , ρN), leading to ˙ρ(t) = (Q −Λ)⊤ρ(t) + M X k=1 Λk X n δ(t −tk n)ρ(t). (4) Equations (2) and (4) can be interpreted as the activity of a linear neural network, where ρi(t) represents the firing rate of neuron i at time t, and the matrix (Q −Λ)⊤represents the synaptic weights (including self-weights); see Figure 1 for a graphical display of the network. Assuming that the tuning functions λk are unimodal, decreasing in all directions from some maximal value (e.g., Gaussian or truncated cosine functions), we observe from (2) that the impact of an input spike at time t is strongest on cell i for which λk(si) is maximal, and decreases significantly for cells j for which sj is ‘far’ from si. This effect can be modelled using excitatory/inhibitory connections, where neurons representing similar states excite each other, while neurons corresponding to very different states inhibit each other (e.g., [2]). This issue will be elaborated on in future work. Figure 1: A graphical depiction of the network implementing optimal filtering of M spike train inputs. Several observations are in place regarding (4). (i) The solution of (4) provides the optimal posterior state estimator given the spike train observations, i.e., no approximation is involved. (ii) The equations are linear even though the equations obeyed by the posterior probabilities pi(t) are nonlinear. (iii) The temporal evolution breaks up neatly into an observationindependent term, which can be conceived of as implementing a Bayesian dynamic prior, and an observation-dependent term, which contributes each time a spike occurs. Note that a similar structure was observed recently in [1]. (iv) The observation process affects the posterior estimate through two terms. First, input processes with strong spiking activity, affect the activity more strongly. Second, the k-th input affects most strongly the components of ρ(t) corresponding to states with large values of the tuning curve λk(si). (v) At this point we assume that the matrix Q is known. In a more general setting, one can expect Q to be learned on a slower time scale, through interaction with the environment. We leave this as a topic for future work. Multi-modal inputs A multi-modal scenario may be envisaged as one in which a subset of the sensory inputs arises from one modality (e.g., visual) while the remaining inputs arise from a different sensory modality (e.g., auditory). These modalities may differ in the shapes of their receptive fields, their response latencies, etc. The framework developed above is sufficiently general to deal with any number of modalities, but consider for simplicity just two modalities, denoted by V and A. It is straightforward to extend the derivation of (4), leading to ˙ρ(t) = (Q −Λv −Λa)⊤ρ(t) + ( Mv X k=1 Λv k X n δ(t −tv,k n ) + Ma X k=1 Λa k X n δ(t −ta,k n ) ) ρ(t). (5) Prediction The framework can easily be extended to prediction, defined as the problem of calculating the future posterior distribution ph i (t) = Pr[Xt+h = si|Y t 0 ]. It is easy to show that the 3 non-normalized probabilities ρh(t) can be calculated using the vector differential equation ˙ρh(t) = (Q −˜Λ)⊤ρh(t) + M X k=1 ˜Λk X n δ(t −tk n)ρh(t), (6) with the initial condition ρh(0) = ehQ⊤ρ(0), and where ˜Λk = ehQ⊤Λke−hQ⊤. Interestingly, the equations obtained are identical to (4), except that the system parameters are modified. Simplified equation When the tuning curves of the sensory cells are uniformly distributed Gaussians (e.g., spatial receptive fields), namely λk(x) = λmax exp(−(x−k∆x)2/2σ2), it can be shown [13] that for small enough ∆x, and a large number of sensory cells, PM k=1 λk(x) ≈β for all x, implying that Λ = P k Λk ≈(β −M)I. Therefore the matrix Λ has no effect on the solution of (4), except for an exponential attenuation that is applied to all the cells simultaneously. Therefore, in cases where the number of sensory cells is large, Λ can be omitted from (4). This means that between spike arrivals, the system behaves solely according to the a-priori knowledge about the world, and when a spike arrives, this information is reshaped according to the firing cell’s tuning curve. 3 Theoretical Implications and Applications Viewing (4) we note that between spike arrivals, the input has no effect on the system. Therefore, the inter-arrival dynamics is simply ˙ρ(t) = (Q −Λ)⊤ρ(t). Defining tn as the n-th arrival time of a spike from any one of the sensors, the solution in the interval (tn, tn+1) is ρ(t) = e(t−tn)(Q−Λ)⊤ρ(tn). When a new spike arrives from the k-th sensory neuron at time tn the system is modified within an infinitesimal window of time as ρi(t+ n ) = ρi(t− n ) + ρi(t− n )(λk(si) −1) = ρi(t− n )λk(si). (7) Thus, at the exact time of a spike arrival from the k-th sensory cell, the vector ρ is reshaped according to the tuning curve of the input cell that fired this spike. Assuming n spikes occurred before time t, we can derive an explicit solution to (4), given by ρ(t) = e(t−tn)(Q−Λ)⊤ n Y i=1 (I + Λk(ti))e(ti−ti−1)(Q−Λ)⊤ρ(0), (8) where k(ti) is the index of the cell that fired at ti, I is the identity matrix, and we assumed initial conditions ρ(0) at t0 = 0. 3.1 Demonstrations We demonstrate the operation of the system on several synthetic examples. First consider a small object moving back and forth on a line, jumping between a set of discrete states, and being observed by a retina with M sensory cells. Each world state si describes the particle’s position, and each sensory cell k generates a Poisson spike train with rate λk(Xt), taken to be a Gaussian λmax exp (−(x −xk)2/2σ2). Figure 2(a) displays the motion of the particle for a specific choice of matrix Q, and 2(b) presents the spiking activity of 10 position sensitive sensory cells. Finally, Figure 2(c) demonstrates the tracking ability of the system, where the width of the gray trace corresponds to the prediction confidence. Note that the system is able to distinguish between 25 different states rather well with only 10 sensory cells. In order to enrich the systems’s estimation capabilities, we can include additional parameters within the state of the world. Considering the previous example, we create a larger set of states: ˜sij = (si, dj), where dj denotes the current movement direction (in this case d1=up, d2=down). We add a population of sensory cells that respond differently to different movement directions. This lends further robustness to the state estimation. As can be seen in Figure 2(d)-(f), when for some reason the input of the sensory cells is blocked (and the sensory cells fire spontaneously) the system estimates a movement that continues in the same direction. When the blockade is removed, the system is resynchronized with the input. It can be seen that even during periods where sensory input is absent, the general trend is well predicted, even though the estimated uncertainty is increased. 4 By expanding the state space it is also possible for the system to track multiple objects simultaneously. In figure 2(g)-(i) we present tracking of two simultaneously moving objects. This is done simply by creating a new state space, sij = (s1 i , s2 j), where sk i denotes the state (location) of the k-th object. 0 1 2 3 4 5 6 7 8 9 0 10 20 (a) Object trajectory t[sec] x[cm] 0 1 2 3 4 5 6 7 8 9 0 5 10 t[sec] cell # (b) Input activity 0 2 4 6 8 10 5 10 15 20 25 t[sec] x[cm] (c) Posterior probability evolution 0 1 2 3 4 5 6 7 8 9 0 10 20 (d) Object trajectory t[sec] x[cm] 0 1 2 3 4 5 6 7 8 9 0 5 10 t[sec] cell # (e) Input activity 0 2 4 6 8 10 5 10 15 20 25 t[sec] x[cm] (f) Posterior probability evolution 0 1 2 3 4 5 6 7 8 9 0 5 10 (g) Object trajectory t[sec] x[cm] Object #1 Object #2 0 1 2 3 4 5 6 7 8 9 0 5 10 t[sec] cell # (h) Input activity 0 2 4 6 8 10 2 4 6 8 10 t[sec] x[cm] (i) Posterior probability evolution Figure 2: Tracking the motion of an object in 1D. (a) The object’s trajectory. (b) Spiking activity of 10 sensory cells. (c) Decoded position estimation with confidence interval. Each of the 10 sensory cells has a Gaussian tuning curve of width σ = 2 and maximal firing rate λmax = 25.(d)-(f) Tracking based on position and direction information. The red bar marks the time when the input was blocked, and the green bar marks the time when the blockade was removed. Here we used 10 place-cells and 4 direction-cells (marked in red). (g)-(i) Tracking of two objets simultaneously. The network activity in (i) represents Pr X1 t = si ∨X2 t = si|Y t 0 . 3.2 Behavior Characterization The solution of the filtering equations (4) depends on two processes, namely the recurrent dynamics due to the first term, and the sensory input arising from the second term. Recall that the connectivity matrix Q is essentially the generator matrix of the state transition process, and as such, incorporates prior knowledge about the world dynamics. The second term, consisting of the sensory input, contributes to the state estimator update every time a spike occurs. Thus, a major question relates to the interplay between the a-priori knowledge embedded in the network through Q and the incoming sensory input. In particular, an important question relates to tailoring the system parameters (e.g., the tuning curves λk), to the properties of the external world. As a simple characterization of the generator matrix Q, we consider the diagonal and non-diagonal terms. The diagonal term qii is related to the average time spent in state i through E[Ti] = −1/qii [5], and thus we define τ(Q) = − q−1 11 + · · · + q−1 NN /N, as a measure of the transition frequency of the process, where small values of τ correspond to a rapidly changing process. A second relevant measure relates to the regularity in the transition between the states. To quantify this consider a state i, and define a probability vector qi consisting of the N −1 elements {Qij}, j ̸= i, normalized so that the sum of the elements is 1. The entropy of qi is a measure for the state transition irregularity from state i, and we define H(Q) as the average of this entropy over all states. In summary, we lump the main properties of Q into τ(Q), related to the rapidity of the process, and H(Q), measuring the transition regularity. Clearly, these variables are but one heuristic choice for characterizing the Markov process dynamics, but they will enable us to relate the ‘world dynamics’ to the system behavior. The sensory input influence on the system is controlled by the tuning curves. To simplify the analysis we assume uniformly placed Gaussian tuning curves, λk(x) = λmax exp (−(x −k∆x)2/2σ2), which can be characterized by two parameters - the maximum value λmax and the width σ. Note, however that our model does not require any special constraints on the tuning curves. Figure 3 examines the system performance under different world setups. We measure the performance using the L1 error of the maximum aposteriori (MAP) estimator built from the posterior distribution generated by the system. The MAP estimator is obtained by selecting the cell with the highest firing activity ρi(t), is optimal under the present setting (leading to the minimal probability of error), and can be easily implemented in a neural network by a Winner-Take-All circuit. The choice of the L1 error is justified in this case since the states {si} represent locations on a line, 5 thereby endowing the state space with a distance measure. In figure 3(a) we can see that as τ(Q) increases, the error diminishes, an expected result, since slower world dynamics are easier to analyze. The effect of H(Q) is opposite - lower entropy implies higher confidence in the next state. Therefore we can see that the error increases with H(Q) (fig. 3(b)). The last issue we examine relates to the behavior of the system when an incorrect Q matrix is used (i.e., the world model is incorrect). It is clear from figure 3(c) that for low values of M (the number of sensory cells), using the wrong Q matrix increases the error level significantly. However as the value of M increases, the differences are reduced. This phenomenon is expected, since the more observations are available about the world, the less weight need be assigned to the a-priori knowledge. 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 τ(Q) L1 Error (a) Effect of state rapidity 1 1.5 2 2.5 3 3.5 4 0.8 1 1.2 1.4 1.6 H(Q) L1 Error (b) Effect of transition entropy 0 100 200 300 400 500 0 2 4 6 8 10 M L1 Error Correct model Wrong Q−matrix, same τ(Q) Wrong Q−matrix, different τ(Q) (c) Effect of misspecification Figure 3: State estimation error for different world dynamics and model misspecification. For (a) and (b) M = 17, N = 17, σ = 3, λmax = 50, and for (c) N = 25, σ = 3, λmax = 50. In figure 4 we examine the effect of the tuning curve parameters on the system’s performance. Given a fixed number of input cells, if the tuning curves are too narrow (fig. 4(a) top), they will not cover the entire state space. On the other hand, if the tuning curves are too wide (fig. 4(a) bottom) the cell’s response is very similar for all states. Therefore we get an error function that has a local minimum (fig. 4(b)). It remains for future work to determine what is the optimal value of σ for a given model. The effect of different values of λmax is obvious - higher values of λmax lead to more spikes per sensory cell which increases the system’s accuracy. Clearly, under physiological conditions, where high firing rates are energetically costly, we would expect a tradeoff between accuracy and energy expenditure. 0 10 20 30 40 50 60 70 80 90 100 0 20 40 low σ x[cm] λk(x) 0 10 20 30 40 50 60 70 80 90 100 0 20 40 medium σ x[cm] λk(x) 0 10 20 30 40 50 60 70 80 90 100 0 20 40 high σ x[cm] λk(x) (a) −2 −1 0 1 2 3 0 2 4 6 8 10 12 14 log(σ) L1 Error λmax = 50 λmax = 25 λmax = 10 (b) Figure 4: The effect of the tuning curves parameters on performance. 4 Mathematical Framework and Derivations We summarize the main mathematical results related to point process filtering, adapted mainly from [3]. Consider a finite-state continuous-time Markov process Xt ∈{s1, s2, . . . , sN} with a generator matrix Q that is being observed via a set of (doubly stochastic) Poisson processes with statedependent rate functions λk(Xt), k = 1, . . . , M. Consider first a single point process observation N t 0 = {Ns}0≤s≤t. We denote the joint probability law for the state and observation process by P1. The objective is to derive a differential equation for the posterior probabilities (1). This is the classic nonlinear filtering problem from systems theory 6 (e.g. [6]). More generally, the problem can be phrased as computing E1[f(Xt)|N t 0], where, in the case of (1), f is a vector function, with components fi(x) = [x = si]. We outline the derivation required to obtain such an equation, using a method referred to as change of measure (e.g., [3]). The basic idea is to replace the computationally hard evaluation of E1[f(Xt)|N t 0], by a tractable computation based on a simple probability law. Consider two probability spaces (Ω, F, P′) and (Ω, F, P∞) that differ only in their probability measures. P1 is said to be absolutely continuous with respect to P0 (denoted by P1 ≪P0), if for all A ∈F, P0(A) = 0 ⇒P1(A) = 0. Assuming P1 ≪P0, it can be proved that there exists a random variable L(ω), ω ∈Ω, such that for all A ∈F, P1(A) = E0[1AL] = Z A L(ω)dP0(ω), (9) where E0 denotes the expectation with regard to P0. The random variable L is called the RadonNykodim derivative of P1 with respect to P0, and is denoted by L(ω) = dP1(ω)/dP0(ω). Consider two continuous-time random processes - Xt,Nt, that have different interpretation under the different probability measures - P0, P1: P0 : Xt is a finite-state Markov process, Xt ∈{s1, s2, . . . , sN}. Nt is a Poisson process with a constant rate of 1, independent of Xt , (10) P1 : Xt is a finite-state Markov process, Xt ∈{s1, s2, . . . , sN}. Nt is a doubly-stochastic Poisson process with rate function: λ(Xt) . (11) The following avatar of Bayes’ formula (eq. 3.5 in chap. 6 of [3]), supplies a way to calculate the conditional expectation E1[f(Xt)|N t 0] based on P1 in terms of an expectation w.r.t.P0, E1[f(Xt)|N t 0] = E0[Ltf(Xt)|N t 0] E0[Lt|N t 0] , (12) where Lt = dP1,t/dP0,t, and P0,t and P1,t are the restrictions of P0 and P1, respectively, to the sigma-algebra generated by {N t 0, X∞ 0 }. We refer the reader to [3] for precise definitions. Using (1) and (12) we have pi(t) = E1[fi(Xt)|N t 0] = E0[Ltfi(Xt)|N t 0] E0[Lt|N t 0] . (13) Since the denominator is independent of i, it can be regarded as a normalization factor. Thus, defining ρi(t) △= E0[Ltfi(Xt)|N t 0], it follows that pi(t) = ρi(t)/ PN j=1 ρj(t). Based on the above derivation, one can show ([3], chap. 6.4) that {ρi(t)} obey the stochastic differential equation (SDE) dρi(t) = N X j=1 Qjiρj(t)dt + (λ(si) −1)ρi(t)(dNt −dt). (14) A SDE of the form dρ(t) = a(t)dt + b(t)dNt should be interpreted as follows. If at time t, no jump occurred in the counting process Nt, then ρ(t + dt) −ρ(t) ≈a(t)dt, where dt denotes an infinitesimal time interval. If a jump occurred at time t then ρ(t + dt) −ρ(t) ≈a(t)dt + b(t). Since the jump locations are random, ρ(t) is a stochastic process, hence the term SDE. Now, this derivation can be generalized to the case where there are M observation processes N (1) t , N (2) t , . . . , N (M) t with different rate functions λ1(Xt), λ2(Xt), . . . , λM(Xt). In this case the differential equations for the non-normalized posterior probabilities is dρi(t) = N X j=1 Qjiρj(t)dt + M X k=1 (λk(si) −1)ρi(t)(dN (k) t −dt) (15) Recalling that N (k) t is a counting process, namely dN (k) t /dt = P n δ(t −tk n), we obtain (2), where tk n is the arrival time of the n-th event in the k-th observation process. 7 5 Discussion In this work we have introduced a linear recurrent neural network model capable of exactly implementing Bayesian state estimation and prediction from input spike trains in real time. The framework is mathematically rigorous and requires few assumptions, is naturally formulated in continuous time, and is based directly on spike train inputs, thereby sacrificing no temporal resolution. The setup is ideally suited to the integration of several sensory modalities, and retains its optimality in this setting as well. The linearity of the system renders an analytic solution possible, and a full characterization in terms of a-priori knowledge and online sensory input. This framework sets the stage for many possible extensions and applications, of which we mention several examples. (i) It is important to find a natural mapping between the current abstract neural model and more standard biological neural network models. One possible approach was mentioned in Section 2, but other options are possible and should be pursued. Additionally, the implementation of the estimation network (namely, the variables ρi(t)) using realistic spiking neurons is still open. (ii) At this point the matrix Q in (4) is assumed to be known. Combining approaches to learning Q and adapting the tuning curves λk in real time will lend further plausibility and robustness to the system. (iii) The present framework, based on doubly stochastic Poisson processes, can be extended to more general point processes, using the filtering framework described in [10]. (iv) Currently, each world state is represented by a single neuron (a grandmother cell). This is clearly a non-robust representation, and it would be worthwhile to develop more distributed and robust representations. Finally, the problem of experimental verification of the framework is a crucial step in future work. Acknowledgments The authors are grateful to Rami Atar his helpful advice on nonlinear filtering. References [1] J.M. Beck and A. Pouget. Exact inferences in a neural implementation of a hidden markov model. Neural Comput, 19(5):1344–1361, 2007. [2] R. Ben-Yishai, R.L. Bar-Or, and H. Sompolinsky. Theory of orientation tuning in visual cortex. Proc Natl Acad Sci U S A, 92(9):3844–8, Apr 1995. 542. [3] P. Br´emaud. Point Processes and Queues: Martingale Dynamics. Springer, New York, 1981. [4] U.T. Eden, L.M. Frank, V. Solo, and E.N. Brown. Dynamic analysis of neural encoding by point process adaptive filtering. Neural Computation, 16:971–998, 2004. [5] G.R. Grimmett and D.R. Stirzaker. Probability and Random Processes. Oxford University Press, third edition, 2001. [6] A.H. Jazwinsky. Stochastic Processes and Filtering Theory. Academic Press, 1970. [7] H.J. Kushner. Dynamical equations for optimal nonlinear filtering. J. Differential Equations, 3:179–190, 1967. [8] R.P.N. Rao. Bayesian computation in recurrent neural circuits. Neural Comput, 16(1):1–38, 2004. 825. [9] R.P.N. Rao. Neural models of Bayesain belief propagation. In K. Doya, S. Ishii, A. Pouget, and R. P. N. Rao, editors, Bayesian Brain, chapter 11. MIT Press, 2006. [10] A. Segall, M. Davis, and T. Kailath. Nonlinear filtering with counting observations. IEEE Tran. Information Theory,, 21(2):143–149, 1975. [11] S. Shoham, L.M. Paninski, M.R. Fellows, N.G. Hatsopoulos, J.P. Donoghue, and R.A. Norman. Statistical encoding model for a primary motor cortical brain-machine interface. IEEE Trans Biomed Eng., 52(7):1312–22, 2005. [12] D. L. Snyder. Filtering and detection for doubly stochastic Poisson processes. IEEE Transactions on Information Theory, IT-18:91–102, 1972. [13] N. Twum-Danso and R. Brockett. Trajectory estimation from place cell data. Neural Netw, 14(6-7):835–844, 2001. [14] W.M. Wonham. Some applications of stochastic differential equations to optimal nonlinear filtering. J. SIAM Control, 2(3):347–369, 1965. [15] M. Zakai. On the optimal filtering of diffusion processes. Z. Wahrscheinlichkeitheorie verw Gebiete, 11:230–243, 1969. 8
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Computational Equivalence of Fixed Points and No Regret Algorithms, and Convergence to Equilibria Elad Hazan IBM Almaden Research Center 650 Harry Road San Jose, CA 95120 hazan@us.ibm.com Satyen Kale Computer Science Department, Princeton University 35 Olden St. Princeton, NJ 08540 satyen@cs.princeton.edu Abstract We study the relation between notions of game-theoretic equilibria which are based on stability under a set of deviations, and empirical equilibria which are reached by rational players. Rational players are modeled by players using no regret algorithms, which guarantee that their payoff in the long run is close to the maximum they could hope to achieve by consistently deviating from the algorithm’s suggested action. We show that for a given set of deviations over the strategy set of a player, it is possible to efficiently approximate fixed points of a given deviation if and only if there exist efficient no regret algorithms resistant to the deviations. Further, we show that if all players use a no regret algorithm, then the empirical distribution of their plays converges to an equilibrium. 1 Introduction We consider a setting where a number of agents need to repeatedly make decisions in the face of uncertainty. In each round, the agent obtains a payoff based on the decision she chose. Each agent would like to be able to maximize her payoff. While this might seem like a natural objective, it may be impossible to achieve without placing restrictions on the kind of payoffs that can arise. For instance, if the payoffs were adversarially chosen, then the agent’s task would become essentially hopeless. In such a situation, one way for the agent to cope with the uncertainty is to aim for a relative benchmark rather an absolute one. The notion of regret minimization captures this intuition. We imagine that the agent has a choice of several well-defined ways to change her decision, and now the agent aims to maximize her payoff relative to what she could have obtained had she changed her decisions in a consistent manner. As an example of what we mean by consistent changes, a possible objective could be to maximize her payoff relative to the most she could have achieved by choosing some fixed decision in all the rounds. The difference between these payoffs is known as external regret in the game theory literature. Another notion is that of internal regret, which arises when the possible ways to change are the ones that switch from some decision i to another, j, whenever the agent chose decision i, leaving all other decisions unchanged. A learning algorithm for an agent is said to have no regret with respect to an associated set of decision modifiers (also called deviations) Φ if the average payoff of an agent using the algorithm converges to the largest average payoff she would have achieved had she changed her decisions using a fixed decision modifier in all the rounds. Based on what set of decision modifiers are under consideration, various no regret algorithms are known (for e.g. Hannan [10] gave algorithms to minimize external regret, and Hart and Mas-Collel [11] give algorithms to minimize internal regret). 1 The reason no regret algorithms are so appealing, apart from the fact that they model rational behavior of agents in the face of uncertainty, is that in various cases it can be shown that using no regret algorithms guides the overall play towards a game theoretic equilibrium. For example, Freund and Schapire [7] show that in a zero-sum game, if all agents use a no external regret algorithm, then the empirical distribution of the play converges to the set of minimax equilibria. Similarly, Hart and Mas-Collel [11] show that if all agents use a no internal regret algorithm, then the empirical distribution of the play converges to the set of correlated equilibria. In general, given a set of decision modifiers Φ, we can define a notion of game theoretic equilibrium that is based on the property of being stable under deviations specified by Φ. This is a joint distribution on the agents’ decisions that ensures that the expected payoff to any agent is no less than the most she could achieve if she decided to unilaterally (and consistently) decided to deviate from her suggested action using any decision modifier in Φ. One can then show that if all agents use a Φ-no regret algorithm, then the empirical distribution of the play converges to the set of Φ-equilibria. This brings us to the question of whether it is possible to design no regret algorithms for various sets of decision modifiers Φ. In this paper, we design algorithms which achieve no regret with respect to Φ for a very general setting of arbitrary convex compact decision spaces, arbitrary concave payoff functions, and arbitrary continuous decision modifiers. Our method works as long as it is possible to compute approximate fixed points for (convex combinations) of decision modifiers in Φ. Our algorithms are based on a connection to the framework of Online Convex Optimization (see, e.g. [18]) and we show how to apply known learning algorithms to obtain Φ-no regret algorithms. The generality of our connection allows us to use various sophisticated Online Convex Optimization algorithms which can exploit various structural properties of the utility functions and guarantee a faster rate of convergence to the equilibrium. Previous work by Greenwald and Jafari [9] gave algorithms for the case when the decision space is the simplex of probability distributions over the agents’ decisions, the payoff functions are linear, and the decision modifiers are also linear. Their algorithm, based on the work of Hart and MasCollel [11], uses a version of Blackwell’s Approachability Theorem, and also needs to computes fixed points of the decision modifiers. Since these modifiers are linear, it is possible to compute fixed points for them by computing the stationary distribution of an appropriate stochastic matrix (say, by computing its top eigenvector). Computing Brouwer fixed points of continuous functions is in general a very hard problem (it is PPAD-complete, as shown by Papadimitriou [15]). Fixed points are ubiquitous in game theory. Most common notions of equilibria in game theory are defined as the set of fixed points of a certain mapping. For example, Nash Equilibria (NE) are the set of fixed points of the best response mapping (appropriately defined to avoid ambiguity). The fact that Brouwer fixed points are hard to compute in general is no reason why computing specific fixed points should be hard (for instance, as mentioned earlier, computing fixed points of linear functions is easy via eigenvector computations). More specifically, could it be the case that the NE, being a fixed point of some well-specified mapping, is easy to compute? These hopes were dashed by the work of [6, 3] who showed that computing NE is as computationally difficult as finding fixed points in a general mapping: they show that computing NE in a two-player game is PPAD-complete. Further work showed that even computing an approximate NE is PPAD-complete [4]. Since our algorithms (and all previous ones as well) depend on computing (approximate) fixed points of various decision modifiers, the above discussion leads us to question whether this is necessary. We show in this paper that indeed it is: a Φ-no-regret algorithm can be efficiently used to compute approximate fixed points of any convex combination of decision modifiers. This establishes an equivalence theorem, which is the main contribution of this paper: there exist efficient Φ-no-regret algorithms if and only it is possible to efficiently compute fixed points of convex combinations of decision modifiers in Φ. This equivalence theorem allows us to translate complexity theoretic lower bounds on computing fixed points to designing no regret algorithms. For instance, a Nash equilibrium can be obtained by applying Brouwer’s fixed point theorem to an appropriately defined continuous mapping from the compact convex set of pairs of the players’ mixed strategies to itself. Thus, if Φ contains this mapping, then it is PPAD-hard to design Φ-no-regret algorithms. It was recently brought to our attention that Stolz and Lugosi [17], building on the work of Hart and Schmeidler [12], have also considered Φ-no-regret algorithms. They also show how to design them 2 from fixed-point oracles, and proved convergence to equilibria under even more general conditions than we consider. Gordon, Greenwald, Marks, and Zinkevich [8] have also considered similar notions of regret and showed convergence to equilibria, in the special case when the deviations in Φ can be represented as the composition of a fixed embedding into a higher dimensional space and an adjustable linear transformation. The focus of our results is on the computational aspect of such reductions, and the equivalence of fixed-points computation and no-regret algorithms. 2 Preliminaries 2.1 Games and Equilibria We consider the following kinds of games. First, the set of strategies for the players of the game is a convex compact set. Second, the utility functions for the players are concave over their strategy sets. To avoid cumbersome notation, we restrict ourselves to two player games, although all of our results naturally extend to multi-player games. Formally, for i = 1, 2, player i plays points from a convex compact set Ki ⊆Rni. Her payoff is given by function ui : K1 × K2 →R, i.e. if x1, x2 is the pair of strategies played by the two players, then the payoff to player i is given by ui(x1, x2). We assume that u1 is a concave function of x1 for any fixed x2, and similarly u2 is a concave function of x2 for any fixed x1. We now define a notion of game theoretic equilibrium based on the property of being stable with respect to consistent deviations. By this, we mean an online game-playing strategy for the players that will guarantee that neither stands to gain if they decided to unilaterally, and consistently, deviate from their suggested moves. To model this, assume that each player i has a set of possible deviations Φi which is a finite1 set of continuous mappings φi : Ki →Ki. Let Φ = (Φ1, Φ2). Let Ψ be a joint distribution on K1 × K2. If it is the case that for any deviation φ1 ∈Φ1, player 1’s expected payoff obtained by sampling x1 using Ψ is always larger than her expected payoff obtained by deviating to φ1(x1), then we call Ψ stable under deviations in Φ1. The distribution Ψ is said to be a Φ-equilibrium if Ψ is stable under deviations in Φ1 and Φ2. A similar definition appears in [12] and [17]. Definition 1 (Φ-equilibrium). A joint distribution Ψ over K1 × K2 is called a Φ-equilibrium if the following holds, for any φ1 ∈Φ1, and for any φ2 ∈Φ2: Z u1(x1, x2)Ψ(x1, x2) ≥ Z u1(φ1(x1), x2)Ψ(x1, x2) Z u2(x1, x2)Ψ(x1, x2) ≥ Z u2(x1, φ2(x2))Ψ(x1, x2) We say that Ψ is a ε-approximate Φ-equilibrium if the inequalities above are satisfied up to an additive error of ε. Intuitively, we imagine a repeated game between the two players, where at equilibrium, the players’ moves are correlated by a signal, which could be the past history of the play, and various external factors. This signal samples a pair of moves from an equilibrium joint distribution over all pairs of moves, and suggests to each player individually only the move she is supposed to play. If no player stands to gain if she unilaterally, but consistently, used a deviation from her suggested move, then the distribution of the correlating signal is stable under the set of deviations, and is hence an equilibrium. Example 1: Correlated Equilibria. A standard 2-player game is obtained when the Ki are the simplices of distributions over some base sets of actions Ai and the utility functions ui are bilinear in x1, x2. If the sets Φi consist of the maps φa,b : Ki →Ki for every pair a, b ∈Ai defined as φa,b(x)[c] = 0 if c = a xa + xb if c = b xc otherwise (1) 1It is highly plausible that the results in this paper extend to the case where Φ is infinite – indeed, our results hold for any set of mappings Φ which is obtained by taking all convex combinations of finitely many mappings – but we restrict to finite Φ in this paper for simplicity. 3 then it can be shown that any Φ-equilibrium can be equivalently viewed as a correlated equilibrium of the game, and vice-versa. Example 2: The Stock Market game. Consider the following setting: there are two investors (the generalization to many investors is straightforward), who invest their wealth in n stocks. In each period, they choose portfolios x1 and x2 over the n stocks, and observe the stock returns. We model the stock returns as a function r of the portfolios x1, x2 chosen by the investors, and it maps the portfolios to the vector of stock returns. We make the assumption that each player has a small influence on the market, and thus the function r is insensitive to the small perturbations in the input. The wealth gain for each investor i is r(x1, x2) · xi. The standard way to measure performance of an investment strategy is the logarithmic growth rate, viz. log(r(x1, x2) · xi). We can now define the utility functions as ui(x1, x2) = log(r(x1, x2) · xi). Intuitively, this game models the setting in which the market prices are affected by the investments of the players. A natural goal for a good investment strategy would be to compare the wealth gain to that of the best fixed portfolio, i.e. Φi is the set of all constant maps. This was considered by Cover in his Universal Portfolio Framework [5]. Another possible goal would be to compare the wealth gained to that achievable by modifying the portfolios using the φa,b maps above, as considered by [16]. In Section 3, we show that the stock market game admits algorithms that converge to an ε-equilibrium in O( 1 ε log 1 ε) rounds, whereas all previous algorithms need O( 1 ε2 ) rounds. 2.2 No regret algorithms The online learning framework we consider is called online convex optimization [18], in which there is a fixed convex compact feasible set K ⊂Rn and an arbitrary, unknown sequence of concave payoff functions f (1), f (2), . . . : K →R. The decision maker must make a sequence of decisions, where the tth decision is a selection of a point x(t) ∈K and obtains a payoff of f (t)(x(t)) on period t. The decision maker can only use the previous points x(1), . . . , x(t−1), and the previous payoff functions f (1), . . . , f (t−1) to choose the point x(t). The performance measure we use to evaluate online algorithms is regret, defined as follows. The decision maker has a finite set of N decision modifiers Φ which, as before, is a set of continuous mappings from K →K. Then the regret for not using some deviation φ ∈Φ is the excess payoff the decision maker could have obtained if she had changed her points in each round by applying φ. Definition 2 (Φ-Regret). Let Φ be a set of continuous functions from K →K. Given a set of T concave utility functions f1, ..., fT , define the Φ-regret as RegretΦ(T) = max φ∈Φ T X t=1 f (t)(φ(x(t))) − T X t=1 f (t)(x(t)). Two specific examples of Φ-regret deserve mention. The first one is “external regret”, which is defined when Φ is the set of all constant mappings from K to itself. The second one is “internal regret”, which is defined when K is the simplex of distributions over some base set of actions A, and Φ is the set of the φa,b functions (defined in (1)) for all pairs a, b ∈A. A desirable property of an algorithm for Online Convex Optimization is Hannan consistency: the regret, as a function of the number of rounds T, is sublinear. This implies that the average per iteration payoff of the algorithm converges to the average payoff of a clairvoyant algorithm that uses the best deviation in hindsight to change the point in every round. For the purpose of this paper, we require a slightly stronger property for an algorithm, viz. that the regret is polynomially sublinear as a function of T. Definition 3 (No Φ-regret algorithm). A no Φ-regret algorithm is one which, given any sequence of concave payoff functions f (1), f (2), . . ., generates a sequence of points x(1), x(2), . . . ∈K such that for all T = 1, 2, . . ., RegretΦ(T) = O(T 1−c) for some constant c > 0. Such an algorithm will be called efficient if it computes x(t) in poly(n, N, t, L) time. In the above definition, L is a description length parameter for K, defined appropriately depending on how the set K is represented. For instance, if K is the n-dimensional probability simplex, then 4 L = n. If K is specified by means of a separation oracle and inner and outer radii r and R, then L = log(R/r), and we allow poly(n, N, t, L) calls to the separation oracle in each iteration. The relatively new framework of Online Convex Optimization (OCO) has received much attention recently in the machine learning community. Our no Φ-regret algorithms can use any of wide variety of algorithms for OCO. In this paper, we will use Exponentiated Gradient (EG) algorithm ([14], [1]), which has the following (external) regret bound: Theorem 1. Let the domain K be the simplex of distributions over a base set of size n. Let G∞be an upper bound on the L∞norm of the gradients of the payoff functions, i.e. G∞≥ supx∈K ∥∇f (t)(x)∥∞. Then the EG algorithm generates points x(1), . . . , x(T ) such that max x∈K T X t=1 f (t)(x) − T X t=1 f (t)(x(t)) ≤O(G∞ p log(n)T) If the utility functions are strictly concave rather than linear, even stronger regret bounds, which depend on log(T) rather than √ T, are known [13]. While most of the literature on online convex optimization focuses on external regret, it was observed that any Online Convex Optimization algorithm for external regret can be converted to an internal regret algorithm (for example, see [2], [16]). 2.3 Fixed Points As mentioned in the introduction, our no regret algorithms depend on computing fixed points of the relevant mappings. For a given set of deviations Φ, denote by CH(Φ) the set of all convex combinations of deviations in Φ, i.e. CH(Φ) = nP φ∈Φαφφ : αφ ≥0 and P φ∈Φαφ = 1 o . Since each map φ ∈CH(Φ) is a continuous function from K →K, and K is a convex compact domain, by Brouwer’s fixed theorem, φ has a fixed point in K, i.e. there exists a point x ∈K such that φ(x) = x. We consider algorithms which approximate fixed points for a given map in the following sense. Definition 4 (FPTAS for fixed points of deviations). Let Φ be a set of N continuous functions from K →K. A fully polynomial time approximation scheme (FPTAS) for fixed points of Φ is an algorithm, which, given any function φ ∈CH(Φ) and an error parameter ε > 0, computes a point x ∈K such that ∥φ(x) −x∥≤ε in poly(n, N, L, 1 ε) time. 3 Convergence of no Φ-regret algorithms to Φ-equilibria In this section we prove that if the players use no Φ-regret algorithms, then the empirical distribution of the moves converges to a Φ-equilibrium. [11] shows that if players use no internal regret algorithms, then the empirical distribution of the moves converges to a correlated equilibrium. This was generalized by [9] to any set of linear transformations Φ. The more general setting of this paper also follows easily from the definitions. A similar theorem was also proved in [17]. The advantage of this general setting is that the connection to online convex optimization allows for faster rates of convergence using recent online learning techniques. We give an example of a natural game theoretic setting with faster convergence rate below. Theorem 2. If each player i chooses moves using a no Φi-regret algorithms, then the empirical game distribution of the players’ moves converges to a Φ-equilibrium. Further, an ε-approximate Φ-equilibrium is reached after T iterations for the first T which satisfies 1 T RegretΦ(T) ≤ε. Proof. Consider the first player. In each game iteration t, let (x1(t), x2(t)) be the pair of moves played by the two players. From player 1’s point of view, the payoff function she obtains, f (t), is the following: ∀x ∈K1 : f (t)(x) ≜u1(x, x2 (t)). 5 Note that this function is concave by assumption. Then we have, by definition 3, RegretΦ1(T) = max φ∈Φ X t f (t)(φ(x1 (t))) − X t f (t)(x1 (t)). Rewriting this in terms of the original utility function, and scaling by the number of iterations we get 1 T T X t=1 u1(x1 (t), x2 (t)) ≥1 T T X t=1 u1(φ(x1 (t)), x2 (t)) −1 T RegretΦ1(T). Denote by Ψ(T ) the empirical distribution of the played strategies till iteration T, i.e. the distribution which puts a probability mass of 1 T on all pairs (x1(t), x2(t)) for t = 1, 2, . . . , T. Then, the above inequality can be rewritten as Z u1(x1, x2)Ψ(T )(x1, x2) ≥ Z u1(φ(x1), x2)Ψ(T )(x1, x2) −1 T RegretΦ1(T). A similar inequality holds for player 2 as well. Now assume that both players use no regret algorithms, which ensure that RegretΦi(T) ≤O(T 1−c) for some constant c > 0. Hence as T →∞, we have 1 T RegretΦi(T) →0. Thus Ψ(T ) converges to a Φ-equilibrium. Also, Ψ(T ) is a ε-approximate equilibrium as soon as T is large enough so that 1 T RegretΦ1(T) and 1 T RegretΦ2(T) are less than ε, i.e. T ≥Ω( 1 ε1/c ). A corollary of Theorem 2 is that we can obtain faster rates of convergence using recent online learning techniques, when the payoff functions are non-linear. This is natural in many situations, since risk aversion is associated with the concavity of utility functions. Corollary 3. For the stock market game as defined in section 2.1, there exists no regret algorithms which guarantee convergence to an ε-equilibrium in O( 1 ε log 1 ε) iterations. Proof sketch. The utility functions observed by the investor i in the stock market game are of the form ui(x1, x2) = log(r(x1, x2) · xi). This logarithmic utility function is exp-concave, by the assumption on the insensitivity of the function r to small perturbations in the input. Thus the online algorithm of [5], or the more efficient algorithms of [13] can be applied. In the full version of this paper, we show that Lemma 6 can be modified to obtain algorithms with RegretΦi(T) = O(log T). By the Theorem 2 above, the investors reach ε-equilibrium in O( 1 ε log 1 ε) iterations. 4 Computational Equivalence of Fixed Points and No Regret algorithms In this section we prove our main result on the computational equivalence of computing fixed points and designing no regret algorithms. By the result of the previous section, players using no regret algorithms converge to equilibria. We assume that the payoff functions f (t) are scaled so that the (L2) norm of their gradients is bounded by 1, i.e. ∥∇f (t)∥≤1. Our main theorem is the following: Theorem 4. Let Φ be a given finite set of deviations. Then there is a FPTAS for fixed points of Φ if and only if there exists an efficient no Φ-regret algorithm. The first direction of the theorem is proved by designing utility functions for which the no regret property will imply convergence to an approximate fixed point of the corresponding transformations. The proof crucially depends on the fact that no regret algorithms have the stringent requirement that their worst case regret, against arbitrary adversarially chosen payoff functions, is sublinear as a function of the number of the rounds. Lemma 5. If there exists a no Φ-regret algorithm then there exists an FPTAS for fixed points of Φ. Proof. Let φ0 ∈CH(Φ) be a given mapping whose fixed point we wish to compute. Let ε be a given error parameter. 6 At iteration t, let x(t) be the point chosen by A. If ∥φ0(x(t)) −x(t)∥≤ε, we can stop, because we have found an approximate fixed point. Else, supply A with the following payoff function: f (t)(x) ≜(φ0(x(t)) −x(t))⊤ ∥φ0(x(t)) −x(t)∥(x −x(t)) This is a linear function, with ∥∇f (t)(x)∥= 1. Also, f (t)(x(t)) = 0, and f (t)(φ0(x(t))) = ∥φ0(x(t)) −x(t)∥≥ε. After T iterations, since φ0 is a convex combination of functions in Φ, and since all the f (t) are linear functions, we have max φ∈Φ T X t=1 f (t)(φ(x(t))) ≥ T X t=1 f (t)(φ0(x(t))) ≥εT. Thus, RegretΦ(T) = max φ∈Φ X t f (t)(φ(x(t))) − X t f (t)(x(t)) ≥εT. (2) Since A is a no-regret algorithm, assume that A ensures that RegretΦ(T) = O(T 1−c) for some constant c > 0. Thus, when T = Ω( 1 ε1/c ) the lower bound (2) on the regret cannot hold unless we have already found an ε-approximate fixed point of φ0. The second direction is on the lines of the algorithms of [2] and [16] which use fixed point computations to obtain no internal regret algorithms. Lemma 6. If there is an FPTAS for fixed points of Φ, then there is an efficient no Φ-regret algorithm. In fact, the algorithm guarantees that RegretΦ(T) = O( √ T). 2 Proof. We reduce the given OCO problem to an “inner” OCO problem. The “outer” OCO problem is the original one. We use a no external regret algorithm for the inner OCO problem to generate points in K for the outer one, and use the payoff functions obtained in the outer OCO problem to generate appropriate payoff functions for the inner one. Let Φ = {φ1, φ2, . . . , φN}. The domain for the inner OCO problem is the simplex of all distributions on Φ, denoted ∆N. For a distribution α ∈∆N, let αi be the probability measure assigned to φi in the distribution α. There is a natural mapping from ∆N →CH(Φ): for any α ∈∆N, denote by φα the function PN i=1 αiφi ∈CH(Φ). Let x(t) ∈K be the point used in the outer OCO problem in the tth round, and let f (t) be the obtained payoff function. Then the payoff functions for the inner OCO problem is the function g(t) : ∆N →R defined as follows: ∀α ∈∆N : g(t)(α) ≜f (t)(φα(x(t))). We now apply the Exponentiated Gradient (EG) algorithm (see Section 2.2) to the inner OCO problem. To analyze the algorithm, we bound ∥∇g(t)∥∞as follows. Let x0 be an arbitrary point in K. We can rewrite g(t) as g(t)(α) = f (t)(x0 + P i αi(φi(x(t)) −x0)), because P i αi = 1. Then, ∇g(t) = X(t)∇f (t)(φα(x(t))), where X(t) is an N × n matrix whose ith row is (φi(x(t)) −x0)⊤. Thus, ∥∇g(t)∥∞= max i |(φi(x(t))−x0)⊤∇f (t)(φα(x(t)))| ≤∥φi(x(t))−x0∥∥∇f (t)(φα(x(t)))∥≤1. The last inequality follows because we assumed that the diameter of K is bounded by 1, and the norm of the gradient of f (t) is also bounded by 1. Let α(t) be the distribution on Φ produced by the EG algorithm at time t. Now, the point x(t) is computed by running the FPTAS for computing an 1 √ t-approximate fixed point of the function φα(t), i.e. we have ∥φα(t)(x(t)) −x(t)∥≤ 1 √ t. 2In the full version of the paper, we improve the regret bound to O(log T) under some stronger concavity assumptions on the payoff functions. 7 Now, using the definition of the g(t) functions, and by the regret bound for the EG algorithm, we have that for any fixed distribution α ∈∆N, T X t=1 f (t)(φα(x(t)))− T X t=1 f (t)(φα(t)(x(t))) = T X t=1 g(t)(α)− T X t=1 g(t)(α(t)) ≤O( p log(N)T). (3) Since ∥∇f (t)∥≤1, f (t)(φα(t)(x(t))) −f (t)(x(t)) ≤∥φα(t)(x(t)) −x(t)∥≤ 1 √ t. (4) Summing (4) from t = 1 to T, and adding to (3), we get that for any distribution α over Φ, T X t=1 f (t)(φα(x(t))) − X t f (t)(x(t)) ≤O( p log(N)T) + T X t=1 1 √ t = O( p log(N)T). In particular, by concentrating α on any given φi, the above inequality implies that PT t=1 f (t)(φi(x(t))) −PT t=1 f (t)(x(t)) ≤O( p log(N)T), and thus we have a no Φ-regret algorithm. References [1] S. Arora, E. Hazan, and S. Kale. The multiplicative weights update method: a meta algorithm and applications. Manuscript, 2005. [2] A. Blum and Y. Mansour. From external to internal regret. In COLT, pages 621–636, 2005. [3] X. Chen and X. Deng. Settling the complexity of two-player nash equilibrium. In 47th FOCS, pages 261–272, 2006. [4] X. Chen, X. Deng, and S-H. Teng. Computing nash equilibria: Approximation and smoothed complexity. focs, 0:603–612, 2006. [5] T. Cover. Universal portfolios. Math. Finance, 1:1–19, 1991. [6] C. Daskalakis, P. W. Goldberg, and C. H. Papadimitriou. The complexity of computing a nash equilibrium. In 38th STOC, pages 71–78, 2006. [7] Y. Freund and R. E. Schapire. Adaptive game playing using multiplicative weights. Games and Economic Behavior, 29:79–103, 1999. [8] G. Gordon, A. Greenwald, C. Marks, and M. Zinkevich. No-regret learning in convex games. Brown University Tech Report CS-07-10, 2007. [9] A. Greenwald and A. Jafari. A general class of no-regret learning algorithms and game-theoretic equilibria, 2003. [10] J. Hannan. Approximation to bayes risk in repeated play. In M. Dresher, A. W. Tucker, and P. Wolfe, editors, Contributions to the Theory of Games, volume III, pages 97–139, 1957. [11] S. Hart and A. Mas-Colell. A simple adaptive procedure leading to correlated equilibrium. Econometrica, 68(5):1127–1150, 2000. [12] S. Hart and D. Schmeidler. Existence of correlated equilibria. Mathematics of Operations Research, 14(1):18–25, 1989. [13] E. Hazan, A. Kalai, S. Kale, and A. Agarwal. Logarithmic regret algorithms for online convex optimization. In 19’th COLT, 2006. [14] J. Kivinen and M. K. Warmuth. Exponentiated gradient versus gradient descent for linear predictors. Inf. 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Random Features for Large-Scale Kernel Machines Ali Rahimi Intel Research Seattle Seattle, WA 98105 ali.rahimi@intel.com Benjamin Recht Caltech IST Pasadena, CA 91125 brecht@ist.caltech.edu Abstract To accelerate the training of kernel machines, we propose to map the input data to a randomized low-dimensional feature space and then apply existing fast linear methods. The features are designed so that the inner products of the transformed data are approximately equal to those in the feature space of a user specified shiftinvariant kernel. We explore two sets of random features, provide convergence bounds on their ability to approximate various radial basis kernels, and show that in large-scale classification and regression tasks linear machine learning algorithms applied to these features outperform state-of-the-art large-scale kernel machines. 1 Introduction Kernel machines such as the Support Vector Machine are attractive because they can approximate any function or decision boundary arbitrarily well with enough training data. Unfortunately, methods that operate on the kernel matrix (Gram matrix) of the data scale poorly with the size of the training dataset. For example, even with the most powerful workstation, it might take days to train a nonlinear SVM on a dataset with half a million training examples. On the other hand, linear machines can be trained very quickly on large datasets when the dimensionality of the data is small [1, 2, 3]. One way to take advantage of these linear training algorithms for training nonlinear machines is to approximately factor the kernel matrix and to treat the columns of the factor matrix as features in a linear machine (see for example [4]). Instead, we propose to factor the kernel function itself. This factorization does not depend on the data, and allows us to convert the training and evaluation of a kernel machine into the corresponding operations of a linear machine by mapping data into a relatively low-dimensional randomized feature space. Our experiments show that these random features, combined with very simple linear learning techniques, compete favorably in speed and accuracy with state-of-the-art kernel-based classification and regression algorithms, including those that factor the kernel matrix. The kernel trick is a simple way to generate features for algorithms that depend only on the inner product between pairs of input points. It relies on the observation that any positive definite function k(x, y) with x, y ∈Rd defines an inner product and a lifting φ so that the inner product between lifted datapoints can be quickly computed as ⟨φ(x), φ(y)⟩= k(x, y). The cost of this convenience is that the algorithm accesses the data only through evaluations of k(x, y), or through the kernel matrix consisting of k applied to all pairs of datapoints. As a result, large training sets incur large computational and storage costs. Instead of relying on the implicit lifting provided by the kernel trick, we propose explicitly mapping the data to a low-dimensional Euclidean inner product space using a randomized feature map z : Rd →RD so that the inner product between a pair of transformed points approximates their kernel evaluation: k(x, y) = ⟨φ(x), φ(y)⟩≈z(x)′z(y). (1) 1 Unlike the kernel’s lifting φ, z is low-dimensional. Thus, we can simply transform the input with z, and then apply fast linear learning methods to approximate the answer of the corresponding nonlinear kernel machine. In what follows, we show how to construct feature spaces that uniformly approximate popular shift-invariant kernels k(x −y) to within ϵ with only D = O(dϵ−2 log 1 ϵ2 ) dimensions, and empirically show that excellent regression and classification performance can be obtained for even smaller D. In addition to giving us access to extremely fast learning algorithms, these randomized feature maps also provide a way to quickly evaluate the machine. With the kernel trick, evaluating the machine at a test point x requires computing f(x) = PN i=1 cik(xi, x), which requires O(Nd) operations to compute and requires retaining much of the dataset unless the machine is very sparse. This is often unacceptable for large datasets. On the other hand, after learning a hyperplane w, a linear machine can be evaluated by simply computing f(x) = w′z(x), which, with the randomized feature maps presented here, requires only O(D + d) operations and storage. We demonstrate two randomized feature maps for approximating shift invariant kernels. Our first randomized map, presented in Section 3, consists of sinusoids randomly drawn from the Fourier transform of the kernel function we seek to approximate. Because this map is smooth, it is wellsuited for interpolation tasks. Our second randomized map, presented in Section 4, partitions the input space using randomly shifted grids at randomly chosen resolutions. This mapping is not smooth, but leverages the proximity between input points, and is well-suited for approximating kernels that depend on the L1 distance between datapoints. Our experiments in Section 5 demonstrate that combining these randomized maps with simple linear learning algorithms competes favorably with state-of-the-art training algorithms in a variety of regression and classification scenarios. 2 Related Work The most popular methods for large-scale kernel machines are decomposition methods for solving Support Vector Machines (SVM). These methods iteratively update a subset of the kernel machine’s coefficients using coordinate ascent until KKT conditions are satisfied to within a tolerance [5, 6]. While such approaches are versatile workhorses, they do not always scale to datasets with more than hundreds of thousands of datapoints for non-linear problems. To extend learning with kernel machines to these scales, several approximation schemes have been proposed for speeding up operations involving the kernel matrix. The evaluation of the kernel function can be sped up using linear random projections [7]. Throwing away individual entries [7] or entire rows [8, 9, 10] of the kernel matrix lowers the storage and computational cost of operating on the kernel matrix. These approximations either preserve the separability of the data [8], or produce good low-rank or sparse approximations of the true kernel matrix [7, 9]. Fast multipole and multigrid methods have also been proposed for this purpose, but, while they appear to be effective on small and low-dimensional problems, they have not been demonstrated on large datasets. Further, the quality of the Hermite or Taylor approximation that these methods rely on degrades exponentially with the dimensionality of the dataset [11]. Fast nearest neighbor lookup with KD-Trees has been used to approximate multiplication with the kernel matrix, and in turn, a variety of other operations [12]. The feature map we present in Section 4 is reminiscent of KD-trees in that it partitions the input space using multi-resolution axis-aligned grids similar to those developed in [13] for embedding linear assignment problems. 3 Random Fourier Features Our first set of random features project data points onto a randomly chosen line, and then pass the resulting scalar through a sinusoid (see Figure 1 and Algorithm 1). The random lines are drawn from a distribution so as to guarantee that the inner product of two transformed points approximates the desired shift-invariant kernel. The following classical theorem from harmonic analysis provides the key insight behind this transformation: Theorem 1 (Bochner [15]). A continuous kernel k(x, y) = k(x −y) on Rd is positive definite if and only if k(δ) is the Fourier transform of a non-negative measure. 2 RD R2 ω x Kernel Name k(∆) p(ω) Gaussian e− ∥∆∥2 2 2 (2π)−D 2 e− ∥ω∥2 2 2 Laplacian e−∥∆∥1 Q d 1 π(1+ω2 d) Cauchy Q d 2 1+∆2 d e−∥∆∥1 Figure 1: Random Fourier Features. Each component of the feature map z(x) projects x onto a random direction ω drawn from the Fourier transform p(ω) of k(∆), and wraps this line onto the unit circle in R2. After transforming two points x and y in this way, their inner product is an unbiased estimator of k(x, y). The table lists some popular shift-invariant kernels and their Fourier transforms. To deal with non-isotropic kernels, the data may be whitened before applying one of these kernels. If the kernel k(δ) is properly scaled, Bochner’s theorem guarantees that its Fourier transform p(ω) is a proper probability distribution. Defining ζω(x) = ejω′x, we have k(x −y) = Z Rd p(ω)ejω′(x−y) dω = Eω[ζω(x)ζω(y)∗], (2) so ζω(x)ζω(y)∗is an unbiased estimate of k(x, y) when ω is drawn from p. To obtain a real-valued random feature for k, note that both the probability distribution p(ω) and the kernel k(∆) are real, so the integrand ejω′(x−y) may be replaced with cos ω′(x −y). Defining zω(x) = [ cos(x) sin(x) ]′ gives a real-valued mapping that satisfies the condition E[zω(x)′zω(y)] = k(x, y), since zω(x)′zω(y) = cos ω′(x −y). Other mappings such as zω(x) = √ 2 cos(ω′x + b), where ω is drawn from p(ω) and b is drawn uniformly from [0, 2π], also satisfy the condition E[zω(x)′zω(y)] = k(x, y). We can lower the variance of zω(x)′zω(y) by concatenating D randomly chosen zω into a column vector z and normalizing each component by √ D. The inner product of points featureized by the 2D-dimensional random feature z, z(x)′z(y) = 1 D PD j=1 zωj(x)zωj(y) is a sample average of zωj(x)zωj(y) and is therefore a lower variance approximation to the expectation (2). Since zω(x)′zω(y) is bounded between -1 and 1, for a fixed pair of points x and y, Hoeffding’s inequality guarantees exponentially fast convergence in D between z(x)′z(y) and k(x, y): Pr [|z(x)′z(y) −k(x, y)| ≥ϵ] ≤2 exp(−Dϵ2/2). Building on this observation, a much stronger assertion can be proven for every pair of points in the input space simultaneously: Claim 1 (Uniform convergence of Fourier features). Let M be a compact subset of Rd with diameter diam(M). Then, for the mapping z defined in Algorithm 1, we have Pr sup x,y∈M |z(x)′z(y) −k(x, y)| ≥ϵ ≤28 σp diam(M) ϵ 2 exp − Dϵ2 4(d + 2) , where σ2 p ≡ Ep[ω′ω] is the second moment of the Fourier transform of k. Further, supx,y∈M |z(x)′z(y) −k(y, x)| ≤ ϵ with any constant probability when D = Ω d ϵ2 log σp diam(M) ϵ . The proof of this assertion first guarantees that z(x)′z(y) is close to k(x −y) for the centers of an ϵ-net over M × M. This result is then extended to the entire space using the fact that the feature map is smooth with high probability. See the Appendix for details. By a standard Fourier identity, the scalar σ2 p is equal to the trace of the Hessian of k at 0. It quantifies the curvature of the kernel at the origin. For the spherical Gaussian kernel, k(x, y) = exp −γ∥x −y∥2 , we have σ2 p = 2dγ. 3 Algorithm 1 Random Fourier Features. Require: A positive definite shift-invariant kernel k(x, y) = k(x −y). Ensure: A randomized feature map z(x) : Rd →R2D so that z(x)′z(y) ≈k(x −y). Compute the Fourier transform p of the kernel k: p(ω) = 1 2π R e−jω′∆k(∆) d∆. Draw D iid samples ω1, · · · , ωD ∈Rd from p. Let z(x) ≡ q 1 D [ cos(ω′ 1x) ··· cos(ω′ Dx) sin(ω′ 1x) ··· sin(ω′ Dx) ]′. 4 Random Binning Features Our second random map partitions the input space using randomly shifted grids at randomly chosen resolutions and assigns to an input point a binary bit string that corresponds to the bin in which it falls (see Figure 2 and Algorithm 2). The grids are constructed so that the probability that two points x and y are assigned to the same bin is proportional to k(x, y). The inner product between a pair of transformed points is proportional to the number of times the two points are binned together, and is therefore an unbiased estimate of k(x, y). 10000000 01000000 00100000 00010000 00001000 00000100 00000010 00000001 ≈ + + + · · · = k(xi, xj) z1(xi)′z1(xj) z2(xi)′z2(xj) z3(xi)′z3(xj) z(xi)′z(xj) Figure 2: Random Binning Features. (left) The algorithm repeatedly partitions the input space using a randomly shifted grid at a randomly chosen resolution and assigns to each point x the bit string z(x) associated with the bin to which it is assigned. (right) The binary adjacency matrix that describes this partitioning has z(xi)′z(xj) in its ijth entry and is an unbiased estimate of kernel matrix. We first describe a randomized mapping to approximate the “hat” kernel khat(x, y; δ) = max 0, 1 −|x−y| δ on a compact subset of R × R, then show how to construct mappings for more general separable multi-dimensional kernels. Partition the real number line with a grid of pitch δ, and shift this grid randomly by an amount u drawn uniformly at random from [0, δ]. This grid partitions the real number line into intervals [u + nδ, u + (n + 1)δ] for all integers n. The probability that two points x and y fall in the same bin in this grid is max 0, 1 −|x−y| δ [13]. In other words, if we number the bins of the grid so that a point x falls in bin ˆx = ⌊x−u δ ⌋and y falls in bin ˆy = ⌊y−u δ ⌋, then Pru[ˆx = ˆy|δ] = khat(x, y; δ). If we encode ˆx as a binary indicator vector z(x) over the bins, z(x)′z(y) = 1 if x and y fall in the same bin and zero otherwise, so Pru[z(x)′z(y) = 1|δ] = Eu[z(x)′z(y)|δ] = khat(x, y; δ). Therefore z is a random map for khat. Now consider shift-invariant kernels that can be written as convex combinations of hat kernels on a compact subset of R × R: k(x, y) = R ∞ 0 khat(x, y; δ)p(δ) dδ. If the pitch δ of the grid is sampled from p, z again gives a random map for k because Eδ,u[z(x)′z(y)] = Eδ [Eu[z(x)′z(y)|δ]] = Eδ[khat(x, y; δ)] = k(x, y). That is, if the pitch δ of the grid is sampled from p, and the shift u is drawn uniformly from [0, δ] the probability that x and y are binned together is k(x, y). Lemma 1 in the appendix shows that p can be easily recovered from k by setting p(δ) = δ¨k(δ). For example, in the case of the Laplacian kernel, kLaplacian(x, y) = exp(−|x −y|), p(δ) is the Gamma distribution δ exp(−δ). For the Gaussian kernel, ¨k is not everywhere positive, so this procedure does not yield a random map. Random maps for separable multivariate shift-invariant kernels of the form k(x −y) = Qd m=1 km(|xm−ym|) (such as the multivariate Laplacian kernel) can be constructed in a similar way if each km can be written as a convex combination of hat kernels. We apply the above binning process over each dimension of Rd independently. The probability that xm and ym are binned together in dimension m is km(|xm −ym|). Since the binning process is independent across dimensions, the 4 probability that x and y are binned together in every dimension is Qd m=1 km(|xm−ym|) = k(x−y). In this multivariate case, z(x) encodes the integer vector [ ˆx1,··· ,ˆxd ] corresponding to each bin of the d-dimensional grid as a binary indicator vector. In practice, to prevent overflows when computing z(x) when d is large, our implementation eliminates unoccupied bins from the representation. Since there are never more bins than training points, this ensures no overflow is possible. We can again reduce the variance of the estimator z(x)′z(y) by concatenating P random binning functions z into a larger list of features z and scaling by p 1/P. The inner product z(x)′z(y) = 1 P PP p=1 zp(x)′zp(y) is the average of P independent z(x)′z(y) and has therefore lower variance. Since z(x)′z(y) is binary, Hoeffding’s inequality guarantees that for a fixed pair of points x and y, z(x)′z(y) converges exponentially quickly to k(x, y) as a function of P. Again, a much stronger claim is that this convergence holds simultaneously for all points: Claim 2. Let M be a compact subset of Rd with diameter diam(M). Let α = E[1/δ] and let Lk denote the Lipschitz constant of k with respect to the L1 norm. With z as above, we have Pr sup x,y∈M |z(x)′z(y) −k(x, y)| ≤ϵ ≥1 −36dPα diam(M) exp − P ϵ2 8 + ln ϵ Lk d + 1 , Note that α = R ∞ 0 1 δ p(δ) dδ = R ∞ 0 ¨k(δ) dδ is 1, and Lk = 1 for the Laplacian kernel. The proof of the claim (see the appendix) partitions M × M into a few small rectangular cells over which k(x, y) does not change much and z(x) and z(y) are constant. With high probability, at the centers of these cells z(x)′z(y) is close to k(x, y), which guarantees that k(x, y) and z(x)′z(y) are close throughout M × M. Algorithm 2 Random Binning Features. Require: A point x ∈Rd. A kernel function k(x, y) = Qd m=1 km(|xm −ym|), so that pm(∆) ≡ ∆¨km(∆) is a probability distribution on ∆≥0. Ensure: A randomized feature map z(x) so that z(x)′z(y) ≈k(x −y). for p = 1 . . . P do Draw grid parameters δ, u ∈Rd with the pitch δm ∼pm, and shift um from the uniform distribution on [0, δm]. Let z return the coordinate of the bin containing x as a binary indicator vector zp(x) ≡ hash(⌈x1−u1 δ1 ⌉, · · · , ⌈xd−ud δd ⌉). end for z(x) ≡ q 1 P [ z1(x)···zP (x) ]′. 5 Experiments The experiments summarized in Table 1 show that ridge regression with our random features is a fast way to approximate the training of supervised kernel machines. We focus our comparisons against the Core Vector Machine [14] because it was shown in [14] to be both faster and more accurate than other known approaches for training kernel machines, including, in most cases, random sampling of datapoints [8]. The experiments were conducted on the five standard large-scale datasets evaluated in [14], excluding the synthetic datasets. We replicated the results in the literature pertaining to the CVM, SVMlight, and libSVM using binaries provided by the respective authors.1 For the random feature experiments, we trained regressors and classifiers by solving the ridge regression problem 1We include KDDCUP99 results for completeness, but note this dataset is inherently oversampled: training an SVM (or least squares with random features) on a random sampling of 50 training examples (0.001% of the training dataset) is sufficient to consistently yield a test-error on the order of 8%. Also, while we were able to replicate the CVM’s 6.2% error rate with the parameters supplied by the authors, retraining after randomly shuffling the training set results in 18% error and increases the computation time by an order of magnitude. Even on the original ordering, perturbing the CVM’s regularization parameter by a mere 15% yields 49% error rate on the test set [16]. 5 Dataset Fourier+LS Binning+LS CVM Exact SVM CPU 3.6% 5.3% 5.5% 11% regression 20 secs 3 mins 51 secs 31 secs 6500 instances 21 dims D = 300 P = 350 ASVM Census 5% 7.5% 8.8% 9% regression 36 secs 19 mins 7.5 mins 13 mins 18,000 instances 119 dims D = 500 P = 30 SVMTorch Adult 14.9% 15.3% 14.8% 15.1% classification 9 secs 1.5 mins 73 mins 7 mins 32,000 instances 123 dims D = 500 P = 30 SVMlight Forest Cover 11.6% 2.2% 2.3% 2.2% classification 71 mins 25 mins 7.5 hrs 44 hrs 522,000 instances 54 dims D = 5000 P = 50 libSVM KDDCUP99 (see footnote) 7.3% 7.3% 6.2% (18%) 8.3% classification 1.5 min 35 mins 1.4 secs (20 secs) < 1 s 4,900,000 instances 127 dims D = 50 P = 10 SVM+sampling Table 1: Comparison of testing error and training time between ridge regression with random features, Core Vector Machine, and various state-of-the-art exact methods reported in the literature. For classification tasks, the percent of testing points incorrectly predicted is reported, and for regression tasks, the RMS error normalized by the norm of the ground truth. 10 0 10 2 10 4 10 6 0.1 0.2 0.3 0.4 0.5 Training set size Testing error 10 20 30 40 50 2 3 4 5 6 P % error 10 20 30 40 50 400 800 1200 P training+testing time (sec) Figure 3: Accuracy on test data continues to improve as the training set grows. On the Forest dataset, using random binning, doubling the dataset size reduces testing error by up to 40% (left). Error decays quickly as P grows (middle). Training time grows slowly as P grows (right). minw ∥Z′w −y∥2 2 + λ∥w∥2 2, where y denotes the vector of desired outputs and Z denotes the matrix of random features. To evaluate the resulting machine on a datapoint x, we can simply compute w′z(x). Despite its simplicity, ridge regression with random features is faster than, and provides competitive accuracy with, alternative methods. It also produces very compact functions because only w and a set of O(D) random vectors or a hash-table of partitions need to be retained. Random Fourier features perform better on the tasks that largely rely on interpolation. On the other hand, random binning features perform better on memorization tasks (those for which the standard SVM requires many support vectors), because they explicitly preserve locality in the input space. This difference is most dramatic in the Forest dataset. Figure 3(left) illustrates the benefit of training classifiers on larger datasets, where accuracy continues to improve as more data are used in training. Figure 3(middle) and (right) show that good performance can be obtained even from a modest number of features. 6 Conclusion We have presented randomized features whose inner products uniformly approximate many popular kernels. We showed empirically that providing these features as input to a standard linear learning algorithm produces results that are competitive with state-of-the-art large-scale kernel machines in accuracy, training time, and evaluation time. It is worth noting that hybrids of Fourier features and Binning features can be constructed by concatenating these features. While we have focused on regression and classification, our features can be applied to accelerate other kernel methods, including semi-supervised and unsupervised learning algorithms. In all of these cases, a significant computational speed-up can be achieved by first computing random features and then applying the associated linear technique. 6 7 Acknowledgements We thank Eric Garcia for help on early versions of these features, Sameer Agarwal and James R. Lee for helpful discussions, and Erik Learned-Miller and Andres Corrada-Emmanuel for helpful corrections. References [1] T. Joachims. Training linear SVMs in linear time. In ACM Conference on Knowledge Discovery and Data Mining (KDD), 2006. [2] M. C. Ferris and T. S. Munson. Interior-point methods for massive Support Vector Machines. SIAM Journal of Optimization, 13(3):783–804, 2003. [3] S. Shalev-Shwartz, Y. Singer, and N. Srebro. Pegasos: Primal Estimated sub-GrAdient SOlver for SVM. In IEEE International Conference on Machine Learning (ICML), 2007. [4] D. DeCoste and D. Mazzoni. Fast query-optimized kernel machine classification via incremental approximate nearest support vectors. In IEEE International Conference on Machine Learning (ICML), 2003. [5] J. Platt. Using sparseness and analytic QP to speed training of Support Vector Machines. In Advances in Neural Information Processing Systems (NIPS), 1999. [6] C.-C. Chang and C.-J. Lin. LIBSVM: a library for support vector machines, 2001. Software available at http://www.csie.ntu.edu.tw/∼cjlin/libsvm. [7] D. Achlioptas, F. McSherry, and B. Sch¨olkopf. Sampling techniques for kernel methods. In Advances in Neural Information Processing Systems (NIPS), 2001. [8] A. Blum. Random projection, margins, kernels, and feature-selection. LNCS, 3940:52–68, 2006. [9] A. Frieze, R. Kannan, and S. Vempala. Fast monte-carlo algorithms for finding low-rank approximations. In Foundations of Computer Science (FOCS), pages 378–390, 1998. [10] P. Drineas and M. W. Mahoney. On the nystrom method for approximating a Gram matrix for improved kernel-based learning. In COLT, pages 323–337, 2005. [11] C. Yang, R. Duraiswami, and L. Davis. Efficient kernel machines using the improved fast gauss transform. In Advances in Neural Information Processing Systems (NIPS), 2004. [12] Y. Shen, A. Y. Ng, and M. Seeger. Fast gaussian process regression using KD-Trees. In Advances in Neural Information Processing Systems (NIPS), 2005. [13] P. Indyk and N. Thaper. Fast image retrieval via embeddings. In International Workshop on Statistical and Computational Theories of Vision, 2003. [14] I. W. Tsang, J. T. Kwok, and P.-M. Cheung. Core Vector Machines: Fast SVM training on very large data sets. Journal of Machine Learning Research (JMLR), 6:363–392, 2005. [15] W. Rudin. Fourier Analysis on Groups. Wiley Classics Library. Wiley-Interscience, New York, reprint edition edition, 1994. [16] G. Loosli and S. Canu. Comments on the ‘Core Vector Machines: Fast SVM training on very large data sets’. Journal of Machine Learning Research (JMLR), 8:291–301, February 2007. [17] F. Cucker and S. Smale. On the mathematical foundations of learning. Bull. Amer. Soc., 39:1–49, 2001. A Proofs Lemma 1. Suppose a function k(∆) : R →R is twice differentiable and has the form R ∞ 0 p(δ) max(0, 1 −∆ δ ) dδ. Then p(δ) = δ¨k(δ). Proof. We want p so that k(∆) = Z ∞ 0 p(δ) max(0, 1 −∆/δ) dδ (3) = Z ∆ 0 p(δ) · 0 dδ + Z ∞ ∆ p(δ)(1 −∆/δ) dδ = Z ∞ ∆ p(δ) dδ −∆ Z ∞ ∆ p(δ)/δ dδ. (4) To solve for p, differentiate twice w.r.t. to ∆to find that ˙k(∆) = − R ∞ ∆p(δ)/δ dδ and ¨k(∆) = p(∆)/∆. 7 Proof of Claim 1. Define s(x, y) ≡z(x)′z(y), and f(x, y) ≡s(x, y) −k(y, x). Since f, and s are shift invariant, as their arguments we use ∆≡x −y ∈M∆for notational simplicity. M∆is compact and has diameter at most twice diam(M), so we can find an ϵ-net that covers M∆ using at most T = (4 diam M/r)d balls of radius r [17]. Let {∆i}T i=1 denote the centers of these balls, and let Lf denote the Lipschitz constant of f. We have |f(∆)| < ϵ for all ∆∈M∆if |f(∆i)| < ϵ/2 and Lf < ϵ 2r for all i. We bound the probability of these two events. Since f is differentiable, Lf = ∥∇f(∆∗)∥, where ∆∗= arg max∆∈M∆∥∇f(∆)∥. We have E[L2 f] = E∥∇f(∆∗)∥2 = E∥∇s(∆∗)∥2 −E∥∇k(∆∗)∥2 ≤E∥∇s(∆∗)∥2 ≤Ep∥ω∥2 = σ2 p, so by Markov’s inequality, Pr[L2 f ≥t] ≤E[L2 f]/t, or Pr h Lf ≥ϵ 2r i ≤ 2rσp ϵ 2 . (5) The union bound followed by Hoeffding’s inequality applied to the anchors in the ϵ-net gives Pr ∪T i=1 |f(∆i)| ≥ϵ/8 ≤2T exp −Dϵ2/2 . (6) Combining (5) and (6) gives a bound in terms of the free variable r: Pr sup ∆∈M∆ |f(∆)| ≤ϵ ≥1 −2 4 diam(M) r d exp −Dϵ2/8 − 2rσp ϵ 2 . (7) This has the form 1 −κ1r−d −k2r2. Setting r = κ1 κ2 1 d+2 turns this to 1 −2κ d d+2 2 κ 2 d+2 1 , and assuming that σp diam(M) ϵ ≥1 and diam(M) ≥1, proves the first part of the claim. To prove the second part of the claim, pick any probability for the RHS and solve for D. Proof of Claim 2. M can be covered by rectangles over each of which z is constant. Let δpm be the pitch of the pth grid along the mth dimension. Each grid has at most ⌈diam(M)/δpm⌉bins, and P overlapping grids produce at most Nm = PP g=1⌈diam(M)/δgm⌉≤ P + diam(M) PP p=1 1 δpm partitions along the mth dimension. The expected value of the right hand side is P +P diam(M)α. By Markov’s inequality and the union bound, Pr ∀d m=1Nm ≤t(P + P diam(M)α) ≥1 −d/t. That is, with probability 1 −d/t, along every dimension, we have at most t(P + P diam(M)α) one-dimensional cells. Denote by dmi the width of the ith cell along the mth dimension and observe that PNm i=1 dmi ≤diam(M). We further subdivide these cells into smaller rectangles of some small width r to ensure that the kernel k varies very little over each of these cells. This results in at most PNm i=1⌈dmi r ⌉≤ Nm+diam(M) r small one-dimensional cells over each dimension. Plugging in the upper bound for Nm, setting t ≥ 1 αP and assuming α diam(M) ≥1, with probability 1 −d/t, M can be covered with T ≤ 3tP α diam(M) r d rectangles of side r centered at {xi}T i=1. The condition |z(x, y) −k(x, y)| ≤ϵ on M × M holds if |z(xi, yi) −k(xi, yi)| ≤ϵ −Lkrd and z(x) is constant throughout each rectangle. With rd = ϵ 2Lk , the union bound followed by Hoeffding’s inequality gives Pr [∪ij|z(xi, yj) −k(xi, yj)| ≥ϵ/2] ≤2T 2 exp −Pϵ2/8 (8) Combining this with the probability that z(x) is constant in each cell gives a bound in terms of t: Pr sup x,y∈M×M |z(x, y) −k(x, y)| ≤ϵ ≥1 −d t −2(3tPα diam(M))d 2Lk ϵ exp −Pϵ2 8 . This has the form 1 −κ1t−1 −κ2td. To prove the claim, set t = κ1 2κ2 1 d+1 , which results in an upper bound of 1 −3κ1κ 1 d+1 2 . 8
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A Constraint Generation Approach to Learning Stable Linear Dynamical Systems Sajid M. Siddiqi Robotics Institute Carnegie-Mellon University Pittsburgh, PA 15213 siddiqi@cs.cmu.edu Byron Boots Computer Science Department Carnegie-Mellon University Pittsburgh, PA 15213 beb@cs.cmu.edu Geoffrey J. Gordon Machine Learning Department Carnegie-Mellon University Pittsburgh, PA 15213 ggordon@cs.cmu.edu Abstract Stability is a desirable characteristic for linear dynamical systems, but it is often ignored by algorithms that learn these systems from data. We propose a novel method for learning stable linear dynamical systems: we formulate an approximation of the problem as a convex program, start with a solution to a relaxed version of the program, and incrementally add constraints to improve stability. Rather than continuing to generate constraints until we reach a feasible solution, we test stability at each step; because the convex program is only an approximation of the desired problem, this early stopping rule can yield a higher-quality solution. We apply our algorithm to the task of learning dynamic textures from image sequences as well as to modeling biosurveillance drug-sales data. The constraint generation approach leads to noticeable improvement in the quality of simulated sequences. We compare our method to those of Lacy and Bernstein [1, 2], with positive results in terms of accuracy, quality of simulated sequences, and efficiency. 1 Introduction Many problems in machine learning involve sequences of real-valued multivariate observations. To model the statistical properties of such data, it is often sensible to assume each observation to be correlated to the value of an underlying latent variable, or state, that is evolving over the course of the sequence. In the case where the state is real-valued and the noise terms are assumed to be Gaussian, the resulting model is called a linear dynamical system (LDS), also known as a Kalman Filter [3]. LDSs are an important tool for modeling time series in engineering, controls and economics as well as the physical and social sciences. Let {λi(M)}n i=1 denote the eigenvalues of an n × n matrix M in decreasing order of magnitude, {νi(M)}n i=1 the corresponding unit-length eigenvectors, and define its spectral radius ρ(M) ≡|λ1(M)|. An LDS with dynamics matrix A is stable if all of A’s eigenvalues have magnitude at most 1, i.e., ρ(A) ≤1. Standard algorithms for learning LDS parameters do not enforce this stability criterion, learning locally optimal values for LDS parameters by gradient descent [4], Expectation Maximization (EM) [5] or least squares on a state sequence estimate obtained by subspace identification methods, as described in Section 3.1. However, when learning from finite data samples, the least squares solution may be unstable even if the system is stable [6]. The drawback of ignoring stability is most apparent when simulating long sequences from the system in order to generate representative data or infer stretches of missing values. We propose a convex optimization algorithm for learning the dynamics matrix while guaranteeing stability. An estimate of the underlying state sequence is first obtained using subspace identification. We then formulate the least-squares problem for the dynamics matrix as a quadratic program (QP) [7], initially without constraints. When this QP is solved, the estimate ˆA obtained may be unstable. However, any unstable solution allows us to derive a linear constraint which we then add to our original QP and re-solve. The above two steps are iterated until we reach a stable solution, which is then refined by a simple interpolation to obtain the best possible stable estimate. Our method can be viewed as constraint generation for an underlying convex program with a feasible set of all matrices with singular values at most 1, similar to work in control systems [1]. However, we terminate before reaching feasibility in the convex program, by checking for matrix stability after each new constraint. This makes our algorithm less conservative than previous methods for enforcing stability since it chooses the best of a larger set of stable dynamics matrices. The difference in the resulting stable systems is noticeable when simulating data. The constraint generation approach also achieves much greater efficiency than previous methods in our experiments. One application of LDSs in computer vision is learning dynamic textures from video data [8]. An advantage of learning dynamic textures is the ability to play back a realistic-looking generated sequence of any desired duration. In practice, however, videos synthesized from dynamic texture models can quickly degenerate because of instability in the underlying LDS. In contrast, sequences generated from dynamic textures learned by our method remain “sane” even after arbitrarily long durations. We also apply our algorithm to learning baseline dynamic models of over-the-counter (OTC) drug sales for biosurveillance, and sunspot numbers from the UCR archive [9]. Comparison to the best alternative methods [1, 2] on these problems yields positive results. 2 Related Work Linear system identification is a well-studied subject [4]. Within this area, subspace identification methods [10] have been very successful. These techniques first estimate the model dimensionality and the underlying state sequence, and then derive parameter estimates using least squares. Within subspace methods, techniques have been developed to enforce stability by augmenting the extended observability matrix with zeros [6] or adding a regularization term to the least squares objective [11]. All previous methods were outperformed by Lacy and Bernstein [1], henceforth referred to as LB-1. They formulate the problem as a semidefinite program (SDP) whose objective minimizes the state sequence reconstruction error, and whose constraint bounds the largest singular value by 1. This convex constraint is obtained by rewriting the nonlinear matrix inequality In −AAT ⪰0 as a linear matrix inequality [12], where In is the n × n identity matrix. Here, ≻0 (⪰0) denotes positive (semi-) definiteness. The existence of this constraint also proves the convexity of the σ1 ≤1 region. A follow-up to this work by the same authors [2], which we will call LB-2, attempts to overcome the conservativeness of LB-1 by approximating the Lyapunov inequalities P −APAT ≻0, P ≻0 with the inequalities P −APAT −δIn ⪰0, P −δIn ⪰0, δ > 0. These inequalities hold iff the spectral radius is less than 1. However, the approximation is achieved only at the cost of inducing a nonlinear distortion of the objective function by a problem-dependent reweighting matrix involving P, which is a variable to be optimized. In our experiments, this causes LB-2 to perform worse than LB-1 (for any δ) in terms of the state sequence reconstruction error, even while obtaining solutions outside the feasible region of LB-1. Consequently, we focus on LB-1 in our conceptual and qualitative comparisons as it is the strongest baseline available. However, LB-2 is more scalable than LB-1, so quantitative results are presented for both. To summarize the distinction between constraint generation, LB-1 and LB-2: it is hard to have both the right objective function (reconstruction error) and the right feasible region (the set of stable matrices). LB-1 optimizes the right objective but over the wrong feasible region (the set of matrices with σ1 ≤1). LB-2 has a feasible region close to the right one, but at the cost of distorting its objective function to an extent that it fares worse than LB-1 in nearly all cases. In contrast, our method optimizes the right objective over a less conservative feasible region than that of any previous algorithm with the right objective, and this combination is shown to work the best in practice. 3 Linear Dynamical Systems The evolution of a linear dynamical system can be described by the following two equations: xt+1 = Axt + wt yt = Cxt + vt (1) Time is indexed by the discrete variable t. Here xt denotes the hidden states in Rn, yt the observations in Rm, and wt and vt are zero-mean normally distributed state and observation noise variables. 100 0 200 Sunspot numbers A. B. C. 0 300 Figure 1: A. Sunspot data, sampled monthly for 200 years. Each curve is a month, the x-axis is over years. B. First two principal components of a 1-observation Hankel matrix. C. First two principal components of a 12-observation Hankel matrix, which better reflect temporal patterns in the data. Assume some initial state x0. The parameters of the system are the dynamics matrix A ∈Rn×n, the observation model C ∈Rm×n, and the noise covariance matrices Q and R. Note that we are learning uncontrolled linear dynamical systems, though, as in previous work, control inputs can easily be incorporated into the objective function and convex program. Linear dynamical systems can also be viewed as probabilistic graphical models. The standard LDS filtering and smoothing inference algorithms [3, 13] are instantiations of the junction tree algorithm for Bayesian Networks (see, for example, [14]). We follow the subspace identification literature in estimating all parameters other than the dynamics matrix. A clear and concise exposition of the required techniques is presented in Soatto et al. [8], which we summarize below. We use subspace identification methods in our experiments for uniformity with previous work we are building on (in the control systems literature) and with work we are comparing to ([8] on the dynamic textures data). 3.1 Learning Model Parameters by Subspace Methods Subspace methods calculate LDS parameters by first decomposing a matrix of observations to yield an estimate of the underlying state sequence. The most straightforward such technique is used here, which relies on the singular value decomposition (SVD) [15]. See [10] for variations. Let Y1:τ = [y1 y2 . . . yτ] ∈Rm×τ and X1:τ = [x1 x2 . . . xτ] ∈Rn×τ. D denotes the matrix of observations which is the input to SVD. One typical choice for D is D = Y1:τ; we will discuss others below. SVD yields D ≈UΣV T where U ∈Rm×n and V ∈Rτ×n have orthonormal columns {ui} and {vi}, and Σ = diag{σ1, . . . , σn} contains the singular values. The model dimension n is determined by keeping all singular values of D above a threshold. We obtain estimates of C and X: ˆC = U ˆX = ΣV T (2) See [8] for an explanation of why these estimates satisfy certain canonical model assumptions. ˆX is referred to as the extended observability matrix in the control systems literature; the tth column of ˆX represents an estimate of the state of our LDS at time t. The least squares estimate of A is: ˆA = arg min A J2(A) = arg min A °°AX0:τ−1 −X1:τ °°2 F = X1:τX† 0:τ−1 (3) where ∥· ∥F denotes the Frobenius norm and † denotes the Moore-Penrose inverse. Eq. (3) asks ˆA to minimize the error in predicting the state at time t + 1 from the state at time t. Given the above estimates ˆA and ˆC, the covariance matrices ˆQ and ˆR can be estimated directly from residuals. 3.2 Designing the Observation Matrix In the decomposition above, we chose each column of D to be the observation vector for a single time step. Suppose that instead we set D to be a matrix of the form D = y1 y2 y3 · · · yτ ... ... ... ... ... yd yd+1 yd+2 · · · yd+τ−1 md×τ A matrix of this form, with each block of rows equal to the previous block but shifted by a constant number of columns, is called a block Hankel matrix [4]. We say “d-observation Hankel matrix of size τ” to mean the data matrix D ∈Rmd×τ with d length-m observation vectors per column. Stacking observations causes each state to incorporate more information about the future, since ˆxt Afinal LB-1 A generated constraint A S A ^ S unstable matrices stable matrices R n 2 * * λ σ α −10 0 10 10 0 10 unstable matrices (stable matrices) S S λ σ β A. B. Figure 2: (A): Conceptual depiction of the space of n × n matrices. The region of stability (Sλ) is non-convex while the smaller region of matrices with σ1 ≤1 (Sσ) is convex. The elliptical contours indicate level sets of the quadratic objective function of the QP. ˆA is the unconstrained least-squares solution to this objective. ALB-1 is the solution found by LB-1 [1]. One iteration of constraint generation yields the constraint indicated by the line labeled ‘generated constraint’, and (in this case) leads to a stable solution A∗. The final step of our algorithm improves on this solution by interpolating A∗with the previous solution (in this case, ˆA) to obtain A∗ final. (B): The actual stable and unstable regions for the space of 2×2 matrices Eα,β = [ 0.3 α ; β 0.3 ], with α, β ∈[−10, 10]. Constraint generation is able to learn a nearly optimal model from a noisy state sequence of length 7 simulated from E0,10, with better state reconstruction error than either LB-1 or LB-2. now represents coefficients reconstructing yt as well as other observations in the future. However the observation model estimate must now be ˆC = U( : , 1:m), i.e., the submatrix consisting of the first m columns of U, because U( : , 1:m)ˆxt = ˆyt for any t, where ˆyt denotes a reconstructed observation. Having multiple observations per column in D is particularly helpful when the underlying dynamical system is known to have periodicity. For example, see Figure 1(A). See [12] for details. 4 The Algorithm The estimation procedure in Section 3.1 does not enforce stability in ˆA. To account for stability, we first formulate the dynamics matrix learning problem as a quadratic program with a feasible set that includes the set of stable dynamics matrices. Then we demonstrate how instability in its solutions can be used to generate constraints that restrict this feasible set appropriately. As a final step, the solution is refined to be as close as possible to the least-squares estimate while remaining stable. The overall algorithm is illustrated in Figure 2(A). We now explain the algorithm in more detail. 4.1 Formulating the Objective The least squares problem in Eq. (3) can be written as follows (see [12] for the derivation): ˆA = arg minA °°AX0:τ−1 −X1:τ °°2 F = arg mina © aTPa −2 qTa + r ª (4) where a ∈Rn2×1, q ∈Rn2×1, P ∈Rn2×n2 and r ∈R are defined as: a = vec(A) = [A11 A21 A31 · · · Ann]T P = In ⊗ ¡ X0:τ−1XT 0:τ−1 ¢ q = vec(X0:τ−1XT 1:τ) r = tr ¡ XT 1:τX1:τ ¢ (5) In is the n × n identity matrix and ⊗denotes the Kronecker product. Note that P is a symmetric nonnegative-definite matrix. The objective function in (4) is a quadratic function of a. 4.2 Generating Constraints The quadratic objective function above is equivalent to the least squares problem of Eq. (3). Its feasible set is the space of all n × n matrices, regardless of their stability. When its solution yields an unstable matrix, the spectral radius of ˆA (i.e. |λ1( ˆA)|) is greater than 1. Ideally we would like to use ˆA to calculate a convex constraint on the spectral radius. However, consider the class of 2 × 2 matrices [16]: Eα,β = [ 0.3 α ; β 0.3 ]. The matrices E10,0 and E0,10 are stable with λ1 = 0.3, but their convex combination γE10,0 + (1 −γ)E0,10 is unstable for (e.g.) γ = 0.5 (Figure 2(B)). This shows that the set of stable matrices is non-convex for n = 2, and in fact this is true for all n > 1. We turn instead to the largest singular value, which is a closely related quantity since σmin( ˆA) ≤|λi( ˆA)| ≤σmax( ˆA) ∀i = 1, . . . , n [15] Therefore every unstable matrix has a singular value greater than one, but the converse is not necessarily true. Moreover, the set of matrices with σ1 ≤1 is convex. Figure 2(A) conceptually depicts the non-convex region of stability Sλ and the convex region Sσ with σ1 ≤1 in the space of all n × n matrices for some fixed n. The difference between Sσ and Sλ can be significant. Figure 2(B) depicts these regions for Eα,β with α, β ∈[−10, 10]. The stable matrices E10,0 and E0,10 reside at the edges of the figure. While results for this class of matrices vary, the constraint generation algorithm described below is able to learn a nearly optimal model from a noisy state sequence of τ = 7 simulated from E0,10, with better state reconstruction error than LB-1 and LB-2. Let ˆA = ˜U ˜Σ ˜V T by SVD, where ˜U = [˜ui]n i=1 and ˜V = [˜vi]n i=1 and ˜Σ = diag{˜σ1, . . . , ˜σn}. Then: ˆA = ˜U ˜Σ ˜V T ⇒ ˜Σ = ˜U T ˆA ˜V ⇒ ˜σ1( ˆA) = ˜uT 1 ˆA˜v1 = tr(˜uT 1 ˆA˜v1) (6) Therefore, instability of ˆA implies that: ˜σ1 > 1 ⇒ tr ³ ˜uT 1 ˆA˜v1 ´ > 1 ⇒ tr ³ ˜v1˜uT 1 ˆA ´ > 1 ⇒ gT ˆa > 1 (7) Here g = vec(˜u1˜vT 1 ). Since Eq. (7) arose from an unstable solution of Eq. (4), g is a hyperplane separating ˆa from the space of matrices with σ1 ≤1. We use the negation of Eq. (7) as a constraint: gT ˆa ≤1 (8) 4.3 Computing the Solution The overall quadratic program can be stated as: minimize aTPa −2 qTa + r subject to Ga ≤h (9) with a, P, q and r as defined in Eqs. (5). {G, h} define the set of constraints, and are initially empty. The QP is invoked repeatedly until the stable region, i.e. Sλ, is reached. At each iteration, we calculate a linear constraint of the form in Eq. (8), add the corresponding gT as a row in G, and augment h with 1. Note that we will almost always stop before reaching the feasible region Sσ. Once a stable matrix is obtained, it is possible to refine this solution. We know that the last constraint caused our solution to cross the boundary of Sλ, so we interpolate between the last solution and the previous iteration’s solution using binary search to look for a boundary of the stable region, in order to obtain a better objective value while remaining stable. An interpolation could be attempted between the least squares solution and any stable solution. However, the stable region can be highly complex, and there may be several folds and boundaries of the stable region in the interpolated area. In our experiments (not shown), interpolating from the LB-1 solution yielded worse results. 5 Experiments For learning the dynamics matrix, we implemented1 least squares, constraint generation (using quadprog), LB-1 [1] and LB-2 [2] (using CVX with SeDuMi) in Matlab on a 3.2 GHz Pentium with 2 GB RAM. Note that these algorithms give a different result from the basic least-squares system identification algorithm only in situations where the least-squares model is unstable. However, least-squares LDSs trained in scarce-data scenarios are unstable for almost any domain, and some domains lead to unstable models up to the limit of available data (e.g. the steam dynamic textures in Section 5.1). The goals of our experiments are to: (1) examine the state evolution and simulated observations of models learned using our method, and compare them to previous work; and (2) compare the algorithms in terms of reconstruction error and efficiency. The error metric used for the quantitative experiments when evaluating matrix A∗is ex(A∗) = 100 × ³ J2(A∗) −J2( ˆA) ´ /J2( ˆA) (10) i.e. percent increase in squared reconstruction error compared to least squares, with J(·) as defined in Eq. (4). We apply these algorithms to learning dynamic textures from the vision domain (Section 5.1), as well as OTC drug sales counts and sunspot numbers (Section 5.2). 1Source code is available at http://www.select.cs.cmu.edu/projects/stableLDS Least Squares LB-1 Constraint Generation A. B. C. −2 0 2 x 10 4 0 500 1000 0 500 1000 t =100 t =200 t =400 t =800 0 500 1000 t =100 t =200 t =400 t =800 t t t −1 0 1 state evolution Figure 3: Dynamic textures. A. Samples from the original steam sequence and the fountain sequence. B. State evolution of synthesized sequences over 1000 frames (steam top, fountain bottom). The least squares solutions display instability as time progresses. The solutions obtained using LB-1 remain stable for the full 1000 frame image sequence. The constraint generation solutions, however, yield state sequences that are stable over the full 1000 frame image sequence without significant damping. C. Samples drawn from a least squares synthesized sequences (top), and samples drawn from a constraint generation synthesized sequence (bottom). Images for LB-1 are not shown. The constraint generation synthesized steam sequence is qualitatively better looking than the steam sequence generated by LB-1, although there is little qualitative difference between the two synthesized fountain sequences. CG LB-1 LB-1∗ LB-2 CG LB-1 LB-1∗ LB-2 steam (n = 10) fountain (n = 10) |λ1| 1.000 0.993 0.993 1.000 0.999 0.987 0.987 0.997 σ1 1.036 1.000 1.000 1.034 1.051 1.000 1.000 1.054 ex(%) 45.2 103.3 103.3 546.9 0.1 4.1 4.1 3.0 time 0.45 95.87 3.77 0.50 0.15 15.43 1.09 0.49 steam (n = 20) fountain (n = 20) |λ1| 0.999 — 0.990 0.999 0.999 — 0.988 0.996 σ1 1.037 — 1.000 1.062 1.054 — 1.000 1.056 ex(%) 58.4 — 154.7 294.8 1.2 — 5.0 22.3 time 2.37 — 1259.6 33.55 1.63 — 159.85 5.13 steam (n = 40) fountain (n = 40) |λ1| 1.000 — 0.989 1.000 1.000 — 0.991 1.000 σ1 1.120 — 1.000 1.128 1.034 — 1.000 1.172 ex(%) 20.24 — 282.7 768.5 3.3 — 4.8 21.5 time 5.85 — 79516.98 289.79 61.9 — 43457.77 239.53 Table 1: Quantitative results on the dynamic textures data for different numbers of states n. CG is our algorithm, LB-1and LB-2 are competing algorithms, and LB-1∗is a simulation of LB-1 using our algorithm by generating constraints until we reach Sσ, since LB-1 failed for n > 10 due to memory limits. ex is percent difference in squared reconstruction error as defined in Eq. (10). Constraint generation, in all cases, has lower error and faster runtime. 5.1 Stable Dynamic Textures Dynamic textures in vision can intuitively be described as models for sequences of images that exhibit some form of low-dimensional structure and recurrent (though not necessarily repeating) characteristics, e.g. fixed-background videos of rising smoke or flowing water. Treating each frame of a video as an observation vector of pixel values yt, we learned dynamic texture models of two video sequences: the steam sequence, composed of 120 × 170 pixel images, and the fountain sequence, composed of 150 × 90 pixel images, both of which originated from the MIT temporal texture database (Figure 3(A)). We use parameters τ = 80, n = 15, and d = 10. Note that the state sequence we learn has no a priori interpretation. An LDS model of a dynamic texture may synthesize an “infinitely” long sequence of images by driving the model with zero mean Gaussian noise. Each of our two models uses an 80 frame training sequence to generate 1000 sequential images in this way. To better visualize the difference between image sequences generated by least-squares, LB-1, and constraint generation, the evolution of each method’s state is plotted over the course of the synthesized sequences (Figure 3(B)). Sequences generated by the least squares models appear to be unstable, and this was in fact the case; both the steam and the fountain sequences resulted in unstable dynamics matrices. Conversely, the constrained subspace identification algorithms all produced well-behaved sequences of states and stable dynamics matrices (Table 1), although constraint generation demonstrates the fastest runtime, best scalability, and lowest error of any stability-enforcing approach. A qualitative comparison of images generated by constraint generation and least squares (Figure 3(C)) indicates the effect of instability in synthesized sequences generated from dynamic texture models. While the unstable least-squares model demonstrates a dramatic increase in image contrast over time, the constraint generation model continues to generate qualitatively reasonable images. Qualitative comparisons between constraint generation and LB-1 indicate that constraint generation learns models that generate more natural-looking video sequences2 than LB-1. Table 1 demonstrates that constraint generation always has the lowest error as well as the fastest runtime. The running time of constraint generation depends on the number of constraints needed to reach a stable solution. Note that LB-1 is more efficient and scalable when simulated using constraint generation (by adding constraints until Sσ is reached) than it is in its original SDP formulation. 5.2 Stable Baseline Models for Biosurveillance We examine daily counts of OTC drug sales in pharmacies, obtained from the National Data Retail Monitor (NDRM) collection [17]. The counts are divided into 23 different categories and are tracked separately for each zipcode in the country. We focus on zipcodes from a particular American city. The data exhibits 7-day periodicity due to differential buying patterns during the week. We isolate a 60-day subsequence where the data dynamics remain relatively stationary, and attempt to learn LDS parameters to be able to simulate sequences of baseline values for use in detecting anomalies. We perform two experiments on different aggregations of the OTC data, with parameter values n = 7, d = 7 and τ = 14. Figure 4(A) plots 22 different drug categories aggregated over all zipcodes, and Figure 4(B) plots a single drug category (cough/cold) in 29 different zipcodes separately. In both cases, constraint generation is able to use very little training data to learn a stable model that captures the periodicity in the data, while the least squares model is unstable and its predictions diverge over time. LB-1 learns a model that is stable but overconstrained, and the simulated observations quickly drift from the correct magnitudes. We also tested the algorithms on the sunspots data (Figure 2(C)) with parameters n = 7, d = 18 and τ = 50, with similar results. Quantitative results on both these domains exhibit similar trends as those in Table 1. 6 Discussion We have introduced a novel method for learning stable linear dynamical systems. Our constraint generation algorithm is more powerful than previous methods in the sense of optimizing over a larger set of stable matrices with a suitable objective function. The constraint generation approach also has the benefit of being faster than previous methods in nearly all of our experiments. One possible extension is to modify the EM algorithm for LDSs to incorporate constraint generation into the M-step in order to learn stable systems that locally maximize the observed data likelihood. Stability could also be of advantage in planning applications. 2See videos at http://www.select.cs.cmu.edu/projects/stableLDS 0 300 Multi-drug sales counts 30 60 0 Multi-zipcode sales counts 30 60 0 Sunspot numbers 100 200 0 0 300 0 300 0 300 0 400 0 400 0 400 0 400 0 1500 0 1500 0 1500 0 1500 A. B. C. Training Data Constraint Generation Least Squares LB-1 Figure 4: (A): 60 days of data for 22 drug categories aggregated over all zipcodes in the city. (B): 60 days of data for a single drug category (cough/cold) for all 29 zipcodes in the city. (C): Sunspot numbers for 200 years separately for each of the 12 months. The training data (top), simulated output from constraint generation, output from the unstable least squares model, and output from the over-damped LB-1 model (bottom). Acknowledgements This paper is based on work supported by DARPA under the Computer Science Study Panel program (authors GJG and BEB), the NSF under Grant Nos. EEC-0540865 (author BEB) and IIS-0325581 (author SMS), and the CDC under award 8-R01-HK000020-02, ”Efficient, scalable multisource surveillance algorithms for Biosense” (author SMS). References [1] Seth L. Lacy and Dennis S. Bernstein. Subspace identification with guaranteed stability using constrained optimization. In Proc. American Control Conference, 2002. [2] Seth L. Lacy and Dennis S. Bernstein. Subspace identification with guaranteed stability using constrained optimization. IEEE Transactions on Automatic Control, 48(7):1259–1263, July 2003. [3] R.E. Kalman. A new approach to linear filtering and prediction problems. Trans. ASME–JBE, 1960. [4] L. Ljung. System Identification: Theory for the user. Prentice Hall, 2nd edition, 1999. [5] Zoubin Ghahramani and Geoffrey E. Hinton. Parameter estimation for Linear Dynamical Systems. Technical Report CRG-TR-96-2, U. of Toronto, Department of Comp. Sci., 1996. [6] N. L. C. Chui and J. M. Maciejowski. Realization of stable models with subspace methods. Automatica, 32(100):1587–1595, 1996. [7] Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, 2004. [8] S. Soatto, G. Doretto, and Y. Wu. Dynamic Textures. Intl. Conf. on Computer Vision, 2001. [9] E. Keogh and T. Folias. The UCR Time Series Data Mining Archive, 2002. [10] P. Van Overschee and B. De Moor. Subspace Identification for Linear Systems: Theory, Implementation, Applications. Kluwer, 1996. [11] T. Van Gestel, J. A. K. Suykens, P. Van Dooren, and B. De Moor. Identification of stable models in subspace identification by using regularization. IEEE Transactions on Automatic Control, 2001. [12] Sajid M. Siddiqi, Byron Boots, and Geoffrey J. Gordon. A Constraint Generation Approach to Learning Stable Linear Dynamical Systems. Technical Report CMU-ML-08-101, CMU, 2008. [13] H. Rauch. Solutions to the linear smoothing problem. In IEEE Transactions on Automatic Control, 1963. [14] Kevin Murphy. Dynamic Bayesian Networks. PhD thesis, UC Berkeley, 2002. [15] Roger Horn and Charles R. Johnson. Matrix Analysis. Cambridge University Press, 1985. [16] Andrew Y. Ng and H. Jin Kim. Stable adaptive control with online learning. In Proc. NIPS, 2004. [17] M. Wagner. A national retail data monitor for public health surveillance. Morbidity and Mortality Weekly Report, 53:40–42, 2004.
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Catching Change-points with Lasso Zaid Harchaoui, C´eline L´evy-Leduc LTCI, TELECOM ParisTech and CNRS 37/39 Rue Dareau, 75014 Paris, France {zharchao,levyledu}@enst.fr Abstract We propose a new approach for dealing with the estimation of the location of change-points in one-dimensional piecewise constant signals observed in white noise. Our approach consists in reframing this task in a variable selection context. We use a penalized least-squares criterion with a ℓ1-type penalty for this purpose. We prove some theoretical results on the estimated change-points and on the underlying piecewise constant estimated function. Then, we explain how to implement this method in practice by combining the LAR algorithm and a reduced version of the dynamic programming algorithm and we apply it to synthetic and real data. 1 Introduction Change-points detection tasks are pervasive in various fields, ranging from audio [10] to EEG segmentation [5]. The goal is to partition a signal into several homogeneous segments of variable durations, in which some quantity remains approximately constant over time. This issue was addressed in a large literature (see [20] [11]), where the problem was tackled both from an online (sequential) [1] and an off-line (retrospective) [5] points of view. Most off-line approaches rely on a Dynamic Programming algorithm (DP), allowing to retrieve K change-points within n observations of a signal with a complexity of O(Kn2) in time [11]. Such a feature refrains practitioners from applying these methods to large datasets. Moreover, one often observes a sub-optimal behavior of the raw DP algorithm on real datasets. We suggest here to slightly depart from this line of research, by focusing on a reformulation of change-point estimation in a variable selection framework. Then, estimating change-point locations off-line turns into performing variable selection on dummy variables representing all possible change-point locations. This allows us to take advantage of the latest theoretical [23], [3] and practical [7] advances in regression with Lasso penalty. Indeed, Lasso provides us with a very efficient method for selecting potential change-point locations. This selection is then refined by using the DP algorithm to estimate the change-point locations. Let us outline the paper. In Section 2, we first describe our theoretical reformulation of off-line change-point estimation as regression with a Lasso penalty. Then, we show that the estimated magnitude of jumps are close in mean, in a sense to be precized, to the true magnitude of jumps. We also give a non asymptotic inequality to upper-bound the ℓ2-loss of the true underlying piecewise constant function and the estimated one. We describe our algorithm in Section 3. In Section 4, we discuss related works. Finally, we provide experimental evidence of the relevance of our approach. 1 2 Theoretical approach 2.1 Framework We describe, in this section, how off-line change-point estimation can be cast as a variable selection problem. Off-line estimation of change-point locations within a signal (Yt) consists in estimating the τ ⋆ k’s in the following model: Yt = µ⋆ k + εt, t = 1, . . . , n such that τ ⋆ k−1 + 1 ≤t ≤τ ⋆ k, 1 ≤k ≤K⋆with τ ⋆ 0 = 0, (1) where εt are i.i.d zero-mean random variables with finite variance. This problem can be reformulated as follows. Let us consider: Yn = Xnβn + εn (2) where Yn is a n × 1 vector of observations, Xn is a n × n lower triangular matrix with nonzero elements equal to one and εn = (εn 1, . . . , εn n)′ is a zero-mean random vector such that the εn j ’s are i.i.d with finite variance. As for βn, it is a n × 1 vector having all its components equal to zero except those corresponding to the change-point instants. The above multiple change-point estimation problem (1) can thus be tackled as a variable selection one: Minimize β ∥Yn −Xnβ∥2 n subject to ∥β∥1 ≤s , (3) where ∥u∥1 and ∥u∥n are defined for a vector u = (u1, . . . , un) ∈Rn by ∥u∥1 = Pn j=1 |uj| and ∥u∥2 n = n−1 Pn j=1 u2 j respectively. Indeed, the above formulation amounts to minimize the following counterpart objective in model (1): Minimize µ1,...,µn 1 n n X t=1 (Yt −µt)2 subject to n−1 X t=1 |µt+1 −µt| ≤s, (4) which consists in imposing an ℓ1-constraint on the magnitude of jumps. The underpinning insight is the sparsity-enforcing property of the ℓ1-constraint, which is expected to give a sparse vector, whose non-zero components would match with those of βn and thus with change-point locations. It is related to the popular Least Absolute Shrinkage eStimatOr (LASSO) in least-square regression of [21], used for efficient variable selection. In the next section, we provide two results supporting the use of the formulation (3) for off-line multiple change-point estimation. We show that estimates of jumps minimizing (3) are consistent in mean, and we provide a non asymptotic upper bound for the ℓ2 loss of the underlying estimated piecewise constant function and the true underlying piecewise function. This inequality shows that, at a precized rate, the estimated piecewise constant function tends to the true piecewise constant function with a probability tending to one. 2.2 Main results In this section, we shall study the properties of the solutions of the problem (3) defined by ˆβn(λ) = Arg min β n ∥Yn −Xnβ∥2 n + λ∥β∥1 o . (5) Let us now introduce the notation sign. It maps positive entry to 1, negative entry to -1 and a null entry to zero. Let A = {k, βn k ̸= 0} and A = {1, . . . , n}\A (6) and let Cn the covariance matrix be defined by Cn = n−1X′ nXn . (7) In a general regression framework, [18] recall that, with probability tending to one, ˆβn(λ) and βn have the same sign for a well-chosen λ, only if the following condition holds element-wise: ¯¯Cn AA(Cn AA)−1sign(βn A) ¯¯ < 1, (8) where Cn IJ is a sub-matrix of Cn obtained by keeping rows with index in the set I and columns with index in J. The vector βn A is defined by βn A = (βn k )k∈A. The condition (8) is not fulfilled in the 2 change-point framework implying that we cannot have a perfect estimation of the change-points as it is already known, see [13]. But, following [18] and [3], we can prove some consistency results, see Propositions 1 and 2 below. In the following, we shall assume that the number of break points is equal to K⋆. The following proposition ensures that for a large enough value of n the estimated change-point locations are close to the true change-points. Proposition 1. Assume that the observations (Yn) are given by (2) and that the εn j ’s are centered. If λ = λn is such that λn √n →0 as n tends to infinity then ∥E(ˆβn(λn)) −βn∥n →0 . Proof. We shall follow the proof of Theorem 1 in [18]. For this, we denote βn(λ) the estimator ˆβn(λ) under the absence of noise and γn(λ) the bias associated to the Lasso estimator: γn(λ) = βn(λ) −βn. For notational simplicity, we shall write γ instead of γn(λ). Note that γ satisfies the following minimization: γ = Arg minζ∈Rn f(ζ) , where f(ζ) = ζ′Cnζ + λ X k∈A |βn k + ζk| + λ X k∈¯ A |ζk| . Since f(γ) ≤f(0), we get γ′Cnγ + λ X k∈A |βn k + γk| + λ X k∈¯ A |γk| ≤λ X k∈A |βn k | . We thus obtain using the Cauchy-Schwarz inequality the following upper bound γ′Cnγ ≤λ X k∈A |γk| ≤λ √ K⋆ à n X k=1 |γk|2 !1/2 . Using that γ′Cnγ ≥n−1 Pn k=1 |γk|2, we obtain: ∥γ∥n ≤λ √ nK⋆. The following proposition ensures, thanks to a non asymptotic result, that the estimated underlying piecewise function is close to the true piecewise constant function. Proposition 2. Assume that the observations (Yn) are given by (2) and that the εn j ’s are centered iid Gaussian random variables with variance σ2 > 0. Assume also that (βn k )k∈A belong to (βmin, βmax) where βmin > 0. For all n ≥1 and A > √ 2 then, with a probability larger than 1 −n1−A2/2, if λn = Aσ p log n/n, ∥Xn(ˆβn(λn) −βn)∥2 n ≤2AσβmaxK⋆ r log n n . Proof. By definition of ˆβn(λ) in (5) as a minimizer of a criterion, we have ∥Yn −Xn ˆβn(λ)∥2 n + λ∥ˆβn(λ)∥1 ≤∥Yn −Xnβn∥2 n + λ∥βn∥1 . Using (2), we get ∥Xn(βn −ˆβn(λ))∥2 n + 2 n(βn −ˆβn(λ))′X′ nεn + λ n X j=1 |ˆβn j (λ)| ≤λ n X j=1 |βn j | . Thus, ∥Xn(βn −ˆβn(λ))∥2 n ≤2 n(ˆβn(λ) −βn)′X′ nεn + λ X j∈A (|βn j | −|ˆβn j (λ)|) −λ X j∈¯ A |ˆβn j (λ)| . Observe that 2 n(ˆβn(λ) −βn)′X′ nεn = 2 n X j=1 (ˆβn j (λ) −βn j ) 1 n n X i=j εn i . 3 Let us define the event E = Tn j=1 n n−1 ¯¯¯Pn i=j εn i ¯¯¯ ≤λ o . Then, using the fact that the εn i ’s are iid zero-mean Gaussian random variables, we obtain P( ¯E) ≤ n X j=1 P n−1 ¯¯¯¯¯¯ n X i=j εn i ¯¯¯¯¯¯ > λ ≤ n X j=1 exp µ − n2λ2 2σ2(n −j + 1) ¶ . Thus, if λ = λn = Aσ p log n/n, P( ¯E) ≤n1−A2/2 . With a probability larger than 1 −n1−A2/2, we get ∥Xn(βn −ˆβn(λ))∥2 n ≤λn n X j=1 |ˆβn j (λ) −βn j | + λn X j∈A (|βn j | −|ˆβn j |) −λn X j∈¯ A |ˆβn j | . We thus obtain with a probability larger than 1 −n1−A2/2 the following upper bound ∥Xn(βn −ˆβn(λ))∥2 n ≤2λn X j∈A |βn j | = 2Aσ r log n n X j∈A |βn j | ≤2AσβmaxK⋆ r log n n . 3 Practical approach The previous results need to be efficiently implemented to cope with finite datasets. Our algorithm, called Cachalot (CAtching CHAnge-points with LassO), can be split into the following three steps described hereafter. Estimation with a Lasso penalty We compute the first Kmax non-null coefficients ˆβτ1, . . . , ˆβτKmax on the regularization path of the LASSO problem (3). The LAR/LASSO algorithm, as described in [7], provides an efficient algorithm to compute the entire regularization path for the LASSO problem. Since P j |βj| ≤s is a sparsity-enforcing constraint, the set {j, ˆβj ̸= 0} = {τj} becomes larger as we run through the regularization path. We shall denote by S the Kmax-selected variables: S = {τ1, . . . , τKmax} . (9) The computational complexity of the Kmax-long regularization path of LASSO solutions is O(K3 max + K2 maxn). Most of the time, we can see that the Lasso effectively catches the true changepoint but also irrelevant change-points at the vicinity of the true ones. Therefore, we propose to refine the set of change-points caught by the Lasso by performing a post-selection. Reduced Dynamic Programming algorithm One can consider several strategies to remove irrelevant change-points from the ones retrieved by the Lasso. Among them, since usually in applications, one is only interested in change-point estimation up to a given accuracy, we could launch the Lasso on a subsample of the signal. Here, we suggest to perform post-selection by using the standard Dynamic Programming algorithm (DP) thoroughly described in [11] (Chapter 12, p. 450) but on the reduced set S instead of {1, . . . , n}. This algorithm allows one to efficiently minimize the following objective for each K in {1, . . . , Kmax}: J(K) = Min τ1<···<τK s.t τ1,...,τK∈S K X k=1 τk X i=τk−1+1 (Yi −ˆµk)2, (10) S being defined in (9) and outputs for each K, the corresponding subset of change-points (ˆτ1, . . . , ˆτK). The DP algorithm has a computational complexity of O(Kmax n2) if we look for at most Kmax change-points within the signal. Here, our reduced DP calculations (rDP) scales as O(Kmax K2 max) where Kmax is the maximum number of change-points/variables selected by LAR/LASSO algorithm. Since typically Kmax ≪n, our method thus provides a reduction of the computational burden associated with the classical change-points detection approach which consists in running the DP algorithm over all the n observations. 4 Selecting the number of change-points The point is now to select the adequate number of change-points. As n →∞, according to [15], the ratio ρk = J(k + 1)/J(k) should show different qualitative behavior when k ⩽K⋆and when k > K⋆, K⋆being the true number of change-points. In particular, ρk ≥Cn for k > K⋆, where Cn →1 as n →∞. Actually we found out that Cn was close to 1, even in small-sample settings, for various experimental designs in terms of noise variance and true number of change-points. Hence, conciliating theoretical guidance in large-sample setting and experimental findings in fixed-sample setting, we suggest the following rule of thumb for selecting the number of change-points ˆK : ˆK = Mink≥1 {ρk ≥1 −ν} , where ρk = J(k + 1)/J(k). Cachalot Algorithm Input • Vector of observations Y ∈Rn • Upper bound Kmax on the number of change-points • Model selection threshold ν Processing 1. Compute the first Kmax non-null coefficients (βτ1, . . . , βτKmax ) on the regularization path with the LAR/LASSO algorithm. 2. Launch the rDP algorithm on the set of potential change-points (τ1, . . . , τKmax). 3. Select the smallest subset of the potential change-points (τ1, . . . , τKmax) selected by the rDP algorithm for which ρk ≥1 −ν. Output Change-point locations estimates ˆτ1, . . . , ˆτ ˆ K. To illustrate our algorithm, we consider observations (Yn) satisfying model (2) with (β30, β50, β70, β90) = (5, −3, 4, −2), the other βj being equal to zero, n = 100 and εn a Gaussian random vector with a covariance matrix equal to Id, Id being a n × n identity matrix. The set of the first nine active variables caught by the Lasso along the regularization path, i.e. the set {k, ˆβk ̸= 0} is given in this case by: S = {21, 23, 28, 29, 30, 50, 69, 70, 90}. The set S contains the true change-points but also irrelevant ones close to the true change-points. Moreover the most significant variables do not necessarily appear at the beginning. This supports the use of the reduced version of the DP algorithm hereafter. Table 1 gathers the J(K), K = 1, . . . , Kmax and the corresponding (ˆτ1, . . . , ˆτK). Table 1: Toy example: The empirical risk J and the estimated change-points as a function of the possible number of change-points K K J(K) (ˆτ1, . . . , ˆτK) 0 696.28 ∅ 1 249.24 30 2 209.94 (30,70) 3 146.29 (30,50,69) 4 120.21 (30,50,70,90) 5 118.22 (30,50,69,70,90) 6 116.97 (21,30,50,69,70,90) 7 116.66 (21,29,30,50,69,70,90) 8 116.65 (21,23,29,30,50,69,70,90) 9 116.64 (21,23,28,29,30,50,69,70,90) The different values of the ratio ρk for k = 0, . . . , 8 of the model selection procedure are given in Table 2. Here we took ν = 0.05. We conclude, as expected, that ˆK = 4 and that the change-points are (30, 50, 70, 90), thanks to the results obtained in Table 1. 4 Discussion Off-line multiple change-point estimation has recently received much attention in theoretical works, both in a non-asymptotic and in an asymptotic setting by [17] and [13] respectively. From a practical point of view, retrieving the set of change-point locations {τ ⋆ 1 , . . . , τ ⋆ K} is challenging, since it is 5 Table 2: Toy example: The values of the ratio (ρk = J(k + 1)/J(k), k = 0, . . . , 8) k 0 1 2 3 4 5 6 7 8 ρk 0.3580 0.8423 0.6968 0.8218 0.9834 0.9894 0.9974 0.9999 1.0000 plagued by the curse of dimensionality. Indeed, all of the n observation times have to be considered as potential change-point instants. Yet, a dynamic programming algorithm (DP), proposed by [9] and [2], allows to explore all the configurations with a complexity of O(n3) in time. Then selecting the number of change-points is usually performed thanks to a Schwarz-like penalty λnK, where λn has to be calibrated on data [13] [12], or a penalty K(a + b log(n/K)) as in [17] [14], where a and b are data-driven as well. We should also mention that an abundant literature tackles both change-point estimation and model selection issues from a Bayesian point of view (see [20] [8] and references therein). All approaches cited above rely on DP, or variants in Bayesian settings, and hence yield a computational complexity of O(n3), which makes them inappropriate for very largescale signal segmentation. Moreover, despite its theoretical optimality in a maximum likelihood framework, raw DP may sometimes have poor performances when applied to very noisy observations. Our alternative framework for multiple change-point estimation was previously elusively mentioned several times, e.g. in [16] [4] [19]. However up to our knowledge neither successful practical implementation nor theoretical grounding was given so far to support such an approach for change-point estimation. Let us also mention [22], where the Fused Lasso is applied in a similar yet different way to perform hot-spot detection. However, this approach includes an additional penalty, penalizing departures from the overall mean of the observations, and should thus rather be considered as an outlier detection method. 5 Comparison with other methods 5.1 Synthetic data We propose to compare our algorithm with a recent method based on a penalized least-squares criterion studied by [12]. The main difficulty in such approaches is the choice of the constants appearing in the penalty. In [12], a very efficient approach to overcome this difficulty has been proposed: the choice of the constants is completely data-driven and has been implemented in a toolbox available online at http://www.math.u-psud.fr/˜lavielle/programs/index.html. In the following, we benchmark our algorithm: A together with the latter method: B. We shall use Recall and Precision as relevant performance measures to analyze the previous two algorithms. More precisely, the Recall corresponds to the ratio of change-points retrieved by a method with those really present in the data. As for the Precision, it corresponds to the number of change-points retrieved divided by the number of suggested change-points. We shall also estimate the probability of false alarm corresponding to the number of suggested change-points which are not present in the signal divided by the number of true change-points. To compute the precision and the recall of methods A and B, we ran Monte-Carlo experiments. More precisely, we sampled 30 configurations of change-points for each real number of change-points K⋆ equal to 5, 10, 15 and 20 within a signal containing 500 observations. Change-points were at least distant of 10 observations. We sampled 30 configurations of levels from a Gaussian distribution. We used the following setting for the noise: for each configuration of change-points and levels, we synthesized a Gaussian white noise such that the standard deviation is set to a multiple of the minimum magnitude jump between two contiguous segments, i.e. σ = m Mink(µ∗ k+1 −µ∗ k), µ⋆ k being the level of the kth segment. The number of noise replications was set to 10. As shown in Tables 3, 4 and 5 below, our method A yields competitive results compared to method B with 1 −ν = 0.99 and Kmax = 50. Performances in recall are comparable whereas method A provides better results than method B in terms of precision and false alarm rate. 5.2 Real data In this section, we propose to apply our method previously described to real data which have already been analyzed by Bayesian methods: the well-log data which are described in [20] and [6] and 6 Table 3: Precision of methods A and B K⋆= 5 K⋆= 10 K⋆= 15 K⋆= 20 Method A B A B A B A B m = 0.1 0.81±0.15 0.71±0.29 0.89±0.08 0.8±0.22 0.95±0.05 0.86±0.13 0.97±0.03 0.91±0.09 m = 0.5 0.8±0.16 0.73±0.29 0.89±0.08 0.8±0.21 0.95±0.05 0.86±0.13 0.97±0.03 0.92±0.09 m = 1.0 0.78±0.17 0.71±0.27 0.88±0.09 0.78±0.21 0.93±0.06 0.85±0.13 0.96±0.04 0.9±0.09 m = 1.5 0.73±0.19 0.66±0.28 0.84±0.1 0.79±0.2 0.93±0.06 0.84±0.13 0.95±0.04 0.9±0.1 Table 4: Recall of methods A and B K⋆= 5 K⋆= 10 K⋆= 15 K⋆= 20 Method A B A B A B A B m = 0.1 0.99±0.02 0.99±0.02 1±0 1±0 0.99±0 0.99±0 0.99±0 1±0 m = 0.5 0.98±0.04 0.99±0.03 0.99±0.01 0.99±0.01 0.99±0.01 0.99±0.01 0.99±0.01 1±0 m = 1.0 0.95±0.08 0.94±0.08 0.96±0.06 0.96±0.05 0.97±0.03 0.97±0.04 0.97±0.03 0.98±0.02 m = 1.5 0.85±0.16 0.87±0.15 0.92±0.07 0.91±0.09 0.94±0.06 0.94±0.06 0.95±0.04 0.96±0.04 Table 5: False alarm rate of methods A and B K⋆= 5 K⋆= 10 K⋆= 15 K⋆= 20 Method A B A B A B A B m = 0.1 0.13±0.03 0.23±0.2 0.24±0.03 0.33±0.19 0.34±0.02 0.42±0.13 0.44±0.02 0.51±0.12 m = 0.5 0.13±0.03 0.22±0.2 0.23±0.03 0.32±0.18 0.33±0.02 0.41±0.13 0.44±0.02 0.5±0.11 m = 1.0 0.13±0.03 0.21±0.18 0.23±0.03 0.32±0.18 0.33±0.02 0.4±0.13 0.43±0.03 0.5±0.12 m = 1.5 0.13±0.03 0.21±0.2 0.23±0.03 0.29±0.16 0.31±0.03 0.4±0.15 0.42±0.03 0.48±0.11 displayed in Figure 1. They consist in nuclear magnetic response measurements expected to carry information about rock structure and especially its stratification. One distinctive feature of these data is that they typically contain a non-negligible amount of outliers. The multiple change-point estimation method should then, either be used after a data cleaning step (median filtering [6]), or explicitly make heavy-tailed noise distribution assumption. We restricted ourselves to a median filtering pre-processing. The results given by our method applied to the welllog data processed with a median filter are displayed in Figure 1 for Kmax = 200 and 1 −ν = 0.99. The vertical lines locate the change-points. We can note that they are close to those found out by [6] (P. 206) who used Bayesian techniques to perform change-points detection. 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 x 10 5 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 x 10 5 Figure 1: Left: Raw well-log data, Right: Change-points locations obtained with our method in well-log data processed with a median filter 7 6 Conclusion and prospects We proposed here to cast the multiple change-point estimation as a variable selection problem. A least-square criterion with a Lasso-penalty yields an efficient primary estimation of change-point locations. Yet these change-point location estimates can be further refined thanks to a reduced dynamic programming algorithm. We obtained competitive performances on both artificial and real data, in terms of precision, recall and false alarm. Thus, Cachalot is a computationally efficient multiple change-point estimation method, paving the way for processing large datasets. References [1] M. Basseville and N. Nikiforov. The detection of abrupt changes. Information and System sciences series. Prentice-Hall, 1993. [2] R. Bellman. On the approximation of curves by line segments using dynamic programming. Communications of the ACM, 4(6), 1961. [3] P. Bickel, Y. Ritov, and A. Tsybakov. Simultaneous analysis of Lasso and Dantzig selector. Preprint 2007. [4] L. Boysen, A. Kempe, A. Munk, V. Liebscher, and O. Wittich. Consistencies and rates of convergence of jump penalized least squares estimators. Annals of Statistics, In revision. [5] B. Brodsky and B. Darkhovsky. Non-parametric statistical diagnosis: problems and methods. Kluwer Academic Publishers, 2000. [6] O. Capp´e, E. Moulines, and T. Ryden. Inference in Hidden Markov Models (Springer Series in Statistics). Springer-Verlag New York, Inc., 2005. [7] B. Efron, T. Hastie, and R. Tibshirani. Least angle regression. Annals of Statistics, 32:407–499, 2004. [8] P. Fearnhead. Exact and efficient bayesian inference for multiple changepoint problems. Statistics and Computing, 16:203–213, 2006. [9] W. D. Fisher. On grouping for maximum homogeneity. Journal of the American Statistical Society, 53:789–798, 1958. [10] O. Gillet, S. Essid, and G. Richard. On the correlation of automatic audio and visual segmentation of music videos. IEEE Transactions on Circuits and Systems for Video Technology, 2007. [11] S. M. Kay. Fundamentals of statistical signal processing: detection theory. Prentice-Hall, Inc., 1993. [12] M. Lavielle. Using penalized contrasts for the change-points problems. Signal Processing, 85(8):1501– 1510, 2005. [13] M. Lavielle and E. Moulines. Least-squares estimation of an unknown number of shifts in a time series. Journal of time series analysis, 21(1):33–59, 2000. [14] E. Lebarbier. Detecting multiple change-points in the mean of a gaussian process by model selection. Signal Processing, 85(4):717–736, 2005. [15] C.-B. L. Lee. Estimating the number of change-points in a sequence of independent random variables. Statistics and Probability Letters, 25:241–248, 1995. [16] E. Mammen and S. Van De Geer. Locally adaptive regression splines. Annals of Statistics, 1997. [17] P. Massart. A non asymptotic theory for model selection. pages 309–323. European Mathematical Society, 2005. [18] N. Meinshausen and B. Yu. Lasso-type recovery of sparse representations for high-dimensional data. Preprint 2006. [19] S. Rosset and J. Zhu. Piecewise linear regularized solution paths. Annals of Statistics, 35, 2007. [20] J. Ruanaidh and W. Fitzgerald. Numerical Bayesian Methods Applied to Signal Processing. Statistics and Computing. Springer, 1996. [21] R. Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B, 58(1):267–288, 1996. [22] R. Tibshirani and P. Wang. Spatial smoothing and hot spot detection for cgh data using the fused lasso. Biostatistics, 9(1):18–29, 2008. [23] P. Zhao and B. Yu. On model selection consistency of lasso. Journal Of Machine Learning Research, 7, 2006. 8
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Feature Selection Methods for Improving Protein Structure Prediction with Rosetta Ben Blum, Michael I. Jordan Department of Electrical Engineering and Computer Science University of California at Berkeley Berkeley, CA 94305 {bblum,jordan}@cs.berkeley.edu David E. Kim, Rhiju Das, Philip Bradley, David Baker Department of Genome Sciences University of Washington Seattle, WA 98195 {dekim, rhiju, pbradley, dabaker}@u.washington.edu Abstract Rosetta is one of the leading algorithms for protein structure prediction today. It is a Monte Carlo energy minimization method requiring many random restarts to find structures with low energy. In this paper we present a resampling technique for structure prediction of small alpha/beta proteins using Rosetta. From an initial round of Rosetta sampling, we learn properties of the energy landscape that guide a subsequent round of sampling toward lower-energy structures. Rather than attempt to fit the full energy landscape, we use feature selection methods—both L1-regularized linear regression and decision trees—to identify structural features that give rise to low energy. We then enrich these structural features in the second sampling round. Results are presented across a benchmark set of nine small alpha/beta proteins demonstrating that our methods seldom impair, and frequently improve, Rosetta’s performance. 1 Introduction Protein structure prediction is one of the most important unsolved problems in biology today. With the wealth of genome data now available, it is of great interest to determine the structures of the proteins that genes encode. Proteins are composed of long chains of amino acid residues, of which there are twenty natural varieties. A gene encodes a specific amino acid sequence, which, when translated, folds into a unique three-dimensional conformation. The protein structure prediction problem is to predict this conformation (the protein’s tertiary structure) from the amino acid sequence (the protein’s primary structure). The biological function of a protein is dependent on its structure, so structure prediction is an important step towards function prediction. Potential applications of structure prediction range from elucidation of cellular processes to vaccine design. Experimental methods for protein structure determination are costly and time-intensive, and the number of known protein sequences now far outstrips the capacity of experimentalists to determine their structures. Computational methods have been improving steadily and are approaching the level of resolution attainable in experiments. Structure prediction methods fall into two broad camps: comparative modeling, in which solved protein structures are known for one or more proteins with sequences similar to the target sequence (“homologs”), and ab initio modeling, in which no homologs are known. In this paper we concentrate on ab initio modeling, and specifically on the Rosetta algorithm [3]. 1 Figure 1: Flowchart of resampling method. Rosetta is one of the leading methods for ab initio protein structure prediction today. Rosetta uses a Monte Carlo search procedure to minimize an energy function that is sufficiently accurate that the conformation found in nature (the “native” conformation) is generally the conformation with lowest Rosetta energy. Finding the global minimum of the energy function is very difficult because of the high dimensionality of the search space and the very large number of local minima. Rosetta employs a number of strategies to combat these issues, but the primary one is to perform a large number of random restarts. Thanks to a very large-scale distributed computing platform called Rosetta@home, composed of more than three hundred thousand volunteer computers around the world, up to several million local minima of the energy function (“decoys,” in Rosetta parlance) can be computed for each target sequence. Our work begins with the observation that a random-restart strategy throws away a great deal of information from previously computed local minima. In particular, previous samples from conformation space might suggest regions of uniformly lower energy; these are regions in which Rosetta may wish to concentrate further sampling. This observation is applicable to many global optimization problems, and past researchers have proposed a variety of methods for exploiting it, including fitting a smoothed response surface to the local minima already gathered [1] and learning to predict good starting points for optimization [2]. Unfortunately, conformation space is very high-dimensional and very irregular, so response surfaces do not generalize well beyond the span of the points to which they are fitted. Generally, the correct (or “native”) structure will not be in the span of the points seen so far—if it were, the first round of Rosetta sampling would already have been successful. We have developed an approach that sidesteps this limitation by explicitly recombining successful features of the models seen so far. No single local minimum computed in the first round of Rosetta search will have all the native features. However, many native features are present in at least some of the decoys. If these features can be identified and combined with each other, then sampling can be improved. Our approach has three steps, each mapping from one structural representation space to another (Figure 1). In the first step, we project the initial set of Rosetta models from continuous conformation space into a discrete feature space. The structural features that we have designed characterize significant aspects of protein structure and are largely sufficient to determine a unique conformation. In the second step, we use feature selection methods including both decision trees and Least Angle Regression (LARS) [4] to identify structural features that best account for energy variation in the initial set of models. We can then predict that certain of these features (generally, those associated with low energy) are present in the native conformation. In the third step, we use constrained Rosetta search to generate a set of models enriched for these key features. 2 Outline In section 3, we outline a response surface approach and its shortcomings, and motivate the move to a feature-based representation. In section 4, we describe the features we use and the way that particular feature values are enforced in Rosetta search. This characterizes the way we map points from our discretized feature space back to continuous conformation space. In section 5, we describe the feature selection techniques we use to determine which features to enforce. In section 6, we 2 (a) (b) Figure 2: (a) Rosetta models (black) and relaxed natives (blue) projected onto the first principal component. (b) Models and natives projected onto the third principal component. show the results of Rosetta search biased towards selected features. In section 7, we conclude with a discussion of the results achieved so far and of further work to be done. 3 Response Surface Methods As an initial attempt at developing resampling methods for protein structure prediction, we investigated a response surface fitting approach. Our goal was to fit a smoothed energy surface to the Rosetta models seen so far and then to minimize this surface to find new starting points for local optimization of the Rosetta energy function. The first task was to define the conformation space. The most natural space is defined in terms of the conformational degrees of freedom. Each residue in an amino acid sequence has two primary degrees of freedom: rotation around the Cα–N bond, referred to as the φ torsion angle, and rotation around the Cα–C bond, referred to as the ψ torsion angle. However, it is difficult to fit a response surface in the space of torsion angles because the energy function is highly irregular in this space; a slight change in a single torsion angle typically causes large global structural changes, which in turn cause large energy changes. Instead, we took the three-dimensional coordinates of the backbone atoms as our conformation space, with all models in the set aligned to a reference model. There are four backbone atoms per residue and three coordinates per backbone atom, so an n-residue protein is represented by a 12n-dimensional vector. Even for small proteins of only around 70 residues this space is very high-dimensional, but we found that most of the structural variation in sets of Rosetta models was captured by the first 10 principal components. Data were sufficient to fit a response surface in these 10 dimensions. Along certain directions, energy gradients were detectable that pointed toward the native structure. One such direction was the first principal component for protein 1n0u (Figure 2.a; in this graph, the native structure is represented as an ensemble of Rosetta-minimized structures that started at the native conformation). However, in most directions the gradient did not point toward the natives (Figure 2.b). A response surface fitted to the Rosetta models shown in these graphs will therefore have high energy in the vicinity of the natives. These observations suggest a new strategy: rather than fitting a response surface to all the dimensions jointly, we should identify a few dimensions that are associated with clear score gradients and make no claims about the other dimensions. This motivates a shift in philosophy: rather than predicting energy and minimizing, we wish to predict features of the native structure and then enforce them independently of each other. 3 (a) (b) Figure 3: (a) Bins in Ramachandran plot. (b) Structure of 1dcj. Two helices are visible behind a beta pleated sheat consisting of four strands, the bottommost three paired in the anti-parallel orientation and the topmost two paired in the parallel orientation. In this “cartoon” representation of structure, individual atoms are not rendered. 4 Structural features For the purpose of the work described in this paper, we make use of two types of structural features: torsion angle features and beta contact features. 4.1 Torsion angle features The observed values of the φ and ψ angles for a single residue are strongly clustered in the database of solved protein structures (the PDB). Their empirical distribution is shown in a Ramachandran plot. In order to discretize the possible torsion angles for each residue, we divide the Ramachandran plot into five regions, referred to as “A,” “B,” “E,” “G,” and “O,” (Figure 3.a). These regions are chosen to correspond roughly to clusters observed in the PDB. A protein with 70 amino acid residues has 70 torsion bin features, each with possible values A, B, E, G, and O. The primary search move in Rosetta is a fragment replacement move: the conformation of a string of three or nine consecutive residues within the target sequence is replaced with the conformation of a similar subsequence from the PDB. A torsion angle feature can be constrained in Rosetta search by limiting the fragments to those which have torsion angles within the given bin at the given residue position. Strings of torsion features are referred to as barcodes in Rosetta, and the apparatus for defining and constraining them was developed in-house by Rosetta developers. 4.2 Beta contact features Proteins exhibit two kinds of secondary structure, characterized by regular hydrogen bond patterns: alpha helices and beta pleated sheets (Figure 3.b). In alpha helices, the hydrogen bonds are all local, and are predicted fairly consistently by Rosetta. In beta sheets, however, the bonds can be between residues that are quite distant along the chain. A beta contact feature for residues i and j indicates the presence of two backbone hydrogen bonds between i and j. We use the same definition of beta pairing as the standard secondary structure assignment algorithm DSSP [5]. The bonding pattern can be either parallel (as between the red residues in Figure 3.b) or antiparallel (as between the blue residues). Furthermore, the pleating can have one of two different orientations. A beta pairing feature is defined for every triple (i, j, o) of residue numbers i and j and orientations o ∈ {parallel, antiparallel}. The possible values of a beta pairing feature are X, indicating no pairing, and P1 or P2, indicating pleating of orientation 1 or 2, respectively. Beta contact features are enforced in Rosetta by means of a technique called “jumping.” A pseudobackbone-bond is introduced between the two residues to be glued together. This introduces a closed loop into the backbone topology of the protein. Torsion angles within the loop can no longer be altered without breaking the loop, so, in order to permit further fragment replacements, a cut (or 4 “chainbreak”) must be introduced somewhere else in the loop. The backbone now takes the form of a tree rather than a chain. After a Rosetta search trajectory terminates, an attempt is made to close the chainbreak with local search over several torsion angles on either side of it. 5 Prediction of native features Let us transform our set of multi-valued features into a set of 0-1 valued features indicating whether or not a particular value for the feature is present. Let us assume that each binary feature f has an independent energetic effect; if present, it brings with it an average energy bonus bf. Under these assumptions, the full energy of a conformation d is modelled as E0 + X f dfbf + N, where E0 is a constant offset, df is either 1 if the feature is present in d or 0 if it is absent, and N is Gaussian noise. This model is partially justified by the fact that the true energy is indeed a sum of energies from local interactions, and our features capture local structural information. Our hypothesis is that native features have lower energy on average even if other native features are not present. In order to identify a small set of potentially native features, we use L1 regularization, or lasso regression [6], to find a sparse model. The minimization performed is argmin(b,E0) X d∈D (E(d) −E0 − X f dfbf)2 + C X f |bf|, where E(d) is the computed Rosetta energy of model d and C is a regularization constant. The small set of features that receive non-zero weights are those that best account for energy variations in the population of decoys. These are the features we can most confidently predict to be native. The Least Angle Regression algorithm [4] allows us to efficiently compute the trajectory of solutions for all values of C simultaneously. Experience with Rosetta has shown that constraining more than ten or fifteen torsion features can hamper search more than it helps; if there are very few fragments available for a given position that satisfy all torsion constraints, the lack of mobility at that position can be harmful. We typically take the point in the LARS trajectory that gives fifteen feature values. 5.1 Feature enforcement strategy LARS gives us a set of feature values that have a strong effect on energy. Our hypothesis is that features strongly associated with lower energies—namely, those selected by LARS and given negative weights—are more likely to be native, and that features given positive weights by LARS are more likely to be non-native. This hypothesis is born out by our experiments on a benchmark set of 9 small alpha/beta proteins. The LARS prediction accuracy is given in Figure 4.a. This chart shows, for each protein, the fraction of LARS-selected features correctly labeled as native or non-native by the sign of the LARS weight. Fifteen LARS features were requested per protein. The more accurate “low energy leaf” predictions will be discussed in the next section. It is clear from Figure 4.a that LARS is informative about native features for most proteins. However, we cannot rely wholly on its predictions. If we were simply to constrain every LARS feature, then Rosetta would never find the correct structure, since some incorrect features would be present in every model. Our resampling strategy is therefore to flip a coin at the beginning of the Rosetta run to decide whether or not to constrain a particular LARS feature. Coins are flipped independently for each LARS feature. Resampling improves on unbiased Rosetta sampling if the number of viable runs (runs in which no non-native features are enforced) is sufficiently high that the benefits from the enforcement of native features are visible. We have achieved some success by enforcing LARS features with probability 30% each, as demonstrated in the results section. 5.2 Decision trees for beta contact features Beta contact features are less suited to the lasso regression approach than torsion angle features, because independence assumptions are not as valid. For instance, contact (i, j, parallel) and contact 5 (a) (b) Figure 4: (a) LARS prediction accuracy when fitted to total decoy population and to the three decision-tree leaves with lowest 10th percentile energies, ordered here by average rmsd. (b) Relation of prediction accuracy to resampling improvement in LARS-only runs. (i + 1, j + 1, parallel) are redundant and will usually co-occur, whereas contact (i, j, parallel) and contact (i −1, j + 1, parallel) are mutually exclusive and will never co-occur. For beta contact features, we employ a decision tree approach to divide the decoy population into non-overlapping clusters each defined by the presence of several beta contacts. Lasso regression is then employed in each cluster separately to determine likely native torsion features. We use decision trees of depth three. At each node, a beta contact feature is selected to use as a split point and a child node is created for each of the three possible values X, P1, and P2. Our strategy is to choose split points which most reduce entropy in the features. The beta contact feature is therefore chosen whose mutual information with the other beta contact features is maximized, as approximated by the sum of the marginal mutual informations with each other feature. Since some clusters are sampled more heavily than others, the lowest energy within a cluster is not a fair measure of its quality, even though, in principle, we care only about the lowest achievable energy. Instead, we use the 10th percentile energy to evaluate clusters. Its advantage as a statistic is that its expectation is not dependent on sample size, but it often gives a reasonably tight upper bound on achievable energy. Our resampling strategy, given a decision tree, is to sample evenly from each of the top three leaves as ranked by 10th percentile energy. Within the subpopulation of decoys defined by each leaf, we select torsion features using LARS. In our benchmark set, the top three low-energy leaves of the decision tree were generally closer to the native than the population at large. Perhaps as a result, LARS generally achieved greater prediction accuracy when restricted to their associated subpopulations, as seen in Figure 3.b. Leaves are sorted by average rmsd, so “low energy leaf 1,” the “best” leaf, consists of decoys which are closest, on average, to the native conformation. The best leaf consisted of only native contacts for all proteins except 1n0u and 1ogw, but in both these cases it contained structures generally lower in rmsd than the population at large and resampling achieved improvements over plain Rosetta sampling. In general, LARS performed better on the leaves that were closer to the native structure, although there were a few notable exceptions. Ideally, we would concentrate our sampling entirely on the best leaf, but since we cannot generally identify which one it is, we have to hedge our bets. Including more leaves in the resampling round increases the chances of resampling a native leaf but dilutes sampling of the best leaf in the pool. This tradeoff is characteristic of resampling methods. 6 Results We tested two Rosetta resampling schemes over a set of 9 alpha/beta proteins of between 59 and 81 residues. In the first scheme (referred to henceforth as “LARS-only”), 15 LARS-predicted torsion features were constrained at 30% frequency. In the second (referred to henceforth as “decision6 RMSD of low-energy decoys Lowest RMSD of 25 low-energy decoys Decision tree LARS only Decision tree LARS only Control Resamp Control Resamp Control Resamp Control Resamp 1di2 2.35 2.14 2.76 0.97 1.78 1.34 1.82 0.73 1dtj 3.20 1.53 5.28 1.88 1.46 1.53 1.95 1.59 1dcj 2.35 3.31 2.34 2.11 2.19 1.86 1.71 1.88 1ogw 5.22 3.99 3.03 2.80 3.12 2.6 2.08 2.48 2reb 1.15 1.17 1.07 1.27 0.89 0.93 0.83 0.86 2tif 5.68 4.57 3.57 6.85 3.32 3.27 3.27 2.61 1n0u 11.89 11.60 11.93 3.54 9.78 3.19 3.54 2.84 1hz6A 2.52 1.06 3.36 4.68 2.38 1.06 1.97 1.19 1mkyA 10.39 8.21 4.60 4.58 3.43 3.25 3.33 4.23 Mean difference -0.8 -1.03 -1.04 -0.23 Median difference -1.11 -0.23 -0.33 -0.36 Number improved 7/9 6/9 7/9 5/9 tree”), three subpopulations were defined for each protein using a decision tree, and within each subpopulation 15 LARS-predicted torsion features were constrained at frequencies heuristically determined on the basis of several meta-level “features of features,” including the rate of the feature’s occurrence in the first round of Rosetta sampling and the magnitude of the regression weight for the feature. Each resampling scheme was compared against a control population generated at the same time. Exactly the same number of models were generated for the control and resampled populations. The control and resampled populations for the LARS-only scheme consist of about 200,000 decoys each. The populations for the decision-tree scheme consist of about 30,000 decoys each, due to limitations in available compute time. The difference in quality between the two control populations is partially explained by the different numbers of samples in each, and partially by changes in Rosetta in the time between the generation of the two datasets. Our two primary measures of success for a resampling run are both based on root-mean-square distance to the native structure. Root-mean-square distance (rmsd) is a standard measure of discrepancy between two structures. It is defined as the square root of the mean of the squared distances between pairs of corresponding backbone atoms in the two structures, under the alignment that minimizes this quantity. Our first measure of success is the rmsd between the native structure and the lowest scoring model. This measures Rosetta’s performance if forced to make a single prediction. Our second measure of success is lowest rmsd among the twenty-five top-scoring models. This is a smoother measure of the quality of the lowest scoring Rosetta models, and gives some indication of the prediction quality if more sophisticated minima-selection methods are used than Rosetta energy ranking. Structures at 1 ˚A from the native have atomic-level resolution—this is the goal. Structures at between 2 ˚A and 4 ˚A generally have several important structural details incorrect. In proteins the size of those in our benchmark, structures more than 5 ˚A from the native are poor predictions. Both resampling schemes achieved some success. The performance measures are shown in table 6. The decision-tree scheme performed more consistently and achieved larger improvements on average; it improved the low-energy rmsd in 7 of the 9 benchmark proteins, with a significant median improvement of 1.11 ˚A. Particularly exciting are the atomic-resolution prediction for 1hz6 and the nearly atomic-resolution prediction for 1dtj. In both these cases, plain Rosetta sampling performed considerably worse. The LARS-only scheme was successful as well, providing improved lowest-energy predictions on 6 of the 9 benchmark proteins with a median improvement of 0.23 ˚A. The LARS-only low-energy prediction for 1di2 is atomic-resolution at 0.97 ˚A away from the native structure, as compared to 2.97 ˚A for the control run. In general, improvements correlated with LARS accuracy (Figure 4.b). The two notable exceptions were 2reb, for which plain Rosetta search performs so well that constraints only hurt sampling, and 1n0u, for which plain Rosetta search concentrates almost entirely on a cluster with incorrect topology at 10 ˚A. Certain LARS-selected features, when enforced, switch sampling over to a cluster at around 3 ˚A. Even when incorrect features are enforced within this cluster, sampling is much improved. The cases in which the decision-tree scheme did not yield improved low-energy predictions are interesting in their own right. In the case of 1dcj, resampling does yield lower rmsd structures—the top 25 low rms prediction is superior, and the minimum rmsd from the set is 1.35, nearly atomic 7 resolution, as compared to 1.95 for the control run—but the Rosetta energy function does not pick them out. This suggests that better decoy-selection techniques would improve our algorithms. In the case of 2reb, the unbiased rounds of Rosetta sampling were so successful that they would have been difficult to improve on. This emphasizes the point that resampling cannot hurt us too much. If a plain Rosetta sampling round of n decoys is followed by a resampling round of n decoys, then no matter how poor the resampled decoys are, sampling efficiency is decreased by at most a factor of 2 (since we could have generated n plain Rosetta samples in the same time). The danger is that resampling may overconverge to broad, false energy wells, achieving lower energies in the resampling round even though rmsd is higher. This appears to occur with 2tif, in which the LARS-only low-energy prediction has significantly lower energy than the control prediction despite being much farther from the native. Once more, better decoy-selection techniques might help. 7 Discussion and Conclusions Our results demonstrate that our resampling techniques improve structure prediction on a majority of the proteins in our benchmark set. Our first resampling method significantly improves Rosetta predictions in 3 of the 9 test cases, and marginally improves two or three more. Our second resampling method expands the set of proteins on which we achieve improvements, including an additional atomic-level prediction. It is important to note that significant improvements over Rosetta on any proteins are hard to achieve; if our methods achieved one or two significantly improved predictions, we would count them a success. Rosetta is the state of the art in protein structure prediction, and it has undergone years of incremental advances and optimizations. Surpassing its performance is very difficult. Furthermore, it doesn’t hurt Rosetta too badly if a resampling scheme performs worse than unbiased sampling on some proteins, since models from the unbiased sampling round that precedes the resampling round can be used as predictions as well. There are a number of avenues of future work to pursue. We have designed a number of other structural features, including per-residue secondary structure features, burial features, and side-chain rotamer features, and we hope to incorporate these into our methods. The primary barrier is that each new feature requires a method for constraining it during Rosetta search. We also plan to further investigate the possibility of detecting which LARS predictions are correct using “features of features,” and to apply these methods to discrimate between decision tree leaves as well. It is possible that, with more sampling, the decision tree runs would yield atomic-resolution predictions. However, computational costs for Rosetta are high; each Rosetta model takes approximately fifteen minutes of CPU time to compute on a 1GHz CPU, and each of the 36 data sets represented here consists of on the order of 100, 000 models. The success of our feature selection techniques suggests that the high dimensionality and multiple minima that make high resolution protein structure prediction difficult to solve using traditional methods provide an excellent application for modern machine learning methods. The intersection between the two fields is just beginning, and we are excited to see further developments. References [1] G. E. P. Box and K. B. Wilson. On the experimental attainment of optimum conditions (with discussion). Journal of the Royal Statistical Society Series B, 13(1):1–45, 1951. [2] Justin Boyan and Andrew W. Moore. Learning evaluation functions to improve optimization by local search. The Journal of Machine Learning Research, 1:77–112, 2001. [3] Phil Bradley, Lars Malmstrom, Bin Qian, Jack Schonbrun, Dylan Chivian, David E. Kim, Jens Meiler, Kira M. Misura, and David Baker. Free modeling with Rosetta in CASP6. Proteins, 61(S7):128–134, 2005. [4] Bradley Efron, Trevor Hastie, Iain Johnstone, and Robert Tibshirani. Least angle regression. Annals of Statistics (with discussion), 32(2):407–499, 2004. [5] Wolfgang Kabsch and Chris Sander. Dictionary of protein secondary structure: pattern recognition of hydrogen-bonded and geometrical features. Biopolymers, 22(12):2577–2637, 1983. [6] Robert Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B, 58(1):267–288, 1996. 8
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Selecting Observations against Adversarial Objectives Andreas Krause SCS, CMU H. Brendan McMahan Google, Inc. Carlos Guestrin SCS, CMU Anupam Gupta SCS, CMU Abstract In many applications, one has to actively select among a set of expensive observations before making an informed decision. Often, we want to select observations which perform well when evaluated with an objective function chosen by an adversary. Examples include minimizing the maximum posterior variance in Gaussian Process regression, robust experimental design, and sensor placement for outbreak detection. In this paper, we present the Submodular Saturation algorithm, a simple and efficient algorithm with strong theoretical approximation guarantees for the case where the possible objective functions exhibit submodularity, an intuitive diminishing returns property. Moreover, we prove that better approximation algorithms do not exist unless NP-complete problems admit efficient algorithms. We evaluate our algorithm on several real-world problems. For Gaussian Process regression, our algorithm compares favorably with state-of-the-art heuristics described in the geostatistics literature, while being simpler, faster and providing theoretical guarantees. For robust experimental design, our algorithm performs favorably compared to SDP-based algorithms. 1 Introduction In tasks such as sensor placement for environmental temperature monitoring or experimental design, one has to select among a large set of possible, but expensive, observations. Often, there are several different objective functions which we want to simultaneously optimize. For example, in the environmental monitoring problem, we want to minimize the marginal posterior variance of our temperature estimate at all locations simultaneously. In experimental design, we often have uncertainty about the model parameters, and we want our experiments to be informative no matter what the true parameters of the model are. These problems can be interpreted as a game: We select a set of observations (sensor locations, experiments), and an adversary selects an objective function (location to evaluate predictive variance, model parameters etc.) to test us on. Often, the individual objective functions (e.g., the marginal variance at one location, or the information gain for a fixed set of parameters [1, 2]) satisfy submodularity, an intuitive diminishing returns property: Adding a new observation helps less if we have already made many observations, and more if we have made few observation thus far. While NP-hard, the problem of selecting an optimal set of k observations maximizing a single submodular objective can be approximately solved using a simple greedy forwardselection algorithm, which is guaranteed to perform near-optimally [3]. However, as we show, this simple myopic algorithm performs arbitrarily badly in the case of an adversarially chosen objective. In this paper, we address this problem. In particular: (1) We present SATURATE, an efficient algorithm for settings where an adversarially-chosen submodular objective function must be optimized. Our algorithm guarantees solutions which are at least as informative as the optimal solution, at only a slightly higher cost. (2) We prove that our approximation guarantee is best possible and cannot be improved unless NP-complete problems admit efficient algorithms. (3) We extensively evaluate our algorithm on several real-world tasks, including minimizing the maximum posterior variance in Gaussian Process regression, finding experiment designs which are robust with respect to parameter uncertainty, and sensor placement for outbreak detection. 2 The adversarial observation selection problem Observation selection with a single submodular objective. Observation selection problems can often be modeled using set functions: We have a finite set V of observations to choose from, and 1 a utility function F which assigns a real number F(A) to each A ⊆V, quantifying its informativeness. In many settings, such as the ones described above, the utility F exhibits the property of submodularity: adding an observation helps more, the fewer observations made so far [2]. Formally, F is submodular [3] if, for all A ⊆B ⊆V and s ∈V \ B, it holds that F(A ∪{s}) −F(A) ≥ F(B ∪{s}) −F(B); F is monotonic if for all A ⊆B ⊆V it holds that F(A) ≤F(B), and F is normalized if F(∅) = 0. Hence, many observation selection problems can be formalized as max A⊆V F(A), subject to |A| ≤k, (2.1) where F is normalized, monotonic and submodular, and k is a bound on the number of observations we can make. Since solving the problem (2.1) is generally NP-hard [4], in practice heuristics are often used. One such heuristic is the greedy algorithm. This algorithm starts with the empty set, and iteratively adds the element s∗= argmaxs∈V\A F(A ∪{s}), until k elements have been selected. Perhaps surprisingly, a fundamental result by Nemhauser et. al. [3] states that for submodular functions, the greedy algorithm achieves a constant factor approximation: The set AG obtained by the greedy algorithm achieves at least a constant fraction (1 −1/e) of the objective value obtained by the optimal solution, i.e., F(AG) ≥(1 −1/e) max|A|≤k F(A). Moreover, no polynomial time algorithm can provide a better approximation guarantee unless P = NP [4]. Observation selection with adversarial objectives. In many applications (such as those discussed below), one wants to simultaneously optimize multiple objectives. Here, we are given a collection of monotonic submodular functions F1, . . . , Fm, and we want to solve max A⊆V min i Fi(A), subject to |A| ≤k. (2.2) Problem (2.2) can be considered a game: First, we (the max-player) select a set of observations A, and then our opponent (the min-player) selects a criterion Fi to test us on. Our goal is to select a set A of observations which performs well against an opponent who chooses the worst possible Fi knowing our choice A. Thereby, we try to find a pure equilibrium to a sequential game on a matrix, with one row per A, and one column per Fi. Note, that even if the Fi are all submodular, G(A) = mini Fi(A) is not submodular. In fact, we show below that, in this setting, the simple greedy algorithm (which performs near-optimally in the single-criterion setting) can perform arbitrarily badly. Examples of adversarial observation selection problems. We consider three instances of adversarial selection problems. Sec. 4 provides more details and experimental results for these domains. Several more examples are presented in the longer version of this paper [5]. Minimizing the maximum Kriging variance. Consider a Gaussian Process (GP) [6] XV defined over a finite set of locations (indices) V. Hereby, XV is a set of random variables, one variable Xs for each location s ∈V. Given a set of locations A ⊆V which we observe, we can compute the predictive distribution P(XV\A | XA = xA), i.e., the distribution of the variables XV\A at the unobserved locations V \ A, conditioned on the measurements at the selected locations, XA = xA. Let σ2 s|A be the residual variance after making observations at A. Let ΣAA be the covariance matrix of the measurements at the chosen locations A, and ΣsA be the vector of cross-covariances between the measurements at s and A. Then, the variance σ2 s|A = σ2 s −ΣsAΣ−1 AAΣAs depends only on the set A, and not on the observed values xA. Assume that the a priori variance σ2 s is constant for all locations s (in Sec. 3, we show our approach generalizes to non-constant marginal variances). We want to select locations A such that the maximum marginal variance is as small as possible. Equivalently, we can define the variance reduction Fs(A) = σ2 s −σ2 s|A, and desire that the minimum variance reduction over all locations s is as large as possible. Das and Kempe [1] show that, in many practical cases, the variance reduction Fs is a monotonic submodular function. Robust experimental designs. Another application is experimental design under nonlinear dynamics [7]. The goal is to estimate a set of parameters θ of a nonlinear function y = f(x, θ) + w, by providing a set of experimental stimuli x, and measuring the (noisy) response y. In many cases, experimental design for linear models (where y = A(x)T θ + w) with Gaussian noise w can be efficiently solved [8]. In the nonlinear case, the common approach is to linearize f around an initial parameter estimate θ0, i.e., y = f(x, θ0)+V (x)(θ −θ0)+w, where V (x) is the Jacobian of f with respect to the parameters θ, evaluated at θ0. In [7], it was shown that the efficiency of the design can be very sensitive with respect to the initial parameter estimates θ0. Consequently, they develop an efficient semi-definite program (SDP) for E-optimal design (i.e., the goal is to minimize the maximum eigenvalue of the error covariance) which is robust against perturbations of the Jacobian 2 V . However, it might be more natural to directly consider robustness with respect to perturbation of the initial parameter estimates θ0, around which the linearization is performed. We show how to find (Bayesian A-optimal) designs which are robust against uncertainty in these parameter estimates. In this setting, the objectives Fθ0(A) are the reductions of the trace of the parameter covariance, Fθ0(A) = tr(Σ(θ0) θ )−tr(Σ(θ0) θ|A ), where Σ(θ0) is the joint covariance of observations and parameters after linearization around θ0; thus, Fθ0 is the sum of marginal parameter variance reductions, which are individually monotonic and (often) submodular [1], and so Fθ0 is monotonic and submodular as well. Hence, in order to find a robust design, we maximize the minimum variance reduction, where the minimum is taken over (a discretization into a finite subset of) all initial parameter values θ0. Sensor placement for outbreak detection. Another class of examples are outbreak detection problems on graphs, such as contamination detection in water distribution networks [9]. Here, we are given a graph G = (V, E), and a phenomenon spreading dynamically over the graph. We define a set of intrusion scenarios I; each scenario i ∈I models an outbreak (e.g., spreading of contamination) starting from a given node s ∈V in the network. By placing sensors at a set of locations A ⊆V, we can detect such an outbreak, and incur a utility Fi(A) (e.g., reduction in detection time or population affected). In [9], it was shown that these utilities Fi are monotonic and submodular for a large class of utility functions. In the adversarial setting, the adversary observes our sensor placement A, and then decides on an intrusion i for which our utility Fi(A) is as small as possible. Hence, our goal is to find a placement A which performs well against such an adversarial opponent. Hardness of the adversarial observation selection problem. Given the near-optimal performance of the greedy algorithm for the single-objective problem, a natural question is if the performance guarantee generalizes to the more complex adversarial setting. Unfortunately, this is far from true. Consider the case with two submodular functions, F1 and F2, where the set of observations is V = {s1, s2, t1, t2}. We set F1(∅) = F2(∅) = 0, and define F1(A) = 1 if s1 ∈A, otherwise ε times the number of ti contained in A. Similarly, if s2 ∈A, we set F2(A) = 1, otherwise ε times the number of ti contained in A. Both F1 and F2 are submodular and monotonic. Optimizing for a set of 2 elements, the greedy algorithm maximizing G(A) = min{F1(A), F2(A)} would choose the set {t1, t2}, since such choice increases G by 2ε, whereas adding si would not increase the score. However, the optimal solution with k = 2 is {s1, s2}, with a score of 1. Hence, as ε →0, the greedy algorithm performs arbitrarily worse than the optimal solution. Our next hope would be to obtain a different good approximation algorithm. However, we can show that most likely this is not possible: Theorem 1. Unless P = NP, there cannot exist any polynomial time approximation algorithm for Problem (2.2). More precisely: Let n be the size of the problem instance, and γ(·) > 0 be any positive function of n. If there exists a polynomial-time algorithm which is guaranteed to find a set A′ of size k such that mini Fi(A′) ≥γ(n) max|A|≤k mini Fi(A), then P = NP. Thus, unless P = NP, there cannot exist any algorithm which is guaranteed to provide, e.g., even an exponentially small fraction (γ(n) = 2−n) of the optimal solution. All proofs can be found in [5]. 3 The Submodular Saturation Algorithm Since Theorem 1 rules out any approximation algorithm which respects the constraint k on the size of the set A, our only hope for non-trivial guarantees requires us to relax this constraint. We now present an algorithm that finds a set of observations which perform at least as well as the optimal set, but at slightly increased cost; moreover, we show that no efficient algorithms can provide better guarantees (under reasonable complexity-theoretic assumptions). For now we assume all Fi take only integral values; this assumption is relaxed later. The key idea is to consider the following alternative formulation: max c,A c, subject to c ≤Fi(A) for 1 ≤i ≤m and |A| ≤αk. (3.1) We want a set A of size at most αk, such that Fi(A) ≥c for all i, and c is as large as possible. Here α ≥1 is a parameter relaxing the constraint on |A|: if α = 1, we recover the original problem (2.2). We solve program (3.1) as follows: For each value c, we find the cheapest set A with Fi(A) ≥c for all i. If this cheapest set has at most αk elements, then c is feasible. A binary search on c allows us to find the optimal solution with the maximum feasible c. We first show how to approximately solve Equation (3.1) for a fixed c. For c > 0 define bFi,c(A) = min{Fi(A), c}, the original function Fi truncated at score level c; these bFi,c functions are also submodular [10]. 3 GPC (F c, c) A ←∅; while F c(A) < c do foreach s ∈V \ A do δs ←F c(A ∪{s}) −F c(A); A ←A ∪{argmaxs δs}; Algorithm 1: The greedy submodular partial cover (GPC) algorithm. SATURATE (F1, . . . , Fm, k, α) cmin ←0; cmax ←mini Fi(V); Abest ←∅; while (cmax −cmin) ≥1 m do c ←(cmin + cmax)/2; ∀A define F c(A) ←1 m P i min{Fi(A), c}; A ←GPC(F c, c); if |A| > αk then cmax ←c; else cmin ←c; Abest = A ; Algorithm 2: The Submodular Saturation algorithm. Let F c(A) = 1 m P i bFi,c(A) be their average value; submodular functions are closed under convex combinations, so F c is submodular and monotonic. Furthermore, Fi(A) ≥c for all 1 ≤i ≤m if and only if F c(A) = c. Hence, in order to determine whether some c is feasible, we solve a submodular covering problem: Ac = argminA⊆V |A|, such that F c(A) = c. (3.2) Such problems are NP-hard in general [4], but in [11] it is shown that the greedy algorithm (c.f., Algorithm 1) achieves near-optimal performance on this problem. Using this result, we find: Lemma 2. Given monotonic submodular functions F1, . . . , Fm and a (feasible) constant c, Algorithm 1 (with input F c) finds a set AG such that Fi(AG) ≥c for all i, and |AG| ≤α|A∗|, where A∗is the optimal solution, and α = 1 + log (maxs∈V P i Fi(s)) ≥1 + log m maxs∈V F c(s) 1. We can compute this approximation guarantee α for any given instance of the adversarial observation selection problem. Hence, if for a given value of c the greedy algorithm returns a set of size greater than αk, there cannot exist a solution A′ with |A′| ≤k with Fi(A′) ≥c for all i; thus, the optimal solution to the adversarial observation selection problem must be less than c. We can use this argument to conduct a binary search to find the optimal value of c. We call Algorithm 2, which formalizes this procedure, the submodular saturation algorithm (SATURATE), as the algorithm considers the truncated objectives bFi,c, and chooses sets which saturate all these objectives. Theorem 3 (given below) states that SATURATE is guaranteed to find a set which achieves adversarial score mini Fi at least as high as the optimal solution, if we allow the set to be logarithmically larger than the optimal solution. Theorem 3. For any integer k, SATURATE finds a solution AS such that mini Fi(AS) ≥ max|A|≤k mini Fi(A) and |AS| ≤αk, for α = 1 + log (maxs∈V P i Fi(s)). The total number of submodular function evaluations is O |V|2m log(P i Fi(V)) . Note, that the algorithm still makes sense for any value of α. However, if α < 1 + log (maxs∈V P i Fi(s)), the guarantee of Theorem 3 does not hold. If we had an exact algorithm for submodular coverage, α = 1 would be the correct choice. Since the greedy algorithm solves submodular coverage very effectively, in our experiments, we call SATURATE with α = 1, which empirically performs very well. The worst-case running time guarantee is quite pessimistic, and in practice the algorithm is much faster: Using a priority queue and lazy evaluations, Algorithm 1 can be sped up drastically (c.f., [12] for details). Furthermore, in practical implementations, one would stop GPC once αk + 1 elements have been selected, which already proves that the optimal solution with k elements cannot achieve score c. Also, Algorithm 2 can be terminated once cmax −cmin is sufficiently small; in our experiments, 10-15 iterations usually sufficed. One might ask, whether the guarantee on the size of the set, α, can be improved. Unfortunately, this is not likely, as the following Theorem shows: Theorem 4. If there were a polynomial time algorithm which, for any integer k, is guaranteed to find a solution AS such that mini Fi(AS) ≥max|A|≤k mini Fi(A) and |AS| ≤βk, where β ≤(1 −ε)(1 + log maxs∈V P i Fi(s)) for some fixed ε > 0, then NP ⊆DTIME(nlog log n). 1This bound is only meaningful for integral Fi, otherwise it could be arbitrarily improved by scaling the Fi. 4 Hereby, DTIME(nlog log n) is a class of deterministic, slightly superpolynomial (but subexponential) algorithms [4]; the inclusion NP ⊆DTIME(nlog log n) is considered unlikely [4]. Extensions. We now show how the assumptions made in our presentation above can be relaxed. Non-integral objectives. Most objective functions Fi in the observation selection setting are not integral (e.g., marginal variances of GPs). If they take rational numbers, we can scale the objectives by multiplying by their common denominator. If we allow small additive error, we can approximate their values by their leading digits. An analysis similar to the one presented in [2] can be used to bound the effect of this approximation on the theoretical guarantees obtained by the algorithm. Non-constant thresholds. Consider the example of Minimax Kriging Designs for GP regression. Here, the Fi(A) = σ2 i −σ2 i|A denote the variance reductions at location i. However, rather than guaranteeing that Fi(A) ≥c for all i (which, in this example, means that the minimum variance reduction is c), we want to guarantee that σ2 i|A ≤c for all i. We can easily adapt our approach to handle this case: Instead of defining bFi,c(A) = min{Fi(A), c}, we define bFi,c(A) = min{Fi(A), σ2 i −c}, and then again perform binary search over c, but searching for the smallest c instead. The algorithm, using objectives modified in this way, will bear the same approximation guarantees. Non-uniform observation costs. We can extend SATURATE to the setting where different observations have different costs. Suppose a cost function g : V →R+ assigns each element s ∈V a positive cost g(s); the cost of a set of observations is then g(A) = P s∈A g(s). The problem is to find A∗= maxA⊂V mini Fi(A) subject to g(A) ≤B, where B > 0 is a budget we can spend on making observations. In this case, we use the rule δs ← F c(A ∪{s}) −F c(A) /g(s) in Algorithm 1. For this modified algorithm, Theorem 3 still holds, with |A| replaced by g(A) and k replaced by B. 4 Experimental Results Minimax Kriging. We use SATURATE to select observations in a GP to minimize the maximum posterior variance. We consider Precipitation data from the Pacific Northwest of the United States [13]. We discretize the space into 167 locations. In order to estimate variance reduction, we consider the empirical covariance of 50 years of data, which we preprocessed as described in [2]. In the geostatistics literature, the predominant choice of optimization algorithms are carefully tuned local search procedures, prominently simulated annealing (c.f., [14, 15]). We compare our SATURATE algorithm against a state-of-the-art implementation of such a simulated annealing (SA) algorithm, first proposed by [14]. We use an optimized implementation described recently by [15]. This algorithm has 7 parameters which need to be tuned, describing the annealing schedule, distribution of iterations among several inner loops, etc. We use the parameter settings as reported by [15], and report the best result of the algorithm among 10 random trials. In order to compare observation sets of the same size, we called SATURATE with α = 1. Fig. 1(a) compares simulated annealing, SATURATE, and the greedy algorithm which greedily selects elements which decrease the maximum variance the most. We also used SATURATE to initialize the simulated annealing algorithm (using only a single run of simulated annealing, as opposed to 10 random trials). SATURATE obtains placements which are drastically better than the placements obtained by the greedy algorithm. Furthermore, the performance is very close to the performance of the simulated annealing algorithm. When selecting 30 and more sensors, SATURATE strictly outperforms the simulated annealing algorithm. Furthermore, as Fig. 1(b) shows, SATURATE is significantly faster than simulated annealing, by factors of 5-10 for larger problems. When using SATURATE in order to initialize the simulated annealing algorithm, the resulting performance almost always resulted in the best solutions we were able to find, while still executing faster than simulated annealing with 10 random restarts as proposed by [15]. These results indicate that SATURATE compares favorably to state-of-the-art local search heuristics, while being faster, requiring no parameters to tune, and providing theoretical approximation guarantees. Optimizing for the maximum variance could potentially be considered too pessimistic. Hence we compared placements obtained by SATURATE, minimizing the maximum marginal posterior variance, with placements obtained by the greedy algorithm, where we minimize the average marginal variance. Note, that, whereas the reduction of the maximum variance is non-submodular, the average variance reduction is (often) submodular [1], and hence the greedy algorithm can be expected to provide near-optimal placements. Fig. 1(c) presents the maximum and average marginal variances for both algorithms. Our results show that if we optimize for the maximum variance we still achieve comparable average variance. If we optimize for average variance however, the 5 0 20 40 60 80 100 0.5 1 1.5 2 2.5 Number of sensors Maximum marginal variance Greedy Saturate Simulated Annealing (SA) Saturate + SA (a) Algorithm comparison 0 10 20 30 40 50 60 0 100 200 300 400 500 Number of observations Running time (s) Simulated Annealing (SA) Saturate+SA Saturate Greedy (b) Running time 0 5 10 15 20 0 0.5 1 1.5 2 2.5 3 Number of sensors Marginal variance Max. var. opt. avg. (Greedy) Max. var. opt. max. (Saturate) Avg. var. opt. max. (Saturate) Avg. var. opt. avg. (Greedy) (c) Avg. vs max. variance Figure 1: (a) SATURATE, greedy and SA on the precipitation data. SATURATE performs comparably with the fine-tuned SA algorithm, and outperforms it for larger placements. (b) Running times for the same experiment. (c) Optimizing for the maximum variance (using SATURATE) leads to low average variance, but optimizing for average variance (using greedy) does not lead to low maximum variance. maximum posterior variance remains much higher. In the longer version of this paper [5], we present results on two more real data sets, which are qualitatively similar to those discussed here. Robust Experimental Design. We consider the robust design of experiments for the MichaelisMenten mass-action kinetics model, as discussed in [7]. The goal is least-square parameter estimation for a function y = f(x, θ), where x is the chosen experimental stimulus (the initial substrate concentration S0), and θ = (θ1, θ2) are two parameters as described in [7]. The stimulus x is chosen from a menu of six options, x ∈{1/8, 1, 2, 4, 8, 16}, each of which can be repeatedly chosen. The goal is to produce a fractional design w = (w1, . . . , w6), where each component wi measures the relative frequency according to which the stimulus xi is chosen. Since f is nonlinear, f is linearized around an initial parameter estimate θ0 = (θ01, θ02), and approximated by its Jacobian Vθ0. Classical experimental design considers the error covariance of the least squares estimate ˆθ, Cov(ˆθ | θ0, w) = σ2(V T θ0WVθ0)−1, where W = diag(w), and aims to find designs w which minimize this error covariance. E-optimality, the criterion adopted by [7], measures smallness in terms of the maximum eigenvalue of the error covariance matrix. The optimal w can be found using Semidefinite Programming (SDP) [8]. The estimate Cov(ˆθ | θ0, w) depends on the initial parameter estimate θ0, where linearization is performed. However, since the goal is parameter estimation, a “certain circularity is involved” [7]. To avoid this problem, [7] find a design wρ(θ0) by solving a robust SDP which minimizes the error size, subject to a worst-case (adversarially-chosen) perturbation ∆on the Jacobian Vθ0; the robustness parameter ρ bounds the spectral norm of ∆. As evaluation criterion, [7] define a notion of efficiency, which is the error size of the optimal design with correct initial parameter estimate, divided by the error when using a robust design obtained at the wrong initial parameter estimates, i.e., efficiency ≡λmax[Cov(ˆθ | θtrue, wopt(θtrue)))]/λmax[Cov(ˆθ | θtrue, wρ(θ0))], where wopt(θ) is the E-optimal design for parameter θ. They show that for appropriately chosen values of ρ, the robust design is more efficient than the optimal design, if the initial parameter θ0 does not equal the true parameter. While their results are very promising, an arguably more natural approach than perturbing the Jacobian would be to perturb the initial parameter estimate, around which linearization is performed. E.g., if the function f describes a process, which behaves characteristically differently in different “phases”, and the parameter θ controls which of the phases the process is in, then a robust design should intuitively “hedge” the design against the behavior in each possible phase. In such a case, the uniform distribution (which the robust SDP chooses for large ρ) would not be the most robust design. If we discretize the space of possible parameter perturbations (within a reasonably chosen interval), we can use SATURATE to find robust experimental designs. While the classical E-optimality is not submodular [2], Bayesian A-optimality is (often) submodular [1, 2]. Here, the goal is to minimize the trace instead of eigenvalue size as error metric. Furthermore, we equip the parameters θ with an uninformative normal prior (which we chose as diag([202, 202])), and then minimize the expected trace of the posterior error covariance, tr(Σθ|A). Hereby, A is a discrete design of 20 experiments, where each option xi can be chosen repeatedly. In order to apply SATURATE, for each θ, we define Fθ(A) as the normalized variance reduction Fθ(A) = 1 Zθ (σ2 θ −σ2 θ|A). The normalization Zθ is chosen such that Fθ(A) = 1 if A = argmax|A′|=20 Fθ(A′), i.e., if A is chosen to maximize only Fθ. SATURATE is then used to maximize the worst-case normalized variance reduction. 6 A B C 10 -1 10 0 10 1 0 0.2 0.4 0.6 0.8 1 Initial parameter estimate θ02 Efficiency (w.r.t. E-optimality) Classical E-optimal design true θ2 SDP: ρ = 16.3 ρ = 10-3 Saturate: large interval small interval (a) Robust experimental design 0 5 10 15 20 25 30 0 500 1000 1500 2000 2500 3000 Number of sensors Maximum detection time (minutes) Greedy Simulated Annealing Saturate (b) [W] algorithms Z1 5 10 15 20 25 30 0 0.5 1 1.5 2 x 10 4 Number of sensors Maximum population affected Greedy Saturate Simulated Annealing Saturate + SA (c) [W] algorithms Z2 Figure 2: (a) Efficiency of robust SDP of [7] and SATURATE on a biological experimental design problem. For a large range of initial parameter estimates, SATURATE outperforms the SDP solutions. (b,c) SATURATE, greedy and SA in the water network setting, when optimizing worst-case detection time (Z1) and affected population (Z2). SATURATE performs comparably to SA for Z2 and strictly outperforms SA for Z1. We reproduced the experiment of [7], where the initial estimate of the second component θ02 of θ0 was varied between 0 and 16, the “true” value being θ2 = 2. For each initial estimate of θ02, we computed a robust design, using the SDP approach and using SATURATE, and compared them using the efficiency metric of [7]. We first optimized designs which are robust against a small perturbation of the initial parameter estimate. For the SDP, we chose a robustness parameter ρ = 10−3, as reported in [7]. For SATURATE, we considered an interval around [θ 1 1+ε, θ(1 + ε)], discretized in a 5 × 5 grid, with ε = .1. Fig. 2(a) shows three characteristically different regions, A, B, C, separated by vertical lines. In region B which contains the true parameter setting, the E-optimal design (which is optimal if the true parameter is known, i.e., θ02 = θ2) performs similar to both robust methods. Hence, in region B (i.e., small deviation from the true parameter), robustness is not really necessary. Outside of region B however, where the standard E-optimal design performs badly, both robust designs do not perform well either. This is an intuitive result, as they were optimized to be robust only to small parameter perturbations. Consequently, we compared designs which are robust against a large parameter range. For SDP, we chose ρ = 16.3, which is the maximum spectral variation of the Jacobian when we consider all initial estimates from θ02 varying between 0 and 16. For SATURATE, we optimized a single design which achieves the maximum normalized variance reduction over all values of θ02 between 0 and 16. Fig. 2(a) shows, that in this case, the design obtained by SATURATE achieves an efficiency of 69%, whereas the efficiency of the SDP design is only 52%. In the regions A and C, the SATURATE design strictly outperforms the other robust designs. This experiment indicates that designs which are robust against a large range of initial parameter estimates, as provided by SATURATE, can be more efficient than designs which are robust against perturbations of the Jacobian (the SDP approach). Outbreak Detection. Consider a city water distribution network, delivering water to households via a system of pipes, pumps, and junctions. Accidental or malicious intrusions can cause contaminants to spread over the network, and we want to select a few locations (pipe junctions) to install sensors, in order to detect these contaminations as quickly as possible. In August 2006, the Battle of Water Sensor Networks (BWSN) [16] was organized as an international challenge to find the best sensor placements for a real (but anonymized) metropolitan water distribution network, consisting of 12,527 nodes. In this challenge, a set of intrusion scenarios is specified, and for each scenario a realistic simulator provided by the EPA [17] is used to simulate the spread of the contaminant for a 48 hour period. An intrusion is considered detected when one selected node shows positive contaminant concentration. BWSN considered a variety of impact measures, including the time to detection (called Z1), and the size of the affected population calculated using a realistic disease model (Z2). The goal of BWSN was to minimize the expectation of the impact measures Z1 and Z2 given a uniform distribution over intrusion scenarios. In this paper, we consider the adversarial setting, where an opponent chooses the contamination scenario with knowledge of the sensor locations. The objective functions Z1 and Z2 are in fact submodular for a fixed intrusion scenario [9], and so the adversarial problem of minimizing the impact of the worst possible intrusion fits into our model. For these experiments, we consider scenarios which affect at least 10% of the network, resulting in a total of 3424 scenarios. Figures 2(b) and 2(c) compare the greedy algorithm, SATURATE and the simulated annealing (SA) algorithm for the problem of maximizing the worst-case detection time (Z1) and worst-case affected population (Z2). Interestingly, the behavior is very different for the two objectives. For the affected population (Z2), greedy performs reasonably, and SA sometimes even outperforms SATURATE. For the detection 7 time (Z1), however, the greedy algorithm did not improve the objective at all, and SA performs poorly. The reason is that for Z2, the maximum achievable scores, Fi(V), vary drastically, since some scenarios have much higher impact than others. Hence, there is a strong “gradient”, as the adversarial objective changes quickly when the high impact scenarios are covered. This gradient allows greedy and SA to work well. On the contrary, for Z1, the maximum achievable scores, Fi(V), are constant, since all scenarios have the same simulation duration. Unless all scenarios are detected, the worst-case detection time stays constant at the simulation length. Hence, many node exchange proposals considered by SA, as well as the addition of a new sensor location by greedy, do not change the adversarial objective, and the algorithms have no useful performance metric. Similarly to the GP Kriging setting, our results show that optimizing the worst-case score leads to reasonable performance in the average case score, but not necessarily vice versa. 5 Conclusions In this paper, we considered the problem of selecting observations which are informative with respect to an objective function chosen by an adversary. We demonstrated how this class of problems encompasses the problem of finding designs which minimize the maximum posterior variance in Gaussian Processes regression, robust experimental design, and detecting events spreading over graphs. In each of these settings, the individual objectives are submodular and can be approximated well using, e.g., the greedy algorithm; the adversarial objective, however, is not submodular. We proved that there cannot exist any approximation algorithm for the adversarial problem if the constraint on the observation set size must be exactly met, unless P = NP. Consequently, we presented an efficient approximation algorithm, SATURATE, which finds observation sets which are guaranteed to be least as informative as the optimal solution, and only logarithmically more expensive. In a strong sense, this guarantee is the best possible. We extensively evaluated our algorithm on several real-world problems. For Gaussian Process regression, we showed that SATURATE compares favorably to state-of-the-art heuristics, while being simpler, faster, and providing theoretical guarantees. For robust experimental design, SATURATE performs favorably compared to SDP based approaches. Acknowledgements This work was partially supported by NSF Grants No. CNS-0509383, CNS0625518, CCF-0448095, CCF-0729022, and a gift from Intel. Anupam Gupta and Carlos Guestrin were partly supported by Alfred P. Sloan Fellowships, Carlos Guestrin by an IBM Faculty Fellowship and Andreas Krause by a Microsoft Research Graduate Fellowship. References [1] A. Das and D. Kempe. Algorithms for subset selection in linear regression. In Manuscript, 2007. [2] A. Krause, A. Singh, and C. Guestrin. Near-optimal sensor placements in Gaussian processes: Theory, efficient algorithms and empirical studies. In To appear in the JMLR, 2007. [3] G. Nemhauser, L. Wolsey, and M. Fisher. An analysis of the approximations for maximizing submodular set functions. Mathematical Programming, 14:265–294, 1978. [4] U. Feige. A threshold of ln n for approximating set cover. J. ACM, 45(4), 1998. [5] A. Krause, B. McMahan, C. Guestrin, and A. Gupta. Robust submodular observation selection. Technical report, CMU-ML-08-100, 2008. [6] C. E. Rasmussen and C. K. I. Williams. Gaussian Process for Machine Learning. Adaptive Computation and Machine Learning. MIT Press, 2006. [7] P. Flaherty, M. Jordan, and A. Arkin. Robust design of biological experiments. In NIPS, 2006. [8] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge UP, March 2004. [9] J. Leskovec, A. Krause, C. Guestrin, C. Faloutsos, J. VanBriesen, and N. Glance. Cost-effective outbreak detection in networks. In KDD, 2007. [10] T. Fujito. Approximation algorithms for submodular set cover with applications. TIEICE, 2000. [11] L.A. Wolsey. An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica, 2:385–393, 1982. [12] T. G. Robertazzi and S. C. Schwartz. An accelerated sequential algorithm for producing D-optimal designs. SIAM Journal of Scientific and Statistical Computing, 10(2):341–358, March 1989. [13] M. Widmann and C. S. Bretherton. 50 km resolution daily precipitation for the pacific northwest. http://www.jisao.washington.edu/data sets/widmann/, May 1999. [14] J. Sacks and S. Schiller. Statistical Decision Theory and Related Topics IV, Vol. 2. Springer, 1988. [15] D. P. Wiens. Robustness in spatial studies ii: minimax design. Environmetrics, 16:205–217, 2005. [16] A. Ostfeld, J. G. Uber, and E. Salomons. Battle of water sensor networks: A design challenge for engineers and algorithms. In 8th Symposium on Water Distribution Systems Analysis, 2006. [17] L. A. Rossman. The epanet programmer’s toolkit for analysis of water distribution systems. In Annual Water Resources Planning and Management Conference, 1999. 8
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Spatial Latent Dirichlet Allocation Xiaogang Wang and Eric Grimson Computer Science and Artificial Intelligence Lab Massachusetts Institute of Technology, Cambridge, MA, 02139, USA xgwang@csail.mit.edu, welg@csail.mit.edu Abstract In recent years, the language model Latent Dirichlet Allocation (LDA), which clusters co-occurring words into topics, has been widely applied in the computer vision field. However, many of these applications have difficulty with modeling the spatial and temporal structure among visual words, since LDA assumes that a document is a “bag-of-words”. It is also critical to properly design “words” and “documents” when using a language model to solve vision problems. In this paper, we propose a topic model Spatial Latent Dirichlet Allocation (SLDA), which better encodes spatial structures among visual words that are essential for solving many vision problems. The spatial information is not encoded in the values of visual words but in the design of documents. Instead of knowing the partition of words into documents a priori, the word-document assignment becomes a random hidden variable in SLDA. There is a generative procedure, where knowledge of spatial structure can be flexibly added as a prior, grouping visual words which are close in space into the same document. We use SLDA to discover objects from a collection of images, and show it achieves better performance than LDA. 1 Introduction Latent Dirichlet Allocation (LDA) [1] is a language model which clusters co-occurring words into topics. In recent years, LDA has been widely used to solve computer vision problems. For example, LDA was used to discover objects from a collection of images [2, 3, 4] and to classify images into different scene categories [5]. [6] employed LDA to classify human actions. In visual surveillance, LDA was used to model atomic activities and interactions in a crowded scene [7]. In these applications, LDA clustered low-level visual words (which were image patches, spatial and temporal interest points or moving pixels) into topics with semantic meanings (which corresponded to objects, parts of objects, human actions or atomic activities) utilizing their co-occurrence information. Even with these promising achievements, however, directly borrowing a language model to solve vision problems has some difficulties. First, LDA assumes that a document is a bag of words, such that spatial and temporal structures among visual words, which are meaningless in a language model but important in many computer vision problems, are ignored. Second, users need to define the meaning of “documents” in vision problems. The design of documents often implies some assumptions on vision problems. For example, in order to cluster image patches, which are treated as words, into classes of objects, researchers treated images as documents [2]. This assumes that if two types of patches are from the same object class, they often appear in the same images. This assumption is reasonable, but not strong enough. As an example shown in Figure 1, even though sky is far from vehicles, if they often exist in the same images in some data set, they would be clustered into the same topic by LDA. Furthermore, since in this image most of the patches are sky and building, a patch on a vehicle is likely to be labeled as building or sky as well. These problems could be solved if the document of a patch, such as the yellow patch in Figure 1, only includes other 1 Figure 1: There will be some problems (see text) if the whole image is treated as one document when using LDA to discover classes of objects. patches falling within its neighborhood, marked by the red dashed window in Figure 1, instead of the whole image. So a better assumption is that if two types of image patches are from the same object class, they are not only often in the same images but also close in space. We expect to utilize spatial information in a flexible way when designing documents for solving vision problems. In this paper, we propose a Spatial Latent Dirichlet Allocation (SLDA) model which encodes the spatial structure among visual words. It clusters visual words (e.g. an eye patch and a nose patch), which often occur in the same images and are close in space, into one topic (e.g. face). This is a more proper assumption for solving many vision problems when images often contain several objects. It is also easy for SLDA to model activities and human actions by encoding temporal information. However the spatial or temporal information is not encoded in the values of visual words, but in the design of documents. LDA and its extensions, such as the author-topic model [8], the dynamic topic model [9], and the correlated topic model [10], all assume that the partition of words into documents is known a priori. A key difference of SLDA is that the word-document assignment becomes a hidden random variable. There is a generative procedure to assign words to documents. When visual words are close in space or time, they have a high probability to be grouped into the same document. Some approaches such as [11, 3, 12, 4] could also capture some spatial structures among visual words. [11] assumed that the spatial distribution of an object class could be modeled as Gaussian and the number of objects in the image was known. Both [3] and [4] first roughly segmented images using graph cuts and added spatial constraint using these segments. [12] modeled the spatial dependency among image patches as Markov random fields. As an example application, we use the SLDA model to discover objects from a collection of images. As shown in Figure 2, there are different classes of objects, such as cows, cars, faces, grasses, sky, bicycles, etc., in the image set. And an image usually contains several objects of different classes. The goal is to segment objects from images, and at the same time, to label these segments as different object classes in an unsupervised way. It integrates object segmentation and recognition. In our approach images are divided into local patches. A local descriptor is computed for each image patch and quantized into a visual word. Using topic models, the visual words are clustered into topics which correspond to object classes. Thus an image patch can be labeled as one of the object classes. Our work is related to [2] which used LDA to cluster image patches. As shown in Figure 2, SLDA achieves much better performance than LDA. We will compare more results of LDA and SLDA in the experimental section. 2 Computation of Visual Words To obtain the local descriptors, images are convolved with the filter bank proposed in [13], which is a combination of 3 Gaussians, 4 Laplacian of Gaussians, and 4 first order derivatives of Gaussians, and was shown to have good performance for object categorization. Instead of only computing visual words at interest points as in [2], we divide an image into local patches on a grid and densely sample a local descriptor for each patch. A codebook of size W is created by clustering all the local descriptors in the image set using K-means. Each local patch is quantized into a visual word according to the codebook. In the next step, these visual words (image patches) will be further clustered into classes of objects. We will compare two clustering methods, LDA and SLDA. 2 Figure 2: Given a collection of images as shown in the first row (which are selected from the MSRC image dataset [13]), the goal is to segment images into objects and cluster these objects into different classes. The second row uses manual segmentation and labeling as ground truth. The third row is the LDA result and the fourth row is the SLDA result. Under the same labeling approach, image patches marked in the same color are in one object cluster, but the meaning of colors changes across different labeling methods. 3 LDA When LDA is used to solve our problem, we treat local patches of images as words and the whole image as a document. The graphical model of LDA is shown in Figure 3 (a). There are M documents (images) in the corpus. Each document j has Nj words (image patches). wji is the observed value of word i in document j. All the words in the corpus will be clustered into K topics (classes of objects). Each topic k is modeled as a multinomial distribution over the codebook. and φ are Dirichlet prior hyperparameters. k, j, and zji are hidden variables to be inferred. The generative process of LDA is: 1. For a topic k, a multinomial parameter k is sampled from Dirichlet prior k Dir(φ). 2. For a document j, a multinomial parameter j over the K topics is sampled from Dirichlet prior j Dir( ). 3. For a word i in document j, a topic label zji is sampled from discrete distribution zji Discrete( j). 4. The value wji of word i in document j is sampled from the discrete distribution of topic zji, wji Discrete( zji). zji can be sampled through a Gibbs sampling procedure which integrates out j and k [14]. p(zji = k| z ji,w, ,φ) n(k) ji,wji + φwji W w=1 n(k) ji,w + φw · n(j) ji,k + k K k =1 n(j) ji,k + k (1) where n(k) ji,w is the number of words in the corpus with value w assigned to topic k excluding word i in document j, and n(j) ji,k is the number of words in document j assigned to topic k excluding word i in document j. Eq 1 is the product of two ratios: the probability of word wji under topic k and the probability of topic k in document j. So LDA clusters the visual words often co-occurring in the same images into one object class. As shown by some examples in Figure 2 (see more results in the experimental section), there are two problems in using LDA for object segmentation and recognition. The segmentation result is 3 Figure 3: Graphical model of LDA (a) and SLDA (b). See text for details. noisy since spatial information is not considered. Although LDA assumes that one image contains multiple topics, from experimental results we observe that the patches in the same image are likely to have the same labels. Since the whole image is treated as one document, if one object class, e.g. car in Figure 2, is dominant in the image, the second ratio in Eq 1 will lead to a large bias towards the car class, and thus the patches of street are also likely to be labeled as car. This problem could be solved if a local patch only considers its neighboring patches as being in the same document. 4 SLDA We assume that if visual words are from the same class of objects, they not only often co-occur in the same images but also are close in space. So we try to group image patches which are close in space into the same documents. One straightforward way is to divide the image into regions as shown in Figure 4 (a). Each region is treated as a document instead of the whole image. However, since these regions are not overlapped, some patches, such as A (red patch) and B (cyan patch) in Figure 4 (a), even though very close in space, are assigned to different documents. In Figure 4 (a), patch A on the cow is likely to be labeled as grass, since most other patches in its document are grass. To solve this problem, we may put many overlapped regions, each of which is a document, on the images as shown in Figure 4 (b). If a patch is inside a region, it “could” belong to that document. Any two patches whose distance is smaller than the region size “could” belong to the same document if the regions are placed densely enough. We use the word “could” because each local patch is covered by several regions, so we have to decide to which document it belongs. Different from the LDA model, in which the word-document relationship is known a priori, we need a generative procedure assigning words to documents. If two patches are closer in space, they have a higher probability to be assigned to the same document since there are more regions covering both of them. Actually we can go even further. As shown in Figure 4 (c), each document can be represented by a point (marked by magenta circle) in the image, assuming its region covers the whole image. If an image patch is close to a document, it has a high probability to be assigned to that document. The graphical model is shown in Figure 3 (b). In SLDA, there are M documents and N words in the corpus. A hidden variable di indicates which document word i is assigned to. For each document j there is a hyperparameter cd j = gd j , xd j, yd j known a priori. gd j is the index of the image where document j is placed and xd j, yd j is the location of the document. For a word i, in addition to the observed word value wi, its location (xi, yi) and image index gi are also observed and stored in variable ci = (gi, xi, yi). The generative procedure of SLDA is: 1. For a topic k, a multinomial parameter φk is sampled from Dirichlet prior φk ∼Dir(β). 4 Figure 4: There are several ways to add spatial information among image patches when designing documents. (a): Divide the image into regions without overlapping. Each region, marked by a dashed window, corresponds to a document. Image patches inside the region are assigned to the corresponding document. (b): densely put overlapped regions over images. One image patch is covered by multiple regions. (c): Each document is associated with a point (marked in magenta color). These points are densely placed over the image. If a image patch is close to a document, it has a high probability to be assigned to that document. 2. For a document j, a multinomial parameter j over the K topics is sampled from Dirichlet prior j Dir( ). 3. For a word (image patch) i, a random variable di is sampled from prior p(di| σ) indicating to which document word i is assigned. We choose p(di| σ) as a uniform prior. 4. The image index and location of word i is sampled from distribution p(ci| cd di, ). We may choose this as a Gaussian kernel. p((gi,xi,yi) | gd di,xd di,yd di , ) πgd di (gi) exp xd di xi 2 + yd di yi 2 2 p(ci| cd di, ) = 0 if the word and the document are not in the same image. 5. The topic label zi of word i is sampled from the discrete distribution of document di, zi Discrete( di). 6. The value wi of word i is sampled from the discrete distribution of topic zi, wi Discrete( zi). 4.1 Gibbs Sampling zi and di can be sampled through a Gibbs sampling procedure integrating out k and j. In SLDA the conditional distribution of zi given di is the same as in LDA. p(zi = k| di = j,d i,z i,w, ,φ) n(k) i,wi + φwi W w=1 n(k) i,w + φw · n(j) i,k + k K k =1 n(j) i,k + k (2) where n(k) i,w is the number of words in the corpus with value w assigned to topic k excluding word i, and n(j) i,k is the number of words in document j assigned to topic k excluding word i. This is easy to understand since if the word-document assignment is fixed, SLDA is the same as LDA. In addition, we also need to sample di from the conditional distribution given zi. p di = j| zi = k,z i,d i,ci,{ cd j } , ,φ,σ, p (di = j| σ) p ci| cd j, p (zi = k,z i| di = j,d i, ) p (zi = k,z i| di = j,d i, ) is obtained by integrating out j . p (zi = k,z i| di = j,d i, ) = M j =1 p( j | )p(zj | ji)d j = M j =1 K k =1 k K k =1 ( k ) · K k =1 n(j ) k + k K k =1 n(j ) k + K k =1 k . 5 We choose p (di = j|η) as a uniform prior and p ci|cd j, σ as a Gaussian kernel. Thus the conditional distribution of di is p di = j|zi = k, z−i, d−i, ci, {cd j′}, α, β, η, σ ∝ δgd j (gi) · e−(xd j −xi) 2+(yd j −yi) 2 σ2 · n(j) −i,k + αk PK k′=1 n(j) −i,k′ + αk′ (3) Word i is likely to be assigned to document j if they are in the same image, close in space and word i has the same topic label as other words in document j. In real applications, we only care about the distribution of zi while dj can be marginalized by simply ignoring its samples. From Eq 2 and 3, we observed that a word tends to have the same topic label as other words in its document and words closer in space are more likely to be assigned to the same documents. So essentially under SLDA a word tends to be labeled as the same topic as other words close to it. This satisfies our assumption that visual words from the same object class are closer in space. Since we densely place many documents over one image, during Gibbs sampling some documents are only assigned a few words and the distributions cannot be well estimated. To solve this problem we replicate each image patch to get many particles. These particles have the same word value and location but can be assigned to different documents and have different labels. Thus each document will have enough samples of words to estimate the distributions. 4.2 Discussion SLDA is a flexible model intended to encode spatial structure among image patches and design documents. If there is only one document placed over one image, SLDA simply reduces to LDA. If p(ci|cd j) is an uniform distribution inside a local region, SLDA implements the scheme described in Figure 4 (b). If these local regions are not overlapped, it is the case of Figure 4 (a). There are also other possible ways to add spatial information by choosing different spatial priors p(ci|cd j). In SLDA, the spatial information is used when designing documents. However the object class model φk, simply a multinomial distribution over the codebook, has no spatial structure. So the objects of a class could be in any shape and anywhere in the images, as long as they smoothly distribute in space. By simply adding a time stamp to ci and cd j, it is easy for SLDA to encode temporal structure among visual words. So SLDA also can be applied to human action and activity analysis. 5 Experiments We test LDA and SLDA on the MSRC image dataset [13] with 240 images. Our codebook size is 200 and the topic number is 15. In Figure 2, we show some examples of results using LDA and SLDA. Colors are used indicate different topics. The results of LDA are noisy and within one image most of the patches are labeled as one topic. SLDA achieves much better results than LDA. The results are smoother and objects are well segmented. The detection rate and false alarm rate of four classes, cows, cars, faces, and bicycles are shown in Table 1. They are counted in pixels. We use the manual segmentation and labeling in [13] as ground truth. The two models are also tested on a tiger video sequence with 252 frames. We treat all the frames in the sequence as an image collection and ignore their temporal order. Figure 5 shows their results on two sampled frames. Please see the result of the whole video sequence from our website [15]. Using LDA, usually there are one or two dominant topics distributed like noise in a frame. Topics change as the video background changes. LDA cannot segment out any objects. SLDA clusters image patches into tigers, rock, water, and grass. If we choose the topic of tiger, as shown in the last row of Figure 5, all the tigers in the video can be segmented out. 6 Conclusion We propose a novel Spatial Latent Dirichlet Allocation model which clusters co-occurring and spatially neighboring visual words into the same topic. Instead of knowing word-document assignment a priori, SLDA has a generative procedure partitioning visual words which are close in space into the same documents. It is also easy to extend SLDA to including temporal information. 6 Figure 5: Discovering objects from a video sequence. The first column shows two frames in the video sequence. In the second column, we label the patches in the two frames as different topics using LDA. The thrid column plots the topic labels using SLDA. The red color indicates the topic of tigers. In the fourth column, we segment tigers out by choosing the topic marked in red. Table 1: Detection(D) rate and False Alarm (FA) rate of LDA and SLDA on the MSRC data set cows cars faces bicycles LDA(D) 0.3755 0.5552 0.7172 0.5563 SLDA(D) 0.5662 0.6838 0.6973 0.5661 LDA(FA) 0.5576 0.3963 0.5862 0.5285 SLDA(FA) 0.0334 0.2437 0.3714 0.4217 7 Acknowledgement The authors wish to acknowledge DSO National Laboratory of Singapore for partially supporting this research. References [1] D. M. Blei, A. Y. Ng, and M. I. Jordan. Latent dirichlet allocation. Journal of Machine Learning Research, 3:993–1022, 2003. [2] J. Sivic, B. C. Russell, A. A. Efros, A. Zisserman, and W. T. Freeman. Discovering object categories in image collections. In Proc. ICCV, 2005. [3] B. C. Russell, A. A. Efros, J. Sivic, W. T. Freeman, and A. Zisserman. Using multiple segmentations to discover objects and their extent in image collections. In Proc. CVPR, 2006. [4] L. Cao and L. Fei-Fei. Spatially coherent latent topic model for concurrent object segmentation and classification. In Proc. ICCV, 2007. [5] L. Fei-Fei and P. Perona. A bayesian hierarchical model for learning natural scene categories. In Proc. CVPR, 2005. [6] J. C. Niebles, H. Wang, and L. Fei-Fei. Unsupervised learning of human action categories using spatialtemporal words. In Proc. BMVC, 2006. [7] X. Wang, X. Ma, and E. Grimson. Unsupervised activity perception by hierarchical bayesian models. In Proc. CVPR, 2007. [8] M. Rosen-Zvi, T. Griffiths, M. Steyvers, and P. Smyth. The author-topic model for authors and documents. In Proc. of Uncertainty in Artificial Intelligence, 2004. [9] D. Blei and J. Lafferty. Dynamic topic models. In Proc. ICML, 2006. [10] D. Blei and J. Lafferty. Correlated topic models. In Proc. NIPS, 2006. [11] E. B. Sudderth, A. Torralba, W. T. Freeman, and A. S. Willsky. Learning hierarchical models of scenes, objects, and parts. In Proc. ICCV, 2005. [12] J. Verbeek and B. Triggs. Region classification with markov field aspect models. In Proc. CVPR, 2007. 7 (a) (b) (c) Figure 6: Examples of experimental results on the MSRC image data set. (a): original images; (b): LDA results; (c) SLDA results. [13] J. Winn, A. Criminisi, and T. Minka. Object categorization by learned universal visual dictionary. In Proc. ICCV, 2005. [14] T. Griffiths and M. Steyvers. Finding scientific topics. In Proc. of the National Academy of Sciences, 2004. [15] http://people.csail.mit.edu/xgwang/slda.html. 8
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Learning Visual Attributes Vittorio Ferrari ∗ University of Oxford (UK) Andrew Zisserman University of Oxford (UK) Abstract We present a probabilistic generative model of visual attributes, together with an efficient learning algorithm. Attributes are visual qualities of objects, such as ‘red’, ‘striped’, or ‘spotted’. The model sees attributes as patterns of image segments, repeatedly sharing some characteristic properties. These can be any combination of appearance, shape, or the layout of segments within the pattern. Moreover, attributes with general appearance are taken into account, such as the pattern of alternation of any two colors which is characteristic for stripes. To enable learning from unsegmented training images, the model is learnt discriminatively, by optimizing a likelihood ratio. As demonstrated in the experimental evaluation, our model can learn in a weakly supervised setting and encompasses a broad range of attributes. We show that attributes can be learnt starting from a text query to Google image search, and can then be used to recognize the attribute and determine its spatial extent in novel real-world images. 1 Introduction In recent years, the recognition of object categories has become a major focus of computer vision and has shown substantial progress, partly thanks to the adoption of techniques from machine learning and the development of better probabilistic representations [1, 3]. The goal has been to recognize object categories, such as a ‘car’, ‘cow’ or ‘shirt’. However, an object also has many other qualities apart from its category. A car can be red, a shirt striped, a ball round, and a building tall. These visual attributes are important for understanding object appearance and for describing objects to other people. Figure 1 shows examples of such attributes. Automatic learning and recognition of attributes can complement category-level recognition and therefore improve the degree to which machines perceive visual objects. Attributes also open the door to appealing applications, such as more specific queries in image search engines (e.g. a spotted skirt, rather than just any skirt). Moreover, as different object categories often have attributes in common, modeling them explicitly allows part of the learning task to be shared amongst categories, or allows previously learnt knowledge about an attribute to be transferred to a novel category. This may reduce the total number of training images needed and improve robustness. For example, learning the variability of zebra stripes under non-rigid deformations tells us a lot about the corresponding variability in striped shirts. In this paper we propose a probabilistic generative model of visual attributes, and a procedure for learning its parameters from real-world images. When presented with a novel image, our method infers whether it contains the learnt attribute and determines the region it covers. The proposed model encompasses a broad range of attributes, from simple colors such as ‘red’ or ‘green’ to complex patterns such as ‘striped’ or ‘checked’. Both the appearance and the shape of pattern elements (e.g. a single stripe) are explicitly modeled, along with their layout within the overall pattern (e.g. adjacent stripes are parallel). This enables our model to cover attributes defined by appearance (‘red’), by shape (‘round’), or by both (the black-and-white stripes of zebras). Furthermore, the model takes into account attributes with general appearance, such as stripes which are characterized by a pattern of alternation ABAB of any two colors A and B, rather than by a specific combination of colors. Since appearance, shape, and layout are modeled explictly, the learning algorithm gains an understanding of the nature of the attribute. As another attractive feature, our method can learn in a weakly supervised setting, given images labeled only by the presence or absence of the attribute, ∗This research was supported by the EU project CLASS. The authors thank Dr. Josef Sivic for fruitful discussions and helpful comments on this paper. unary binary red round black/white stripes generic stripes Figure 1: Examples of different kinds of attributes. On the left we show two simple attributes, whose characteristic properties are captured by individual image segments (appearance for red, shape for round). On the right we show more complex attributes, whose basic element is a pair of segments. without indication of the image region it covers. The presence/absence labels can be noisy, as the training method can tolerate a considerable number of mislabeled images. This enables attributes to be learnt directly from a text specification by collecting training images using a web image search engine, such as Google-images, and querying on the name of the attribute. Our approach is inspired by the ideas of Jojic and Caspi [4], where patterns have constant appearance within an image, but are free to change to another appearance in other images. We also follow the generative approach to learning a model from a set of images used by many authors, for example LOCUS [10]. Our parameter learning is discriminative – the benefits of this have been shown before, for example for training the constellation model of [3]. In term of functionality, the closest works to ours are those on the analysis of regular textures [5, 6]. However, they work with textures covering the entire image and focus on finding distinctive appearance descriptors. In constrast, here textures are attributes of objects, and therefore appear in complex images containing many other elements. Very few previous works appeared in this setting [7, 11]. The approach of [7] focuses on colors only, while in [11] attributes are limited to individual regions. Our method encompasses also patterns defined by pairs of regions, allowing to capture more complex attributes. Moreover, we take up the additional challenge of learning the pattern geometry. Before describing the generative model in section 3, in the next section we briefly introduce image segments, the elementary units of measurements observed in the model. 2 Image segments – basic visual representation The basic units in our attribute model are image segments extracted using the algorithm of [2]. Each segment has a uniform appearance, which can be either a color or a simple texture (e.g. sand, grain). Figure 2a shows a few segments from a typical image. Inspired by the success of simple patches as a basis for appearance descriptors [8, 9], we randomly sample a large number of 5 × 5 pixel patches from all training images and cluster them using kmeans [8]. The resulting cluster centers form a codebook of patch types. Every pixel is soft-assigned to the patch types. A segment is then represented as a normalized histogram over the patch types of the pixels it contains. By clustering the segment histograms from the training images we obtain a codebook A of appearances (figure 2b). Each entry in the codebook is a prototype segment descriptor, representing the appearance of a subset of the segments from the training set. Each segment s is then assigned the appearance a ∈A with the smallest Bhattacharya distance to the histogram of s. In addition to appearance, various geometric properties of a segment are measured, summarizing its shape. In our current implementation, these are: curvedness, compactness, elongation (figure 2c), fractal dimension and area relative to the image. We also compute two properties of pairs of segments: relative orientation and relative area (figure 2d). a b d A P A P 2 M m M − θ 1 θ 2 1 θ1 θ2 A A2) m c C C P A1 ( ln A2 Figure 2: Image segments as visual features. a) An image with a few segments overlaid, including two pairs of adjacent segments on a striped region. b) Each row is an entry from the appearance codebook A (i.e. one appearance; only 4 out of 32 are shown). The three most frequent patch types for each appearance are displayed. Two segments from the stripes are assigned to the white and black appearance respectively (arrows). c) Geometric properties of a segment: curvedness, which is the ratio between the number of contour points C with curvature above a threshold and the total perimeter P; compactness; and elongation, which is the ratio between the minor and major moments of inertia. d) Relative geometric properties of a pair of segments: relative area and relative orientation. Notice how these measures are not symmetric (e.g. relative area is the area of the first segment wrt to the second). 3 Generative models for visual attributes Figure 1 shows various kinds of attributes. Simple attributes are entirely characterized by properties of a single segment (unary attributes). Some unary attributes are defined by their appearance, such as colors (e.g. red, green) and basic textures (e.g. sand, grainy). Other unary attributes are defined by a segment shape (e.g. round). All red segments have similar appearance, regardless of shape, while all round segments have similar shape, regardless of appearance. More complex attributes have a basic element composed of two segments (binary attributes). One example is the black/white stripes of a zebra, which are composed of pairs of segments sharing similar appearance and shape across all images. Moreover, the layout of the two segments is characteristic as well: they are adjacent, nearly parallel, and have comparable area. Going yet further, a general stripe pattern can have any appearance (e.g. blue/white stripes, red/yellow stripes). However, the pairs of segments forming a stripe pattern in one particular image must have the same appearance. Hence, a characteristic of general stripes is a pattern of alternation ABABAB. In this case, appearance is common within an image, but not across images. The attribute models we present in this section encompass all aspects discussed above. Essentially, attributes are found as patterns of repeated segments, or pairs of segments, sharing some properties (geometric and/or appearance and/or layout). 3.1 Image likelihood. We start by describing how the model M explains a whole image I. An image I is represented by a set of segments {s}. A latent variable f is associated with each segment, taking the value f = 1 for a foreground segment, and f = 0 for a background segment. Foreground segments are those on the image area covered by the attribute. We collect f for all segments of I into the vector F. An image has a foreground appearance a, shared by all the foreground segments it contains. The likelihood of an image is p(I|M; F, a) = Y x∈I p(x|M; F, a) (1) where x is a pixel, and M are the model parameters. These include α ⊂A, the set of appearances allowed by the model, from which a is taken. The other parameters are used to explain segments and are dicussed below. The probability of pixels is uniform within a segment, and independent across segments: p(x|M; F, a) = p(sx|M; f, a) (2) with sx the segment containing x. Hence, the image likelihood can be expressed as a product over the probability of each segment s, counted by its area Ns (i.e. the number of pixels it contains) p(I|M; F, a) = Y x∈I p(sx|M; f, a) = Y s∈I p(s|M; f, a)Ns (3) s f iS G D λ a α β (a) f i a α s Si D R G c β C λ λ γ 2 1 δ (b) Figure 3: a) Graphical model for unary attributes. D is the number of images in the dataset, Si is the number of segments in image i, and G is the total number of geometric properties considered (both active and inactive). b) Graphical model for binary attributes. c is a pair of segments. Φ1,2 are the geometric distributions for each segment a pair. Ψ are relative geometric distributions (i.e. measure properties between two segments in a pair, such as relative orientation), and there are R of them in total (active and inactive). δ is the adjacency model parameter. It tells whether only adjacent pairs of segments are considered (so p(c|δ = 1) is one only iff c is a pair of adjacent segments). Note that F and a are latent variables associated with a particular image, so there is a different F and a for each image. In contrast, a single model M is used to explain all images. 3.2 Unary attributes Segments are the only observed variables in the unary model. A segment s = (sa, {sj g}) is defined by its appearance sa and shape, captured by a set of geometric measurements {sj g}, such as elongation and curvedness. The graphical model in figure 3a illustrates the conditional probability of image segments p(s|M; f, a) = p(sa|a) · Q j p(sj g|Φj)vj if f = 1 β if f = 0 (4) The likelihood for a segment depends on the model parameters M = (α, β, {λj}), which specify a visual attribute. For each geometric property λj = (Φj, vj), the model defines its distribution Φj over the foreground segments and whether the property is active or not (vj = 1 or 0). Active properties are relevant for the attribute (e.g. elongation is relevant for stripes, while orientation is not) and contribute substantially to its likelihood in (4). Inactive properties instead have no impact on the likelihood (exponentiation by 0). It is the task of the learning stage to determine which properties are active and their foreground distribution. The factor p(sa|a) = [sa = a] is 1 for segments having the foreground appearance a for this image, and 0 otherwise (thus it acts as a selector). The scalar value β represents a simple backgroundmodel: all segments assigned to the background have likelihood β. During inference and learning we want to maximize the likelihood of an image given the model over F, which is achieved by setting f to foreground when the f = 1 case of equation (4) is greater than β. As an example, we give the ideal model parameters for the attribute ‘red’. α contains the red appearance only. β is some low value, corresponding to how likely it is for non-red segments to be assigned the red appearance. No geometric property {λj} is active (i.e. all vj = 0). 3.3 Binary attributes The basic element of binary attributes is a pair of segments. In this section we extend the unary model to describe pairs of segments. In addition to duplicating the unary appearance and geometric properties, the extended model includes pairwise properties which do not apply to individual segments. In the graphical model of figure 3b, these are relative geometric properties γ (area, orientation) and adjacency δ, and together specify the layout of the attribute. For example, the orientation of a segment with respect to the other can capture the parallelism of subsequent stripe segments. Adjacency expresses whether the two segments in the pair are adjacent (like in stripes) or not (like the maple leaf and the stripes in the canadian flag). We consider two segments adjacent if they share part of the boundary. A pattern characterized by adjacent segments is more distinctive, as it is less likely to occur accidentally in a negative image. Segment likelihood. An image is represented by a set of segments {s}, and the set of all possible pairs of segments {c}. The image likelihood p(I|M; F, a) remains as defined in equation (3), but now a = (a1, a2) specifies two foreground appearances, one for each segment in the pair. The likelihood of a segment s is now defined as the maximum over all pairs containing it p(s|M; f, a) = max{c|s∈c} p(c|M, t) if f = 1 β if f = 0 (5) Pair likelihood. The observed variables in our model are segments s and pairs of segments c. A pair c = (s1, s2, {ck r}) is defined by two segments s1, s2 and their relative geometric measurements {ck r} (relative orientation and relative area in our implementation). The likelihood of a pair given the model is p(c|M, a) = p(s1,a, s2,a|a) | {z } appearance · Y j p(sj 1,g|Φj 1)vj 1 · p(sj 2,g|Φj 2)vj 2 | {z } shape · Y k p(ck r|Ψk)vk r · p(c|δ) | {z } layout (6) The binary model parameters M = (α, β, δ, {λj 1}, {λj 2}, {γk}) control the behavior of the pair likelihood. The two sets of λj i = (Φj i, vj i ) are analogous to their counterparts in the unary model, and define the geometric distributions and their associated activation states for each segment in the pair respectively. The layout part of the model captures the interaction between the two segments in the pair. For each relative geometric property γk = (Ψk, vk r ) the model gives its distribution Ψk over pairs of foreground segments and its activation state vk r . The model parameter δ determines whether the pattern is composed of pairs of adjacent segments (δ = 1) or just any pair of segments (δ = 0). The factor p(c|δ) is defined as 0 iff δ = 1 and the segments in c are not adjacent, while it is 1 in all other cases (so, when δ = 1, p(c|δ) acts as a pair selector). The appearance factor p(s1,a, s2,a|a) = [s1,a = a1 ∧s2,a = a2] is 1 when the two segments have the foreground appearances a = (a1, a2) for this image. As an example, the model for a general stripe pattern is as follows. α = (A, A) contains all pairs of appearances from A. The geometric properties λelong 1 , λcurv 1 are active (vj 1 = 1) and their distributions Φj 1 peaked at high elongation and low curvedness. The corresponding properties {λj 2} have similar values. The layout parameters are δ = 1, and γrel area, γrel orient are active and peaked at 0 (expressing that the two segments are parallel and have the same area). Finally, β is a value very close to 0, as the probability of a random segment under this complex model is very low. 4 Learning the model Image Likelihood. The image likelihood defined in (3) depends on the foreground/background labels F and on the foreground appearance a. Computing the complete likelihood, given only the model M, involves maximizing a over the appearances α allowed by the model, and over F: p(I|M) = max a∈α max F p(I|M; F, a) (7) The maximization over F is easily achieved by setting each f to the greater of the two cases in equation (4) (equation (5) for a binary model). The maximization over a requires trying out all allowed appearances α. This is computationally inexpensive, as typically there are about 32 entries in the appearance codebook. Training data. We learn the model parameters in a weakly supervised setting. The training data consists of positive I+ = {Ii +} and negative images I−= {Ii −}. While many of the positive images contain examples of the attribute to be learnt (figure 4), a considerable proportion don’t. Conversely, some of the negative images do contain the attribute. Hence, we must operate under a weak assumption: the attribute occurs more frequently on positive training images than on negative. Moreover, only the (unreliable) image label is given, not the location of the attribute in the image. As demonstrated in section 5, our approach is able to learn from this noisy training data. Although our attribute models are generative, learning them in a discriminative fashion greatly helps given the challenges posed by the weakly supervised setting. For example, in figure 4 most of the overall surface for images labeled ‘red’ is actually white. Hence, a maximum likelihood estimator over the positive training set alone would learn white, not red. A discriminative approach instead positive training images negative training images Figure 4: Advantages of discriminative training. The task is to learn the attribute ‘red’. Although the most frequent color in the positive training images is white, white is also common across the negative set. notices that white occurs frequently also on the negative set, and hence correctly picks up red, as it is most discriminative for the positive set. Formally, the task of learning is to determine the model parameters M that maximize the likelihood ratio p(I+|M) p(I−|M) = Q Ii +∈I+ p(Ii +|M) Q Ii −∈I−p(Ii −|M) (8) Learning procedure. The parameters of the binary model are M = (α, β, δ, {λj 1}, {λj 2}, {γk}), as defined in the previous sections. Since the binary model is a superset of the unary one, we only explain here how to learn the binary case. The procedure for the unary model is derived analogously. In our implementation, α can contain either a single appearance, or all appearances in the codebook A. The former case covers attributes such as colors, or patterns with specific colors (such as zebra stripes). The latter case covers generic patterns, as it allows each image to pick a different appearance a ∈α, while at the same time it properly constrains all segments/pairs within an image to share the same appearance (e.g. subsequent pairs of stripe segments have the same appearance, forming a pattern of alternation ABABAB). Because of this definition, α can take on (1 + |A|)2/2 different values (sets of appearances). As typically a codebook of |A| ≤32 appearances is sufficient to model the data, we can afford exhaustive search over all possible values of α. The same goes for δ, which can only take on two values. Given a fixed α and δ, the learning task reduces to estimating the background probability β, and the geometric properties {λj 1}, {λj 2}, {γk}. To achieve this, we need determine the latent variable F for each training image, as it is necessary for estimating the geometric distributions over the foreground segments. These are in turn necessary for estimating β. Given β and the geometric properties we can estimate F (equation (6)). This particular circular dependence in the structure of our model suggests a relatively simple and computationally cheap approximate optimization algorithm: 1. For each I ∈{I+ S I−}, estimate an initial F and a via equation (7), using an initial β = 0.01, and no geometry (i.e. all activation variables set to 0). 2. Estimate all geometric distributions Φj 1, Φj 2, Ψk over the foreground segments/pairs from all images, according to the initial estimates {F}. 3. Estimate β and the geometric activations v iteratively: (a) Update β as the average probability of segments from I−. This is obtained using the foreground expression of (5) for all segments of I−. (b) Activate the geometric property which most increases the likelihood-ratio (8) (i.e. set the corresponding v to 1). Stop iterating when no property increases (8). 4. The above steps already yield a reasonable estimate of all model parameters. We use it as initialization for the following EM-like iteration, which refines β and Φj 1, Φj 2, Ψk (a) Update {F} given the current β and geometric properties (set each f to maximize (5)) (b) Update Φj 1, Φj 2, Ψk given the current {F}. (c) Update β over I−using the current Φj 1, Φj 2, Ψk. The algorithm is repeated over all possible α and δ, and the model maximizing (8) is selected. Notice how β is continuously re-estimated as more geometric properties are added. This implicitly offers to the selector the probability of an average negative segment under the current model as an up-to-date baseline for comparison. It prevents the model from overspecializing as it pushes it to only pick up properties which distinguish positive segments/pairs from negative ones. Segment 2 <.33 >.67 0 curvedness elongation 0 1 area 1 compactness elongation >.67 <.33 −4 curvedness relative orientation 0 1 1 0 <.33 >.67 Segment 1 Layout (a) (b) (c) 0.4 0 0 relative area 0 4 0 1 <.33 >.67 −4 4 −π/2 0 π/2 Figure 5: a) color models learnt for red, green, blue, and yellow. For each, the three most frequent patch types are displayed. Notice how each model covers different shades of a color. b+c) geometric properties of the learned models for stripes (b) and dots (c). Both models are binary, have general appearance, i.e. α = (A, A), and adjacent segments, i.e. δ = 1. The figure shows the geometric distributions for the activated geometric properties. Lower elongation values indicate more elongated segments. A blank slot means the property is not active for that attribute. See main text for discussion. One last, implicit, parameter is the model complexity: is the attribute unary or binary ? This is tackled through model selection: we learn the best unary and binary models independently, and then select the one with highest likelihood-ratio. The comparison is meaningful because image likelihood is measured in the same way in both unary and binary cases (i.e. as the product over the segment probabilities, equation (3)). 5 Experimental results Learning. We present results on learning four colors (red, green, blue, and yellow) and three patterns (stripes, dots, and checkerboard). The positive training set for a color consists of the 14 images in the first page returned by Google-images when queried by the color name. The proportion of positive images unrelated to the color varies between 21% and 36%, depending on the color (e.g. figure 4). The negative training set for a color contains all positive images for the other colors. Our approach delivers an excellent performance. In all cases, the correct model is returned: unary, no active geometric property, and the correct color as a specific appearance (figure 5a). Stripes are learnt from 74 images collected from Google-images using ‘striped’, ‘stripe’, ‘stripes’ as queries. 20% of them don’t contain stripes. The positive training set for dots contains 35 images, 29% of them without dots, collected from textile vendors websites and Google-images (keywords ‘dots’, ‘dot’, ‘polka dots’). For both attributes, the 56 images for colors act as negative training set. As shown in figure 5, the learnt models capture well the nature of these attributes. Both stripes and dots are learnt as binary and with general appearance, while they differ substantially in their geometric properties. Stripes are learnt as elongated, rather straight pairs of segments, with largely the same properties for the two segments in a pair. Their layout is meaningful as well: adjacent, nearly parallel, and with similar area. In contrast, dots are learnt as small, unelongated, rather curved segments, embedded within a much larger segment. This can be seen in the distribution of the area of the first segment, the dot, relative to the area of the second segment, the ‘background’ on which dots lie. The background segments have a very curved, zigzagging outline, because they circumvent several dots. In contrast to stripes, the two segments that form this dotted pattern are not symmetric in their properties. This characterisic is modeled well by our approach, confirming its flexibility. We also train a model from the first 22 Google-images for the query ‘checkerboard’, 68% of which show a black/white checkerboard. The learnt model is binary, with one segment for a black square and the other for an adjacent white square, demonstrating the learning algorithm correctly infers both models with specific and generic appearance, adapting to the training data. Recognition. Once a model is learnt, it can be used to recognize whether a novel image contains the attribute, by computing the likelihood (7). Moreover, the area covered by the attribute is localized by the segments with f = 1 (figure 6). We report results for red, yellow, stripes, and dots. All test images are downloaded from Yahoo-images, Google-images, and Flickr. There are 45 (red), 39 (yellow), 106 (stripes), 50 (dots) positive test images. In general, the object carrying the attribute stands against a background, and often there are other objects in the image, making the localization task non-trivial. Moreover, the images exhibit extreme variability: there are paintings as well as photographs, stripes appear in any orientation, scale, and appearance, and they are often are deformed Figure 6: Recognition results. Top row: red (left) and yellow (right). Middle rows: stripes. Bottom row: dots. We give a few example test images and the corresponding localizations produced by the learned models. Segments are colored according to their foreground likelihood, using matlab’s jet colormap (from dark blue to green to yellow to red to dark red). Segments deemed not to belong to the attribute are not shown (black). In the case of dots, notice how the pattern is formed by the dots themselves and by the uniform area on which they lie. The ROC plots shows the image classification performance for each attribute. The two lower curves in the stripes plot correspond to a model without layout, and without either layout nor any geometry respectively. Both curves are substantially lower, confirming the usefulness of the layout and shape components of the model. (human body poses, animals, etc.). The same goes for dots, which can vary in thickness, spacing, and so on. Each positive set is coupled with a negative one, in which the attribute doesn’t appear, composed of 50 images from the Caltech-101 ‘Things’ set [12]. Because these negative images are rich in colors, textures and structure, they pose a considerable challenge for the classification task. As can be seen in figure 6, our method achieves accurate localizations of the region covered by the attribute. The behavior on stripe patterns composed of more than two appearances is particularly interesting (the trousers in the rightmost example). The model explains them as disjoint groups of binary stripes, with the two appearances which cover the largest image area. In terms of recognizing whether an image contains the attribute, the method performs very well for red and yellow, with ROC equal-error rates above 90%. Performance is convincing also for stripes and dots, especially since these attributes have generic appearance, and hence must be recognized based only on geometry and layout. In contrast, colors enjoy a very distinctive, specific appearance. References [1] N. Dalal and B. Triggs, Histograms of Oriented Gradients for Human Detection, CVPR, 2005. [2] P. Felzenszwalb and D Huttenlocher, Efficient Graph-Based Image Segmentation, IJCV, (50):2, 2004. [3] R. Fergus, P. Perona, and A. Zisserman, Object Class Recognition by Unsupervised Scale-Invariant Learning, CVPR, 2003. [4] N. Jojic and Y. Caspi, Capturing image structure with probabilistic index maps, CVPR, 2004 [5] S. Lazebnik, C. Schmid, and J. Ponce, A Sparse Texture Representation Using Local Affine Regions, PAMI, (27):8, 2005 [6] Y. Liu, Y. Tsin, and W. Lin, The Promise and Perils of Near-Regular Texture, IJCV, (62):1, 2005 [7] J. Van de Weijer, C. Schmid, and J. Verbeek, Learning Color Names from Real-World Images, CVPR, 2007. [8] M. Varma and A. Zisserman, Texture classification: Are filter banks necessary?, CVPR, 2003. [9] J. Winn, A. Criminisi, and T. Minka, Object Categorization by Learned Universal Visual Dictionary, ICCV, 2005. [10] J. Winn and N. Jojic. LOCUS: Learning Object Classes with Unsupervised Segmentation, ICCV, 2005. [11] K. Yanai and K. Barnard, Image Region Entropy: A Measure of ”Visualness” of Web Images Associated with One Concept, ACM Multimedia, 2005. [12] Caltech 101 dataset: www.vision.caltech.edu/Image Datasets/Caltech101/Caltech101.html
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Collapsed Variational Inference for HDP Yee Whye Teh Gatsby Unit University College London ywteh@gatsby.ucl.ac.uk Kenichi Kurihara Dept. of Computer Science Tokyo Institute of Technology kurihara@mi.cs.titech.ac.jp Max Welling ICS UC Irvine welling@ics.uci.edu Abstract A wide variety of Dirichlet-multinomial ‘topic’ models have found interesting applications in recent years. While Gibbs sampling remains an important method of inference in such models, variational techniques have certain advantages such as easy assessment of convergence, easy optimization without the need to maintain detailed balance, a bound on the marginal likelihood, and side-stepping of issues with topic-identifiability. The most accurate variational technique thus far, namely collapsed variational latent Dirichlet allocation, did not deal with model selection nor did it include inference for hyperparameters. We address both issues by generalizing the technique, obtaining the first variational algorithm to deal with the hierarchical Dirichlet process and to deal with hyperparameters of Dirichlet variables. Experiments show a significant improvement in accuracy. 1 Introduction Many applications of graphical models have traditionally dealt with discrete state spaces, where each variable is multinomial distributed given its parents [1]. Without strong prior knowledge on the structure of dependencies between variables and their parents, the typical Bayesian prior over parameters has been the Dirichlet distribution. This is because the Dirichlet prior is conjugate to the multinomial, leading to simple and efficient computations for both the posterior over parameters and the marginal likelihood of data. When there are latent or unobserved variables, the variational Bayesian approach to posterior estimation, where the latent variables are assumed independent from the parameters, has proven successful [2]. In recent years there has been a proliferation of graphical models composed of a multitude of multinomial and Dirichlet variables interacting in various inventive ways. The major classes include the latent Dirichlet allocation (LDA) [3] and many other topic models inspired by LDA, and the hierarchical Dirichlet process (HDP) [4] and many other nonparametric models based on the Dirichlet process (DP). LDA pioneered the use of Dirichlet distributed latent variables to represent shades of membership to different clusters or topics, while the HDP pioneered the use of nonparametric models to sidestep the need for model selection. For these Dirichlet-multinomial models the inference method of choice is typically collapsed Gibbs sampling, due to its simplicity, speed, and good predictive performance on test sets. However there are drawbacks as well: it is often hard to access convergence of the Markov chains, it is harder still to accurately estimate the marginal probability of the training data or the predictive probability of test data (if latent variables are associated with the test data), averaging topic-dependent quantities based on samples is not well-defined because the topic labels may have switched during sampling and avoiding local optima through large MCMC moves such as split and merge algorithms are tricky to implement due to the need to preserve detailed balance. Thus there seems to be a genuine need to consider alternatives to sampling. For LDA and its cousins, there are alternatives based on variational Bayesian (VB) approximations [3] and on expectation propagation (EP) [5]. [6] found that EP was not efficient enough for large scale applications, while VB suffered from significant bias resulting in worse predictive performance than Gibbs sampling. [7] addressed these issues by proposing an improved VB approximation based on the idea of collapsing, that is, integrating out the parameters while assuming that other latent variables are independent. As for nonparametric models, a number of VB approximations have been proposed for DP mixture models [8, 9], while to our knowledge none has been proposed for the HDP thus far ([10] derived a VB inference for the HDP, but dealt only with point estimates for higher level parameters). In this paper we investigate a new VB approach to inference for the class of Dirichlet-multinomial models. To be concrete we focus our attention on an application of the HDP to topic modeling [4], though the approach is more generally applicable. Our approach is an extension of the collapsed VB approximation for LDA (CV-LDA) presented in [7], and represents the first VB approximation to the HDP1. We call this the collapsed variational HDP (CV-HDP). The advantage of CV-HDP over CV-LDA is that the optimal number of variational components is not finite. This implies, apart from local optima, that we can keep adding components indefinitely while the algorithm will take care removing unnecessary clusters. Ours is also the first variational algorithm to treat full posterior distributions over the hyperparameters of Dirichlet variables, and we show experimentally that this results in significant improvements in both the variational bound and test-set likelihood. We expect our approach to be generally applicable to a wide variety of Dirichlet-multinomial models beyond what we have described here. 2 A Nonparametric Hierarchical Bayesian Topic Model We consider a document model where each document in a corpus is modelled as a mixture over topics, and each topic is a distribution over words in the vocabulary. Let there be D documents in the corpus, and W words in the vocabulary. For each document d = 1, . . . , D, let θd be a vector of mixing proportions over topics. For each topic k, let φk be a vector of probabilities for words in that topic. Words in each document are drawn as follows: first choose a topic k with probability θdk, then choose a word w with probability φkw. Let xid be the ith word token in document d, and zid its chosen topic. We have, zid | θd ∼Mult(θd) xid | zid, φzid ∼Mult(φzid) (1) We place Dirichlet priors on the parameters θd and φk, θd | π ∼Dir(απ) φk | τ ∼Dir(βτ) (2) where π is the corpus-wide distribution over topics, τ is the corpus-wide distribution over the vocabulary, and α and β are concentration parameters describing how close θd and φk are to their respective prior means π and τ. If the number of topics K is finite and fixed, the above model is LDA. As we usually do not know the number of topics a priori, and would like a model that can determine this automatically, we consider a nonparametric extension reposed on the HDP [4]. Specifically, we have a countably infinite number of topics (thus θd and π are infinite-dimensional vectors), and we use a stick-breaking representation [11] for π: πk = ˜πk Qk−1 l=1 (1 −˜πl) ˜πk|γ ∼Beta(1, γ) for k = 1, 2, . . . (3) In the normal Dirichlet process notation, we would equivalently have Gd ∼DP(α, G0) and G0 ∼ DP(γ, Dir(βτ)), where Gd = P∞ k=1 θdkδφk and G0 = P∞ k=1 πkδφk are sums of point masses, and Dir(βτ) is the base distribution. Finally, in addition to the prior over π, we place priors over the other hyperparameters α, β, γ and τ of the model as well, α ∼Gamma(aα, bα) β ∼Gamma(aβ, bβ) γ ∼Gamma(aγ, bγ) τ ∼Dir(aτ) (4) The full model is shown graphically in Figure 1(left). 1In this paper, by HDP we shall mean the two level HDP topic model in Section 2. We do not claim to have derived a VB inference for the general HDP in [4], which is significantly more difficult; see final discussions. topics k=1...∞ document d=1...D words i=1...nd π zid xid θd γ α β τ κ φk topics k=1...∞ document d=1...D words i=1...nd π zid xid β τ α γ ηd sd ξk tk κ Figure 1: Left: The HDP topic model. Right: Factor graph of the model with auxiliary variables. 3 Collapsed Variational Bayesian Inference for HDP There is substantial empirical evidence that marginalizing out variables is helpful for efficient inference. For instance, in [12] it was observed that Gibbs sampling enjoys better mixing, while in [7] it was shown that variational inference is more accurate in this collapsed space. In the following we will build on this experience and propose a collapsed variational inference algorithm for the HDP, based upon first replacing the parameters with auxiliary variables, then effectively collapsing out the auxiliary variables variationally. The algorithm is fully Bayesian in the sense that all parameter posteriors are treated exactly and full posterior distributions are maintained for all hyperparameters. The only assumptions made are independencies among the latent topic variables and hyperparameters, and that there is a finite upper bound on the number of topics used (which is found automatically). The only inputs required of the modeller are the values of the top-level parameters aα, bα, .... 3.1 Replacing parameters with auxiliary variables In order to obtain efficient variational updates, we shall replace the parameters θ = {θd} and φ = {φk} with auxiliary variables. Specifically, we first integrate out the parameters; this gives a joint distribution over latent variables z = {zid} and word tokens x = {xid} as follows: p(z, x|α, β, γ, π, τ) = D Y d=1 Γ(α) Γ(α+nd··) QK k=1 Γ(απk+ndk·) Γ(απk) K Y k=1 Γ(β) Γ(β+n·k·) QW w=1 Γ(βτw+n·kw) Γ(βτw) (5) with ndkw = #{i : xid = w, zid = k}, dot denoting sum over that index, and K denoting an index such that zid ≤K for all i, d. The ratios of gamma functions in (5) result from the normalization constants of the Dirichlet densities of θ and φ, and prove to be nuisances for updating the hyperparameter posteriors. Thus we introduce four sets of auxiliary variables: ηd and ξk taking values in [0, 1], and sdk and tkw taking integral values. This results in a joint probability distribution over an expanded system, p(z, x, η, ξ, s, t|α, β, γ, π, τ) = D Y d=1 ηα−1 d (1−ηd)nd··−1 QK k=1[ndk· sdk ](απk)sdk Γ(nd··) K Y k=1 ξβ−1 k (1−ξk)n·k·−1 QW w=1[n·kw tkw ](βτw)tkw Γ(n·k·) (6) where [ n m] are unsigned Stirling numbers of the first kind, and bold face letters denote sets of the corresponding variables. It can be readily verified that marginalizing out η, ξ, s and t reduces (6) to (5). The main insight is that conditioned on z and x the auxiliary variables are independent and have well-known distributions. Specifically, ηd and ξk are Beta distributed, while sdk (respectively tkw) is the random number of occupied tables in a Chinese restaurant process with ndk· (respectively n·kw) customers and a strength parameter of απk (respectively βτw) [13, 4]. 3.2 The Variational Approximation We assume the following form for the variational posterior over the auxiliary variables system: q(z, η, ξ, s, t, α, β, γ, τ, π) = q(α)q(β)q(γ)q(τ)q(π)q(η, ξ, s, t|z) D Y d=1 nd·· Y i=1 q(zid) (7) where the dependence of auxiliary variables on z is modelled exactly. [7] showed that modelling exactly the dependence of a set of variables on another set is equivalent to integrating out the first set. Thus we can interpret (7) as integrating out the auxiliary variables with respect to z. Given the above factorization, q(π) further factorizes so that the ˜πk’s are independent, as do the posterior over auxiliary variables. For computational tractability, we also truncated our posterior representation to K topics. Specifically, we assumed that q(zid > K) = 0 for every i and d. A consequence is that observations have no effect on ˜πk and φk for all k > K, and these parameters can be exactly marginalized out. Notice that our approach to truncation is different from that in [8], who implemented a truncation at T by instead fixing the posterior for the stick weight q(vT = 1) = 1, and from that in [9], who assumed that the variational posteriors for parameters beyond the truncation level are set at their priors. Our truncation approximation is nested like that in [9], and unlike that in [8]. Our approach is also simpler than that in [9], which requires computing an infinite sum which is intractable in the case of HDPs. We shall treat K as a parameter of the variational approximation, possibly optimized by iteratively splitting or merging topics (though we have not explored these in this paper; see discussion section). As in [9], we reordered the topic labels such that E[n·1·] > E[n·2·] > · · · . An expression for the variational bound on the marginal log-likelihood is given in appendix A. 3.3 Variational Updates In this section we shall derive the complete set of variational updates for the system. In the following E[y] denotes the expectation of y, G[y] = eE[log y] the geometric expectation, and V[y] = E[y2] − E[y]2 the variance. Let Ψ(y) = ∂log Γ(y) ∂y be the digamma function. We shall also employ index summation shorthands: · sums out that index, while >l sums over i where i > l. Hyperparameters. Updates for the hyperparameters are derived using the standard fully factorized variational approach, since they are assumed independent from each other and from other variables. For completeness we list these here, noting that α, β, γ are gamma distributed in the posterior, ˜πk’s are beta distributed, and τ is Dirichlet distributed: q(α) ∝αaα+E[s··]−1e−α(bα−P d E[log ηd]) q(β) ∝βaβ+E[t··]−1e−β(bβ−P k E[log ξk]) q(γ) ∝γaγ+K−1e−γ(bγ−PK k=1 E[log(1−˜πk)] q(˜πk) ∝˜πE[s·k] k (1 −˜πk)E[γ]+E[s·>k]−1 q(τ) ∝QW w=1 τ aτ +E[t·w]−1 w (8) In subsequent updates we will need averages and geometric averages of these quantities which can be extracted using the following identities: p(x) ∝xa−1e−bx ⇒E[x] = a/b, G[x] = eΨ(a)/b, p(x) ∝ Q k xak−1 k ⇒G[xk] = eΨ(ak)/eΨ(P k ak). Note also that the geometric expectations factorizes: G[απk] = G[α]G[πk], G[βτw] = G[β]G[τw] and G[πk] = G[˜πk] Qk−1 l=1 G[1 −˜πl]. Auxiliary variables. The variational posteriors for the auxiliary variables depend on z through the counts ndkw. ηd and ξk are beta distributed. If ndk· = 0 then q(sdk = 0) = 1 otherwise q(sdk) > 0 only if 1 ≤sdk ≤ndk·. Similarly for tkw. The posteriors are: q(ηd|z) ∝ηE[α]−1 d (1 −ηd)nd··−1 q(sdk = m|z) ∝[ndk· m ] (G[απk])m (9) q(ξk|z) ∝ξE[β]−1 k (1 −ξk)n·k·−1 q(tkw = m|z) ∝[n·kw m ] (G[βτw])m To obtain expectations of the auxiliary variables in (8) we will have to average over z as well. For ηd this is E[log ηd] = Ψ(E[α]) −Ψ(E[α] + nd··) where nd·· is the (fixed) number of words in document d. For the other auxiliary variables these expectations depend on counts which can take on many values and a na¨ıve computation can be expensive. We derive computationally tractable approximations based upon an improvement to the second-order approximation in [7]. As we see in the experiments these approximations are very accurate. Consider E[log ξk]. We have, E[log ξk|z] = Ψ(E[β]) −Ψ(E[β] + n·k·) (10) and we need to average over n·k· as well. [7] tackled a similar problem with log instead of Ψ using a second order Taylor expansion to log. Unfortunately such an approximation failed to work in our case as the digamma function Ψ(y) diverges much more quickly than log y at y = 0. Our solution is to treat the case n·k· = 0 exactly, and apply the second-order approximation when n·k· > 0. This leads to the following approximation: E[log ξk] ≈P+[n·k·] Ψ(E[β]) −Ψ(E[β] + E+[n·k·]) −1 2V+[n·k·]Ψ′′(E[β] + E+[n·k·]) (11) where P+ is the “probability of being positive” operator: P+[y] = q(y > 0), and E+[y], V+[y] are the expectation and variance conditional on y > 0. The other two expectations are derived similarly, making use of the fact that sdk and tkw are distributionally equal to the random numbers of tables in Chinese restaurant processes: E[sdk] ≈G[απk]P+[ndk·] Ψ(G[απk]+E+[ndk·])−Ψ(G[απk])+ V+[ndk·]Ψ′′(G[απk]+E+[ndk·]) 2 (12) E[tkw] ≈G[βτw]P+[n·kw] Ψ(G[βτw]+E+[n·kw])−Ψ(G[βτw])+ V+[n·kw]Ψ′′(G[βτw]+E+[n·kw]) 2 As in [7], we can efficiently track the relevant quantities above by noting that each count is a sum of independent Bernoulli variables. Consider ndk· as an example. We keep track of three quantities: E[ndk·] = P i q(zid =k) V[ndk·] = P i q(zid =k)q(zid ̸=k) Z[ndk·] = P i log q(zid ̸=k) (13) Some algebraic manipulations now show that: P+[ndk·] = 1 −eZ[ndk·] E+[ndk·] = E[ndk·] P+[ndk·] V+[ndk·] = V[ndk·] P+[ndk·] −eZ[ndk·]E+[ndk·] (14) Topic assignment variables. [7] showed that if the dependence of a set of variables, say A, on another set of variables, say z, is modelled exactly, then in deriving the updates for z we may equivalently integrate out A. Applying to our situation with A = {η, ξ, s, t}, we obtain updates similar to those in [7], except that the hyperparameters are replaced by either their expectations or their geometric expectations, depending on which is used in the updates for the corresponding auxiliary variables: q(zid = k) ∝G G[απk] + n¬id dk· G G[βτxid] + n¬id ·kxid G E[β] + n¬id ·k· −1 ≈∝ G[απk] + E[n¬id dk· ] G[βτxid] + E[n¬id ·kxid] E[β] + E[n¬id ·k· ] −1 exp − V[n¬id dk· ] 2(G[απk]+E[n¬id dk· ])2 − V[n¬id ·kxid] 2(G[βτxid]+E[n¬id ·kxid])2 + V[n¬id ·k· ] 2(E[β]+E[n¬id ·k· ])2 (15) 4 Experiments We implemented and compared performances for 5 inference algorithms for LDA and HDP: 1) variational LDA (V-LDA) [3], collapsed variational LDA (CV-LDA) [7], collapsed variational HDP (CV-HDP, this paper), collapsed Gibbs sampling for LDA (G-LDA) [12] and the direct assignment Gibbs sampler for HDP (G-HDP) [4]. We report results on the following 3 datasets: i) KOS (W = 6906, D = 3430, number of wordtokens N = 467, 714), ii) a subset of the Reuters dataset consisting of news-topics with a number of documents larger than 300 (W = 4593, D = 8433, N = 566, 298), iii) a subset of the 20Newsgroups dataset consisting of the topics ‘comp.os.ms-windows.misc’, ‘rec.autos’, ‘rec.sport.baseball’, ‘sci.space’ and ‘talk.politics.misc’ (W = 8424, D = 4716, N = 437, 850). For G-HDP we use the released code at http://www.gatsby.ucl.ac.uk/∼ywteh/research/software.html. The variables β, τ are not adapted in that code, so we fixed them at β = 100 and τw = 1/W for all algorithms (see below for discussion regarding adapting these in CV-HDP). G-HDP was initialized with either 1 topic (G-HDP1) or with 100 topics (G-HDP100). For CV-HDP we use the following initialization: E[β] = G[β] = 100 and G[τw] = 1/W (kept fixed to compare with G-HDP), E[α] = aα/bα, G[α] = eΨ(aα)/bα, E[γ] = aγ/bγ, G[πk] = 1/K and q(zij = k) ∝1 + u with u ∼U[0, 1]. We set2 hyperparameters aα, bα, aβ, bβ in the range between [2, 6], while aγ, bγ was chosen in the range [5, 10] and aτ in [30 −50]/W. The number of topics used in CV-HDP was truncated at 40, 80, and 120 topics, corresponding to the number of topics used in the LDA algorithms. Finally, for all LDA algorithms we used α = 0.1, π = 1/K. 2We actually set these values using a fixed but somewhat elaborate scheme which is the reason they ended up different for each dataset. Note that this scheme simply converts prior expectations about the number of topics and amount of sharing into hyperparameter values, and that they were never tweaked. Since they always ended up in these compact ranges and since we do not expect a strong dependence on their values inside these ranges we choose to omit the details. Performance was evaluated by comparing i) the in-sample (train) variational bound on the loglikelihood for all three variational methods and ii) the out-of-sample (test) log-likelihood for all five methods. All inference algorithms were run on 90% of the words in each document while testset performance was evaluated on the remaining 10% of the words. Test-set log-likelihood was computed as follows for the variational methods: p(xtest) = Q ij P k ¯θjk ¯φkxtest ij ¯θjk = απk+Eq[njk·] α+Eq[nj··] ¯φkw = βτw+Eq[n·kw] β+Eq[n·k·] (16) Note that we used estimated mean values of θjk and φkw [14]. For CV-HDP we replaced all hyperparameters by their expectations. For the Gibbs sampling algorithms, given S samples from the posterior, we used: p(xtest) = Q ij 1 S PS s=1 P k θs jkφs kxtest ij θs jk = αsπs k+ns jk· αs+ns j·· φs kw = βτw+ns ·kw β+ns ·k· (17) We used all samples obtained by the Gibbs sampling algorithms after an initial burn-in period; each point in the predictive probabilities plots below is obtained from the samples collected thus far. The results, shown in Figure 2, display a significant improvement in accuracy of CV-HDP over CV-LDA, both in terms of the bound on the training log-likelihood as well as for the test-set loglikelihood. This is caused by the fact that CV-HDP is learning the variational distributions over the hyperparameters. We note that we have not trained β or τ for any of these methods. In fact, initial results for CV-HDP show no additional improvement in test-set log-likelihood, in some cases even a deterioration of the results. A second observation is that convergence of all variational methods is faster than for the sampling methods. Thirdly, we see significant local optima effects in our simulations. For example, G-HDP100 achieves the best results, better than G-HDP1, indicating that pruning topics is a better way than adding topics to escape local optima in these models and leads to better posterior modes. In further experiments we have also found that the variational methods benefit from better initializations due to local optima. In Figure 3 we show results when the variational methods were initialized at the last state obtained by G-HDP100. We see that indeed the variational methods were able to find significantly better local optima in the vicinity of the one found by G-HDP100, and that CV-HDP is still consistently better than the other variational methods. 5 Discussion In this paper we have explored collapsed variational inference for the HDP. Our algorithm is the first to deal with the HDP and with posteriors over the parameters of Dirichlet distributions. We found that the CV-HDP performs significantly better than the CV-LDA on both test-set likelihood and the variational bound. A caveat is that CV-HDP gives slightly worse test-set likelihood than collapsed Gibbs sampling. However, as discussed in the introduction, we believe there are advantages to variational approximations that are not available to sampling methods. A second caveat is that our variational approximation works only for two layer HDPs—a layer of group-specific DPs, and a global DP tying the groups together. It would be interesting to explore variational approximations for more general HDPs. CV-HDP presents an improvement over CV-LDA in two ways. Firstly, we use a more sophisticated variational approximation that can infer posterior distributions over the higher level variables in the model. Secondly, we use a more sophisticated HDP based model with an infinite number of topics, and allow the model to find an appropriate number of topics automatically. These two advances are coupled, because we needed the more sophisticated variational approximation to deal with the HDP. Along the way we have also proposed two useful technical tricks. Firstly, we have a new truncation technique that guarantees nesting. As a result we know that the variational bound on the marginal log-likelihood will reach its highest value (ignoring local optima issues) when K →∞. This fact should facilitate the search over number of topics or clusters, e.g. by splitting and merging topics, an aspect that we have not yet fully explored, and for which we expect to gain significantly from in the face of the observed local optima issues in the experiments. Secondly, we have an improved secondorder approximation that is able to handle the often encountered digamma function accurately. An issue raised by the reviewers and in need of more thought by the community is the need for better evaluation criteria. The standard evaluation criteria in this area of research are the variational bound 40 80 120 !8 !7.8 !7.6 !7.4 !7.2 K 40 80 120 !8.4 !8.2 !8 !7.8 !7.6 K 0 4000 8000 12000 !8 !7.8 !7.6 !7.4 !7.2 #steps 40 80 120 !6.6 !6.4 !6.2 !6 !5.8 K 40 80 120 !7 !6.8 !6.6 !6.4 K 0 4000 8000 12000 !7 !6.8 !6.6 !6.4 !6.2 !6 !5.8 #steps 40 80 120 !7.4 !7.2 !7 !6.8 K 40 80 120 !8.2 !8 !7.8 !7.6 !7.4 K 0 4000 8000 12000 !8 !7.8 !7.6 !7.4 !7.2 !7 !6.8 #steps GHDP100 GHDP1 GLDA CVHDP CVLDA VLDA CVHDP CVLDA VLDA GHDP100 GHDP1 GLDA CVHDP CVLDA VLDA Figure 2: Left column: KOS, Middle column: Reuters and Right column: 20Newsgroups. Top row: log p(xtest) as a function of K, Middle row: log p(xtest) as a function of number of steps (defined as number of iterations multiplied by K) and Bottom row: variational bounds as a function of K. Log probabilities are on a per word basis. Shown are averages and standard errors obtained by repeating the experiments 10 times with random restarts. The distribution over the number of topics found by G-HDP1 are: KOS: K = 113.2 ± 11.4, Reuters: K = 60.4 ± 6.4, 20News: K = 83.5 ± 5.0. For G-HDP100 we have: KOS: K = 168.3 ± 3.9, Reuters: K = 122.2 ± 5.0, 20News: K = 128.1 ± 6.6. 0 5000 #0000 !9 !8&5 !8 !7&5 !7 #ste,s variational bound 0 5000 10000 !7.8 !7.6 !7.4 !7.2 !7 !6.8 !6.6 #steps log p(test) / N GHDP100 Gibbs init. CVHDP Gibbs init. CVLDA Gibbs init. VLDA random init. CVHDP random init. CVLDA random init. VLDA Figure 3: G-HDP100 initialized variational methods (K = 130), compared against variational methods initialized in the usual manner with K = 130 as well. Results were averaged over 10 repeats. and the test-set likelihood. However both confound improvements to the model and improvements to the inference method. An alternative is to compare the computed posteriors over latent variables on toy problems with known true values. However such toy problems are much smaller than real world problems, and inferential quality on such problems may be of limited interest to practitioners. We expect the proliferation of Dirichlet-multinomial models and their many exciting applications to continue. For some applications variational approximations may prove to be the most convenient tool for inference. We believe that the methods presented here are applicable to many models of this general class and we hope to provide general purpose software to support inference in these models in the future. A Variational lower bound E[log p(z,x|α,π,τ)−log q(z)]−KL[q(α)∥p(α)]−KL[q(β)∥p(β)]−PK k=1 KL[q(˜πk)∥p(˜πk)]−KL[q(τ)∥p(τ)] (18) = P d log Γ(E[α]) Γ(E[α]+nd··) +P dk F h log Γ(G[α]G[πk]+ndk·) Γ(G[α]G[πk]) i +P k F h log Γ(E[β]) Γ(E[β]+n·k·) i +P kw F h log Γ(G[β]G[τw]+n·kw) Γ(G[β]G[τw]) i −log (bα−P d E[log ηd])aα+E[s··] baα α Γ(aα) Γ(aα+E[s··]) G[α]E[s··]eE[α] P d E[log ηd]−P dk Pnd i=1 q(zid=k) log q(zid=k) −log (bβ −P k E[log ξk])aβ +E[t··] b aβ β Γ(aβ ) Γ(aβ +E[t··]) G[β]E[t··]eE[β] P k E[log ξk] −P k log Γ(1+γ+E[s·k]+E[s·>k]) γΓ(1+E[s·k])Γ(γ+E[s·>k]) G[˜πk]E[s·k]G[1−˜πk]E[s·>k]−log Γ(κ+E[t··]) Γ(κ) Q w Γ(κτw) Γ(κτw+E[t·w]) G[τw]E[t·w] where F[f(n)]=P+[n](f(E+[n])+ 1 2 V+[n]f ′′(E+[n])) is the improved second order approximation. Acknowledgements We thank the reviewers for thoughtful and constructive comments. MW was supported by NSF grants IIS-0535278 and IIS-0447903. References [1] R. G. Cowell, A. P. Dawid, S. L. Lauritzen, and D. J. Spiegelhalter. Probabilistic Networks and Expert Systems. Springer-Verlag, 1999. [2] M. J. Beal and Z. Ghahramani. Variational Bayesian learning of directed graphical models with hidden variables. Bayesian Analysis, 1(4), 2006. [3] D. M. Blei, A. Y. Ng, and M. I. Jordan. Latent Dirichlet allocation. Journal of Machine Learning Research, 3:993–1022, 2003. [4] 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. [5] T. P. Minka and J. Lafferty. Expectation propagation for the generative aspect model. In Proceedings of the Conference on Uncertainty in Artificial Intelligence, volume 18, 2002. [6] W. Buntine and A. Jakulin. Applying discrete PCA in data analysis. In Proceedings of the Conference on Uncertainty in Artificial Intelligence, volume 20, 2004. [7] Y. W. Teh, D. Newman, and M. Welling. A collapsed variational Bayesian inference algorithm for latent Dirichlet allocation. In Advances in Neural Information Processing Systems, volume 19, 2007. [8] D. M. Blei and M. I. Jordan. Variational inference for Dirichlet process mixtures. Bayesian Analysis, 1(1):121–144, 2006. [9] K. Kurihara, M. Welling, and N. Vlassis. Accelerated variational DP mixture models. In Advances in Neural Information Processing Systems, volume 19, 2007. [10] P. Liang, S. Petrov, M. I. Jordan, and D. Klein. The infinite PCFG using hierarchical Dirichlet processes. In Proceedings of the Conference on Empirical Methods in Natural Language Processing, 2007. [11] J. Sethuraman. A constructive definition of Dirichlet priors. Statistica Sinica, 4:639–650, 1994. [12] T.L. Griffiths and M. Steyvers. A probabilistic approach to semantic representation. In Proceedings of the 24th Annual Conference of the Cognitive Science Society, 2002. [13] C. E. Antoniak. Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Annals of Statistics, 2(6):1152–1174, 1974. [14] M. J. Beal. Variational Algorithms for Approximate Bayesian Inference. PhD thesis, Gatsby Computational Neuroscience Unit, University College London, 2003.
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Progressive mixture rules are deviation suboptimal Jean-Yves Audibert Willow Project - Certis Lab ParisTech, Ecole des Ponts 77455 Marne-la-Vall´ee, France audibert@certis.enpc.fr Abstract We consider the learning task consisting in predicting as well as the best function in a finite reference set G up to the smallest possible additive term. If R(g) denotes the generalization error of a prediction function g, under reasonable assumptions on the loss function (typically satisfied by the least square loss when the output is bounded), it is known that the progressive mixture rule ˆg satisfies ER(ˆg) ≤ming∈G R(g) + Cst log |G| n , (1) where n denotes the size of the training set, and E denotes the expectation w.r.t. the training set distribution.This work shows that, surprisingly, for appropriate reference sets G, the deviation convergence rate of the progressive mixture rule is no better than Cst /√n: it fails to achieve the expected Cst /n. We also provide an algorithm which does not suffer from this drawback, and which is optimal in both deviation and expectation convergence rates. 1 Introduction Why are we concerned by deviations? The efficiency of an algorithm can be summarized by its expected risk, but this does not precise the fluctuations of its risk. In several application fields of learning algorithms, these fluctuations play a key role: in finance for instance, the bigger the losses can be, the more money the bank needs to freeze in order to alleviate these possible losses. In this case, a “good” algorithm is an algorithm having not only low expected risk but also small deviations. Why are we interested in the learning task of doing as well as the best prediction function of a given finite set? First, one way of doing model selection among a finite family of submodels is to cut the training set into two parts, use the first part to learn the best prediction function of each submodel and use the second part to learn a prediction function which performs as well as the best of the prediction functions learned on the first part of the training set. This scheme is very powerful since it leads to theoretical results, which, in most situations, would be very hard to prove without it. Our work here is related to the second step of this scheme. Secondly, assume we want to predict the value of a continuous variable, and that we have many candidates for explaining it. An input point can then be seen as the vector containing the prediction of each candidate. The problem is what to do when the dimensionality d of the input data (equivalently the number of prediction functions) is much higher than the number of training points n. In this setting, one cannot use linear regression and its variants in order to predict as well as the best candidate up to a small additive term. Besides, (penalized) empirical risk minimization is doomed to be suboptimal (see the second part of Theorem 2 and also [1]). As far as the expected risk is concerned, the only known correct way of predicting as well as the best prediction function is to use the progressive mixture rule or its variants. These algorithms are introduced in Section 2 and their main good property is given in Theorem 1. In this work we prove that they do not work well as far as risk deviations are concerned (see the second part of Theorem 1 3). We also provide a new algorithm for this ’predict as well as the best’ problem (see the end of Section 4). 2 The progressive mixture rule and its variants We assume that we observe n pairs of input-output denoted Z1 = (X1, Y1), . . . , Zn = (Xn, Yn) and that each pair has been independently drawn from the same unknown distribution denoted P. The input and output spaces are denoted respectively X and Y, so that P is a probability distribution on the product space Z ≜X × Y. The quality of a (prediction) function g : X →Y is measured by the risk (or generalization error): R(g) = E(X,Y )∼P ℓ[Y, g(X)], where ℓ[Y, g(X)] denotes the loss (possibly infinite) incurred by predicting g(X) when the true output is Y . We work under the following assumptions for the data space and the loss function ℓ: Y × Y →R ∪{+∞}. Main assumptions. The input space is assumed to be infinite: |X| = +∞. The output space is a non-trivial (i.e. infinite) interval of R symmetrical w.r.t. some a ∈R: for any y ∈Y, we have 2a −y ∈Y. The loss function is • uniformly exp-concave: there exists λ > 0 such that for any y ∈Y, the set y′ ∈R : ℓ(y, y′) < +∞ is an interval containing a on which the function y′ 7→e−λℓ(y,y′) is concave. • symmetrical: for any y1, y2 ∈Y, ℓ(y1, y2) = ℓ(2a −y1, 2a −y2), • admissible: for any y, y′ ∈Y∩]a; +∞[, ℓ(y, 2a −y′) > ℓ(y, y′), • well behaved at center: for any y ∈Y∩]a; +∞[, the function ℓy : y′ 7→ℓ(y, y′) is twice continuously differentiable on a neighborhood of a and ℓ′ y(a) < 0. These assumptions imply that • Y has necessarily one of the following form: ] −∞; +∞[, [a −ζ; a + ζ] or ]a −ζ; a + ζ[ for some ζ > 0. • for any y ∈Y, from the exp-concavity assumption, the function ℓy : y′ 7→ℓ(y, y′) is convex on the interval on which it is finite1. As a consequence, the risk R is also a convex function (on the convex set of prediction functions for which it is finite). The assumptions were motivated by the fact that they are satisfied in the following settings: • least square loss with bounded outputs: Y = [ymin; ymax] and ℓ(y1, y2) = (y1−y2)2. Then we have a = (ymin + ymax)/2 and may take λ = 1/[2(ymax −ymin)2]. • entropy loss: Y = [0; 1] and ℓ(y1, y2) = y1 log y1 y2 + (1 −y1) log 1−y1 1−y2 . Note that ℓ(0, 1) = ℓ(1, 0) = +∞. Then we have a = 1/2 and may take λ = 1. • exponential (or AdaBoost) loss: Y = [−ymax; ymax] and ℓ(y1, y2) = e−y1y2. Then we have a = 0 and may take λ = e−y2 max. • logit loss: Y = [−ymax; ymax] and ℓ(y1, y2) = log(1 + e−y1y2). Then we have a = 0 and may take λ = e−y2 max. Progressive indirect mixture rule. Let G be a finite reference set of prediction functions. Under the previous assumptions, the only known algorithms satisfying (1) are the progressive indirect mixture rules defined below. For any i ∈{0, . . . , n}, the cumulative loss suffered by the prediction function g on the first i pairs of input-output is Σi(g) ≜Pi j=1 ℓ[Yj, g(Xj)], 1Indeed, if ξ denotes the function e−λℓy, from Jensen’s inequality, for any probability distribution, Eℓy(Y ) = E −1 λ log ξ(Y ) ≥−1 λ log Eξ(Y ) ≥−1 λ log ξ(EY ) = ℓy(EY ). 2 where by convention we take Σ0 ≡0. Let π denote the uniform distribution on G. We define the probability distribution ˆπi on G as ˆπi ∝e−λΣi · π equivalently for any g ∈G, ˆπi(g) = e−λΣi(g)/(P g′∈G e−λΣi(g′)). This distribution concentrates on functions having low cumulative loss up to time i. For any i ∈{0, . . . , n}, let ˆhi be a prediction function such that ∀(x, y) ∈Z ℓ[y, ˆhi(x)] ≤−1 λ log Eg∼ˆπi e−λℓ[y,g(x)]. (2) The progressive indirect mixture rule produces the prediction function ˆgpim = 1 n+1 Pn i=0 ˆhi. From the uniform exp-concavity assumption and Jensen’s inequality, ˆhi does exist since one may take ˆhi = Eg∼ˆπi g. This particular choice leads to the progressive mixture rule, for which the predicted output for any x ∈X is ˆgpm(x) = P g∈G 1 n+1 Pn i=0 e−λΣi(g) P g′∈G e−λΣi(g′) g(x). Consequently, any result that holds for any progressive indirect mixture rule in particular holds for the progressive mixture rule. The idea of a progressive mean of estimators has been introduced by Barron ([2]) in the context of density estimation with Kullback-Leibler loss. The form ˆgpm is due to Catoni ([3]). It was also independently proposed in [4]. The study of this procedure was made in density estimation and least square regression in [5, 6, 7, 8]. Results for general losses can be found in [9, 10]. Finally, the progressive indirect mixture rule is inspired by the work of Vovk, Haussler, Kivinen and Warmuth [11, 12, 13] on sequential prediction and was studied in the “batch” setting in [10]. Finally, in the upper bounds we state, e.g. Inequality (1), one should notice that there is no constant larger than 1 in front of ming∈G R(g), as opposed to some existing upper bounds (e.g. [14]). This work really studies the behaviour of the excess risk, that is the random variable R(ˆg) −ming∈G R(g). The largest integer smaller or equal to the logarithm in base 2 of x is denoted by ⌊log2 x⌋. 3 Expectation convergence rate The following theorem, whose proof is omitted, shows that the expectation convergence rate of any progressive indirect mixture rule is (i) at least (log |G|)/n and (ii) cannot be uniformly improved, even when we consider only probability distributions on Z for which the output has almost surely two symmetrical values (e.g. {-1;+1} classication with exponential or logit losses). Theorem 1 Any progressive indirect mixture rule satisfies ER(ˆgpim) ≤min g∈G R(g) + log |G| λ(n+1). Let y1 ∈Y −{a} and d be a positive integer. There exists a set G of d prediction functions such that: for any learning algorithm, there exists a probability distribution generating the data for which • the output marginal is supported by 2a −y1 and y1: P(Y ∈{2a −y1; y1}) = 1, • ER(ˆg) ≥min g∈G R(g) + e−1κ 1 ∧⌊log2 |G|⌋ n+1 , with κ ≜sup y∈Y [ℓ(y1, a) −ℓ(y1, y)] > 0. The second part of Theorem 1 has the same (log |G|)/n rate as the lower bounds obtained in sequential prediction ([12]). From the link between sequential predictions and our “batch” setting with i.i.d. data (see e.g. [10, Lemma 3]), upper bounds for sequential prediction lead to upper bounds for i.i.d. data, and lower bounds for i.i.d. data leads to lower bounds for sequential prediction. The converse of this last assertion is not true, so that the second part of Theorem 1 is not a consequence of the lower bounds of [12]. 3 The following theorem, whose proof is also omitted, shows that for appropriate set G: (i) the empirical risk minimizer has a p (log |G|)/n expectation convergence rate, and (ii) any empirical risk minimizer and any of its penalized variants are really poor algorithms in our learning task since their expectation convergence rate cannot be faster than p (log |G|)/n (see [5, p.14] and [1] for results of the same spirit). This last point explains the interest we have in progressive mixture rules. Theorem 2 If B ≜supy,y′,y′′∈Y[ℓ(y, y′) −ℓ(y, y′′)] < +∞, then any empirical risk minimizer, which produces a prediction function ˆgerm in argming∈G Σn, satisfies: ER(ˆgerm) ≤min g∈G R(g) + B q 2 log |G| n . Let y1, ˜y1 ∈Y∩]a; +∞[ and d be a positive integer. There exists a set G of d prediction functions such that: for any learning algorithm producing a prediction function in G (e.g. ˆgerm) there exists a probability distribution generating the data for which • the output marginal is supported by 2a −y1 and y1: P(Y ∈{2a −y1; y1}) = 1, • ER(ˆg) ≥min g∈G R(g) + δ 8 q ⌊log2 |G|⌋ n ∧2 , with δ ≜ℓ(y1, 2a −˜y1) −ℓ(y1, ˜y1) > 0. The lower bound of Theorem 2 also says that one should not use cross-validation. This holds for the loss functions considered in this work, and not for, e.g., the classification loss: ℓ(y, y′) = 1y̸=y′. 4 Deviation convergence rate The following theorem shows that the deviation convergence rate of any progressive indirect mixture rule is (i) at least 1/√n and (ii) cannot be uniformly improved, even when we consider only probability distributions on Z for which the output has almost surely two symmetrical values (e.g. {-1;+1} classication with exponential or logit losses). Theorem 3 If B ≜supy,y′,y′′∈Y[ℓ(y, y′) −ℓ(y, y′′)] < +∞, then any progressive indirect mixture rule satisfies: for any ϵ > 0, with probability at least 1 −ϵ w.r.t. the training set distribution, we have R(ˆgpim) ≤min g∈G R(g) + B q 2 log(2ϵ−1) n+1 + log |G| λ(n+1) Let y1 and ˜y1 in Y∩]a; +∞[ such that ℓy1 is twice continuously differentiable on [a; ˜y1] and ℓ′ y1( ˜y1) ≤0 and ℓ′′ y1( ˜y1) > 0. Consider the prediction functions g1 ≡˜y1 and g2 ≡2a −˜y1. For any training set size n large enough, there exist ϵ > 0 and a distribution generating the data such that • the output marginal is supported by y1 and 2a −y1 • with probability larger than ϵ, we have R(ˆgpim) − min g∈{g1,g2} R(g) ≥ c q log(eϵ−1) n where c is a positive constant depending only on the loss function, the symmetry parameter a and the output values y1 and ˜y1. Proof 1 See Section 5. This result is quite surprising since it gives an example of an algorithm which is optimal in terms of expectation convergence rate and for which the deviation convergence rate is (significantly) worse than the expectation convergence rate. In fact, despite their popularity based on their unique expectation convergence rate, the progressive mixture rules are not good algorithms since a long argument essentially based on convexity shows that the following algorithm has both expectation and deviation convergence rate of order 1/n. Let 4 ˆgerm be the minimizer of the empirical risk among functions in G. Let ˜g be the minimizer of the empirical risk in the star ˆG = ∪g∈G [g; ˆgerm]. The algorithm producing ˜g satisfies for some C > 0, for any ϵ > 0, with probability at least 1 −ϵ w.r.t. the training set distribution, we have R(˜g) ≤min g∈G R(g) + C log(ϵ−1|G|) n . This algorithm has also the benefit of being parameter-free. On the contrary, in practice, one will have recourse to cross-validation to tune the parameter λ of the progressive mixture rule. To summarize, to predict as well as the best prediction function in a given set G, one should not restrain the algorithm to produce its prediction function among the set G. The progressive mixture rules satisfy this principle since they produce a prediction function in the convex hull of G. This allows to achieve (log |G|)/n convergence rates in expectation. The proof of the lower bound of Theorem 3 shows that the progressive mixtures overfit the data: the deviations of their excess risk are not PAC bounded by C log(ϵ−1|G|)/n while an appropriate algorithm producing prediction functions on the edges of the convex hull achieves the log(ϵ−1|G|)/n deviation convergence rate. Future work might look at whether one can transpose this algorithm to the sequential prediction setting, in which, up to now, the algorithms to predict as well as the best expert were dominated by algorithms producing a mixture expert inside the convex hull of the set of experts. 5 Proof of Theorem 3 5.1 Proof of the upper bound Let Zn+1 = (Xn+1, Yn+1) be an input-output pair independent from the training set Z1, . . . , Zn and with the same distribution P. From the convexity of y′ 7→ℓ(y, y′), we have R(ˆgpim) ≤ 1 n+1 Pn i=0 R(ˆhi). (3) Now from [15, Theorem 1] (see also [16, Proposition 1]), for any ϵ > 0, with probability at least 1 −ϵ, we have 1 n+1 Pn i=0 R(ˆhi) ≤ 1 n+1 Pn i=0 ℓ Yi+1, ˆh(Xi+1) + B q log(ϵ−1) 2(n+1) (4) Using [12, Theorem 3.8] and the exp-concavity assumption, we have Pn i=0 ℓ Yi+1, ˆh(Xi+1) ≤min g∈G Pn i=0 ℓ Yi+1, g(Xi+1) + log |G| λ (5) Let ˜g ∈argminG R. By Hoeffding’s inequality, with probability at least 1 −ϵ, we have 1 n+1 Pn i=0 ℓ Yi+1, ˜g(Xi+1) ≤R(˜g) + B q log(ϵ−1) 2(n+1) (6) Merging (3), (4), (5) and (6), with probability at least 1 −2ϵ, we get R(ˆgpim) ≤ 1 n+1 Pn i=0 ℓ Yi+1, ˜g(Xi+1) + log |G| λ(n+1) + B q log(ϵ−1) 2(n+1) ≤ R(˜g) + B q 2 log(ϵ−1) n+1 + log |G| λ(n+1). 5.2 Sketch of the proof of the lower bound We cannot use standard tools like Assouad’s argument (see e.g. [17, Theorem 14.6]) because if it were possible, it would mean that the lower bound would hold for any algorithm and in particular for ˜g, and this is false. To prove that any progressive indirect mixture rule have no fast exponential deviation inequalities, we will show that on some event with not too small probability, for most of the i in {0, . . . , n}, π−λΣi concentrates on the wrong function. The proof is organized as follows. First we define the probability distribution for which we will prove that the progressive indirect mixture rules cannot have fast deviation convergence rates. Then we define the event on which the progressive indirect mixture rules do not perform well. We lower bound the probability of this excursion event. Finally we conclude by lower bounding R(ˆgpim) on the excursion event. Before starting the proof, note that from the “well behaved at center” and exp-concavity assumptions, for any y ∈Y∩]a; +∞[, on a neighborhood of a, we have: ℓ′′ y ≥λ(ℓ′ y)2 and since ℓ′ y(a) < 0, y1 and ˜y1 exist. Due to limited space, some technical computations have been removed. 5 5.2.1 Probability distribution generating the data and first consequences. Let γ ∈]0; 1] be a parameter to be tuned later. We consider a distribution generating the data such that the output distribution satisfies for any x ∈X P(Y = y1|X = x) = (1 + γ)/2 = 1 −P(Y = y2|X = x), where y2 = 2a −y1. Let ˜y2 = 2a −˜y1. From the symmetry and admissibility assumptions, we have ℓ(y2, ˜y2) = ℓ(y1, ˜y1) < ℓ(y1, ˜y2) = ℓ(y2, ˜y1). Introduce δ ≜ℓ(y1, ˜y2) −ℓ(y1, ˜y1) > 0. (7) We have R(g2) −R(g1) = 1+γ 2 [ℓ(y1, ˜y2) −ℓ(y1, ˜y1)] + 1−γ 2 [ℓ(y2, ˜y2) −ℓ(y2, ˜y1)] = γδ. (8) Therefore g1 is the best prediction function in {g1, g2} for the distribution we have chosen. Introduce Wj ≜1Yj=y1 −1Yj=y2 and Si ≜Pi j=1 Wj. For any i ∈{1, . . . , n}, we have Σi(g2) −Σi(g1) = Pi j=1[ℓ(Yj, ˜y2) −ℓ(Yj, ˜y1)] = Pi j=1 Wjδ = δ Si The weight given by the Gibbs distribution π−λΣi to the function g1 is π−λΣi(g1) = e−λΣi(g1) e−λΣi(g1)+e−λΣi(g2) = 1 1+eλ[Σi(g1)−Σi(g2)] = 1 1+e−λδSi . (9) 5.2.2 An excursion event on which the progressive indirect mixture rules will not perform well. Equality (9) leads us to consider the event: Eτ = ∀i ∈{τ, . . . , n}, Si ≤−τ , with τ the smallest integer larger than (log n)/(λδ) such that n −τ is even (for convenience). We have log n λδ ≤τ ≤log n λδ + 2. (10) The event Eτ can be seen as an excursion event of the random walk defined through the random variables Wj = 1Yj=y1 −1Yj=y2, j ∈{1, . . . , n}, which are equal to +1 with probability (1+γ)/2 and −1 with probability (1 −γ)/2. From (9), on the event Eτ, for any i ∈{τ, . . . , n}, we have π−λΣi(g1) ≤ 1 n+1. (11) This means that π−λΣi concentrates on the wrong function, i.e. the function g2 having larger risk (see (8)). 5.2.3 Lower bound of the probability of the excursion event. This requires to look at the probability that a slightly shifted random walk in the integer space has a very long excursion above a certain threshold. To lower bound this probability, we will first look at the non-shifted random walk. Then we will see that for small enough shift parameter, probabilities of shifted random walk events are close to the ones associated to the non-shifted random walk. Let N be a positive integer. Let σ1, . . . , σN be N independent Rademacher variables: P(σi = +1) = P(σi = −1) = 1/2. Let si ≜Pi j=1 σi be the sum of the first i Rademacher variables. We start with the following lemma for sums of Rademacher variables (proof omitted). Lemma 1 Let m and t be positive integers. We have P max 1≤k≤N sk ≥t; sN ̸= t; sN −t ≤m = 2P t < sN ≤t + m (12) Let σ′ 1, . . . , σ′ N be N independent shifted Rademacher variables to the extent that P(σ′ i = +1) = (1 + γ)/2 = 1 −P(σ′ i = −1). These random variables satisfy the following key lemma (proof omitted) 6 Lemma 2 For any set A ⊂ (ϵ1, . . . , ϵN) ∈{−1, 1}n : PN i=1 ϵi ≤M where M is a positive integer, we have P (σ′ 1, . . . , σ′ N) ∈A ≥ 1−γ 1+γ M/2 1 −γ2N/2P (σ1, . . . , σN) ∈A (13) We may now lower bound the probability of the excursion event Eτ. Let M be an integer larger than τ. We still use Wj ≜1Yj=y1 −1Yj=y2 for j ∈{1, . . . , n}. By using Lemma 2 with N = n −2τ, we obtain P(Eτ) ≥ P W1 = −1, . . . , W2τ = −1; ∀2τ < i ≤n, Pi j=2τ+1 Wj ≤τ = 1−γ 2 2τP ∀i ∈{1, . . . , N} Pi j=1 σ′ j ≤τ ≥ 1−γ 2 2τ 1−γ 1+γ M/2 1 −γ2 N 2 P |sN| ≤M; ∀i ∈{1, . . . , N} si ≤τ By using Lemma 1, since τ ≤M, the r.h.s. probability can be lower bounded, and after some computations, we obtain P(Eτ) ≥ τ 1−γ 2 2τ 1−γ 1+γ M/2 1 −γ2 N 2 [P(sN = τ) −P(sN = M)] (14) where we recall that τ have the order of log n, N = n −2τ has the order of n and that γ > 0 and M ≥τ have to be appropriately chosen. To control the probabilities of the r.h.s., we use Stirling’s formula nne−n√ 2πn e1/(12n+1) < n! < nne−n√ 2πn e1/(12n), (15) and get for any s ∈[0; N] such that N −s even, P(sN = s) ≥ q 2 πN 1 −s2 N2 −N 2 1−s N 1+ s N s 2 e− 1 6(N+s) − 1 6(N−s) (16) and similarly P(sN = s) ≤ q 2 πN 1 −s2 N2 −N 2 1−s N 1+ s N s 2 e 1 12N+1 . (17) These computations and (14) leads us to take M as the smallest integer larger than √n such that n −M is even. Indeed, from (10), (16) and (17), we obtain limn→+∞ √n[P(sN = τ) −P(sN = M)] = c, where c = p 2/π 1 −e−1/2 > 0. Therefore for n large enough we have P(Eτ) ≥ cτ 2√n 1−γ 2 2τ 1−γ 1+γ M/2 1 −γ2 N 2 (18) The last two terms of the r.h.s. of (18) leads us to take γ of order 1/√n up to possibly a logarithmic term. We obtain the following lower bound on the excursion probability Lemma 3 If γ = p C0(log n)/n with C0 a positive constant, then for any large enough n, P(Eτ) ≥ 1 nC0 . 5.2.4 Behavior of the progressive indirect mixture rule on the excursion event. From now on, we work on the event Eτ. We have ˆgpim = (Pn i=0 ˆhi)/(n + 1). We still use δ ≜ ℓ(y1, ˜y2)−ℓ(y1, ˜y1) = ℓ(y2, ˜y1)−ℓ(y2, ˜y2). On the event Eτ, for any x ∈X and any i ∈{τ, . . . , n}, by definition of ˆhi, we have ℓ[y2, ˆhi(x)] −ℓ(y2, ˜y2) ≤ −1 λ log Eg∼π−λΣi e−λ{ℓ[y2,g(x)]−ℓ(y2, ˜ y2)} = −1 λ log e−λδ + (1 −e−λδ)π−λΣi(g2) ≤ −1 λ log 1 −(1 −e−λδ) 1 n+1 In particular, for any n large enough, we have ℓ[y2, ˆhi(x)] −ℓ(y2, ˜y2) ≤Cn−1, with C > 0 independent from γ. From the convexity of the function y 7→ℓ(y2, y) and by Jensen’s inequality, we obtain ℓ[y2, ˆgpim(x)] −ℓ(y2, ˜y2) ≤ 1 n+1 Pn i=0 ℓ[y2, ˆhi(x)] −ℓ(y2, ˜y2) ≤ τδ n+1 + Cn−1 < C1 log n n 7 for some constant C1 > 0 independent from γ. Let us now prove that for n large enough, we have ˜y2 ≤ˆgpim(x) ≤˜y2 + C q log n n ≤˜y1, (19) with C > 0 independent from γ. From (19), we obtain R(ˆgpim) −R(g1) = 1+γ 2 ℓ(y1, ˆgpim) −ℓ(y1, ˜y1) + 1−γ 2 ℓ(y2, ˆgpim) −ℓ(y2, ˜y1) = 1+γ 2 ℓy1(ˆgpim) −ℓy1( ˜y1) + 1−γ 2 ℓy1(2a −ˆgpim) −ℓy1( ˜y2) = 1+γ 2 δ + ℓy1(ˆgpim) −ℓy1( ˜y2) + 1−γ 2 −δ + ℓy1(2a −ˆgpim) −ℓy1( ˜y1) ≥ γδ −(ˆgpim −˜y2)|ℓ′ y1( ˜y2)| ≥ γδ −C2 q log n n , (20) with C2 independent from γ. We may take γ = 2C2 δ p (log n)/n and obtain: for n large enough, on the event Eτ, we have R(ˆgpim) −R(g1) ≥C p log n/n. From Lemma 3, this inequality holds with probability at least 1/nC4 for some C4 > 0. To conclude, for any n large enough, there exists ϵ > 0 s.t. with probability at least ϵ, R(ˆgpim) −R(g1) ≥c q log(eϵ−1) n . where c is a positive constant depending only on the loss function, the symmetry parameter a and the output values y1 and ˜y1. References [1] G. Lecu´e. Suboptimality of penalized empirical risk minimization in classification. In Proceedings of the 20th annual conference on Computational Learning Theory, 2007. [2] A. Barron. 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Experience-Guided Search: A Theory of Attentional Control Michael C. Mozer Department of Computer Science and Institute of Cognitive Science University of Colorado mozer@colorado.edu David Baldwin Department of Computer Science Indiana University Bloomington, IN 47405 baldwind@indiana.edu Abstract People perform a remarkable range of tasks that require search of the visual environment for a target item among distractors. The Guided Search model (Wolfe, 1994, 2007), or GS, is perhaps the best developed psychological account of human visual search. To prioritize search, GS assigns saliency to locations in the visual field. Saliency is a linear combination of activations from retinotopic maps representing primitive visual features. GS includes heuristics for setting the gain coefficient associated with each map. Variants of GS have formalized the notion of optimization as a principle of attentional control (e.g., Baldwin & Mozer, 2006; Cave, 1999; Navalpakkam & Itti, 2006; Rao et al., 2002), but every GS-like model must be ’dumbed down’ to match human data, e.g., by corrupting the saliency map with noise and by imposing arbitrary restrictions on gain modulation. We propose a principled probabilistic formulation of GS, called Experience-Guided Search (EGS), based on a generative model of the environment that makes three claims: (1) Feature detectors produce Poisson spike trains whose rates are conditioned on feature type and whether the feature belongs to a target or distractor; (2) the environment and/or task is nonstationary and can change over a sequence of trials; and (3) a prior specifies that features are more likely to be present for target than for distractors. Through experience, EGS infers latent environment variables that determine the gains for guiding search. Control is thus cast as probabilistic inference, not optimization. We show that EGS can replicate a range of human data from visual search, including data that GS does not address. 1 Introduction Human visual activity often involves search. We search for our keys on a cluttered desk, a face in a crowd, an exit sign on the highway, a key paragraph in a paper, our favorite brand of cereal at the supermarket, etc. The flexibility of the human visual system stems from the endogenous (or internal) control of attention, which allows for processing resources to be directed to task-relevant regions and objects in the visual field. How is attention directed based on an individual’s goals? To what sort of features of the visual environment can attention be directed? These two questions are central to the study of how humans explore their environment. Visual search has traditionally been studied in the laboratory using cluttered stimulus displays containing artificial objects. The objects are defined by a set of primitive visual features, such as color, shape, and size. For example, an experimental task might be to search for a red vertical line segment—the target—among green verticals and red horizontals—the distractors. Performance is typically evaluated as the response latency to detect the presence or absence of a target with high accuracy. The efficiency of visual search is often characterized by the search slope—the increase 1 Figure 1: The architecture of Guided Search visual saliency map vertical horizontal green red primitive-feature top-down bottom-up activations attentional selection attentional state image gains noise contrast maps process + in response latency with each additional distractor in the display. An inefficient search can often require an additional 25–35 ms/item (or more, if eye movements are required). Many computational models of visual search have been proposed to explain data from the burgeoning experimental literature (e.g., Baldwin & Mozer, 2006; Cave, 1999; Itti & Koch, 2001; Mozer, 1991; Navalpakkam & Itti, 2006; Sandon, 1990; Wolfe, 1994). Despite differences in their details, they share central assumptions, perhaps most plainly emphasized by Wolfe’s (1994) Guided Search 2.0 (GS) model. We describe the central assumptions of GS, taking some liberty in ignoring details and complications of GS that obfuscate the similarities within this class of models and that are not essential for the purpose of this paper.1 As depicted Figure 1, GS posits that primitive visual features are detected across the retina in parallel along dimensions such as color and orientation, yielding a set of feature activity maps. Feature activations are scalars in [0, 1]. The feature maps represent each dimension via a coarse coding. That is, the maps for a particular dimension are highly overlapping and broadly tuned. For example, color might be represented by maps tuned to red, green, blue, and yellow; orientation might be represented by maps tuned to left, right, steep, and shallow-sloped edges. The feature activity maps are passed through a differencing mechanism that enhances local contrast and texture discontinuities, yielding a bottom-up activation. The bottom-up activations from all feature maps are combined to form a saliency map in which activation at a location indicates the priority of that location for the task at hand. Attention is directed to locations in order from most salient to least, and the object at each location is identified. GS supposes that response latency is linear in the number of locations that need to be searched before a target is found. (The model includes rules for terminating search if no target is found after a reasonable amount of effort.) Consider the task of searching for a red vertical bar among green vertical bars and red horizontal bars. Ideally, attention should be drawn to red and vertical bars, not to green or horizontal bar. To allow for guidance of attention, GS posits that a weight or top-down gain is associated with each feature map, and the contribution of given feature map to the saliency map is scaled by the gain. It is the responsibility of control processes to determining gains that emphasize task-relevant features. Although gain modulation is a sensible means of implementing goal-directed action, it yields behavior than is more efficient than people appear to be. To elaborate, consider again the task of searching for a red vertical. If the gains on the red and vertical maps are set to 1, and the gains on green and horizontal maps are set to 0, then a target (red vertical) will have two units of activation in the saliency map, whereas each distractor (red horizontal or green vertical) will have only one unit of activation. Because the target is the most salient item and GS assumes that response time is monotonically related to the saliency ranking of the target, the target should be located quickly, in a time independent of the number of distractors. In contrast, human response times increase linearly with the number of distractors in conjunction search. To reduce search efficiency, GS assumes noise corruption of the saliency map. In the case of GS, the signal-to-noise ratio is roughly 2:1. Baldwin and Mozer (2006) also require noise corruption for the same reason, although the corruption is to the low-level feature representation not the saliency map. Although Navalpakkam and Itti (2006) do not explicitly introduce noise in their model, they do so implicitly via a selection rule that the probability of attending an item is proportional to its saliency. To further reduce search efficiency, GS includes a complex set of rules that limit gain control. For example, gain modulation is allowed for only one feature map per dimension. Other attentional models 1Although Guided Search has undergone refinement (Wolfe, 2007), the key claims summarized here are unchanged. Recent extensions to GS consider eye movements and acuity changes with retinal eccentricity. 2 place similar, somewhat arbitrary limitations on gain modulation. Baldwin and Mozer (2006) impose the restriction P i |gi −1| < c, where gi is the gain of feature map i and c is a constant. Navalpakkam and Itti (2006) prefer the constraints P i gi = c and gi > 0. Finally, we mention one other key property the various models have in common. Gain tuning is cast as an optimization problem: the goal of the model is to adjust the gains so as to maximize the target saliency relative to distractor saliency for the task at hand. Baldwin and Mozer (2006) define the criterion in terms of the target saliency ranking. Navalpakkam and Itti (2006) use the expected target to distractor saliency ratio. Wolfe (1994) sets gains according to rules that he describes as performing optimization. 2 Experience-Guided Search The model we introduce in this paper makes three contributions over the class of Guided Search models previously proposed. (1) GS uses noise or nondeterminism to match human data. In reality, noise and nondeterminism serve to degrade the model’s performance over what it could otherwise be. In contrast, all components of our model are justified on computational grounds, leading to a more elegant, principled account. (2) GS imposes arbitrary limitations on gain modulation that also result in the model performing worse than it otherwise could. Although limitations on gain modulation might be neurobiologically rationalized, a more elegant account would characterize these limitations in terms of trade offs: constraints on gain modulation may limit performance, but they yield certain benefits. Our model offers such a trade-off account. (3) In GS, attentional control is achieved by tuning gains to optimize performance. In contrast, our model is designed to infer the structure of its environment through experience, and gain modulation is a byproduct of this inference. Consequently, our model has no distinct control mechanism, leading to a novel perspective on executive control processes in the brain. Our approach begins with the premise that attention is fundamentally task based: a location in the visual field is salient if a target is likely at that location. We define saliency as the target probability, P(Tx = 1|Fx), where Fx is the local feature activity vector at retinal location x and Tx is a binary random variable indicating if location x contains a target. Torralba et al. (2006) and Zhang and Cottrell (submitted) have also suggested that saliency should reflect target probability, although they propose approaches to computing the target probability very different from ours. Our approach is to compute the target probability using statistics obtained from recent experience performing the task. Consequently, we refer to our model as experience-guided search or EGS. To expand P(Tx|Fx), we make the naive-Bayes assumption that the feature activities are independent of one another, yielding P(Tx|Fx, ρ) = P(Tx) Q i P(Fxi|Tx, ρ)/ P1 t=0 P(Tx = t) Q i P(Fxi|Tx = t, ρ), (1) where ρ is a vector of parameters that characterize the current stimulus environment in the current task, and Fxi encodes the activity of feature i. Consider Fxi to be a rate-coded representation of a neural spike train. Specifically, Fxi denotes the count of the number of spikes that occurred in a window of n time intervals, where at most one spike can occur in each interval. We propose a generative model of the environment in which Fxi is a binomial random variable, Fxi|{Tx = t, ρ} ∼Binomial(ρit, n), where a spike rate ρit is associated with feature i for target (t = 1) and nontarget (t = 0) items. As n becomes large—i.e., the spike count is obtained over a larger period of time—the binomial is well approximated by a Gaussian: Fxi|{Tx = t, ρ} ∼ N(nρit, nρit(1 −ρit)). Using the Gaussian approximation, Equation 1 can be rewritten in the form of a logistic function: P(Tx|Fx, ρ) = 1/(1 + e−(rx+ n 2 sx)), where rx = ln P(Tx = 1) P(Tx = 0) −1 2 X i ln ρi1(1 −ρi1) ρi0(1 −ρi0) and sx = X i 1 X t=0 1 −2t ρit(1 −ρit)( ˜fxi−ρit)2 (2) and ˜fxi = fxi/n denotes the observed spike rate for a feature detector. Because of the logistic relationship, P(Tx|Fx, ρ) is monotonic in rx + n 2 sx. Consequently, if attentional priority is given to locations in order of their target probability, P(Tx|Fx, ρ), then it is 3 equivalent to rank using rx + n 2 sx. Further, if we assume that the target is equally likely in any location, then rx is constant across locations, and sx can substitute for P(Tx|Fx, ρ) as an equivalent measure of saliency. This saliency measure, sx, makes intuitive sense. Saliency at a location increases if feature i’s activity is distant from the mean activity observed in the past for a distractor (ρi0) and decreases if feature i’s activity is distant from the mean activity observed in the past for a target (ρi1). These saliency increases (decreases) are scaled by the variance of the distractor (target) activities, such that high-variance features have less impact on saliency. Expanding the numerator terms in the definition of sx (Equation 2), we observe that sx can be written as a linear combination of terms involving the feature activities, ˜fxi, and the squared activities, ˜f 2 xi (along with a constant term that can be ignored for ranking by saliency). The saliency measure sx in EGS is thus quite similar to the saliency measure in GS, sGS x = P i gi ˜fxi. The differences are: first, EGS incorporates quadratic terms, and second, gain coefficients of EGS are not free parameters but are derived from statistics of targets and distractors in the current task and stimulus environment. In this fact lies the virtue of EGS relative to GS: The control parameters are obtained not by optimization, but are derived directly from statistics of the environment. 2.1 Uncertainty in the Environment Statistics The model parameters, ρ, could be maximum likelihood estimates obtained by observing target and distractor activations over a series of trials. That is, suppose that each item in the display is identified as a target or distractor. The set of activations of feature i at all locations containing a target could be used to estimate ρi1, and likewise with locations containing a distractor to estimate ρi0. Alternatively, one could adopt a Bayesian approach and treat ρit as a random variable, whose uncertainty is reduced by the evidence obtained on each trial. Because feature spike rates lie in [0, 1], we define ρit as a beta random variable, ρit ∼Beta(αit, βit). This Bayesian approach also allows us to specify priors over ρit in terms of imaginary counts, α0 it and β0 it. For example, in the absence of any task experience, a conservative assumption is that all feature activations are predictive of a target, i.e., ρi1 should be drawn from a distribution biased toward 1, and ρi0 should be drawn from a distribution biased toward 0. To compute the target probabilities, we must marginalize over ρ, i.e., P(Tx|Fx) = R ρ P(Tx|Fx, ρ)p(ρ)dρ. Unfortunately, this integral is impossible to evaluate analytically. We instead compute the expectation of sx over ρ, ¯sx ≡Eρ(sx), which has the solution ¯sx = X i 1 X t=0 (1 −2t) (αit + βit −1)(αit + βit −2) (αit −1)(βit −1) ˜f 2 xi −2(αit + βit −1) βit −1 ˜fxi + αit βit −1 (3) Note that, like the expression for sx, ¯sx is a weighted sum of linear and quadratic feature-activity terms. When αit and βit are large, the distribution of ρit is sharply peaked, and ¯sx approaches sx with ρit = αit/(αit + βit). When this condition is satisfied, ranking by ¯sx is equivalent to ranking by P(Tx|Fx). Although the equivalence is not guaranteed for smaller αit and βit, we have found the equivalence to hold in empirical tests. Indeed, in our simulations, we find that defining saliency as either sx or ¯sx yields similar results, reinforcing the robustness of our approach. 2.2 Modeling the Stimulus Environment The parameter vectors α and β maintain a model of the stimulus environment in the context of the current task. Following each trial, these parameters must be updated to reflect the statistics of the trial. We assume that following a trial, each item in the display has been identified as either a target or distractor. (All other adaptive attention models such as GS make this assumption.) Consider a location x that has been labeled as type t (1 for target, 0 for distractor), and some feature i at that location, Fxi. We earlier characterized Fxi as a binomial random variable reflecting a spike count; that is, during n time intervals, fxi spikes are observed. Each time interval provides evidence as to the value ρit. Given prior distribution ρit ∼Beta(αit, βit), the posterior is ρit|Fxi ∼ Beta(αit + fxi, βit + n −fxi). However, to limit the evidence provided from each item, we scale it 4 by a factor of n. When all locations are considered, the resulting posterior is: ρit|Fi ∼Beta αit + P x∈χt ˜fxi, βit + P x∈χt 1 −˜fxi (4) where Fi is feature map i and χt is the set of locations containing elements of type t. With the approach we’ve described, evidence concerning the value of ρit accumulates over a sequence of trials. However, if an environment is nonstationary, this build up of evidence is not adaptive. We thus consider a switching model of the environment that specifies with probability λ, the environment changes and all evidence should be discarded. The consequence of this assumption is that the posterior distribution is a mixture of Equation 4 and the prior distribution, Beta(α0 it, β0 it). Modeling the mixture distribution is problematic because the number of mixture components grows linearly with the number of trials. We could approximate the mixture distribution by the beta distribution that best approximates the mixture, in the sense of Kullback-Leibler divergence. However, we chose to adopt a simpler, more intuitive solution: to interpolate between the two distributions. The update rule we use is therefore ρit|Fi ∼Beta λα0 it + (1 −λ) " αit + X x∈χt ˜fxi # , λβ0 it + (1 −λ) " βit + X x∈χt 1 −˜fxi #! . (5) 3 Simulation Methodology We present a step-by-step description of how the model runs to simulate experimental subjects performing a visual search task. We start by generating a sequence of experimental trials with the properties studied in an experiment. The model is initialized with αit = α0 it and βit = β0 it. On each simulation trial, the following sequence occurs. (1) Feature extraction is performed on the display to obtain firing rates, ˜fxi for each location x and feature type i. (2) Saliency, ¯sx, is computed for each location according to Equation 3. (3) The saliency rank of each location is assessed, and the number of locations that need to be searched in order to identify the target is assumed to be equal to the target rank. Response time should then be linear in target rank. (4) Following each trial, target and distractor statistics, αit and βit, are updated according to Equation 5. EGS has potentially many free parameters: {α0 it} and {β0 i1}, and λ. However, with no reason to believe that one feature behaves differently than another, we assign all the features the same priors. Further, we impose symmetry such that α0 i0 = β0 j1 = ν and α0 i1 = β0 j0 = µ for all i and j, reducing the total number of free parameters to three. Because we are focused on the issue of attentional control, we wanted to sidestep other issues, such as feature extraction. Consequently, EGS uses the front-end preprocessing of GS. GS takes as input an 8 × 8 array of locations, each of which can contain a single colored bar. As described earlier, GS analyzes the input via four broadly tuned features for color, and four for orientation. After a local contrast-enhancement operator, GS yields activation values in [0, 1] at each of 8 × 8 locations for each of eight feature dimensions. We treat the activation produced by GS for feature i at location x as the firing rate ˜fxi needed to simulate EGS. Like GS, the response time of EGS is linear in the target ranking. A scaling factor is required to convert rank to response time; we chose 25 msec/item, which is a fourth free parameter of GS. 4 Results We simulated EGS on a series of tasks that Wolfe (1994) used to evaluate GS. Because GS is limited to processing displays containing colored, oriented lines, some of the tasks constructed by Wolfe did not have an exact correspondence in the experimental literature. Rather, Wolfe, the leading expert in visual search, identified key findings that he wanted GS to replicate. Because EGS shares frontend processing with GS, EGS is limited to the same set of tasks as GS. Consequently, we present a comparison of GS and EGS. We began by replicating Wolfe’s results on GS. This replication was nontrivial, because GS contains many parameters, rules, and special cases, and published descriptions of GS do not provide a crisp 5 algorithmic description of the model. To implement EGS, we simply removed much of the complexity of GS—including the distinction between bottom-up and top-down weights, heuristics for setting the weights, and the injection of high-amplitude noise into the saliency map—and replaced it with Equations 3 and 5. Each simulation begins with a sequence of 100 practice trials, followed by a sequence of 1000 trials for each blocked condition. Displays on each trial are generated according to the constraints of the task with random variation with respect to unconstrained aspects of the task (e.g., locations of display elements, distractor identities, etc.). In typical search tasks, the participant is asked to detect the presence or absence of a target. We omit results for target-absent trials, since GS and EGS make identical predictions for these trials. The qualitative performance of EGS does not depend on its free parameters when two conditions are met: λ > 0 and µ > ν. The latter condition yields E[ρi1] > E[ρi0] for all i, and corresponds to the bias that features are more likely to be present for a target than for a distractor. This bias is rational in order to prevent cognition from suppressing information that could potentially be critical to behavior. All simulation results reported here used λ = 0.3, µ = 25, and ν = 10. Figure 2 shows simulation results on six sets of tasks, labeled A–F. The first and third columns (thin lines) are data from our replication of GS; the second and fourth columns (thick lines) are data from our implementation of EGS. The key feature to note is that results from EGS are qualitatively and quantitatively similar to results from GS. As should become clear when we explain the individual tasks, EGS probably produces a better qualitative fit to the human data. (Unfortunately, it is not feasible to place the human data side-by-side with the simulation results. Although the six sets of tasks were chosen by Wolfe to represent key experiments in the literature, most are abstractions of the original experimental tasks because the retina of GS—and its descendent EGS—is greatly simplified and cannot accommodate the stimulus arrays used in human studies. Thus, Wolfe never intended to quantitatively model specific experimental studies.) We briefly describe the six tasks. The first four involve displays of a homogeneous color, and search for a target orientation among distractors of different orientations. Task A explores search for a vertical (defined as 0◦) target among homogeneous distractors of a different orientation. The graph plots the slope of the line relating display size to response latency, as a function of the distractor orientation. Search slopes become more efficient as the target-distractor similarity decreases. Task B explores search for a target among two types of distractors as a function of display size. The distractors are 100◦apart, and the target is 40◦and 60◦from the distractors, but in one case the target differs from the distractors in that it is the only nearly vertical item, allowing pop out via the vertical feature detector. Note that pop out is not wired into EGS, but emerges because EGS identifies vertical-feature activity as a reliable predictor of the target. Task C examines search efficiency for a target among heterogeneous distractors, for two target orientations and two degrees of targetdistractor similarity. Search is more efficient when the target and distractors are dissimilar. (EGS obtains results better matched to the human data than GS.) Task D explores an asymmetry in search: it is more efficient to find a tilted bar among verticals than a vertical among tilted. This effect arises from the same mechanism that yielded efficient search in task B: a unique feature is highly activated when the target is tilted but not when it is vertical. And search is better guided to features that are present than to features that are absent in EGS, due to the ρ priors. Task E involves conjunction search. The target is a red vertical among green vertical and red tilted distractors. The red item’s tilt can be either 90◦(i.e., horizontal) or 40◦. Both distractor environments yield inefficient search, but—consistent with human data—conjunction searches can vary in their relative difficulty. Task F examines search efficiency for a red vertical among red 60◦and yellow vertical distractors, as a function of the ratio of the two distractor types. The result shows that search can be guided: response times become faster as either the target color or target orientation becomes sparse, because a relatively unique feature serves as a reliable cue to the target. Figure 3a depicts how EGS adapts differently for the extreme conditions in which the distractors are mostly vertical (dark bars) or mostly red (light bars). The bars represent E[ρi0]; the lower the value, the more a feature is viewed as reliably discriminating targets and distractors. (E[ρi1] is independent of the experimental condition.) When distractors are mostly vertical, the red feature is a better cue, and vice versa. The standard explanation for this phenomenon in the psychological literature is that subjects operate in two stages, first filtering out based on the more discriminative feature, and then serially searching the remaining 6 10 20 30 40 400 600 800 1000 1200 Display Size RT (msec) T: 20°; D: 0° T: 0°; D: 20° 10 20 30 40 400 600 800 1000 1200 (D) Feature Search Asymmetry Display Size RT (msec) 4 8 12 400 500 600 700 Display Size RT (msec) T: 0°; D: −20°; 20° T: 0°; D: −40°; 40° T: 20°; D: 0°; 40° T: 20°; D: −20°; 60° 4 8 12 400 500 600 700 (C) Target−Distractor Similarity Display Size RT (msec) 0 .25 .50 .75 1 600 800 1000 Proportion of Red Distractors RT (msec) 0 .25 .50 .75 1 600 800 1000 (F) Conjunction Search Varying Distractor Ratio Proportion of Red Distractors RT (msec) 10 20 30 40 400 600 800 1000 Display Size RT (msec) T: 10°; D: −50°; 50° T: 20°; D: −20°; 80° T: 10°; D: −30°; 70° 10 20 30 40 400 600 800 1000 Display Size RT (msec) (B) Categorical Search 10 20 30 40 600 800 1000 T: 0° R; D: 0° G; 40° R T: 0° R; D: 0° G; 90° R Display Size RT (msec) 10 20 30 40 600 800 1000 (E) Conjunction Search Varying Distractor Confusability Display Size RT (msec) 10 20 30 40 50 0 5 10 15 Distractor Orientation RT Slope (msec/item) 10 20 30 40 50 0 5 10 15 (A) Vertical Bar Among Homogeneous Distractors Distractor Orientation RT Slope (msec/item) Figure 2: Simulation results on six sets of tasks, labeled A–F, for GS (thin lines, 1st and 3d columns) and EGS (thick lines, 2nd and 4th columns). Simulation details are explained in the text. items. EGS provides a single-stage account that does not need to invoke specialized mechanisms for adaptation to the environment, because all attentional control is adaptation of this sort. To summarize, EGS predicts the key factors in visual search that determine search efficiency. Most efficient search is for a target defined by the presence of a single categorical feature among homogeneous distractors that do not share the categorical feature. Least efficient search is for target and distractors that share features (e.g., T among L’s, or red verticals among red horizontals and green verticals) and/or when distractors are heterogeneous. Wolfe, Cave, & Franzel (1989) conducted an experiment to demonstrate that people can benefit from guidance. This experiment, which oddly has never been modeled by GS, involves search for a conjunction target defined by a triple of features, e.g., a big red vertical bar. The target might be presented among heterogeneous distractors that share two features with it, such as a big red horizontal bar, or distractors that share only one feature with it, such as a small green vertical bar. Performance in these two conditions, denoted T3-D2 and T3-D1, respectively, is compared to performance in a standard conjunction search task, denoted T2-D1, involving targets defined by two features and sharing one feature with each distractor. Wolfe et al. reasoned that if search can be guided, saliency at a location should be proportional to the number of target-relevant features at that location, and the ratio of target to distractor salience should be x/y in condition Tx-Dy. Because x > y, the target is always more salient than any distractor, but GS assumes less efficient search due to noise corruption of the saliency map, thereby predicting search slopes that are inversely related to x/y. The human data show exactly this pattern, producing almost flat search slopes for T3-D1. EGS replicates the human data (Figure 3b) without employing GS’s arbitrary assumption that prioritization is corrupted by noise. Instead, x/y reflects the amount of evidence available on each trial about features that discriminate targets from distractors. Essentially, EGS suggests that x/y determines the availability of discriminative statistics in the environment. Thus, the limitation is on learning, not on performance. 7 Figure 3: (a) Values of E[ρi0] in task F. (b) EGS performance on the triple-conjunction task of Wolfe, Cave, & Franzel (1989) vertical red 0 0.05 0.1 0.15 0.2 0.25 Feature Activation (a) mostly vert distractors mostly red distractors 0 10 20 30 40 400 600 800 1000 1200 Display Size Reaction Time (b) T3−D2 T2−D1 T3−D1 5 Discussion We presented a model, EGS, that guides visual search via statistics collected over the course of experience in a task environment. The primary contributions of EGS are as follows. First, EGS is a significantly more elegant and parsimonious theory than its predecessors. In contrast to EGS, GS is a complex model under the hood with many free parameters and heuristic assumptions. We and other groups have spent many months reverse engineering GS to determine how exactly it works, because published descriptions do not have the specificity of an algorithm. Second, to explain human data, GS and its ancestors are “retarded” by injecting noise or arbitrarily limiting gains. Although it may ultimately be determined that the brain suffers from these conditions, one would prefer theories that cast performance of the brain as ideal or rational. EGS achieves this objective via explicit assumptions about the generative model of the environment embodied by cognition. In particular, the dumbing-down of GS and its variants is replaced in EGS by the claim that environments are nonstationary. If the environment can change from one trial to the next, the cognitive system does well not to turn up gains on one feature dimension at the expense of other feature dimensions. The result is a sensible trade off: attentional control can be rapidly tuned as the task or environment changes, but this flexibility restricts EGS’s search efficiency when the task and environment remain constant. Third, EGS suggests a novel perspective on attentional control, and executive control more generally. All other modern perspectives we are aware of treat control as optimization, whereas in EGS, control arises directly from statistical inference on the task environment. Our current research is exploring the implications of this intriguing perspective. Acknowledgments This research was supported by NSF BCS 0339103 and NSF CSE-SMA 0509521. Support for the second author comes from an NSF Graduate Fellowship. References Baldwin, D., & Mozer, M. C. (2006). Controlling attention with noise: The cue-combination model of visual search. In R. Sun & N. Miyake (Eds.), Proc. of the 28th Ann. Conf. of the Cog. Sci. Society (pp. 42-47). Hillsdale, NJ: Erlbaum. Cave, K. R. (1999). The FeatureGate model of visual selection. Psychol. Res., 62, 182–194. Itti, L., & Koch, C. (2001). Computational modeling of visual attention. Nature Rev. Neurosci., 2, 194–203. Mozer, M. C. (1991). The perception of multiple objects: A connectionist approach. Cambridge, MA: MIT. Navalpakkam, V., & Itti, L. (2006). Optimal cue selection strategy. In Advances in Neural Information Processing Systems Vol. 19 (pp. 1-8). Cambridge, MA: MIT Press. Rao, R., Zelinsky, G., Hayhoe, M., & Ballard, D. (2002). Eye movements in iconic visual search. Vis. Res., 42, 1447–1463. Sandon, P. A. (1990). Simulating visual attention. Journal of Cog. Neuro., 2, 213–231. Sandon, 1990 Torralba, A., Oliva, A., Castelhano, M.S., & Henderson, J. M. (2006). Contextual guidance of eye movements and attention in real-world scenes: The role of global features on objects search. Psych. Rev., 113, 766–786. Wolfe, J. M., Cave, K. R., & Franzel, S. L. (1989). Guided search: An alternative to the feature integration model for visual search. Jnl. Exp. Psych.: Hum. Percep. & Perform., 15, 419–433. Wolfe, J. M. (1994). Guided Search 2.0: A revised model of visual search. Psych. Bull. & Rev., 1, 202–238. Wolfe, J. M. (2007). Guided Search 4.0: Current progress with a model of visual search. In. W. Gray (Ed.), Integrated Models of Cognitive Systems. NY: Oxford. Zhang, L., & Cottrell, G. W. (submitted). Probabilistic search: A new theory of visual search. Submitted for publication. 8
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Hierarchical Penalization Marie Szafranski 1, Yves Grandvalet 1, 2 and Pierre Morizet-Mahoudeaux 1 Heudiasyc 1, UMR CNRS 6599 Universit´e de Technologie de Compi`egne BP 20529, 60205 Compi`egne Cedex, France IDIAP Research Institute 2 Av. des Pr´es-Beudin 20 P.O. Box 592, 1920 Martigny, Switzerland marie.szafranski@hds.utc.fr Abstract Hierarchical penalization is a generic framework for incorporating prior information in the fitting of statistical models, when the explicative variables are organized in a hierarchical structure. The penalizer is a convex functional that performs soft selection at the group level, and shrinks variables within each group. This favors solutions with few leading terms in the final combination. The framework, originally derived for taking prior knowledge into account, is shown to be useful in linear regression, when several parameters are used to model the influence of one feature, or in kernel regression, for learning multiple kernels. Keywords – Optimization: constrained and convex optimization. Supervised learning: regression, kernel methods, sparsity and feature selection. 1 Introduction In regression, we want to explain or to predict a response variable y from a set of explanatory variables x = (x1, . . . , xj, . . . , xd), where y ∈R and ∀j, xj ∈R. For this purpose, we use a model such that y = f(x) + ϵ, where f is a function able to characterize y when x is observed and ϵ is a residual error. Supervised learning consists in estimating f from the available training dataset S = {(xi, yi)}n i=1. It can be achieved in a predictive or a descriptive perspective: to predict accurate responses for future observations, or to show the correlations that exist between the set of explanatory variables and the response variable, and thus, give an interpretation to the model. In the linear case, the function f consists of an estimate β = (β1, . . . , βj, . . . , βd) t applied to x, that is to say f(x) = xβ. In a predictive perspective, xβ produces an estimate of y, for any observation x. In a descriptive perspective, |βj| can be interpreted as a degree of relevance of variable xj. Ordinary Least Squares (OLS) minimizes the sum of the residual squared error. When the explanatory variables are numerous and many of them are correlated, the variability of the OLS estimate tends to increase. This leads to reduced prediction accuracy, and an interpretation of the model becomes tricky. Coefficient shrinkage is a major approach of regularization procedures in linear regression models. It overcomes the drawbacks described above by adding a constraint on the norm of the estimate β. According to the chosen norm, coefficients associated to variables with little predictive information may be shrunk, or even removed when variables are irrelevant. This latest case is referred to as variable selection. In particular, ridge regression shrinks coefficients with regard to the ℓ2-norm, while the lasso (Least Absolute Shrinkage and Selection Operator) [1] and the lars (Least Angle Regression Stepwise) [2] both shrink and remove coefficients using the ℓ1-norm. 1 @ @ @ @@ R | l QQQQQ s | l 3 QQQQQ s | l x1 x2 x3 x4 x5 x6 @ @ @ @@ R | l 0 QQQQQ s | l 3 3 QQQQQ s | l 2 | l 1 x1 x2 x3 x4 x5 x6 σ1,1 σ1,2 σ1,3 σ2,1 σ2,2 σ2,3 σ2,4 σ2,5 σ2,6 o J1 J2 J3 Figure 1: left: toy-example of the original structure of variables; right: equivalent tree structure considered for the formalization of the scaling problem. In some applications, explanatory variables that share a similar characteristic can be gathered into groups – or factors. Sometimes, they can be organized hierarchically. For instance, in genomics, where explanatory variables are (products of) genes, some factors can be identified from the prior information available in the hierarchies of Gene Ontology. Then, it becomes necessary to find methods that retain meaningful factors instead of individual variables. Group-lasso and group-lars [3] can be considered as hierarchical penalization methods, with trees of height two defining the hierarchies. They perform variable selection by encouraging sparseness over predefined factors. These techniques seem perfectible in the sense that hierarchies can be extended to more than two levels and sparseness integrated within groups. This papers proposes a penalizer, derived from an adaptive penalization formulation [4], that highlights factors of interest by balancing constraints on each element, at each level of a hierarchy. It performs soft selection at the factor level, and shrinks variables within groups, to favor solutions with few leading terms. Section 2 introduces the framework of hierarchical penalization and the associated algorithm is presented in Section 3. Section 4 shows how this framework can be applied to linear and kernel regression. We conclude with a general survey of our future works. 2 Hierarchical Penalization 2.1 Formalization We introduce hierarchical penalization by considering problems where the variables are organized in a tree structure of height two, such as the example displayed in figure 1. The nodes of height one are labelled in {1, . . . , K}. The set of children (that is, leaves) of node k is denoted Jk and its cardinality is dk. As displayed on the right-hand-side of figure 1, a branch stemming from the root and going to node k is labelled by σ1,k, and the branch reaching leaf j is labelled by σ2,j. We consider the problem of minimizing a differentiable loss function L(·), subject to sparseness constraints on β and the subsets of β defined in a tree hierarchy. This reads min β,σ L(β) + λ K X k=1 X j∈Jk β2 j √σ1,k σ2,j , subject to K X k=1 dk σ1,k = 1 , d X j=1 σ2,j = 1 , σ1,k ≥0 k = 1, . . . , K , σ2,j ≥0 j = 1, . . . , d , (1a) (1b) (1c) where λ > 0 is a Lagrangian parameter that controls the amount of shrinkage, x/y is defined by continuation at zero as x/0 = ∞if x ̸= 0 and 0/0 = 0. 2 The second term of expression (1a) penalizes β, according to the tree structure, via scaling factors σ1 and σ2. The constraints (1b) shrink the coefficients β at group level and inside groups. In what follows, we show that problem (1) is convex and that this joint shrinkage encourages sparsity at the group level. 2.2 Two important properties We first prove that the optimization problem (1) is tractable and moreover convex. Then, we show an equivalence with another optimization problem, which exhibits the exact nature of the constraints applied to the coefficients β. Proposition 1 Provided L(·) is convex, problem (1) is convex. Proof: A problem minimizing a convex criterion on a convex set is convex. Since L(·) is convex and λ is positive, the criterion (1a) is convex provided f(x, y, z) = x2 √yz is convex. To show this, we compute the Hessian: 4(yz) 1 2 ∇2f(x, y, z) = 8 −4 x y −4 x z −4 x y 3 x2 y2 x2 yz −4 x z x2 yz 3 x2 z2 = 2 2 −x y −x z 2 −x y −x z t + 0 x y −x z 0 x y −x z t . Hence, the Hessian is positive semi-definite, and criterion (1a) is convex. Next, constraints (1c) define half-spaces for σ1 and σ2, which are convex sets. Equality constraints (1b) define linear subspaces of dimension K −1 and d−1 which are also convex sets. The intersection of convex sets being a convex set, the constraints define a convex admissible set, and problem (1) is convex. □ Proposition 2 Problem (1) is equivalent to min β L(β) + λ K X k=1 d 1 4 k X j∈Jk |βj| 4 3 3 4 2 . (2) Sketch of proof: The Lagrangian of problem (1) is L = L(β) + λ K X k=1 X j∈Jk β2 j √σ1,k σ2,j + ν1 K X k=1 dk σ1,k −1 ! + ν2 d X j=1 σ2,j −1 − K X k=1 ξ1,k σ1,k − d X j=1 ξ2,j σ2,j . Hence, the optimality conditions for σ1,k and σ2,j are ∂L ∂σ1,k = 0 ∂L ∂σ2,j = 0 ⇒ −λ 2 X j∈Jk β2 j σ 3 2 1,kσ 1 2 2,j + ν1dk −ξ1,k = 0 −λ 2 β2 j σ 1 2 1,kσ 3 2 2,j + ν2 −ξ2,j = 0 . After some tedious algebra, the optimality conditions for σ1,k and σ2,j can be expressed as σ1,k = d −3 4 k (sk) 3 4 K P k=1 d 1 4 k (sk) 3 4 , and σ2,j = d 1 4 k |βj| 4 3 (sk) 1 4 K P k=1 d 1 4 k (sk) 3 4 for j ∈Jk , where sk = P j∈Jk |βj| 4 3 . Plugging these conditions in criterion (1a) yields the claimed result. □ 3 2.3 Sparseness Proposition 2 shows how the penalization influences the groups of variables and each variable in each group. Note that, thanks to the positivity of the squared term in (2), the expression can be further simplified to min β L(β) + ν K X k=1 d 1 4 k X j∈Jk |βj| 4 3 3 4 , (3) where, for any L(β), there is a one-to-one mapping from λ in (2) to ν in (3). This expression can be interpreted as the Lagrangian formulation of a constrained optimization problem, where the admissible set for β is defined by the multiplicand of ν. We display the shape of the admissible set in figure 2, and compare it to ridge regression, which does not favor sparsity, lasso, which encourages sparsity for all variables but does not take into account the group structure, and group-lasso, which is invariant to rotations of within-group variables. One sees that hierarchical penalization combines some features of lasso and group-lasso. ridge regression lasso group-lasso hierarchical penalization β2 1+β2 2+β2 3≤1 |β1|+|β2|+|β3|≤1 2(β2 1+β2 2) 1 2 +|β3|≤1 2 1 4 “ |β1| 4 3 +|β2| 4 3 ” 3 4 +|β3|≤1 Figure 2: Admissible sets for various penalties, the two horizontal axes are the (β1, β2) plane (first group) and the vertical axis is for β3 (second group). By looking at the curvature of these sets when they meet axes, one gets a good intuition on why ridge regression does not suppress variables, why lasso does, why group-lasso suppresses groups of variables but not within-group variables, and why hierarchical penalization should do both. This intuition is however not correct for hierarchical penalization because the boundary of the admissible set is differentiable in the within-group hyper-plane (β1, β2) at β1 = 0 and β2 = 0. However, as its curvature is very high, solutions with few leading terms in the within-group variables are encouraged. To go beyond the hints provided by these figures, we detail here the optimality conditions for β minimizing (3). The first-order optimality conditions are 1. for βj = 0, j ∈Jk and P j∈Jk |βj| = 0, ∂L(β) ∂βj + ν d 1 4 k vj = 0, where vj ∈[−1, 1]; 2. for βj = 0, j ∈Jk and P j∈Jk |βj| ̸= 0, ∂L(β) ∂βj = 0; 3. for βj ̸= 0, j ∈Jk, ∂L(β) ∂βj + ν d 1 4 k sign(βj) 1 + 1 |βj| 4 3 X ℓ∈Jk ℓ̸=j |βℓ| 4 3 −1 4 = 0. These equations signify respectively that 1. the variables belonging to groups that are estimated to be irrelevant are penalized with the highest strength, thus limiting the number of groups influencing the solution; 2. when a group has some non-zero relevance, all variables enter the set of active variables provided they influence the fitting criterion; 3. however, the penalization strength increases very rapidly (as a smooth step function) for small values of |βj|, thus limiting the number of βj with large magnitude. 4 Overall, hierarchical penalization is thus expected to provide solutions with few active groups and few leading variables within each group. 3 Algorithm To solve problem (3), we use an active set algorithm, based on the approach proposed by Osborne et al. [5] for the lasso. This algorithm iterates two phases: first, the optimization problem is solved with a sub-optimal set of active variables, that is, non-zero variables: we define A = {j | βj ̸= 0}, the current active set of variables, γ = {βj}j∈A, the vector of coefficients associated to A, and Gk = {Jk ∩A}, the subset of coefficients γ associated to group k. Then, at each iteration, we solve the problem min γ L(γ) = L(γ) + ν K X k=1 d 1 4 k X j∈Gk |γj| 4 3 3 4 , (4) by alternating steps A and B described below. Second, the set of active variables is incrementally updated as detailed in steps C and D. A Compute a candidate update from an admissible vector γ The goal is to solve min h L(γ + h), where γ is the current estimate of the solution and h ∈R|A|. The difficulties in solving (4) stem from the discontinuities of the derivative due to the absolute values. These difficulties are circumvented by replacing |γj + hj| by sign(γj)(γj + hj). This enables the use of powerful continuous optimizers based either on the Newton, quasi-Newton or conjugate gradient methods according to the size of the problem. B Obtain a new admissible vector γ† Let γ† = γ + h. If for all j, sign(γ† j) = sign(γj), then γ is sign-feasible, and we go to step C, otherwise: B.1 Let S be the set of indices m such that sign(γ+ m) ̸= sign(γm). Let µ = min m∈S −γm hm , that is, µ is the largest step in direction h such that sign(γm + µhm) = sign(γm), except for one variable, ℓ= arg min m −γm hm , for which γℓ+ µhℓ= 0. B.2 Set γ = γ + µh and sign(γℓ) = −sign(γℓ), and compute a new direction h as in step A. If, for the new solution γ†, sign(γ† ℓ) ̸= sign(γℓ), then ℓis removed from A. Go to step A. B.3 Iterate step B until γ is sign-feasible. C Test optimality of γ If the appropriate optimality condition holds for all inactive variables βℓ(βℓ= 0), that is C.1 for ℓ∈Jk, where P j∈Jk |βj| = 0, then ∂L(β) ∂βℓ ≤ν d 1 4 k , C.2 for ℓ∈Jk, where P j∈Jk |βj| ̸= 0, then ∂L(β) ∂βℓ = 0, then γ is a solution. Else, go to step D. D Select the variable that enters the active set D.1 Select variable ℓ, ℓ/∈A that maximizes d −1 4 k ∂L(β) ∂βℓ , where k is the group of variable ℓ. D.2 Update the active set: A ←A ∪{ℓ}, with initial vector: γ = [γ, 0]t where the sign of the new zero component is −sign ∂L(β) ∂βℓ . D.3 Go to step A. The algorithm is initialized with A = ∅, and the first variable is selected with the process described at step D. 5 4 Experiments We illustrate on two datasets how hierarchical penalization can be useful in exploratory analysis and in prediction. Then, we show how the algorithm can be applied for multiple kernel learning in kernel regression. 4.1 Abalone Database The Abalone problem [6] consists in predicting the age of abalone from physical measurements. The dataset is composed of 8 attributes. One concerns the sex of abalone, and has been encoded with dummy variables, that is xsex i = (100) for male, xsex i = (010) for female, or xsex i = (001) for infant. This variable defines the first group. The second group is composed of 3 attributes concerning size parameters (length, diameter and height), and the last group is composed of weight parameters (whole, shucked, viscera and shell weight). We randomly selected 2920 examples for training, including the tuning of ν by 10-fold cross validation, and left the 1257 other for testing. The mean squared test error is at par with lasso (4.3). The coefficients estimated on the training set are reported in table 4.1. Weight parameters are a main contributor to the estimation of the age of an abalon, while sex is not essential, except for infant. sex 0.051 0.036 -0.360 0.516 size -0.044 1.134 0.358 1.7405 weight 4.370 -4.499 -1.110 1.399 11.989 Table 1: Coefficients obtained on the Abalone dataset. The last column represents the value d 1 4 k 0 @ P j∈Jk |βj| 4 3 1 A 3 4 . 4.2 Delve Census Database The Delve Census problem [7] consists in predicting the median price of a house in different survey regions. Each 22732 survey region is represented by 134 demographic information measurements. Several prototypes are available. We focussed on the prototype “house-price-16L”, composed of 16 variables. We derived this prototype by including all the other variables related to these 16 variables. The final dataset is then composed of 37 variables, split up into 10 groups1. We randomly selected 8000 observations for training and left the 14732 for testing. We divided the training observations into 10 distinct datasets. For each dataset, the parameter ν was selected by a 10-fold cross validation, and the mean squared error was computed on the testing set. We reported on table 4.2 the mean squared test errors obtained with the hierarchical penalization (hp), the group-lasso (gl) and the lasso estimates. Datasets 1 2 3 4 5 6 7 8 9 10 mean error hp (×109) 2.363 2.745 2.289 4.481 2.211 2.364 2.460 2.298 2.461 2.286 2.596 gl (×109) 2.429 2.460 2.289 4.653 2.230 2.364 2.472 2.308 2.454 2.291 2.595 lasso (×109) 2.380 2.716 2.293 4.656 2.216 2.368 2.490 2.295 2.483 2.288 2.618 Table 2: Mean squared test errors obtained with different methods for the 10 datasets. Hierarchical penalization performs better than lasso on 8 datasets. It also performs better than group-lasso on 6 datasets, and obtains equal results on 2 datasets. However the lowest overall mean error is achieved by group-lasso. 4.3 Multiple Kernel Learning Multiple Kernel Learning has drawn much interest in classification with support vector machines (SVMs) starting from the work of Lanckriet et al. [8]. The problem consists in learning a convex 1 A description of the dataset is available at http://www.hds.utc.fr/˜mszafran/nips07/. 6 combination of kernels in the SVM optimization algorithm. Here, we show that hierarchical penalization is well suited for this purpose for other kernel predictors, and we illustrate its effect on kernel smoothing in the regression setup. Kernel smoothing has been studied in nonparametric statistics since the 60’s [9]. Here, we consider the model where the response variable y is estimated by a sum of kernel functions yi = n X j=1 βj κh(xi, xj) + ϵi , where κh is the kernel with scale factor (or bandwidth) h, and ϵi is a residual error. For the purpose of combining K bandwidths, the general criterion (3) reads min {βk}K k=1 n X i=1 yi − K X k=1 n X j=1 βk,j κhk(xi, xj) 2 + ν K X k=1 n 1 4 k n X j=1 |βk,j| 4 3 3 4 . (5) The penalized model (5) has been applied to the motorcycle dataset [9]. This uni-dimensional problems enables to display the contribution of each bandwidth to the solution. We used Gaussian kernels, with 7 bandwidths ranging from 10−1 to 102. Figure 3 displays the results obtained for different penalization parameters: the estimated function obtained by the combination of the selected bandwidths, and the contribution of each bandwidth to the model. We display three settings for the penalization parameter ν, corresponding to slight overfitting, good fit and slight under-fitting. The coefficients of bandwidths h2, h6 and h7 were always null and are thus not displayed. As expected, when the penalization parameter ν increases, the fit becomes smoother, and the number of contributing bandwidths decreases. We also observe that the effective contribution of some bandwidths is limited to a few kernels: there are few leading terms in the expansion. 5 Conclusion and further works Hierarchical penalization is a generic framework enabling to process hierarchically structured variables by usual statistical models. The structure is provided to the model via constraints on the subgroups of variables defined at each level of the hierarchy. The fitted model is then biased towards statistical explanations that are “simple” with respect to this structure, that is, solutions which promote a small number of groups of variables, with a few leading components. In this paper, we detailed the general framework of hierarchical penalization for tree structures of height two, and discussed its specific properties in terms of convexity and parsimony. Then, we proposed an efficient active set algorithm that incrementally builds an optimal solution to the problem. We finally illustrated how the approach can be used when groups of features, or when discrete variables exist, after being encoded by several binary variables, result in groups of variables. Finally, we also shown how the algorithm can be used to learn from multiple kernels in regression. We are now performing quantitative empirical evaluations, with applications to regression, classification and clustering, and comparisons to other regularization schemes, such as the group-lasso. We then plan to extend the formalization to hierarchies of arbitrary height, whose properties are currently under study. We will then be able to tackle new applications, such as genomics, where the available gene ontologies are hierarchical structures that can be faithfully approximated by trees. References [1] R. Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society. Series B, 58(1):267–288, 1996. [2] B. Efron, T. Hastie, I. Johnstone, and R. Tibshirani. Least angle regression. Annals of Statistics, 32(2):407–499, 2004. [3] M. Yuan and Y. Lin. Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society. Series B, 68(1):49–67, 2006. 7 combined h1=10−1 h3=1 h4=10 1 2 h5=10 ν=10 ν=25 ν=50 Figure 3: Hierarchical penalization applied to kernel smoothing on the motorcycle data. Combined: the points represent data and the solid line the function of estimated responses. Isolated bandwidths: the points represent partial residuals and the solid line represents the contribution of the bandwidth to the model. [4] Y. Grandvalet and S. Canu. Adaptive scaling for feature selection in SVMs. In Advances in Neural Information Processing Systems, volume 15. MIT Press, 2003. [5] M. R. Osborne, B. Presnell, and B. A. Turlach. On the lasso and its dual. Journal of Computational and Graphical Statistics, 9(2):319–337, June 2000. [6] C.L. Blake D.J. Newman, S. Hettich and C.J. Merz. UCI repository of machine learning databases, 1998. URL http://www.ics.uci.edu/˜mlearn/MLRepository.html. [7] Delve: Data for evaluating learning in valid experiments. URL http://www.cs.toronto. edu/˜delve/. [8] G. Lanckriet, T. De Bie, N. Cristianini, M. Jordan, and W. Noble. A statistical framework for genomic data fusion. Bioinformatics, 20:2626–2635, 2004. [9] W. H¨ardle. Applied Nonparametric Regression, volume 19. Economic Society Monographs, 1990. 8
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Linear Programming Analysis of Loopy Belief Propagation for Weighted Matching Sujay Sanghavi, Dmitry M. Malioutov and Alan S. Willsky Laboratory for Information and Decision Systems Massachusetts Institute of Technology Cambridge, MA 02139 {sanghavi,dmm,willsky}@mit.edu Abstract Loopy belief propagation has been employed in a wide variety of applications with great empirical success, but it comes with few theoretical guarantees. In this paper we investigate the use of the max-product form of belief propagation for weighted matching problems on general graphs. We show that max-product converges to the correct answer if the linear programming (LP) relaxation of the weighted matching problem is tight and does not converge if the LP relaxation is loose. This provides an exact characterization of max-product performance and reveals connections to the widely used optimization technique of LP relaxation. In addition, we demonstrate that max-product is effective in solving practical weighted matching problems in a distributed fashion by applying it to the problem of self-organization in sensor networks. 1 Introduction Loopy Belief Propagation (LBP) and its variants [6, 9, 13] have been shown empirically to be effective in solving many instances of hard problems in a wide range of fields. These algorithms were originally designed for exact inference (i.e. calculation of marginals/MAP estimates) in probability distributions whose associated graphical models are tree-structured. While some progress has been made in understanding their convergence and accuracy on general “loopy” graphs (see [8, 12, 13] and their references), it still remains an active research area. In this paper we study the application of the widely used max-product form of LBP (or simply max-product (MP) algorithm), to the weighted matching problem. Given a graph G = (V, E) with non-negative weights we on its edges e ∈E, the weighted matching problem is to find the heaviest set of mutually disjoint edges (i.e. a set of edges such that no two edges share a node). Weighted matching is a classic problem that has played a central role in computer science and combinatorial optimization, with applications in resource allocation, scheduling in communications networks [10], and machine learning [5]. It has often been perceived to be the “easiest non-trivial problem”, and one whose analysis and solution has inspired methods (or provided insights) for a variety of other problems. Weighted matching thus naturally suggests itself as a good candidate for the study of convergence and correctness of algorithms like max-product. Weighted matching can be naturally formulated as an integer program. The technique of linear programming (LP) relaxation involves replacing the integer constraints with linear inequality constraints. This relaxation is tight if the resulting linear program has an integral optimum. LP relaxation is not always tight for the weighted matching problem. The primary contribution of this paper is an exact characterization of max-product performance for the matching problem, which also establishes a link to LP relaxation. We show that (i) if the LP relaxation is tight then max-product 1 converges to the correct answer, and (ii) if the LP relaxation is not tight then max-product does not converge. Weighted matching is a special case of the weighted b-matching problem, where there can be up to bi edges touching node i (setting all bi = 1 reduces to simple matching). All the results of this paper hold for the general case of b-matchings on arbitrary graphs. However, in the interests of clarity, we provide proofs only for the conceptually easier case of simple matchings where bi = 1. The minor modifications needed for general b-matchings will appear in a longer publication. In prior work, Bayati et. al [2] established that max-product converges for weighted matching in bipartite graphs, and [5] extended this result to b-matching. These results are implied by our result1, as for bipartite graphs, the LP relaxation is always tight. In Section 2 we set up the weighted matching problem and its LP relaxation. We describe the maxproduct algorithm for weighted matching in Section 3. The main result of the paper is established in Section 4. Finally, in Section 5 we apply b-matching to a sensor-network self-organization problem and show that max-product provides an effective way to solve the problem in a distributed fashion. 2 Weighted Matching and its LP Relaxation Suppose that we are given a graph G with weights we, we also positive integers bi for each node i ∈V . A b-matching is any set of edges such that the total number of edges in the set incident to any node i is at most bi. The weighted b-matching problem is to find the b-matching of largest weight. Weighted b-matching can be naturally formulated as the following integer program (setting all bi = 1 gives an integer program for simple matching): IP : max X e∈E wexe, s.t. X e∈Ei xe ≤bi for all i ∈V, xe ∈{0, 1} for all e ∈E Here Ei is the set of edges incident to node i. The linear programming (LP) relaxation of the above problem is to replace the constraint xe ∈{0, 1} with the constraint xe ∈[0, 1], for each e ∈E. We denote the corresponding linear program by LP. Throughout this paper, we will assume that LP has a unique optimum. The LP relaxation is said to be tight if the unique optimum is integral (i.e. one in which all xe ∈{0, 1}). Note that the LP relaxation is not tight in general. Apart from the bipartite case, the tightness of LP relaxation is a function of both the weights and the graph structure2. 3 Max-Product for Weighted Matching We now formulate weighted b-matching on G as a MAP estimation problem by constructing a suitable probability distribution. This construction is naturally suggested by the form of the integer program IP. Associate a binary variable xe ∈{0, 1} with each edge e ∈E, and consider the following probability distribution: p(x) ∝ Y i∈V ψ(xEi) Y e∈E exp(wexe), (1) which contains a factor ψ(xEi) for each node i ∈V , the value of which is ψ(xEi) = 1 if P e∈Ei xe ≤bi, and 0 otherwise. Note that we use i to refer both to the nodes of G and factors of p, and e to refer both to the edges of G and variables of p. The factor ψ(xEi) enforces the constraint that at most one edge incident to node i can be assigned the value “1”. It is easy to see that, for any x, p(x) = exp(P e wexe) if the set of edges {e|xe = 1} constitute a b-matching in G, and p(x) = 0 otherwise. Thus the max-weight b-matching of G corresponds to the MAP estimate of p. 1However, [2] uses a graphical model which is different from ours to represent weighted matching. 2A simple example: G is a cycle of length 3, all the bi = 1. If all we = 1, LP relaxation is loose: setting each xe = 1 2 is the optimum. However, if instead the weights are {1, 1, 3}, then LP relaxation is tight. 2 The factor-graph version of the max-product algorithm [6] passes messages between variables and the factors that contain them (for the formulation in (1), each variable is a member of exactly two factors). The output is an estimate ˆx of the MAP of p. We now present the max-product update equations simplified for p in (1). In the following e and (i, j) denote the same edge. Also, for two sets A and B the set difference is denoted by the notation A −B. Max-Product for Weighted Matching (INIT) Set t = 0 and initialize each message to be uniform. (ITER) Iteratively compute new messages until convergence as follows: Variable to Factor: mt+1 e→i[xe] = exp(xewe) × mt j→e[xe] Factor to Variable: mt+1 i→e[xe] = max xEi−e ( ψ(xEi) Y e′∈Ei−e mt e′→i[xe′] ) Also, at each t compute beliefs nt e[xe] = exp(wexe) × mt i→e[xe] × mt j→e[xe] (ESTIM) Upon convergence, output estimate ˆx: for each edge set ˆxe = 1 if ne[1] > ne[0], and ˆxe = 0 otherwise. Remark: If the degree |Ei| of a node is large, the corresponding factor ψ(xEi) will depend on many variables. In general, for very large factors it is intractable to compute the “factor to variable” update (and even to store the factors in memory). However, for our problem the special form of ψ makes this step easy even for large degrees: for each edge e ∈Ei compute ae = max ³ 1, mt e→i[1] mt e→i[0] ´ . Then, if all bi = 1, we have that mt+1 i→e[1] = Y e′∈Ei−e mt e′→i[0] , mt+1 i→e[0] = max e′∈Ei−e ae′ × Y e′∈Ei−e mt e′→i[0] The simplification for general b is as follows: let Fe ⊂Ei −e be the set of bi variables in Ei −e with the largest values of ae, and let Ge ⊂Ei −e be the set of bi −1 variables with largest ae. Then, mt+1 i→e[1] = Y e′∈Ge ae′ Y e′∈Ei−e mt e′→i[0] , mt+1 i→e[0] = Y e′∈Fe ae′ Y e′∈Ei−e mt e′→i[0] These updates are linear in the degree |Ei|. The Computation Tree for Weighted Matching Our proofs rely on the computation tree interpretation [12, 11] of loopy max-product beliefs, which we now describe for the special case of simple matching (bi = 1). Recall the variables of p correspond to edges in G, and nodes in G correspond to factors. For any edge e, the computation tree Te(1) at time 1 is just the edge e, the root of the tree. Both endpoints of the root are leaves. The tree Te(t) at time t is generated from Te(t −1) by adding to each leaf of Te(t −1) a copy of each of its neighbors in G, except for the neighbor that is already present in Te(t −1). The weights of the edges in Te are copied from the corresponding edges in G. Suppose M is a matching on the original graph G, and Te is a computation tree. Then, the image of M in Te is the set of edges in Te whose corresponding copy in G is a member of M. We now illustrate the ideas of this section with a simple example. Example 1: Consider the figure above. G appears on the left, the numbers are the edge weights and the letters are node labels. The max-weight matching M ∗= {(a, b), (c, d)} is depicted in bold. In the center plot we show T(a,b)(4), the computation tree at time t = 4 rooted at edge (a, b). Each node is labeled in accordance to its copy in G. The bold edges in the middle tree depict the matching which is the image of M ∗onto T(a,b)(4). The weight of this matching is 6.6, and it is easy to see that any matching on T(a,b)(4) that includes the root edge will have weight at most 6.6. On the right we depict M, the max-weight matching on the tree T(a,b)(4). M has weight 7.3. In this example we see that even though (a, b) is in the unique optimal matching in G, the beliefs at the root are such that n4 (a,b)[0] > n4 (a,b)[1]. Note also that the dotted edges are not an image of any matching in the original graph G. This example thus illustrates how “spurious” matchings in the computation tree can lead to incorrect beliefs, and estimates. 3 a b c d a b d c d b a a a b c a d a a b d c d b a a a b c d a 1.1 1 1 1.1 1 b b c d c a d 4 Main Result: Equivalence of LP Relaxation and Loopy Max-product In this section we formally state the main result of this paper, and give an outline of the proofs. Theorem 1 Let G = (V, E) be a graph with nonnegative real weights we on the edges e ∈E. Assume the linear programming relaxation LP has a unique optimal solution. Then, the following holds: 1. If the LP relaxation is tight, i.e. if the unique solution is integral, then the max-product converges and the resulting estimate is the optimal matching. 2. If the LP relaxation is not tight, i.e. if the unique solution contains fractional values, then the max-product does not converge. The above theorem implies that LP relaxation and Max-product will both succeed, or both fail, on the same problem instances, and thus are equally powerful for the weighted matching problem. We now prove the two parts of the theorem. In the interest of brevity and clarity, the theorem and the proofs are presented for the conceptually easier case of simple matchings, in which all bi = 1. Also, for the purposes of the proofs we will assume that “convergence” means that there exists a τ < ∞ such that the maximizing assignment arg maxxe nt e(xe) remains constant for all t > τ. Proof of Part 1: Max-Product is as Powerful as LP Relaxation Suppose LP has an integral optimum. Consider now the linear-programming dual of LP, denoted below as DUAL. DUAL : min X i∈V zi s.t. wij ≤zi + zj for all (i, j) ∈E, zi ≥0 for all i ∈V The following lemma states that the standard linear programming properties of complimentary slackness hold in the strict sense for the weighted matching problem (this is a special case of [3, ex. 4.20]). Lemma 1 (strict complimentary slackness) If the solution to LPis unique and integral, and M ∗ is the optimal matching, then there exists an optimal dual solution z to DUAL such that 1. For all (i, j) ∈M ∗, we have wij = zi + zj 2. There exists ϵ > 0 such that for all (i, j) /∈M ∗we have wij ≤zi + zj −ϵ 3. if no edge in M ∗is incident on node i, then zi = 0 4. zi ≤maxe we for all i Let t ≥2wmax ϵ , where wmax = maxe we is the weight of the heaviest edge, and ϵ is as in part 2 of Lemma 1 above. Suppose now that there exists an edge e /∈M ∗for which the belief at time t is incorrect, i.e nt e[1] > nt e[0]. We now show that this leads to a contradiction. 4 Recall that nt e[1] > nt e[0] means that there is a matching M in Te(t) such that (a) the root e ∈M, and (b) M is a max-weight matching on Te(t). Let M ∗ T be the image of M ∗onto Te(t). By definition, e /∈M ∗ T . From e, build an alternating path P by successively adding edges as follows: first add e, then add all edges adjacent to e that are in M ∗ T , then all their adjacent edges that are in M, and so forth until no more edges can be added – this will occur either because no edges are available that maintain the alternating structure, or a leaf of Te(t) has been reached. Note that this will be a path, because M and M ∗ T are matchings and so any node in Te(t) can have at most one edge adjacent to it in each of the two matchings. For illustration, consider Example 1 of section 3. M ∗ T is in the center plot and M is on the right. The above procedure for building P would yield the path adcabcda that goes from the left-most leaf to the right-most leaf. It is easy to see that this path alternates between edges in M and M ∗ T . We now show that w(P ∩M ∗ T ) > w(P ∩M). Let z be the dual optimum corresponding to the ϵ above. Suppose first that neither endpoint of P is a leaf of Te(t). Then, from parts 1 and 3 of Lemma 1 it follows that w(P ∩M ∗ T ) = X (i,j)∈P ∩M ∗ T wij = X (i,j)∈P ∩M ∗ T zi + zj = X i∈P zi. On the other hand, from part 2 of Lemma 1 it follows that w(P ∩M) = X (i,j)∈P ∩M wij ≤ X (i,j)∈P ∩M zi + zj −ϵ = ÃX i∈P zi ! −ϵ|P ∩M|. Now by construction the root e ∈P ∩S, and hence w(P ∩M ∗ T ) > w(P ∩M). A similar argument, with minor modifications, holds for the case when one or both endpoints of P are leaves of Te. For these cases we would need to make explicit use of the fact that t ≥2wmax ϵ . We now show that M cannot be an optimal matching in Te(t). We do so by “flipping” the edges in P to obtain a matching with higher weight. Specifically, let M ′ = M −(P ∩M) + (P ∩M ∗ T ) be the matching containing all edges in M except the ones in P, which are replaced by the edges in P ∩M ∗ T . It is easy to see that M ′ is a matching in Te(t), and that w(M ′) > w(M). This contradicts the choice of M, and shows that for e /∈M ∗the beliefs satisfy nt e[1] ≤nt e[0] for all t large enough. This means that the estimate has converged and is correct for e. A similar argument can be used to show that the max-product estimate converges to the correct answer for e ∈M ∗as well. Hence max-product converges globally to the correct M ∗. Proof of Part 2: LP Relaxation is as Powerful as Max-Product Suppose the optimum solution of LP contains fractional values. We now show that in this case max-product does not converge. As a first step we have the following lemma. Lemma 2 If Max-Product converges, the resulting estimate is M ∗. The proof of this lemma uses the result in [12], that states that if max-product converges then the resulting estimates are “locally optimal”: the posterior probability of the max-product assignment can not be increased by changing values in any induced subgraph in which each connected component contains at most one loop. For the weighted matching problem this local optimality implies global optimality, because the symmetric difference of any two matchings is a union of disjoint paths and cycles. The above lemma implies that, for the proof of part 2 of the theorem, it is sufficient to show that max-product does not converge to the correct answer M ∗. We do this by showing that for any given τ, there exists a t ≥τ such that the max-product estimate at time t will not be M ∗. We first provide a combinatorial characterization of when the LP relaxation is loose. Let M ∗be the max-weight matching on G. An alternating path in G is a path in which every alternate edge is in M ∗, and each node appears at most once. A blossom is an alternating path that wraps onto itself, such that the result is a single odd cycle C and a path R leading out of that cycle3. The importance of blossoms for matching problems is well-known [4]. A bad blossom is a blossom in which the edge weights satisfy w(C ∩M ∗) + 2w(R ∩M ∗) < w(C −M ∗) + 2w(R −M ∗). 3The path may be of zero length, in which case the blossom is just the odd cycle. 5 Example: On the right is a bad blossom: bold edges are in M ∗, numbers are edge weights and alphabets are node labels. Cycle C in this case is abcdu, and path R is cfghi. 3 3 3 3 3 1 1 1 0.5 b a c d u f g h i A dumbbell is an alternating path that wraps onto itself twice, such that the result is two disjoint odd cycles C1 and C2 and an alternating path R connecting the two cycles. In a bad dumbbell the edge weights satisfy w(C1 ∩M ∗) + w(C2 ∩M ∗) + 2w(R ∩M ∗) < w(C1 −M ∗) + w(C2 −M ∗) + 2w(R −M ∗). Example: On the right is a bad dumbbell. Cycles C1 and C2 are abcdu and fghij, and (c, f) is the path R. 3 3 3 3 3 1 3 3 3 3 3 a b c d u f g h i j Proposition 1 If LP relaxation is loose, then in G there exists either a bad blossom, or a bad dumbbell. Proof. The proof of this proposition will appear in a longer version of this paper. (It is also in the appendix submitted along with the paper). Suppose now that max-product converges to M ∗by iteration τ, and suppose also there exists a bad blossom B1 in G. For an edge e ∈B1 ∩M ∗consider the computation tree Te(τ + |V |) for e at time τ + |V |. Let M be the optimal matching on the tree. From the definition of convergence, it follows that near the root e, M will be the image of M ∗onto Te: for any edge e′ in the tree at distance less than |V | from the root, e′ ∈M if and only if its copy in G is in M ∗. This means that the copies in Te of the edges in B1 will contain an alternating path P in Te: every alternate edge of P will be in M. For the bad blossom example above, the alternating path is ihgfcbaudcfghi (it will go once around the cycle and twice around the path of the blossom). Make a new matching M ′ on Te(τ + |V |) by “switching” the edges in this path: M ′ = M −(M ∩P) + (P −M). Then, it is easy to see that w(M) −w(M ′) = w(C ∩M ∗) + 2w(R ∩M ∗) −w(C −M ∗) −2w(R −M ∗). By assumption B1 is a bad blossom, and hence we have that w(M) < w(M ′), which violates the optimality of M. Thus, max-product does not converge to M∗if there exists a bad blossom. A similar proof precludes convergence to M ∗for the case when there is a bad dumbbell. It follows from Proposition 1 that if LP relaxation is loose, then max-product cannot converge to M ∗. 5 Sensor network self-organization via b-matching We now consider the problem of sensor network self-organization. Suppose a large number of lowcost sensors are deployed randomly over an area, and as a first step of any communication or remote sensing application the sensors have to organize themselves into a network [1]. The network should be connected, and robust to occasional failing links, but at the same time it should be sparse (i.e. have nodes with small degrees) due to severe limitations on power available for communication. Simply connecting every pair of sensors that lie within some distance d of each other (close enough to communicate reliably) may lead to large clusters of very densely connected components, and nodes with high degrees. Hence, sparser networks with fewer edges are needed [7]. The throughput of a link drops fast with distance, so the sparse network should mostly contain short edges. The sparsest connected network is achieved by a spanning tree solution. However, a spanning tree may have nodes with large degrees, and a single failed link disconnects it. An interesting set of sparse subgraph constructions with various tradeoffs addressing power efficiency in wireless networks is proposed in [7]. 6 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (a) (b) Figure 1: Network with N = 100 nodes. (a) Nodes within d = 0.5 are connected by an edge. (b) Sparse network obtained by b-matching with b = 5. 0 10 20 30 40 50 60 0 0.02 0.04 0.06 0.08 0.1 0.12 N = 50 N = 100 N = 200 40 60 80 100 120 140 160 180 200 0.984 0.986 0.988 0.99 0.992 0.994 0.996 0.998 1 1.002 BP, b=3 BP, b=5 BP, b=10 LP, b=3 LP, b=5 LP, b=10 (a) (b) Figure 2: (a) Histogram of node degrees versus node density. (b) Average fraction of the LP upper bound on optimal cost obtained using LP relaxation and max-product. We consider using b-matching to find a sparse power-efficient subgraph. We assign edge weights to be proportional to the throughput of the link. For typical sensor network applications the received power (which can be used as a measure of throughput) decays as d−p with distance, where p ∈[2, 4]. We set p = 3 for concreteness, and let the edge weights be we = d−p e . The b-matching objective is now to maximize the total throughput (received power) among sparse subgraphs with degree at most b. We use the max-product algorithm to solve weighted b-matching in a distributed fashion. For our experiments we randomly disperse N nodes in a square region [−1, 1] × [−1, 1]. First we create the adjacency graph for nodes that are close enough to communicate, we set the threshold to be d = 0.5. In Figure 2(a) we plot the histogram over a 100 trials of resulting node degrees. Clearly, as N increases, nodes have increasingly higher degrees. Next we apply max-product (MP) and LP relaxation4 to solve the b-matching objective. As we have established earlier, the performance of LP relaxation, and hence, of MP for b-matching depends on the existence of ’bad blossoms’, i.e. odd-cycles where the weights on the edges are quite similar. We show in simulations that bad blossoms appear rarely for the random graphs and weights in our construction, and LP-relaxation and MP produce nearly optimal b-matchings. For the cases where LP relaxation has fractional edges, and MP has oscillating (or non-converged) edges, we erase them from the final matching and ensure that LP and MP solutions are valid matchings. Also, instead of comparing LP and MP costs to the optimal b-matching cost, we compare them to the LP upper bound on the cost (the cost of the fractional LP solution). This avoids the need to find optimal b-matchings. In figure 1 we plot the dense adjacency graph for N = 100 nodes, and the much sparser b-matching subgraph with b = 5 obtained by MP. Now, consider figure 2(b). We plot the percentage of the LP 4LP is not practical for sensor networks, as it is not easily distributed. 7 0 5 10 15 20 25 30 0.9 0.95 1 b = 5 b = 7 b = 10 Fraction disconnected 5/100 0/100 0/100 Mean power stretch 3.64 1.45 1.06 (a) (b) Figure 3: (a) Average fraction of the LP upper bound on optimal cost obtained using T iterations of max-product. (b) A table showing probability of disconnect, and the power stretch factor for N = 100 averaged over 100 trials. upper bound obtained by MP and by rounded LP relaxation. It can be seen that both LP and MP produce nearly optimal b-matchings, with more than 98 percent of the optimal cost. The percentage decreases slowly with sensor density (with higher N), but improves for larger b. An important performance metric for sensor network self-organization is the power-stretch factor5, which compares the weights of shortest paths in G to weights of shortest paths in the sparse subgraph. In figure 3(b) we display the maximum power stretch factor over all pairs of nodes, averaged over 100 trials. For b = 10 there is almost no loss in power by using the sparse subgraph. A limitation of the b-matching solution is that connectedness of the subgraph is not guaranteed. In fact, for b = 1 it is always disconnected. However, as b increases, the graph gets rarely disconnected. In figure 3(b) we display probability of disconnect over 100 trials. For b = 10 and N = 100 in a longer simulation, the sparse subgraph got disconnected twice over 500 trials. In figure 3(a) we study the performance of MP versus the number of iterations. We run MP for a fixed number of iterations, remove oscillating edges to get a valid matching, and plot the average fraction of the LP upper bound that the solution gets. We set b = 5, and N = 100. Quite surprisingly, MP achieves a large percentage of the optimal cost even with as few as 3 iterations. After 20 this figure exceeds 99 percent. References [1] I.F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, “A survey on sensor networks,” IEEE Communications Magazine, vol. 40, no. 8, pp. 102–114, Aug. 2002. [2] M. Bayati, D. Shah, and M. Sharma, “Maximum weight matching via max-product belief propagation,” in ISIT, Sept. 2005, pp. 1763 – 1767. [3] D. Bertsimas and J. Tsitsiklis. Linear Opitimization. Athena Scientific, 1997. [4] J. Edmonds, “Paths, trees and flowers,” Canadian Journal of Mathematics, vol. 17, pp. 449–467, 1965. [5] B. Huang and T. Jebara, “Loopy belief propagation for bipartite maximum weight b-matching,” in Artificial Intelligence and Statistics (AISTATS), March 2007. [6] F. Kschischang, B. Frey, and H. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Transactions on Information Theory, vol. 47, no. 2, pp. 498–519, Feb. 2001. [7] X. Y. Li, P. J. Wan, Y. Wang, and O. Frieder, “Sparse power efficient topology for wireless networks,” in Proc. IEEE Hawaii Int. Conf. on System Sciences, Jan. 2002. [8] D. Malioutov, J. Johnson, and A. Willsky, “Walk-sums and belief propagation in Gaussian graphical models,” Journal of Machine Learning Research, vol. 7, pp. 2031–2064, Oct. 2006. [9] J. Pearl. Probabilistic inference in intelligent systems. Morgan Kaufmann, 1988. [10] L. Tassiulas and A. Ephremides Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks IEEE Trans. on Automatic Control, vol. 37, no. 12, Dec. 1992. [11] S. Tatikonda and M. Jordan, “Loopy belief propagation and Gibbs measures,” in Uncertainty in Artificial Intelligence, vol. 18, 2002, pp. 493–500. [12] Y. Weiss and W. Freeman, “On the optimality of solutions of the max-product belief-propagation algorithm in arbitrary graphs,” IEEE Trans. on Information Theory, vol. 47, no. 2, pp. 736–744, Feb. 2001. [13] J. Yedidia, W. Freeman, and Y. Weiss. Understanding belief propagation and its generalizations. Exploring AI in the new millennium, pages 239–269, 2003. 5To compute the power-stretch the edges are weighted by d3, i.e. the power needed to get a fixed throughput. 8
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An online Hebbian learning rule that performs Independent Component Analysis Claudia Clopath School of Computer Science and Brain Mind Institute Ecole polytechnique federale de Lausanne 1015 Lausanne EPFL claudia.clopath@epfl.ch Andre Longtin Center for Neural Dynamics University of Ottawa 150 Louis Pasteur, Ottawa alongtin@uottawa.ca Wulfram Gerstner School of Computer Science and Brain Mind Institute Ecole polytechnique federale de Lausanne 1015 Lausanne EPFL wulfram.gerstner@epfl.ch Abstract Independent component analysis (ICA) is a powerful method to decouple signals. Most of the algorithms performing ICA do not consider the temporal correlations of the signal, but only higher moments of its amplitude distribution. Moreover, they require some preprocessing of the data (whitening) so as to remove second order correlations. In this paper, we are interested in understanding the neural mechanism responsible for solving ICA. We present an online learning rule that exploits delayed correlations in the input. This rule performs ICA by detecting joint variations in the firing rates of pre- and postsynaptic neurons, similar to a local rate-based Hebbian learning rule. 1 Introduction The so-called cocktail party problem refers to a situation where several sound sources are simultaneously active, e.g. persons talking at the same time. The goal is to recover the initial sound sources from the measurement of the mixed signals. A standard method of solving the cocktail party problem is independent component analysis (ICA), which can be performed by a class of powerful algorithms. However, classical algorithms based on higher moments of the signal distribution [1] do not consider temporal correlations, i.e. data points corresponding to different time slices could be shuffled without a change in the results. But time order is important since most natural signal sources have intrinsic temporal correlations that could potentially be exploited. Therefore, some algorithms have been developed to take into account those temporal correlations, e.g. algorithms based on delayed correlations [2, 3, 4, 5] potentially combined with higher-order statistics [6], based on innovation processes [7], or complexity pursuit [8]. However, those methods are rather algorithmic and most of them are difficult to interpret biologically, e.g. they are not online or not local or require a preprocessing of the data. Biological learning algorithms are usually implemented as an online Hebbian learning rule that triggers changes of synaptic efficacy based on the correlations between pre- and postsynaptic neurons. A Hebbian learning rule, like Oja’s learning rule [9], combined with a linear neuron model, has been shown to perform principal component analysis (PCA). Simply using a nonlinear neuron combined with Oja’s learning rule allows one to compute higher moments of the distributions which yields ICA if the signals have been preprocessed (whitening) at an earlier stage [1]. In this paper, we are 1 s x y C W Mixing ICA Hebbian Learning Figure 1: The sources s are mixed with a matrix C, x = Cs, x are the presynaptic signals. Using a linear neuron y = Wx, we want to find the matrix W which allows the postsynaptic signals y to recover the sources, y = Ps, where P is a permutation matrix with different multiplicative constants. interested in exploiting the correlation of the signals at different time delays, i.e. a generalization of the theory of Molgedey and Schuster [4]. We will show that a linear neuron model combined with a Hebbian learning rule based on the joint firing rates of the pre- and postsynaptic neurons of different time delays performs ICA by exploiting the temporal correlations of the presynaptic inputs. 2 Mathematical derivation of the learning rule 2.1 The problem We assume statistically independent autocorrelated source signals si with mean < si >= 0 (<> means averaging over time) and correlations < si(t)sj(t′) >= Ki(|t −t′|)δij. The sources s are mixed by a matrix C x = Cs, (1) where x are the mixed signals recorded by a finite number of receptors (bold notation refers to a vector). We think of the receptors as presynaptic neurons that are connected via a weight matrix W to postsynaptic neurons. We consider linear neurons [9], so that the postsynaptic signals y can be written y = Wx. (2) The aim is to find a learning rule that adjusts the appropriate weight matrix W to W ∗(* denotes the value at the solution) so that the postsynaptic signals y recover the independent sources s (Fig 1), i.e. y = Ps where P is a permutation matrix with different multiplicative constants (the sources are recovered in a different order up to a multiplicative constant), which means that, neglecting P, W ∗= C−1. (3) To solve this problem we extend the theory of Molgedey and Schuster [4] in order to derive an online biological hebbian rule. 2.2 Theory of Molgedey and Schuster and generalization The paper of Molgedey and Schuster [4] focuses on the instantaneous correlation matrix but also the time delayed correlations Mij =< xi(t)xj(t + τ) > of the incoming signals. Since the correlation matrix Mij is symmetric, it has up to n(n + 1)/2 independent elements. However, the unknown mixing matrix C has potentially n2 elements (for n sources and n detectors). Therefore, we need to evaluate two delayed correlation matrices M and ¯ M with two different time delays defined as Mij =< xi(t)xj(t + τ2) > ¯ Mij =< xi(t)xj(t + τ1) > (4) to get enough information about the mixing process [10]. From equation 1, we obtain the relation Mij = P l CilCjlΛll and similarly ¯ Mij = P l CilCjl ¯Λll where Λij = δijKi(τ2) and ¯Λij = δijKi(τ1) are diagonal matrices. Since M = CΛCT and ¯ M = C ¯ΛCT , we have (M ¯ M −1)C = C(Λ¯Λ−1). (5) 2 It follows that C can be found from an eigenvalue problem. Since C is the mixing matrix, a simple algorithmic inversion allows Molgedey and Schuster to recover the original sources [4]. 2.3 Our learning rule In order to understand the putative neural mechanism performing ICA derived from the formalism developed above, we need to find an online learning rule describing changes of the synapses as a function of pre- and postsynaptic activity. Taking the inverse of (5), we have C−1 ¯ MM −1 = ¯ΛΛ−1C−1. Therefore, for weights that solve the ICA problem we expect because of (3) that W ∗¯ M = ¯ΛΛ−1W ∗M, (6) which defines the weight matrix W ∗at the solution. For the sake of simplicity, consider only one linear postsynaptic neuron. The generalization to many postsynaptic neurons is straightforward (see section 4). The output signal y of the neuron can be written as y = w∗Tx, where w∗T is a row of the matrix W ∗. Then equation 6 can be written as w∗T ¯ M = λw∗TM, (7) where λ is one element of the diagonal matrix ¯ΛΛ−1. In order to solve this equation, we can use the following iterative update rule with update parameter γ. ˙w = γ[wT ¯ M −λwTM]. (8) The fixed point of this update rule is giving by (7), i.e. w = w∗. Furthermore, multiplication of (7) with w yields λ = wT ¯ Mw wTMw. If we insert the definition of M from (2), we obtain the following rule ˙w = γ[< y(t)x(t + τ1) > −λ < y(t)x(t + τ2) >], (9) with a parameter λ given by λ = < y(t)y(t + τ1) > < y(t)y(t + τ2) >. It is possible to show that ˙w is orthogonal to w. This implies that to first order (in | ˙w/w|), w will keep the same norm during iterations of (9). The rule 9 we derived is a batch-rule, i.e. it averages over all sample signals. We convert this rule into an online learning rule by taking a small learning rate γ and using an online estimate of λ. ˙w = γ[y(t)x(t + τ1) −λ1 λ2 y(t)x(t + τ2)] (10) τλ ˙λ1 = −λ1 + y(t)y(t + τ1) τλ ˙λ2 = −λ2 + y(t)y(t + τ2). Note that the rule defined in (10) uses information on the correlated activity xy of pre- and postsynaptic neurons as well as an estimate of the autocorrelation < yy > of the postsynaptic neuron. τλ is taken sufficiently long so as to average over a representative sample of the signals and |γ| ≪1 is a small learning rate. Stability properties of updates under rule (10) are discussed in section 4. 3 Performances of the learning rule A simple example of a cocktail party problem is shown in Fig 2 where two signals, a sinus and a ramp (saw-tooth signal), have been mixed. The learning rule converges to a correct set of synaptic 3 A B time signals time autocorrelation Ki(t-t’) 0 5 10 -5 -10 C D time signals time signals Figure 2: A. Two periodic source signals, a sinus (thick solid line) and a ramp (thin solid line), are mixed into the presynaptic signals (dotted lines). B. The autocorrelation functions of the two source signals are shown (the sinus in thick solid line and the ramp in thin solid line). The sources are normalized so that Λ(0) = 1 for both. C. The learning rule with τ1 = 3 and τ2 = 0 extracts the sinusoidal output signal (dashed) composed to the two input signals. In agreement with the calculation of stability, γ > 0 , the output is recovering the sinus source because Λsin(3) > Λramp(3). D. The learning rule with τ1 = 10, τ2 = 0, converges to the other signal (dashed line), i.e. the ramp, because Λramp(10) > Λsin(10). Note that the signals have been rescalled since the learning rule recovers the signals up to a multiplicative factor. weights so that the postsynaptic signal recovers correctly one of the sources. Postsynaptic neurons with different combinations of τ1 and τ2 are able to recover different signals (see the section 4 on Stability). In the simulations, we find that the convergence is fast and the performance is very accurate and stable. Here we show only a two-sources problem for the sake of visual clarity. However, the rule can easily recover several mixed sources that have different temporal characteristics. Fig 3 shows an ICA problem with sources s(t) generated by an Ornstein-Uhlenbeck process of the form τsi ˙si = −si + ξ, where ξ is some gaussian noise. The different sources are characterized by different time constants. The learning rule is able to decouple these colored noise signals with gaussian amplitude distribution since they have different temporal correlations. Finally, Fig 4 shows an application with nine different sounds. We used 60 postsynaptic neurons with time delays τ1 chosen uniformly in an interval [1,30ms] and τ2 = 0 . Globally 52 of the 60 neurons recovered exactly 1 source (A, B) and the remaining 8 recovered mixtures of 2 sources (E). One postsynaptic neuron is recovering one of the sources depending on the source’s autocorrelation at time τ1 and τ2 (.i.e. the source with the biggest autocorrelation at time τ1 since τ2 = 0 for all neurons, see section Stability). A histogram (C) shows how many postsynaptic neurons recover each source. However, as it will become clear from the stability analysis below, a few specific postsynaptic neurons tuned to time delays, where the autocorrelation functions intersect (D, at time τ1 = 3ms and τ2 = 0), cannot recover one of the sources precisely (E). 4 A B time singals time singals Figure 3: A. The 3 source signals (solid lines generated with the equation τsi ˙si = −si + ξ with different time constants, where ξ is some gaussian noise) are plotted together with the output signal (dashed). The learning rule is converging to one of the sources. B. Same as before, but only the one signal (solid) that was recovered is shown together with the neuronal output (dashed). A B 1 2 3 4 5 time [s] signals time signals 10 ms C D 1 2 3 4 5 6 7 8 9 0 5 10 15 sources # # of ouput −4 −2 0 2 4 time [ms] autocorrelation E 1 2 3 4 5 time [s] signals Figure 4: Nine different sound sources from [11] were mixed with a random matrix. 60 postsynaptic neurons tuned to different τ1 and τ2 were used in order to recover the sources, i.e. τ1 varies from 1ms to 30ms by steps of 0.5ms and τ2 = 0 for all neurons. A. One source signal (below) is recovered by one of the postsynaptic neurons (above, for clarity reason, the output is shifted upward). B. Zoom on one source (solid line) and one output (dashed line). C. Histogram of the number of postsynaptic neurons recovering each sources. D. Autocorrelation of the different sources. There are several sources with the biggest autocorrelation at time 3ms. E. The postsynaptic neuron tuned to a τ1 = 3ms and τ2 = 0 (above) is not able to recover properly one of the sources even though it still performs well except for the low amplitude parts of the signal (below). 5 4 Stability of the learning rule In principle our online learning rule (10) could lead to several solutions corresponding to different fixed points of the dynamics. Fixed points will be denoted by w∗= ek, which are by construction the row vectors of the decoupling matrix W ∗(see (5) and (7)). The rule 10 has two parameters, i.e. the delays τ1 and τ2 (the τλ is considered fixed). We assume that in our architecture, these delays characterize different properties of the postsynaptic neuron. Neurons with different choices of τ1 and τ2 will potentially recover different signals from the same mixture. The stability analysis will show which fixed point is stable depending on the autocorrelation functions of the signals and the delays τ1 and τ2. We analyze the stability, assuming small perturbation of the weights, i.e. w = ei + ϵej where {ek}, the basis of the matrix C−1, are the fixed points. We obtain the expression (see Appendix for calculation details) ˙ϵ = γϵΛjj(τ1)Λii(τ2) −Λii(τ1)Λjj(τ2) Λii(τ2) , (11) where Λ(τ)ij =< si(t)sj(t + τ) > is the diagonal correlation matrix. To illustrate the stability equation (11), let us take τ1 = 0 and assume that Λii(0) = Λjj(0), i.e. all signals have the same zero-time-lag autocorrelation. In this case (11) reduces to ˙ϵ = γϵ[Λjj(τ1) − Λii(τ1)]. That is the solution ei is stable if Λjj(τ1) < Λii(τ1) for all directions ej (with biggest autocorrelation at time τ1) for γ > 0. If γ < 0, the solution ei is stable for Λjj(τ1) > Λii(τ1). This stability relation is verified in the simulations. Fig 2 shows two signals with different autocorrelation functions. In this example, we chose τ1 = 0 and Λ(0) = I, i.e. the signals are normalized. The learning rule is recovering the signal with the biggest autocorrelation at time τ1, Λkk(τ1), for a positive learning rate. 5 Comparison between Spatial ICA and Temporal ICA One of the algorithms most used to solve ICA is FastICA [1]. It is based on an approximation of negentropy and is purely spatial, i.e. it takes into account only the amplitude distribution of the signal, but not it’s temporal structure. Therefore we show an example (Fig. 5), where three signals generated by Ornstein-Uhlenbeck processes have the same spatial distribution but different time constants of the autocorrelation. With a spatial algorithm data points corresponding to different time slices can be shuffled without any change in the results. Therefore, it cannot solve this example. We tested our example with FastICA downloaded from [11] and it failed to recover the original sources (Fig. 5). However, to our surprise, FastICA could for very few trial solve this problem even though the convergence was not stable. Indeed, since FastICA algorithm is an iterative online algorithm, it takes the signals in the temporal order in which they arrive. Therefore temporal correlations can in some cases be taken into account even though this is not part of the theory of FastICA. 6 Discussions and conclusions We presented a powerful online learning rule that performs ICA by computing joint variations in the firing rates of pre- and postsynaptic neurons at different time delays. This is very similar to a standard Hebbian rule with exception of an additional factor λ which is an online estimate of the output correlations at different time delays. The different delay times τ1, τ2 are necessary to recover different sources. Therefore properties varying between one postsynaptic neuron and the next could lead to different time delays used in the learning rule. We could assume that the time delays are intrinsic properties of each postsynaptic neuron due to for example the distance on the dendrites where the synapse is formed [12], i.e. due to different signal propagation time. The calculation of stability shows that a postsynaptic neuron will recover the signal with the biggest autocorrelation at the considered delay time or the smallest depending of the sign of the learning rates. We assume that for biological signals autocorrelation functions cross so that it’s possible with different postsynaptic neurons to recover all the signals. 6 A B signals distribution time delay autocorrelation C D time signals time signals Figure 5: Two signals generated by an Ornstein-Uhlenbeck process are mixed. A. The signals have the same spatial distributions. B. The time constants of the autocorrelations are different. C. Our learning rule converges to an output (dashed line) recovering one of the signals source (solid line). D. FastICA (dashed line) doesn’t succeed to recover the sources (solid line). The algorithm assumes centered signals. However for a complete mapping of those signals to neural rates, we have to consider positive signals. Nevertheless we can easily compute an online estimate of the mean firing rate and remove this mean from the original rates. This way the algorithm still holds taking neural rates as input. Hyvaerinen proposed an ICA algorithm [8] based on complexity pursuit. It uses the nongaussianity of the residuals once the part of the signals that is predictable from the temporal correlations has been removed. The update step of this algorithm has some similarities with our learning rule even though the approach is completely different since we want to exploit temporal correlations directly rather than formally removing them by a ”predictor”. We also do not assume pre-whitened data and are not considering nongaussianity. Our learning rule considers smooth signals that are assumed to be rates. However, it is commonly accepted that synaptic plasticity takes into account spike trains of pre- and postsynaptic neurons looking at the precise timing of the spikes, i.e. Spike Timing Dependent Plasticity (STDP) [13, 14, 15]. Therefore a spike-based description of our algorithm is currently under study. Appendix: Stability calculation By construction, the row vectors {ek, k = 1,..,n} of W ∗= C−1, the inverse of the mixing matrix, are solutions of the batch learning rule 9 (n is the number of sources). Assume one of these row vectors eT i , (i.e. a fixed point of the dynamic), and consider w = ei + ϵej a small perturbation in direction eT j . Note that {ek} is a basis because det(C) ̸= 0 (the matrix must be invertible). The rule (9) becomes: 7 ˙ϵei =γ[< x(t + τ1)(ei + ϵej)T x(t) > (12) −< (ei + ϵej)T x(t)(ei + ϵej)T x(t + τ1) > < (ei + ϵej)T x(t)(ei + ϵej)T x(t + τ2 >) < x(t + τ2 >)(ei + ϵej)T x(t) >]. We can expand the terms on the righthand side to first order in ϵ. Multiplying the stability expression by eT j (here we can assume that eT j ej = 1 since the recovering of the sources are up to a multiplicative constant), we find: ˙ϵ =γϵ[eT j CΛ(τ1)CT ej][eT i CΛ(τ2)CT ei] −[eT i CΛ(τ1)CT ei][eT j CΛ(τ2)CT ej] eT i CΛ(τ2)CT ei (13) −ϵ4[eT i CΛ(τ1)CT ej][eT j CΛ(τ2)CT ei] eT i CΛ(τ2)CT ei . where Λ(τ)ij =< si(t)sj(t + τ) > is the diagonal matrix. This expression can be simplified because eT i is a row of W ∗= C−1, so that eT i C is the unit vector of the form (0,0,...,1,0,...) where the position of the ”1” indicates the solution number 0. Therefore, we have eT i CΛ(τ)CT ek = Λ(τ)ik. The expression of stability becomes ˙ϵ = γϵΛjj(τ1)Λii(τ2) −Λii(τ1)Λjj(τ2) Λii(τ2) (14) References [1] A. Hyvaerinen, J. Karhunen, and E. Oja. Independent Component Analysis. Wiley-Interscience, 2001. [2] L Tong, R Liu, VC Soon, and YF Huang. Indeterminacy and identifiability of blind identification. IEEE Trans. on Circuits and Systems, 1991. [3] A. Belouchrani, KA. Meraim, JF. Cardoso, and E. Moulines. A blind source separation technique based on second order statistics. IEEE Trans. on Sig. Proc., 1997. [4] L. Molgedey and H.G. Schuster. Separation of a mixture of independent signals using time delayed correlations. Phys. Rev. Lett., 72:3634–37, 1994. [5] A. Ziehe and K. Muller. Tdsep – an efficient algorithm for blind separation using time structure. [6] KR. Mueller, P. Philips, and A. Ziehe. Jade td : Combining higher-order statistics and temporal information for blind source separation (with noise). Proc. Int. Workshop on ICA, 1999. [7] A. Hyvaerinen. Independent component analysis for time-dependent stochastic processes. Proc. Int. Conf. on Art. Neur. Net., 1998. [8] A. Hyvaerinen. Complexity pursuit: Separating interesting components from time-series. Neural Computation, 13:883–898, 2001. [9] E. Oja. A simplified neuron model as principal component analyzer. J. Math. Biol., 15:267 –273, 1982. [10] J.J. Hopfield. Olfactory computation and object perception. PNAS, 88:6462–6466, 1991. [11] H. Gavert, J. Hurri, J. Sarela, and A. Hyvarinen. Fastica and cocktail party demo. http://www.cis.hut.fi/projects/ica/. [12] R. C. Froemke, M. Poo, and Y. Dan. Spike-timing dependent synaptic plasticity depends on dentritic location. Nature, 434:221–225, 2005. [13] G. Bi and M. Poo. Synaptic modification by correlated activity: Hebb’s postulate revisited. Annual Review of Neuroscience, 2001. [14] H. Markram, J. L¨ubke, M. Frotscher, and B. Sakmann. Regulation of synaptic efficacy by coincidence of postsynaptic APs and EPSPs. Science, 275:213–215, 1997. [15] W. Gerstner, R. Kempter, JL. van Hemmen, and H. Wagner. A neuronal learning rule for sub-millisecond temporal coding. Nature, 383:76–78, 1996. 8
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Learning Horizontal Connections in a Sparse Coding Model of Natural Images Pierre J. Garrigues Department of EECS Redwood Center for Theoretical Neuroscience Univ. of California, Berkeley Berkeley, CA 94720 garrigue@eecs.berkeley.edu Bruno A. Olshausen Helen Wills Neuroscience Inst. School of Optometry Redwood Center for Theoretical Neuroscience Univ. of California, Berkeley Berkeley, CA 94720 baolshausen@berkeley.edu Abstract It has been shown that adapting a dictionary of basis functions to the statistics of natural images so as to maximize sparsity in the coefficients results in a set of dictionary elements whose spatial properties resemble those of V1 (primary visual cortex) receptive fields. However, the resulting sparse coefficients still exhibit pronounced statistical dependencies, thus violating the independence assumption of the sparse coding model. Here, we propose a model that attempts to capture the dependencies among the basis function coefficients by including a pairwise coupling term in the prior over the coefficient activity states. When adapted to the statistics of natural images, the coupling terms learn a combination of facilitatory and inhibitory interactions among neighboring basis functions. These learned interactions may offer an explanation for the function of horizontal connections in V1 in terms of a prior over natural images. 1 Introduction Over the last decade, mathematical explorations into the statistics of natural scenes have led to the observation that these scenes, as complex and varied as they appear, have an underlying structure that is sparse [1]. That is, one can learn a possibly overcomplete basis set such that only a small fraction of the basis functions is necessary to describe a given image, where the operation to infer this sparse representation is non-linear. This approach is known as sparse coding. Exploiting this structure has led to advances in our understanding of how information is represented in the visual cortex, since the learned basis set is a collection of oriented, Gabor-like filters that resemble the receptive fields in primary visual cortex (V1). The approach of using sparse coding to infer sparse representations of unlabeled data is useful for classification as shown in the framework of self-taught learning [2]. Note that classification performance relies on finding “hard-sparse” representations where a few coefficients are nonzero while all the others are exactly zero. An assumption of the sparse coding model is that the coefficients of the representation are independent. However, in the case of natural images, this is not the case. For example, the coefficients corresponding to quadrature pair or colinear Gabor filters are not independent. This has been shown and modeled in the early work of [3], in the case of the responses of model complex cells [4], feedforward responses of wavelet coefficients [5, 6, 7] or basis functions learned using independent component analysis [8, 9]. These dependencies are informative and exploiting them leads to improvements in denoising performance [5, 7]. We develop here a generative model of image patches that does not make the independence assumption. The prior over the coefficients is a mixture of a Gaussian when the corresponding basis 1 function is active, and a delta function centered at zero when it is silent as in [10]. We model the binary variables or “spins” that control the activation of the basis functions with an Ising model, whose coupling weights model the dependencies among the coefficients. The representations inferred by this model are also “hard-sparse”, which is a desirable feature [2]. Our model is motivated in part by the architecture of the visual cortex, namely the extensive network of horizontal connections among neurons in V1 [11]. It has been hypothesized that they facilitate contour integration [12] and are involved in computing border ownership [13]. In both of these models the connections are set a priori based on geometrical properties of the receptive fields. We propose here to learn the connection weights in an unsupervised fashion. We hope with our model to gain insight into the the computations performed by this extensive collateral system and compare our findings to known physiological properties of these horizontal connections. Furthermore, a recent trend in neuroscience is to model networks of neurons using Ising models, and it has been shown to predict remarkably well the statistics of groups of neurons in the retina [14]. Our model gives a prediction for what is expected if one fits an Ising model to future multi-unit recordings in V1. 2 A non-factorial sparse coding model Let x ∈Rn be an image patch, where the xi’s are the pixel values. We propose the following generative model: x = Φa + ν = m X i=1 aiϕi + ν, where Φ = [ϕ1 . . . ϕm] ∈Rn×m is an overcomplete transform or basis set, and the columns ϕi are its basis functions. ν ∼N(0, ǫ2In) is small Gaussian noise. Each coefficient ai = si+1 2 ui is a Gaussian scale mixture (GSM). We model the multiplier s with an Ising model, i.e. s ∈{−1, 1}m has a Boltzmann-Gibbs distribution p(s) = 1 Z e 1 2 sT Ws+bT s, where Z is the normalization constant. If the spin si is down (si = −1), then ai = 0 and the basis function ϕi is silent. If the spin si is up (si = 1), then the basis function is active and the analog value of the coefficient ai is drawn from a Gaussian distribution with ui ∼N(0, σ2 i ). The prior on a can thus be described as a “hard-sparse” prior as it is a mixture of a point mass at zero and a Gaussian. The corresponding graphical model is shown in Figure 1. It is a chain graph since it contains both undirected and directed edges. It bears similarities to [15], which however does not have the intermediate layer a and is not a sparse coding model. To sample from this generative model, one first obtains a sample s from the Ising model, then samples coefficients a according to p(a | s), and then x according to p(x | a) ∼N(Φa, ǫ2In). s1 s2 sm a1 a2 am x1 x2 xn Φ W1m W2m Figure 1: Proposed graphical model The parameters of the model to be learned from data are θ = (Φ, (σ2 i )i=1..m, W, b). This model does not make any assumption about which linear code Φ should be used, and about which units should exhibit dependencies. The matrix W of the interaction weights in the Ising model describes these dependencies. Wij > 0 favors positive correlations and thus corresponds to an excitatory connection, whereas Wij < 0 corresponds to an inhibitory connection. A local magnetic field bi < 0 favors the spin si to be down, which in turn makes the basis function ϕi mostly silent. 2 3 Inference and learning 3.1 Coefficient estimation We describe here how to infer the representation a of an image patch x in our model. To do so, we first compute the maximum a posteriori (MAP) multiplier s (see Section 3.2). Indeed, a GSM model reduces to a linear-Gaussian model conditioned on the multiplier s, and therefore the estimation of a is easy once s is known. Given s = ˆs, let Γ = {i : ˆsi = 1} be the set of active basis functions. We know that ∀i /∈Γ, ai = 0. Hence, we have x = ΦΓaΓ + ν, where aΓ = (ai)i∈Γ and ΦΓ = [(ϕi)i∈Γ]. The model reduces thus to linear-Gaussian, where aΓ ∼N(0, H = diag((σ2 i )i∈Γ)). We have aΓ | x, ˆs ∼N(µ, K), where K = (ǫ−2ΦΓΦT Γ + H−1)−1 and µ = ǫ−2KΦT Γx. Hence, conditioned on x and ˆs, the Bayes Least-Square (BLS) and maximum a posteriori (MAP) estimators of aΓ are the same and given by µ. 3.2 Multiplier estimation The MAP estimate of s given x is given by ˆs = arg maxs p(s | x). Given s, x has a Gaussian distribution N(0, Σ), where Σ = ǫ2In + P i : si=1 σ2 i ϕiϕT i . Using Bayes’ rule, we can write p(s | x) ∝p(x | s)p(s) ∝e−Ex(s), where Ex(s) = 1 2xT Σ−1x + 1 2 log det Σ −1 2sT Ws −bT s. We can thus compute the MAP estimate using Gibbs sampling and simulated annealing. In the Gibbs sampling procedure, the probability that node i changes its value from si to ¯si given x, all the other nodes s¬i and at temperature T is given by p(si →¯si|s¬i, x) = 1 + exp −∆Ex T −1 , where ∆Ex = Ex(si, s¬i) −Ex( ¯si, s¬i). Note that computing Ex requires the inverse and the determinant of Σ, which is expensive. Let ¯Σ and Σ be the covariance matrices corresponding to the proposed state ( ¯si, s¬i) and current state (si, s¬i) respectively. They differ only by a rank 1 matrix, i.e. ¯Σ = Σ + αϕiϕT i , where α = 1 2( ¯si −si)σ2 i . Therefore, to compute ∆Ex we can take advantage of the Sherman-Morrison formula ¯Σ−1 = Σ−1 −αΣ−1ϕi(1 + αϕT i Σ−1ϕi)−1ϕT i Σ−1 (1) and of a similar formula for the log det term log det ¯Σ = log det Σ + log 1 + αϕT i Σ−1ϕi . (2) Using (1) and (2) ∆Ex can be written as ∆Ex = 1 2 α(xT Σ−1ϕi)2 1 + αϕT i Σ−1ϕi −1 2 log 1 + αϕT i Σ−1ϕi + ( ¯si −si) X j̸=i Wijsj + bi . The transition probabilities can thus be computed efficiently, and if a new state is accepted we update Σ and Σ−1 using (1). 3.3 Model estimation Given a dataset D = {x(1), . . . , x(N)} of image patches, we want to learn the parameters θ = (Φ, (σ2 i )i=1..m, W, b) that offer the best explanation of the data. Let p∗(x) = 1 N PN i=1 δ(x −x(i)) be the empirical distribution. Since in our model the variables a and s are latent, we use a variational expectation maximization algorithm [16] to optimize θ, which amounts to maximizing a lower bound on the log-likelihood derived using Jensen’s inequality log p(x | θ) ≥ X s Z a q(a, s | x) log p(x, a, s | θ) q(a, s | x) da, 3 where q(a, s | x) is a probability distribution. We restrict ourselves to the family of point mass distributions Q = {q(a, s | x) = δ(a −ˆa)δ(s −ˆs)}, and with this choice the lower bound on the log-likelihood of D can be written as L(θ, q) = Ep∗[log p(x, ˆa, ˆs | θ)] (3) = Ep∗[log p(x | ˆa, Φ)] | {z } LΦ + Ep∗[log p(ˆa | ˆs, (σ2 i )i=1..m)] | {z } Lσ + Ep∗[log p(ˆs | W, b)] | {z } LW,b . We perform coordinate ascent in the objective function L(θ, q). 3.3.1 Maximization with respect to q We want to solve maxq∈Q L(θ, q), which amounts to finding arg maxa,s log p(x, a, s) for every x ∈D. This is computationally expensive since s is discrete. Hence, we introduce two phases in the algorithm. In the first phase, we infer the coefficients in the usual sparse coding model where the prior over a is factorial, i.e. p(a) = Q i p(ai) ∝Q i exp{−λS(ai)}. In this setting, we have ˆa = arg max a p(x|a) Y i e−λS(ai) = arg min a 1 2ǫ2 ∥x −Φa∥2 2 + λ X i S(ai). (4) With S(ai) = |ai|, (4) is known as basis pursuit denoising (BPDN) whose solution has been shown to be such that many coefficient of ˆa are exactly zero [17]. This allows us to recover the sparsity pattern ˆs, where ˆsi = 2.1[ ˆai ̸= 0] −1 ∀i. BPDN can be solved efficiently using a competitive algorithm [18]. Another possible choice is S(ai) = 1[ai ̸= 0] (p(ai) is not a proper prior though), where (4) is combinatorial and can be solved approximately using orthogonal matching pursuits (OMP) [19]. After several iterations of coordinate ascent and convergence of θ using the above approximation, we enter the second phase of the algorithm and refine θ by using the GSM inference described in Section 3.1 where ˆs = arg max p(s|x) and ˆa = E[a | ˆs, x]. 3.3.2 Maximization with respect to θ We want to solve maxθ L(θ, q). Our choice of variational posterior allowed us to write the objective function as the sum of the three terms LΦ, Lσ and LW,b (3), and hence to decouple the variables Φ, (σ2 i )i=1..m and (W, b) of our optimization problem. Maximization of LΦ. Note that LΦ is the same objective function as in the standard sparse coding problem when the coefficients a are fixed. Let {ˆa(i), ˆs(i)} be the coefficients and multipliers corresponding to x(i). We have LΦ = −1 2ǫ2 N X i=1 ∥x(i) −Φˆa(i)∥2 2 −Nn 2 log 2πǫ2. We add the constraint that ∥ϕi∥2 ≤1 to avoid the spurious solution where the norm of the basis functions grows and the coefficients tend to 0. We solve this ℓ2 constrained least-square problem using the Lagrange dual as in [20]. Maximization of Lσ. The problem of estimating σ2 i is a standard variance estimation problem for a 0-mean Gaussian random variable, where we only consider the samples ˆai such that the spin ˆsi is equal to 1, i.e. σ2 i = 1 card{k : ˆsi (k) = 1} X k : ˆsi(k)=1 ( ˆai (k))2. Maximization of LW,b. This problem is tantamount to estimating the parameters of a fully visible Boltzmann machine [21] which is a convex optimization problem. We do gradient ascent in LW,b, where the gradients are given by ∂LW,b ∂Wij = −Ep∗[sisj] + Ep[sisj] and ∂LW,b ∂bi = −Ep∗[si] + Ep[si]. We use Gibbs sampling to obtain estimates of Ep[sisj] and Ep[si]. 4 Note that since computing the parameters (ˆa, ˆs) of the variational posterior in phase 1 only depends on Φ, we first perform several steps of coordinate ascent in (Φ, q) until Φ has converged, which is the same as in the usual sparse coding algorithm. We then maximize Lσ and LW,b, and after that we enter the second phase of the algorithm. 4 Recovery of the model parameters Although the learning algorithm relies on a method where the family of variational posteriors q(a, s | x) is quite limited, we argue here that if data D = {x(1), . . . , x(N)} is being sampled according to parameters θ0 that obey certain conditions that we describe now, then our proposed learning algorithm is able to recover θ0 with good accuracy using phase 1 only. Let η be the coherence parameter of the basis set which equals the maximum absolute inner product between two distinct basis functions. It has been shown that given a signal that is a sparse linear combination of p basis functions, BP and OMP will identify the optimal basis functions and their coefficients provided that p < 1 2(η−1 + 1), and the sparsest representation of the signal is unique [19]. Similar results can be derived when noise is present (ǫ > 0) [22], but we restrict ourselves to the noiseless case for simplicity. Let ∥s∥↑be the number of spins that are up. We require (W0, b0) to be such that Pr ∥s∥↑< 1 2(η−1 + 1) ≈1, which can be enforced by imposing strong negative biases. A data point x(i) ∈D thus has a high probability of yielding a unique sparse representation in the basis set Φ. Provided that we have a good estimate of Φ we can recover its sparse representation using OMP or BP, and therefore identify s(i) that was used to originally sample x(i). That is we recover with high probability all the samples from the Ising model used to generate D, which allows us to recover (W0, b0). We provide for illustration a simple example of model recovery where n = 7 and m = 8. Let (e1, . . . , e7) be an orthonormal basis in R7. We let Φ0 = [e1, . . . e7, 1 √ 7 P i ei]. We fix the biases b0 at −1.2 such that the model is sufficiently sparse as shown by the histogram of ∥s∥↑in Figure 2, and the weights W0 are sampled according to a Gaussian distribution. The variance parameters σ0 are fixed to 1. We then generate synthetic data by sampling 100000 data from this model using θ0. We then estimate θ from this synthetic data using the variational method described in Section 3 using OMP and phase 1 only. We found that the basis functions are recovered exactly (not shown), and that the parameters of the Ising model are recovered with high accuracy as shown in Figure 2. 0 1 2 3 4 5 6 7 0 2 4 6 8 10 12 14x 10 4 sparsity histogram 1 2 3 4 5 6 7 −2 −1 0 b0 1 2 3 4 5 6 7 −2 −1 0 b W0 −0.2 −0.1 0 0.1 0.2 W −0.2 −0.1 0 0.1 0.2 Figure 2: Recovery of the model. The histogram of ∥s∥↑is such that the model is sparse. The parameters (W, b) learned from synthetic data are close to the parameters (W0, b0) from which this data was generated. 5 Results for natural images We build our training set by randomly selecting 16 × 16 image patches from a standard set of 10 512 × 512 whitened images as in [1]. It has been shown that change of luminance or contrast have little influence on the structure of natural scenes [23]. As our goal is to uncover this structure, we subtract from each patch its own mean and divide it by its standard deviation such that our dataset is contrast normalized (we do not consider the patches whose variance is below a small threshold). We fix the number of basis functions to 256. In the second phase of the algorithm we only update Φ, and we have found that the basis functions do not change dramatically after the first phase. Figure 3 shows the learned parameters Φ, σ and b. The basis functions resemble Gabor filters at a variety of orientations, positions and scales. We show the weights W in Figure 4 according to 5 Φ 0 50 100 150 200 250 0 1 2 σ 0 50 100 150 200 250 −1 −0.5 0 b Figure 3: On the left is shown the entire set of basis functions Φ learned on natural images. On the right are the learned variances (σ2 i )i=1..m (top) and the biases b in the Ising model (bottom). the spatial properties (position, orientation, length) of the basis functions that are linked together by them. Each basis function is denoted by a bar that indicates its position, orientation, and length within the 16 × 16 patch. (a) 10 most positive weights (b) 10 most negative weights ϕi ϕj ϕk (c) Weights visualization Wij < 0 Wik > 0 (d) Association fields Figure 4: (a) (resp. (b)) shows the basis function pairs that share the strongest positive (resp. negative) weights ordered from left to right. Each subplot in (d) shows the association field for a basis function ϕi whose position and orientation are denoted by the black bar. The horizontal connections (Wij)j̸=i are displayed by a set of colored bars whose orientation and position denote those of the basis functions ϕj to which they correspond, and the color denotes the connection strength (see (c)). We show a random selection of 36 association fields, see www.eecs.berkeley.edu/ garrigue/nips07.html for the whole set. We observe that the connections are mainly local and connect basis functions at a variety of orientations. The histogram of the weights (see Figure 5) shows a long positive tail corresponding to a bias toward facilitatory connections. We can see in Figure 4a,b that the 10 most “positive” pairs have similar orientations, whereas the majority of the 10 most “negative” pairs have dissimilar orientations. We compute for a basis function the average number of basis functions sharing with it a weight larger than 0.01 as a function of their orientation difference in four bins, which we refer to as the “orientation profile” in Figure 5. The error bars are a standard deviation. The resulting orientation profile is consistent with what has been observed in physiological experiments [24, 25]. We also show in Figure 5 the tradeoff between the signal to noise ratio (SNR) of an image patch x and its reconstruction Φˆa, and the ℓ0 norm of the representation ∥ˆa∥0. We consider ˆa inferred using both the Laplacian prior and our proposed prior. We vary λ (see Equation (4)) and ǫ respectively, and average over 1000 patches to obtain the two tradeoff curves. We see that at similar SNR the representations inferred by our model are more sparse by about a factor of 2, which bodes well for compression. We have also compared our prior for tasks such as denoising and filling-in, and have found its performance to be similar to the factorial Laplacian prior even though it does not exploit the dependencies of the code. One possible explanation is that the greater sparsity of our inferred representations makes them less robust to noise. Thus we are currently investigating whether this 6 property may instead have advantages in the self-taught learning setting in improving classification performance. −0.05 0 0.05 0.1 0.15 0 1000 2000 3000 4000 5000 6000 7000 weights coupling weights histogram (W,ΦTΦ) correlation |ϕT i ϕj| Wij 0 0.1 0.2 0.3 0.4 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 1 2 3 4 −2 0 2 4 6 8 10 12 14 orientation bins average # of connections orientation profile −π / 4 π / 4 π / 2 0 5 6 7 8 9 10 11 12 13 20 30 40 50 60 70 80 90 100 110 SNR sparsity tradeoff SNR−sparsity Laplacian prior proposed prior Figure 5: Properties of the weight matrix W and comparison of the tradeoff curve SNR - ℓ0 norm between a Laplacian prior over the coefficients and our proposed prior. To access how much information is captured by the second-order statistics, we isolate a group (ϕi)i∈Λ of 10 basis functions sharing strong weights. Given a collection of image patches that we sparsify using (4), we obtain a number of spins (ˆsi)i∈Λ from which we can estimate the empirical distribution pemp, the Boltzmann-Gibbs distribution pIsing consistent with first and second order correlations, and the factorial distribution pfact (i.e. no horizontal connections) consistent with first order correlations. We can see in Figure 6 that the Ising model produces better estimates of the empirical distribution, and results in better coding efficiency since KL(pemp||pIsing) = .02 whereas KL(pemp||pfact) = .1. 10 −5 10 −4 10 −3 10 −2 10 −1 10 −5 10 −4 10 −3 10 −2 10 −1 Model probability Empirical probaility factorial model Ising model all spins up all spins down 3 spins up Figure 6: Model validation for a group of 10 basis functions (right). The empirical probabilities of the 210 patterns of activation are plotted against the probabilities predicted by the Ising model (red), the factorial model (blue), and their own values (black). These patterns having exactly three spins up are circled. The prediction of the Ising model is noticably better than that of the factorial model. 6 Discussion In this paper, we proposed a new sparse coding model where we include pairwise coupling terms among the coefficients to capture their dependencies. We derived a new learning algorithm to adapt the parameters of the model given a data set of natural images, and we were able to discover the dependencies among the basis functions coefficients. We showed that the learned connection weights are consistent with physiological data. Furthermore, the representations inferred in our model have greater sparsity than when they are inferred using the Laplacian prior as in the standard sparse coding model. Note however that we have not found evidence that these horizontal connections facilitate contour integration, as they do not primarily connect colinear basis functions. Previous models in the literature simply assume these weights according to prior intuitions about the function of horizontal connections [12, 13]. It is of great interest to develop new models and unsupervised learning schemes possibly involving attention that will help us understand the computational principles underlying contour integration in the visual cortex. 7 References [1] 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, June 1996. [2] R. Raina, A. Battle, H. Lee, B. Packer, and A.Y. Ng. Self-taught learning: Transfer learning from unlabeled data. Proceedings of the Twenty-fourth International Conference on Machine Learning, 2007. [3] G. Zetzsche and B. Wegmann. The atoms of vision: Cartesian or polar? J. Opt. Soc. Am., 16(7):1554– 1565, 1999. [4] P. Hoyer and A. Hyv¨arinen. A multi-layer sparse coding network learns contour coding from natural images. 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COFIRANK Maximum Margin Matrix Factorization for Collaborative Ranking Markus Weimer∗ Alexandros Karatzoglou† Quoc Viet Le‡ Alex Smola§ Abstract In this paper, we consider collaborative filtering as a ranking problem. We present a method which uses Maximum Margin Matrix Factorization and optimizes ranking instead of rating. We employ structured output prediction to optimize directly for ranking scores. Experimental results show that our method gives very good ranking scores and scales well on collaborative filtering tasks. 1 Introduction Collaborative filtering has gained much attention in the machine learning community due to the need for it in webshops such as those of Amazon, Apple and Netflix. Webshops typically offer personalized recommendations to their customers. The quality of these suggestions is crucial to the overall success of a webshop. However, suggesting the right items is a highly nontrivial task: (1) There are many items to choose from. (2) Customers only consider very few (typically in the order of ten) recommendations. Collaborative filtering addresses this problem by learning the suggestion function for a user from ratings provided by this and other users on items offered in the webshop. Those ratings are typically collected on a five star ordinal scale within the webshops. Learning the suggestion function can be considered either a rating (classification) or a ranking problem. In the context of rating, one predicts the actual rating for an item that a customer has not rated yet. On the other hand, for ranking, one predicts a preference ordering over the yet unrated items. Given the limited size of the suggestion shown to the customer, both (rating and ranking) are used to compile a top-N list of recommendations. This list is the direct outcome of a ranking algorithm, and can be computed from the results of a rating algorithm by sorting the items according to their predicted rating. We argue that rating algorithms solve the wrong problem, and one that is actually harder: The absolute value of the rating for an item is highly biased for different users, while the ranking is far less prone to this problem. One approach is to solve the rating problem using regression. For example for the Netflix prize which uses root mean squared error as an evaluation criterion,1 the most straightforward approach is to use regression. However, the same arguments discussed above apply to regression. Thus, we present an algorithm that solves the ranking problem directly, without first computing the rating. For collaborative rating, Maximum Margin Matrix Factorization (MMMF) [11, 12, 10] has proven to be an effective means of estimating the rating function. MMMF takes advantage of the collaborative effects: rating patterns from other users are used to estimate ratings for the current user. One key ∗Telecooperation Group, TU Darmstadt, Germany, mweimer@tk.informatik.tu-darmstadt.de †Department of Statistics, TU Wien, alexis@ci.tuwien.ac.at ‡Computer Science Department, Stanford University, Stanford, CA 94305, quoc.le@stanford.edu §SML, NICTA, Northbourne Av. 218, Canberra 2601, ACT, Australia, alex.smola@nicta.com.au 1We conjecture that this is the case in order to keep the rules simple, since ranking scores are somewhat nontrivial to define, and there are many different ways to evaluate a ranking, as we will see in the following. 1 advantage of this approach is that it works without feature extraction. Feature extraction is domain specific, e.g. the procedures developed for movies cannot be applied to books. Thus, it is hard to come up with a consistent feature set in applications with many different types of items, as for example at Amazon. Our algorithm is based on this idea of MMMF, but optimizes ranking measures instead of rating measures. Given that only the top ranked items will actually be presented to the user, it is much more important to rank the first items right than the last ones. In other words, it is more important to predict what a user likes than what she dislikes. In more technical terms, the value of the error for estimation is not uniform over the ratings. All of above reasonings lead to the following goals: • The algorithm needs to be able to optimize ranking scores directly. • The algorithm needs to be adaptable to different scores. • The algorithm should not require any features besides the actual ratings. • The algorithm needs to scale well and parallelize such as to deal with millions of ratings arising from thousands of items and users with an acceptable memory footprint. We achieve these goals by combining (a) recent results in optimization, in particular the application of bundle methods to convex optimization problems [14], (b) techniques for representing functions on matrices, in particular maximum margin matrix factorizations [10, 11, 12] and (c) the application of structured estimation for ranking problems. We describe our algorithm COFIRANK in terms of optimizing the ranking measure Normalized Discounted Cumulative Gain (NDCG). 2 Problem Definition Assume that we have m items and u users. The ratings are stored in the sparse matrix Y where Yi,j ∈{0, . . . , r} is the rating of item j by user i and r is some maximal score. Yi,j is 0 if user i did not rate item j. In rating, one estimates the missing values in Y directly while we treat this as a ranking task. Additionally, in NDCG [16], the correct order of higher ranked items is more important than that of lower ranked items: Definition 1 (NDCG) Denote by y ∈{1, . . . , r}n a vector of ratings and let π be a permutation of that vector. πi denotes the position of item i after the permutation. Moreover, let k ∈N be a truncation threshold and πs sorts y in decreasing order. In this case the Discounted Cumulative Gains (DCG@k) score [5] and its normalized variant (NDCG@k) are given by DCG@k(y, π) = k X i=1 2yπi −1 log(i + 2) and NDCG@k(y, π) = DCG@k(y, π) DCG@k(y, πs) DCG@k is maximized for π = πs. The truncation threshold k reflects how many recommendations users are willing to consider. NDCG is a normalized version of DCG so that the score is bounded by [0, 1]. Unlike classification and regression measures, DCG is defined on permutations, not absolute values of the ratings. Departing from traditional pairwise ranking measures [4], DCG is positiondependent: Higher positions have more influence on the score than lower positions. Optimizing DCG has gained much interest in the machine learning and information retrieval (e.g. [2]) communities. However, we present the first effort to optimize this measure for collaborative filtering. To perform estimation, we need a recipe for obtaining the permutations π. Since we want our system to be scalable, we need a method which scales not much worse than linearly in the number of the items to be ranked. The avenue we pursue is to estimate a matrix F ∈Rm×u and to use the values Fij for the purpose of ranking the items j for user i. Given a matrix Y of known ratings we are now able to define the performance of F: R(F, Y ) := u X i=1 NDCG@k(Πi, Y i), (1) 2 where Πi is argsort(−F i), it sorts F i in decreasing order.2 While we would like to maximize R(F, Ytest) we only have access to R(F, Ytrain). Hence, we need to restrict the complexity of F to ensure good performance on the test set when maximizing the score on the training set. 3 Structured Estimation for Ranking However, R(F, Y ) is non-convex. In fact, it is piecewise constant and therefore clearly not amenable to any type of smooth optimization. To address this issue we take recourse to structured estimation [13, 15]. Note that the scores decompose into a sum over individual users’ scores, hence we only need to show how minimizing −NDCG(π, y) can be replaced by minimizing a convex upper bound on the latter. Summing over the users then provides us with a convex bound for all of the terms.3 Our conversion works in three steps: 1. Converting NDCG(π, y) into a loss by computing the regret with respect to the optimal permutation argsort(−y). 2. Denote by π a permutation (of the n items a user might want to see) and let f ∈Rn be a estimated rating. We design a mapping ψ(π, f) →R which is linear in f in such a way that maximizing ψ(π, f) with respect to π yields argsort(f). 3. We use the convex upper-bounding technique described by [15] to combine regret and linear map into a convex upper bound which we can minimize efficiently. Step 1 (Regret Conversion) Instead of maximizing NDCG(π, y) we may also minimize ∆(π, y) := 1 −NDCG(π, y). (2) ∆(π, y) is nonnegative and vanishes for π = πs. Step 2 (Linear Mapping) Key in our reasoning is the use of the Polya-Littlewood-Hardy inequality: For any two vectors a, b ∈Rn their inner product is maximized by sorting a and b in the same order, that is ⟨a, b⟩≤⟨sort(a), sort(b)⟩. This allows us to encode the permuation π = argsort(f) in the following fashion: denote by c ∈Rn a decreasing nonnegative sequence, then the function ψ(π, f) := ⟨c, fπ⟩ (3) is linear in f and maximized with respect to π for argsort(f). Since ci is decreasing by construction, the Polya-Littlewood-Hardy inequality applies. We found that choosing ci = (i + 1)−0.25 produced good results in our experiments. However, we did not formally optimize this parameter. Step 3 (Convex Upper Bound) We adapt a result of [15] which describes how to find convex upper bounds on nonconvex optimization problems. Lemma 2 Assume that ψ is defined as in (3). Moreover let π∗:= argsort(−f) be the ranking induced by f. Then the following loss function l(f, y) is convex in f and it satisfies l(f, y) ≥ ∆(y, π∗). l(f, y) := max π h ∆(π, y) + ⟨c, fπ −f⟩ i (4) Proof We show convexity first. The argument of the maximization over the permutations π is a linear and thus convex function in f. Taking the maximum over a set of convex functions is convex itself, which proves the first claim. To see that it is an upper bound, we use the fact that l(f, y) ≥∆(π∗, y) + ⟨c, fπ∗−f⟩≥∆(π∗, y). (5) The second inequality follows from the fact that π∗maximizes ⟨c, fπ∗⟩. 2M i denotes row i of matrix M. Matrices are written in upper case, while vectors are written in lower case. 3This also opens the possibility for parallelization in the implementation of the algorithm. 3 4 Maximum Margin Matrix Factorization Loss The reasoning in the previous section showed us how to replace the ranking score with a convex upper bound on a regret loss. This allows us to replace the problem of maximizing R(F, Y ) by that of minimizing a convex function in F, namely L(F, Y ) := u X i=1 l(F i, Y i) (6) Matrix Regularization Having addressed the problem of non-convexity of the performance score we need to find an efficient way of performing capacity control of F, since we only have L(F, Ytrain) at our disposition, whereas we would like to do well on L(F, Ytest). The idea to overcome this problem is by means of a regularizer on F, namely the one proposed for Maximum Margin Factorization by Srebro and coworkers[10, 11, 12]. The key idea in their reasoning is to introduce a regularizer on F via Ω[F] := 1 2 min M,U [tr MM ⊤+ tr UU ⊤] subject to UM = F. (7) More specifically, [12] show that the above is a proper norm on F. While we could use a semidefinite program as suggested in [11], the latter is intractable for anything but the smallest problems.4 Instead, we replace F by UM and solve the following problem: minimize M,U L(UM, Ytrain) + λ 2 tr MM ⊤+ tr UU ⊤ (8) Note that the above matrix factorization approach effectively allows us to learn an item matrix M and a user matrix U which will store the specific properties of users and items respectively. This approach learns the features of the items and the users. The dimension d of M ∈Rd×m and U ∈Rd×u is chosen mainly based on computational concerns, since a full representation would require d = min(m, u). On large problems the storage requirements for the user matrix can be enormous and it is convenient to choose d = 10 or d = 100. Algorithm While (8) may not be jointly convex in M and U any more, it still is convex in M and U individually, whenever the other term is kept fixed. We use this insight to perform alternating subspace descent as proposed by [10]. Note that the algorithm does not guarantee global convergence, which is a small price to pay for computational tractability. repeat For fixed M minimize (8) with respect to U. For fixed U minimize (8) with respect to M. until No more progress is made or a maximum iteration count has been reached. Note that on problems of the size of Netflix the matrix Y has 108 entries, which means that the number of iterations is typically time limited. We now discuss a general optimization method for solving regularized convex optimization problems. For more details see [14]. 5 Optimization Bundle Methods We discuss the optimization over the user matrix U first, that is, consider the problem of minimizing R(U) := L(UM, Ytrain) + λ 2 tr UU ⊤ (9) The regularizer tr UU ⊤is rather simple to compute and minimize. On the other hand, L is expensive to compute, since it involves maximizing l for all users. Bundle methods, as proposed in [14] aim to overcome this problem by performing successive Taylor approximations of L and by using them as lower bounds. In other words, they exploit the fact that L(UM, Ytrain) ≥L(UM ′, Ytrain) + tr(M −M ′)⊤∂ML(UM ′, Y )∀M, M ′. 4In this case we optimize over » A F F ⊤ B – ⪰0 where Ω[F] is replaced by 1 2[tr A + tr B]. 4 Algorithm 1 Bundle Method(ϵ) Initialize t = 0, U0 = 0, b0 = 0 and H = ∞ repeat Find minimizer Ut and value L of the optimization problem minimize U max 0≤j≤t tr U ⊤ j M + bj + λ 2 tr U ⊤U. Compute Ut+1 = ∂UL(UtM, Ytrain) Compute bt+1 = L(UtM, Ytrain) −tr Ut+1Mt if H′ := tr U ⊤ t+1Mt + bt+1 + λ 2 tr UU ⊤≤H then Update H ←H′ end if until H −L ≤ϵ Since this holds for arbitrary M ′, we may pick a set of Mi and use the maximum over the Taylor approximations at locations Mi to lower-bound L. Subsequently, we minimize this piecewise linear lower bound in combination with λ 2 tr UU ⊤to obtain a new location where to compute our next Taylor approximation and iterate until convergence is achieved. Algorithm 1 provides further details. As we proceed with the optimization, we obtain increasingly tight lower bounds on L(UM, Ytrain). One may show [14] that the algorithm converges to ϵ precision with respect to the minimizer of R(U) in O(1/ϵ) steps. Moreover, the initial distance from the optimal solution enters the bound only logarithmically. After solving the optimization problem in U we switch to optimizing over the item matrix M. The algorithm is virtually identical to that in U, except that we now need to use the regularizer in M instead of that in U. We find experimentally that a small number of iterations (less than 10) is more than sufficient for convergence. Computing the Loss So far we simply used the loss l(f, y) of (4) to define a convex loss without any concern to its computability. To implement Algorithm 1, however, we need to be able to solve the maximization of l with respect to the set of permutations π efficiently. One may show that computing the π which maximizes l(f, y) is possible by solving the inear assignment problem min P i P j Ci,jXi,j with the cost matrix: Ci,j = κi 2Y [j] −1 DCG(Y, k, πs)log(i + 1) −cifj with κi = 1 if i < k, 0 otherwise Efficient algorithms [7] based on the Hungarian Marriage algorithm (also referred to as the KuhnMunkres algorithm) exist for this problem [8]: it turns out that this integer programming problem can be solved by invoking a linear program. This in turn allows us to compute l(f, y) efficiently. Computing the Gradients The second ingredient needed for applying the bundle method is to compute the gradients of L(F, Y ) with respect to F, since this allows us to compute gradients with respect to M and U by applying the chain rule: ∂ML(UM, Y ) = U ⊤∂F L(X, F, Y ) and ∂UL(UM, Y ) = ∂F L(X, F, Y )⊤M L decomposes into losses on individual users as described in (6). For each user i only row i of F matters. It follows that ∂F L(F, Y ) is composed of the gradients of l(F i, Y i). Note that for l defined as in (4) we know that ∂F il(F i, Y i) = [c −c¯π−1]. Here we denote by ¯π the maximizer of of the loss and c¯π−1 denotes the application of the inverse permutation ¯π−1 to the vector c. 5 6 Experiments We evaluated COFIRANK with the NDCG loss just defined (denoted by COFIRANK-NDCG) as well as with loss functions which optimize ordinal regression (COFIRANK-Ordinal) and regression (COFIRANK-Regression). COFIRANK-Ordinal applies the algorithm described above to preference ranking by optimizing the preference ranking loss. Similarly, COFIRANK-Regression optimizes for regression using the root mean squared loss. We looked at two real world evaluation settings: “weak” and “strong” [9] generalization on three publicly available data sets: EachMovie, MovieLens and Netflix. Statistics for those can be found in table 1. Dataset Users Movies Ratings EachMovie 61265 1623 2811717 MovieLens 983 1682 100000 Netflix 480189 17770 100480507 Table 1: Data set statistics Weak generalization is evaluated by predicting the rank of unrated items for users known at training time. To do so, we randomly select N = 10, 20, 50 ratings for each user for training and and evaluate on the remaining ratings. Users with less then 20, 30, 60 rated movies where removed to ensure that the we could evaluate on at least 10 movies per user We compare COFIRANK-NDCG, COFIRANK-Ordinal, COFIRANK-Regression and MMMF [10]. Experimental results are shown in table 2. For all COFIRANK experiments, we choose λ = 10. We did not optimize for this parameter. The results for MMMF were obtained using MATLAB code available from the homepage of the authors of [10]. For those, we used λ = 1 1.9 for EachMovie, and λ = 1 1.6 for MovieLens as it is reported to yield the best results for MMMF. In all experiments, we choose the dimensionality of U and M to be 100. All COFIRANK experiments and those of MMMF on MovieLens were repeated ten times. Unfortunately, we underestimated the runtime and memory requirements of MMMF on EachMovie. Thus, we cannot report results on this data set using MMMF. Additionally, we performed some experiments on the Netflix data set. However, we cannot compare to any of the other methods on that data set as to the best of our knowledge, COFIRANK is the first collaborative ranking algorithm to be applied to this data set, supposedly because of its large size. Strong generalization is evaluated on users that were not present at training time. We follow the procedure described in [17]: Movies with less than 50 ratings are discarded. The 100 users with the most rated movies are selected as the test set and the methods are trained on the remaining users. In evaluation, 10, 20 or 50 ratings from those of the 100 test users are selected. For those ratings, the user training procedure is applied to optimize U. M is kept fixed in this process to the values obtained during training. The remaining ratings are tested using the same procedure as for the weak Method N=10 N=20 N=50 EachMovie COFIRANK-NDCG 0.6562 ± 0.0012 0.6644 ± 0.0024 0.6406 ± 0.0040 COFIRANK-Ordinal 0.6727 ± 0.0309 0.7240 ± 0.0018 0.7214 ± 0.0076 COFIRANK-Regression 0.6114 ± 0.0217 0.6400 ± 0.0354 0.5693 ± 0.0428 MovieLens COFIRANK-NDCG 0.6400 ± 0.0061 0.6307 ± 0.0062 0.6076 ± 0.0077 COFIRANK-Ordinal 0.6233 ± 0.0039 0.6686 ± 0.0058 0.7169 ± 0.0059 COFIRANK-Regression 0.6420 ± 0.0252 0.6509 ± 0.0190 0.6584 ± 0.0187 MMMF 0.6061 ± 0.0037 0.6937 ± 0.0039 0.6989 ± 0.0051 Netflix COFIRANK-NDCG 0.6081 0.6204 COFIRANK-Regression 0.6082 0.6287 Table 2: Results for the weak generalization setting experiments. We report the NDCG@10 accuracy for various numbers of training ratings used per user. For most results we report the mean over ten runs and the standard deviation. We also report the p-values for the best vs. second best score. 6 Method N=10 N=20 N=50 EachMovie COFIRANK-NDCG 0.6367 ± 0.001 0.6619 ± 0.0022 0.6771 ± 0.0019 GPR 0.4558 ± 0.015 0.4849 ± 0.0066 0.5375 ± 0.0089 CGPR 0.5734 ± 0.014 0.5989 ± 0.0118 0.6341 ± 0.0114 GPOR 0.3692 ± 0.002 0.3678 ± 0.0030 0.3663 ± 0.0024 CGPOR 0.3789 ± 0.011 0.3781 ± 0.0056 0.3774 ± 0.0041 MMMF 0.4746 ± 0.034 0.4786 ± 0.0139 0.5478 ± 0.0211 MovieLens COFIRANK-NDCG 0.6237 ± 0.0241 0.6711 ± 0.0065 0.6455 ± 0.0103 GPR 0.4937 ± 0.0108 0.5020 ± 0.0089 0.5088 ± 0.0141 CGPR 0.5101 ± 0.0081 0.5249 ± 0.0073 0.5438 ± 0.0063 GPOR 0.4988 ± 0.0035 0.5004 ± 0.0046 0.5011 ± 0.0051 CGPOR 0.5053 ± 0.0047 0.5089 ± 0.0044 0.5049 ± 0.0035 MMMF 0.5521 ± 0.0183 0.6133 ± 0.0180 0.6651 ± 0.0190 Table 3: The NGDC@10 accuracy over ten runs and the standard deviation for the strong generalization evaluation. generalization. We repeat the whole process 10 times and again use λ = 10 and a dimensionality of 100. We compare COFIRANK-NDCG to Gaussian Process Ordinal Regression (GPOR) [3] Gaussian Process Regression (GPR) and the collaborative extensions (CPR, CGPOR) [17]. Table 3 shows our results compared to the ones from [17]. COFIRANK performs strongly compared to most of the other tested methods. Particularly in the strong generalization setting COFIRANK outperforms the existing methods in almost all the settings. Note that all methods except COFIRANK and MMMF use additional extracted features which are either provided with the dataset or extracted from the IMDB. MMMF and COFIRANK only rely on the rating matrix. In the weak generalization experiments on the MovieLens data, COFIRANK performs better for N = 20 but is marginally outperformed by MMMF for the N = 10 and N = 50 cases. We believe that with proper parameter tuning, COFIRANK will perform better in these cases. 7 Discussion and Summary COFIRANK is a novel approach to collaborative filtering which solves the ranking problem faced by webshops directly. It can do so faster and at a higher accuracy than approaches which learn a rating to produce a ranking. COFIRANK is adaptable to different loss functions such as NDCG, Regression and Ordinal Regression in a plug-and-play manner. Additionally, COFIRANK is well suited for privacy concerned applications, as the optimization itself does not need ratings from the users, but only gradients. Our results, which we obtained without parameters tuning, are on par or outperform several of the most successful approaches to collaborative filtering like MMMF, even when they are used with tuned parameters. COFIRANK performs best on data sets of realistic sizes such as EachMovie and significantly outperforms other approaches in the strong generalization setting. In our experiments, COFIRANKshows to be very fast. For example, training on EachMovie with N = 10 can be done in less than ten minutes and uses less than 80MB of memory on a laptop. For N = 20, COFIRANK obtained a NDCG@10 of 0.72 after the first iteration, which also took less than ten minutes. This is the highest NDCG@10 score on that data set we are aware of (apart from the result of COFIRANK after convergence). A comparison to MMMF in that regard is difficult, as it is implemented in MATLAB and COFIRANK in C++. However, COFIRANK is more than ten times faster than MMMF while using far less memory. In the future, we will exploit the fact that the algorithm is easily parallelizable to obtain even better performance on current multi-core hardware as well as computer clusters. Even the current implementation allows us to report the first results on the Netflix data set for direct ranking optimization. Acknowledgments: Markus Weimer is funded by the German Research Foundation as part of the Research Training Group 1223: “Feedback-Based Quality Management in eLearning”. Software: COFIRANK is available from http://www.cofirank.org 7 References [2] C. J. Burges, Q. V. Le, and R. Ragno. Learning to rank with nonsmooth cost functions. In B. Sch¨olkopf, J. Platt, and T. Hofmann, editors, Advances in Neural Information Processing Systems 19, 2007. [3] W. Chu and Z. Ghahramani. Gaussian processes for ordinal regression. J. Mach. Learn. Res., 6:1019–1041, 2005. [4] R. Herbrich, T. Graepel, and K. Obermayer. Large margin rank boundaries for ordinal regression. In A. J. Smola, P. L. Bartlett, B. Sch¨olkopf, and D. Schuurmans, editors, Advances in Large Margin Classifiers, pages 115–132, Cambridge, MA, 2000. MIT Press. [5] K. Jarvelin and J. Kekalainen. IR evaluation methods for retrieving highly relevant documents. In ACM Special Interest Group in Information Retrieval (SIGIR), pages 41–48. New York: ACM, 2002. [7] R. Jonker and A. Volgenant. A shortest augmenting path algorithm for dense and sparse linear assignment problems. Computing, 38:325–340, 1987. [8] H.W. Kuhn. The Hungarian method for the assignment problem. Naval Research Logistics Quarterly, 2:83–97, 1955. [9] B. Marlin. Collaborative filtering: A machine learning perspective. Masters thesis, University of Toronto, 2004. [10] J. Rennie and N. Srebro. Fast maximum margin matrix factoriazation for collaborative prediction. In Proc. Intl. Conf. Machine Learning, 2005. [11] N. Srebro, J. Rennie, and T. Jaakkola. Maximum-margin matrix factorization. In L. K. Saul, Y. Weiss, and L. Bottou, editors, Advances in Neural Information Processing Systems 17, Cambridge, MA, 2005. MIT Press. [12] N. Srebro and A. Shraibman. Rank, trace-norm and max-norm. In P. Auer and R. Meir, editors, Proc. Annual Conf. Computational Learning Theory, number 3559 in Lecture Notes in Artificial Intelligence, pages 545–560. Springer-Verlag, June 2005. [13] B. Taskar, C. Guestrin, and D. Koller. Max-margin Markov networks. In S. Thrun, L. Saul, and B. Sch¨olkopf, editors, Advances in Neural Information Processing Systems 16, pages 25–32, Cambridge, MA, 2004. MIT Press. [14] C.H. Teo, Q. Le, A.J. Smola, and S.V.N. Vishwanathan. A scalable modular convex solver for regularized risk minimization. In Conference on Knowledge Discovery and Data Mining, 2007. [15] I. Tsochantaridis, T. Joachims, T. Hofmann, and Y. Altun. Large margin methods for structured and interdependent output variables. J. Mach. Learn. Res., 6:1453–1484, 2005. [16] E. Voorhees. Overview of the TREC 2001 question answering track. In Text REtrieval Conference (TREC) Proceedings. Department of Commerce, National Institute of Standards and Technology, 2001. NIST Special Publication 500-250: The Tenth Text REtrieval Conference (TREC 2001). [17] S. Yu, K. Yu, V. Tresp, and H. P. Kriegel. Collaborative ordinal regression. In W.W. Cohen and A. Moore, editors, Proc. Intl. Conf. Machine Learning, pages 1089–1096. ACM, 2006. 8
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Infinite State Bayesian Networks Max Welling∗, Ian Porteous, Evgeniy Bart† Donald Bren School of Information and Computer Sciences University of California Irvine Irvine, CA 92697-3425 USA {welling,iporteou}@ics.uci.edu, bart@caltech.edu Abstract A general modeling framework is proposed that unifies nonparametric-Bayesian models, topic-models and Bayesian networks. This class of infinite state Bayes nets (ISBN) can be viewed as directed networks of ‘hierarchical Dirichlet processes’ (HDPs) where the domain of the variables can be structured (e.g. words in documents or features in images). We show that collapsed Gibbs sampling can be done efficiently in these models by leveraging the structure of the Bayes net and using the forward-filtering-backward-sampling algorithm for junction trees. Existing models, such as nested-DP, Pachinko allocation, mixed membership stochastic block models as well as a number of new models are described as ISBNs. Two experiments have been performed to illustrate these ideas. 1 Introduction Bayesian networks remain the cornerstone of modern AI. They have been applied to a wide range of problems both in academia as well as in industry. A recent development in this area is a class of Bayes nets known as topic models (e.g. LDA [1]) which are well suited for structured data such as text or images. A recent statistical sophistication of topic models is a nonparametric extension known as HDP [2], which adaptively infers the number of topics based on the available data. This paper has the goal of bridging the gap between these three developments. We propose a general modeling paradigm, the “infinite state Bayes net” (ISBN), that incorporates these three aspects. We consider models where the variables may have the nested structure of documents and images, may have infinite discrete state spaces, and where the random variables are related through the intuitive causal dependencies of a Bayes net. ISBN’s can be viewed as collections of HDP “modules” connected together to form a network. Inference in these networks is achieved through a two-stage Gibbs sampler, which combines the “forward-filtering-backward-sampling algorithm” [3] extended to junction trees and the direct assignment sampler for HDPs [2]. 2 Bayes Net Structure for ISBN Consider observed random variables xA ≜{xa}, a = 1..A. These variables can take values in an arbitrary domain. In the following we will assume that xa is sampled from a (conditional) distribution in the exponential family. We will also introduce hidden (unobserved, latent) variables {zb}, b = 1..B which will always take discrete values. The indices a, b thus index the nodes of the Bayesian network. We will introduce a separate index, e.g. na, to label observations. In the simplest setting we assume IID data n = i, i.e. N independent identically distributed observations for each variable. We will ∗On sabbatical at Radboud University Nijmegen, Netherlands, Dept. of Biophysics. †Joint appointment at California Institute of Technology, USA, Dept. of Electrical Engineering. 1 1 iz 1 ix 1 1 2 | x z z I 1 N 2 iz 3 iz 3 z S 3 D 2 z W 2 4 | z z S 2 D 2 J 1 z W 1 3 4 | z z z S 1 D 1 J 4 iz 2 ix 4 z S 4 D 2 2 | x z I 2 N ji z ji x | x z I | z j S z W N D J kji z kji x | x z I | z kj S z W N D | z k U E J (a) (b) (c) Figure 1: Graphical representation of (a) Unstructured infinite state Bayesian network, (b) HDP, (c) H2DP. however also be interested in more structured data, such as words in documents, where the index n can be decomposed into e.g. n = (j, ij). In this notation we think of j as labelling a document and ij as labelling a word in document j. To simplify notation we will often write n = (ji). It is straightforward to generalize to deeper nestings of indices, e.g. n = (k, jk, ijk) = (kji) where k can index e.g. books, j chapters and i words. We interpret this as the observed structure in the data, as opposed to the latent structure which we seek to infer. The unobserved structure is labelled with the discrete “assignment variables” za n which assign the object indexed by n to latent groups (a.k.a. topics, factors, components). The assignment variables z together with the observed variables x are organized into a Bayes net, where dependencies are encoded by the usual “conditional probability tables” (CPT), which we denote with φa xa|℘a for observed variables and πb zb|℘b for latent variables1. Here, ℘a denotes the joint state of all the parent variables of xa or zb. When a vertical bar is present we normalize over the variables to the left of it, e.g. P xa φa xa|℘a = 1, ∀a, ℘a. Note that CPTs are considered random variables and may themselves be indexed by (a subset of) n, e.g. φxa|℘aj. We assume that each πb is sampled from a Dirichlet prior: e.g. πzb|℘b ∼D[αbτ zb] independently and identically for all values of ℘b. The distribution τ itself is Dirichlet distributed, τ za ∼D[γa/Ka], where Ka is the number of states for variable za. We can put gamma priors on αa, γa and consider them as random variables as well, but to keep things simple we will consider them fixed variables here. We refer to [4] for algorithms to learn them from data and to [5] and [2] for ways to infer them through sampling. In section 5 we further discuss these hierarchical priors. In drawing BNs we will not include the plates to avoid cluttering the figures. However, it is always possible to infer the number of times variables in the BN are replicated by looking at its indices. For instance, the variable node labelled with πz1|z2j in Fig.3a stands for K(2) × J IID copies of π1 sampled from τ 1. 3 Networks of HDPs In Fig.1b we have drawn the finite version of the HDP. Here φ is a distribution over words, one for each topic value z, and is often referred to a “topic distribution”. Topic values are generated from a document specific distribution π which in turn are generated from a “mother distribution” over topics τ. As was shown in [2] one can take the infinite limit K →∞in this model and arrive at the HDP. We will return to this infinite limit when we describe Gibbs sampling. In the following we will use the same graphical model for finite and infinite versions of ISBNs. 1We will often avoid writing the super-indices a, b when it is clear from the context, e.g. φa xa|℘a = φxa|℘a. 2 1 ji z ji x 1 2 | x z z I 1 z W N 2 ji z 2 z W 1 J 1| z i S 2| z j S 1 D 2 J 2 D j i z o ji x | , ' x z z I z W N J | z j S D 1 jw z 1 jw x 1 1 | x z I 1| z j S 1 z W 1 N 1 D 1 J 2 jf z 2 jf x 2 z W 2 N 2 D 2 J 0 j z 0 z S 0 D 2| z j S 2 2 | x z I (a) (b) (c) Figure 2: Graphical representation for (a) BiHDP, (b) Mixed membership stochastic block model and (c) the “multimedia” model. One of the key features of the HDP is that topics are shared across all documents indexed by j. The reason for this is the distribution τ: new states are “invented” at this level and become available to all other documents. In other words, there is a single state space for all copies of π. One can interpret j is an instantiation of a dummy, fully observed random variable ι. We could add this node to the BN as a parent of z (since π depends on it) and reinterpret the statement of sharing topics as a fully connected transition matrix between states of ι to states of z. This idea can be extended to a combination of fully observed parent variables and multiple unobserved parent variables, e.g. ι →z2, z3, ι. Moreover, the child variables do not have to be observed either, so we can also replace x →z. In this fashion we can connect together multiple vertical stacks τ →φ →z where each such module is part of a “virtual-HDP” where the joint child states act as virtual data and the joint parent states act as virtual document labels. Examples are given in Figs. 1a (infinite extension of a Bayes net with IID data items) and 3a (infinite extension of Pachinko Allocation). 4 Inference To simplify the presentation we will now restrict attention to a Bayesian network where all CPTs are shared across all data-items (see Fig.1a). In this case data is unstructured, assumed IID and indexed by a flat index n = i. Instead of going through the detailed derivation, which is an extension of the derivation in [2] for HDP, we will describe the sampling process in the following. There is a considerable body of empirical evidence which confirms that marginalizing out the variables π, φ will result in improved inference (e.g. [6, 7]). In this collapsed space, we sample two sets of variables alternatingly, {z} on the hand and {τ} on the other. First, we focus on the latter given z and notice that all τ are conditionally independent given z, x. Sampling τ|(z, x): Given x, z we can compute count matrices2 Nzb|℘b and Nxa|℘a as Nzb=k|℘b=l = P i I[zb i = k ∧℘b i = l] and similarly for Nxa|℘a. Given these counts, for each value of k, l, we now create the following vector: vkl = ατ k/(ατ k +nk|l −1) with nk|l = [1, 2, .., Nk|l]. We then draw a number Nk|l Bernoulli random variables with probability of success given by the elements of v, which we call3 st k|l and compute their sum across t: Sk|l = P t st k|l. This procedure is equivalent to running a Chinese restaurant process (CRP) with Nk|l customers and only keeping track of how many tables become occupied. We will denote it with Sk|l ∼A[Nk|l, ατ k] after Antoniak [8]. Next we compute Sk = P l Sk|l and sample τ from a Dirichlet distribution, τ ∼D[γ, S1, .., Sk]. Note that τ is a distribution over Ka + 1 states, where we now denote with Ka the number of occupied states. If the state corresponding to γ is picked, a new state is created and we increment Ka ←Ka + 1. If on the other hand a state becomes empty, we remove it from 2Note that these can be also used to compute Rao-Blackwellised estimates of π a φ, i.e. E[πzb|℘b] = (αbτ b z + Nzb|℘b)/(αb + N℘b) and similarly for φ. 3These variables are so called auxiliary variables to facilitate sampling τ . 3 1 ji z ji x 1 | x z I 1 z W N 2 ji z 2 z W 3 ji z 3 z W 1 2 | z z j S 2 3 | z z j S 3| z j S 1 D 3 J 2 D 3 D 2 J 1 J 1 n z n x 1 | x z I N 2 n z 3 n z 1 2 | z z S 2 3 | z z S 3 z S 1 D 2 D 3 D (a) (b) Figure 3: Graphical representation for (a) Pachinko Allocation and (b) Nested DP. the list and we decrement Ka ←Ka −1. This will allow assignment variables to add or remove states adaptively4. Sampling z|(τ, x): The conditional probability of all {zi, xi} variables jointly (for fixed i) is given by, P(xi, zi|z¬i, x¬i, τ, α) = Y a F(xa i |xa ¬i, ℘a ¬i) Y b αbτ zb i + N ¬i zb i |℘b i αb + N ¬i ℘b i (1) where N ¬i zb i |℘b i is the number data-cases assigned to group zb i for variable b and its parents assigned to group ℘b i, where we exclude data-case i from this count. Also, F(xa i |xa ¬i, ℘a ¬i = k) = R dφk P(xa i |φk) Q i′\i:℘a i′=k P(xa i′|φk) P(φk) R dφk Q i′\i:℘a i′=k P(xa i′|φk) P(φk) (2) Importantly, equation 1 follows the structure of the original Bayes net, where each term has the form of a conditional distribution P(za i |℘a i ) and is based on sufficient statistics collected over all the other data-cases. Hence, we can use the structure of the Bayes net to sample the assignment variables jointly across the BN (for data-case i). The general technique that allows one to exploit network structure is ‘forward-filtering-backward-sampling’ (FFBS) [3]. Assume for instance that the network is a tree. In that case we first propagate information from the leaves to the root, computing the probabilities P(zb|{xb↓}) as we go where ‘↓’ means that we compute a marginal conditioned on ‘downstream’ evidence. When we reach the root we draw a sample from P(zroot|{xb}). Finally, we work our way back to the leaves, conditioning on drawn samples (which summarize upstream information) and using the marginal probabilities P(zb|{xb↓}) cashed during the filtering phase to represent downstream evidence. For networks with higher treewidth we can extend this technique to junction trees. Alternatively, one can use cut-set sampling techniques [9]. 5 ISBN for Structured Data In section 2 we introduced an index n to label the known structure of the data. The simplest nontrivial example is given by the HDP, where n = (ji) indexing e.g. documents and words. In this case the CPT πz|j is not shared across all data, but is specific to a document. Next consider Fig.1c where n = (kji) is labelling for instance words (i) in chapters (j) in books (k). The first level CPT πz|kj is specific to chapters (and hence books) and is sampled from a Dirichlet distribution with mean given 4We adopted the name ‘infinite state Bayesian network’ because the (Ka + 1)th state actually represents an infinite pool of indistinguishable states. 4 by a second level CPT ρz|k specific to books, which in turn is sampled from a Dirichlet distribution with mean τ z, which finally is sampled from a Dirichlet prior with parameters γ. Sampling occurs again in two phases: sampling ρ, τ|x, z and z|ρ, τ, x while marginalizing out π, φ. To sample from ρ, τ we compute counts Nu|m,jk which is the number of times words were assigned in chapter j and book k to the joint state z = u, ℘= m. We then work our way up the stack, sampling new count arrays S, R as we go, and then down again sampling the CPTs (τ, ρ) using these count arrays5. Note that this is just one step of Gibbs sampling from P(τ, ρ|z, x) and does not (unlike the other phase for z) generate an equilibrium sample from this conditional distribution. ↑: su|jkm ∼A[Nu|jkm, αρu|k] →Su|k = X j,m su|jkm →ru|k ∼A[Su|k, βτ u] →Ru = X k tu|k ↓: τ u ∼D[(γ, Ru)] →ρu|k ∼D[βτ u + Su|k] (3) A similar procedure is defined for the priors of φ and extensions to deeper stacks are straightforward. If all z variables carry the same index n, sampling zn given the hierarchical priors is very similar to the FFBS procedure described in the previous section, except that the count arrays may carry a subset of the indices from n, e.g. N ¬ijk z|℘jk. Since these counts are specific to a chapter they are typically smaller resulting in a higher variance for the samples z. If two neighboring z variables carry different subsets of n labels, e.g. node z0 j in Fig.2c, the conditional distributions are harder to compute. The general rule is to identify and remove all z′ variables that are impacted by changing the value for z under consideration, e.g. {z1 jw, ∀w ∪z2 jf, ∀f} in Fig.2c if we resample z0 j. To compute the conditional probability we set z = k and add the impacted variables z′ back into the system, one-by-one in an arbitrary order and assigning them to their old values. It is also instructive to place DP priors (instead of HDP priors) of the form D[αb/Kb] directly on π (skipping τ). In taking the infinite limit the conditional distribution for existing states zb becomes directly proportional to Nzb|℘b (the αbτ zb term is missing). This has the effect that a new state zb = k that was discovered for some parent state ℘b = l will not be available to other parent states, simply because Nk|l′ = 0, l′ ̸= l. The result is that the state space forks into a tree structure as we move down the Bayes net. When the network structure is a linear chain, this model is equivalent to the ‘nested-DP’ introduced in [10] as a prior on tree-structures. The corresponding Bayes net is depicted in Fig.3b. A chain of length 1 is of course just a Dirichlet process mixture model. A DP prior is certainly appropriate for nodes zb with CPTs that do not depend on other parents or additional labels, e.g. nodes z3 and z4 in Fig.1a. Interestingly, an HDP would also be possible and would result in a different model. We will however follow the convention that we will use the minimum depth necessary for modelling the structure of the data. 6 Examples Example: HDP Perhaps the simplest example is an HDP itself, see Fig.1b. It consists of a single topic node and a single observation node. If we make φ depend on the item index i, i.e. φx|z,i, we obtain the infinite version of the ’user rating profile’ (URP) model [11]. If we make φ depend on j instead and add a prior: ψx|z →φx|z,j, we obtain an “HDP with random effects” [12] which has the benefit that shared topics across documents can vary slightly relative to each other. Example: Infinite State Chains The ‘Pachinko allocation model’ (PAM) [13] consists of a linear chain of assignment variables with document specific transition probabilities, see Fig.3a. It was proposed to model correlations between topics. The infinite version of this is clearly an example of an ISBN. An equivalent Chinese restaurant process formulation was published in [14]. A slight variation on this architecture was described in [15] (POM). Here, images are modeled as mixtures over parts and parts were modeled as mixtures over visual words. Finally, a visual word is a distribution over features. POM is only subtly different from PAM (see Fig.3a) in that parts are not image-specific distributions over words, and so the distribution πz1|z2 does not depend on j. Example: BiHDP This model, depicted in Fig.2a has a data variable xji and two parent topic variables z1 ji and z2 ji. One can think of j as the customer index and i as the product index (and no IID repeated index). The value of x is the rating of that customer for that product. The hidden variables 5Teh’s code npbayes-r21, (available from his web-site) does in fact implement this sampling process. 5 z1 ji and z2 ji represent product groups and customer groups. Every data entry is assigned to both a customer group and a product group which together determine the factor from which we sample the rating. Note that the difference between the assignment variables is that their corresponding CPTs πz1,j and πz2,i depend on j and i respectively. Extensions are easily conceived. For instance, instead of two modalities, we can model multiple modalities (e.g. customers, products, year). Also, single topics can be generalized to hierarchies of topics, so every branch becomes a PAM. Note that for unobserved xji values (not all products have been rated by all customers) the corresponding za ji, zb ji are “dangling” and can be integrated out. The result is that we should skip that variable in the Gibbs sampler. Example: The Mixed-Membership Stochastic Block Model[16] This model is depicted in Fig.2b. The main difference with HDP is that (like BiHDP) π depends on two parent states zi→j and zj→i by which we mean that item i has chosen topic zi→j to interact with item j and vice versa. However, (unlike BiHDP) those topic states share a common distribution π. Indices only run over distinct pairs i > j. These features make the model suitable for modeling social interaction networks or protein-protein interaction networks. The hidden variables jointly label the type of interaction that was used to generate ‘matrix-element’ xij. Example: The Multimedia Model In the above examples we had a single observed variable in the graphical model (repeated over ij). The model depicted in Fig.2c has two observed variables and an assignment variable that is not repeated over items. We can think of the middle node z0 j as the class label for a web-page j. The left branch can then model words on the web-page while the right branch can model visual features on the web-page. Since no sharing is required for z0 j we used a Dirichlet prior. The other variables have the usual HDP priors. 7 Experiments To illustrate the ideas we implemented two models: BiHDP of Fig.2a and the “probabilistic object model” (POM), explained in the previous section. Market Basket Data In this experiment we investigate the performance of BiHDP on a synthetic market basket dataset. We used the IBM Almaden association and sequential patterns generator to create this dataset [17]. This is a standard synthetic transaction dataset generator often used by the association research community. The generated data consists of purchases from simulated groups of customers who have similar buying habits. Similarity of buying habits refers to the fact that customers within a group buy similar groups of items. For example, items like strawberries and cream are likely to be in the same item group and thus are likely to be purchased together in the same market basket. The following parameters were used to generate data for our experiments: 1M transactions, 10K customers, 1K different items, 4 items per transaction on average, 4 item groups per customer group on average, 50 market basket patterns, 50 customer patterns. Default values were used for the remaining parameters. The two assignment variables correspond to customers and items respectively. For a given pair of customer and item groups, a binomial distribution was used to model the probability of a customer group making a purchase from that item group. A collapsed Gibbs sampler was used to fit the model. After 1000 epochs the system converged to 278 customer groups and 39 item factors. Fig.4 shows the results. As can be seen, most item groups correspond directly to the hidden ground truth data. The conclusion is that the model can learn successfully the hidden structure in the data. Learning Visual Vocabularies LDA has also gained popularity to model images as collections of features. The visual vocabulary is usually determined in a preprocessing step where k-means is run to cluster features collected from the training set. In [15] a different approach was proposed in which the visual word vocabulary was learned jointly with fitting the parameters of the model. This can have the benefit that the vocabulary is better adapted to suit the needs of the model. Our extension of their PLSA-based model is the infinite state model given by Fig.3a with 2 hidden variables (instead of 3) and πz1|z2 independent of j. x is modeled as a Gaussian-distributed random variable over feature values, z1 represents the word identity and z2 is the topic index. We used the Harris interest-point detector and 21×21 patches centered on each interest point as input to the algorithm. We normalized the patches to have zero mean. Next we reduced the dimensionality of detections from 441 to 100 using PCA. The procedure described above generates a set of between 6 Learned: 223, 619, 271, 448, 39, 390 True: 223, 271, 448, 39, 427, 677 Learned: 364, 250, 718, 952, 326, 802 True: 364, 718, 952, 326, 542, 98 Learned: 159, 563, 780, 995, 103, 216, 598, 72 True: 159, 563, 780, 995, 103, 216, 542, 72 Learned: 227, 130, 862, 991, 904, 213 True: 227, 130, 862, 991, 904, 213 Learned: 953, 175, 956, 385, 269, 14, 64 True: 953, 175, 956, 385, 269, 14, 956 Learned: 49, 657, 906, 604, 229 True: 49, 657, 906, 604, 229 Learned: 295, 129, 662, 922, 705, 210 True: 295, 129, 662, 922, 705, 68 Learned: 886, 460, 471, 933, 544 True: 886, 460, 471, 933, 917 Learned: 489, 818, 927, 378, 64, 710 True: 489, 818, 927, 378, 64, 247 Learned: 776, 224, 139, 379 True: 776, 224, 139, 379 Figure 4: The 10 most popular item groups learned by the BiHDP model (left) compared to ground truth item groups for market basket data (right). Learned items are ordered by decreasing popularity. Ground truth items have no associated weight; therefore, they were ordered to facilitate comparison with the left row. Nonmatching items are shown in boldface. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Figure 5: Precision Recall curves for Caltech-4 dataset (left) and turntable dataset (right). Solid curve represents POM and dashed curve represents LDA. 50 and 400, 100-dimensional detections per image. Experiments were performed using the Caltech4 and ‘turntable’ datasets. For Caltech-4 we used 130 randomly sampled images from each of the 4 categories for training. LDA was fit using 500 visual words and 50 parts (which we found to give the best results). The turntable database contains images of 15 toy objects. The objects were placed on a turntable and photographed every 5 degrees. We have used angles 15, 20, 25, 35, 40, and 45 for training, and angles 10, 30, and 50 for testing. LDA used 15 topics and 200 visual words (which again was optimal). LDA was then fitted to both datasets using Gibbs sampling. We initialized POM with the output of LDA to make sure the comparison involved similar modes of the distribution. The precision-recall curves for this dataset are shown in Fig.5. Images were labelled by choosing the majority class across the 11 most similar retrieved images. Similarity was measured as the probability of the query image given the part probabilities of the retrieved image. These experiments show that ISBNs can be successfully implemented. We are not interested in claiming superiority of ISBNs, but rather hope to convey that ISBNs are a convenient tool to design models and to facilitate the search for the number of latent states. 8 Discussion We have presented a unified framework to organize the fast growing class of ‘topic models’. By merging ideas from Bayes nets, nonparametric Bayesian statistics and topic models we have arrived at a convenient framework to 1) extend existing models to infinite state spaces, 2) reason about and design new models and 3) derive efficient inference algorithms that exploit the structure of the underlying Bayes net. Not every topic model naturally fits the suit of an ISBN. For instance, the infinite HMM [18] is like a POM model with emission states, but with a single transition probability shared across time. When marginalizing out π this has the effect of coupling all z variables. An efficient sampler for this model was introduced in [19]. Also, in [10, 20] models were studied where a word can be emitted at 7 any node corresponding to a topic variable z. We would need an extra switching variable to fit this into the ISBN framework. We are currently working towards a graphical interface where one can design ISBN models by attaching together HkDP modules and where the system will automatically perform the inference necessary for the task at hand. Acknowledgements This material is based upon work supported by the National Science Foundation under Grant No. 0447903 and No. 0535278 and by ONR under Grant No. 00014-06-1-0734. References [1] D. M. Blei, A. Y. Ng, and M. I. Jordan. Latent Dirichlet allocation. Journal of Machine Learning Research, 3:993–1022, 2003. [2] Y. W. Teh, M. I. Jordan, M. J. Beal, and D. M. Blei. Hierarchical Dirichlet processes. To appear in Journal of the American Statistical Association, 2006. [3] S. L. Scott. Bayesian methods for hidden Markov models, recursive computing in the 21st century. volume 97, pages 337–351, 2002. [4] T. Minka. Estimating a dirichlet distribution. Technical report, 2000. [5] M.D. Escobar and M. West. Bayesian density estimation and inference using mixtures. Journal of the American Statistical Association, 90:577–588, 1995. [6] T.L. Griffiths and M. Steyvers. A probabilistic approach to semantic representation. In Proceedings of the 24th Annual Conference of the Cognitive Science Society, 2002. [7] Y.W. Teh, D. Newman, and M. Welling. A collapsed variational bayesian inference algorithm for latent dirichlet allocation. In NIPS, volume 19, 2006. [8] C.E. Antoniak. Mixtures of Dirichlet processes with applications to bayesian nonparametric problems. The Annals of Statistics, 2:1152–1174, 1974. [9] B. Bidyuk and R. Dechter. Cycle-cutset sampling for Bayesian networks. In Sixteenth Canadian Conf. on AI, 2003. [10] David Blei, Thomas L. Griffiths, Michael I. Jordan, and Joshua B. Tenenbaum. Hierarchical topic models and the nested chinese restaurant process. In Neural Information Processing Systems 16, 2004. [11] B. Marlin. Modeling user rating profiles for collaborative filtering. In Advances in Neural Information Processing Systems 16. 2004. [12] S. Kim and P. Smyth. Hierarchical dirichlet processes with random effects. In NIPS, volume 19, 2006. [13] W. Li and A. McCallum. Pachinko allocation: Dag-structured mixture models of topic correlations. In Proceedings of the 23rd international conference on Machine learning, pages 577–584, 2006. [14] W. Li, A. McCallum, and D. Blei. Nonparametric bayes pachinko allocation. In UAI, 2007. [15] D. Larlus and F. Jurie. Latent mixture vocabularies for object categorization. In British Machine Vision Conference, 2006. [16] E. Airoldi, D. Blei, E. Xing, and S. Fienberg. A latent mixed membership model for relational data. In LinkKDD ’05: Proceedings of the 3rd international workshop on Link discovery, pages 82–89, 2005. [17] R. Agrawal, T. Imielinski, and A. Swami. Mining associations between sets of items in massive databases. In Proc. of the ACM-SIGMOD 1993 Intl Conf on Management of Data, 1993. [18] M.J. Beal, Z. Ghahramani, and C.E. Rasmussen. The infinite hidden markov model. In NIPS, pages 577–584, 2001. [19] Y. W. Teh, D. G¨or¨ur, and Z. Ghahramani. Stick-breaking construction for the Indian buffet process. In Proceedings of the International Conference on Artificial Intelligence and Statistics, volume 11, 2007. [20] W. Li D. Mimno and A. McCallum. Mixtures of hierarchical topics with pachinko allocation. In Proceedings of the 21st International Conference on Machine Learning, 2007. 8
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Regularized Boost for Semi-Supervised Learning Ke Chen and Shihai Wang School of Computer Science The University of Manchester Manchester M13 9PL, United Kingdom {chen,swang}@cs.manchester.ac.uk Abstract Semi-supervised inductive learning concerns how to learn a decision rule from a data set containing both labeled and unlabeled data. Several boosting algorithms have been extended to semi-supervised learning with various strategies. To our knowledge, however, none of them takes local smoothness constraints among data into account during ensemble learning. In this paper, we introduce a local smoothness regularizer to semi-supervised boosting algorithms based on the universal optimization framework of margin cost functionals. Our regularizer is applicable to existing semi-supervised boosting algorithms to improve their generalization and speed up their training. Comparative results on synthetic, benchmark and real world tasks demonstrate the effectiveness of our local smoothness regularizer. We discuss relevant issues and relate our regularizer to previous work. 1 Introduction Semi-supervised inductive learning concerns the problem of automatically learning a decision rule from a set of both labeled and unlabeled data, which has received a great deal of attention due to enormous demands of real world learning tasks ranging from data mining to medical diagnosis [1]. From different perspectives, a number of semi-supervised learning algorithms have been proposed [1],[2], e.g., self-training, co-training, generative models along with the EM algorithm, transductive learning models and graph-based methods. In semi-supervised learning, the ultimate goal is to find out a classification function which not only minimizes classification errors on the labeled training data but also must be compatible with the input distribution by inspecting their values on unlabeled data. To work towards the goal, unlabeled data can be exploited to discover how data is distributed in the input space and then the information acquired from the unlabeled data is used to find a good classifier. As a generic framework, regularization has been used in semi-supervised learning to exploit unlabeled data by working on well known semi-supervised learning assumptions, i.e., the smoothness, the cluster, and the manifold assumptions [1], which leads to a number of regularizers applicable to various semi-supervised learning paradigms, e.g., the measure-based [3], the manifold-based [4], the information-based [5], the entropy-based [6], harmonic mixtures [7] and graph-based regularization [8]. As a generic ensemble learning framework [9] , boosting works by sequentially constructing a linear combination of base learners that concentrate on difficult examples, which results in a great success in supervised learning. Recently boosting has been extended to semi-supervised learning with different strategies. Within the universal optimization framework of margin cost functional [9], semi-supervised MarginBoost [10] and ASSEMBLE [11] were proposed by introducing the “pseudo-classes” to unlabeled data for characterizing difficult unlabeled examples. In essence, such extensions work in a self-training way; the unlabeled data are assigned pseudo-class labels based on the constructed ensemble learner so far, and in turn the pseudo-class labels achieved will be used to find out a new proper learner to be added to the ensemble. The co-training idea was extended to boosting, e.g. CoBoost [12]. More recently, the Agreement Boost algorithm [13] has been developed with a theoretic justification of benefits from the use of multiple boosting learners within the co-training framework. To our knowledge, however, none of the aforementioned semi-supervised boosting algorithms has taken the local smoothness constraints into account. In this paper, we exploit the local smoothness constraints among data by introducing a regularizer to semi-supervised boosting. Based on the universal optimization framework of margin cost functional for boosting [9], our regularizer is applicable to existing semi-supervised boosting algorithms [10]-[13]. Experimental results on the synthetic, benchmark and real world classification tasks demonstrate its effectiveness of our regularizer in semi-supervised boosting learning. In the reminder of this paper, Sect. 2 briefly reviews semi-supervised boosting learning and presents our regularizer. Sect. 3 reports experimental results and the behaviors of regularized semi-supervised boosting algorithms. Sect. 4 discusses relevant issues and the last section draws conclusions. 2 Semi-supervised boosting learning and regularization In the section, we first briefly review the basic idea behind existing semi-supervised boosting algorithms within the universal optimization framework of margin cost functional [9] for making it self-contained. Then we present our Regularized Boost based on the previous work. 2.1 Semi-supervised boosting learning Given a training set, S = L ∪U, of |L| labeled examples, L = {(x1, y1), · · · , (x|L|, y|L|)}, and |U| unlabeled examples, U = {x|L|+1, · · · , x|L|+|U|}, we wish to construct an ensemble learner F(x) = P t wtft(x), where wt is coefficients for linear combination and ft(x) is a base learner, so that P(F(x) ̸= y) is small. Since there exists no label information available for unlabeled data, the critical idea underlying semi-supervised boosting is introducing a pseudo-class [11] or a pseudo margin [10] concept within the universal optimization framework [9] to unlabeled data. Similar to an approach in supervised learning, e.g., [14], a multi-class problem can be converted into binary classification forms. Therefore, our presentation below focuses on the binary classification problem only; i.e. y ∈{−1, 1}. The pseudo-class of an unlabeled example, x, is typically defined as y = sign[F(x)] [11] and its corresponding pseudo margin is yF(x) = |F(x)| [10],[11]. Within the universal optimization framework of margin cost functional [9], the semi-supervised boosting learning is to find F such that the cost of functional C(F) = X xi∈L αiC[yiF(xi)] + X xi∈U αiC[|F(xi)|] (1) is minimized for some non-negative and monotonically decreasing cost function C : R →R and the weight αi ∈R+. In the universal optimization framework [9], constructing an ensemble learner needs to choose a base learner, f(x), to maximize the inner product −⟨∇C(F), f⟩. For unlabeled data, a subgradient of C(F) in (1) has been introduced to tackle its non-differentiable problem [11] and then unlabeled data of pseudo-class labels can be treated in the same way as labeled data in the optimization problem. As a result, finding a proper f(x) amounts to maximizing −⟨∇C(F), f⟩= X i:f(xi)̸=yi αiC′[yiF(xi)] − X i:f(xi)=yi αiC′[yiF(xi)], (2) where yi is the true class label if xi is a labeled example or a pseudo-class label otherwise. After dividing through by −P i∈S αiC′[yiF(xi)] on both sides of (2), finding f(x) to maximize −⟨∇C(F), f⟩is equivalent to searching for f(x) to minimize X i:f(xi)̸=yi D(i) − X i:f(xi)=yi D(i) = 2 X i:f(xi)̸=yi D(i) −1, (3) where D(i), for 1 ≤i ≤|L| + |U|, is the empirical data distribution defined as D(i) = αiC′[yiF (xi)] P i∈S αiC′[yiF (xi)]. From (3), a proper base learner, f(x), can be found by minimizing weighted errors P i:f(xi)̸=yi D(i). Thus, any boosting algorithms specified for supervised learning [9] are now applicable to semi-supervised learning with the aforementioned treatment. For co-training based semi-supervised boosting algorithms [12],[13], the above semi-supervised boosting procedure is applied to each view of data to build up a component ensemble learner. Instead of self-training, the pseudo-class label of an unlabeled example for a specific view is determined by ensemble learners trained on other views of this example. For example, the Agreement Boost [13] defines the co-training cost functional as C(F 1, · · · , F J) = J X j=1 X xi∈L C[yiF j(xi)] + η X xi∈U C[−V (xi)]. (4) Here J views of data are used to train J ensemble learners, F 1, · · · , F J, respectively. The disagreement of J ensemble learners for an unlabeled example, xi ∈U, is V (xi) = 1 J PJ j=1[F j(xi)]2 − £ 1 J PJ j=1 F j(xi) ¤2 and the weight η ∈R+. In light of view j, the pseudo-class label of an unlabeled example, xi, is determined by yi = sign £ 1 J PJ j=1 F j(xi) −F j(xi) ¤ . Thus, the minimization of (3) with such pseudo-class labels leads to a proper base learner f j(x) to be added to F j(x). 2.2 Boosting with regularization Motivated by the work on the use of regularization in semi-supervised learning [3]-[8], we introduce a local smoothness regularizer to semi-supervised boosting based on the universal optimization framework of margin cost functional [9], which results in a novel objective function: T (F, f) = −⟨∇C(F), f⟩− X i:xi∈S βiR(i), (5) where βi ∈R+ is a weight, determined by the input distribution to be discussed in Sect. 4, associated with each training example and the local smoothness around an example, xi, is measured by R(i) = X j:xj∈S,j̸=i Wij ˜C(−Iij). (6) Here, Iij is a class label compatibility function for two different examples xi, xj ∈S and defined as Iij = |yi −yj| where yi and yj are the true labels of xi and xj for labeled data or their pseudo-class labels otherwise. ˜C : R →R is a monotonically decreasing function derived from the cost function adopted in (1) so that ˜C(0)=0. Wij is an affinity measure defined by Wij = exp(−||xi−xj||2/2σ2) where σ is a bandwidth parameter. To find a proper base learner, f(x), we now need to maximize T (F, f) in (5) so as to minimize not only misclassification errors as before (see Sect. 2.1) but also the local class label incompatibility cost for smoothness. In order to use the objective function in (5) for boosting learning, we need to have the new empirical data distribution and the termination condition. Inserting (2) into (5) results in T (F, f) = X i:f(xi)̸=yi αiC′[yiF(xi)] − X i:f(xi)=yi αiC′[yiF(xi)] − X i:xi∈S βiR(i). (7) Since an appropriate cost function used in (1) is non-negative and monotonically decreasing, C′[yiF(xi)] is always negative and R(i) is non-negative according to its definition in (6). Therefore, we can define our empirical data distribution as ˜D(i) = αiC′[yiF(xi)] −βiR(i) P k:xk∈S © αkC′[ykF(xk)] −βkR(k) ª, 1 ≤i≤|L| + |U|. (8) ˜D(i) is always non-negative based on definitions of cost function in (1) and R(i) in (6). Applying (8) to (7) with some mathematical development similar to that described in Sect. 2.1, we can show that finding a proper base learner f(x) to maximize T (F, f) is equivalent to finding f(x) to minimize X i:f(xi)̸=yi ˜D(i) − X i:f(xi)=yi ˜D(i) −2 X i:f(xi)=yi βiR(i) P k:xk∈S © αkC′[ykF(xk)] −βkR(k) ª, which is equal to 2 X i:f(xi)̸=yi ˜D(i) | {z } misclassification errors + 2 X i:f(xi)=yi −βiR(i) P k:xk∈S © αkC′[ykF(xk)] −βkR(k) ª | {z } local class label incompatibility −1. (9) In (9), the first term refers to misclassification errors while the second term corresponds to the class label incompatibility of a data point with its nearby data points even though this data point itself fits well. In contrast to (3), finding a proper base learner, f(x), now needs to minimize not only the misclassification errors but also the local class label incompatibility in our Regularized Boost. Accordingly, a new termination condition of our Regularized Boost is derived from (9) as ϵ ≥1 2 where ϵ = P i:f(xi)̸=yi ˜D(i) + P i:f(xi)=yi −βiR(i) P k:xk∈S © αkC′[ykF (xk)]−βkR(k) ª. Once finding an optimal base learner, ft+1(x), at step t+1, we need to choose a proper weight, wt+1, to form a new ensemble, Ft+1(x) = Ft(x) + wt+1ft+1(x). In our Regularized Boost, we choose wt+1 = 1 2 log ¡ 1−ϵ ϵ ¢ by simply treating pseudo-class labels for unlabeled data as same as true labels of labeled data, as suggested in [11]. 3 Experiments In this section, we report experimental results on synthetic, benchmark and real data sets. Although our regularizer is applicable to existing semi-supervised boosting [10]-[13], we mainly apply it to the ASSEMBLE [11], a winning algorithm from the NIPS 2001 Unlabeled Data Competition, on a variety of classification tasks. In addition, our regularizer is also used to train component ensemble learners of the Agreement Boost [13] for binary classification benchmark tasks since the algorithm [13] in its original form can cope with binary classification only. In our experiments, we use C(γ) = e−γ in (1) and ˜C(γ) = C(γ) −1 in (6) and set αi = 1 in (1) and βi = 1 2 in (5). For synthetic and benchmark data sets, we always randomly select 20% of examples as testing data except that a benchmark data set has pre-defined a training/test split. Accordingly, the remaining examples used as a training set or those in a pre-defined training set, S, are randomly divided into two subsets, i.e., labeled data (L) and unlabeled data (U), and the ratio between labeled and unlabeled data is 1:4 in our experiments. For reliability, each experiment is repeated for ten times. To test the effectiveness of our Regularized Boost across different base learners, we perform all experiments with K nearest-neighbors (KNN) classifier, a local classifier, and multi-layer perceptron (MLP), a global classifier, where 3NN and a single hidden layered MLP are used in our experiments. For comparison, we report results of a semi-supervised boosting algorithm (i.e., ASSEMBLE [11] or Agreement Boost [13]) and its regularized version (i.e., Regularized Boost). In addition, we also provide results of a variant of Adaboost [14] trained on the labeled data only for reference. The above experimental method conforms to those used in semi-supervised boosting methods [10]-[13] as well as other empirical studies of semi-supervised learning methods, e.g., [15]. 3.1 Synthetic data set We use a Gaussian mixture model of four components to generate a data set of four categories in the 2-D space; 200 examples are in each category, as illustrated in Figure 1(a). We wish to test our regularizer on this intuitive multi-class classification task of a high optimal Bayes error. −5 −4 −3 −2 −1 0 1 2 3 4 −5 −4 −3 −2 −1 0 1 2 3 4 5 (a) AdaBoost ASSEMBLE RegularizedBoost 20 25 30 35 Error Rate(%) with KNN with MLP 32.50 31.88 28.75 27.37 26.87 26.25 (b) Figure 1: Synthetic data classification task. (a) The data set. (b) Classification results From Figure 1(b), it is observed that the use of unlabeled data improves the performance of Adaboost and the use of our regularizer further improves the generalization performance of the ASSEMBLE by achieving an averaging error rate closer to the optimal Bayes error no matter what kind of a base learner is used. Our further observation via visualization with the ground truth indicates that the use of our regularizer leads to smoother decision boundaries than the original ASSEMBLE, which yields the better generalization performance. 3.2 Benchmark data sets To assess the performance of our regularizer for semi-supervised boosting algorithms, we perform a series of experiments on benchmark data sets from the UCI machine learning repository [16] without any data transformation. In our experiments, we use the same initialization conditions for all boosting algorithms. Our empirical work suggests that a maximum number of 100 boosting steps is sufficient to achieve the reasonable performance for those benchmark tasks. Hence, we set such a maximum number of boosting steps to stop all boosting algorithms for a sensible comparison. We first apply our regularizer to the ASSEMBLE [11] on five UCI benchmark classification tasks of different categories[16]: BUPA liver disorders (BUPA), Wisconsin Diagnostic Breast Cancer (WDBC), Balance Scale Weight & Distance (BSWD), Car Evaluation Database (CAR), and Optical Recognition of Handwritten Digits (OPTDIGITS) where its data set has been split into the fixed training and testing subsets in advance by the data collector. Table 1: Error rates (mean±dev.)% of AdaBoost, ASSEMBLE and Regularized Boost (RegBoost) with different base learners on five UCI classification data sets. Data Set KNN MLP AdaBoost ASSEMBLE RegBoost AdaBoost ASSEMBLE RegBoost BUPA 37.7±3.4 36.1±3.0 34.9±3.1 35.1±1.1 31.2±6.7 28.8±5.6 WDBC 8.3±1.9 4.1±1.0 3.7±2.0 9.7±2.0 3.5±0.9 3.2±0.8 BSWD 22.2±0.9 18.7±0.4 17.4±0.9 16.8±2.8 14.4±2.4 13.6±2.6 CAR 31.3±1.2 24.4±0.7 23.2 ±1.1 30.6±3.0 20.5±0.9 17.7±1.1 OTIDIGITS 4.9±0.1 3.1±0.5 2.7±0.7 6.3±0.2 5.2±0.2 5.0±0.2 Table 1 tabulates the results of different boosting learning algorithms. It is evident from Table 1 that in general the use of unlabeled data constantly improves the generalization performance in contrast to the performance of AdaBoost and the use of our regularizer in the ASSEMBLE always further reduces its error rates on all five data sets no matter what kind of a base learner is used. It is also observed that the use of different base learners results in various performance on five data sets; the use of KNN as a base learner yields better performance on the WDBC and OPTDIGITS data set whereas the use of MLP as a base learner outperforms its KNN counterpart on other three data sets. Apparently the nature of a base learner, e.g., global vs. local classifiers, may determine if it is suitable for a classification task. It is worth mentioning that for the OPTDIGITS data set the lowest error rate achieved by 3NN with the entire training set, i.e., using all 3823 examples as training prototypes, is around 2.2% on the testing set, as reported in the literature [16]. In contrast, the ASSEMBLE [11] on 3NN equipped with our regularizer yields an error rate of 2.7% on average despite the fact that our Regularized Boost algorithm simply uses 765 labeled examples. Table 2: Error rates (mean±dev.)% of AdaBoost, Agreement Boost and Regularized Boost (RegBoost) on five UCI binary classification data sets. Data Set AdaBoost-KNN AdaBoost-MLP AgreementBoost RegBoost BUPA 37.7±3.4 35.1±1.1 30.4±7.5 28.9±5.8 WDBC 8.3±1.9 9.7±2.0 3.3±0.7 3.0±0.8 VOTE 9.0±1.5 10.6±0.5 4.4±0.8 2.8±0.6 AUSTRALIAN 37.7±1.2 21.0±3.4 16.7±2.1 15.2±2.8 KR-vs-KP 15.6±0.7 7.1±0.2 6.3±1.3 5.2±1.6 We further apply our regularizer to the Agreement Boost [13]. Due to the limitation of this algorithm [13], we can use only the binary classification data sets to test the effectiveness. As a result, we use BUPA and WDBC mentioned above and three additional UCI binary classification data sets [16]: 1984 U.S. Congressional Voting Records (VOTE), Australian Credit Approval (AUSTRALIAN) and Chess End-Game King Rook versus King Pawn (KR-vs-KP). As required by the Agreement Boost [13], the KNN and the MLP classifiers as base learners are used to construct two component ensemble learners without and with the use of our regularizer in experiments, which corresponds to its original and regularized version of the Agreement Boost. Table 2 tabulates results produced by different boosting algorithms. It is evident from Table 2 that the use of our regularizer in its component ensemble learners always leads the Agreement Boost to improve its generalization on five benchmark tasks while its original version trained with labeled and unlabeled data considerably outperforms the Adaboost trained with labeled data only. 0 10 20 30 40 50 60 70 80 90 100 2.6 2.8 3 3.2 3.4 3.6 3.8 4 Number of base learners Error Rate(%) RegularizedBoost ASSEMBLE (a) 0 10 20 30 40 50 60 70 80 90 100 5 5.5 6 6.5 7 7.5 Error Rate(%) Number of base learners RegularizedBoost ASSEMBLE (b) 0 10 20 30 40 50 60 70 80 90 100 5 5.5 6 6.5 7 7.5 8 8.5 Error Rate(%) Number of base learners RegularizedBoost AgreementBoost (c) Figure 2: Behaviors of semi-supervised boosting algorithms: the original version vs. the regularized version. (a) The ASSEMBLE with KNN on the OPTDIGITS. (b) The ASSEMBLE with MLP on the OPTDIGITS. (c) The Agreement Boost on the KR-vs-KP. We investigate behaviors of regularized semi-supervised boosting algorithms on two largest data sets, OPTDIGITS and VR-vs-VP. Figure 2 shows the averaging generalization performance achieved by stopping a boosting algorithm at different boosting steps. From Figure 2, the use of our regularizer in the ASSEMBLE regardless of base learners adopted and the Agreement Boost always yields fast training. As illustrated in Figures 2(a) and 2(b), the regularized version of the ASSEMBLE with KNN and MLP takes only 22 and 46 boosting steps on average to reach the performance of the original ASSEMBLE after 100 boosting steps, respectively. Similarly, Figure 2(c) shows that the regularized Agreement Boost takes only 12 steps on average to achieve the performance of its original version after 100 boosting steps. 3.3 Facial expression recognition Facial expression recognition is a typical semi-supervised learning task since labeling facial expressions is an extremely expensive process and very prone to errors due to ambiguities. We test the effectiveness of our regularizer by using a facial expression benchmark database, JApanese Female Facial Expression (JAFFE) [17] where there are 10 female expressers who posed 3 or 4 examples for each of seven universal facial expressions (anger, disgust, fear, joy, neutral, sadness and surprise), as exemplified in Figure 3(a), and 213 pictures of 256 × 256 pixels were collected totally. (a) AdaBoost ASSEMBLE RegularizedBoost 15 20 25 30 35 40 45 Error Rate(%) 34.27 32.19 26.37 (b) Figure 3: Facial expression recognition on the JAFFE. (a) Exemplar pictures corresponding to seven universal facial expressions. (b) Classification results of different boosting algorithms. In our experiments, we first randomly choose 20% images (balanced to seven classes) as testing data and the rest of images constitute a training set (S) randomly split into labeled (L) and unlabeled (U) data of equal size in each trial. We apply the independent component analysis and then the principal component analysis (PCA) to each image for feature extraction and use only first 40 PCA coefficients to form a feature vector. A single hidden layered MLP of 30 hidden neurons is used as the based learner. We set a maximum number of 1000 boosting rounds to stop the algorithms if their termination conditions are not met while the same initialization is used for all boosting algorithms. For reliability, the experiment is repeated 10 times. From Figure 3(b), it is evident that the ASSEMBLE with our regularizer yields 5.82% error reduction on average; an averaging error rate of 26.37% achieved is even better than that of some supervised learning methods on the same database, e.g., [18] where around 70% images were used to train a convolutional neural network and an averaging error rate of 31.5% was achieved on the remaining images. 4 Discussions In this section, we discuss issues concerning our regularizer and relate it to previous work in the context of regularization in semi-supervised learning. As defined in (5), our regularizer has a parameter, βi, associated with each training point, which can be used to encode the information of the marginal or input distribution, P(x), by setting βi = λP(x) where λ is a tradeoff or regularization parameter. Thus, the use of βi would make the regularization take effect only in dense regions although our experiments reported were carried out by setting βi = 1 2; i.e., we were using a weak assumption that data are scattered uniformly throughout the whole space. In addition, (6) uses an affinity metric system to measure the proximity of data points and can be extended by incorporating the manifold information, if available, into our regularizer. Our local smoothness regularizer plays an important role in re-sampling all training data including labeled and unlabeled data for boosting learning. As uncovered in (9), the new empirical distribution based on our regularizer not only assigns a large probability to a data point misclassified but also may cause a data point even classified correctly in the last round of boosting learning but located in a “non-smoothing” region to be assigned a relatively large probability, which distinguishes our approach from existing boosting algorithms where the distribution for re-sampling training data is determined solely by misclassification errors. For unlabeled data, such an effect always makes sense to work on the smoothness and the cluster assumptions [1] as performed by existing regularization techniques [3]-[8]. For labeled data, it actually has an effect that the labeled data points located in a “non-smoothing” region is more likely to be retained in the next round of boosting learning. As exemplified in Figure 1, such points are often located around boundaries between different classes and therefore more informative in determining a decision boundary, which would be another reason why our regularizer improves the generalization of semi-supervised boosting algorithms. The use of manifold smoothness in a special form of Adaboost, marginal Adaboost, has been attempted in [19] where the graph Laplacian regularizer was applied to select base learners by the adaptive penalization of base learners according to their decision boundaries and the actual manifold structural information. In essence, the objective of using manifold smoothness in our Regularized Boost is identical to theirs in [19] but we accomplish it in a different way. We encode the manifold smoothness into the empirical data distribution used in boosting algorithms for semi-supervised learning, while their implementation adaptively adjusts the edge offset in the marginal Adaboost algorithm for a weight decay used in the linear combination of based learners [19]. In contrast, our implementation is simpler yet applicable to any boosting algorithms for semi-supervised learning, while theirs needs to be fulfilled via the marginal Adaboost algorithm even though their regularized marginal Adaboost is applicable to both supervised and semi-supervised learning indeed. By comparison with existing regularization techniques used in semi-supervised learning, our Regularized Boost is closely related to graph-based semi-supervised learning methods, e.g., [8]. In general, a graph-based method wants to find a function to simultaneously satisfy two conditions [2]: a) it should be close to given labels on the labeled nodes, and b) it should be smooth on the whole graph. In particular, the work in [8] develops a regularization framework to carry out the above idea by defining the global and local consistency terms in their cost function. Similarly, our cost function in (9) has two terms explicitly corresponding to global and local consistency though true labels of labeled data never change during our boosting learning, which resembles theirs [8]. Nevertheless, a graph-based algorithm is an iterative label propagation process on a graph where a regularizer directly gets involved in label modification over the graph, whereas our Regularized Boost is an iterative process that runs a base learner on various distributions over training data where our regularizer simply plays a role in determining distributions. In general, a graph-based algorithm is applicable to transductive learning only although it can be combined with other methods, e.g. a mixture model [7], for inductive learning. In contrast, our Regularized Boost is developed for inductive learning. Finally it is worth stating that unlike most of existing regularization techniques used in semi-supervised learning, e.g., [5],[6], our regularization takes effect on both labeled and unlabeled data while theirs are based on unlabeled data only. 5 Conclusions We have proposed a local smoothness regularizer for semi-supervising boosting learning and demonstrated its effectiveness on different types of data sets. In our ongoing work, we are working for a formal analysis to justify the advantage of our regularizer and explain the behaviors of Regularized Boost, e.g. fast training, theoretically. References [1] Chapelle, O., Sch¨olkopf, B., & Zien, A. (2006) Semi-Supervised Learning. Cambridge, MA: MIT Press. [2] Zhu, X. (2006) Semi-supervised learning literature survey. Computer Science TR-1530, University of Wisconsin - Madison, U.S.A. [3] Bousquet, O., Chapelle, O., & Hein, M. (2004) Measure based regularization. In Advances in Neural Information Processing Systems 16. Cambridge, MA: MIT Press. [4] Belkin, M., Niyogi, P., & Sindhwani, V. (2004) Manifold regularization: a geometric framework for learning from examples. Technical Report, University of Michigan, U.S.A. [5] Szummer, M., & Jaakkola, T. (2003) Information regularization with partially labeled data. In Advances in Neural Information Processing Systems 15. Cambridge, MA: MIT Press. [6] Grandvalet, Y., & Begio, Y. (2005) Semi-supervised learning by entropy minimization. In Advances in Neural Information Processing Systems 17. Cambridge, MA: MIT Press. [7] Zhu, X., & Lafferty, J. (2005) Harmonic mixtures: combining mixture models and graph-based methods for inductive and scalable semi-supervised learning. In Proc. Int. Conf. Machine Learning, pp. 1052-1059. [8] Zhou, D., Bousquet, O., Lal, T., Weston, J., & Sch¨lkopf, B. (2004) Learning with local and global consistency. In Advances in Neural Information Processing Systems 16. Cambridge, MA: MIT Press. [9] Mason, L., Bartlett, P., Baxter, J., & Frean, M. (2000) Functional gradient techniques for combining hypotheses. In Advances in Large Margin Classifiers. Cambridge, MA: MIT Press. [10] d’Alch´e-Buc, F., Grandvalet, Y., & Ambroise, C. (2002) Semi-supervised MarginBoost. In Advances in Neural Information Processing Systems 14. Cambridge, MA: MIT Press. [11] Bennett, K., Demiriz, A., & Maclin, R. (2002) Expoliting unlabeled data in ensemble methods. In Proc. ACM Int. Conf. Knowledge Discovery and Data Mining, pp. 289-296. [12] Collins, M., & Singer, Y. (1999) Unsupervised models for the named entity classification. In Proc. SIGDAT Conf. Empirical Methods in Natural Language Processing and Very Large Corpora. [13] Leskes, B. (2005) The value of agreement, a new boosting algorithm. In Proc. Int. Conf. Algorithmic Learning Theory (LNAI 3559), pp. 95-110, Berlin: Springer-Verlag. [14] G¨unther, E., & Pfeiffer, K.P. (2005) Multiclass boosting for weak classifiers. Journal of Machine Learning Research 6:189-210. [15] Nigam, K., McCallum, A., Thrum, S., & Mitchell, T. (2000) Using EM to classify text from labeled and unlabeled documents. Machine Learning 39:103-134. [16] Blake, C., Keogh, E., & Merz, C.J. (1998) UCI repository of machine learning databases. University of California, Irvine. [on-line] http://www.ics.uci.edu/ mlearn/MLRepository.html [17] The JAFFE Database. [Online] http://www.kasrl.org/jaffe.html [18] Fasel, B. (2002) Robust face analysis using convolutional neural networks. In Proc. Int. Conf. Pattern Recognition, vol. 2, pp. 40-43. [19] K´egl, B., & Wang, L. (2004) Boosting on manifolds: adaptive regularization of base classifier. In Advances in Neural Information Processing Systems 16. 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Consistent Minimization of Clustering Objective Functions Ulrike von Luxburg Max Planck Institute for Biological Cybernetics ulrike.luxburg@tuebingen.mpg.de S´ebastien Bubeck INRIA Futurs Lille, France sebastien.bubeck@inria.fr Stefanie Jegelka Max Planck Institute for Biological Cybernetics stefanie.jegelka@tuebingen.mpg.de Michael Kaufmann University of T¨ubingen, Germany mk@informatik.uni-tuebingen.de Abstract Clustering is often formulated as a discrete optimization problem. The objective is to find, among all partitions of the data set, the best one according to some quality measure. However, in the statistical setting where we assume that the finite data set has been sampled from some underlying space, the goal is not to find the best partition of the given sample, but to approximate the true partition of the underlying space. We argue that the discrete optimization approach usually does not achieve this goal. As an alternative, we suggest the paradigm of “nearest neighbor clustering”. Instead of selecting the best out of all partitions of the sample, it only considers partitions in some restricted function class. Using tools from statistical learning theory we prove that nearest neighbor clustering is statistically consistent. Moreover, its worst case complexity is polynomial by construction, and it can be implemented with small average case complexity using branch and bound. 1 Introduction Clustering is the problem of discovering “meaningful” groups in given data. Many algorithms try to achieve this by minimizing a certain quality function Qn, for example graph cut objective functions such as ratio cut or normalized cut, or various criteria based on some function of the within- and between-cluster similarities. The objective of clustering is then stated as a discrete optimization problem. Given a data set Xn = {X1, . . . , Xn} and a clustering quality function Qn, the ideal clustering algorithm should take into account all possible partitions of the data set and output the one that minimizes Qn. The implicit understanding is that the “best” clustering can be any partition out of the set of all possible partitions of the data set. The algorithmic challenge is to construct an algorithm which is able to find this clustering. We will call this approach the “discrete optimization approach to clustering”. If we look at clustering from the perspective of statistical learning theory we assume that the finite data set has been sampled from an underlying data space X according to some probability measure. The ultimate goal in this setting is not to discover the best possible partition of the data set Xn, but to learn the “true clustering” of the underlying space. In an approach based on quality functions, this “true clustering” can be defined easily. We choose a clustering quality function Q on the set of partitions of the entire data space X, and define the true clustering f ∗to be the partition minimizing Q. In this setting, a very important property of a clustering algorithm is consistency. Denoting the clustering constructed on the finite sample by fn, we require that Q(fn) converges to Q(f ∗) when n→∞. The most important insight of statistical learning theory is that in order to be consistent, learning algorithms have to choose their functions from some “small” function space only. To measure the size of a function space F one uses the quantity NF(x1, .., xn) which denotes the number 1 of ways in which the points x1, . . . , xn can be partitioned by functions in F. One can prove that in the standard setting of statistical learning theory, a necessary condition for consistency is that E log NF(x1, .., xn)/n→0 (cf. Theorem 2.3 in Vapnik, 1995, Section 12.4 of Devroye et al., 1996). Stated like this, it becomes apparent that the two viewpoints described above are not compatible with each other. While the discrete optimization approach on any given sample attempts to find the best of all (exponentially many) partitions, the statistical learning theory approach restricts the set of candidate partitions to have sub-exponential size. Hence, from the statistical learning theory perspective, an algorithm which is considered ideal in the discrete optimization setting is likely to overfit. One can construct simple examples (cf. Bubeck and von Luxburg, 2007) which show that this indeed can happen: here the partitions constructed on the finite sample do not converge to the true clustering of the data space. In practice, for most cases the discrete optimization approach cannot be performed perfectly as the corresponding optimization problem is NP hard. Instead, people resort to heuristics. One approach is to use local optimization procedures potentially ending in local minima only (this is what happens in the k-means algorithm). Another approach is to construct a relaxation of the original problem which can be solved efficiently (spectral clustering is an example for this). In both cases, one usually cannot guarantee how close the heuristic solution is to the global finite sample optimum. This situation is clearly unsatisfactory: for most clustering algorithms, we neither have guarantees on the finite sample behavior of the algorithm, nor on its statistical consistency in the limit. The following alternative approach looks much more promising. Instead of attempting to solve the discrete optimization problem over the set of all partitions, and then resorting to relaxations due to the NP-hardness of this problem, we turn the tables. Directly from the outset, we only consider candidate partitions in some restricted class Fn containing only polynomially many functions. Then the discrete optimization problem of minimizing Qn over Fn is no longer NP hard – it can trivially be solved in polynomially many steps by trying all candidates in Fn. From a theoretical point of view this approach has the advantage that the resulting clustering algorithm has the potential of being consistent. In addition, it also leads to practical benefits: rather than dealing with uncontrolled relaxations of the original problem, we restrict the function class to some small enough subset Fn of “reasonable” partitions. Within this subset, we then have complete control over the solution of the optimization problem and can find the global optimum. Put another way, one can also interpret this approach as some controlled way of sparsifying the NP hard optimization problem, with the positive side effect of obeying the rules of statistical learning theory. 2 Nearest neighbor clustering In the following we assume that we are given a set of data points Xn = {X1, . . . , Xn} and pairwise distances dij = d(Xi, Xj) or pairwise similarities sij = s(Xi, Xj). Let Qn be the finite sample quality function to optimize on the sample. To follow the approach outlined above we have to optimize Qn over a “small” set Fn of partitions of Xn. Essentially, we have three requirements on Fn: First, the number of functions in Fn should be at most polynomial in n. Second, in the limit of n →∞the class Fn should be rich enough to approximate any measurable partition of the underlying space. Third, in order to perform the optimization we need to be able to enumerate all members of this class, that is the function class Fn should be “constructive” in some sense. A convenient choice satisfying all those properties is the class of “nearest neighbor partitions”. This class contains all functions which can be generated as follows. Fix a subset of m ≪n “seed points” Xs1, . . . , Xsm among the given data points. Assign all other data points to their closest seed points, that is for all j = 1, . . . , m define the set Zj as the subset of data points whose nearest seed point is Xsj. Then consider all partitions of Xn which are constant on the sets Zj. More formally, for given seeds we define the set Fn as the set of all functions f : X →{1, . . . , K} which are constant on the cells of the Voronoi partition induced by the seeds. Here K denotes the number of clusters we want to construct. The function class Fn contains Km functions, which is polynomial in n if the number m of seeds satisfies m = O(log n). Given Fn, the simplest polynomial-time optimization algorithm is then to evaluate Qn(f) for all f ∈Fn and choose the solution fn = argminf∈Fn Qn(f). We call the resulting clustering the nearest neighbor clustering and denote it by NNC(Qn). In practice, the seeds will be chosen randomly among the given data points. 2 3 Consistency of nearest neighbor clustering In this section we prove that nearest neighbor clustering is statistically consistent for many clustering quality functions. Due to the complexity of the proofs and the page restriction we can only present sketches of the proofs. All details can be found in von Luxburg et al. (2007). Let us start with some notation. For any clustering function f : Rd →{1, . . . , K} we denote by the predicate A(f) a property of the function which can either be true or false. As an example, define A(f) to be true if all clusters have at least a certain minimal size. Moreover, we need to introduce a predicate An(f) which will be an “estimator” of A(f) based on the finite sample only. Let m := m(n) ≤n be the number of seeds used in nearest neighbor clustering. To simplify notation we assume in this section that the seeds are the first m data points; all results remain valid for any other (even random) choice of seeds. As data space we use X = Rd. We define: NNm(x) := NNm(n)(x) := argminy∈{X1,...,Xm} ∥x −y∥ ( for x ∈Rd) F := {f : Rd →{1, . . . , K} | f continuous P-a.e. and A(f) true} Fn := FX1,...,Xn := {f : Rd →{1, . . . , K} | f satisfies f(x) = f(NNm(x)), and An(f) is true} eFn := S X1,...,Xn∈Rd FX1,...,Xn Furthermore, let Q : F →R be the quality function we aim to minimize, and Qn : Fn →R an estimator of this quality function on a finite sample. With this notation, the true clustering f ∗on the underlying space and the nearest neighbor clustering fn introduced in the last section are given by f ∗∈argminf∈F Q(f) and fn ∈argminf∈Fn Qn(f). Later on we will also need to work with the functions f ∗ n ∈argminf∈Fn Q(f) and ef ∗(x) := f ∗(NNm(x)). As distance function between different clusterings f, g we will use Ln(f, g) := P(f(X) ̸= g(X) | X1, . . . , Xn) (we need the conditioning in case f or g depend on the data, it has no effect otherwise). Theorem 1 (Consistency of nearest neighbor clustering) Let (Xi)i∈N be a sequence of points drawn i.i.d. according to some probability measure P on Rd, and m := m(n) the number of seed points used in nearest neighbor clustering. Let Q : F →R be a clustering quality function, Qn : eFn →R its estimator, and A(f) and An(f) some predicates. Assume that: 1. Qn(f) is a consistent estimator of Q(f) which converges sufficiently fast: ∀ε > 0, Km(2n)(d+1)m2 supf∈e Fn P(|Qn(f) −Q(f)| > ε) →0. 2. An(f) is an estimator of A(f) which is “consistent” in the following way: P(An( ef ∗) true) →1 and P(A(fn) true) →1. 3. Q is uniformly continuous with respect to the distance Ln between F and Fn: ∀ε > 0 ∃δ(ε) > 0 ∀f ∈F ∀g ∈Fn : Ln(f, g) ≤δ(ε) =⇒|Q(f) −Q(g)| ≤ε. 4. limn→∞m(n) = +∞. Then nearest neighbor clustering as introduced in Section 2 is weakly consistent, that is Q(fn) → Q(f ∗) in probability. Proof. (Sketch, for details see von Luxburg et al. (2007)). We split the term P(|Q(fn)−Q(f ∗)| ≥ε) into its two sides P(Q(fn) −Q(f ∗) ≤−ε) and P(Q(fn) −Q(f ∗) ≥ε). It is a straightforward consequence of Condition (2) that the first term converges to 0. The main work consists in bounding the second term. As usual we consider the estimation and approximation errors P Q(fn) −Q(f ∗) ≥ε ≤P Q(fn) −Q(f ∗ n) ≥ε/2 + P Q(f ∗ n) −Q(f ∗) ≥ε/2 . 3 First we bound the estimation error. In a few lines one can show that P(Q(fn) −Q(f ∗ n) ≥ε/2) ≤P(supf∈Fn |Qn(f) −Q(f)| ≥ε/4). Note that even though the right hand side resembles the standard quantities often considered in statistical learning theory, it is not straightforward to bound as we do not assume that Q(f) = EQn(f). Moreover, note that the function class Fn is data dependent as the seed points used in the Voronoi partition are data points. To circumvent this problem, we replace the function class Fn by the larger class eFn, which is not data dependent. Using symmetrization by a ghost sample (cf. Section 12.3 of Devroye et al., 1996), we then move the supremum out of the probability: P sup f∈Fn |Qn(f) −Q(f)| ≥ε/4 ≤2SK( eFn, 2n) supf∈e Fn P |Qn(f) −Q(f)| ≥ε/16 inff∈e Fn P |Qn(f) −Q(f)| ≤ε/8 (1) Note that the unusual denominator in Eq. (1) emerges in the symmetrization step as we do not assume Q(f) = EQn(f). The quantity SK( eFn, 2n) denotes the shattering coefficient, that is the maximum number of ways that 2n points can be partitioned into K sets using the functions in eFn. It is well known (e.g., Section 21.5 of Devroye et al., 1996) that the number of Voronoi partitions of n points using m cells in Rd is bounded by n(d+1)m2, hence the number of nearest neighbor clusterings into K classes is bounded by SK( eFn, n) ≤Kmn(d+1)m2. Under Condition (1) of the Theorem we now see that for fixed ε and n →∞the right hand side of (1) converges to 0. Thus the same holds for the estimation error. To deal with the approximation error, observe that if An( ef ∗) is true, then ef ∗∈Fn, and by the definition of f ∗ n we have Q(f ∗ n) −Q(f ∗) ≤Q( ef ∗) −Q(f ∗) and thus P Q(f ∗ n) −Q(f ∗) ≥ε ≤P(An( ef ∗) false) + P ef ∗∈Fn and Q( ef ∗) −Q(f ∗) ≥ε . (2) The first expression on the right hand side converges to 0 by Condition (2) in the theorem. Using Condition (3), we can bound the second expression in terms of the distance Ln to obtain P ef ∗∈Fn, Q( ef ∗) −Q(f ∗) ≥ε ≤P Q( ef ∗) −Q(f ∗) ≥ε ≤P Ln(f ∗, ef ∗) ≥δ(ε) . Now we use techniques from Fritz (1975) to show that if n is large enough, then the distance between a function f ∈F evaluated at x and the same function evaluated at NNm(x) is small. Namely, for any f ∈F and any ε>0 there exists some b(δ(ε))>0 which does not depend on n and f such that P(Ln(f, f(NNm(·))) > δ(ε)) ≤(2/δ(ε)) e−mb(δ(ε)). The quantity δ(ε) has been introduced in Condition (3). For every fixed ε, this term converges to 0 due to Condition (4), thus the approximation error vanishes. , Now we want to apply our general theorem to particular objective functions. We start with the normalized cut. Let s : Rd×Rd →R+ be a similarity function which is upper bounded by a constant C. For a clustering f : Rd →{1, . . . , K} denote by fk(x) := 1f(x)=k the indicator function of the k-th cluster. Define the empirical and true cut, volume, and normalized cut as follows: cutn(fk) := 1 n(n−1) Pn i,j=1 fk(Xi)(1 −fk(Xj))s(Xi, Xj) cut(fk) := EX,Y fk(X)(1 −fk(Y ))s(X, Y ) voln(fk) := 1 n(n−1) Pn i,j=1 fk(Xi)s(Xi, Xj) vol(fk) := EX,Y fk(X)s(X, Y ) Ncutn(f) := PK k=1 cutn(fk) voln(fk) Ncut(f) := PK k=1 cut(fk) vol(fk) Note that E Ncutn(f) ̸= Ncut(f), but E cutn(f) = cut(f) and E voln(f) = vol(f). We fix a constant a > 0, a sequence (an)n∈N with an ≥an+1 and an →a and define the predicates A(f) is true : ⇐⇒vol(fk) > a ∀k = 1, . . . , K An(f) is true : ⇐⇒voln(fk) > an ∀k = 1, . . . , K (3) Theorem 2 (Consistency of NNC(Ncutn)) Let (Xi)i∈N be a sequence of points drawn i.i.d. according to some probability measure P on Rd and s : Rd × Rd →R+ be a similarity function which is upper bounded by a constant C. Let m := m(n) be the number of seed points used in nearest neighbor clustering, a > 0 an arbitrary constant, and (an)n∈N a monotonically decreasing sequence with an →a. Then nearest neighbor clustering using Q := Ncut, Qn := Ncutn, and A and An as defined in (3) is weakly consistent if m(n) →∞and m2 log n/(n(a −an)2) →0. 4 Proof. We will check that all conditions of Theorem 1 are satisfied. First we establish that {| cutn(fk) −cut(fk)| ≤aε} ∩{| voln(fk) −vol(fk)| ≤aε} ⊂{|cutn(fk) voln(fk) −cut(fk) vol(fk) | ≤2ε}. Applying the McDiarmid inequality to cutn and voln, respectively, we obtain that for all f ∈eFn P(| Ncut(f) −Ncutn(f)| > ε) ≤4K exp −na2ε2 8C2K2 . Together with m2 log n/(n(a −an)2) →0 this shows Condition (1) of Theorem 1. The proof of Condition (2) is rather technical, but in the end also follows by applying the McDiarmid inequality to voln(f). Condition (3) follows by establishing that for f ∈F and g ∈Fn we have | Ncut(f) −Ncut(g)| ≤4CK a Ln(f, g). , In fact, Theorem 1 can be applied to a large variety of clustering objective functions. As examples, consider ratio cut, within-sum of squares, and the ratio of between- and within-cluster similarity: RatioCutn(f) := PK k=1 cutn(fk) nk RatioCut(f) := PK k=1 cut(fk) Efk(X) WSSn(f) := 1 n Pn i=1 PK k=1 fk(Xi)∥Xi −ck,n∥2 WSS(f) := E PK k=1 fk(X)∥X −ck∥2 BWn := PK k=1 cutn(fk) voln(fk)−cutn(fk) BW := PK k=1 cut(fk) vol(fk)−cut(fk) Here nk := P i fk(Xi)/n is the fraction of points in the k-th cluster, and ck,n := P i fk(Xi)Xi/(nnk) and ck := Efk(X)X/Efk(X) are the empirical and true cluster centers. Theorem 3 (Consistency of NNC(RatioCutn), NNC(WSSn), and NNC(BWn)) Let fn and f ∗be the empirical and true minimizers of nearest neighbor clustering using RatioCutn, WSSn, or BWn, respectively. Then, under conditions similar to the ones in Theorem 2, we have RatioCut(fn) →RatioCut(f ∗), WSS(fn) →WSS(f ∗), and BW(fn) →BW(f ∗) in probability. See von Luxburg et al. (2007) for details. 4 Implementation using branch and bound It is an obvious question how nearest neighbor clustering can be implemented in a more efficient way than simply trying all functions in Fn. Promising candidates are branch and bound methods. They are guaranteed to achieve an optimal solution, but in most cases are much more efficient than a naive implementation. As an example we introduce a branch and bound algorithm for solving NNC(Ncut) for K = 2 clusters. For background reading see Brusco and Stahl (2005). First of all, observe that minimizing Ncutn over the nearest neighbor function set Fn is the same as minimizing Ncutm over all partitions of a contracted data set consisting of m “super-points” Z1, . . . , Zm (super-point Zi contains all data points assigned to the i-th seed point), endowed with the “super-similarity” function ¯s(Zs, Zt) := P Xi∈Zs,Xj∈Zt s(Xi, Xj). Hence nearest neighbor clustering on the original data set with n points can be performed by directly optimizing Ncut on the contracted data set consisting of only m super-points. Assume we already determined the labels l1, . . . , li−1 ∈{±1} of the first i−1 super-points. For those points we introduce the sets A = {Z1, . . . , Zi−1}, A−:= {Zj | j < i, lj = −1}, A+ := {Zj | j < i, lj = +1}, for the remaining points the set B = {Zi, . . . , Zm}, and the set V := A ∪B of all points. By default we label all points in B with −1 and, in recursion level i, decide about moving Zi to cluster +1. Analogously to the notation fk of the previous section, in case K =2 we can decompose Ncut(f) = cut(f+1) · (1/ vol(f+1) + 1/ vol(f−1)); we call the first term the “cut term” and the second term the “volume term”. As it is standard in branch and bound we have to investigate whether the “branch” of clusterings with the specific fixed labels on A could contain a solution which is better than all the previously considered solutions. We use two criteria for this purpose. The first one is very simple: assigning at least one point in B to +1 can only lead to an improvement if this either decreases the cut term or the volume term of Ncut. Necessary conditions for this are maxj≥i ¯s(Zj, A+) −¯s(Zj, A−) ≥0 or vol(A+) ≤vol(V )/2, respectively. If neither is satisfied, we retract. The second criterion involves a lower bound θl on the Ncut value of 5 Branch and bound algorithm for Ncut: f ∗= bbncut( ¯S, i, f, θu){ 1. Set g := f; set A−, A+, and B as described in the text 2. // Deal with special cases: • If i = m and A−= ∅then return f. • If i = m and A−̸= ∅: – Set gi = +1. – If Ncut(g) < Ncut(f) return g, else return f. 3. // Pruning: • If vol(A+) > vol(A∪B)/2 and maxj≥i(¯s(j, A+)−¯s(j, A−)) ≤0 return f. • Compute lower bound θl as described in the text. • If θl ≥θu then return f. 4. // If no pruning possible, recursively call bbncut: • Set gi = +1, θ′ u := min{Ncut(g), θu}, call g′ := bbncut( ¯S, g, i + 1, θ′ u) • Set gi = −1, θ′′ u := min{Ncut(g′), θ′ u}, call g′′ := bbncut( ¯S, g, i + 1, θ′′ u) • If Ncut(g′) ≤Ncut(g′′) then return g′, else return g′′. } Figure 1: Branch and bound algorithm for NNC(Ncut) for K = 2. The algorithm is initially called with the super-similarity matrix ¯S, i = 2, f = (+1, −1, . . . , −1), and θu the Ncut value of f. all solutions in the current branch. It compares θl to an upper bound θu on the optimal Ncut value, namely to the Ncut value of the best function we have seen so far. If θl ≥θu then no improvement is possible by any clustering in the current branch of the tree, and we retract. To compute θl, assume we assign a non-empty set B+ ⊂B to label +1 and the remaining set B−= B \ B+ to label -1. Using the conventions ¯s(A, B) = P Zi∈A,Zj∈B ¯sij and ¯s(A, ∅) = 0, the cut term is bounded by cut(A+ ∪B+, A−∪B−) ≥ minj≥i s(Zj, A+) if A−= ∅ ¯s(A+, A−) + minj≥i ¯s(Zj, A−) otherwise. (4) The volume term can be maximally decreased in case vol(A+) < vol(V )/2, when choosing B+ such that vol(A+ ∪B+) = vol(A−∪B−) = vol(V )/2. If vol(A+) > vol(V )/2, then an increase of the volume term is unavoidable; this increase is minimal when we move one vertex only to A+: 1 vol(A+∪B+) + 1 vol(A−∪B−) ≥ ( 4/ vol(V ) if vol(A+) ≤vol(V )/2 vol(V )/max j≥i (vol(A+ ∪Zj) vol(A−∪B \ Zj)) otherw. (5) Combining both bounds we can now define the lower bound θl as the product of Eq. (4) and (5). The entire algorithm is presented in Fig. 1. On top of the basic algorithm one can apply various heuristics to improve the retraction behavior and thus the average running time of the algorithm. For example, in our experience it is of advantage to sort the super-points by decreasing degree, and from one recursion level to the next one alternate between first visiting branch gi = 1 and gi = −1. 5 Experiments The main point about nearest neighbor clustering is its statistical consistency: for large n it reveals an approximately correct clustering. In this section we want to show that it also behaves reasonably on smaller samples. Given an objective function Qn (such as WSS or Ncut) we compare the NNC results to heuristics designed to optimize Qn directly (such as k-means or spectral clustering). As numeric data sets we used classification benchmark data sets from different repositories (UCI repository, repository by G. R¨atsch) and microarray data from Spellman et al. (1998). Moreover, we use graph data sets of the internet graph and of biological, social, and political networks: COSIN collection, collection by M. Newman, email data by Guimer`a et al. (2003), electrical power network by Watts and Strogatz (1998), and protein interaction networks of Jeong et al. (2001) and Tsuda et al. (2005). Due to space constraints we focus on the case of constructing K = 2 clusters using the objective functions WSS and Ncut. We always set the number m of seed points for NNC to m = log n. In case of WSS, we compare the result of the k-means algorithm to the result of NNC using the WSS objective function and the Euclidean distance to assign data points to seed points. 6 Numeric WSS Ncut data sets K-means NNC SC NNC breast-c. 6.95 ± 0.19 7.04 ± 0.21 0.11 ± 0.02 0.09 ± 0.02 7.12 ± 0.20 7.12 ± 0.22 0.22 ± 0.07 0.21 ± 0.07 diabetis 6.62 ± 0.22 6.71 ± 0.22 0.03 ± 0.02 0.03 ± 0.02 6.72 ± 0.22 6.72 ± 0.22 0.04 ± 0.03 0.05 ± 0.05 german 18.26 ± 0.27 18.56 ± 0.28 0.02 ± 0.02 0.02 ± 0.02 18.35 ± 0.30 18.45 ± 0.32 0.04 ± 0.08 0.03 ± 0.03 heart 10.65 ± 0.46 10.77 ± 0.47 0.18 ± 0.03 0.17 ± 0.02 10.75 ± 0.46 10.74 ± 0.46 0.28 ± 0.03 0.30 ± 0.07 splice 68.99 ± 0.24 69.89 ± 0.24 0.36 ± 0.10 0.44 ± 0.16 69.03 ± 0.24 69.18 ± 0.25 0.58 ± 0.09 0.66 ± 0.18 bcw 3.97 ± 0.26 3.98 ± 0.26 0.02 ± 0.01 0.02 ± 0.01 3.98 ± 0.26 3.98 ± 0.26 0.04 ± 0.01 0.08 ± 0.07 ionosph. 25.72 ± 1.63 25.77 ± 1.63 0.06 ± 0.03 0.04 ± 0.01 25.76 ± 1.63 25.77 ± 1.63 0.12 ± 0.11 0.14 ± 0.12 pima 6.62 ± 0.22 6.73 ± 0.23 0.03 ± 0.03 0.03 ± 0.03 6.73 ± 0.23 6.73 ± 0.23 0.05 ± 0.04 0.09 ± 0.13 cellcycle 0.78 ± 0.03 0.78 ± 0.03 0.12 ± 0.02 0.10 ± 0.01 0.78 ± 0.03 0.78 ± 0.02 0.16 ± 0.02 0.15 ± 0.03 Network data NNC SC ecoli.interact 0.06 0.06 ecoli.metabol 0.03 0.04 helico 0.16 0.16 beta3s 0.00 0.00 AS-19971108 0.02 0.02 AS-19980402 0.01 1.00 AS-19980703 0.02 0.02 AS-19981002 0.04 0.04 AS-19990114 0.08 0.05 AS-19990402 0.11 0.10 netscience 0.01 0.01 polblogs 0.11 0.11 power 0.00 0.00 email 0.27 0.27 yeastProtInt 0.04 0.06 protNW1 0.00 0.00 protNW2 0.08 1.00 protNW3 0.01 0.80 protNW4 0.03 0.76 Table 1: Left: Numeric data. Results for K-means algorithm, NNC(WSS) with Euclidean distance; spectral clustering (SC); NNC(Ncut) with commute distance. The top line always shows the results on the training set, the second line the extended results on the test set. Right: Network data. NNC(Ncut) with commute distance and spectral clustering, both trained on the entire graph. Note that one cannot run K-means on pure network data, which does not provide coordinates. In case of Ncut, we use the Gaussian kernel as similarity function on the numeric data sets. The kernel width σ is set to the mean distance of a data point to its k-th nearest neighbor. We then build the k-nearest neighbor graph (both times using k=ln n). On the network data, we directly use the given graph. For both types of data, we use the commute distance on the graph (e.g., Gutman and Xiao, 2004) as distance function to determine the nearest seed points for NNC. In the first experiment we compare the values obtained by the different algorithms on the training sets. From the numeric data sets we generated z = 40 training sets by subsampling n/2 points. On each training set, we repeated all algorithms r = 50 times with different random initializations (the seeds in NNC; the centers in K-means; the centers in the K-means post-processing step in spectral clustering). Denoting the quality of an individual run of the algorithm by q, we then report the values meanz(minrq) ± standarddevz(minrq). For the network data sets we ran spectral clustering and NNC on the whole graph. Again we use r=50 different initializations, and we report minrq. All results can be found in Table 1. For both the numeric data sets (left table, top lines) and the network data sets (right table) we see that the training performance of NNC is comparable to the other algorithms. This is what we had hoped, and we find it remarkable as NNC is in fact a very simple clustering algorithm. In the second experiment we try to measure the amount of overfitting induced by the different algorithms. For each of the numeric data sets we cluster n/2 points, extend the clustering to the other n/2 points, and then compute the objective function on the test set. For the extensions we proceed in a greedy way: for each test point, we add this test point to the training set and then give it the label +1 or -1 that leads to the smaller quality value on the augmented training set. We also tried several other extensions suggested in the literature, but the results did not differ much. To compute the test error, we then evaluate the quality function on the test set labeled according to the extension. For Ncut, we do this based on the k-nearest neighbor graph on the test set only. Note that this experiment does not make sense on the network data, as there is no default procedure to construct the subgraphs for training and testing. The results on the numeric data sets are reported in Table 1 (left table, bottom lines). We see that NNC performs roughly comparably to the other algorithms. This is not really what we wanted to obtain, our hope was that NNC obtains better test values as it is less prone to overfitting. The most likely explanation is that both K-means and spectral clustering have already reasonably good extension properties. This can be due to the fact that as NNC, both algorithms consider only a certain subclass of all partitions: Voronoi partitions for K-means, and partitions induced by eigenvectors for spectral clustering. See below for more discussion. 7 6 Discussion In this paper we investigate clustering algorithms which minimize quality functions. Our main point is that, as soon as we require statistical consistency, we have to work with “small” function classes Fn. If we even choose Fn to be polynomial, then all problems due to NP hardness of discrete optimization problems formally disappear as the remaining optimization problems become inherently polynomial. From a practical point of view, the approach of using a restricted function class Fn can be seen as a more controlled way of simplifying NP hard optimization problems than the standard approaches of local optimization or relaxation. Carefully choosing the function class Fn such that overly complex target functions are excluded, we can guarantee to pick the best out of all remaining target functions. This strategy circumvents the problem that solutions of local optimization or relaxation heuristics can be arbitrarily far away from the optimal solution. The generic clustering algorithm we studied in this article is nearest neighbor clustering, which produces clusterings that are constant on small local neighborhoods. We have proved that this algorithm is statistically consistent for a large variety of popular clustering objective functions. Thus, as opposed to other clustering algorithms such as the K-means algorithm or spectral clustering, nearest neighbor clustering is guaranteed to converge to a minimizer of the true global optimum on the underlying space. This statement is much stronger than the results already known for K-means or spectral clustering. For K-means it has been proved that the global minimizer of the WSS objective function on the sample converges to a global minimizer on the underlying space (e.g., Pollard, 1981). However, as the standard K-means algorithm only discovers a local optimum on the discrete sample, this result does not apply to the algorithm used in practice. A related effect happens for spectral clustering, which is a relaxation attempting to minimize Ncut (see von Luxburg (2007) for a tutorial). It has been shown that under certain conditions the solution of the relaxed problem on the finite sample converges to some limit clustering (e.g., von Luxburg et al., to appear). However, it has been conjectured that this limit clustering is not necessarily the optimizer of the Ncut objective function. So for both cases, our consistency results represent an improvement: our algorithm provably converges to the true limit minimizer of K-means or Ncut, respectively. The same result also holds for a large number of alternative objective functions used for clustering. References M. Brusco and S. Stahl. Branch-and-Bound Applications in Combinatorial Data Analysis. Springer, 2005. S. Bubeck and U. von Luxburg. Overfitting of clustering and how to avoid it. Preprint, 2007. Data repository by G. 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Pollard. Strong consistency of k-means clustering. Annals of Statistics, 9(1):135 – 140, 1981. P. Spellman, G. Sherlock, M. Zhang, V. Iyer, M. Anders, M. Eisen, P. Brown, D. Botstein, and B. Futcher. Comprehensive identification of cell cycle-regulated genes of the yeast saccharomyces cerevisiae by microarray hybridization. Mol Biol Cell, 9(12):3273–97, 1998. K. Tsuda, H. Shin, and B. Sch¨olkopf. Fast protein classification with multiple networks. Bioinformatics, 21 (Supplement 1):ii59 – ii65, 2005. V. Vapnik. The Nature of Statistical Learning Theory. Springer, 1995. U. von Luxburg. A tutorial on spectral clustering. Statistics and Computing, 17(4), 2007. U. von Luxburg, S. Bubeck, S. Jegelka, and M. Kaufmann. Supplementary material to ”Consistent minimization of clustering objective functions”, 2007. http://www.tuebingen.mpg.de/˜ule. U. von Luxburg, M. Belkin, and O. Bousquet. Consistency of spectral clustering. Annals of Statistics, to appear. D. Watts and S. Strogatz. Collective dynamics of small world networks. Nature, 393:440–442, 1998. 8
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The pigeon as particle filter Nathaniel D. Daw Center for Neural Science and Department of Psychology New York University daw@cns.nyu.edu Aaron C. Courville Département d’Informatique et de recherche opérationnelle Université de Montréal aaron.courville@gmail.com Abstract Although theorists have interpreted classical conditioning as a laboratory model of Bayesian belief updating, a recent reanalysis showed that the key features that theoretical models capture about learning are artifacts of averaging over subjects. Rather than learning smoothly to asymptote (reflecting, according to Bayesian models, the gradual tradeoff from prior to posterior as data accumulate), subjects learn suddenly and their predictions fluctuate perpetually. We suggest that abrupt and unstable learning can be modeled by assuming subjects are conducting inference using sequential Monte Carlo sampling with a small number of samples — one, in our simulations. Ensemble behavior resembles exact Bayesian models since, as in particle filters, it averages over many samples. Further, the model is capable of exhibiting sophisticated behaviors like retrospective revaluation at the ensemble level, even given minimally sophisticated individuals that do not track uncertainty in their beliefs over trials. 1 Introduction A central tenet of the Bayesian program is the representation of beliefs by distributions, which assign probability to each of a set of hypotheses. The prominent theoretical status accorded to such ambiguity seems rather puzzlingly at odds with the all-or-nothing nature of our everyday perceptual lives. For instance, subjects observing ambiguous or rivalrous visual displays famously report experiencing either percept alternately and exclusively; for even the most fervent Bayesian, it seems impossible simultaneously to interpret the Necker cube as potentially facing either direction [1]. A longstanding laboratory model for the formation of beliefs and their update in light of experience is Pavlovian conditioning in animals, and analogously structured prediction tasks in humans. There is a rich program of reinterpreting data from such experiments in terms of statistical inference [2, 3, 4, 5, 6]. The data do appear in a number of respects to reflect key features of the Bayesian ideal — specifically, that subjects represent beliefs as distributions with uncertainty and appropriately employ it in updating them in light of new evidence. Most notable in this respect are retrospective revaluation phenomena (e.g., [7]), which demonstrate that subjects are able to revise previously favored beliefs in a way suggesting that they had entertained alternative hypotheses all along [6]. However, the data addressed by such models are, in almost all cases, averages over large numbers of subjects. This raises the question whether individuals really exhibit the sophistication attributed to them, or if it instead somehow emerges from the ensemble. Recent work by Gallistel and colleagues [8] frames the problem particularly sharply. Whereas subject-averaged responses exhibit smooth learning curves approaching asymptote (interpreted by Bayesian modelers as reflecting the gradual tradeoff from prior to posterior as data accumulate), individual records exhibit neither smooth learning nor steady asymptote. Instead responding emerges abruptly and fluctuates perpetually. These analyses soundly refute all previous quantitative theories of learning in these tasks: both Bayesian and traditional associative learning. 1 Here we suggest that individuals’ behavior in conditioning might be understood in terms of Monte Carlo methods for sequentially sampling different hypotheses (e.g., [9]). Such a model preserves the insights of a statistical framing while accounting for the characteristics of individual records. Through the metaphor of particle filtering, it also explains why exact Bayesian reasoning is a good account of the ensemble. Finally, it addresses another common criticism of Bayesian models: that they attribute wildly intractable computations to the individual. A similar framework has also recently been used to characterize human categorization learning [10]. To make our point in the most extreme way, and to explore the most novel corner of the model space, we here develop as proof of concept the idea that (as with percepts in the Necker cube) subjects sample only a single hypothesis at a time. That is, we treat them as particle filters employing only one particle. We show that even given individuals of such minimal capacity, sophisticated effects like retrospective revaluation can emerge in the ensemble. Clearly intermediate models are possible, either employing more samples or mixtures of sampling and exact methods within the individual, and the insights developed here will extend to those cases. We therefore do not mean to defend the extreme claim that subjects never track or employ uncertainty — we think this would be highly maladaptive — but instead intend to explore the role of sampling and also point out how poor is the evidentiary record supporting more sophisticated accounts, and how great is the need for better experimental and analytical methods to test them. 2 Model 2.1 Conditioning as exact filtering In conditioning experiments, a subject (say, a dog) experiences outcomes (“reinforcers,” say, food) paired with stimuli (say, a bell). That subjects learn thereby to predict outcomes on the basis of antecedent stimuli is demonstrated by the finding that they emit anticipatory behaviors (such as salivation to the bell) which are taken directly to reflect the expectation of the outcome. Human experiments are analogously structured, but using various cover stories (such as disease diagnosis) and with subjects typically simply asked to state their beliefs about how much they expect the outcome. A standard statistical framing for such a problem [5], which we will adopt here, is to assume that subjects are trying to learn the conditional probability P(r | x) of (real-valued) outcomes r given (vector-valued) stimuli x. One simple generative model is to assume that each stimulus xi (bells, lights, tones) produces reinforcement according to some unknown parameter wi; that the contributions of multiple stimuli sum; and that the actual reward is Gaussian in the the aggregate. That is, P(r | x) = N(x · w, σ2 o), where we take the variance parameter as known. The goal of the subject is then to infer the unknown weights in order to predict reinforcement. If we further assume the weights w can change with time, and take that change as Gaussian diffusion, P(wt+1 | wt) = N(wt, σ2 dI) (1) then we complete the well known generative model for which Bayesian inference about the weights can be accomplished using the Kalman filter algorithm [5]. Given a Gaussian prior on w0, the posterior distribution P(wt | x1..t, r1...t) also takes a Gaussian form, N( ˆwt, Σt), with the mean and covariance given by the recursive Kalman filter update equations. Returning to conditioning, a subject’s anticipatory responding to test stimulus xt is taken to be proportional to her expectation about rt conditional on xt, marginalizing out uncertainty over the weights. E(rt | xt, ˆwt, Σt) = xt · ˆwt. 2.2 Conditioning as particle filtering Here we assume instead that subjects do not maintain uncertainty in their posterior beliefs, via covariance Σt, but instead that subject L maintains a point estimate ewL t and treats it as true with certainty. Even given such certainty, because of diffusion intervening between t and t + 1, ewL t+1 will be uncertain; let us assume that she recursively samples her new point estimate ewL t+1from the posterior given this diffusion and the new observation xt+1, rt+1: ewL t+1 ∼P(wL t+1 | wt = ewL t , xt+1, rt+1) (2) 2 This is simply a Gaussian given by the standard Kalman filter equations. In particular, the mean of the sampling distribution is ewL t +xt+1κ(rt+1−xt+1· ewt). Here the Kalman gain κ = σ2 d/(σ2 d+σ2 o) is constant; the expected update in ew, then, is just that given by the Rescorla-Wagner [11] model. Such seemingly peculiar behavior may be motivated by the observation that, assuming that the initial ewL 0 is sampled according to the prior, this process also describes the evolution of a single sample in particle filtering by sequential importance sampling, with Equation 2 as the optimal proposal distribution [9]. (In this algorithm, particles evolve independently by sequential sampling, and do not interact except for resampling.) Of course, the idea of such sampling algorithms is that one can estimate the true posterior over wt by averaging over particles. In importance sampling, the average must be weighted according to importance weights. These (here, the product of P(rt+1 | xt+1, wt = ewL t ) over each t) serve to squelch the contribution of particles whose trajectories turn out to be conditionally more unlikely given subsequent observations. If subjects were to behave in accord with this model, then this would give us some insight into the ensemble average behavior, though if computed without importance reweighting, the ensemble average will appear to learn more slowly than the true posterior. 2.3 Resampling and jumps One reason why subjects might employ sampling is that, in generative models more interesting than the toy linear, Gaussian one used here, Bayesian reasoning is notoriously intractable. However, the approximation from a small number of samples (or in the extreme case considered here, one sample) would be noisy and poor. As we can see by comparing the particle filter update rule of Equation 2 to the Kalman filter, because the subject-as-single-sample does not carry uncertainty from trial to trial, she is systematically overconfident in her beliefs and therefore tends to be more reluctant than optimal in updating them in light of new evidence (that is, the Kalman gain is low). This is the individual counterpart to the slowness at the ensemble level, and at the ensemble level, it can be compensated for by importance reweighting and also by resampling (for instance, standard sequential importance resampling; [12, 9]). Resampling kills off conditionally unlikely particles and keeps most samples in conditionally likely parts of the space, with similar and high importance weights. Since optimal reweighting and resampling both involve normalizing importance weights over the ensemble, they are not available to our subject-as-sample. However, there are some generative models that are more forgiving of these problems. In particular, consider Yu and Dayan’s [13] diffusion-jump model, which replaces Equation 1 with P(wt+1 | wt) = (1 −π)N(wt, σ2 dI) + πN(0, σ2 j I) (3) with σj ≫σd. Here, the weights usually diffuse as before, but occasionally (with probability π) are regenerated anew. (We refer to these events as “jumps” and the previous model of Equation 1 as a “no-jump” model, even though, strictly speaking, diffusion is accomplished by smaller jumps.) Since optimal inference in this model is intractable (the number of modes in the posterior grows exponentially) Yu and Dayan [13] propose maintaining a simplified posterior: they make a sort of maximum likelihood determination whether a jump occurred or not; conditional on this the posterior is again Gaussian and inference proceeds as in the Kalman filter. If we use Equation 3 together with the one-sample particle filtering scheme of Equation 2, then we simplify the posterior still further by not carrying over uncertainty from trial to trial, but instead only a point estimate. As before, at each step, we sample from the posterior P(wL t+1 | wt = ewL t , xt+1, rt+1) given total confidence in our previous estimate. This distribution now has two modes, one representing the posterior given that a jump occurred, the other representing the posterior given no jump. Importantly, we are more likely to infer a jump, and resample from scratch, if the observation rt+1 is far from that expected under the hypothesis of no jump, xt+1 · ewL t . Specifically, the probability that no jump occurred (and that we therefore resample according to the posterior distribution given drift — effectively, the chance that the sample “survives” as it would have in the no-jump Kalman filter) — is proportional to P(rt+1 | xt+1, wt =ewL t , no jump). This is also the factor that the trial would contribute to the importance weight in the no-jump Kalman filter model of the previous section. The importance weight, in turn, is also the factor that would determine the chance that a particle would be selected during an exact resampling step [12, 9]. 3 Figure 1: Aggregate versus individual behavior in conditioning, figures adapted with permission from [8], copyright 2004 by The National Academy of Sciences of the USA. (a) Mean over subjects reveals smooth, slow acquisition curve (timebase is in sessions). (b) Individual records are noisier and with more abrupt changes (timebase is in trials). (c) Examples of fits to individual records assuming the behavior is piecewise Poisson with abrupt rate shifts. 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 trial average P(r) kalman jumps no jumps (a) 0 50 100 0 1.5 jumps 0 50 100 0 1.5 0 50 100 0 1 no jumps 0 50 100 0 1 (b) (c) 1 50 >100 0 0.05 0.1 0.15 0.2 0.25 dynamic interval probability (d) Figure 2: Simple acquisition in conditioning, simulations using particle filter models. (a) Mean behavior over samples for jump (π = 0.075; σj = 1; σd = 0.1; σo = 0.5) and no-jump (π = 0) particle filter models of conditioning, plotted against exact Kalman filter for same parameters (and π = 0). (b) Two examples of individual subject traces for the no-jump particle filter model. (c) Two examples of individual subject traces for the particle filter model incorporating jumps. (d) Distribution over individuals using the jump model of the “dynamic interval” of acquisition, that is the number of trials over which responding grows from negligible to near-asymptotic levels. There is therefore an analogy between sampling in this model and sampling with resampling in the simpler generative model of Equation 1. Of course, this cannot exactly accomplish optimal resampling, both because the chance that a particle survives should be normalized with respect to the population, and because the distribution from which a non-surviving particle resamples should also depend on the ensemble distribution. However, it has a similar qualitative effect of suppressing conditionally unlikely samples and replacing them ultimately with conditionally more likely ones. We can therefore view the jumps of Equation 3 in two ways. First, they could correctly model a jumpy world; by periodically resetting itself, such a world would be relatively forgiving of the tendency for particles in sequential importance sampling to turn out conditionally unlikely. Alternatively, the jumps can be viewed as a fiction effectively encouraging a sort of resampling to improve the performance of low-sample particle filtering in the non-jumpy world of Equation 1. Whatever their interpretation, as we will show, they are critical to explaining subject behavior in conditioning. 3 Acquisition In this and the following section, we illustrate the behavior of individuals and of the ensemble in some simple conditioning tasks, comparing particle filter models with and without jumps (Equations 1 and 3). Figure 1 reproduces some data reanalyzed by Gallistel and colleagues [8], who quantify across a number of experiments what had long been anecdotally known about conditioning: that individual 4 records look nothing like the averages over subjects that have been the focus of much theorizing. Consider the simplest possible experiment, in which a stimulus A is paired repeatedly with food. (We write this as A+.) Averaged learning curves slowly and smoothly climb toward asymptote (Figure 1a, here the anticipatory behavior measured is pigeons pecking), just as does the estimate of the mean, ˆwA, in the Kalman filter models. Viewed in individual records (Figure 1b), the onset of responding is much more abrupt (often it occurred in a single trial), and the subsequent behavior much more variable. The apparently slow learning results from the average over abrupt transitions occurring at a range of latencies. Gallistel et al. [8] characterized the behavior as piecewise Poisson with instantaneous rate changes (Figure 1c). These results present a challenge to the bulk of models of conditioning — not just Bayesian ones, but also associative learning theories like the seminal model of Rescorla & Wagner [11] ubiquitously produce smooth, asymptoting learning curves of a sort that these data reveal to be essentially an artifact of averaging. One further anomaly with Bayesian models even as accounts for the average curves is that acquisition is absurdly slow from a normative perspective — it emerges long after subjects using reasonable priors would be highly certain to expect reward. This was pointed out by Kakade and Dayan [5], who also suggested an account for why the slow acquisition might actually be normative due to unaccounted priors caused by pretraining procedures known as hopper training. However, Balsam and colleagues later found that manipulating the hopper pretraining did not speed learning [14]. Figure 2 illustrates individual and group behavior for the two particle filter models. As expected, at the ensemble level (Figure 2a), particle filtering without jumps learns slowly, when averaged without importance weighting or resampling and compared to the optimal Kalman filter for the same parameters. As shown, the inclusion of jumps can speed this up. In individual traces using the jumps model (Figure 2c) frequent sampled jumps both at and after acquisition of responding capture the key qualitative features of the individual records: the abrupt onset and ongoing instability. The inclusion of jumps in the generative model is key to this account: as shown in Figure 2b, without these, behavior changes more smoothly. In the jump model, when a jump is sampled, the posterior distribution conditional on the jump having occurred is centered near the observed rt, meaning that the sampled weight will most likely arrive immediately near its asymptotic level. Figure 2d shows that such an abrupt onset of responding is the modal behavior of individuals. Here (after [8]), we have fit each individual run from the jump-model simulations with a sigmoidal Weibull function, and defined the “dynamic interval” over which acquisition occurs as the number of trials during which this fit function rises from 10% to 90% of its asymptotic level. Of course, the monotonic Weibull curve is not a great characterization of the individual’s noisy predictions, and this mismatch accounts for the long tail of the distribution. Nevertheless, the cumulative distribution from our simulations closely matches the proportions of animals reported as achieving various dynamic intervals when the same analysis was performed on the pigeon data [8]. These simulations demonstrate, first, how sequential sampling using a very low number of samples is a good model of the puzzling features of individual behavior in acquisition, and at the same time clarify why subject-averaged records resemble the results of exact inference. Depending on the presumed frequency of jumps (which help to compensate for this problem) the fact that these averages are of course computed without importance weighting may also help to explain the apparent slowness of acquisition. This could be true regardless of whether other factors, such as those posited by Kakade and Dayan [5], also contribute. 4 Retrospective revaluation So far, we have shown that sequential sampling provides a good qualitative characterization of individual behavior in the simplest conditioning experiments. But the best support for sophisticated Bayesian models of learning comes from more demanding tasks such as retrospective revaluation. These tasks give the best indication that subjects maintain something more than a point estimate of the weights, and instead strongly suggest that they maintain a full joint distribution over them. However, as we will show here, this effect can actually emerge due to covariance information being implicitly represented in the ensemble of beliefs over subjects, even if all the individuals are one-particle samplers. 5 weight A weight B after AB+ −1 0 1 −1 0 1 (a) after B+ weight A weight B −1 0 1 −1 0 1 0 50 100 0 0.5 1 A B expected P(r) trials B+→ AB+→ after AB+ weight B weight A −1 0 1 −1 0 1 (b) after B+ weight A weight B −1 0 1 −1 0 1 0 50 100 0 0.5 1 A B average P(r) trials B+→ AB+→ Figure 3: Simulations of backward blocking effect, using exact Kalman filter (a) and particle filter model with jumps (b). Left, middle: Joint distributions over wAand wB following first-phase AB+ training (left) and second phase B+ training (middle). For the particle filter, these are derived from the histogram of individual particles’ joint point beliefs about the weights. Right: Mean beliefs about wA and wB, showing development of backward blocking. Parameters as in Figure 2. Retrospective revaluation refers to how the interpretation of previous experience can be changed by subsequent experience. A typical task, called backward blocking [7], has two phases. First, two stimuli, A and B, are paired with each other and reward (AB+), so that both develop a moderate level of responding. In the second phase, B alone is paired with reward (B+), and then the prediction to A alone is probed. The typical finding is that responding to A is attenuated; the intuition is that the B+ trials suggested that B alone was responsible for the reward received in the AB+ trials, so the association of A with reward is retrospectively discounted. Such retrospective revaluation phenomena are hard to demonstrate in animals (though see [15]) but robust in humans [7]. Kakade and Dayan [6] gave a more formal analysis of the task in terms of the Kalman filter model. In particular they point out that conditonal on the initial AB+ trials, the model will infer an anticorrelated joint distribution over wA and wB — i.e., that they together add up to about one. This is represented in the covariance Σ; the joint distribution is illustrated in Figure 3a (left). Subsequent B+ training indicates that wB is high, which means, given its posterior anticorrelation with wA, that the latter is likely low. Note that this explanation seems to turn crucially on the representation of the full joint distribution over the weights, rather than just a point estimate. Contrary to this intuition, Figure 3b demonstrates the same thing in the particle filter model with jumps. At the end of AB+ training, the subjects as an ensemble represent the anti-correlated joint distribution over the weights, even though each individual maintains only a particular point belief. Moreover, B+ training causes an aggregate backward blocking effect. This is because individuals who believe that wA is high tend also to believe that wB is low, which makes them most likely to sample that a jump has occurred during subsequent B+ training. The samples most likely to stay in place already have ewA low and ewB high; beliefs about wA are, on average, thereby reduced, producing the backward blocking effect in the ensemble. Note that this effect depends on the subjects sampling using a generative model that admits of jumps (Equation 3). Although the population implicitly represents the posterior covariance between wA and wB even using the diffusion model with no jumps (Equation 1; simulations not illustrated), sub6 sequent B+ training has no tendency to suppress the relevant part of the posterior, and no backward blocking effect is seen. Again, this traces to the lack of a mechanism for downweighting samples that turn out to be conditionally unlikely. 5 Discussion We have suggested that individual subjects in conditioning experiments behave as though they are sequentially sampling hypotheses about the underlying weights: like particle filters using a single sample. This model reproduces key and hitherto theoretically troubling features of individual records, and also, rather more surprisingly, has the ability to reproduce more sophisticated behaviors that had previously been thought to demonstrate that subjects represented distributions in a fully Bayesian fashion. One practical problem with particle filtering using a single sample is the lack of distributional information to allow resampling or reweighting; we have shown that use of a particular generative model previously proposed by Yu and Dayan [13] (involving sudden shocks that effectively accomplish resampling) helps to compensate qualitatively if not quantitatively for this failing. This mechanism is key to all of our results. The present work echoes and formalizes a long history of ideas in psychology about hypothesis testing and sudden insight in learning, going back to Thorndike’s puzzle boxes. It also complements a recent model of human categorization learning [10], which used particle filters to sample (sparsely or even with a single sample) over possible clusterings of stimuli. That work concentrated on trial ordering effects arising from the sparsely represented posterior (see also [16]); here we concentrate on a different set of phenomena related to individual versus ensemble behavior. Gallistel and colleagues’ [8] demonstration that individual learning curves exhibit none of the features of the ensemble average curves that had previously been modeled poses rather a serious challenge for theorists: After all, what does it mean to model only the ensemble? Surely the individual subject is the appropriate focus of theory — particularly given the evolutionary rationale often advanced for Bayesian modeling, that individuals who behave rationally will have higher fitness. The present work aims to refocus theorizing on the individual, while at the same time clarifying why the ensemble may be of interest. (At the group level, there may also be a fitness advantage to spreading different beliefs — say, about productive foraging locations — across subjects rather than having the entire population gravitate toward the “best” belief. This is similar to the phenomenon of mixed strategy equilibrium in multiplayer games, and may provide an additional motivation for sampling.) Previous models fail to predict any intersubject variability because they incorporate no variation in either the subjects’ beliefs or in their responses given their beliefs. We have suggested that the structure in response timeseries suggests a prominent role for intersubject variability in the beliefs, due to sampling. There is surely also noise in the responding, which we do not model, but for this alone to rescue previous models, one would have to devise some other explanation for the noise’s structure. (For instance, if learning is monotonic, simple IID output noise would not predict sustained excursions away from asymptote as in Fig 1c.) Similarly, nonlinearity in the performance function relating beliefs to response rates might help to account for the sudden onset of responding even if learning is smooth, but would not address the other features of the data. In addition to addressing the empirical problem of fit to the individual, sampling also answers an additional problem with Bayesian models: that they attribute to subjects the capacity for radically intractable calculations. While the simple Kalman filter used here is tractable, there has been a trend in modeling human and animal learning toward assuming subjects perform inference about model structure (e.g., recovering structural variables describing how different latent causes interact to produce observations; [4, 3, 2]). Such inference cannot be accomplished exactly using simple recursive filtering like the Kalman filter. Indeed, it is hard to imagine any approach other than sequentially sampling one or a small number of hypothetical model structures, since even with the structure known, there typically remains a difficult parametric inference problem. The present modeling is therefore motivated, in part, toward this setting. While in our model, subjects do not explicitly carry uncertainty about their beliefs from trial to trial, they do maintain hyperparameters (controlling the speed of diffusion, the noise of observations, and the probability of jumps) that serve as a sort of constant proxy for uncertainty. We might expect them 7 to adjust these so as to achieve the best performance; because the inference is anyway approximate, the veridical, generative settings of these parameters will not necessarily perform the best. Of course, the present model is only the simplest possible sketch, and there is much work to do in developing it. In particular, it would be useful to develop less extreme models in which subjects either rely on sampling with more particles, or on some combination of sampling and exact inference. We posit that many of the insights developed here will extend to such models, which seem more realistic since exclusive use of low-sample particle filtering would be extremely brittle and unreliable. (The example of the Necker cube also invites consideration of Markov Chain Monte Carlo sampling for exploration of multimodal posteriors even in nonsequential inference [1] — such methods are clearly complementary.) However, there is very little information available about individual-level behavior to constrain the details of approximate inference. The present results on backward blocking stress again the perils of averaging and suggest that data must be analyzed much more delicately if they are ever to bear on issues of distributions and uncertainty. In the case of backward blocking, if our account is correct, there should be a correlation, over individuals, between the degree to which they initially exhibited a low ewB and the degree to which they subsequently exhibited a backward blocking effect. This would be straightforward to test. More generally, there has been a recent trend [17] toward comparing models against raw trial-by-trial data sets according to the cumulative loglikelihood of the data. Although this measure aggregates over trials and subjects, it measures the average goodness of fit, not the goodness of fit to the average, making it much more sensitive for purposes of studying the issues discussed in this article. References [1] P Schrater and R Sundareswara. Theory and dynamics of perceptual bistability. In NIPS 19, 2006. [2] TL Griffiths and JB Tenenbaum. Structure and strength in causal induction. Cognit Psychol, 51:334–384, 2005. [3] AC Courville, ND Daw, and DS Touretzky. Similarity and discrimination in classical conditioning: A latent variable account. In NIPS 17, 2004. [4] AC Courville, ND Daw, GJ Gordon, and DS Touretzky. Model uncertainty in classical conditioning. In NIPS 16, 2003. [5] S Kakade and P Dayan. Acquisition and extinction in autoshaping. Psychol Rev, 109:533–544, 2002. [6] S Kakade and P Dayan. Explaining away in weight space. In NIPS 13, 2001. [7] DR Shanks. Forward and backward blocking in human contingency judgement. Q J Exp Psychol B, 37:1–21, 1985. [8] CR Gallistel, S Fairhurst, and P Balsam. The learning curve: Implications of a quantitative analysis. Proc Natl Acad Sci USA, 101:13124–13131, 2004. [9] A Doucet, S Godsill, and C Andrieu. On sequential Monte Carlo sampling methods for Bayesian filtering. Stat Comput, 10:197–208, 2000. [10] AN Sanborn, TL Griffiths, and DJ Navarro. A more rational model of categorization. In CogSci 28, 2006. [11] RA Rescorla and AR Wagner. A theory of Pavlovian conditioning: The effectiveness of reinforcement and non-reinforcement. In AH Black and WF Prokasy, editors, Classical Conditioning, 2: Current Research and Theory, pages 64–69. 1972. [12] DB Rubin. Using the SIR algorithm to simulate posterior distributions. In JM Bernardo, MH DeGroot, DV Lindley, and AFM Smith, editors, Bayesian Statistics, Vol. 3, pages 395–402. 1988. [13] AJ Yu and P Dayan. Expected and unexpected uncertainty: ACh and NE in the neocortex. In NIPS 15, 2003. [14] PD Balsam, S Fairhurst, and CR Gallistel. Pavlovian Contingencies and Temporal Information. J Exp Psychol Anim Behav Process, 32:284–295, 2006. [15] RR Miller and H Matute. Biological significance in forward and backward blocking: Resolution of a discrepancy between animal conditioning and human causal judgment. J Exp Psychol Gen, 125:370–386, 1996. [16] ND Daw, AC Courville, and P Dayan. Semi-rational models of cognition: The case of trial order. In N Chater and M Oaksford, editors, The Probabilistic Mind. 2008. (in press). [17] ND Daw and K Doya. The computational neurobiology of learning and reward. Curr Opin Neurobiol, 16:199–204, 2006. 8
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Non-Parametric Modeling of Partially Ranked Data Guy Lebanon Department of Statistics, and School of Elec. and Computer Engineering Purdue University - West Lafayette, IN lebanon@stat.purdue.edu Yi Mao School of Elec. and Computer Engineering Purdue University - West Lafayette, IN ymao@ecn.purdue.edu Abstract Statistical models on full and partial rankings of n items are often of limited practical use for large n due to computational consideration. We explore the use of non-parametric models for partially ranked data and derive efficient procedures for their use for large n. The derivations are largely possible through combinatorial and algebraic manipulations based on the lattice of partial rankings. In particular, we demonstrate for the first time a non-parametric coherent and consistent model capable of efficiently aggregating partially ranked data of different types. 1 Introduction Rankers such as humans, search engines, and classifiers, output full or partial rankings representing preference relations over n items. The absence of numeric scores or the lack of calibration between existing numeric scores output by the rankers necessitates modeling rankings rather than numeric scores. To effectively analyze ranked data, a statistical model has the following desiderata. (1) Handle efficiently a very large number of items n by reverting to partial rather than full rankings. (2) Probability assignment to full and partial rankings should be coherent and contradiction-free. (3) Conduct inference based on training data consisting of partial rankings of different types. (4) Correct retrieval of the underlying process as training data increases (statistical consistency). (5) In the case of large n convergence of the estimator to the underlying process can be extremely slow for fully ranked data but should be much faster when restricted to simpler partial rankings. In this paper, we present a model achieving the above requirements without any parametric assumptions on the underlying generative process. The model is based on the non-parametric Parzen window estimator with a Mallows kernel on permutations. By considering partial rankings as censored data we are able to define the model on both full and partial rankings in a coherent and contradictionfree manner. Furthermore, we are able to estimate the underlying structure based on data containing partial rankings of different types. We demonstrate computational efficiency for partial rankings, even in the case of a very large n, by exploiting the combinatorial and algebraic structure of the lattice of partial rankings. We start below by reviewing basic concepts concerning partially ranked data (see [1] for further details) and the Mallows model and then proceed to define our non-parametric estimator. We conclude by demonstrating computational efficiency and some experiments. 2 Permutations and Cosets A permutation π is a bijective function π : {1, . . . , n} →{1, . . . , n} associating with each item i ∈{1, . . . , n} a rank π(i) ∈{1, . . . , n}. In other words, π(i) denotes the rank given to item i 1 and π−1(i) denotes the item assigned to rank i. We denote a permutation π using the following vertical bar notation π−1(1)|π−1(2)| · · · |π−1(n). For example, the permutation π(1) = 2, π(2) = 3, π(3) = 1 would be denoted as 3|1|2. In this notation the numbers correspond to items and the locations of the items in their corresponding compartments correspond to their ranks. The collection of all permutations of n items forms the non-Abelian symmetric group of order n, denoted by Sn, using function composition as the group operation πσ = π ◦σ. We denote the identity permutation by e. The concept of inversions and the result below, taken from [7], will be of great use later on. Definition 1. The inversion set of a permutation π is the set of pairs U(π) def= {(i, j) : i < j, π(i) > π(j)} ⊂{1, . . . , n} × {1, . . . , n} whose cardinality is denoted by i(π) def= |U(π)|. For example, i(e) = |∅| = 0, and i(3|2|1|4) = |{(1, 2), (1, 3), (2, 3)}| = 3. Proposition 1 (e.g., [7]). The map π 7→U(π) is a bijection. When n is large, the enormous number of permutations raises difficulties in using the symmetric group for modeling rankings. A reasonable solution is achieved by considering partial rankings which correspond to cosets of the symmetric group. For example, the subgroup of Sn consisting of all permutations that fix the top k positions is denoted S1,...,1,n−k = {π ∈Sn : π(i) = i, i = 1, . . . , k}. The right coset S1,...,1,n−kπ = {σπ : σ ∈S1,...,1,n−k} is the set of permutations consistent with the ordering of π on the k top-ranked items. It may thus be interpreted as a partial ranking of the top k items, that does not contain any information concerning the relative ranking of the bottom n−k items. The set of all such partial rankings forms the quotient space Sn/S1,...,1,n−k. Figure 1 (left) displays the set of permutations that corresponds to a partial ranking of the top 2 out of 4 items. We generalize this concept to arbitrary partial rankings using the concept of composition. Definition 2. A composition of n is a sequence γ =(γ1, . . . , γr) of positive integers whose sum is n. Note that in contrast to a partition, in a composition the order of the integers matters. A composition γ = (γ1, . . . , γr) corresponds to a partial ranking with γ1 items in the first position, γ2 items in the second position and so on. For such a partial ranking it is known that the first set of γ1 items are to be ranked before the second set of γ2 items etc., but no further information is conveyed about the orderings within each set. The partial ranking S1,...,1,n−kπ of the top k items is a special case corresponding to γ = (1, . . . , 1, n −k). More formally, let N1 = {1, . . . , γ1}, N2 = {γ1 + 1, . . . , γ1 + γ2}, · · · , Nr = {γ1 + · · · + γr−1 + 1, . . . , n}. Then the subgroup Sγ contains all permutations π for which the set equalities π(Ni) = Ni, ∀i holds (all permutations that only permute within each Ni). A partial ranking of type γ is equivalent to a coset Sγπ = {σπ:σ∈Sγ, π∈Sn} and the set of such partial rankings forms the quotient space Sn/Sγ. The vertical bar notation described above is particularly convenient for denoting partial rankings. We list items 1, . . . , n separated by vertical bars, indicating that items on the left side of each vertical bar are preferred to (ranked higher than) items on the right side of the bar. For example, the partial ranking displayed in Figure 1 (left) is denoted by 3|1|2, 4. In the notation above, the ordering of items not separated by a vertical line is meaningless, and for consistency we use the conventional ordering e.g., 1|2, 3|4 rather than 1|3, 2|4. The set of all partial rankings Wn def= {Sγπ : π ∈Sn, ∀γ} (1) which includes all full rankings π ∈Sn, is a subset of all possible partial orders on {1, . . . , n}. While the formalism of partial rankings in Wn cannot realize all partial orderings, it is sufficiently powerful to include many useful naturally occurring orderings as special cases. Furthermore, as demonstrated in later sections, it enables simplification of the otherwise overwhelming computational difficulty. Special cases include the following partial rankings. • π ∈Sn corresponds to permutation or a full ordering e.g. 3|2|4|1. • S1,n−1π e.g. 3|1, 2, 4, corresponds to selection of the top alternative such as a multiclass classification. • S1,...,1,n−kπ e.g. 1|3|2, 4, corresponds to top k ordering such as the ranked list of top k webpages output by search engines. 2 • Sk,n−kπ e.g. 1, 2, 4|3, 5, corresponds to a more preferred and a less preferred dichotomy such as a multilabel classification. In the cases above, we often have a situation where n is large (or even approaching infinity as in the third example above) but k is of manageable size. Traditionally, data from each one of the special cases above was modeled using different tools and was considered fundamentally different. That problem was aggravated as different special cases were usually handled by different communities such as statistics, computer science, and information retrieval. In constructing a statistical model on permutations or cosets, it is essential to relate one permutation to another. We do this using a distance function on permutations d : Sn ×Sn →R that satisfies the usual metric function properties, and in addition is invariant under item relabeling or right action of the symmetric group [1] d(π, σ) = d(πτ, στ) ∀π, σ, τ ∈Sn. There have been many propositions for such right-invariant distance functions, the most popular of them being Kendall’s tau [3] d(π, σ) = n−1 X i=1 X l>i I(πσ−1(i) −πσ−1(l)) (2) where I(x) = 1 for x > 0 and I(x) = 0 otherwise. Kendall’s tau d(π, σ) can be interpreted as the number of pairs of items for which π and σ have opposing orderings (called disconcordant pairs) or the minimum number of adjacent transpositions needed to bring π−1 to σ−1 (adjacent transposition flips a pair of items having adjacent ranks). By right invariance, d(π, σ) = d(πσ−1, e) which, for Kendall’s tau equals the number of inversions i(πσ−1). This is an important observation that will allow us to simplify many expressions concerning Kendall’s tau using the theory of permutation inversions from the combinatorics literature. 3 The Mallows Model and its Extension to Partial Rankings The Mallows model [5] is a simple model on permutations based on Kendall’s tau distance using a location parameter κ and a spread parameter c (which we often treat as a constant) pκ(π) = exp (−cd(π, κ) −log ψ(c)) π, κ ∈Sn c ∈R+. (3) The normalization term ψ doesn’t depend on κ and has the closed form ψ(c) = X π∈Sn e−c d(π,κ) = (1 + e−c)(1 + e−c + e−2c) · · · (1 + e−c + · · · + e−(n−1)c) (4) as shown by the fact that d(π, σ) = i(πσ−1) and the following proposition. Proposition 2 (e.g., [7]). For q > 0, P π∈Sn qi(π) = Qn−1 j=1 Pj k=0 qk. Model (3) has been motivated on axiomatic grounds by Mallows and has been a major focus of statistical modeling on permutations. A natural extension to partially ranked data is to consider a partial ranking as censored data equivalent to the set of permutations in its related coset: pκ(Sγπ) def= X τ∈Sγπ pκ(τ) = ψ−1(c) X τ∈Sγπ exp (−c d(τ, κ)) . (5) Fligner and Verducci [2] have shown that in the case of γ = (1, . . . , 1, n −k) the above summation has a closed form expression. However, the apparent absence of a closed form formula for more general partial rankings prevented the widespread use of the above model for large n and encouraged more ad-hoc and heuristic models [1, 6]. This has become especially noticeable due to a new surge of interest, especially in the computer science community, in partial ranking models for large n. The ranking lattice presented next enables extending Fligner and Verducci’s closed form to a more general setting which is critical to the practicality of our non-parametric estimator. 4 The Ranking Lattice Partial rankings Sγπ relate to each other in a natural way by expressing more general, more specific or inconsistent ordering. We define below the concepts of partially ordered sets and lattices and then relate them to partial rankings by considering the set of partial rankings Wn as a lattice. Some of the definitions below are taken from [7], where a thorough introduction to posets can be found. 3 Items: web pages movies labels etc. Ranks 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 PSfrag replacements S1,1,2π = {σ1π, σ2π} = 3|1|2, 4 σ1π π σ2π asdf
PSfrag replacements 1|2|3 1|3|2 2|1|3 3|1|2 1|2,3 1,3|2 2|1,3 3|1,2 2,3|1 1,2,3 2|3|1 3|2|1 1,2|3 Figure 1: A partial ranking corresponds to a coset or a set of permutations (left). The Hasse diagram of W3. Some lines are dotted for 3D visualization purposes (right). Definition 3. A partially ordered set or poset (Q, ⪯), is a set Q endowed with a binary relation ⪯ satisfying ∀x, y, z ∈Q (i) reflexibility: x ⪯x, (ii) anti-symmetry: x ⪯y and y ⪯x ⇒x = y, and (iii) transitivity: x ⪯y and y ⪯z ⇒x ⪯z. We write x ≺y when x ⪯y and x ̸= y. We say that y covers x when x ≺y and there is no z ∈Q such that x ≺z ≺y. A finite poset is completely described by its covering relation. The planar Hasse diagram of (Q, ⪯) is the graph connecting the elements of Q as nodes using edges that correspond to the covering relation. An additional requirement is that if y covers x then y is drawn higher than x. Two elements x, y are comparable if x ⪯y or y ⪯x and otherwise are incomparable. The set of partial rankings Wn defined in (1) is naturally endowed with the partial order of ranking refinement i.e. π ≺σ if π refines σ or alternatively if we can get from π to σ by dropping vertical lines [4]. Figure 1 (right) shows the Hasse diagram of W3. A lower bound z of two elements in a poset x, y satisfies z ⪯x and z ⪯y. The greatest lower bound of x, y or infimum is a lower bound of x, y that is greater than or equal to any other lower bound of x, y. Infimum, and the analogous concept of supremum are denoted by x ∧y and x ∨y or V{x1, . . . , xk} and W{x1, . . . , xk} respectively. Two elements x, y ∈Wn are consistent if there exists a lower bound in Wn. Note that consistency is a weaker relation than comparability. For example, 1|2, 3|4 and 1, 2|3, 4 are consistent but incomparable while 1|2, 3|4 and 2|1, 3|4 are both inconsistent and incomparable. Using the vertical bar notation, two elements are inconsistent iff there exists two items i, j that appear on opposing sides of a vertical bar in x, y i.e. x = · · · i|j · · · while y = · · · j|i · · · . A poset for which ∧and ∨always exist is called a lattice. Lattices satisfy many useful combinatorial properties - one of which is that they are completely described by the ∧ and ∨operations. While the ranking poset is not a lattice, it may be turned into one by augmenting it with a minimum element ˆ0. Proposition 3. The union ˜Wn def= Wn∪{ˆ0} of the ranking poset and a minimum element is a lattice. Proof. Since ˜Wn is finite, it is enough to show existence of ∧, ∨for pairs of elements [7]. We begin by showing existence of x ∧y. If x, y are inconsistent, there is no lower bound in Wn and therefore the unique lower bound ˆ0 is also the infimum x ∧y. If x, y are consistent, their infimum may be obtained as follows. Since x and y are consistent, we do not have a pair of items i, j appearing as i|j in x and j|i in y. As a result we can form a lower bound z to x, y by starting with a list of numbers and adding the vertical bars that are in either x or y, for example for x = 3|1, 2, 5|4 and y = 3|2|1, 4, 5 we have z = 3|2|1, 5|4. The resulting z ∈Wn, is smaller than x and y since by construction it contains all preferences (encoded by vertical bars) in x and y. It remains to show that for every other lower bound z′ to x and y we have z′ ⪯z. If z′ is comparable to z, z′ ⪯z since removing any vertical bar from z results in an element that is not a lower bound. If z′ is not comparable to z, then both z, z′ contain the vertical bars in x and vertical bars in y possibly with some additional ones. By construction z contains only the essential vertical bars to make it a lower bound and hence z′ ≺z, contradicting the assumption that z, z′ are non-comparable. By Proposition 3.3.1 of [7] a poset for which an infimum is always defined and that has a supremum element is necessarily a lattice. Since we just proved that ∧always exists for ˜Wn and 1, . . . , n = W ˜Wn, the proof is complete. 4 PSfrag replacements 1, . . . , n ˆ0 Sγπ Sλσ PSfrag replacements 1, . . . , n ˆ0 Sγπ Sλσ Figure 2: Censored data in the Hasse diagram of ˜Wn corresponding to two partial rankings with the same (left) and different (right) number of vertical bars. The two big triangles correspond to the Hasse diagram of Figure 1 (right) with permutations occupying the bottom level. 5 Non-Parametric Models on the Ranking Lattice The censored data approach to partial ranking described by Equation (5) may be generalized to arbitrary probability models p on Sn. Extending a probability model p on Sn to ˜Wn by defining it to be zero on ˜Wn \ Sn and considering the partial ranking model g(Sγπ) = X σ∈Sγπ p(σ) = X τ⪯Sγπ p(τ), τ ∈˜Wn. (6) The function g, when restricted to partial rankings of the same type G = {Sγπ : π ∈Sn} constitutes a distribution over G. The relationship between p and g may be more elegantly described through M¨obius inversion on lattices: for the functions p, g : ˜Wn →[0, 1] defined above we have g(τ) = X τ ′⪯τ p(τ ′) iff p(τ) = X τ ′⪯τ g(τ ′)µ(τ ′, τ) τ, τ ′ ∈˜Wn (7) where µ : ˜Wn × ˜Wn →R is the M¨obius function of the lattice ˜Wn [7]. For large n, modeling partial, rather than full rankings is a computational necessity. It is tempting to construct a statistical model on partial rankings directly without reference to an underlying permutation model, e.g. [1, 6]. However, doing so may lead to contradicting probabilities in the permutation level i.e. there exists no distribution p on Sn consistent with the specified values of g at g(Sγπ) and g(Sλσ), γ ̸= λ. Figure 2 illustrates this problem for partial rankings with the same (left) and different (right) number of vertical bars. Verifying that no contradictions exist involves solving a lengthy and complicated set of equations. The alternative we present of starting with a permutation model p : Sn →R and extending it to g via the M¨obius inversion is a simple and effective way of avoiding such lack of coherence. Identifying partially ranked training data D = {Sγiπi : i = 1, . . . , m} as censored data, a nonparametric Parzen window estimator based on the Mallows kernel is ˆp(π) = 1 m ψ(c) m X i=1 1 |Sγi| X τ∈Sγiπi exp(−cd(π, τ)) π ∈Sn (8) where we used the fact that |Sγiπi| = |Sγie| = |Sγi|, or its censored data extension ˆg(Sλπ) = 1 m ψ(c) m X i=1 1 |Sγi| X κ∈Sλπ X τ∈Sγiπi exp(−cd(κ, τ)) Sγπ ∈˜Wn. (9) Model (8) and its partial ranking extension (9) satisfy requirement 3 in Section 1 since D contains partial rankings of possibly different types. Similarly, by the censored data interpretation of partial rankings, they satisfy requirement 2. Requirement 4 holds as m, c →∞by standard properties of the Parzen window estimator. Requirement 5 holds since ˆg in (9) restricted to G = {Sγπ : π ∈Sn} becomes a consistent model on a much smaller probability space. Requirement 1 is demonstrated in the next section by deriving an efficient computation of (9). In the case of a very large number of items reverting to partial ranking of type γ is a crucial element. The coherence between ˆp, ˆg and 5 the nature of D are important factors in modeling partially ranked data. In the next section we show that even for n →∞(as is nearly the case for web-search), the required computation is feasible as it depends only on the complexity of the composition γ characterizing the data D and the partial rankings on which ˆg is evaluated. 6 Efficient Computation and Inversion Combinatorics Computational efficiency of the inner summations in Equations (8)-(9) is crucial to the practical application of the estimators ˆp, ˆg. By considering how the pairs constituting i(τ) decompose with respect to certain cosets we can obtain efficient computational schemes for (8),(9). Proposition 4. The following decomposition of i(τ) with respect to a composition γ holds i(τ) = r X k=1 aγ k(τ) + r X k=1 r X l=k+1 bγ kl(τ) ∀τ ∈Sn where (10) aγ k(τ) def= (s, t) : s < t , k−1 X j=1 γj < τ −1(t) < τ −1(s) ≤ k X j=1 γj (11) bγ kl(τ) def= (s, t) : s < t , k−1 X j=1 γj < τ −1(t) ≤ k X j=1 γj ≤ l−1 X j=1 γj < τ −1(s) ≤ l X j=1 γj . (12) Proof. First note that by the right invariance of Kendall’s tau d(τ, σ) = i(τσ−1), we have i(τ) = i(τ −1) and we may decompose i(τ −1) instead of i(τ). The set appearing in the definition of aγ k(τ) contains all label pairs (s, t) that are inversions of τ −1 and that appear in the k-compartment of the decomposition γ. The set appearing in the definition of bγ kl(τ) contains label pairs (s, t) that are inversions of τ −1 and for which s and t appear in the l and k compartments of γ respectively. Since any inversion pair appear in either one or two compartments, the decomposition holds. Decomposition (10) is actually a family of decompositions as it holds for all possible compositions γ. For example, i(τ) = 4 for τ = 4|1|3|2 ∈S4−2π = 1, 4|2, 3, with inversions (4, 1), (4, 3), (4, 2), (3, 2) for τ −1. The first compartment 1, 4 contains the inversion (4, 1) and so aγ 1(τ) = 1. The second compartment 2, 3 contains the inversion (3, 2) and so aγ 2(τ) = 1. The cross compartment inversions are (4, 3), (4, 2) making bγ 12(τ) = 2. The significance of (10) is that as we sum over all representatives of the coset τ ∈Sγπ the cross compartmental inversions bγ kl(τ) remain constant while the within-compartmental inversions aγ k(τ) vary over all possible combinations. This leads to powerful extensions of Proposition 2 which in turn lead to efficient computation of (8), (9). Proposition 5. For π ∈Sn, q > 0, and a composition γ we have X τ∈Sγπ qi(τ) = q P r k=1 P r l=k+1 bγ kl(π) r Y s=1 γs−1 Y j=1 j X k=0 qk. (13) Proof. X τ∈Sγπ qi(τ) = X τ∈Sγπ q P r k=1 aγ k(τ)+P r k=1 P r l=k+1 bγ kl(τ) = q P r k=1 P r l=k+1 bγ kl(π) X τ∈Sγπ q P r k=1 aγ k(τ) = q P r k=1 P r l=k+1 bγ kl(π) r Y s=1 X τ∈Sγs qi(τ) = q P r k=1 P r l=k+1 bγ kl(π) r Y s=1 γs−1 Y j=1 j X k=0 qk. Above, we used two ideas: (i) disconcordant pairs between two different compartments of the coset Sγπ are invariant under change of the coset representative, and (ii) the number of disconcordant pairs within a compartment varies over all possible choices enabling the replacement of the summation by a sum over a lower order symmetric group. An important feature of (13) is that only the first and relatively simple term q P r k=1 P r l=k+1 bγ kl(π) depends on π. The remaining terms depend only on the partial ranking type γ and thus may be pre-computed and tabulated for efficient computation. The following two corollaries generalize the well known Proposition 2 to arbitrary cosets enabling efficient computation of (8), (9). 6 λ⧹γ (1, n −1) (1, · · · , 1, n −t) (t, n −t) (1, n −1) O(1) O(1) O(1) (1, · · · , 1, n −k) O(k) O(k + t) O(k + t) (k, n −k) O(k) O(k + t) O(k + t) Table 1: Computational complexity for computing Equation (9) for each training example. Notice the independence of the complexity terms from n. Corollary 1. P τ∈Sγπ qi(τκ) = q P r k=1 P r l=k+1 bγ kl(πκ) Qr s=1 Qγs−1 j=1 Pj k=0 qk κ ∈Sn. Proof. Using group theory, it can be shown that the set equality (Sγπ)κ = Sγ(πκ) holds. As a result, P τ∈Sγπ qi(τκ) = P τ ′∈Sγ(πκ) qi(τ ′). Proposition 5 completes the proof. Corollary 2. The partial ranking extension corresponding to the Mallows model pκ is pκ(Sγπ) = Qr s=1 Qγs−1 j=1 Pj k=0 e−kc Qn−1 j=1 Pj k=0 e−kc e−c P r k=1 P r l=k+1 bγ kl(πκ−1) ∝e−c P r k=1 P r l=k+1 bγ kl(πκ−1) Proof. Using Corollary 1 we have pκ(Sγπ) = X τ∈Sγπ pκ(τ) = P τ∈Sγπ exp(−c d(τ, κ)) P τ∈Sn exp(−c d(τ, κ)) = P τ∈Sγπ exp(−c i(τκ−1)) Qn−1 j=1 Pj k=0 e−kc = P τ∈Sγπ(exp(−c))i(τκ−1) Qn−1 j=1 Pj k=0 e−kc = e−c P r k=1 P r l=k+1 bγ kl(πκ−1) Qr s=1 Qγs−1 j=1 Pj k=0 e−kc Qn−1 j=1 Pj k=0 e−kc Despite its daunting appearance, the expression in Corollary 2 can be computed relatively easily. The fraction does not depend on π or κ and in fact may be considered as a normalization constant that may be easily pre-computed and tabulated. The remaining term is relatively simple and depends on the location parameter κ and the coset representative π. Corollary 2 and Proposition 6 below (whose proof is omitted due to lack of space), provide efficient computation for the estimators (8), (9). The complexity of computing (14) and (8), (9) for some popular partial ranking types appears in Table 1. Proposition 6. X σ∈Sλπ1 X τ∈Sγπ2 e−c d(σ,τ) = X τ∈π1π−1 2 Sγ r Y k=1 r Y l=k+1 e−c bλ kl(τ) r Y s=1 λs−1 Y j=1 j X k=0 e−kc . (14) 7 Applications Figure 3 (top left) compares the average test log-likelihood between the Mallows model and the nonparametric model with different c as a function of training size averaged over 10 cross validations. We use fully ranked APA election data (rankings are ballots for five APA presidential candidates), and during each iteration, 30% of the examples are randomly selected for testing. The parameters of the Mallows model are estimated by maximum likelihood. The figure illustrates the advantage of using a non-parametric estimator over the parametric Mallows model given enough training data. Also note when c increases, the non-parametric model approaches the empirical histogram thus performing worse for small datasets and better for large datasets. To visualize the advantage of the non-parametric model over the Mallows model we display in Figure 3 (bottom row) their estimated probabilities by scaling the vertices of the permutation polytope proportionally. The displayed polytope has vertices corresponding to rankings of 4 items and whose edges correspond to an adjacent transposition (Kendall’s tau distance is the shortest path between two vertices). In this case the four ranked items are movies no. 357, 1356, 440, 25 from the EachMovie dataset containing rankings of 1628 movies. Note how the probabilities assigned by the Mallows model (left) form a unimodal function centered at 2|1|3|4 while the non-parametric estimator (right) discovers the true modes 2|3|1|4 and 4|1|2|3 that were undetected by the Mallows model. 7 Figure 3 (top right) demonstrates modeling partial rankings of a much larger n. We used 10043 rankings from the Jester dataset which contains user rankings of n = 100 jokes. We kept the partial ranking type of the testing data fixed at (5, n −5) and experimented with different censoring of the training data. The figure illustrates the slower consistency rate for fully ranked training data and the statistical benefit in censoring full rankings in the training data. This striking statistical advantage demonstrates the achievement of property 5 in Section 1 and is independent of the computational advantage obtained from censoring the training data. 800 1600 2400 3200 4000 −4.9 −4.85 −4.8 −4.75 −4.7 −4.65 # of samples average log−likelihood mallows c=1 c=2 c=5 −18.5 −18 −17.5 1000 2000 3000 4000 5000 6000 7000 −80 −60 −40 −20 # of samples average log−likelihood (k(6),n−k(6)) (k(0),n−k(0)) (1,1,n−2) (k(8),n−k(8)) (1,n−1) (1,1,1,n−3) (1,1,1,1,1,n−5) fully ranked ¼ 1423 4132 1432 4321 4123 3421 3412 3241 3124 3142 2341 2431 2413 1324 4312 4231 4213 2134 2143 1234 2314 1243 3214 1342 1243 1423 1234 1432 4321 3421 3412 3241 3214 3124 3142 2431 2413 1324 1342 4312 4231 4213 4123 4132 2341 2314 2143 2134 Figure 3: Top row: Average test log-likelihood as a function of the training size: Mallows model vs. non-parametric model for APA election data (left) and non-parametric model with different partial ranking types for Jester data (right). Bottom row: Visualizing estimated probabilities for EachMovie data by permutation polytopes: Mallows model (left) and non-parametric model for c = 2 (right). 8 Discussion In this paper, we demonstrate for the first time a non-trivial effective modeling framework satisfying properties 1-5 in Section 1. The key component is our ability to efficiently compute (14) for simple partial ranking types and large n. Table 1 indicates the resulting complexity scales up with complexity of the composition k but is independent of n which is critical for modeling practical situations of k ≪n partial rankings. Experiments show the statistical advantage of the non-parametric partial ranking modeling in addition to its computational feasibility. References [1] D. E. Critchlow. Metric Methods for Analyzing Partially Ranked Data. Springer, 1986. [2] M. A. Fligner and J. S. Verducci. Distance based ranking models. Journal of the Royal Statistical Society B, 43:359–369, 1986. [3] M. G. Kendall. A new measure of rank correlation. Biometrika, 30, 1938. [4] G. Lebanon and J. Lafferty. Conditional models on the ranking poset. In Advances in Neural Information Processing Systems, 15, 2003. [5] C. L. Mallows. Non-null ranking models. Biometrika, 44:114–130, 1957. [6] J. I. Marden. Analyzing and modeling rank data. CRC Press, 1996. [7] R. P. Stanley. Enumerative Combinatorics, volume 1. Cambridge University Press, 2000. 8
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Multiple-Instance Pruning For Learning Efficient Cascade Detectors Cha Zhang and Paul Viola Microsoft Research One Microsoft Way, Redmond, WA 98052 {chazhang,viola}@microsoft.com Abstract Cascade detectors have been shown to operate extremely rapidly, with high accuracy, and have important applications such as face detection. Driven by this success, cascade learning has been an area of active research in recent years. Nevertheless, there are still challenging technical problems during the training process of cascade detectors. In particular, determining the optimal target detection rate for each stage of the cascade remains an unsolved issue. In this paper, we propose the multiple instance pruning (MIP) algorithm for soft cascades. This algorithm computes a set of thresholds which aggressively terminate computation with no reduction in detection rate or increase in false positive rate on the training dataset. The algorithm is based on two key insights: i) examples that are destined to be rejected by the complete classifier can be safely pruned early; ii) face detection is a multiple instance learning problem. The MIP process is fully automatic and requires no assumptions of probability distributions, statistical independence, or ad hoc intermediate rejection targets. Experimental results on the MIT+CMU dataset demonstrate significant performance advantages. 1 Introduction The state of the art in real-time face detection has progressed rapidly in recently years. One very successful approach was initiated by Viola and Jones [11]. While some components of their work are quite simple, such as the so called “integral image”, or the use of AdaBoost, a great deal of complexity lies in the training of the cascaded detector. There are many required parameters: the number and shapes of rectangle filters, the number of stages, the number of weak classifiers in each stage, and the target detection rate for each cascade stage. These parameters conspire to determine not only the ROC curve for the resulting system but also its computational complexity. Since the Viola-Jones training process requires CPU days to train and evaluate, it is difficult, if not impossible, to pick these parameters optimally. The conceptual and computational complexity of the training process has led to many papers proposing improvements and refinements [1, 2, 4, 5, 9, 14, 15]. Among them, three are closely related to this paper: Xiao, Zhu and Zhang[15], Sochman and Matas[9], and Bourdev and Brandt[1]. In each paper, the original cascade structure of distinct and separate stages is relaxed so that earlier computation of weak classifier scores can be combined with later weak classifiers. Bourdev and Brandt coined the term, “soft-cascade”, where the entire detector is trained as a single strong classifier without stages (with 100’s or 1000’s of weak classifiers sometimes called “features”). The score assigned to a detection window by the soft cascade is simply a weighted sum of the weak classifiers: sk(T) = P j≤T αjhj(xk), where T is the total number of weak classifiers; hj(xk) is the jth feature computed on example xk; αj is the vote on weak classifier j. Computation of the sum is terminated early whenever the partial sum falls below a rejection threshold: sk(t) < θ(t). Note the soft cascade 1 is similar to, but simpler than both the boosting chain approach of Xiao, Zhu, and Zhang and the WaldBoost approach of Sochman and Matas. The rejection thresholds θ(t), t ∈{1, · · · , T −1} are critical to the performance and speed of the complete classifier. However, it is difficult to set them optimally in practice. One possibility is to set the rejection thresholds so that no positive example is lost; this leads to very conservative thresholds and a very slow detector. Since the complete classifier will not achieve 100% detection (Note, given practical considerations, the final threshold of the complete classifier is set to reject some positive examples because they are difficult to detect. Reducing the final threshold further would admit too many false positives.), it seems justified to reject positive examples early in return for fast detection speed. The main question is which positive examples can be rejected and when. A key criticism of all previous cascade learning approaches is that none has a scheme to determine which examples are best to reject. Viola-Jones attempted to reject zero positive examples until this become impossible and then reluctantly gave up on one positive example at a time. Bourdev and Brandt proposed a method for setting rejection thresholds based on an ad hoc detection rate target called a “rejection distribution vector”, which is a parameterized exponential curve. Like the original Viola-Jones proposal, the soft-cascade gradually gives up on a number of positive examples in an effort to aggressively reduce the number of negatives passing through the cascade. Perhaps a particular family of curves is more palatable, but it is still arbitrary and non-optimal. SochmanMatas used a ratio test to determine the rejection thresholds. While this has statistical validity, distributions must be estimated, which introduces empirical risk. This is a particular problem for the first few rejection thresholds, and can lead to low detection rates on test data. This paper proposes a new mechanism for setting the rejection thresholds of any soft-cascade which is conceptually simple, has no tunable parameters beyond the final detection rate target, yet yields a cascade which is both highly accurate and very fast. Training data is used to set all reject thresholds after the final classifier is learned. There are no assumptions about probability distributions, statistical independence, or ad hoc intermediate targets for detection rate (or false positive rate). The approach is based on two key insights that constitute the major contributions of this paper: 1) positive examples that are rejected by the complete classifier can be safely rejected earlier during pruning; 2) each ground-truth face requires no more than one matched detection window to maintain the classifier’s detection rate. We propose a novel algorithm, multiple instance pruning (MIP), to set the reject thresholds automatically, which results in a very efficient cascade detector with superior performance. The rest of the paper is organized as follows. Section 2 describes an algorithm which makes use of the final classification results to perform pruning. Multiple instance pruning is presented in Section 3. Experimental results and conclusions are given in Section 4 and 5, respectively. 2 Pruning Using the Final Classification We propose a scheme which is simultaneously simpler and more effective than earlier techniques. Our key insight is quite simple: the reject thresholds are set so that they give up on precisely those positive examples which are rejected by the complete classifier. Note that the score of each example, sk(t) can be considered a trajectory through time. The full classifier rejects a positive example if its final score sk(T) falls below the final threshold θ(T). In the simplest version of our threshold setting algorithm, all trajectories from positive windows which fall below the final threshold are removed. Each rejection threshold is then simply: θ(t) = min © k ¯¯sk(T )>θ(T ),yk=1 ª sk(t) where {xk, yk} is the training set in which yk = 1 indicates positive windows and yk = −1 indicates negative windows. These thresholds produce a reasonably fast classifier which is guaranteed to produce no more errors than the complete classifier (on the training dataset). We call this pruning algorithm direct backward pruning (DBP). One might question whether the minimum of all retained trajectories is robust to mislabeled or noisy examples in the training set. Note that the final threshold of the complete classifier will often reject mislabeled or noisy examples (though they will be considered false negatives). These rejected 2 0 5 10 Positive Windows Negative Windows Cumulative Score Final Threshold 0 100 200 300 400 500 600 700 -20 -15 -10 -5 Feature Index Cumulative Score Final Threshold Positive windows but below threshold Positive windows above threshold Positive windows retained after pruning Figure 1: Traces of cumulative scores of different windows in an image of a face. See text. examples play no role in setting the rejection thresholds. We have found this procedure very robust to the types of noise present in real training sets. In past approaches, thresholds are set to reject the largest number of negative examples and only a small percentage of positive examples. These approaches justify these thresholds in different ways, but they all struggle to determine the correct percentage accurately and effectively. In the new approach, the final threshold of the complete soft-cascade is set to achieve the require detection rate. Rejection thresholds are then set to reject the largest number of negative examples and retain all positive examples which are retained by the complete classifier. The important difference is that the particular positive examples which are rejected are those which are destined to be rejected by the final classifier. This yields a fast classifier which labels all positive examples in exactly the same way as the complete classifier. In fact, it yields the fastest possible soft-cascade with such property (provided the weak classifiers are not re-ordered). Note, some negative examples that eventually pass the complete classifier threshold may be pruned by earlier rejection thresholds. This has the satisfactory side benefit of reducing false positive rate as well. In contrast, although the detection rate on the training set can also be guaranteed in Bourdev-Brandt’s algorithm, there is no guarantee that false positive rate will not increase. Bourdev-Brandt propose reordering the weak classifiers based on the separation between the mean score of the positive examples and the mean score of the negative examples. Our approach is equally applicable to a reordered soft-cascade. Figure 1 shows 293 trajectories from a single image whose final score is above -15. While the rejection thresholds are learned using a large set of training examples, this one image demonstrates the basic concepts. The red trajectories are negative windows. The single physical face is consistent with a set of positive detection windows that are within an acceptable range of positions and scales. Typically there are tens of acceptable windows for each face. The blue and magenta trajectories correspond to acceptable windows which fall above the final detection threshold. The cyan trajectories are potentially positive windows which fall below the final threshold. Since the cyan trajectories are rejected by the final classifier, rejection thresholds need only retain the blue and magenta trajectories. In a sense the complete classifier, along with a threshold which sets the operating point, provides labels on examples which are more valuable than the ground-truth labels. There will always be a set of “positive” examples which are extremely difficult to detect, or worse which are mistakenly labeled positive. In practice the final threshold of the complete classifier will be set so that these particular examples are rejected. In our new approach these particular examples can be rejected early in the computation of the cascade. Compared with existing approaches, that set the reject thresholds in a heuristic manner, our approach is data-driven and hence more principled. 3 Multiple Instance Pruning The notion of an “acceptable detection window” plays a critical role in an improved process for setting rejection thresholds. It is difficult to define the correct position and scale of a face in an image. 3 For a purely upright and frontal face, one might propose the smallest rectangle which includes the chin, forehead, and the inner edges of the ears. But, as we include a range of non-upright and non-frontal faces these rectangles can vary quite a bit. Should the correct window be a function of apparent head size? Or is eye position and interocular distance more reliable? Even given clear instructions, one finds that two subjects will differ significantly in their “ground-truth” labels. Recall that the detection process scans the image generating a large, but finite, collection of overlapping windows at various scales and locations. Even in the absence of ambiguity, some slop is required to ensure that at least one of the generated windows is considered a successful detection for each face. Experiments typically declare that any window which is within 50% in size and within a distance of 50% (of size) be considered a true positive. Using typical scanning parameters this can lead to tens of windows which are all equally valid positive detections. If any of these windows is classified positive then this face is consider detected. Even though all face detection algorithms must address the “multiple window” issue, few papers have discussed it. Two papers which have fundamentally integrated this observation into the training process are Nowlan and Platt [6] and more recently by Viola, Platt, and Zhang [12]. These papers proposed a multiple instance learning (MIL) framework where the positive examples are collected into “bags”. The learning algorithm is then given the freedom to select at least one, and perhaps more examples, in each bag as the true positive examples. In this paper, we do not directly address soft-cascade learning, though we will incorporate the “multiple window” observation into the determination of the rejection thresholds. One need only retain one “acceptable” window for each face which is detected by the final classifier. A more aggressive threshold is defined as: θ(t) = min i∈P max © k ¯¯k∈Fi∩Ri,yk=1 ª sk(t) where i is the index of ground-truth faces; Fi is the set of acceptable windows associated with ground-truth face i and Ri is the set of windows which are “retained” (see below). P is the set of ground-truth faces that have at least one acceptable window above the final threshold: P = © i ¯¯ max © k ¯¯k∈Fi ª sk(T) > θ(T) ª In this new procedure the acceptable windows come in bags, only one of which must be classified positive in order to ensure that each face is successfully detected. This new criteria for success is more flexible and therefore more aggressive. We call this pruning method multiple instance pruning (MIP). Returning to Figure 1 we can see that the blue, cyan, and magenta trajectories actually form a “bag”. Both in this algorithm, and in the simpler previous algorithm, the cyan trajectories are rejected before the computation of the thresholds. The benefit of this new algorithm is that the blue trajectories can be rejected as well. The definition of “retained” examples in the computation above is a bit more complex than before. Initially the trajectories from the positive bags which fall above the final threshold are retained. The set of retained examples is further reduced as the earlier thresholds are set. This is in contrast to the simpler DBP algorithm where the thresholds are set to preserve all retained positive examples. In the new algorithm the partial score of an example can fall below the current threshold (because it is in a bag with a better example). Each such example is removed from the retained set Ri and not used to set subsequent thresholds. The pseudo code of the MIP algorithm is shown in Figure 2. It guarantees the same face detection rate on the training dataset as the complete classifier. Note that the algorithm is greedy, setting earlier thresholds first so that all positive bags are retained and the fewest number of negative examples pass. Theoretically it is possible that delaying the rejection of a particular example may result in a better threshold at a later stage. Searching for the optimal MIP pruned detector, however, may be quite expensive. The MIP algorithm is however guaranteed to generate a soft-cascade that is at least as fast as DBP, since the criteria for setting the thresholds is less restrictive. 4 Input • A cascade detector. • Threshold θ(T) at the final stage of the detector. • A large training set (the whole training set to learn the cascade detector can be reused here). Initialize • Run the detector on all rectangles that match with any ground-truth faces. Collect all windows that are above the final threshold θ(T). Record all intermediate scores as s(i, j, t), where i = 1, · · · , N is the face index; j = 1, · · · , Mi is the index of windows that match with face i; t = 1, · · · , T is the index of the feature node. • Initialize flags f(i, j) as true. MIP For t = 1, · · · , T: 1. For i = 1, · · · , N: find ˆs(i, t) = max{j|f(i,j)=true} s(i, j, t). 2. Set θ(t) = mini ˆs(i, t) −ϵ as the rejection threshold of node t, ϵ = 10−6. 3. For i = 1, · · · , N, j = 1, · · · , Mi: set f(i, j) as false if s(i, j, t) < θ(t). Output Rejection thresholds θ(t), t = 1, · · · , T. Figure 2: The MIP algorithm. (a) (b) Figure 3: (a) Performance comparison with existing works on MIT+CMU frontal face dataset. (b) ROC curves of the detector after MIP pruning using the original training set. No performance degradation is found on the MIT+CMU testing dataset. 4 Experimental Results More than 20,000 images were collected from the web, containing roughly 10,000 faces. Over 2 billion negative examples are generated from the same image set. A soft cascade classifier is learned through a new framework based on weight trimming and bootstrapping (see Appendix). The training process was conducted on a dual core AMD Opteron 2.2 GHz processor with 16 GB of RAM. It takes less than 2 days to train a classifier with 700 weak classifiers based on the Haar features [11]. The testing set is the standard MIT+CMU frontal face database [10, 7], which consists of 125 grayscale images containing 483 labeled frontal faces. A detected rectangle is considered to be a true detection if it has less than 50% variation in shift and scale from the ground-truth. It is difficult to compare the performance of various detectors, since every detector is trained on a different dataset. Nevertheless, we show the ROC curves of a number of existing detectors and ours in Figure 3(a). Note there are two curves plotted for soft cascade. The first curve has very good performance, at the cost of slow speed (average 37.1 features per window). The classification accuracy dropped significantly in the second curve, which is faster (average 25 features per window). 5 Final Threshold -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 Detection Rate 95.2% 94.6% 93.2% 92.5% 91.7% 90.3% 88.8% # of False Positive 95 51 32 20 8 7 5 DBP 36.13 35.78 35.76 34.93 29.22 28.91 26.72 MIP 16.11 16.06 16.80 18.60 16.96 15.53 14.59 Approach Viola-Jones Boosting chain FloatBoost WaldBoost Wu et al. Soft cascade Total # of features 6061 700 2546 600 756 4943 Slowness 10 18.1 18.9 13.9 N/A 37.1 (25) (a) (b) Figure 4: (a) Pruning performance of DBP and MIP. The bottom two rows indicate the average number of features visited per window on the MIT+CMU dataset. (b) Results of existing work. Figure 4(a) compares DBP and MIP with different final thresholds of the strong classifier. The original data set for learning the soft cascade is reused for pruning the detector. Since MIP is a more aggressive pruning method, the average number of features evaluated is much lower than DBP. Note both DBP and MIP guarantee that no positive example from the training set is lost. There is no similar guarantee for test data, though. Figure 3(b) shows that there is no practical loss in classification accuracy on the MIT+CMU test dataset for various applications of the MIP algorithm (note that the MIT+CMU data is not used by the training process in any way). Speed comparison with other algorithms are subtle (Figure 4(b)). The first observation is that higher detection rates almost always require the evaluation of additional features. This is certainly true in our experiments, but it is also true in past papers (e.g., the two curves of Bourdev-Brandt soft cascade in Figure 3(a)). The fastest algorithms often cannot achieve very high detection rates. One explanation is that in order to achieve higher detection rates one must retain windows which are “ambiguous” and may contain faces. The proposed MIP-based detector yields a much lower false positive rate than the 25-feature Bourdev-Brandt soft cascade and nearly 35% improvement on detection speed. While the WaldBoost algorithm is quite fast, detection rates are measurably lower. Detectors such as Viola-Jones, boosting chain, FloatBoost, and Wu et al. all requires manual tuning. We can only guess how much trial and error went into getting a fast detector that yields good results. The expected computation time of the DBP soft-cascade varies monotonically in detection rate. This is guaranteed by the algorithm. In experiments with MIP we found a surprising quirk in the expected computation times. One would expect that if the required detection rate is higher, it world be more difficult to prune. In MIP, when the detection rate increases, there are two conflicting factors involved. First, the number of detected faces increases, which increases the difficulty of pruning. Second, for each face the number of retained and acceptable windows increases. Since we are computing the maximum of this larger set, MIP can in some cases be more aggressive. The second factor explains the increase of speed when the final threshold changes from -1.5 to -2.0. The direct performance comparison between MIP and Bourdev-Brandt (B-B) was performed using the same soft-cascade and the same data. In order to better measure performance differences we created a larger test set containing 3,859 images with 3,652 faces collected from the web. Both algorithms prune the strong classifier for a target detection rate of 97.2% on the training set, which corresponds to having a final threshold of −2.5 in Figure 4(a). We use the same exponential function family as [1] for B-B, and adjust the control parameter α in the range between −16 and 4. The results are shown in Figure 5. It can be seen that the MIP pruned detector has the best detection performance. When a positive α is used (e.g., α = 4), the B-B pruned detector is still worse than the MIP pruned detector, and its speed is 5 times slower (56.83 vs. 11.25). On the other hand, when α is negative, the speed of B-B pruned detectors improves and can be faster than MIP (e.g., when α = −16). Note, adjusting α leads to changes both in detection time and false positive rate. In practice, both MIP and B-B can be useful. MIP is fully automated and guarantees detection rate with no increase in false positive rate on the training set. The MIP pruned strong classifier is usually fast enough for most real-world applications. On the other hand, if speed is the dominant factor, one can specify a target detection rate and target execution time and use B-B to find a solution. 6 0.901 0.903 0.905 0.907 0.909 Detection Rate MIP, T=-2.5, #f=11.25 B-B, alpha=-16, #f=8.46 0.895 0.897 0.899 1000 1100 1200 1300 1400 1500 1600 Detection Rate Number of False Positive B-B, alpha=-16, #f=8.46 B-B, alpha=-8, #f=10.22 B-B, alpha=-4, #f=13.17 B-B, alpha=0, #f=22.75 B-B, alpha=4, #f=56.83 Figure 5: The detector performance comparison after applying MIP and Bourdev-Brandt’s method [1]. Note, this test was done using a much larger, and more difficult, test set than MIT+CMU. In the legend, symbol #f represents the average number of weak classifiers visited per window. Note such a solution is not guaranteed, and the false positive rate may be unacceptably high (The performance degradation of B-B heavily depends on the given soft-cascade. While with our detector the performance of B-B is acceptable even when α = −16, the performance of the detector in [1] drops significantly from 37 features to 25 features, as shown in Fig. 3 (a).). 5 Conclusions We have presented a simple yet effective way to set the rejection thresholds of a given soft-cascade, called multiple instance pruning (MIP). The algorithm begins with a conventional strong classifier and an associated final threshold. MIP then adds a set of rejection thresholds to construct a cascade detector. The rejection thresholds are determined so that every face which was detected by the original strong classifier is guaranteed to be detected by the soft cascade. The algorithm also guarantees that the false positive rate on the training set will not increase. There is only one parameter used throughout the cascade training process, the target detection rate for the final system. Moreover, there are no required assumptions about probability distributions, statistical independence, or ad hoc intermediate targets for detection rate or false positive rate. Appendix: Learn Soft Cascade with Weight Trimming and Bootstrapping We present an algorithm for learning a strong classifier from a very large set of training examples. In order to deal with the many millions of examples, the learning algorithm uses both weight trimming and bootstrapping. Weight trimming was proposed by Friedman, Hastie and Tibshirani [3]. At each round of boosting it ignores training examples with the smallest weights, up to a percentage of the total weight which can be between 1% and 10%. Since the weights are typically very skewed toward a small number of hard examples, this can eliminate a very large number of examples. It was shown that weight trimming can dramatically reduce computation for boosted methods without sacrificing accuracy. In weight trimming no example is discarded permanently, therefore it is ideal for learning a soft cascade. The algorithm is described in Figure 6. In step 4, a set A is predefined to reduce the number of weight updates on the whole training set. One can in theory update the scores of the whole training set after each feature is learned if computationally affordable, though the gain in detector performance may not be visible.Note, a set of thresholds are also returned by this process (making the result a softcascade). These preliminary rejection thresholds are extremely conservative, retaining all positive examples in the training set. They result in a very slow detector – the average number of features visited per window is on the order of hundreds. These thresholds will be replaced with the ones derived by the MIP algorithm. We set the preliminary thresholds only to moderately speed up the computation of ROC curves before MIP. 7 Input • Training examples (x1, y1), · · · , (xK, yK), where yk = −1, 1 for negative and positive examples. K is on the order of billions. • T is the total number of weak classifiers, which can be set through cross-validation. Initialize • Take all positive examples and randomly sample negative examples to form a subset of Q examples. Q = 4 × 106 in the current implementation. • Initialize weights ω1,i to guarantee weight balance between positive and negative examples on the sampled dataset. • Define A as the set {2, 4, 8, 16, 32, 64, 128, 192, 256, · · ·}. Adaboost Learning For t = 1, · · · , T: 1. For each rectangle filter in the pool, construct a weak classifier that minimizes the Z score [8] under the current set of weights ωt,i, i ∈Q. 2. Select the best classifier ht with the minimum Z score, find the associated confidences αt. 3. Update weights of all Q sampled examples. 4. If t ∈A, • Update weights of the whole training set using the previously selected classifiers h1, · · · , ht. • Perform weight trimming [3] to trim 10% of the negative weights. • Take all positive examples and randomly sample negative examples from the trimmed training set to form a new subset of Q examples. 5. Set preliminary rejection threshold θ(t) of Pt j=1 αjhj as the minimum score of all positive examples at stage t. Output Weak classifiers ht, t = 1, · · · , T, the associated confidences αt and preliminary rejection thresholds θ(t). Figure 6: Adaboost learning with weight trimming and booststrapping. References [1] L. Bourdev and J. Brandt. Robust object detection via soft cascade. In Proc. of CVPR, 2005. [2] S. C. Brubaker, M. D. Mullin, and J. M. Rehg. Towards optimal training of cascaded detectors. In Proc. of ECCV, 2006. [3] J. Friedman, T. Hastie, and R. Tibshirani. Additive logistic regression: a statistical view of boosting. Technical report, Dept. of Statistics, Stanford University, 1998. [4] S. Li, L. Zhu, Z. Zhang, A. Blake, H. Zhang, and H. Shum. Statistical learning of multi-view face detection. In Proc. of ECCV, 2002. [5] H. Luo. Optimization design of cascaded classifiers. In Proc. of CVPR, 2005. [6] S. J. Nowlan and J. C. Platt. A convolutional neural network hand tracker. In Proc. of NIPS, volume 7, 1995. [7] H. Rowley, S. Baluja, and T. Kanade. Neural network-based face detection. IEEE Trans. on PAMI, 20:23–38, 1998. [8] R. E. Schapire and Y. Singer. Improved boosting algorithms using confidence-rated predictions. Machine Learning, 37:297–336, 1999. [9] J. Sochman and J. Matas. Waldboost - learning for time constrained sequential detection. In Proc. of CVPR, 2005. [10] K. Sung and T. Poggio. Example-based learning for view-based face detection. IEEE Trans. on PAMI, 20:39–51, 1998. [11] P. Viola and M. Jones. Rapid object detection using a boosted cascade of simple features. In Proc. of CVPR, 2001. [12] P. Viola, J. C. Platt, and C. Zhang. Multiple instance boosting for object detection. In Proc. of NIPS, volume 18, 2006. [13] B. Wu, H. Ai, C. Huang, and S. Lao. Fast rotation invariant multi-view face detection based on real adaboost. In Proc. of IEEE Automatic Face and Gesture Recognition, 2004. [14] J. Wu, J. M. Rehg, and M. D. Mullin. Learning a rare event detection cascade by direct feature selection. In Proc. of NIPS, volume 16, 2004. [15] R. Xiao, L. Zhu, and H. Zhang. Boosting chain learning for object detection. In Proc. of ICCV, 2003. 8
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Modeling Natural Sounds with Modulation Cascade Processes Richard E. Turner and Maneesh Sahani Gatsby Computational Neuroscience Unit 17 Alexandra House, Queen Square, London, WC1N 3AR, London Abstract Natural sounds are structured on many time-scales. A typical segment of speech, for example, contains features that span four orders of magnitude: Sentences (∼1 s); phonemes (∼10−1 s); glottal pulses (∼10−2 s); and formants (≲10−3 s). The auditory system uses information from each of these time-scales to solve complicated tasks such as auditory scene analysis [1]. One route toward understanding how auditory processing accomplishes this analysis is to build neuroscienceinspired algorithms which solve similar tasks and to compare the properties of these algorithms with properties of auditory processing. There is however a discord: Current machine-audition algorithms largely concentrate on the shorter time-scale structures in sounds, and the longer structures are ignored. The reason for this is two-fold. Firstly, it is a difficult technical problem to construct an algorithm that utilises both sorts of information. Secondly, it is computationally demanding to simultaneously process data both at high resolution (to extract short temporal information) and for long duration (to extract long temporal information). The contribution of this work is to develop a new statistical model for natural sounds that captures structure across a wide range of time-scales, and to provide efficient learning and inference algorithms. We demonstrate the success of this approach on a missing data task. 1 Introduction Computational models for sensory processing are still in their infancy, but one promising approach has been to compare aspects of sensory processing with aspects of machine-learning algorithms crafted to solve the same putative task. A particularly fruitful approach in this vein uses the generative modeling framework to derive these learning algorithms. For example, Independent Component Analysis (ICA) and Sparse Coding (SC), Slow Feature Analysis (SFA), and Gaussian Scale Mixture Models (GSMMs) are examples of algorithms corresponding to generative models that show similarities with visual processing [3]. In contrast, there has been much less success in the auditory domain, and this is due in part to the paucity of flexible models with an explicit temporal dimension (although see [2]). The purpose of this paper is to address this imbalance. This paper has three parts. In the first we review models for the short-time structure of sound and argue that a probabilistic time-frequency model has several distinct benefits over traditional timefrequency representations for auditory modeling. In the second we review a model for the long-time structure in sounds, called probabilistic amplitude demodulation. In the third section these two models are combined with the notion of auditory features to produce a full generative model for sounds called the Modulation Cascade Process (MCP). We then show how to carry out learning and inference in such a complex hierarchical model, and provide results on speech for complete and missing data tasks. 1 2 Probabilistic Time-Frequency Representations Most representations of sound focus on the short temporal structures. Short segments (<10−1 s) are frequently periodic and can often be efficiently represented in a Fourier basis as the weighted sum of a few sinusoids. Of course, the spectral content of natural sounds changes slowly over time. This is handled by time-frequency representations, such as the Short-Time Fourier Transform (STFT) and spectrogram, which indicate the spectral content of a local, windowed section of the sound. More specifically, the STFT (xd,t) and spectrogram (sd,t) of a discretised sound (yt′) are given by, xd,t = T ′ X t′=1 rt−t′yt′ exp (−iωdt′) , sd,t = log |xd,t|. (1) The (possibly frequency dependent) duration of the window (rt−t′) must be chosen carefully, as it controls whether features are represented in the spectrum or in the time-variation of the spectra. For example, the window for speech is typically chosen to last for several pitch periods, so that both pitch and formant information is represented spectrally. The first stage of the auditory pathway derives a time-frequency-like representation mechanically at the basilar membrane. Subsequent stages extract progressively more complex auditory features, with structure extending over more time. Thus, computational models of auditory processing often begin with a time-frequency (or auditory-filter bank) decomposition, deriving new representations from the time-frequency coefficients [4]. Machine-learning algorithms also typically operate on the timefrequency coefficients, and not directly on the waveform. The potential advantage lies in the ease with which auditory features may be extracted from the STFT representation. There are, however, associated problems. For example, time-frequency representations tend to be over-complete (e.g. the number of STFT coefficients tends to be larger than the number of samples of the original sound T × D > T ′). This means that realisable sounds live on a manifold in the time-frequency space (for the STFT this manifold is a hyper-plane). Algorithms that solve tasks like filling-in missing data or denoising must ensure that the new coefficients lie on the manifold. Typically this is achieved in an ad hoc manner by projecting time-frequency coefficients back onto the manifold according to an arbitrary metric [5]. For generative models of time-frequency coefficients, it is difficult to force the model to generate only on the realisable manifold. An alternative is to base a probabilistic model of the waveform on the same heuristics that led to the original time-frequency representation. Not only does this side-step the generation problem, but it also allows parameters of the representation, like the “window”, to be chosen automatically. The heuristic behind the STFT – that sound comprises sinusoids in slowly-varying linear superposition – led Qi et al [6] to propose a probabilistic algorithm called Bayesian Spectrum Estimation (BSE), in which the sinusoid coefficients (xd,t) are latent variables. The forward model is, p(xd,t|xd,t−1) = Norm λdxd,t−1, σ2 d , (2) p(yt|xt) = Norm X d xd,t sin (ωdt + φd) , σ2 y ! . (3) The prior distribution over the coefficients is Gaussian and auto-regressive, evolving at a rate controlled by the dynamical parameters λd and σ2 d. Thus, as λd →1 and σ2 d →0 the processes become very slow, and as λd →0 and σ2 d →∞they become very fast. More precisely, the length-scale of the coefficients is given by λd = −log(λd). The observations are generated by a weighted sum of sinusoids, plus Gaussian noise. This model is essentially a Linear Gaussian State Space System with time varying weights defined by the sinusoids. Thus, inference is simple, proceeding via the Kalman Smoother recursions with time-varying weights. In effect, these recursions dynamically adjust the window used to derive the coefficients, based on the past history of the stimulus. BSE is a model for the short-time structure of sounds and it will essentially form the bottom level of the MCP. In the next section we turn our attention to a model of the longer-time structure. 3 Probabilistic Demodulation Cascade A salient property of the long-time statistics of sounds is the persistence of strong amplitude modulation [7]. Speech, for example, contains power in isolated regions corresponding to phonemes. 2 The phonemes themselves are localised into words, and then into sentences. Motivated by these observations, Anonymous Authors [8] have proposed a model for the long-time structures in sounds using a demodulation cascade. The basic idea of the demodulation cascade is to represent a sound as a product of processes drawn from a hierarchy, or cascade, of progressively longer time-scale modulators. For speech this might involve three processes: representing sentences on top, phonemes in the middle, and pitch and formants at the bottom (e.g. fig. 1A and B). To construct such a representation, one might start with a traditional amplitude demodulation algorithm, which decomposes a signal into a quickly-varying carrier and more slowly-varying envelope. The cascade could then be built by applying the same algorithm to the (possibly transformed) envelope, and then to the envelope that results from this, and so on. This procedure is only stable, however, if both the carrier and the envelope found by the demodulation algorithm are well-behaved. Unfortunately, traditional methods (like the Hilbert Transform, or low-pass filtering a non-linear transformation of the stimulus) return a suitable carrier or envelope, but not both. A new approach to amplitude demodulation is thus called for. In a nutshell, the new approach is to view amplitude demodulation as a task of probabilistic inference. This is natural, as demodulation is fundamentally ill-posed — there are infinitely many decompositions of a signal into a positive envelope and real valued carrier — and so prior information must always be leveraged to realise such a decomposition. The generative model approach makes this information explicit. Furthermore, it not necessary to use the recursive procedure (just described) to derive a modulation cascade: the whole hierarchy can be estimated at once using a single generative model. The generative model for Probabilistic Amplitude Demodulation (PAD) is p z(m) 0 = Norm (0, 1) , p z(m) t |z(m) t−1 = Norm λmz(m) t−1, σ2 m ∀t > 0, (4) x(m) t = fa(m) z(m) t ∀m > 1, x(1) t = Norm (0, 1) , yt = M Y m=1 x(m) t . (5) A set of modulators (X2:M) are drawn in a two stage process: First a set of slowly varying processes (Z2:M) are drawn from a one-step linear Gaussian prior (identical to Eq. 2). The effective lengthscales of these processes, inherited by the modulators, are ordered such that λm > λm−1. Second the modulators are formed by passing these variables through a point-wise non-linearity to enforce positivity. A typical choice might be fa(m) z(m) t = log exp z(m) t + a(m) + 1 , (6) which is logarithmic for large negative values of z(m) t , and linear for large positive values. This transforms the Gaussian distribution over z(m) t into a sparse, non-negative, distribution, which is a good match to the marginal distributions of natural envelopes. The parameter a(m) controls exactly where the transition from log to linear occurs, and consequently alters the degree of sparsity. These positive signals modulate a Gaussian white-noise carrier, to yield observations y1:T by a simple point-wise product. A typical draw from this generative model can be seen in Fig. 1C. This model is a fairly crude one for natural sounds. For example, as described in the previous section, we expect that the carrier process will be structured and yet it is modelled as Gaussian white noise. The surprising observation is that this very simple model is excellent at demodulation. Inference in this model typically proceeds by a zero-temperature EM-like procedure. Firstly the carrier (x(1) t ) is integrated out and then the modulators are found by maximum a posteriori (MAP). Slower, more Bayesian algorithms that integrate over the modulators using MCMC indicate that this approximation is not too severe, and the results are compelling. 4 Modulation Cascade Processes We have reviewed two contrasting models: The first captures the local harmonic structure of sounds, but has no long-time structure; The second captures long-time amplitude modulations, but models the short-time structure as white noise. The goal of this section is to synthesise both to form a new model. We are guided by the observation that the auditory system might implement a similar synthesis. In the well-known psychophysical phenomenon of comodulation masking release (see [9] for a review), a tone masked by noise with a bandwidth greater than an auditory filter becomes audible 3 x(1) 1:T x(2) 1:T x(3) 1:T 2 4 6 y1:T time /s 0.5 1 1.5 time /s A. B. C. Figure 1: An example of a modulation-cascade representation of speech (A and B) and typical samples from generative models used to derive that representation (C). A) The spoken-speech waveform (black) is represented as the product of a carrier (blue), a phoneme modulator (red) and a sentence modulator (magenta). B) A close up of the first sentence (2 s) additionally showing the derived envelope (x(2) t x(3) t ) superposed onto the speech (red, bottom panel). C) The generative model (M = 3) with a carrier (blue), a phoneme modulator (red) and a sentence modulator (magenta). if the noise masker is amplitude modulated. This suggests that long-time envelope information is processed and analysed across (short-time) frequency channels in the auditory system. A simple way to combine the two models would be to express each filter coefficient of the timefrequency model as a product of processes (e.g. xd,t = x(1) d,tx(2) d,t). However, power across even widely seperated channels of natural sounds can be strongly correlated [7]. Furthermore, comodulation masking release suggests that amplitude-modulation is processed across frequency channels and not independently in each channel. Presumably this reflects the collective modulation of wideband (or harmonic) sounds, with features that span many frequencies. Thus, a synthesis of BSE and PAD should incorporate the notion of auditory features. The forward model. The Modulation Cascade Process (MCP) is given by p z(m) km,t|z(m) km,t−1, θ = Norm λ(m)z(m) km,t−1, σ2 (m) m = 1 : 3, t > 0, (7) p z(m) km,0 = Norm (0, 1) , x(m) km,t = f(z(m) km,t, a(m)) m = 1 : 3, t ≥0, (8) p yt|x(m) t , θ = Norm µyt, σ2 y , µyt = X d,k1,k2 gd,k1,k2x(1) k1,tx(2) k2,tx(3) t sin (ωdt + φd) . (9) Once again, latent variables are arranged in a hierarchy according to their time-scales (which depend on m). At the top of the hierarchy is a long-time process which models slow structures, like the sentences of speech. The next level models more quickly varying structure (like phonemes). Finally, the bottom level of the hierarchy captures short-time variability (intra-phoneme variability for instance). Unlike in PAD, the middle and lower levels now contain multiple process. So, for example if K1 = 4 and K2 = 2, there would be four quickly varying modulators in the lower level, two modulators in the middle level, and one slowly varying modulator at the top (see fig. 2A). The idea is that the modulators in the first level independently control the presence or absence of individual spectral features (given by P d gd,k1,k2 sin (ωdt + φd)). For example, in speech a typical phoneme might be periodic, but this periodicity might change systematically as the speaker alters their pitch. This change in pitch might be modeled using two spectral features: one for the start of the phoneme and one for the end, with a region of coactivation in the middle. Indeed it is because speech 4 x(3) t x(2) t x(1) t 0 0.5 1 1.5 2 yt x(3) t x(2) t x(1) t 0.84 0.86 0.88 0.9 0.92 0.94 yt time /s × × x(3) t x(2) t x(1) t × × × × × × × × + + + × = yt A. B. Figure 2: A. Schematic representation of the MCP forward model in the simple case when K1 = 4, K2 = 2 and D = 6. The hierarchy of latent variables moves from the slowest modulator at the top (magenta) to the fastest (blue) with an intermediate modulator between (red). The outer-product of the modulators multiplies the generative weights (black and white, only 4 of the 8 shown). In turn, these modulate sinusoids (top right) which are summed to produce the observations (bottom right). B. A draw from the forward model using parameters learned from a spoken-sentence (see the results section for more details of the model). The grey bars on the top four panels indicate the region depicted in the bottom four panels. and other natural sounds are not precisely stationary even over short time-scales that we require the lowest layer of the hierarchy. The role of the modulators in the second level is to simultaneously turn on groups of similar features. For example, one modulator might control the presence of all the harmonic features and the other the broad-band features. Finally the top level modulator gates all the auditory features at once. Fig. 2B shows a draw from the forward model for a more complicated example. Promisingly the samples share many features of natural sounds. Relationship to other models. This model has an interesting relationship to previous statistical models and in particular to the GSMMs. It is well known that when ICA is applied to data from natural scenes the inferred filter coefficients tend not to be independent (see [3, 10]), with coefficients corresponding to similar filters sharing power. GSMMs model dependencies using a hierarchical framework, in which the distribution over the coefficients depends on a set of latent variables that introduce correlations between their powers. The MCP is similar, in that the higher level latent variables alter the power of similar auditory features. Indeed, we suggest that the correlations in the power of ICA coefficients are a sign that AM is prevalent in natural scenes. The MCP can be seen as a generalisation of the GSMMs to include time-varying latent variables, a deeper hierarchy and a probabilistic time-frequency representation. Inference and learning algorithms. Any type of learning in the MCP is computationally demanding. Motivated by speed, and encouraged by the results from PAD, the aim will therefore be to find a joint MAP estimate of the latent variables and the weights, that is X, G = arg max X,G log p(X, Y, G|θ). (10) 5 Note that we have introduced a prior over the generative tensor. This prevents an undesirable feature of combined MAP and ML inference in such models: namely that the weights grow without bound, enabling the modal values of latent variables to shrink towards zero, increasing their density under the prior. The resulting cost function is, log p(X, Y, G|θ) = T X t=1 log p(yt|x(1) t , x(2) t ) + 3 X m=1 X km T X t=1 log p(z(m) km,t|z(m) km,t−1) + T X t=0 log dz(m) km,t dx(m) km,t + log p(z(m) km,0) + X k1,k2,d log p(gd,k1,k2) (11) We would like to optimise this objective-function with respect to the latent variables (x(m) km,t) and the generative tensor (gd,k1,k2). There are, however, two main obstacles. The first is that there are a large number of latent variables to estimate (T × (K1 + K2)), making inference slow. The second is that the generative tensor contains a large number of elements D × K1 × K2, making learning slow too. The solution is to find a good initialisation procedure, and then to fine-tune using a slow EM-like algorithm that iterates between updating the latents and the weights. First we outline the initialisation procedure. The key to learning complicated hierarchical models is to initialise well, and so the procedure developed for the MCP will be explained in some detail. The main idea is to learn the model one layer at a time. This is achieved by clamping the upper layers of the hierarchy that are not being learned to unity. In the first stage of the initialisation, for example, the top and middle levels of the hierarchy are clamped and the mean of the emission distribution becomes µyt = X d,k1 γd,k1x(1) k1,t sin (ωdt + φd) , (12) where γd,k1 = P k2 gd,k1,k2. Learning and inference then proceed by gradient based optimisation of the cost-function (log p(X, Y, G|θ)) with respect to the un-clamped latents (x(1) k1,t) and the contracted generative weights (γd,k1). This is much faster than the full optimisation as there are both fewer latents and fewer parameters to estimate. When this process is complete, the second layer of latent variables is un-clamped, and learning of these variables commences. This requires the full generative tensor, which must be initialised from the contracted generative weights learned at the previous stage. One choice is to set the individual weights to their averages gd,k1,k2 = 1 K2 γd,k1 and this works well, but empirically slows learning. An alternative is to use small chunks of sounds to learn the lower level weights. These chunks are chosen to be relatively stationary segments that have a time-scale similar to the second-level modulators. This allows us to make the simplifying assumption that just one second-level modulator was active during the chunk. The generative tensor can be therefore be initialised using gd,k1,k2 = γd,k1δk2,j. Typically this method causes the second stage of learning to converge faster, and to a similar solution. In contrast to the initialisation, the fine tuning algorithm is simple. In the E-Step the latent variables are updated simultaneously using gradient based optimisation of Eq. 11. In the M-Step, the generative tensor is updated using co-ordinate ascent. That is to say that we sequentially update each gk1,k2 using gradient based optimisation of Eq. 11 and iterate over k1 and k2. In principle, joint optimisation of the generative tensor and latent variables is possible, but the memory requirements are prohibitive. This is also why co-ordinate ascent is used to learn the generative tensor (rather than using the usual linear regression solution which involves a prohibitive matrix inverse). 5 Results The MCP was trained on a spoken sentence, lasting 2s and sampled at 8000Hz, using the algorithm outlined in the previous section. The time-scales of the modulators were chosen to be {20 ms, 200 ms, 2 s}. The time-frequency representation had D/2 = 100 sines and D/2 = 100 cosines spaced logarithmically from 100 −4000Hz. The model was given K1 = 18 latent variables in the first level of the hierarchy, and K2 = 6 in the second. Learning took 16hrs to run on a 2.2 GHz Opteron with 2Gb of memory. 6 x(3) t x(2) t,k 2 x(1) t,k 1 0 0.5 1 1.5 time /s y 0 1000 2000 3000 4000 frequency /HZ [t] Figure 3: Application of the MCP to speech. Left panels: The inferred latent variable hierarchy. At top is the sentence modulator (magenta). Next are the phoneme modulators, followed by the intra-phoneme modulators. These are coloured according to which of the phoneme modulators they interact most strongly with. The speech waveform is shown in the bottom panel. Right panels: The learned spectral features ( p g2 sin + g2cos) coloured according to phoneme modulator. For example, the top panel show the spectra from gk1=1:18,k2=1. Spectra corresponding to one phoneme modulator look similar and offer the features only differ in their phase. The results can be seen in Fig. 3. The MCP recovers a sentence modulator, phoneme modulators, and intra-phoneme modulators. Typically a pair of features are used to model a phoneme, and often they have similar spectra as expected. The spectra fall into distinct classes: those which are harmonic (modelling voiced features) and those which are broad-band (modelling unvoiced features). One way of assessing which features of speech the model captures is to sample from the forward model using the learned parameters. This can be seen in Fig. 2B. The conclusion is that the model is capturing structure across a wide range of time-scales: formants and pitch structure, phoneme structure, and sentence structure. There are, however, two noticeable differences between the real and generated data. The first is that the generated data contain fewer transients and noise segments than natural speech, and more vowellike components. The reason for this is that at the sampling rates used, many of the noisy segments are indistinguishable from white-noise and are explained using observation noise. These problems are alleviated by moving to higher sampling rates, but the algorithm is then markedly slower. The second difference concerns the inferred and generated latent variables in that the former are much sparser than the latter. The reason is that learned generative tensor contains many gk1,k2 which are nearly zero. In generation, this means that significant contributions to the output are only made when particular pairs of phoneme and intra-phoneme modulators are active. So although many modulators are active at one time, only one or two make sizeable contributions. Conversely, in inference, we can only get information about the value of a modulator when it is part of a contributing pair. If this is not the case, the inference goes to the maximum of the prior which is zero. In effect there are large error-bars on the non-contributing components’ estimates. Finally, to indicate the improvement of the MCP over PAD and BSE, we compare the algorithms abilities to fill in missing sections of a spoken sentence. The average root-mean-squared (RMS) error per sample is used as a metric to compare the algorithms. In order to use the MCP to fill in the missing data, it is first necessary to learn a set of auditory features. The MCP was therefore trained on a different spoken sentence from the same speaker, before inference was carried out on the test data. To make the comparison fair, BSE is given an identical set of sinusoidal basis functions as MCP, and the associated smoothness priors were learned on the same training data. Typical results can be seen in fig. 4. On average the RMS errors for MCP, BSE and PAD were: {0.10, 0.30, 0.41}. As PAD models the carrier as white noise it predicts zeros in the missing regions 7 original MCP 0.08 0.1 0.12 0.14 0.16 0.18 time /s BSE 0.08 0.1 0.12 0.14 0.16 time /s 0.15 0.2 0.25 0.3 time /s Figure 4: A selection of typical missing data results for three phonemes (columns). The top row shows the original speech segement with the missing regions shown in red. The middle row shows the predictions made by the MCP and the bottom row those made by BSE. and therefore it merely serves as a baseline in these experiments. Both MCP and BSE smoothly interpolate their latent variables over the missing region. However, whereas BSE smoothly interpolates each sinusoidal component independently, MCP interpolates the set of learned auditory features in a complex manner determined by the interaction of the modulators. It is for this reason that it improves over BSE by such a large margin. 6 Conclusion We have introduced a neuroscience-inspired generative model for natural sounds that is capable of capturing structure spanning a wide range of temporal scales. The model is a marriage between a probabilistic time-frequency representation (that captures the short-time structure) and a probabilistic demodulation cascade (that captures the long-time structure). When the model is trained on a spoken sentence, the first level of the hierarchy learns auditory features (weighted sets of sinusoids) that capture structures like different voiced sections of speech. The upper levels comprise a temporally ordered set of modulators are used to represent sentence structure, phoneme structure and intra-phoneme variability. The superiority of the new model over its parents was demonstrated in a missing data experiment where it out-performed the Bayesian time-frequency analysis by a large margin. References [1] Bregman, A.S. (1990) Auditory Scene Analysis. MIT Press. [2] Smith E. & Lewicki, M.S. (2006) Efficient Auditory Coding. Nature 439 (7079). [3] Simoncelli, E.P. (2003) Vision and the statistics of the visual environment. Curr Opin Neurobi 13(2):144-9. [4] Patterson, R.D. (2000) Auditory images: How complex sounds are represented in the auditory system. J Acoust Soc Japan (E) 21(4):183-190. [5] Griffin, D. & Lim J. (1984) Signal estimation from modified short-time Fourier transform. IEEE Trans. on ASSP 32(2):236-243. [6] Qi, Y., Minka, T. & Picard, R.W. (2002) Bayesian Spectrum Estimation of Unevenly Sampled Nonstationary Data. MIT Media Lab Technical Report Vismod-TR-556. [7] Attias, H. & Schreiner, C.E. (1997) Low-Order Temporal Statistics of Natural Sounds. Adv in Neural Info Processing Sys 9. MIT Press. [8] Anonymous Authors (2007) Probabilistic Amplitude Demodulation. ICA 2007 Conference Proceedings. Springer, in press. [9] Moore, B.C.J. (2003) An Introduction to the Psychology of Hearing. Academic Press. [10] Karklin, Y. & Lewicki, M.S. (2005) A hierarchical Bayesian model for learning nonlinear statistical regularities in nonstationary natural signals. Neural Comput 17(2):397-423. 8
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Fast and Scalable Training of Semi-Supervised CRFs with Application to Activity Recognition Maryam Mahdaviani Computer Science Department University of British Columbia Vancouver, BC, Canada Tanzeem Choudhury Intel Research 1100 NE 45th Street Seattle, WA 98105,USA Abstract We present a new and efficient semi-supervised training method for parameter estimation and feature selection in conditional random fields (CRFs). In real-world applications such as activity recognition, unlabeled sensor traces are relatively easy to obtain whereas labeled examples are expensive and tedious to collect. Furthermore, the ability to automatically select a small subset of discriminatory features from a large pool can be advantageous in terms of computational speed as well as accuracy. In this paper, we introduce the semi-supervised virtual evidence boosting (sVEB) algorithm for training CRFs – a semi-supervised extension to the recently developed virtual evidence boosting (VEB) method for feature selection and parameter learning. The objective function of sVEB combines the unlabeled conditional entropy with labeled conditional pseudo-likelihood. It reduces the overall system cost as well as the human labeling cost required during training, which are both important considerations in building real-world inference systems. Experiments on synthetic data and real activity traces collected from wearable sensors, illustrate that sVEB benefits from both the use of unlabeled data and automatic feature selection, and outperforms other semi-supervised approaches. 1 Introduction Conditional random fields (CRFs) are undirected graphical models that have been successfully applied to the classification of relational and temporal data [1]. Training complex CRF models with large numbers of input features is slow, and exact inference is often intractable. The ability to select the most informative features as needed can reduce the training time and the risk of over-fitting of parameters. Furthermore, in complex modeling tasks, obtaining the large amount of labeled data necessary for training can be impractical. On the other hand, large unlabeled datasets are often easy to obtain, making semi-supervised learning methods appealing in various real-world applications. The goal of our work is to build an activity recognition system that is not only accurate but also scalable, efficient, and easy to train and deploy. An important application domain for activity recognition technologies is in health-care, especially in supporting elder care, managing cognitive disabilities, and monitoring long-term health. Activity recognition systems will also be useful in smart environments, surveillance, emergency and military missions. Some of the key challenges faced by current activity inference systems are the amount of human effort spent in labeling and feature engineering and the computational complexity and cost associated with training. Data labeling also has privacy implications because it often requires human observers or recording of video. In this paper, we introduce a fast and scalable semi-supervised training algorithm for CRFs and evaluate its classification performance on extensive real world activity traces gathered using wearable sensors. In addition to being computationally efficient, our proposed method reduces the amount of labeling required during training, which makes it appealing for use in real world applications. 1 Several supervised techniques have been proposed for feature selection in CRFs. For discrete features, McCallum [2] suggested an efficient method for feature induction by iteratively increasing conditional log-likelihood. Dietterich [3] applied gradient tree boosting to select features in CRFs by combining boosting with parameter estimation for 1D linear-chain models. Boosted random fields (BRFs) [4] combine boosting and belief propagation for feature selection and parameter estimation for densely connected graphs that have weak pairwise connections. Recently, Liao et.al. [5] developed a more general version of BRFs, called virtual evidence boosting (VEB) that does not make any assumptions about graph connectivity or the strength of pairwise connections. The objective function in VEB is a soft version of maximum pseudo-likelihood (MPL), where the goal is to maximize the sum of local log-likelihoods given soft evidence from its neighbors. This objective function is similar to that used in boosting, which makes it suitable for unified feature selection and parameter estimation. This approximation applies to any CRF structures and leads to a significant reduction in training complexity and time. Semi-supervised training techniques have been extensively explored in the case of generative models and naturally fit under the expectation maximization framework [6]. However, it is not straight forward to incorporate unlabeled data in discriminative models using the traditional conditional likelihood criteria. A few semi-supervised training methods for CRFs have been proposed that introduce dependencies between nearby data points [7, 8]. More recently, Grandvalet and Bengio [9] proposed a minimum entropy regularization framework for incorporating unlabeled data. Jiao et.al. [10] used this framework and proposed an objective function that combines the conditional likelihood of the labeled data with the conditional entropy of the unlabeled data to train 1D CRFs, which was extended to 2D lattice structures by Lee et.al. [11]. In our work, we combine the minimum entropy regularization framework for incorporating unlabeled data with VEB for training CRFs. The contributions of our work are: (i) semi-supervised virtual evidence boosting (sVEB) - an efficient technique for simultaneous feature selection and semi-supervised training of CRFs, which to the best of our knowledge is the first method of its kind, (ii) experimental results that demonstrate the strength of sVEB, which consistently outperforms other training techniques on synthetic data and real-world activity classification tasks, and (iii) analysis of the time and complexity requirements of our algorithm, and comparison with other existing techniques that highlight the significant computational advantages of our approach. The sVEB algorithm is fast and easy to implement and has the potential of being broadly applicable. 2 Approaches to training of Conditional Random Fields Maximum likelihood parameter estimation in CRFs involves maximizing the overall conditional log-likelihood, where x is the observation sequence and y is the hidden state sequence: L(θ) = log(p(y|x, θ)) −∥θ∥/2 = log exp( K P k=1 θkfk(x, y)) P y′ exp( K P k=1 θkfk(x, y′)) −∥θ∥/2 (1) The conditional distribution is defined by a log-linear combination of k features functions fk associated with weight θk. A regularizer on θ is used to keep the weights from getting too large and to avoid overfitting1. For large CRFs exact inference is often intractable and approximate methods such as mean field approximation or loopy belief propagation [12, 13] are used. An alternative to approximating the conditional likelihood is to change the objective function. MPL [14] and VEB [5] are such techniques. For MPL the CRF is cut into a set of independent patches; each patch consists of a hidden node or class label yi, the true value of its direct neighbors and the observations, i.e., the Markov Blanket(MByi) of the node. The parameter estimation then becomes maximizing the pseudo log-likelihood: Lpseudo(θ) = N P i=1 log(p(yi|MByi, θ)) = N P i=1 log exp( K P k=1 θkfk(MByi,yi)) P y′ i exp( K P k=1 θkfk(MBy′ i,y′ i)) MPL has been known to over-estimate the dependency parameters in some cases and there is no general guideline on when it can be safely used [15]. 1When a prior is used in the maximum likelihood objective function as a regularizer – the second term in eq. (1), the method is in fact called maximum a posteriori. 2 2.1 Virtual evidence boosting By extending the standard LogitBoost algorithm [16], VEB integrates boosting based feature selection into CRF training. The objective function used in VEB is very similar to MPL, except that VEB uses the messages from the neighboring nodes as virtual evidence instead of using the true labels of neighbors. The use of virtual evidence helps to reduce over-estimation of neighborhood dependencies. We briefly explain the approach here but please refer to [5] for more detail. VEB incorporates two types of observations nodes: (i) hard evidence corresponding to the observations ve(xi), which are indicator functions at the observation values and (ii) soft evidence, corresponding to the messages from neighboring nodes ve(n(yi)), which are discrete distributions over the hidden states. Let vei ≜{ve(xi), ve(n(yi))}. The objective function of VEB is as follows: LV EB(θ) = N X i=1 log(p(yi|vei, θ)), where p(yi|vei, θ) = P vei vei exp( K P k=1 θkfk(vei, yi)) P y′ i P vei vei exp( K P k=1 θkfk(vei, y′ i)) (2) VEB learns a set weak learners fts iteratively and estimates the combined feature Ft = Ft−1 + ft by solving the following weighted least square error(WLSE) problem: ft(vei) = arg min f N X i=1 wiE(f(vei) −zi)2 = arg min f [ N X i=1 X vei wip(yi|vei)(f(vei) −zi)2] (3) where wi = p(yi|vei)(1 −p(yi|vei)), zi = yi −0.5 p(yi|vei) (4) The wi and zi in equation 4 are the boosting weight and working response respectively for the ith data point, exactly as in LogitBoost. However, the least square problem for VEB (eq.3) involves NX points because of virtual evidence as opposed to N points in LogitBoost. Although eq. 4 is given for the binary case (i.e. yi ∈{0, 1}), it is easily extendible to the multi-class case and we have done that in our experiments. At each iteration, vei is updated as messages from n(yi) changes with the addition of new features. We run belief propagation (BP) to obtain the virtual evidence before each iteration. The CRF feature weights, θ’s are computed by solving the WLSE problem, where the local features, nki is the count of feature k in data instance i and the compatibility features, nki is the virtual evidence from the neighbors.: θk = N P i=1 wizinki/ N P i=1 winki. 2.2 Semi-supervised training For semi-supervised training of CRFs, Jiao et.al. [10] have proposed an algorithm that utilizes unlabeled data via entropy regularization – an extension of the approach proposed by [9] to structured CRF models. The objective function that is maximized during semi-supervised training of CRFs is given below, where (xl, yl) and (xu, yu) represent the labeled and unlabeled data respectively: LSS(θ) = log p(yl|xl, θ) + α P yu p(yu|xu, θ)log p(yu|xu, θ) −∥θ∥/2 By minimizing the conditional entropy of the unlabeled data, the algorithm will generally find labeling of the unlabeled data that mutually reinforces the supervised labels. One drawback of this objective function is that it is no longer concave and in general there will be local maxima. The authors [10] showed that this method is still effective in improving an initial supervised model. 3 Semi-supervised virtual evidence boosting In this work, we develop semi-supervised virtual evidence boosting (sVEB) that combines feature selection with semi-supervised training of CRFs. sVEB extends the VEB framework to take advantage of unlabeled data via minimum entropy regularization similar to [9, 10, 11]. The new objective function LsV EB we propose is as follows, where (i = 1 · · · N) are labeled and (i = N + 1 · · · M) are unlabled examples: LsV EB = N X i=1 log p(yi|vei) + α M X i=N+1 X y′ i p(y′ i|vei) log p(y′ i|vei) (5) 3 The sVEB aglorithm, similar to VEB, maximizes the conditional soft pseudo-likelihood of the labeled data but in addition minimizes the conditional entropy over unlabeled data. The α is a tuning parameter for controlling how much influence the unlabeled data will have. By considering the soft pseudo-likelihood in LsV EB and using BP to estimate p(yi|vei), sVEB can use boosting to learn the parameters of CRFs. The virtual evidence from the neighboring nodes captures the label dependencies. There are three different types of feature functions fs that’s used: for continuous observations f1(xi) is a linear combination of decision stumps, for discrete observations the learner f2(xi) is expressed as indicator functions, and for virtual evidences the weak learner f3(xi) is the weighted sum of two indicator functions (for binary case). These functions are computed as follows, where δ is an indicator function, h is a threshold for the decision stump, and D is the number of dimensions of the observations: f1(xi) = θ1δ(xi ≥h) + θ2δ(xi < h), f2(xi) = D X k=1 θkδ(xi = d), f3(yi) = 1 X k=0 θkδ(yi = k) (6) Similar to LogitBoost and VEB, the sVEB algorithm estimates a combined feature function F that maximizes the objective by sequentially learning a set of weak learners, ft’s (i.e. iteratively selecting features). In other words, sVEB solves the following weighted least-square error (WLSE) problem to learn fts: ft = arg min f [ N X i=1 X vei wip(yi|vei)(f(xi) −zi)2 + M X i=N+1 X y′ i X vei wip(y′ i|vei)(f(xi) −zi)2] (7) For labeled data (first term in eq.7), boosting weights, wi’s, and working responses, zi’s, are computed as described in equation 4. But for the case of unlabeled data the expression for wi and zi becomes more complicated because of the entropy term. We present the equations for wi and zi below, please refer to the Appendix for the derivations: wi = α2(1 −p(yi|vei))[p(yi|vei)(1 −p(yi|vei)) + log p(yi|vei)] zi = (yi −0.5)p(yi|vei)(1 −log p(yi|vei)) α[p(yi|vei)(1 −p(yi|vei)) + log p(yi|vei)] (8) The soft evidence corresponding to messages from the neighboring nodes is obtained by running BP on the entire training dataset (labeled and unlabeled). The CRF feature weights θks are computed by solving the WLSE problem (e.q.(7)), θk = M P i=1 P yi wizinki/ M P i=1 P yi winki Algorithm 1 gives the pseudo-code for sVEB. The main difference between VEB and sVEB are steps 7 −10, where we compute wi’s and zi’s for all possible values of yi based on the virtual evidence and observations of unlabeled training cases. The boosting weights and working responses are computed using equation (8). The weighted least-square error (WLSE) equation (eq. 7) in step 10 of sVEB is different from that of VEB and the solution results in slightly different CRF feature weights, θ’s. One of the major advantages of VEB and sVEB over ML and sML is that the parameter estimation is done by mainly performing feature counting. Unlike ML and sML, we do not need to use an optimizer to learn the model parameters which results in a huge reduction in the time required to train the CRF models. Please refer to the complexity analysis section for details. 4 Experiments We conduct two sets of experiments to evaluate the performance of the sVEB method for training CRFs and the advantage of performing feature selection as part of semi-supervised training. In the first set of experiments, we analyze how much the complexity of the underlying CRF and the tuning parameter α effect the performance using synthetic data. In the second set of experiments, we evaluate the benefit of feature selection and using unlabeled data on two real-world activity datasets. We compare the performance of the semi-supervised virtual evidence boosting(sVEB) presented in this paper to the semi-supervised maximum likelihood (sML) method [10]. In addition, for the activity datasets, we also evaluate an alternative approach (sML+Boost), where a subset of features is selected in advance using boosting. To benchmark the performance of the semi-supervised techniques, we also evaluate three different supervised training approaches, namely maximum likelihood 4 Algorithm 1: Training CRFs using semi-supervised VEB inputs : structure of CRF and training data (xi, yi), with yi ∈{0, 1}, 1 ≤i ≤M, and F0 = 0 output: Learned FT and their corresponding weights, θ for t = 1, 2, · · · , T do 1 Run BP using Ft to get virtual evidences vei; 2 for i = 1, 2, · · · , N do 3 Compute likelihood p(yi|vei); 4 Compute wi and zi using equation (4) 5 end 6 for i = N + 1, ..., M and yi = 0, 1 do 7 Compute likelihood p(yi|vei); 8 Compute wi and zi using equation (8) 9 end 10 Obtain “best” weak learner ft according to equation (7) and update Ft = Ft−1 + ft ; 11 end 12 0 10 20 30 40 0.6 0.65 0.7 0.75 0.8 0.85 0.9 Number of states Accuracy sML sVEB (b) 0 100 200 300 400 500 0.5 0.6 0.7 0.8 0.9 Dimension of Observations Accuracy sML sVEB (a) 0 2 4 6 8 10 0.7 0.75 0.8 0.85 0.9 0.95 1 Values of α sML sVEB Accuracy (c) Figure 1: Accuracy of sML and sVEB for different number of states, local features and different values of α. method using all observed features(ML), (ML+Boost) using a subset of features selected in advance, and virtual evidence boosting (VEB). All the learned models are tested using standard maximum a posteriori(MAP) estimate and belief propagation. We used a l2-norm shrinkage prior as a regularizer for the ML and sML methods. 4.1 Synthetic data The synthetic data is generated using a first-order Markov Chain with self-transition probabilities set to 0.9. For each model, we generate five sequences of length 4,000 and divide each trace into sequences of length 200. We randomly choose 50% of them as the labeled and the other 50% as unlabeled training data. We perform leave-one-out cross-validation and report the average accuracies. To measure how the complexity of the CRFs affects the performance of the different semi-supervised methods, we vary the number of local features and the number of states. First, we compare the performance of sVEB and sML on CRFs with increasing the number of features. The number of states is set to 10 and the number of observation features is varied from 20 to 400 observations. Figure (1a) shows the average accuracy for the two semi-supervised training methods and their confidence intervals. The experimental results demonstrate that sVEB outperforms sML as we increase the dimension of observations (i.e. the number of local features). In the second experiment, we increase the number of classes and keep the dimension of observations fixed to 100. Figure (1b) demonstrates that sVEB again outperforms sML as we increase the number of states. Given the same amount of training data, sVEB is less likely to overfit because of the feature selection step. In both these experiments we set the value of tuning parameter, α, to 1.5. To explore the effect of tuning parameter α, we vary the value of α from 0.1 to 10 , while setting the number of states to 10 and the number of dimensions to 100. Figure (1c) shows that the performance of both sML and sVEB depends on the value of α but the accuracy decreases for large α’s similar to the sML results presented in [10]. 5 Time Classes 1000 2000 3000 4000 5000 1 2 3 4 5 6 7 8 Sensor Traces Time Ground truth Inference Figure 2: An example of a sensor trace and a classification trace Labeled Average Accuracy (%) - Dataset 1 ML+all obs ML+Boost VEB 60% 62.7 ± 6.6 69.4 ± 3.9 82.6 ± 7.3 80% 73.0 ± 4.2 81.8 ± 4.7 90.3 ± 4.7 100% 77.8 ± 3.4 87.0 ± 2.3 91.5 ± 3.8 Labeled Average Accuracy (%) - Dataset 2 ML+all obs ML+Boost VEB 60% 74.3 ± 3.7 75.8 ± 3.3 88.5 ± 5.1 80% 80.6 ± 2.9 84.8 ± 2.9 93.4 ± 3.8 100% 86.2 ± 3.1 87.5 ± 3.1 93.8 ± 4.6 Table 1: Accuracy ± 95% confidence interval of the supervised algorithms on activity datasets 1 and 2 4.2 Activity dataset We collected two activity datasets using wearable sensors, which include audio, acceleration, light, temperature, pressure, and humidity. The first dataset contains instances of 8 basic physical activities (e.g. walking, running, going up/down stairs, going up/down elevator, sitting, standing, and brushing teeth) from 7 different users. There is on average 30 minutes of data per user and a total of 3.5 hours of data that is manually labeled for training and testing purposes. The data is segmented into 0.25s chunks resulting in a total of 49613 data points. For each chunk, we compute 651 features, which include signal energy in log and linear frequency bands, autocorrelation, different entropy measures, mean, variances etc. The features are chosen based on what is used in existing activity recognition literature and a few additional ones that we felt could be useful. During training, the data from each person is divided into sequences of length 200 and fed into linear chain CRFs as observations. The second dataset contains instances of 5 different indoor activities (e.g. computer usage, meal, meeting, watching TV and sleeping) from a single user. We recorded 15 hours of sensor traces over 12 days. As this set contains longer time-scale activities, the data is segmented into 1 minute chunks and 321 different features are computed, similar to the first dataset. There are a total of 907 data points. These features are fed into CRFs as observations, one linear chain CRF is created per day. We evaluate the performance of supervised and semi-supervised training algorithms on these two datasets. For the semi-supervised case, we randomly select 40% of the sequences for a given person or a given day as labeled and a different subset as the unlabeled training data. We compare the performance of sML and sVEB as we incorporate more unlabeled data (20%, 40% and 60%) into the training process. We also compare the supervised techniques, ML, ML+Boost, and VEB, with increasing amount of labeled data. For all the experiments, the tuning parameter α is set to 1.5. We perform leave-one-person-out cross-validation on dataset 1 and leave-one-day-out cross-validation on dataset 2 and report the average the accuracies. The number of features chosen (i. e. through the boosting iterations) is set to 50 for both datasets – including more features did not significantly improve the classification performance. For both datasets, incorporating more unlabeled data improves accuracy. The sML estimate of the CRF parameters performs the worst. Even with the shrinkage prior, the high dimensionality can still cause over-fitting and lower the accuracy. Whereas parameter estimation and feature selection via sVEB consistently results in the highest accuracy. The (sML+Boost) method performs better than sML but does not perform as well as when feature selection and parameter estimation is done within a unified framework as in sVEB. Table 2 summarize our results. The results of supervised learnUnAverage Accuracy (%) - Dataset 1 labeled sML+all obs sML+Boost sVEB 20% 60.8 ± 5.4 66.4 ± 4.2 72.6 ± 2.3 40% 68.1 ± 4.8 76.8 ± 3.4 78.5 ± 3.4 60% 74.9 ± 3.1 81.3 ± 3.9 85.3 ± 4.1 UnAverage Accuracy (%) - Dataset 2 labeled sML+all obs sML+Boost sVEB 20% 71.4 ± 3.2 70.5 ± 5.3 79.9 ± 4.2 40% 73.5 ± 5.8 74.1 ± 4.6 83.5 ± 6.3 60% 75.6 ± 3.9 77.8 ± 3.2 87.4 ± 4.7 Table 2: Accuracy ± 95% confidence interval of semi-supervised algorithms on activity datasets 1 and 2 6 Labeled Average Accuracy (%) - Dataset 2 ML+all obs ML+Boost VEB 5% 59.2 ± 6.5 65.7 ± 8.3 71.2 ± 5.7 20% 66.9 ± 5.9 67.3 ± 8.5 77.4 ± 3.6 Labeled Average Accuracy (%) - Dataset 2 ML+all obs ML+Boost VEB 5% 71.2 ± 4.1 68.3 ± 6.7 79.7 ± 7.9 20% 71.4 ± 6.3 73.8 ± 5.2 83.1 ± 6.4 Table 3: Accuracy ± 95% confidence interval of semi-supervised algorithms on activity datasets 1 and 2 ing algorithms are presented in Table 1. Similar to the semi-supervised results, the VEB method performs the best, the ML is the worst performer, and the accuracy numbers for the (ML+Boost) method is in between. The accuracy increases if we incorporate more labeled data during training. To evaluate sVEB when a small amount of labeled data is available, we performed another set of experiments on datasets 1 and 2, where only 5% and 20% of the training data is labeled respectively. We used all the available unlabeled data during training. The results are shown in table 3. These experiments clearly demonstrate that although adding more unlabeled data is not as helpful as incorporating more labeled data, the use of cheap unlabeled data along with feature selection can significantly boost the performance of the models. 4.3 Complexity Analysis The sVEB and VEB algorithm are significantly faster than ML and sML because they do not need to use optimizers such as quasi-newton methods to learn the weight parameters. For each training iteration in sML the cost of running BP is O(clns2 +cun2s3) [10] whereas the cost of each boosting iteration in sVEB is O((cl +cu)ns2). An efficient entropy gradient computation is proposed in [17], which reduces the cost of sML to O((cl + cu)ns2) but still requires an optimizer to maximize the log-likelihood. Moreover, the number of training iterations needed is usually much higher than the number of boosting iterations because optimizers such as L-BFGS require many more iterations to reach convergence in high dimensional spaces. For example, for dataset 1, we needed about 1000 iterations for sML to converge but we ran sVEB for only 50 iterations. Table 4 shows the time for performing the experiments on activity datasets (as described in the previous section) 2. On the other hand the space complexity of sVEB is linearly smaller than sML and ML. Similar to ML, sML has the space complexity of O(ns2D) in the best case [10]. VEB and sVEB have a lower space cost of O(ns2Db), because of the feature selection step Db ≪D usually. Therefore, the difference becomes significant when we are dealing with high dimensional data, particularly if they include a large number of redundant features. Time (hours) ML ML+Boost VEB sML sML+Boost sVEB Dataset 1 34 18 2.5 96 48 4 Dataset 2 7.5 4.25 0.4 10.5 8 0.6 n length of training sequence cl number of labeled training sequences cu number of unlabeled training sequences s number of states D, Db dimension of observations Table 4: Training time for the different algorithms. 5 Conclusion We presented sVEB, a new semi-supervised training method for CRFs, that can simultaneously select discriminative features via modified LogitBoost and utilize unlabeled data via minimumentropy regularization. Our experimental results demonstrate the sVEB significantly outperforms other training techniques in real-world activity recognition problems. The unified framework for feature selection and semi-supervised training presented in this paper reduces the computational and human labeling costs, which are often the major bottlenecks in building large classification systems. Acknowledgments The authors would like to thank Nando de Freitas and Lin Liao for many helpful discussions. This work was supported by the NSF under grant number IIS 0433637 and NSERC Canada Graduate Scholarship. References [1] J. Lafferty, A. McCallum, and F. Pereira. Conditional random fields: Probabilistic models for segmenting and labeling sequence data. In Proc. of the International Conference on Machine Learning (ICML), 2001. 2The experiments were run in Matlab environment and as a result they took longer. 7 [2] Andrew McCallum. Efficiently inducing features or conditional random fields. In Proc. of the Conference on Uncertainty in Artificial Intelligence (UAI), 2003. [3] T. Dietterich, A. Ashenfelter, and Y. Bulatov. Training conditional random fields via gradient tree boosting. In Proc. of the International Conference on Machine Learning (ICML), 2004. [4] A. Torralba, K. P. Murphy, and W. T. Freeman. Contextual models for object detection using boosted random fields. In Advances in Neural Information Processing Systems (NIPS), 2004. [5] L. Liao, T. Choudhury, D. Fox, and H Kautz. Training conditional random fields using virtual evidence boosting. In Proc. of the International Joint Conference on Artificial Intelligence (IJCAI), 2007. [6] K. Nigam, A. McCallum, A. Thrun, and T. Mitchell. Text classification from labeled and unlabeled documents using em. Machine learning, 2000. [7] A. Zhu, Z. Ghahramani, and J. Lafferty. Semi-supervised learning using gaussian fields and harmonic functions. In Proc. of the International Conference on Machine Learning (ICML), 2003. [8] W. Li and M. Andrew. Semi-supervised sequence modeling with syntactic topic models. In Proc. of the National Conference on Artificial Intelligence (AAAI), 2005. [9] Y. Grandvalet and Y. Bengio. Semi-supervised learning by entropy minimization. In Advances in Neural Information Processing Systems (NIPS), 2004. [10] F. Jiao, W. Wang, C. H. Lee, R. Greiner, and D. Schuurmans. Semi-supervised conditional random fields for improved sequence segmentation and labeling. In International Committee on Computational Linguistics and the Association for Computational Linguistics, 2006. [11] C. Lee, S. Wang, F. Jiao, Schuurmans D., and R. Greiner. Learning to Model Spatial Dependency: SemiSupervised Discriminative Random Fields. In NIPS, 2006. [12] J.S. Yedidia, W.T. Freeman, and Y. Weiss. Constructing free-energy approximations and generalized belief propagation algorithms. IEEE Transactions on Information Theory, 51(7):2282–2312, 2005. [13] Y. Weiss. Comparing mean field method and belief propagation for approximate inference in mrfs. 2001. [14] J. Besag. Statistical analysis of non-lattice data. The Statistician, 24, 1975. [15] C. J. Geyer and E. A. Thompson. Constrained Monte Carlo Maximum Likelihood for dependent data. Journal of Royal Statistical Society, 1992. [16] Jerome Friedman, Trevor Hastie, and Robert Tibshirani. Additive logistic regression: a statistical view of boosting. The Annals of Statistics, 38(2):337–374, 2000. [17] G. Mann and A. McCullum. Efficient computation of entropy gradient for semi-supervised conditional random fields. In Human Language Technologies, 2007. 6 Appendix In this section, we show how we derived the equations for wi and zi (eq. 8): LF = LsV EB = LV EB −αHemp = N P i=1 log p(yi|vei) + α M P i=N+1 P y′ i p(y′ i|vei) log p(y′ i|vei) As in LogitBoost, the likelihood function LF is maximized by learning an ensemble of weak learners. We start with an empty ensemble F = 0 and iteratively add the next best weak learner, ft, by computing the Newton update s H , where s and H are the first and second derivative respectively of LF with respect to f(vei, yi). F(vei, yi)) ←F(vei, yi) −s H , where s = ∂LF +f ∂f |f=0 and H = ∂2LF +f ∂f2 |f=0 s = N P i=1 2(2yi −1)(1 −p(yi|vei)) + α M P i=N+1 P y′ i [2(2y′ i −1)(1 −p(y′ i|vei))p(y′ i|vei)(1 −log p(y′ i|vei))] H = − N P i=1 4p(yi|vei)(1 −p(yi|vei))(2yi −1)2 + α2 M P i=N+1 P y′ i 4(2y′ i −1)2(1 −p(y′ i|vei))[p(y′ i|vei)(1 − p(y′ i|vei)) + log p(y′ i|vei)] F ←F+ N P i=1 ziwi+ M P i=N+1 P y′ i ziwi N P i=1 wi+ M P i=N+1 P y′ i wi where zi = ( yi−0.5 p(yi|vei) if 1 ≤i ≤N eq. (4) (y′ i−0.5)p(y′ i|vei)(1−log p(y′ i|vei)) α[p(y′ i|vei)(1−p(y′ i|vei))+log p(y′ i|vei)] if N < i ≤M eq. (8) and wi = p(yi|vei)(1 −p(yi|vei)) if 1 ≤i ≤N eq. (4) α2(1 −p(y′ i|vei))[p(y′ i|vei)(1 −p(y′ i|vei)) + log p(y′ i|vei)] if N < i ≤M eq. (8) At iteration t we get the best weak learner, ft, by solving the WLSE problem in eq. 7. 8
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How SVMs can estimate quantiles and the median Ingo Steinwart Information Sciences Group CCS-3 Los Alamos National Laboratory Los Alamos, NM 87545, USA ingo@lanl.gov Andreas Christmann Department of Mathematics Vrije Universiteit Brussel B-1050 Brussels, Belgium andreas.christmann@vub.ac.be Abstract We investigate quantile regression based on the pinball loss and the ǫ-insensitive loss. For the pinball loss a condition on the data-generating distribution P is given that ensures that the conditional quantiles are approximated with respect to ∥· ∥1. This result is then used to derive an oracle inequality for an SVM based on the pinball loss. Moreover, we show that SVMs based on the ǫ-insensitive loss estimate the conditional median only under certain conditions on P. 1 Introduction Let P be a distribution on X × Y , where X is an arbitrary set and Y ⊂R is closed. The goal of quantile regression is to estimate the conditional quantile, i.e., the set valued function F ∗ τ,P(x) := t ∈R : P (−∞, t] | x ≥τ and P [t, ∞) | x ≥1 −τ , x ∈X, where τ ∈(0, 1) is a fixed constant and P( · | x), x ∈X, is the (regular) conditional probability. For conceptual simplicity (though mathematically this is not necessary) we assume throughout this paper that F ∗ τ,P(x) consists of singletons, i.e., there exists a function f ∗ τ,P : X →R, called the conditional τ-quantile function, such that F ∗ τ,P(x) = {f ∗ τ,P(x)}, x ∈X. Let us now consider the so-called τ-pinball loss Lτ : R × R →[0, ∞) defined by Lτ(y, t) := ψτ(y −t), where ψτ(r) = (τ −1)r, if r < 0, and ψτ(r) = τr, if r ≥0. Moreover, given a (measurable) function f : X →R we define the Lτ-risk of f by RLτ ,P(f) := E(x,y)∼PLτ(y, f(x)). Now recall that f ∗ τ,P is up to zero sets the only function that minimizes the Lτ-risk, i.e. RLτ ,P(f ∗ τ,P) = inf RLτ ,P(f) =: R∗ Lτ ,P, where the infimum is taken over all f : X →R. Based on this observation several estimators minimizing a (modified) empirical Lτ-risk were proposed (see [5] for a survey on both parametric and non-parametric methods) for situations where P is unknown, but i.i.d. samples D := ((x1, y1), . . . , (xn, yn)) drawn from P are given. In particular, [6, 4, 10] proposed an SVM that finds a solution fD,λ ∈H of arg min f∈H λ∥f∥2 H + 1 n n X i=1 Lτ(yi, f(xi)) , (1) where λ > 0 is a regularization parameter and H is a reproducing kernel Hilbert space (RKHS) over X. Note that this optimization problem can be solved by considering the dual problem [4, 10], but since this technique is nowadays standard in machine learning we omit the details. Moreover, [10] contains an exhaustive empirical study as well some theoretical considerations. Empirical methods estimating quantiles with the help of the pinball loss typically obtain functions fD for which RLτ ,P(fD) is close to R∗ Lτ ,P with high probability. However, in general this only implies that fD is close to f ∗ τ,P in a very weak sense (see [7, Remark 3.18]), and hence there is so far only little justification for using fD as an estimate of the quantile function. Our goal is to address this issue by showing that under certain realistic assumptions on P we have an inequality of the form ∥f −f ∗ τ,P∥L1(PX) ≤cP q RLτ ,P(f) −R∗ Lτ ,P . (2) We then use this inequality to establish an oracle inequality for SVMs defined by (1). In addition, we illustrate how this oracle inequality can be used to obtain learning rates and to justify a datadependent method for finding the hyper-parameter λ and H. Finally, we generalize the methods for establishing (2) to investigate the role of ǫ in the ǫ-insensitive loss used in standard SVM regression. 2 Main results In the following X is an arbitrary, non-empty set equipped with a σ-algebra, and Y ⊂R is a closed non-empty set. Given a distribution P on X × Y we further assume throughout this paper that the σ-algebra on X is complete with respect to the marginal distribution PX of P, i.e., every subset of a PX-zero set is contained in the σ-algebra. Since the latter can always be ensured by increasing the original σ-algebra in a suitable manner we note that this is not a restriction at all. Definition 2.1 A distribution Q on R is said to have a τ-quantile of type α > 0 if there exists a τ-quantile t∗∈R and a constant cQ > 0 such that for all s ∈[0, α] we have Q (t∗, t∗+ s) ≥cQ s and Q (t∗−s, t∗) ≥cQ s . (3) It is not difficult to see that a distribution Q having a τ-quantile of some type α has a unique τquantile t∗. Moreover, if Q has a Lebesgue density hQ then Q has a τ-quantile of type α if hQ is bounded away from zero on [t∗−α, t∗+α] since we can use cQ := inf{hQ(t) : t ∈[t∗−α, t∗+α]} in (3). This assumption is general enough to cover many distributions used in parametric statistics such as Gaussian, Student’s t, and logistic distributions (with Y = R), Gamma and log-normal distributions (with Y = [0, ∞)), and uniform and Beta distributions (with Y = [0, 1]). The following definition describes distributions on X × Y whose conditional distributions P( · |x), x ∈X, have the same τ-quantile type α. Definition 2.2 Let p ∈(0, ∞], τ ∈(0, 1), and α > 0. A distribution P on X ×Y is said to have a τ-quantile of p-average type α, if Qx := P( · |x) has PX-almost surely a τ-quantile type α and b : X →(0, ∞) defined by b(x):=cP( · |x), where cP( · |x) is the constant in (3), satisfies b−1 ∈Lp(PX). Let us now give some examples for distributions having τ-quantiles of p-average type α. Example 2.3 Let P be a distribution on X × R with marginal distribution PX and regular conditional probability Qx (−∞, y] := 1/(1+e−z), y ∈R, where z := y−m(x) /σ(x), m : X →R describes a location shift, and σ : X →[β, 1/β] describes a scale modification for some constant β ∈(0, 1]. Let us further assume that the functions m and σ are measurable. Thus Qx is a logistic distribution having the positive and bounded Lebesgue density hQx(y) = e−z/(1 + e−z)2, y ∈R. The τ-quantile function is t∗(x) := f ∗ τ,Qx = m(x) + σ(x) log( τ 1−τ ), x ∈X, and we can choose b(x) = inf{hQx(t) : t ∈[t∗(x) −α, t∗(x) + α]}. Note that hQx(m(x) + y) = hQx(m(x) −y) for all y ∈R, and hQx(y) is strictly decreasing for y ∈[m(x), ∞). Some calculations show b(x) = min hQx(t∗(x)−α), hQx(t∗(x)+α) = min n u1(x) (1+u1(x))2 , u2(x) (1+u2(x))2 o ∈ cα,β , 1 4 , where u1(x) := 1−τ τ e−α/σ(x), u2(x) := 1−τ τ eα/σ(x) and cα,β > 0 can be chosen independent of x, because σ(x) ∈[β, 1/β]. Hence b−1 ∈L∞(PX) and P has a τ-quantile of ∞-average type α. Example 2.4 Let ˜P be a distribution on X × Y with marginal distribution ˜PX and regular conditional probability ˜Qx := ˜P(· | x) on Y . Furthermore, assume that ˜Qx is ˜PX-almost surely of τ-quantile type α. Let us now consider the family of distributions P with marginal distribution ˜PX and regular conditional distributions Qx := ˜P (· −m(x))/σ(x) x , x ∈X, where m : X →R and σ : X →(β, 1/β) are as in the previous example. Then Qx has a τ-quantile f ∗ τ,Qx = m(x) + σ(x)f ∗ τ, ˜Qx of type αβ, because we obtain for s ∈[0, αβ] the inequality Qx (f ∗ τ,Qx, f ∗ τ,Qx + s) = ˜Qx (f ∗ τ, ˜Qx, f ∗ τ, ˜Qx + s/σ(x)) ≥b(x)s/σ(x) ≥b(x)βs . Consequently, P has a τ-quantile of p-average type αβ if and only if ˜P does have a τ-quantile of p-average type α. The following theorem shows that for distributions having a quantile of p-average type the conditional quantile can be estimated by functions that approximately minimize the pinball risk. Theorem 2.5 Let p ∈(0, ∞], τ ∈(0, 1), α > 0 be real numbers, and q := p p+1 . Moreover, let P be a distribution on X × Y that has a τ-quantile of p-average type α. Then for all f : X →R satisfying RLτ ,P(f) −R∗ Lτ ,P ≤2−p+2 p+1 α 2p p+1 we have ∥f −f ∗ τ,P∥Lq(PX) ≤ √ 2 ∥b−1∥1/2 Lp(PX) q RLτ ,P(f) −R∗ Lτ ,P . Our next goal is to establish an oracle inequality for SVMs defined by (1). To this end let us assume Y = [−1, 1]. Then we have Lτ(y, ¯t) ≤Lτ(y, t) for all y ∈Y , t ∈R, where ¯t denotes t clipped to the interval [−1, 1], i.e., ¯t := max{−1, min{1, t}}. Since this yields RLτ ,P( ¯f) ≤RLτ ,P(f) for all functions f : X →R we will focus on clipped functions ¯f in the following. To describe the approximation error of SVMs we need the approximation error function A(λ) := inff∈H λ∥f∥2 H + RLτ ,P(f) −R∗ Lτ ,P, λ > 0. Recall that [8] showed limλ→0 A(λ) = 0 if the RKHS H is dense in L1(PX). We also need the covering numbers which for ε > 0 are defined by N BH, ε, L2(µ) := min n ≥1 : ∃x1, . . . , xn ∈L2(µ) with BH ⊂∪n i=1(xi + εBL2(µ)) , (4) where µ is a distribution on X, and BH and BL2(µ) denote the closed unit balls ofH and the Hilbert space L2(µ), respectively. Given a finite sequence D = ((x1, y1), . . . , (xn, yn)) ∈(X × Y )n we write DX := (x1, . . . , xn), and N(BH, ε, L2(DX)) := N(BH, ε, L2(µ)) if µ is the empirical measure defined by DX. Finally, we write Lτ ◦f for the function (x, y) 7→Lτ(y, f(x)). With these preparations we can now recall the following oracle inequality shown in more generality in [9]. Theorem 2.6 Let P be a distribution on X×[−1, 1] for which there exist constants v ≥1, ϑ ∈[0, 1] with EP Lτ ◦¯f −Lτ ◦f ∗ τ,P 2 ≤v EP(Lτ ◦¯f −Lτ ◦f ∗ τ,P) ϑ (5) for all f : X →R. Moreover, let H be a RKHS over X for which there exist ̺ ∈(0, 1) and a ≥1 with sup D∈(X×Y )n log N BH, ε, L2(DX) ≤aε−2̺ , ε > 0 . (6) Then there exists a constant K̺,v depending only on ̺ and v such that for all ς ≥1, n ≥1, and λ > 0 we have with probability not less than 1 −3e−ς that RLτ ,P( ¯fD,λ)−R∗ Lτ ,P ≤8A(λ)+30 r A(λ) λ ς n + K̺,va λ̺n 1 2−ϑ+̺(ϑ−1) + K̺,va λ̺n +5 32vς n 1 2−ϑ . Moreover, [9] showed that oracle inequalities of the above type can be used to establish learning rates and to investigate data-dependent parameter selection strategies. For example if we assume that there exist constants c > 0 and β ∈(0, 1] such that A(λ) ≤cλβ for all λ > 0 then RLτ ,P( ¯fT,λn) converges to R∗ Lτ ,P with rate n−γ where γ := min { β β(2−ϑ+̺(ϑ−1))+̺, 2β β+1} and λn = n−γ/β. Moreover, [9] shows that this rate can also be achieved by selecting λ in a data-dependent way with the help of a validation set. Let us now consider how these learning rates in terms of risks translate into rates for ∥¯fT,λ −f ∗ τ,P∥Lq(PX). To this end we assume that P has a τ-quantile of p-average type α for τ ∈(0, 1). Using the Lipschitz continuity of Lτ and Theorem 2.5 we then obtain EP Lτ ◦¯f−Lτ ◦f ∗ τ,P 2 ≤EP| ¯f−f ∗ τ,P|2 ≤∥¯f−f ∗ τ,P∥2−q ∞EP| ¯f−f ∗ τ,P|q ≤c RLτ ,P( ¯f)−R∗ Lτ ,P q/2 for all f satisfying RLτ ,P( ¯f)−R∗ Lτ ,P ≤2−p+2 p+1 α 2p p+1 , i.e. we have a variance bound (5) for ϑ := q/2 and clipped functions with small excess risk. Arguing carefully to handle the restriction on ¯f we then see that ∥¯fT,λ −f ∗ τ,P∥Lq(PX) can converge as fast as n−γ, where γ := min n β β(4−q+̺(q−2))+2̺, β β+1 o . To illustrate the latter let us assume that H is a Sobolev space W m(X) of order m ∈N over X, where X is the unit ball in Rd. Recall from [3] that H satisfies (6) for ̺ := d/(2m) if m > d/2 and in this case H also consists of continuous functions. Furthermore, assume that we are in the ideal situation f ∗ τ,P ∈W m(X) which implies β = 1. Then the learning rate for ∥¯fT,λ −f ∗ τ,P∥Lq(PX) becomes n−1/(4−q(1−̺)), which for ∞-average type distributions reduces to n−2m/(6m+d) ≈n−1/3. Let us finally investigate whether the ǫ-insensitive loss defined by L(y, t) := max{0, |y −t| −ǫ} for y, t ∈R and fixed ǫ > 0, can be used to estimate the median, i.e. the (1/2)-quantile. Theorem 2.7 Let L be the ǫ-insensitive loss for some ǫ > 0 and P be a distribution on X ×R which has a unique median f ∗ 1/2,P . Furthermore, assume that all conditional distributions P(·|x), x ∈X, are atom-free, i.e. P({y}|x) = 0 for all y ∈R, and symmetric, i.e. P(h(x)+A|x) = P(h(x)−A|x) for all measurable A ⊂R and a suitable function h : X →R. If for the conditional distributions have a positive mass concentrated around f ∗ 1/2,P ± ǫ then f ∗ 1/2,P is the only minimizer of RL,P. Note that using [7] one can show that for distributions specified in the above theorem the SVM using the ǫ-insensitive loss approximates f ∗ 1/2,P whenever the SVM is RL,P-consistent, i.e. RL,P(fT,λ) →R∗ L,P in probability, see [2]. More advanced results in the sense of Theorem 2.5 seem also possible, but are out of the scope of this paper. 3 Proofs Let us first recall some notions from [7] who investigated surrogate losses in general and the question how approximate risk minimizers approximate exact risk minimizers in particular. To this end let L : X × Y × R →[0, ∞) be a measurable function which we call a loss in the following. For a distribution P and an f : X →R the L-risk is then defined by RL,P(f) := E(x,y)∼PL(x, y, f(x)), and, as usual, the Bayes L-risk, is denoted by R∗ L,P := inf RL,P(f), where the infimum is taken over all (measurable) f : X →R. In addition, given a distribution Q on Y the inner L-risks were defined by CL,Q,x(t) := R Y L(x, y, t) dQ(y), x ∈X, t ∈R, and the minimal inner L-risks were denoted by C∗ L,Q,x := inf CL,Q,x(t), x ∈X, where the infimum is taken over all t ∈R. Moreover, following [7] we usually omit the indexes x or Q if L is independent of x or y, respectively. Obviously, we have RL,P(f) = Z X CL,P( · |x),x f(x) dPX(x) , (7) and [7, Theorem 3.2] further shows that x 7→C∗ L,P( · |x),x is measurable if the σ-algebra on X is complete. In this case it was also shown that the intuitive formula R∗ L,P = R X C∗ L,P( · |x),x dPX(x) holds, i.e. the Bayes L-risk is obtained by minimizing the inner risks and subsequently integrating with respect to the marginal distribution PX. Based on this observation the basic idea in [7] is to consider both steps separately. In particular, it turned out that the sets of ε-approximate minimizers ML,Q,x(ε) := t ∈R : CL,Q,x(t) < C∗ L,Q,x + ε , ε ∈[0, ∞], and the set of exact minimizers ML,Q,x(0+) := T ε>0 ML,Q,x(ε) play a crucial role. As in [7] we again omit the subscripts x and Q in these definitions if L happens to be independent of x or y, respectively. Now assume we have two losses Ltar : X × Y × R →[0, ∞] and Lsur : X × Y × R →[0, ∞], and that our goal is to estimate the excess Ltar-risk by the excess Lsur-risk. This issue was investigated in [7], where the main device was the so-called calibration function δmax ( · , Q, x) defined by δmax (ε, Q, x) := ( inft∈R\MLtar,Q,x(ε) CLsur,Q,x(t) −C∗ Lsur,Q,x if C∗ Lsur,Q,x < ∞, ∞ if C∗ Lsur,Q,x = ∞, for all ε ∈[0, ∞]. In the following we sometimes write δmax,Ltar,Lsur(ε, Q, x) := δmax (ε, Q, x) whenever we need to explicitly mention the target and surrogate losses. In addition, we follow our convention which omits x or Q whenever this is possible. Now recall that [7, Lemma 2.9] showed δmax CLtar,Q,x(t) −C∗ Ltar,Q,x, Q, x ≤CLsur,Q,x(t) −C∗ Lsur,Q,x , t ∈R (8) if both C∗ Ltar,Q,x < ∞and C∗ Lsur,Q,x < ∞. Before we use (8) to establish an inequality between the excess risks of Ltar and Lsur, we finally recall that the Fenchel-Legendre bi-conjugate g∗∗: I →[0, ∞] of a function g : I →[0, ∞] defined on an interval I is the largest convex function h : I →[0, ∞] satisfying h ≤g. In addition, we write g∗∗(∞) := limt→∞g∗∗(t) if I = [0, ∞). With these preparations we can now establish the following generalization of [7, Theorem 2.18]. Theorem 3.1 Let P be a distribution on X × Y with R∗ Ltar,P < ∞and R∗ Lsur,P < ∞and assume that there exist p ∈(0, ∞] and functions b : X →[0, ∞] and δ : [0, ∞) →[0, ∞) such that δmax(ε, P( · |x), x) ≥b(x) δ(ε) , ε ≥0, x ∈X, (9) and b−1 ∈Lp(PX). Then for q := p p+1, ¯δ := δq : [0, ∞) →[0, ∞), and all f : X →R we have ¯δ∗∗ RLtar,P(f) −R∗ Ltar,P ≤∥b−1∥q Lp(PX) RLsur,P(f) −R∗ Lsur,P q . Proof: Let us first consider the case RLtar,P(f) < ∞. Since ¯δ∗∗is convex and satisfies ¯δ∗∗(ε) ≤ ¯δ(ε) for all ε ∈[0, ∞) we see by Jensen’s inequality that ¯δ∗∗ RLtar,P(f) −R∗ Ltar,P ≤ Z X ¯δ CLtar,P( · |x),x(t) −C∗ Ltar,P( · |x),x dPX(x) (10) Moreover, using (8) and (9) we obtain b(x) δ CLtar,P( · |x),x(t) −C∗ Ltar,P( · |x),x ≤CLsur,P( · |x),x(t) −C∗ Lsur,P( · |x),x for PX-almost all x ∈X and all t ∈R. By (10), the definition of ¯δ, and H¨older’s inequality in the form of ∥· ∥q ≤∥· ∥p · ∥· ∥1, we thus find that ¯δ∗∗ RLtar,P(f) −R∗ Ltar,P is less than or equal to Z X b(x) −q CLsur,P( · |x),x f(x) −C∗ Lsur,P( · |x),x q dPX(x) q/q ≤ Z X b−pdPX q/pZ X CLsur,P( · |x),x f(x) −C∗ Lsur,P( · |x),x dPX(x) q ≤ ∥b−1∥q Lp(PX) RLsur,P(f) −R∗ Ltar,P q . Let us finally deal with the case RLtar,P(f) = ∞. If ¯δ∗∗(∞) = 0 there is nothing to prove and hence we assume ¯δ∗∗(∞) > 0. Following the proof of [7, Theorem 2.13] we then see that there exist constants c1, c2 ∈(0, ∞) satisfying t ≤c1δ∗∗(t) + c2 for all t ∈[0, ∞]. From this we obtain ∞ = RLtar,P(f) −R∗ Ltar,P ≤c1 Z X ¯δ∗∗ CLtar,P( · |x),x(t) −C∗ Ltar,P( · |x),x dPX(x) + c2 ≤ c1 Z X b(x) −q CLsur,P( · |x),x f(x) −C∗ Lsur,P( · |x),x q dPX(x) + c2 , where the last step is analogous to our considerations for RLtar,P(f) < ∞. By b−1 ∈Lp(PX) and H¨older’s inequality we then conclude RLsur,P(f) −R∗ Lsur,P = ∞. Our next goal is to determine the inner risks and their minimizers for the pinball loss. To this end recall (see, e.g., [1, Theorem 23.8]) that given a distribution Q on R and a non-negative function g : X →[0, ∞) we have Z R g dQ = Z ∞ 0 Q(g ≥s) ds . (11) Proposition 3.2 Let τ ∈(0, 1) and Q be a distribution on R with C∗ Lτ ,Q < ∞and t∗be a τ-quantile of Q. Then there exist q+, q−∈[0, ∞) with q+ + q−= Q({t∗}), and for all t ≥0 we have CLτ ,Q(t∗+ t) −CLτ ,Q(t∗) = tq+ + Z t 0 Q (t∗, t∗+ s) ds , and (12) CLτ ,Q(t∗−t) −CLτ ,Q(t∗) = tq−+ Z t 0 Q (t∗−s, t∗) ds . (13) Proof: Let us consider the distribution Q(t∗) defined by Q(t∗)(A) := Q(t∗+ A) for all measurable A ⊂R. Then it is not hard to see that 0 is a τ-quantile of Q(t∗). Moreover, we obviously have CLτ ,Q(t∗+ t) = CLτ ,Q(t∗)(t) and hence we may assume without loss of generality that t∗= 0. Then our assumptions together with Q((−∞, 0]) + Q([0, ∞)) = 1 + Q({0}) yield τ ≤Q((−∞, 0]) ≤ τ + Q({0}), i.e., there exists a q+ satisfying 0 ≤q+ ≤Q({0}) and Q((−∞, 0]) = τ + q+ . (14) Let us now compute the inner risks of Lτ. To this end we first assume t ≥0. Then we have Z y<t (y −t) dQ(y) = Z y<0 y dQ(y) −tQ((−∞, t)) + Z 0≤y<t y dQ(y) and R y≥t(y −t) dQ(y) = R y≥0 y dQ(y) −tQ([t, ∞)) − R 0≤y<t y dQ(y) and hence we obtain CLτ ,Q(t) = (τ −1) Z y<t (y −t) dQ(y) + τ Z y≥t (y −t) dQ(y) = CLτ ,Q(0) −τt + tQ((−∞, 0)) + tQ([0, t)) − Z 0≤y<t y dQ(y) . Moreover, using (11) we find tQ([0, t)) − Z 0≤y<t y dQ(y) = Z t 0 Q([0, t))ds − Z t 0 Q([s, t)) ds = tQ({0}) + Z t 0 Q((0, s))ds , and since (14) implies Q((−∞, 0)) + Q({0}) = Q((−∞, 0]) = τ + q+ we thus obtain (12). Now (13) can be derived from (12) by considering the pinball loss with parameter 1 −τ and the distribution ¯Q defined by ¯Q(A) := Q(−A), A ⊂R measurable. This further yields a q−satisfying 0 ≤q−≤Q({0}) and Q([0, ∞) = 1 −τ + q−. By (14) we then find q+ + q−= Q({0}). For the proof of Theorem 2.5 we recall a few more concepts from [7]. To this end let us now assume that our loss is independent of x, i.e. we consider a measurable function L : Y × R →[0, ∞]. We write Qmin(L) := Q ∈Qmin(L) : ∃t∗ L,Q ∈R such that ML,Q(0+) = {t∗ L,Q} , i.e. Qmin(L) contains the distributions on Y whose inner L-risks have exactly one exact minimizer. Furthermore, note that this definition immediately yields C∗ L,Q < ∞for all Q ∈Qmin(L). Following [7] we now define the self-calibration loss of L by ˘L(Q, t) := |t −t∗ L,Q| , Q ∈Qmin(L), t ∈R . (15) This loss is a so-called template loss in the sense of [7], i.e., for a given distribution P on X × Y , where X has a complete σ-algebra and P( · |x) ∈Qmin(L) for PX-almost all x ∈X, the P-instance ˘LP(x, t) := |t −t∗ L,P( · |x)| is measurable and hence a loss. [7] extended the definition of inner risks to the self-calibration loss by setting C˘L,Q(t) := ˘L(Q, t), and based on this the minimal inner risks and their (approximate) minimizers were defined in the obvious way. Moreover, the self-calibration function was defined by δmax,˘L,L(ε, Q) = inft∈R; |t−t∗ L,Q|≥ε CL,Q(t) −C∗ L,Q. As shown in [7] this self-calibration function has two important properties: first it satisfies δmax,˘L,L |t −t∗ L,Q|, Q ≤CL,Q(t) −C∗ L,Q , t ∈R, (16) i.e. it measures how well approximate L-risk minimizers t approximate the true minimizer t∗ L,Q, and second it equals the calibration function of the P-instance ˘LP, i.e. δmax,˘LP,L(ε, P( · |x), x) = δmax,˘L,L(ε, P( · |x)) , ε ∈[0, ∞], x ∈X. (17) In other words, the self-calibration function can be utilized in Theorem 3.1. Proof of Theorem 2.5: Let Q be a distribution on R with C∗ L,Q < ∞and t∗be the only τ-quantile of Q. Then the formulas of Proposition 3.2 show δmax,˘L,L(ε, Q) = min n εq+ + Z ε 0 Q (t∗, t∗+ s) ds, εq−+ Z ε 0 Q (t∗−s, t∗) ds o , ε ≥0, where q+ and q−are the real numbers defined in Proposition 3.2. Let us additionally assume that the τ-quantile t∗is of type α. For the Huber type function δ(ε) := ε2/2 if ε ∈[0, α], and δ(ε) := αε −α2/2 if ε > α, a simple calculation then yields δmax,˘L,L(ε, Q) ≥cQδ(ε), where cQ is the constant satisfying (3). Let us further define ¯δ : [0, ∞) →[0, ∞) by ¯δ(ε) := δq(ε1/q), ε ≥0. In view of Theorem 3.1 we then need to find a convex function ˆδ : [0, ∞) →[0, ∞) such that ˆδ ≤¯δ. To this end we define ˆδ(ε) := sp pε2 if ε ∈ 0, spap and ˆδ(ε) := ap ε −sp+2 p ap if ε > spap, where ap := αq and sp := 2−q/p. Then ˆδ : [0, ∞) →[0, ∞) is continuously differentiable and its derivative is increasing, and thus ˆδ is convex. Moreover, we have ˆδ′ ≤¯δ′ and hence ˆδ ≤¯δ which in turn implies ˆδ ≤¯δ∗∗. Now we find the assertion by (16), (17), and Theorem 3.1. The proof of Theorem 2.7 follows immediately from the following lemma. Lemma 3.3 Let Q be a symmetric, atom-free distribution on R with median t∗= 0. Then for ǫ > 0 and L being the ǫ-insensitive loss we have CL,Q(0) = C∗ L,Q = 2 R ∞ ǫ Q[s, ∞)ds and if CL,Q(0) < ∞ we further have CL,Q(t) −CL,Q(0) = Z ǫ ǫ−t Q[s, ǫ] ds + Z ǫ+t ǫ Q[ǫ, s] ds, if t ∈[0, ǫ], CL,Q(t) −CL,Q(ǫ) = t−ǫ Z 0 Q[s, ∞) ds − ǫ+t Z 2ǫ Q[s, ∞) ds + 2 t−ǫ Z 0 Q[0, s] ds ≥0, if t > ǫ. In particular, if Q[ǫ −δ, ǫ + δ] = 0 for some δ > 0 then CL,Q(δ) = C∗ L,Q. Proof: Because L(y, t) = L(−y, −t) for all y, t ∈R we only have to consider t ≥0. For later use we note that for 0 ≤a ≤b ≤∞Equation (11) yields Z b a y dQ(y) = aQ([a, b]) + Z b a Q([s, b])ds . (18) Moreover, the definition of L implies CL,Q(t) = Z t−ǫ −∞ t −y −ǫ dQ(y) + Z ∞ t+ǫ y −ǫ −t dQ(y) . Using the symmetry of Q yields − R t−ǫ −∞y dQ(y) = R ∞ ǫ−t y dQ(y) and hence we obtain CL,Q(t) = Z t−ǫ 0 Q(−∞, t −ǫ]ds − Z t+ǫ 0 Q[t + ǫ, ∞)ds + Z t+ǫ ǫ−t y dQ(y) + 2 Z ∞ t+ǫ y dQ(y) . (19) Let us first consider the case t ≥ǫ. Then the symmetry of Q yields R t+ǫ ǫ−t y dQ(y) = R t+ǫ t−ǫ y dQ(y), and hence (18) implies CL,Q(t) = Z t−ǫ 0 Q[ǫ −t, ∞)ds + Z t−ǫ 0 Q[t−ǫ, t+ǫ] ds + Z t+ǫ t−ǫ Q[s, t+ǫ] ds +2 Z ∞ t+ǫ Q[s, ∞) ds + Z t+ǫ 0 Q[t+ǫ, ∞) ds. Using Z t+ǫ t−ǫ Q[s, t + ǫ) ds = Z t+ǫ 0 Q[s, t + ǫ) ds − Z t−ǫ 0 Q[s, t + ǫ) ds we further obtain t+ǫ Z t−ǫ Q[s, t + ǫ) ds + t+ǫ Z 0 Q[t + ǫ, ∞) ds + ∞ Z t+ǫ Q[s, ∞) ds = ∞ Z 0 Q[s, ∞) ds − t−ǫ Z 0 Q[s, t + ǫ) ds . From this and R t−ǫ 0 Q[t −ǫ, t + ǫ] ds − R t−ǫ 0 Q[s, t + ǫ] ds = − R t−ǫ 0 Q[s, t −ǫ] ds follows CL,Q(t)=− Z t−ǫ 0 Q[s, t −ǫ] ds+ Z t−ǫ 0 Q[ǫ −t, ∞) ds+ Z ∞ t+ǫ Q[s, ∞) ds+ Z ∞ 0 Q[s, ∞) ds . The symmetry of Q implies R t−ǫ 0 Q[ǫ −t, t −ǫ] ds = 2 R t−ǫ 0 Q[0, t −ǫ] ds, and we get − Z t−ǫ 0 Q[s, t −ǫ] ds + Z t−ǫ 0 Q[ǫ −t, ∞) ds = 2 Z t−ǫ 0 Q[0, s) ds + Z t−ǫ 0 Q[s, ∞) ds . This and Z ∞ t+ǫ Q[s, ∞) ds + Z ∞ 0 Q[s, ∞) ds = 2 Z ∞ t+ǫ Q[s, ∞) ds + Z t+ǫ 0 Q[s, ∞) ds yields CL,Q(t) = 2 Z t−ǫ 0 Q[0, s) ds + Z t−ǫ 0 Q[s, ∞) ds + 2 Z ∞ t+ǫ Q[s, ∞) ds + Z t+ǫ 0 Q[s, ∞) ds . By Z t−ǫ 0 Q[s, ∞) ds + Z t+ǫ 0 Q[s, ∞) ds = 2 Z t−ǫ 0 Q[s, ∞) ds + Z t+ǫ t−ǫ Q[s, ∞) ds we obtain CL,Q(t) = 2 Z t−ǫ 0 Q[0, ∞) ds + 2 Z ∞ t+ǫ Q[s, ∞) ds + Z t+ǫ t−ǫ Q[s, ∞) ds if t ≥ǫ. Let us now consider the case t ∈[0, ǫ]. Analogously we obtain from (19) that CL,Q(t) = Z ǫ−t 0 Q[ǫ −t, t + ǫ] ds + Z ǫ+t ǫ−t Q[s, t + ǫ] ds + 2 Z ∞ ǫ+t Q[s, ∞) ds +2 Z ǫ+t 0 Q[ǫ + t, ∞) ds − Z ǫ−t 0 Q[ǫ −t, ∞) ds − Z ǫ+t 0 Q[ǫ + t, ∞) ds . Combining this with Z ǫ−t 0 Q[ǫ −t, t + ǫ] ds − Z ǫ−t 0 Q[ǫ −t, ∞) ds = − Z ǫ−t 0 Q[ǫ + t, ∞) ds and R ǫ+t 0 Q[ǫ + t, ∞) ds − R ǫ−t 0 Q[ǫ + t, ∞) ds = R ǫ+t ǫ−t Q[ǫ + t, ∞) ds we get CL,Q(t) = Z ǫ+t ǫ−t Q[ǫ + t, ∞) ds + Z ǫ+t ǫ−t Q[s, t + ǫ] ds + 2 Z ∞ ǫ+t Q[s, ∞) ds = Z ǫ+t ǫ−t Q[s, ∞) ds + 2 Z ∞ ǫ+t Q[s, ∞) ds = Z ∞ ǫ−t Q[s, ∞) ds + Z ∞ ǫ+t Q[s, ∞) ds. Hence CL,Q(0) = 2 R ∞ ǫ Q[s, ∞) ds. The expressions for CL,Q(t)−CL,Q(0), t ∈(0, ǫ], and CL,Q(t)− CL,Q(ǫ), t > ǫ, given in Lemma 3.3 follow by using the same arguments. Hence one exact minimizer of CL,Q(·) is the median t∗= 0. The last assertion is a direct consequence of the formula for CL,Q(t) −CL,Q(0) in the case t ∈(0, ǫ]. References [1] H. Bauer. Measure and Integration Theory. De Gruyter, Berlin, 2001. [2] A. Christmann and I. Steinwart. Consistency and robustness of kernel based regression. Bernoulli, 15:799–819, 2007. [3] D.E. Edmunds and H. Triebel. Function Spaces, Entropy Numbers, Differential Operators. Cambridge University Press, 1996. [4] C. Hwang and J. Shim. A simple quantile regression via support vector machine. In Advances in Natural Computation: First International Conference (ICNC), pages 512 –520. Springer, 2005. [5] R. Koenker. Quantile Regression. Cambridge University Press, 2005. [6] B. Sch¨olkopf, A. J. Smola, R. C. Williamson, and P. L. Bartlett. New support vector algorithms. Neural Computation, 12:1207–1245, 2000. [7] I. Steinwart. How to compare different loss functions. Constr. Approx., 26:225–287, 2007. [8] I. Steinwart, D. Hush, and C. Scovel. Function classes that approximate the Bayes risk. In Proceedings of the 19th Annual Conference on Learning Theory, COLT 2006, pages 79–93. Springer, 2006. [9] I. Steinwart, D. Hush, and C. Scovel. An oracle inequality for clipped regularized risk minimizers. In Advances in Neural Information Processing Systems 19, pages 1321–1328, 2007. [10] I. Takeuchi, Q.V. Le, T.D. Sears, and A.J. Smola. Nonparametric quantile estimation. J. Mach. Learn. Res., 7:1231–1264, 2006.
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Random Projections for Manifold Learning Chinmay Hegde ECE Department Rice University ch3@rice.edu Michael B. Wakin EECS Department University of Michigan wakin@eecs.umich.edu Richard G. Baraniuk ECE Department Rice University richb@rice.edu Abstract We propose a novel method for linear dimensionality reduction of manifold modeled data. First, we show that with a small number M of random projections of sample points in RN belonging to an unknown K-dimensional Euclidean manifold, the intrinsic dimension (ID) of the sample set can be estimated to high accuracy. Second, we rigorously prove that using only this set of random projections, we can estimate the structure of the underlying manifold. In both cases, the number of random projections required is linear in K and logarithmic in N, meaning that K < M ≪N. To handle practical situations, we develop a greedy algorithm to estimate the smallest size of the projection space required to perform manifold learning. Our method is particularly relevant in distributed sensing systems and leads to significant potential savings in data acquisition, storage and transmission costs. 1 Introduction Recently, we have witnessed a tremendous increase in the sizes of data sets generated and processed by acquisition and computing systems. As the volume of the data increases, memory and processing requirements need to correspondingly increase at the same rapid pace, and this is often prohibitively expensive. Consequently, there has been considerable interest in the task of effective modeling of high-dimensional observed data and information; such models must capture the structure of the information content in a concise manner. A powerful data model for many applications is the geometric notion of a low-dimensional manifold. Data that possesses merely K “intrinsic” degrees of freedom can be assumed to lie on a K-dimensional manifold in the high-dimensional ambient space. Once the manifold model is identified, any point on it can be represented using essentially K pieces of information. Thus, algorithms in this vein of dimensionality reduction attempt to learn the structure of the manifold given highdimensional training data. While most conventional manifold learning algorithms are adaptive (i.e., data dependent) and nonlinear (i.e., involve construction of a nonlinear mapping), a linear, nonadaptive manifold dimensionality reduction technique has recently been introduced that employs random projections [1]. Consider a K-dimensional manifold M in the ambient space RN and its projection onto a random subspace of dimension M = CK log(N); note that K < M ≪N. The result of [1] is that the pairwise metric structure of sample points from M is preserved with high accuracy under projection from RN to RM. (a) (b) (c) (d) Figure 1: Manifold learning using random projections. (a) Input data consisting of 1000 images of a shifted disk, each of size N = 64×64 = 4096. (b) True θ1 and θ2 values of the sampled data. (c,d) Isomap embedding learned from (c) original data in RN, and (d) a randomly projected version of the data into RM with M = 15. This result has far reaching implications. Prototypical devices that directly and inexpensively acquire random projections of certain types of data (signals, images, etc.) have been developed [2,3]; these devices are hardware realizations of the mathematical tools developed in the emerging area of Compressed Sensing (CS) [4,5]. The theory of [1] suggests that a wide variety of signal processing tasks can be performed directly on the random projections acquired by these devices, thus saving valuable sensing, storage and processing costs. The advantages of random projections extend even to cases where the original data is available in the ambient space RN. For example, consider a wireless network of cameras observing a scene. To perform joint image analysis, the following steps might be executed: 1. Collate: Each camera node transmits its respective captured image (of size N) to a central processing unit. 2. Preprocess: The central processor estimates the intrinsic dimension K of the underlying image manifold. 3. Learn: The central processor performs a nonlinear embedding of the data points – for instance, using Isomap [6] – into a K-dimensional Euclidean space, using the estimate of K from the previous step. In situations where N is large and communication bandwidth is limited, the dominating costs will be in the first transmission/collation step. On the one hand, to reduce the communication needs one may perform nonlinear image compression (such as JPEG) at each node before transmitting to the central processing. But this requires a good deal of processing power at each sensor, and the compression would have to be undone during the learning step, thus adding to overall computational costs. On the other hand, every camera could encode its image by computing (either directly or indirectly) a small number of random projections to communicate to the central processor. These random projections are obtained by linear operations on the data, and thus are cheaply computed. Clearly, in many situations it will be less expensive to store, transmit, and process such randomly projected versions of the sensed images. The question now becomes: how much information about the manifold is conveyed by these random projections, and is any advantage in analyzing such measurements from a manifold learning perspective? In this paper, we provide theoretical and experimental evidence that reliable learning of a Kdimensional manifold can be performed not just in the high-dimensional ambient space RN but also in an intermediate, much lower-dimensional random projection space RM, where M = CK log(N). See, for example, the toy example of Figure 1. Our contributions are as follows. First, we present a theoretical bound on the minimum number of measurements per sample point required to estimate the intrinsic dimension (ID) of the underlying manifold, up to an accuracy level comparable to that of the Grassberger-Procaccia algorithm [7,8], a widely used geometric approach for dimensionality estimation. Second, we present a similar bound on the number of measurements M required for Isomap [6] – a popular manifold learning algorithm – to be “reliably” used to discover the nonlinear structure of the manifold. In both cases, M is shown to be linear in K and logarithmic in N. Third, we formulate a procedure to determine, in practical settings, this minimum value of M with no a priori information about the data points. This paves the way for a weakly adaptive, linear algorithm (ML-RP) for dimensionality reduction and manifold learning. The rest of the paper is organized as follows. Section 2 recaps the manifold learning approaches we utilize. In Section 3 presents our main theoretical contributions, namely, the bounds on M required to perform reliable dimensionality estimation and manifold learning from random projections. Section 4 describes a new adaptive algorithm that estimates the minimum value of M required to provide a faithful representation of the data so that manifold learning can be performed. Experimental results on a variety of real and simulated data are provided in Section 5. Section 6 concludes with discussion of potential applications and future work. 2 Background An important input parameter for all manifold learning algorithms is the intrinsic dimension (ID) of a point cloud. We aim to embed the data points in as low-dimensional a space as possible in order to avoid the curse of dimensionality. However, if the embedding dimension is too small, then distinct data points might be collapsed onto the same embedded point. Hence a natural question to ask is: given a point cloud in N-dimensional Euclidean space, what is the dimension of the manifold that best captures the structure of this data set? This problem has received considerable attention in the literature and remains an active area of research [7,9,10]. For the purposes of this paper, we focus our attention on the Grassberger-Procaccia (GP) [7] algorithm for ID estimation. This is a widely used geometric technique that takes as input the set of pairwise distances between sample points. It then computes the scale-dependent correlation dimension of the data, defined as follows. Definition 2.1 Suppose X = (x1, x2, ..., xn) is a finite dataset of underlying dimension K. Define Cn(r) = 1 n(n −1) X i̸=j I∥xi−xj∥<r, where I is the indicator function. The scale-dependent correlation dimension of X is defined as bDcorr(r1, r2) = log Cn(r1) −log Cn(r2) log r1 −log r2 . The best possible approximation to K (call this bK) is obtained by fixing r1 and r2 to the biggest range over which the plot is linear and the calculating Dcorr in that range. There are a number of practical issues involved with this approach; indeed, it has been shown that geometric ID estimation algorithms based on finite sampling yield biased estimates of intrinsic dimension [10, 11]. In our theoretical derivations, we do not attempt to take into account this bias; instead, we prove that the effect of running the GP algorithm on a sufficient number of random projections produces a dimension estimate that well-approximates the GP estimate obtained from analyzing the original point cloud. The estimate bK of the ID of the point cloud is used by nonlinear manifold learning algorithms (e.g., Isomap [6], Locally Linear Embedding (LLE) [12], and Hessian Eigenmaps [13], among many others) to generate a bK-dimensional coordinate representation of the input data points. Our main analysis will be centered around Isomap. Isomap attempts to preserve the metric structure of the manifold, i.e., the set of pairwise geodesic distances of any given point cloud sampled from the manifold. In essence, Isomap approximates the geodesic distances using a suitably defined graph and performs classical multidimensional scaling (MDS) to obtain a reduced K-dimensional representation of the data [6]. A key parameter in the Isomap algorithm is the residual variance, which is equivalent to the stress function encountered in classical MDS. The residual variance is a measure of how well the given dataset can be embedded into a Euclidean space of dimension K. In the next section, we prescribe a specific number of measurements per data point so that performing Isomap on the randomly projected data yields a residual variance that is arbitrarily close to the variance produced by Isomap on the original dataset. We conclude this section by revisiting the results derived in [1], which form the basis for our development. Consider the effect of projecting a smooth K-dimensional manifold residing in RN onto a random M-dimensional subspace (isomorphic to RM). If M is sufficiently large, a stable near-isometric embedding of the manifold in the lower-dimensional subspace is ensured. The key advantage is that M needs only to be linear in the intrinsic dimension of the manifold K. In addition, M depends only logarithmically on other properties of the manifold, such as its volume, curvature, etc. The result can be summarized in the following theorem. Theorem 2.2 [1] Let M be a compact K-dimensional manifold in RN having volume V and condition number 1/τ. Fix 0 < ǫ < 1 and 0 < ρ < 1. Let Φ be a random orthoprojector1 from RN to RM and M ≥O K log(NV τ −1) log(ρ−1) ǫ2 . (1) Suppose M < N. Then, with probability exceeding 1 −ρ, the following statement holds: For every pair of points x, y ∈M, and i ∈{1, 2}, (1 −ǫ) r M N ≤di(Φx, Φy) di(x, y) ≤(1 + ǫ) r M N . (2) where d1(x, y) (respectively, d2(x, y)) stands for the geodesic (respectively, ℓ2) distance between points x and y. The condition number τ controls the local, as well as global, curvature of the manifold – the smaller the τ, the less well-conditioned the manifold with higher “twistedness” [1]. Theorem 2.2 has been proved by first specifying a finite high-resolution sampling on the manifold, the nature of which depends on its intrinsic properties; for instance, a planar manifold can be sampled coarsely. Then the Johnson-Lindenstrauss Lemma [14] is applied to these points to guarantee the so-called “isometry constant” ǫ, which is nothing but (2). 3 Bounds on the performance of ID estimation and manifold learning algorithms under random projection We saw above that random projections essentially ensure that the metric structure of a highdimensional input point cloud (i.e., the set of all pairwise distances between points belonging to the dataset) is preserved up to a distortion that depends on ǫ. This immediately suggests that geometrybased ID estimation and manifold learning algorithms could be applied to the lower-dimensional, randomly projected version of the dataset. The first of our main results establishes a sufficient dimension of random projection M required to maintain the fidelity of the estimated correlation dimension using the GP algorithm. The proof of the following is detailed in [15]. Theorem 3.1 Let M be a compact K-dimensional manifold in RN having volume V and condition number 1/τ. Let X = {x1, x2, ...} be a sequence of samples drawn from a uniform density supported on M. Let bK be the dimension estimate of the GP algorithm on X over the range (rmin, rmax). Let β = ln(rmax/rmin) . Fix 0 < δ < 1 and 0 < ρ < 1. Suppose the following condition holds: rmax < τ/2 (3) Let Φ be a random orthoprojector from RN to RM with M < N and M ≥O K log(NV τ −1) log(ρ−1) β2δ2 . (4) Let bKΦ be the estimated correlation dimension on ΦX in the projected space over the range (rmin p M/N, rmax p M/N). Then, bKΦ is bounded by: (1 −δ) bK ≤bKΦ ≤(1 + δ) bK (5) with probability exceeding 1 −ρ. Theorem 3.1 is a worst-case bound and serves as a sufficient condition for stable ID estimation using random projections. Thus, if we choose a sufficiently small value for δ and ρ, we are guaranteed estimation accuracy levels as close as desired to those obtained with ID estimation in the original signal space. Note that the bound on bKΦ is multiplicative. This implies that in the worst case, the 1Such a matrix is formed by orthogonalizing M vectors of length N having, for example, i.i.d. Gaussian or Bernoulli distributed entries. number of projections required to estimate bKΦ very close to bK (say, within integer roundoff error) becomes higher with increasing manifold dimension K. The second of our main results prescribes the minimum dimension of random projections required to maintain the residual variance produced by Isomap in the projected domain within an arbitrary additive constant of that produced by Isomap with the full data in the ambient space. This proof of this theorem [15] relies on the proof technique used in [16]. Theorem 3.2 Let M be a compact K-dimensional manifold in RN having volume V and condition number 1/τ. Let X = {x1, x2, ..., xn} be a finite set of samples drawn from a sufficiently fine density supported on M. Let Φ be a random orthoprojector from RN to RM with M < N. Fix 0 < ǫ < 1 and 0 < ρ < 1. Suppose M ≥O K log(NV τ −1) log(ρ−1) ǫ2 . Define the diameter Γ of the dataset as follows: Γ = max 1≤i,j≤n diso(xi, xj) where diso(x, y) stands for the Isomap estimate of the geodesic distance between points x and y. Define R and RΦ to be the residual variances obtained when Isomap generates a K-dimensional embedding of the original dataset X and projected dataset ΦX respectively. Under suitable constructions of the Isomap connectivity graphs, RΦ is bounded by: RΦ < R + CΓ2ǫ with probability exceeding 1 −ρ. C is a function only on the number of sample points n. Since the choice of ǫ is arbitrary, we can choose a large enough M (which is still only logarithmic in N) such that the residual variance yielded by Isomap on the randomly projected version of the dataset is arbitrarily close to the variance produced with the data in the ambient space. Again, this result is derived from a worst-case analysis. Note that Γ acts as a measure of the scale of the dataset. In practice, we may enforce the condition that the data is normalized (i.e., every pairwise distance calculated by Isomap is divided by Γ). This ensures that the K-dimensional embedded representation is contained within a ball of unit norm centered at the origin. Thus, we have proved that with only an M-dimensional projection of the data (with M ≪N) we can perform ID estimation and subsequently learn the structure of a K-dimensional manifold, up to accuracy levels obtained by conventional methods. In Section 4, we utilize these sufficiency results to motivate an algorithm for performing practical manifold structure estimation using random projections. 4 How many random projections are enough? In practice, it is hard to know or estimate the parameters V and τ of the underlying manifold. Also, since we have no a priori information regarding the data, it is impossible to fix bK and R, the outputs of GP and Isomap on the point cloud in the ambient space. Thus, often, we may not be able fix a definitive value for M. To circumvent this problem we develop the following empirical procedure that we dub it ML-RP for manifold learning using random projections. We initialize M to a small number, and compute M random projections of the data set X = {x1, x2, ..., xn} (here n denotes the number of points in the point cloud). Using the set ΦX = {Φx : x ∈X}, we estimate the intrinsic dimension using the GP algorithm. This estimate, say bK, is used by the Isomap algorithm to produce an embedding into bK-dimensional space. The residual variance produced by this operation is recorded. We then increment M by 1 and repeat the entire process. The algorithm terminates when the residual variance obtained is smaller than some tolerance parameter δ. A full length description is provided in Algorithm 1. The essence of ML-RP is as follows. A sufficient number M of random projections is determined by a nonlinear procedure (i.e., sequential computation of Isomap residual variance) so that conventional Algorithm 1 ML-RP M ←1 Φ ←Random orthoprojector of size M × N. while residual variance ≥δ do Run the GP algorithm on ΦX. Use ID estimate ( bK) to perform Isomap on ΦX. Calculate residual variance. M ←M + 1 Add one row to Φ end while return M return bK (a) (b) Figure 2: Performance of ID estimation using GP as a function of random projections. Sample size n = 1000, ambient dimension N = 150. (a) Estimated intrinsic dimension for underlying hyperspherical manifolds of increasing dimension. The solid line indicates the value of the ID estimate obtained by GP performed on the original data. (b) Minimum number of projections required for GP to work with 90% accuracy as compared to GP on native data. manifold learning does almost as well on the projected dataset as the original. On the other hand, the random linear projections provide a faithful representation of the data in the geodesic sense. In this manner, ML-RP helps determine the number of rows that Φ requires in order to act as an operator that preserves metric structure. Therefore, ML-RP can be viewed as an adaptive method for linear reduction of data dimensionality. It is only weakly adaptive in the sense that only the stopping criterion for ML-RP is determined by monitoring the nature of the projected data. The results derived in Section 3 can be viewed as convergence proofs for ML-RP. The existence of a certain minimum number of measurements for any chosen error value δ ensures that eventually, M in the ML-RP algorithm is going to become high enough to ensure “good” Isomap performance. Also, due to the built-in parsimonious nature of ML-RP, we are ensured to not “overmeasure” the manifold, i.e., just the requisite numbers of projections of points are obtained. 5 Experimental results This section details the results of simulations of ID estimation and subsequent manifold learning on real and synthetic datasets. First, we examine the performance of the GP algorithm on random projections of K-dimensional dimensional hyperspheres embedded in an ambient space of dimension N = 150. Figure 2(a) shows the variation of the dimension estimate produced by GP as a function of the number of projections M. The sampled dataset in each of the cases is obtained from drawing n = 1000 samples from a uniform distribution supported on a hypersphere of corresponding dimension. Figure 2(b) displays the minimum number of projections per sample point required to estimate the scale-dependent correlation dimension directly from the random projections, up to 10% error, when compared to GP estimation on the original data. We observe that the ID estimate stabilizes quickly with increasing number of projections, and indeed converges to the estimate obtained by running the GP algorithm on the original data. Figure 2(b) illustrates the variation of the minimum required projection dimension M vs. K, the intrinsic dimenFigure 3: Standard databases. Ambient dimension for the face database N = 4096; ambient dimension for the hand rotation databases N = 3840. Figure 4: Performance of ML-RP on the above databases. (left) ML-RP on the face database (N = 4096). Good approximates are obtained for M > 50. (right) ML-RP on the hand rotation database (N = 3840). For M > 60, the Isomap variance is indistinguishable from the variance obtained in the ambient space. sion of the underlying manifold. We plot the intrinsic dimension of the dataset against the minimum number of projections required such that bKΦ is within 10% of the conventional GP estimate bK (this is equivalent to choosing δ = 0.1 in Theorem 3.1). We observe the predicted linearity (Theorem 3.1) in the variation of M vs K. Finally, we turn our attention to two common datasets (Figure 3) found in the literature on dimension estimation – the face database2 [6], and the hand rotation database [17].3 The face database is a collection of 698 artificial snapshots of a face (N = 64 × 64 = 4096) varying under 3 degrees of freedom: 2 angles for pose and 1 for lighting dimension. The signals are therefore believed to reside on a 3D manifold in an ambient space of dimension 4096. The hand rotation database is a set of 90 images (N = 64 × 60 = 3840) of rotations of a hand holding an object. Although the image appearance manifold is ostensibly one-dimensional, estimators in the literature always overestimate its ID [11]. Random projections of each sample in the databases were obtained by computing the inner product of the image samples with an increasing number of rows of the random orthoprojector Φ. We note that in the case of the face database, for M > 60, the Isomap variance on the randomly projected points closely approximates the variance obtained with full image data. This behavior of convergence of the variance to the best possible value is even more sharply observed in the hand rotation database, in which the two variance curves are indistinguishable for M > 60. These results are particularly encouraging and demonstrate the validity of the claims made in Section 3. 6 Discussion Our main theoretical contributions in this paper are the explicit values for the lower bounds on the minimum number of random projections required to perform ID estimation and subsequent manifold learning using Isomap, with high guaranteed accuracy levels. We also developed an empirical greedy algorithm (ML-RP) for practical situations. Experiments on simple cases, such as uniformly generated hyperspheres of varying dimension, and more complex situations, such as the image databases displayed in Figure 3, provide sufficient evidence of the nature of the bounds described above. 2http://isomap.stanford.edu 3http://vasc.ri.cmu.edu//idb/html/motion/hand/index.html. Note that we use a subsampled version of the database used in the literature, both in terms of resolution of the image and sampling of the manifold. The method of random projections is thus a powerful tool for ensuring the stable embedding of lowdimensional manifolds into an intermediate space of reasonable size. The motivation for developing results and algorithms that involve random measurements of high-dimensional data is significant, particularly due to the increasing attention that Compressive Sensing (CS) has received recently. It is now possible to think of settings involving a huge number of low-power devices that inexpensively capture, store, and transmit a very small number of measurements of high-dimensional data. ML-RP is applicable in all such situations. In situations where the bottleneck lies in the transmission of the data to the central processing node, ML-RP provides a simple solution to the manifold learning problem and ensures that with minimum transmitted amount of information, effective manifold learning can be performed. The metric structure of the projected dataset upon termination of MLRP closely resembles that of the original dataset with high probability; thus, ML-RP can be viewed as a novel adaptive algorithm for finding an efficient, reduced representation of data of very large dimension. References [1] R. G. Baraniuk and M. B. Wakin. Random projections of smooth manifolds. 2007. To appear in Foundations of Computational Mathematics. [2] M. B. Wakin, J. N. Laska, M. F. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. F. Kelly, and R. G. Baraniuk. An architecture for compressive imaging. In IEEE International Conference on Image Processing (ICIP), pages 1273–1276, Oct. 2006. [3] S. Kirolos, J.N. Laska, M.B. Wakin, M.F. Duarte, D.Baron, T. Ragheb, Y. Massoud, and R.G. Baraniuk. Analog-to-information conversion via random demodulation. In Proc. IEEE Dallas Circuits and Systems Workshop (DCAS), 2006. [4] E. J. Cand`es, J. Romberg, and T. Tao. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Info. Theory, 52(2):489–509, Feb. 2006. [5] D. L. Donoho. Compressed sensing. IEEE Trans. Info. Theory, 52(4):1289–1306, September 2006. [6] J. B. Tenenbaum, V.de Silva, and J. C. Landford. A global geometric framework for nonlinear dimensionality reduction. Science, 290:2319–2323, 2000. [7] P. Grassberger and I. Procaccia. Measuring the strangeness of strange attractors. Physica D Nonlinear Phenomena, 9:189–208, 1983. [8] J. Theiler. Statistical precision of dimension estimators. Physical Review A, 41(6):3038–3051, 1990. [9] F. Camastra. Data dimensionality estimation methods: a survey. Pattern Recognition, 36:2945– 2954, 2003. [10] J. A. Costa and A. O. Hero. Geodesic entropic graphs for dimension and entropy estimation in manifold learning. IEEE Trans. Signal Processing, 52(8):2210–2221, August 2004. [11] E. Levina and P. J. Bickel. Maximum likelihood estimation of intrinsic dimension. In Advances in NIPS, volume 17. MIT Press, 2005. [12] S. Roweis and L. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290:2323–2326, 2000. [13] D. Donoho and C. Grimes. Hessian eigenmaps: locally linear embedding techniques for high dimensional data. Proc. of National Academy of Sciences, 100(10):5591–5596, 2003. [14] Sanjoy Dasgupta and Anupam Gupta. An elementary proof of the JL lemma. Technical Report TR-99-006, University of California, Berkeley, 1999. [15] C. Hegde, M. B. Wakin, and R. G. Baraniuk. Random projections for manifold learning proofs and analysis. Technical Report TREE 0710, Rice University, 2007. [16] M. Bernstein, V. de Silva, J. Langford, and J. Tenenbaum. Graph approximations to geodesics on embedded manifolds, 2000. Technical report, Stanford University. [17] B. K´egl. Intrinsic dimension estimation using packing numbers. In Advances in NIPS, volume 14. MIT Press, 2002.
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Hippocampal Contributions to Control: The Third Way M´at´e Lengyel Collegium Budapest Institute for Advanced Study 2 Szenth´aroms´ag u, Budapest, H-1014, Hungary and Computational & Biological Learning Lab Cambridge University Engineering Department Trumpington Street, Cambridge CB2 1PZ, UK lmate@gatsby.ucl.ac.uk Peter Dayan Gatsby Computational Neuroscience Unit, UCL 17 Queen Square, London WC1N 3AR, UK dayan@gatsby.ucl.ac.uk Abstract Recent experimental studies have focused on the specialization of different neural structures for different types of instrumental behavior. Recent theoretical work has provided normative accounts for why there should be more than one control system, and how the output of different controllers can be integrated. Two particlar controllers have been identified, one associated with a forward model and the prefrontal cortex and a second associated with computationally simpler, habitual, actor-critic methods and part of the striatum. We argue here for the normative appropriateness of an additional, but so far marginalized control system, associated with episodic memory, and involving the hippocampus and medial temporal cortices. We analyze in depth a class of simple environments to show that episodic control should be useful in a range of cases characterized by complexity and inferential noise, and most particularly at the very early stages of learning, long before habitization has set in. We interpret data on the transfer of control from the hippocampus to the striatum in the light of this hypothesis. 1 Introduction What use is an episodic memory? It might seem that the possibility of a fulminant recreation of a former experience plays a critical role in enabling us to act appropriately in the world [1]. However, why should it be better to act on the basis of the recollection of single happenings, rather than the seemingly normative use of accumulated statistics from multiple events? The task of building such a statistical model is normally the dominion of semantic memory [2], the other main form of declarative memory. Issues of this kind are frequently discussed under the rubric of multiple memory systems [3, 4]; here we consider it from a normative viewpoint in which memories are directly used for control. Our answer to the initial question is the computational challenge of using a semantic memory as a forward model in sequential decision tasks in which many actions must be taken before a goal is reached [5]. Forward and backward search in the tree of actions and consequent states (ie modelbased reinforcement learning [6]) in such domains impose crippling demands on working memory 1 (to store partial evaluations) and it may not even be possible to expand out the tree in reasonable times. If we think of the inevitable resulting errors in evaluation as a form of computational noise or uncertainty, then the use of the semantic memory for control will be expected to be subject to substantial error. The main task for this paper is to explore and understand the circumstances under which episodic control, although seemingly less efficient in its use of experience, should be expected to be more accurate, and therefore be evident both psychologically and neurally. This argument about episodic control exactly parallels one recently made for habitual or cached control [5]. Model-free reinforcement learning methods, such as Q-learning [7] are computationally trivial (and therefore accurate) at the time of use, since they learn state-value functions or stateaction-value functions that cache the results of the expensive and difficult search. However, modelfree methods learn through a form of bootstrapping, which is known to be inefficient in the use of experience. It is therefore optimal to employ cached control rather than model-based control only after sufficient experience, when the inaccuracy of the former over learning is outweighed by the computational noise induced in using the latter. The exact tradeoff depends on the prior statistics over the possible tasks. We will show that in general, just as model-free control is better than model-based control after substantial experience, episodic control is better than model-based control after only very limited experience. For some classes of environments, these two other controllers significantly squeeze the domain of optimal use of semantic control. This analysis is purely computational. However, it has psychological and neural implications and associations. It was argued [5] that the transition from model-based to model-free control explains a wealth of psychological observations about the transition over the course of learning from goaldirected control (which is considered to be model-based) to habitual control (which is model-free). In turn, this is associated with an apparent functional segregation between the dorsolateral prefrontal cortex and dorsomedial striatum, implementing model-based control, and the dorsolateral striatum (and its neuromodulatory inputs), implementing model-free control. Exactly how the uncertainties associated with these two types of control are calculated is not clear, although it is known that the prelimbic and infralimbic cortices are somehow involved in arbitration. The psychological construct for episodic control is obvious; its neural realization is likely to be the hippocampus and medial temporal cortical regions. How arbitration might work for this third controller is also not clear, although there have been suggestions on how uncertainty may be represented neurally in the hippocampus [8]. There is also evidence for the transfer of control from hippocampal to striatal structures over the course of learning [9, 10] suggesting that arbitration might happen, but unfortunately, in these studies, the possibility of an additional step via dorsolateral prefrontal cortex was not fully tested. In this paper, we explore the nature and (f)utility of episodic control. Section 2 describes the simple tree-structured Markov decision problems that we use to illustrate and quantitate our arguments. Section 3 provides a detailed, albeit approximate, analysis of uncertainty and learning in these environments. Finally, section 4 uses these analytical methods and simulations to study the episodic/forward model tradeoff. 2 Paradigm for analysis We seek to analyse computational and statistical trade-offs that arise in choosing actions that maximize long-term rewards in sequential decision making problems. The trade-offs originate in uncertainties associated with learning and inference. We characterize these tasks as Markov decision processes (MDPs) [6] whose transition and reward structure are initially unknown by the subject, but are drawn from a parameterized prior that is known. The key question is how well different possible control strategies can perform given this prior and a measured amount of experience. Like [11], we simplify exploration using a form of parallel sampling model in order to focus on the ability of controllers to exploit knowledge extracted about an environment. Performance is naturally measured using the average reward that would be collected in a trial; this average is then itself averaged over draws of the MDP and the stochasticity associated with the exploratory actions. We analyse three controllers: a model-based controller without computational noise, which provides a theoretical upper limit on performance, a realistic model-based 2 A !10 0 10 0 0.3 0.6 terminal state reward probability 0 0.5 1 1 2 3 4 non!terminal state actions transition probabilities B 0 2 4 6 8 10 0 0.5 1 1.5 2 2.5 3 Learning time Performance numerical simulations analytical approximations Figure 1: A, An example tree-structured MDP, with depth D = 2, branching factor B = 3, and A = 4 available actions in each non-terminal state. The horizontal stacked bars in the boxes of the left and middle column show the transition probabilities for different actions at non-terminal states, color coded by the successor states to which they lead (matching the color of the corresponding arrows). Transition probability distributions are iid. according to a Dirichlet distribution whose parameters are all 1. Gaussians in the right column show the reward distributions at terminal states. Each has unit variance and a mean which is drawn iid. from a normal distribution of mean µ¯r = 0 and standard deviation σ¯r = 5. All parameters in later figures are the same, unless otherwise noted. B, Validating the analytical approximations by numerical simulations (A = 3). controller with computational noise that we regard as the model of semantic memory-based control, and an ‘episodic controller’. We concentrate on a simple subset of MDPs, namely ‘tree-structured MDPs’ (tMDPs), which are illustrated in Figure 1A (and defined formally in the Supporting Material). We expect the qualitative characteristics of our findings to apply for general MDPs; however, we used tMDPs since they represent a first-order, analytically tractable, approximation of the general problem presented by any MDP at a given decision point if it is unfolded in time (ie a decision tree with finite time-horizon). Actions lead to further states (and potentially rewards), from where further possible actions and thus states become available, and so on. The key difference is that in a general MDP, a state can be revisited several times even within the same episode, which is impossible in a tMDP. Thus, our approach neglects correlations between values of future states. This is formally correct in the limit of infinitely diluted MDPs, but is otherwise just an approximation. 3 The model-based controller In our paradigm, the task for the model-based controllers is to use the data from the exploratory trials to work out posterior distributions over the unknown transition and reward structure of the tMDP, and then report the best action at each state. It is well known that actually doing this is radically intractable. However, to understand the tradeoffs between different controllers, we only need to analyze the expected return from doing so, averaging over all the random quantities. One of the technical contributions of this work is the set of analytically- and empirically-justified approximations to those averages (which are presented in the Supplementary Material), based on the assumed knowledge of the parameters governing the generation of the tMDP, and as a function of the amount of exploratory experience. We proceed in three stages. First, we consider the model-based controller in the case that it has experienced so many samples that the parameters of the tMDP are known exactly. This provides an (approximate) upper bound on the expected performance of any controller. Second, we approximate 3 the impact of incomplete exploration by corrupting the controller by an aliquot of noise whose magnitude is determined by the parameters of the problem. Finally, we approximate the additionally deleterious effect of limited computational resources by adding an assumed induced bias and extra variance. The first step is to calculate the asymptotic performance when infinitely many data have been collected. In this limit, transition probabilities and reward distributions can be treated as known quantities. Critical to our analysis is that the independence and symmetry properties of regular tMDPs imply that we mostly need only analyze a single ‘sub-treelet’ of the tree (one non-terminal state and its successor states), from which the results generalise to the whole tree by recursion. In the case of the asymptotic analysis, this recursive formulation turns out to allow for a closed-form solution for the mean µ and variance σ2 of an approximate Gaussian distribution characterizing the average value of one full tree traversal starting from the root node: µ = µ¯r + 1 −λD/2 2 1 −λ1/2 2 λ1 σ¯r σ2 = λD 2 σ2 ¯r (1) where µ¯r and σ2 ¯r are the mean and variance of the normal distribution from which the means of the reward distributions at the terminal states are drawn, and 0 ≤λ1, λ2 ≤1 are constants that depend on the other parameters of the tMDP. This calculation depends on characterizing order statistics of multivariate Gaussian distributions which are equicorrelated (all the off-diagonal terms of the covariance matrix are the same) [12]. Equation 1 is actually an interesting result in and of itself – it indicates the extent to which the controller can take advantage of the variability µ −µ¯r ∝σ¯r in boosting its expected return from the root node as a function of the depth of the tree. The second step is to observe that we expect the benefits of episodic control to be most apparent given very limited exploratory experience. To make analytical progress, we are forced to make the significant assumption that the effects of this can be modeled by assuming that the controller does not have access to the true values of actions, but only to ‘noisy’ versions. This ‘noise’ comes from the fact that computing the values of different actions is based on estimates of transition probability and reward distributions. These estimates are inherently stochastic themselves, as they are based on stochastic experience. We have been able to show that the form of the resulting ‘noise’ in the action values can have the effect of scaling down the true values of actions at states by a factor φ1 and adding extra noise φ2. Although we were unable to find a closed-form solution for the effects of φ1 and φ2 on the performance of the controller, a recursive analytical formulation, though involved, is still possible (see Supporting Material). Figure 1B shows the learning curve for the model-based controller computed using our analytical predictions (blue line) and using exhaustive numerical simulations (red line, average performance in 100 sample tMDPs, with the learning process rerun 100 times in each). The inaccuracies entailed by our approximations are tolerable (also for other parameters; simulations not shown), and so from this point we use those to analyse the performance of the optimal, model-based controller. The dark blue solid curve in figure 2A (labelled η2 = 0) shows the performance of model-based control as a function of the number of exploration samples (the equivalent of the dark blue curve in figure 1B, but for A = 4 rather than A = 3). For comparison, the dashed line shows the asymptotic expected value. The slight decrease in the approximate value arises because the approximations become slightly looser as the noise gets less; however, once again we have been able to show (simulations not shown) that our analysis is highly accurate compared with extensive actual samples. The final step is to model the effects of the computational complexity of the model-based controller on performance arising from the severe demands it places on such facets as working memory. These necessitate pruning (ie ignoring parts of the decision tree), or sub-sampling, or some other such approximation. We treat the effects of all approximations by forcing the controller to have access to only noisy versions of the (exploration-limited) action values. Just as for incomplete exploration, we model the noise as a combination of downscaling the true action values by a parameter η1 and adding excess variability η2. Note that whereas the terms φ1, φ2 characterizing the effects of learning are determined by the number of samples; η1, η2 are set by hand to capture the assumed effects on inference of the computational complexity. The asymptotic values of the curves in figure 2A for various values of η2 (for all of them, η1 = 1) demonstrate the effects of inferential noise on performance. 4 A 0 20 40 60 80 100 0 0.5 1 1.5 2 2.5 3 3.5 Learning time Performance !2=0 ! asymptotic !2=0 !2=1 !2=2 !2=3 B 0 20 40 60 80 100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Learning time Relative performance !2=1 !2=2 !2=3 !2=2, D=12 Figure 2: A, Learning curves for the model-based controller at different levels of computational noise: η1 = 1, η2 is increased from 0 to 3. The approximations used for computing these curves are less accurate in the low-noise limit, hence the paradoxical slight decrease in the performance of the perfect controller (without noise) at the end of learning. The dashed line shows the asymptotic approximation which is more accurate in this limit, demonstrating that the inaccuracy of the experience-dependent approximation is not disastrous. B, Performance of noisy controllers normalized by that of the perfect controller in the same environment at the same amount of experience. The brown line corresponds to a more difficult environment with greater depth. Note that ‘learning time’ is measured by the number of times every state-action pair has been sampled. Thus decreased performance shown in the more complex environment is not due to the increased sparsity of experience. So far, we have separately considered the effects of computational noise and uncertainty due to limited experience. In reality, both factors affect the model-based controller. The full plots in figure 2A, B show the interaction of these two factors (figure 2B shows the same data as figure 2A, but scaled to the performance of the noise-free controller for the given amount of experience). Computational noise not only makes the asymptotic performance worse, by simply down-scaling average rewards, but it also makes learning effectively slower. This is because the adverse effects of computational noise depend on the differences between the values of possible actions. If these values appear to be widely different, then computational noise will still preserve their order, and thus the one that is truly best is still likely to be chosen. However, if action values appear roughly the same, then a little noise can easily change their ordering and make the controller choose a suboptimal one. Little experience only licenses small apparent differences between values, and this boosts the corrupting effect of the inferential noise. Given more experience, the controller increasingly learns to make distinctions between different actions that looked the same a priori. Thus, while earlier work suggested that model-based control will be superior at the limit of few exploratory samples due to the unsurpassable data-efficiency of optimal statistical inference [5], we show here that in the really low data limit another factor cripples its performance: the amplified influence of computational noise. How much experience constitutes ‘little’ and how much noise counts as ’much’ is of course relative to the complexity of the environment. 4 Episodic control If model-based control is indeed crippled by computational noise given limited exploration, could there be an effective alternative? Although outside the scope of our formal analysis, this is particularly important given the ubiquity of non-stationary environments [13], for which the effects of continual change bound the effective number of exploratory samples. That the cache-based or habitual controller is even worse in this limit (since it learns by bootstrapping) was a main rationale for the uncertainty-based account of the transfer from goal-directed to habitual control suggested by Daw et al [5]. Thus the habitual controller cannot step into the breach. It is here that we expect episodic control to be most useful. Intuitively, if a subject has experienced a complex environment just a few times, and found a sequence of actions that works reasonably well, then, provided that exploitation is at a premium over exploration, it seems obvious for the subject 5 A 0 5 10 15 0 0.5 1 1.5 2 2.5 3 3.5 4 Learning time Performance model!based perfect model!based nois; episodic B 0 5 10 15 0 0.5 1 1.5 2 2.5 3 3.5 Learning time Performance model!based perfect model!based nois: episodic C 0 5 10 15 0 0.5 1 1.5 2 2.5 3 3.5 Learning time Performance model!based perfect model!based nois: episodic Figure 3: Episodic vs. model-based control. Solid red line shows the performance of noisy modelbased control (η2 = 2), blue line shows that of episodic control. Dashed red line shows the case of perfect model-based control which constitutes the best performance that could possibly be achieved. The branching factor of the environment increased from B = 2 (A), B = 3 (B) to B = 4 (C). just to repeat exactly those actions, rather than trying to build and use a complex model. This act of replaying a particular sequence of events from the past is exactly an instance of episodic control. More specifically, we employ an extremely simple model of episodic memory, and assume that each time the subject experiences a reward that is considered large enough (larger than expected a priori) it stores the specific sequence of state-action pairs leading up to this reward, and tries to follow such a sequence whenever it stumbles upon a state included in it. If multiple successful sequences are available for the same state, the one that yielded maximal reward is followed. We expect such a strategy to be useful in the low data limit because, unlike in cache-based control, there is no issue of bootstrapping and temporal credit assignment, and unlike in model-based control, there is no exhaustive tree-search involved in action selection. Of course its advantages will be ultimately counteracted by the haphazardness of using single samples that are ‘adequate’, but by that time the other controllers can take over. Although we expect our approximate analytical methods to provide some insight into its characteristics, we have so far only been able to use simulations to study the episodic controller in the usual class of tMDPs. Comparing the blue (episodic) and red (model-based, but noisy; η2 = 2) curves, in figure 3A-C, it is apparent that episodic control indeed outperforms noisy model-based control in the low data limit. The dashed curves show the performance of the idealized model-based controller that is noise-free. This emphasizes the arbitrariness of our choice of noise level – the greater the noise, the longer the dominance of episodic control. However, in complicated environments, even very small amounts of noise are near catastrophic for model-based control (see brown line in Fig. 2B), and so this issue is not nugatory. The progression of the learning curves in figure 3A-C make the same point a different way. They show what happens as the complexity of the environment is increased by increasing the branching factor. At the same level of computational noise, episodic control supplants model-based control for increasing volumes of exploratory samples. We expect that the same is true if the complexity of the environment is increased by increasing the depth of the tree (D) instead, or as well. Figure 3A-C also makes the point that the asymptotic performance of the episodic controller is rather poor, and is barely improved by extra learning. A smarter episodic strategy, perhaps involving reconsolidation to eliminate unfortunate sample trajectories, might perform more competently. 5 Discussion An episodic controller operates by remembering for each state the single action that led to the best outcome so far observed. Here, we studied the nature and benefits of episodic control. This controller is statistically inefficient for solving Markov decision problems compared with the normative strategy of building a statistical forward model of the transitions and outcomes, and searching for the optimal action. However, episodic control is computationally very straightforward, and therefore does not suffer from any excess uncertainty or noise arising from the severe calculational and search complexities of the forward model. This implies that it can best forward model control under various circumstances. 6 To explore this, we first characterized a class of regular tree-structured Markov decision problems using four critical parameters – the depth of the tree; the fan-out from each state; the number of actions per state, and the characteristic (Dirichlet) statistics of the transitions consequent on each action. We then used theoretical and empirical methods to analyze the statistical structure of control based on a forward model in the face of limited data. We showed that this control can readily be outperformed by an episodic controller which does not suffer from computational inaccuracy, at least in the particular limits of high task complexity and significant inferential noise in the modelbased controller. We also showed how the noise in the latter has a particularly pernicious effect on the course of learning, corrupting the choice between actions whose values appear, because of limited experience, closer than they actually are. Our results are obviously partial. In particular, the constraint of using a regular tree-structured MDP is much too severe – given the intuition from the results above, we can now consider more conventional MDPs that better model the classes of experiments that have probed the transfer of control. Further, it would be important to consider models of exploration more general than the parallel sampler, which provides homogeneous sampling of state-action pairs. The particular challenge is when exploration and exploitation are coupled, as then all the samples become interdependent in a gordian manner. Our analysis paralleled that of [5], who showed that the noisy forward-model controller is also beaten by a cached (actor-critic-like) controller in the opposite limit of substantial experience in an environment. The cached controller is also computationally straightforward, but relies on a completely different structure of learning and inference. In psychological terms, the episodic controller is best thought of as being goal-directed, since the ultimate outcome forms part of the episode that is recalled. Unfortunately, this makes it difficult to distinguish behaviorally from goal-directed control resulting from the forward model. In neural terms, the episodic controller is likely to rely on the very well investigated systems involved in episodic memory, namely the hippocampus and medial temporal cortices. Importantly, there is direct evidence of the transfer of control from hippocampal to striatal structures over the course of learning [9, 10], and there is some evidence that episodic and habitual control can be simultaneously active. Unfortunately, there are few data [14] on structures that might control the competition or transfer process, and no test as to whether there is an intermediate phase in which prefrontal mechanisms instantiating the forward model might be dominant. Predictions from our work associated with this are the most ripe for experimental test. This paper is an extended answer to the question of the computational benefit of episodic memory, which, crudely speaking, stores particular samples, over semantic memory, which stores probability distributions. It is, of course, not the only answer – for instance, subjects that cache are obviously better off remembering exactly where in particular they stored food [15] than having to search all the places that are likely under a (semantic) prior. Equally, in game theoretic interactions between competitors, Nash equilibria are typically stochastic, and therefore seemingly excellent candidates for control based on a semantic memory. However, taking advantage of the flaws in an opponent require exactly remembering how its actions deviate from stationary statistics, for which an episodic memory is a most useful tool [16]. One potential caveat to our results is that methods associated with memory-based reasoning [17] (such as kernel density estimation) create a semantic memory from an episodic one, for instance by recalling all episodes close to a cue, and weighting them by a statistically-appropriate measure of their distance. This form of semantic memory can be seen as arising without any consolidation process whatsoever. However, although this method has its computational attractions, it is psychologically implausible since phenomena such as priming make it extremely difficult to recall multiple closely related samples from an episodic memory, let alone to do so in a statistically unbiased way (but see [18]). In sum, we have provided a normative justification from the perspective of appropriate control for the episodic component of a multiple memory system. Pressing from a theoretical perspective is a richer understanding of the integration beyond mere competition, of the information residing in, and the decisions made by, all the systems involved in choice. 7 Acknowledgements We are very grateful to Nathaniel Daw and Quentin Huys for helpful discussions. Funding was from the Gatsby Charitable Foundation (ML and PD), and the EU Framework 6 (IST-FET 1940) (ML). References [1] Dudai, Y. & Carruthers, M. The Janus face of Mnemosyne. Nature 434, 567 (2005). [2] K´ali, S. & Dayan, P. Off-line replay maintains declarative memories in a model of hippocampalneocortical interactions. Nat. Neurosci. 7, 286–294 (2004). [3] McClelland, J.L., McNaughton, B.L. & O’Reilly, R.C. Why there are complementary learning systems in the hippocampus and neocortex: insights from the successes and failures of connectionist models of learning and memory. Psychol. Rev. 102, 419–457 (1995). [4] White, N.M. & McDonald, R.J. Multiple parallel memory systems in the brain of the rat. Neurobiol Learn Mem 77, 125–184 (2002). [5] Daw, N.D., Niv, Y. & Dayan, P. Uncertainty-based competition between prefrontal and dorsolateral striatal systems for behavioral control. Nat. Neurosci. 8, 1704–1711 (2005). [6] Sutton, R.S. & Barto, A.G. Reinforcement Learning (MIT Press, 1998). [7] Watkins, C.J.C.H. Learning from Delayed Rewards. PhD thesis, Cambridge University, (1989). [8] Lengyel, M. & Dayan, P. Uncertainty, phase and oscillatory hippocampal recall. in Advances in Neural Information Processing Systems 19 (eds. Sch¨olkopf, B., Platt, J. & Hoffman, T.) 833–840 (MIT Press, Cambridge, MA, 2007). [9] Packard, M.G. & McGaugh, J.L. Double dissociation of fornix and caudate nucleus lesions on acquisition of two water maze tasks: further evidence for multiple memory systems. Behav. Neurosci. 106, 439–446 (1992). [10] Poldrack, R.A. et al. Interactive memory systems in the human brain. Nature 414, 546–550 (2001). [11] Kearns, M. & Singh, S. Finite-sample convergence rates for Q-learning and indirect algorithms. in Advances in Neural Information Processing Systems Vol. 11 (eds. Kearns, M.S., Solla, S.A. & Cohn, D.A.), Vol. 11, 996–1002 (MIT Press, Cambridge, MA, 1999). [12] Owen, D.B. & Steck, G.P. Moments of order statistics from the equicorrelated multivariate normal distribution. Ann Math Stat 33, 1286–1291 (1962). [13] Kording, K.P., Tenenbaum, J.B. & Shadmehr, R. The dynamics of memory as a consequence of optimal adaptation to a changing body. Nat. Neurosci. 10, 779–786 (2007). [14] Poldrack, R.A. & Rodriguez, P. How do memory systems interact? Evidence from human classification learning. Neurobiol Learn Mem 82, 324–332 (2004). [15] Clayton, N.S. & Dickinson, A. Episodic-like memory during cache recovery by scrub jays. Nature 395, 272–274 (1998). [16] Clayton, N.S., Dally, J.M. & Emery, N.J. Social cognition by food-caching corvids. the western scrub-jay as a natural psychologist. Philos. Trans. R. Soc. Lond. B Biol. Sci. 362, 507–522 (2007). [17] Stanfill, C. & Waltz, D. Toward memory-based reasoning. Communications of the ACM 29, 1213–1228 (1986). [18] Hintzman, D.L. MINERVA 2: A simulation model of human memory. Behav Res Methods Instrum Comput 16, 96–101 (1984). 8
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Rapid Inference on a Novel AND/OR graph for Object Detection, Segmentation and Parsing Yuanhao Chen Department of Automation University of Science and Technology of China yhchen4@ustc.edu.cn Long (Leo) Zhu Department of Statistics University of California, Los Angeles lzhu@stat.ucla.edu Chenxi Lin Microsoft Research Asia chenxil@microsoft.com Alan Yuille Department of Statistics, Psychology and Computer Science University of California, Los Angeles yuille@stat.ucla.edu Hongjiang Zhang Microsoft Advanced Technology Center hjzhang@microsoft.com Abstract In this paper we formulate a novel AND/OR graph representation capable of describing the different configurations of deformable articulated objects such as horses. The representation makes use of the summarization principle so that lower level nodes in the graph only pass on summary statistics to the higher level nodes. The probability distributions are invariant to position, orientation, and scale. We develop a novel inference algorithm that combined a bottom-up process for proposing configurations for horses together with a top-down process for refining and validating these proposals. The strategy of surround suppression is applied to ensure that the inference time is polynomial in the size of input data. The algorithm was applied to the tasks of detecting, segmenting and parsing horses. We demonstrate that the algorithm is fast and comparable with the state of the art approaches. 1 Introduction Most problems in machine intelligence can be formulated as probabilistic inference using probabilistic models defined on structured knowledge representations. Important examples include stochastic grammars [11] and, in particular, AND/OR graphs [8],[4],[10]. In practice, the nature of the representations is constrained by the types of inference algorithms which are available. For example, probabilistic context free grammars for natural language processing have a natural one-dimensional structure which makes it practical to use dynamic programming (DP) for inference [11]. But DP can not be directly applied to vision problems which lack this one-dimensional structure. In this paper, we address the problem of detecting, segmenting and parsing articulated deformable objects, such as horses, in cluttered backgrounds. Formulating these tasks as statistical inference requires a representation that can deal with all the different possible configurations of the object including: (a) the different appearances of sub-configurations (e.g. the variable number of visible legs of a horse) and (b) the unknown location, size, and orientation of the object. In addition, we must specify a fast inference algorithm that can rapidly search over all the possible configurations of the object. 1 We first specify a novel AND/OR graph representation that efficiently allows for all the different configurations of an articulated deformable object (i.e. only a small number of nodes are required). The design of this graph uses the principle of summarization, so that lower level nodes in the graph only pass on summary statistics (abstract) to the higher level nodes. More precisely, the nodes of the AND/OR graph specify the position, orientation and scale of sub-configurations of the object (together with an index variable which specifies which sub-configurations of the object are present). The probability distribution defined on this representation obeys the Markov condition. It is designed to be invariant to the position, pose, and size of the object. In this paper, the representation and probability distributions are specified by hand. We next describe an algorithm for performing inference over this representation. This is a challenging task since the space of possible configurations is enormous and there is no natural ordering to enable dynamic programming. Our algorithm combines a bottom-up process that makes proposals for the possible configurations of the object followed by a top-down process that refines and validates (or rejects) these proposals. The bottom-up process is based on the principle of compositionality, where we combine proposals for sub-configurations together to form proposals for bigger configurations. To avoid a combinational explosion of proposals, we prune out proposals in two ways: (i) removing proposals whose goodness of fit is poor, and (ii) performing surround suppression to represent local clusters of proposals by a single max-proposal. The top-down process refines and validates (or rejects) proposals for the entire configuration by allowing max-proposals to be replaced by other proposals from their local clusters if these leads to a better overall fit. In addition, the top-down process estimates the boundary of the object and performs segmentation. Surround suppression ensures that the computional complexity of the inference algorithm is polynomial in the size of image (input data). The algorithm was tested for the task of detecting horses in cluttered backgrounds, using a standard dataset [2]. The input to the algorithm are the set of oriented edgelets detected in the image. The results show that the algorithm is very fast (approximately 13 seconds) for detecting, parsing, and segmenting the horses. Detection and segmentation are tested on 328 images and we obtain very good results using performance measures compared to ground truth. Parsing is tested on 100 images and we also obtain very good performance results (there are fewer test images for this task because it is harder to obtain datasets with ground truth parsing). 2 Background Detection, segmentation and parsing are all challenging problems. Most computer vision systems only address one of these tasks. There has been influential work on detection [6], [9] and on the related problem of registration [5],[1]. Work on segmentation includes [12], [13], [3], [7], [14], [18], [17] and [16]. Much of this work is formulated, or can be reformulated, in terms of probabilistic inference. But the representations are fixed graph structures defined at a single scale. This restricted choice of representation enables the use of standard inference algorithms (e.g. the hungarian algorithm, belief propagation) but it puts limitations on the types of tasks that can be addressed (e.g. it makes parsing impossible), the number of different object configurations that can be addressed, and on the overall performance of the systems. In the broader context of machine learning, there has been a growing use of probabilistic models defined over variable graph structures. Important examples include stochastic grammars which are particularly effective for natural language processing [11]. In particular, vision researchers have advocated the use of probability models defined over AND/OR graphs [4],[10] where the OR nodes enable the graph to have multiple structures. Similar AND/OR graphs have been used in other machine learning problems [8]. But the representational power of AND/OR graphs comes at the price of increased computational demand for performing inference (or learning). For one dimensional problems, such as natural language processing, this can be handled by dynamic programming. But computation becomes considerably harder for vision problems and it is not clear how to efficiently search over the large number of configurations of an AND/OR graph. The inference problem simplifies significantly if the OR nodes are restricted to lie at certain levels of the graph (e.g. [15], [20]), but these simplifications are not suited to the problem we are addressing. 2 3 The AND/OR Graph Representation 3.1 The topological structure of the AND/OR graph The structure of an AND/OR graph is represented by a graph G = (V, E) where V and E denote the set of vertices and edges respectively. The vertex set V contains three types of nodes,“OR”,“AND” and “LEAF” nodes which are depicted in figure (1) by circles, rectangles and triangles respectively. These nodes have attributes including position, scale, and orientation. The edge set E contains vertical edges defining the topological structure and horizontal edges defining spatial constraints on the node attributes. For each node ν ∈V , the set of its child nodes is defined by Tν. The directed (vertical) edges connect nodes at successive levels of the tree. They connect: (a) the AND nodes to the OR nodes, (b) the OR nodes to the AND nodes, and (c) the AND nodes to the LEAF nodes. The LEAF nodes correspond directly to points in the image. Connection types (a) and (c) have fixed parent-child relationships, but type (b) has switchable parent-child relationship (i.e. the parent is connected to only one of its children, and this connection can switch). The horizontal edges only appear relating the children of the AND nodes. They correspond to Markov Random Fields (MRF’s) and define spatial constraints on the node attributes. These constraints are defined to be invariant to translation, rotation, and scaling of the attributes of the children. ... ... ... ... Figure 1: The AND/OR representation of the object. The AND/OR graph we use in this paper is represented more visually in figure (2). The top node shows all the possible configurations of the horse (there are 40 in this paper). These configurations are obtained by AND-ing sub-configurations corresponding to the head, back, lower torso, and back legs of the horse (see circular nodes in the second row). Each of these sub-configurations has different aspects as illustrated by the AND nodes (rectangles in the third row). These sub-configurations, in turn, are composed by AND-ing more elementary configurations (see fourth row) which can have different aspects (see fifth row). (The topological structure of this representation is specified by the authors. Future work will attempt to learn it from examples). 3.2 The state variables defined on the AND/OR graph A configuration of the AND/OR graph is an assignment of state variables z = {zν} with zν = (xν, yν, θν, sν, tν) to each node ν, where (x, y), θ and s denote image position, orientation, and scale respectively. The t = {tν} variable defines the specific topology of the graph and tν ∈Tν . More precisely, tν defines the vertical parent-child relations by indexing the children of node ν. tν is fixed and tν = Tν if ν is an AND node (because the node is always connected to all its children), ... ... ... ... Figure 2: The AND/OR graph is an efficient way to representation different appearances of an object. The bottom level of the graph indicates points in the image. The higher levels indicating combinations of elementary configurations. The graph that we used contains eight levels (three lower levels are not depicted here due to lack of space). 3 but tν is a variable for an OR node ν (to enable sub-configurations to switch their appearances), see figure (2). We use the notation Zν to denote the state zν at node ν, together with the states of all the descendent nodes of ν (i.e. the children of ν, their children, and so on). The input to the graph is the data d = {dν} defined on the image lattice (at the lowest level of the hierarchy). We define V LEAF (t), V AND(t),V OR(t) to be the set of LEAF, AND, and OR nodes which are active for a specific choice of the topology t. These sets can be computed recursively from the root node, see figure (2). The AND nodes in the second row (i.e. the second highest level of the graph) are always activated and so are the OR nodes in the third row. The AND nodes activated in the fourth row, and their OR node children in the fifth row, are specified by the t variables assigned to their parent OR nodes. This process repeats till we reach the lowest level of the graph. A novel feature of this AND/OR representation is that the node variables are the same at all levels of the hierarchy. We call this the summarization principle. It means that the state of an AND node will be a simple deterministic function of the state variables of the children (see section (3.3)). This differs from other AND/OR graphs [4],[10] where the node variables at different levels of the graph may be at different levels of abstraction. The use of the summarization principle enables us to define a successful inference algorithm. 3.3 The probability distribution for the AND/OR graph The joint distribution on the states and the data is given by: P(z, d) = 1 Z exp{−E(d, z) −Eh(z) −Ev(z))}. (1) where d is the input data and Z is the partition function. The data term E(d, z) is given by: E(d, z) = X ν∈V LEAF (t) f(dν, zν), (2) where V LEAF (t) is the set of the LEAF nodes and f(., .) is (negative) logarithm of Gaussian defined over grey-scale intensity gradient (i.e. magnitude and orientation). It encourages large intensity gradients in the image at locations of the nodes with the orientation roughly aligned to the orientation of the boundary. The next two terms make use of the hierarchical structure. The horizontal component of the hierarchical shape prior is used to impose the horizontal connections at a range of scales and defined by Eh(z) = X ν∈V AND(t) X (µ,ρ,τ)∈tν g(zµ, zρ, zτ), (3) where V AND(t) is the set of AND nodes whose children are OR nodes and g(zµ, zρ, zτ) is a (negative) logarithm of Gaussian distribution defined on the invariant shape vector l(zµ, zρ, zτ) constructed from triple of childs nodes (zµ, zρ, zτ) [20]. (This shape vector depends only on variables of the triple, such as the internal angles, that are invariant to the translation, rotation, and scaling of the triple. This ensures that the full probability distribution is also invariant to these transformations). The summation is over all triples formed by the child nodes of each parent, see figures (2). (Each node has at most four children, which restricts the set of triplets). The parameters of the Gaussian are fixed. The vertical component Ev(z) is used to hold the structure together by relating the state of the parent nodes to the state of its children. Ev(z) is divided into three vertical energy terms denoted by Ea v(z), Eb v(z) and Ec v(z) which correspond to type(a), type(b) and type(c) vertical connections respectively. Hence we have Ev(z) = Ea v(z) + Eb v(z) + Ec v(z) (4) Ea v(z) specifies the coupling from the AND node to the OR node. The state of the parent node is determined precisely by the states of the child nodes. This is defined by: Ea v(z) = X ν∈V AND(t) h(zν; {zµ s.t.µ ∈tν}), (5) 4 where h(., .) = 0 if the average orientations and positions of the child nodes are equal to the orientation and position of the parent node (i.e. the vertical constraints are “hard”). If they are not consistent, then h(., .) = κ, where κ is a large positive number. Eb v(z) accounts for the probability of the assignments of the connections from OR nodes to AND nodes. Eb v(z) = X ν∈V OR(t) λν(tν), (6) where λν() is the potential function which encodes the weights of the assignments determined by tν. The energy term Ec v(z) defines the connection from the lowest AND nodes to the LEAF nodes. This is similar to the definition of Ea v(z), and Ec v(z) is given by: Ec v(z) = X tν∈V LEAF (t) h(zν; ztν), (7) where h(., .) = 0 if the orientation and position of the child (LEAF) node is equal to the orientation and position of the parent (AND) node. If they are not consistent, then h(., .) = κ. Finally, we can compute the energy of the sub-tree for a particular node ν as root node. The sub-tree energy is useful when performing inference, see section (4). This is computed by summation over all the potential functions associating to the node ν and its descendants. This energy is defined by: Eν(Zν) = E(d, z) + Eh(z) + Ev(z). (8) where z ∈Zν and V LEAF (t), V AND(t), V OR(t) in the summation of each term are defined in the set of the node ν and its descendants. Now we have specified a complete probability distribution for the graph. But this model is unable to do segmentation (since it has a limited number of nodes at the lowest level). To obtain a closed boundary based on the states of the leaf nodes, an extra energy term E0(d, z) at level l = 0 must be added to the exponent in equation (1). E0(d, z) is constructed similarly to that of Coughlan et al [6]. It is of form: E0(z) = X ν∈V LEAF X ρ∈C(ν,ν′) {f(dρ, zρ) + g(zρ, zρ′)}, (9) where ν and ν′ are neighbors at level 1, C(ν, ν′) is a curve connecting ν to ν′ containing a fixed number of points, and ρ′ is the neighbor of ρ. The function g(., .) takes the (negative) logarithm of Gaussian form to define the prior on the orientation and scale. This energy term ensures that the leaf nodes are connected by a closed boundary which is used for segmentation. 4 Inference: Bottom-up and Top-down Processing The task of the inference algorithm is to find a maximum a posteriori estimate of the state variables z: z∗= arg max p(z|d) = arg max p(d|z)p(z), (10) where p(d|z)p(z) = p(d, z) is defined in equation (1). The inference algorithm (see the pseudo code in figure (3)) contains a compositional bottom-up stage which makes proposals for the node variables in the tree. This is followed by a top-down stage which refines and validates the proposals. We use the following notation. Each node νl at level l has a set of proposals {P l ν,a} where a indexes the proposals (see table (2) for the typical number of proposals). There are also max-proposals {MP l ν,a}, indexed by a, each associated with a local cluster {CLl ν,a} of proposals (see table (2) for the typical number of max-proposals). Each proposal, or max-proposal, is described by a state vector {zl ν,a : a = 1, ..., M l ν}, the state vectors for it and its descendants {Zl ν,a : a = 1, ..., M l ν}, and an energy function score {El ν(Zl ν,a) : a = 1, ..., M l ν}. We obtain the proposals by a bottom-up strategy starting at level l = 2 (AND node) of the tree. For a node ν2 we define windows {W 2 ν,a} in space, orientation, and scale. We exhaustively search for all configurations within this window which have a score (goodness of fit criterion) E2 ν(P 2 ν,a) < K2, where K2 is a fixed threshold. For each window W 2 ν,a, we select the configuration with smallest 5 • Bottom-Up(MP 1) Loop : l = 2 to L, for each node ν at level l – IF ν is an OR node 1. Union: {MP l ν,b} = S ρ∈Tν ,a=1,...,Ml−1 ρ MP l−1 ρ,a – IF ν is an AND node 1. Composition: {P l ν,b} = ⊕ρ∈Tν ,a=1,...,Ml−1 ρ MP l−1 ρ,a 2. Pruning: {P l ν,a} = {P l ν,a|Eν(P l ν,a) < Kl} 3. Surround Suppression: {(MP l ν,a, CLl ν,a)} = SurroundSuppression({P l ν,a}, ϵW ) where ϵW is the size of the window W l ν defined in space, orientation, and scale. • Top-Down(MP L, CLL): MP ∗= arg mina=1,...,ML ν , ˜ P =MP L ν,a ChangeP roposal( ˜ P , MP L ν,a, CLL ν,a) • ChangeP roposal( ˜ P , MP l ν,a, CLl ν,a) – IF ν is an OR node 1. ChangeP roposal( ˜ P , MP l−1 tν ,a, CLl−1 tν ,a) – IF ν is an AND node 1. ˙P = ˜ P ⊖MP l ν,a 2. ¨ P = arg minP l ν,a′ ∈CLlν,a Eν(P l ν,a′ ⊕˙P ) + E0(P l ν,a′ ⊕˙P ) (E0() is obtained by dynamic programming) 3. ˜ P = ˙P ⊕¨ P 4. Loop: for each ρ ∈Tν, ρ ∋V LEAF and b s.t. MP l−1 ρ,b ∈¨ P ∗ChangeP roposal( ˜ P , MP l−1 ρ,b , CLl−1 ρ,b ) – Return ˜ P and its score Eν( ˜ P ) + E0( ˜ P ) Figure 3: Bottom-up and Top-down Processing. ⊕denotes the operation of combining two proposals. ⊖denotes the operation of removing a part from a proposal. score to be the proposal MP 2 ν,a and store the remaining proposals below threshold in the associated cluster CL2 ν,a. This window enforces surround suppression which performs clustering to keep the proposal with the maximum score in any local window. Surround suppression grantees the number of the remaining proposals at each level is proportional to the size of image (input data). This strategy ensures that we do not obtain too many proposals in the hierarchy and avoid a combinatorial explosion of proposals. We will analyze this property empirically in section 6. The procedure is repeated as we go up the hierarchy. Each parent node νl+1 produces proposals {P l+1 ν,a }, and associated clusters {CLl+1 ν,a }, by combining the proposals from its children. All proposals are required to have scores El+1 ν (Zl+1 ν ) < Kl+1, where Kl is a threshold. The bottom-up process provides us with a set of proposals at the root (top) node. These proposals give a set of state vectors for the hierarchy for all nodes down to level l = 1, {ZL ν0,a : a = 1, ..., M L 0 }, where ν0 denotes the root node. In the top-down processing, for each proposal a at the root node, we fix the state vectors of ZL ν0,a and obtain the state of the level l = 0 variables (on the image lattice) by minimizing Eν(ZL ν0,a)+E0(ZL ν0,a) which is performed by dynamic programming, with the constraint that the level l = 1 nodes are fixed and the boundary contour must pass through them. The output of dynamic programming is a dense boundary contour. Next we refine the solution for each proposal at the root node by recursively changing the parts of the proposal. This is performed using the clusters associated with the proposals at each node. Each element of the cluster is an alternative proposal for the state of that node. The use of these clusters enables us to perform a set of transformations which may give a lower-energy configuration. The basic moves are to change the state vector of a node in the tree hierarchy to another state in the same proposal cluster, and then to determine the zeroth level nodes – for the appropriate segment of the contour – by dynamic programming. Change to the state vectors at high levels of the hierarchy will cause large changes to the boundary contour. Changes at the lower levels will only cause small changes. The procedure is repeated as we examine each node in the hierarchy recursively. 5 Complexity of Representation and Inference We now quantify the representational power of the AND/OR graph and the complexity of the inference algorithm. These complexity measures depend on the following quantities: (i) the number M of AND nodes connecting to OR nodes, (ii) the maximum number K of children of AND nodes (we restrict K ≤4), (iii) the maximum number W of children of OR nodes, (iv) the number h of levels 6 Table 1: Performance for parsing, segmentation and detection. The table compares the results for the hierarchial model (without OR nodes) and AND/OR graph with two inference algorithms, i.e. (a) bottom-up only. (b) bottom-up and top-down. Model Testing Size Parsing Segmentation Detection Time Hierarchical Model (a) 328 18.7 81.3% / 73.4% 86.0 (282 / 328) 3.1s Hierarchical Model (b) 328 17.6 83.3% / 74.2% 88.4 (290 / 328) 6.1s And/Or Graph (a) 328 13.2 81.3% / 74.5% 84.5 (277 / 328) 4.5s And/Or Graph (b) 328 12.5 87.1% / 75.8% 91.2 (299 / 328) 13.2s [16] 172 86.2% / 75.0% Table 2: Complexity Analysis. Level Nodes Aspects Max-Proposals Proposals Time 8 1 12 11.1 2058.8 1.206s 6 8 1.5 30.6 268.9 1.338s 4 27 1 285.1 1541.5 1.631s 2 68 1 172.2 1180.7 0.351s containing OR nodes with more than one child node, (v) the number S of clusters for AND nodes (recall that the cluster is defined over image position, orientation and scale). In the experiments reported in this paper we have K = 4, W = 3, h = 2, M = 36. The number of proposals is linearly proportional to the size of the image. The representational power is given by the number of different topological configurations of the AND/OR graph. It is straightforward to show that this is bounded above by W Kh. In our experiments, the number of different topological configurations is 40. The complexity of our algorithm can also be bounded above by M ×W K ×SK. This shows that the algorithm speed is polynomial in W and S (and hence in the image size). The complexity for our experiments is reported in section (6). 6 Results The experimental results are obtained on 328 horse images [2] for detection and segmentation. We use 100 images for parsing (which requires more work to get groundtruth). The AND/OR model has 40 possible configurations. Some typical parsed and segmentation results are shown in figure (4). In table (1) we compare the performances between the AND/OR graph with 40 configurations and a simple hierarchical model with only one configuration (each OR node has one child node). Column 3 gives the parsing accuracy – the average error of the position of leaf node is 10 pixels. Column 4 gives the precision and recall at the pixel level respectively (i.e. is the pixel inside the object or not). Column 5 quantifies the detection. We rate detection as a success if the area of intersection of the detected object region and the true object region is greater than half the area of the union of these regions. The last column shows the average time taken for one image. The AND/OR graph outperforms the simple hierarchical model in all tasks with two times more cost. Hierarchical model is only capable of locating the main body while AND/OR graph catches more details like legs, heads under different poses. Compared to [16] where the training and evaluation are performed with half of the data set, our method (evaluated on the whole data set) achieves better performance of segmentation with simpler feature. (Their method is unable to do parsing and detection.) Table (2) shows the complexity properties of the algorithm. We described the AND levels only (the model has 8 levels). The computation for the OR-nodes is almost instantaneous (you just need to list the proposals from all its children AND nodes) so we do not include it. Column 2 gives the number of nodes at each level. Column 3 states the average number of aspects 1 of the AND nodes at each level. Column 4 states the average number of max-proposals for each node. Column 5 gives the average number of proposals. Column 6 gives the time. Observe that the number of proposals increases by an order of magnitude from level 6 to level 8. This is mostly due to the similar increase in the number of aspects (the more the number of aspects, the more the number of proposals needed to cover them). But surround suppression is capable of reducing the number of proposals greatly (compare the numbers of Max-proposals and proposals in Table (2)). 1The definition of aspects. Let AND node ν have children OR nodes {ρi : i ∈tν}. This gives a set of grandchildren AND nodes S i∈tν tρi. The aspect of ν is Q i∈tν |tρi|. The aspect of an AND node is an important concept. When passing up the proposals to an AND node we must take into account the number of aspects of this node. We can, in theory, have proposals for all possible aspects. The notion of aspects only goes down two levels. 7 Figure 4: The parsed results. From left to right: original image, edge map, parsed result and segmentation. In the edge map, one can observe that some parts are missing or very ambiguous with low level cues. The colored dots correspond to the leaf nodes of the object. 7 Conclusion We formulated a novel AND/OR graph representation capable of describing the different configurations of deformable articulated objects. The representation makes use of the summarization principle. We developed a novel inference algorithm that combined a bottom-up process for proposing configurations for horses together with a top-down process for refining and validating these proposals. Surround suppression ensures that the inference time is polynomial in the size of image. We demonstrated that the algorithm was fast and effective as evaluated by performance measures on a large dataset. 8 Acknowledgments This research was supported by NSF grant 0413214 and the W.M. Keck foundation. References [1] S. Belongie, J. Malik, and J. Puzicha. Shape Matching and Object Recognition Using Shape Contexts. PAMI, 2002. [2] E. Borenstein and S. Ullman. Class-specific, top-down segmentation. ECCV, 2002. [3] E. Borenstein and J. Malik. Shape Guided object segmentation. CVPR 06 [4] H. Chen, Z.J. Xu, Z.Q. Liu, and S.C. Zhu. Composite Templates for Cloth Modeling and Sketching. CVPR, 2006. [5] H. Chui and A. Rangarajan, A New Algorithm for Non-Rigid Point Matching. CVPR, 2000. [6] J.M. Coughlan, and S. Ferreira. Finding Deformable Shapes using Loopy Belief Propagation. ECCV, 2002. [7] T. Cour and J. Shi. Recognizing Objects by Piecing Together the Segmentation Puzzle. CVPR, 2007. [8] H. Dechter and Robert Mateescu. AND/OR Search Spaces for Graphical Models. In Artificial Intelligence, 2006. [9] R. Fergus, P. Perona and A. Zisserman. Object Class Recognition by Unsupervised Scale-Invariant Learning. CVPR, 2003. [10] Y. Jin, S. Geman. Context and Hierarchy in a Probabilistic Image Model. CVPR 2006. [11] D. Klein and C. Manning. Natural Language Grammar Induction Using a Constituent-Context Model. NIPS, 2001. [12] M. P. Kumar, P. H. S. Torr and A. Zisserman. OBJ CUT, CVPR, 2005. [13] B. Leibe, A. Leonardis and B. Schiele. Combined object categorization and segmentation with an implicit shape model. ECCV, 2004. [14] A. Levin and Y. Weiss. Learning to Combine Bottom-up and Top-down Segmentation. ECCV, 2006 [15] M. Meila and M. I. Jordan. Mixture of Trees: Learning with mixtures of trees. Journal of Machine Learning Research, 1, 1-48, 2000. [16] X. Ren, C. Fowlkes, and J. Malik, Cue integration in figure/ground labeling. NIPS, 2005. [17] P. Srinivasan and J. Shi. Bottom-up Recognition and Parsing of the Human Body. CVPR, 2007. [18] J. Winn and N. Jojic. LOCUS: Learning Object Classes with Unsupervised Segmentation. ICCV, 2005 [19] L. Zhu, and A. Yuille. A Hierarchical Compositional System for Rapid Object Detection. NIPS 2006. [20] L. Zhu, Y. Chen, and A. Yuille. Unsupervised Learning of a Probabilistic Grammar for Object Detection and Parsing. NIPS 2007. 8
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Convex Learning with Invariances Choon Hui Teo Australian National University choonhui.teo@anu.edu.au Amir Globerson CSAIL, MIT gamir@csail.mit.edu Sam Roweis Department of Computer Science University of Toronto roweis@cs.toronto.edu Alexander J. Smola NICTA Canberra, Australia alex.smola@gmail.com Abstract Incorporating invariances into a learning algorithm is a common problem in machine learning. We provide a convex formulation which can deal with arbitrary loss functions and arbitrary losses. In addition, it is a drop-in replacement for most optimization algorithms for kernels, including solvers of the SVMStruct family. The advantage of our setting is that it relies on column generation instead of modifying the underlying optimization problem directly. 1 Introduction Invariances are one of the most powerful forms of prior knowledge in machine learning; they have a long history [9, 1] and their application has been associated with some of the major success stories in pattern recognition. For instance, the insight that in vision tasks, one should be often be designing detectors that are invariant with respect to translation, small degrees of rotation & scaling, and image intensity has led to best-in-class algorithms including tangent-distance [13], virtual support vectors [5] and others [6]. In recent years a number of authors have attempted to put learning with invariances on a solid mathematical footing. For instance, [3] discusses how to extract invariant features for estimation and learning globally invariant estimators for a known class of invariance transforms (preferably arising from Lie groups). Another mathematically appealing formulation of the problem of learning with invariances casts it as a second order cone programming [8]; unfortunately this is neither particularly efficient to implement (having worse than cubic scaling behavior) nor does it cover a wide range of invariances in an automatic fashion. A different approach has been to pursue “robust” estimation methods which, roughly speaking, aim to find estimators whose performance does not suffer significantly when the observed inputs are degraded in some way. Robust estimation has been applied to learning problems in the context of missing data [2] and to deal with specific type of data corruption at test time [7]. The former approach again leads to a second order cone program, limiting its applicability to very small datasets; the latter is also computationally demanding and is limited to only specific types of data corruption. Our goal in this work is to develop a computationally scalable and broadly applicable approach to supervised learning with invariances which is easily adapted to new types of problems and can take advantage of existing optimization infrastructures. In this paper we propose a method which has what we believe are many appealing properties: 1. It formulates invariant learning as a convex problem and thus can be implemented directly using any existing convex solver, requiring minimal additional memory and inheriting the convergence properties/guarantees of the underlying implementation. 1 2. It can deal with arbitrary invariances, including gradual degradations, provided that the user provides a computational recipe to generate invariant equivalents efficiently from a given data vector. 3. It provides a unifying framework for a number of previous approaches, such as the method of Virtual Support Vectors [5] and is broadly applicable not just to binary classification but in fact to any structured estimation problem in the sense of [16]. 2 Maximum Margin Loss with Invariances We begin by describing a maximum margin formulation of supervised learning which naturally incorporates invariance transformations on the input objects. We assume that we are given input patterns x ∈X from from some space X and that we want to estimate outputs y ∈Y. For instance Y = {±1} corresponds to binary classification; Y = An corresponds to sequence prediction over the alphabet A.1 We denote our prediction by ¯y(x), which is obtained by maximizing our learned function f : X × Y →R, i.e. ¯y(x) := argmaxy∈Y f(x, y). For instance, if we are training a (generative or discriminative) probabilistic model, f(x, y) = log p(y|x) then our prediction is the maximum a-posteriori estimate of the target y given x. In many interesting cases ¯y(x) is obtained by solving a nontrivial discrete optimization problem, e.g. by means of dynamic programming. In kernel methods f(x, y) = ⟨φ(x, y), w⟩for a suitable feature map φ and weight vector w. For the purpose of our analysis the precise form of f is immaterial, although our experiments focus on the kernel machines, due to the availability of scalable optimizers for that class of estimators. 2.1 Invariance Transformations and Invariance Sensitive Cost The crucial ingredient to formulating invariant learning is to capture the domain knowledge that there exists some class S of invariance transforms s which can act on the input x while leaving the target y essentially unchanged. We denote by (s(x), y) s ∈S the set of valid transformations of the pair (x, y). For instance, we might believe that slight rotation (in pixel coordinates) of an input image in a pattern recognition problem do not change the image label. For text classification problems such as spam filtering, we may believe that certain editing operations (such as changes in capitalization or substitutions like Viagra →V1agra,V!agra) should not affect our decision function. Of course, most invariances only apply “locally”, i.e. in the neighborhood of the original input vector. For instance, rotating an image of the digit 6 too far might change its label to 9; applying both a substitution and an insertion can change Viagra →diagram. Furthermore, certain invariances may only hold for certain pairs of input and target. For example, we might believe that horizontal reflection is a valid invariance for images of digits in classes 0 and 8 but not for digits in class 2. The set s(x) s ∈S incorporates both the locality and applicability constraints. (We have introduced a slight abuse of notation since s may depend on y but this should always be clear in context.) To complete the setup, we adopt the standard assumption that the world or task imposes a cost function such that if the true target for an input x is y and our prediction is ¯y(x) we suffer a cost ∆(y, ¯y(x)).2 For learning with invariances, we extend the definition of ∆to include the invariance function s(x), if any, which was applied to the input object: ∆(y, ¯y(s(x)), s). This allows the cost to depend on the transformation, for instance we might suffer less cost for poor predictions when the input has undergone very extreme transformations. In a image labeling problem, for example, we might believe that a lighting/exposure invariance applies but we might want to charge small cost for extremely over-exposed or under-exposed images since they are almost impossible to label. Similarly, we might assert that scale invariance holds but give small cost to severely spatially downsampled images since they contain very little information. 2.2 Max Margin Invariant Loss Our approach to the invariant learning problem is very natural, yet allows us to make a surprising amount of analytical and algorithmic progress. A key quantity is the cost under the worst case transformation for each example, i.e. the transformation under which our predicted target suffers 1For more nontrivial examples see, e.g. [16, 14] and the references therein. 2Normally ∆= 0 if ¯y(x) = y but this is not strictly necessary. 2 the maximal cost compared with the true target: C(x, y, f) = sup s∈S ∆(y, ¯y(s(x)), s) (1) The objective function (loss) that we advocate minimizing during learning is essentially a convex upper bound on this worst case cost which incorporates a notion of (scaled) margin: l(x, y, f) := sup y′∈Y,s∈S Γ(y, y′)(f(s(x), y′) −f(s(x), y)) + ∆(y, y′, s) (2) This loss function finds the combination of invariance transformation and predicted target for which the sum of (scaled) “margin violation” plus the cost is maximized. The function Γ(y, y′) is a nonnegative margin scaling which allows different target/prediction pairs to impose different amounts of loss on the final objective function.3 The numerical scale of Γ also sets the regularization tradeoff between margin violations and the prediction cost ∆. This loss function has two mathematically important properties which allow us to develop scalable and convergent algorithms as proposed above. Lemma 1 The loss l(x, y, f) is convex in f for any choice of Γ, ∆and S. Proof For fixed (y′, s) the expression Γ(y, y′)(f(s(x), y′) −f(s(x), y)) + ∆(y, y′, s) is linear in f, hence (weakly) convex. Taking the supremum over a set of convex functions yields a convex function. This means that we can plug l into any convex solver, in particular whenever f belongs to a linear function class, as is the case with kernel methods. The primal (sub)gradient of l is easy to write: ∂fl(x, y, f) = Γ(y, y∗)(φ(s∗(x), y∗) −φ(s∗(x), y)) (3) where s∗, y∗are values of s, y for which the supremum in Eq. (2) is attained and φ is the evaluation functional of f, that is ⟨f, φ(x, y)⟩= f(x, y). In kernel methods φ is commonly referred to as the feature map with associated kernel k((x, y), (x′, y′)) = ⟨φ(x, y), φ(x′, y′)⟩. (4) Note that there is no need to define S formally. All we need is a computational recipe to obtain the worst case s ∈S in terms of the scaled margin in Eq. 2. Nor is there any requirement for ∆(y, y′, s) or (s(x), y) to have any particularly appealing mathematical form, such as the polynomial trajectory required by [8], or the ellipsoidal shape described by [2]. Lemma 2 The loss l(x, y, f) provides an upper bound on C(x, y, f) = sups∈S ∆(y, ¯y(s(x)), s). Proof Denote by (s∗, y∗) the values for which the supremum of C(x, y, f) is attained. By construction f(s∗(x), y∗) ≥f(s∗(x), y). Plugging this inequality into Eq. (2) yields l(x, y, f) ≥Γ(y, y∗)(f(s∗(x), y∗) −f(s∗(x), y)) + ∆(y, y∗, s∗) ≥∆(y, y∗, s∗). Here the first inequality follows by substituting (s∗, y∗) into the supremum. The second inequality follows from the fact that Γ ≥0 and that (s∗, y∗) are the maximizers of the empirical loss. This is essentially a direct extension of [16]. The main modifications are the inclusion of a margin scale Γ and the use of an invariance transform s(x). In section 4 we clarify how a number of existing methods for dealing with invariances can be viewed as special cases of Eq. (2). In summary, Eq. (2) penalizes estimation errors not only for the observed pair (x, y) but also for patterns s(x) which are “near” x in terms of the invariance transform s. Recall, however, that the cost function ∆may assign quite a small cost to a transformation s which takes x very far away from the original. Furthermore, the transformation class is restricted only by the computational consideration that we can efficiently find the “worst case” transformation; S does not have to have a specific analytic form. Finally, there is no specific restriction on y, thus making the formalism applicable to any type of structured estimation. 3Such scaling has been shown to be extremely important and effective in many practical problems especially in structured prediction tasks. For example, the key difference between the large margin settings of [14] and [16] is the incorporation of a sequence-length dependent margin scaling. 3 3 Learning Algorithms for Minimizing Invariant Loss We now turn to the question of learning algorithms for our invariant loss function. We assume that we are given a training set of input patterns X = {x1, . . . , xm} and associated labels Y = {y1, . . . , ym}. We follow the common approach of minimizing, at training time, our average training loss plus a penalty for model complexity. In the context of kernel methods this can be viewed as a regularized empirical risk functional of the form R[f] = 1 m m X i=1 l(xi, yi, f) + λ 2 ∥f∥2 H where f(x, y) = ⟨φ(x, y), w⟩. (5) A direct extension of the derivation of [16] yields that the dual of (5) is given by minimize α m X i,j=1 X y,y′∈Y X s,s′∈S αiysαjy′s′Kiys,jy′s′ + m X i=1 X y∈Y X s∈S ∆(yi, y, s)αiys (6a) subject to λm X y∈Y X s∈S αiys = 1 for all i and αiys ≥0. (6b) Here the entries of the kernel matrix K are given by Kiys,jy′s′ = Γ(yi, y)Γ(yj, y′) ⟨φ(s(xi), y) −φ(s(xi), yi), φ(s′(xj), y′) −φ(s′(xj), yj)⟩ (7) This can be expanded into four kernel functions by using Eq. (4). Moreover, the connection between the dual coefficients αiys and f is given by f(x′, y′) = m X i=1 X y∈Y X s∈S αiys [k((s(xi), y), (x′, y′)) −k((s(xi), yi), (x′, y′))] . (8) There are many strategies for attempting to minimize this regularized loss, either in the primal formulation or the dual, using either batch or online algorithms. In fact, a number of previous heuristics for dealing with invariances can be viewed as heuristics for approximately minimizing an approximation to an invariant loss similar to l. For this reason we believe a discussion of optimization is valuable before introducing specific applications of the invariance loss. Whenever the are an unlimited combination of valid transformations and targets (i.e. the domain S × Y is infinite), the optimization above is a semi-infinite program, hence exact minimization of R[f] or of its dual are essentially impossible. However, even is such cases it is possible to find approximate solutions efficiently by means of column generation. In the following we describe two algorithms exploiting this technique, which are valid for both infinite and finite programs. One based on a batch scenario, inspired by SVMStruct [16], and one based on an online setting, inspired by BMRM/Pegasos [15, 12]. 3.1 A Variant of SVMStruct The work of [16, 10] on SVMStruct-like optimization methods can be used directly to solve regularized risk minimization problems. The basic idea is to compute gradients of l(xi, yi, f), either one observation at a time, or for the entire set of observations simultaneously and to perform updates in the dual space. While bundle methods work directly with gradients, solvers of the SVMStruct type are commonly formulated in terms of column generation on individual observations. We give an instance of SVMStruct for invariances in Algorithm 1. The basic idea is that instead of checking the constraints arising from the loss functions only for y we check them for (y, s), that is, an invariance in combination with a corresponding label which violates the margin most. If we view the tuple (s, y) as a “label” it is straightforward to see that the convergence results of [16] apply. That is, this algorithm converges to ϵ precision in O(ϵ−2) time. In fact, one may show, by solving the difference equation in the convergence proof of [16] that the rate can be improved to O(ϵ−1). We omit technical details here. 4 Algorithm 1 SVMStruct for Invariances 1: Input: data X, labels Y , sample size m, tolerance ϵ 2: Initialize Si = ∅for all i, and w = 0. 3: repeat 4: for i = 1 to m do 5: f(x′, y′) = P i P (s,y)∈Si αiz [k((s(xi), y), (x′, y′)) −k((s(xi), yi), (x′, y′))] 6: (s∗, y∗) = argmaxs∈S,y∈Y Γ(yi, y)[f(s(xi), y) −f(s(xi), yi)] + ∆(yi, y, s) 7: ξi = max(0, max(s,y)∈Si Γ(yi, y)[f(s(xi), y) −f(s(xi), yi)] + ∆(yi, y, s)) 8: if Γ(yi, y∗)[f(s∗(xi), y∗) −f(s∗(xi), yi)] + ∆(yi, y∗, s∗) > ξi + ϵ then 9: Increase constraint set Si ←Si ∪{(s∗, y∗)} 10: Optimize (6) using only αiz where z ∈Si. 11: end if 12: end for 13: until S has not changed in this iteration 3.2 An Application of Pegasos Recently, Shalev-Shwartz et al. [12] proposed an online algorithm for learning optimization problems of type Eq. (5). Algorithm 2 is an adaptation of their method to learning with our convex invariance loss. In a nutshell, the algorithm performs stochastic gradient descent on the regularized version of the instantaneous loss while using a learning rate of 1 λt and while projecting the current weight vector back to a feasible region ∥f∥≤ q 2R[0] λ , should it exceed it. Algorithm 2 Pegasos for Invariances 1: Input: data X, labels Y , sample size m, iterations T, 2: Initialize f1 = 0 3: for t = 1 to T do 4: Pick (x, y) := (xt mod m, yt mod m) 5: Compute constraint violator (s∗, y∗) := argmax ¯s∈S,¯y∈Y Γ(y, ¯y) [f(¯s(x), ¯y) −f(¯s(x), y)] + ∆(y, ¯y, ¯s) 6: Update ft+1 = 1 −1 t ft + Γ(y,y∗) λt [k((s∗(x), y), (·, ·)) −k((s∗(x), y∗), (·, ·))] 7: if ∥ft+1∥> q 2R[0] λ then 8: Update ft+t ← q 2R[0] λ ft+1/ ∥ft+1∥ 9: end if 10: end for We can apply the convergence result from [12] directly to Algorithm 2. In this context note that the gradient with respect to l is bounded by twice the norm of Γ(y, y∗) [φ(s(x), y∗) −φ(s(x), y)], due to Eq. (3). We assume that the latter is given by R. We can apply [12, Lemma 1] immediately: Theorem 3 Denote by Rt[f] := l(xt mod m, yt mod m, f) + λ 2 ∥f∥2 the instantaneous risk at step t. In this case Algorithm 2 satisfies the following bound: 1 T T X t=1 Rt[ 1 T T X ¯t f¯t] ≤1 T T X t=1 Rt[ft] ≤ min ∥f∥≤ q 2R[0] λ 1 T T X t=1 Rt[f] + R2(1 + log T) 2λT . (9) In particular, if T is a multiple of m we obtain bounds for the regularized risk R[f]. 4 Related work and specific invariances While the previous sections gave a theoretical description of the loss, we now discuss a number of special cases which can be viewed as instances of a convex invariance loss function presented here. 5 Virtual Support Vectors (VSVs): The most straightforward approach to incorporate prior knowledge is by adding “virtual” (data) points generated from existing dataset. An extension of this approach is to generate virtual points only from the support vectors (SVs) obtained from training on the original dataset [5]. The advantage of this approach is that it results in far fewer SV than training on all virtual points. However, it is not clear which objective it optimizes. Our current loss based approach does optimize an objective, and generates the required support vectors in the process of the optimization. Second Order Cone Programming for Missing and Uncertain Data: In [2], the authors consider the case where the invariance is in the form of ellipsoids around the original point. This is shown to correspond to a second order cone program (SOCP). Instead of solving SOCP, we can solve an equivalent but unconstrained convex problem. Semidefinite Programming for Invariances: Graepel and Herbrich [8] introduce a method for learning when the invariances are polynomial trajectories. They show that the problem is equivalent to an semidefinite program (SDP). Their formulation is again an instance of our general loss based approach. Since SDPs are typically hard to solve for large problems, it it is likely that the optimization scheme we suggest will perform considerably faster than standard SDP solvers. Robust Estimation: Globerson and Roweis [7] address the case where invariances correspond to deletion of a subset of the features (i.e., setting their values to zero). This results in a quadratic program (QP) with a variables for each data point and feature in the training set. Solving such a large QP (e.g., 107 variables for the MNIST dataset) is not practical, and again the algorithm presented here can be much more efficient. In fact, in the next section we introduce a generalization of the invariance in [7] and show how it can be optimized efficiently. 5 Experiments Knowledge about invariances can be useful in a wide array of applications such as image recognition and document processing. Here we study two specific cases: handwritten digit recognition on the MNIST data, and spam filtering on the ECML06 dataset. Both examples are standard multiclass classification tasks, where ∆(y, y′, s) is taken to be the 0/1 loss. Also, we take the margin scale Γ(y, y′) to be identically one. We used SVMStruct and BMRM as the solvers for the experiments. 5.1 Handwritten Digits Recognition Humans can recognize handwritten digits even when they are altered in various ways. To test our invariant SVM (Invar-SVM) in this context, we used handwritten digits from the MNIST dataset [11] and modeled 20 invariance transformations: 1-pixel and 2-pixel shifts in 4 and 8 directions, rotations by ±10 degrees, scaling by ±0.15 unit, and shearing in vertical or horizontal axis by ±0.15 unit. To test the effect of learning with these invariances we used small training samples of 10, 20, . . . , 50 samples per digit. In this setting invariances are particularly important since they can compensate for the insufficient training data. We compared Invar-SVM to a related method where all possible transformations were applied in advance to each data point to create virtual samples. The virtual and original samples were used to train a multiclass SVM (VIR-SVM). Finally, we also trained a multiclass SVM that did not use any invariance information (STD-SVM). All of the aforementioned SVMs were trained using RBF kernel with well-chosen hyperparameters. For evaluation we used the standard MNIST test set. Results for the three methods are shown in Figure 1. It can be seen that Invar-SVM and VIR-SVM, which use invariances, significantly improve the recognition accuracy compared to STD-SVM. This comes at a certain cost of using more support vectors, but for Invar-SVM the number of support vectors is roughly half of that in the VIR-SVM. 5.2 SPAM Filtering The task of detecting spam emails is a challenging machine learning problem. One of the key difficulties with such data is that it can change over time as a result of attempts of spam authors to outwit spam filters [4]. In this context, the spam filter should be invariant to the ways in which a spam authors will change their style. One common mechanism of style alteration is the insertion of common words, and avoiding using specific keywords consistently over time. If documents are 6 Figure 1: Results for the MNIST handwritten digits recognition task, comparing SVM trained on original samples (STD-SVM), SVM trained on original and virtual samples (VIR-SVM), and our convex invariance-loss method (Invar-SVM). Left figure shows the classification error as a function of the number of original samples per digit used in training. Right figure shows the number of support vectors corresponding to the optimum of each method. represented using a bag-of-words, these two strategies correspond to incrementing the counts for some words, or setting it to zero [7]. Here we consider a somewhat more general invariance class (FSCALE) where word counts may be scaled by a maximum factor of u (e.g., 1.5) and a minimum factor of l (e.g., 0.5), and the maximum number of words subject to such perturbation is limited at K. Note that by setting l = 0 and u = 1 we specialize it to the feature deletion case (FDROP) in [7]. The invariances we consider are thus defined by s(x) = {x ◦α : α ∈[l, u]d, l ≤1 ≤u, #{i : αi ̸= 1} ≤K}, (10) where ◦denotes element-wise product, d is the number of features, and #{·} denotes the cardinality of the set. The set S is large so exhaustive enumeration is intractable. However, the search for optimal perturbation s∗is a linear program and can be computed efficiently by Algorithm 3 in O(d log d) time. We evaluated the performance of our invariance loss FSCALE and its special case FDROP as well as the standard hinge loss on ECML’06 Discovery Challenge Task A dataset.4 This dataset consists of two subsets, namely evaluation set (ecml06a-eval) and tuning set (ecml06a-tune). ecml06a-eval has 4000/7500 training/testing emails with dimensionality 206908, and ecml06a-tune has 4000/2500 training/testing emails with dimensionality 169620. We selected the best parameters for each methods on ecml06a-tune and used them for the training on ecml06a-eval. Results and parameter sets are shown in Table 1. We also performed McNemar’s Tests and rejected the null hypothesis that there is no difference between hinge and FSCALE/FDROP with p-value < 10−32. Algorithm 3 FSCALE loss 1: Input: datum x, label y, weight vector w ∈Rd, invariance-loss parameters (K, l, u) 2: Initialize i := 1, j := d 3: B := y ∗w ◦x 4: I := IndexSort(B), such that B(I) is in ascending order 5: for k = 1 to K do 6: if B[I[i]] ∗(1 −u) > B[I[j]] ∗(1 −l) then 7: x[I[i]] := x[I[i]] ∗u and i := i + 1 8: else 9: x[I[j]] := x[I[j]] ∗l and j := j −1 10: end if 11: end for 4http://www.ecmlpkdd2006.org/challenge.html 7 Loss Average Accuracy % Average AUC % Parameters (λ, K, l, u) Hinge 74.75 83.63 (0.005,-,-,-) FDROP 81.73 87.79 (0.1,14,0,1) FSCALE 83.71 89.14 (0.01,10,0.5,8) Table 1: SPAM filtering results on ecml06a-eval averaged over 3 testing subsets. λ is regularization constant, (K, l, u) are parameters for invariance-loss methods. The loss FSCALE and its special case FDROP statistically significantly outperform the standard hinge loss (Hinge). 6 Summary We have presented a general approach for learning using knowledge about invariances. Our cost function is essentially a worst case margin loss, and thus its optimization only relies on finding the worst case invariance for a given data point and model. This approach can allow us to solve invariance problems which previously required solving very large optimization problems (e.g. a QP in [7]). We thus expect it to extend the scope of learning with invariances both in terms of the invariances used and efficiency of optimization. Acknowledgements: We thank Carlos Guestin and Bob Williamson for fruitful discussions. Part of the work was done when CHT was visiting NEC Labs America. NICTA is funded through the Australian Government’s Backing Australia’s Ability initiative, in part through the ARC. This work was supported in part by the IST Programme of the European Community, under the PASCAL Network of Excellence, IST-2002-506778. References [1] Y. Abu-Mostafa. A method for learning from hints. In S. J. Hanson, J. D. Cowan, and C. L. Giles, editors, NIPS 5, 1992. [2] C. Bhattacharyya, K. S. Pannagadatta, and A. J. Smola. A second order cone programming formulation for classifying missing data. In L. K. Saul, Y. Weiss, and L. Bottou, editors, NIPS 17, 2005. [3] C. J. C. Burges. Geometry and invariance in kernel based methods. In B. Sch¨olkopf, C. J. C. Burges, and A. J. Smola, editors, Advances in Kernel Methods — Support Vector Learning, pages 89–116, Cambridge, MA, 1999. MIT Press. [4] N. Dalvi, P. Domingos, Mausam, S. Sanghai, and D. Verma. Adversarial classification. In KDD, 2004. [5] D. DeCoste and B. Sch¨olkopf. Training invariant support vector machines. Machine Learning, 46:161– 190, 2002. [6] M. Ferraro and T. M. Caelli. Lie transformation groups, integral transforms, and invariant pattern recognition. Spatial Vision, 8:33–44, 1994. [7] A. Globerson and S. Roweis. Nightmare at test time: Robust learning by feature deletion. In ICML, 2006. [8] T. Graepel and R. Herbrich. Invariant pattern recognition by semidefinite programming machines. In S. Thrun, L. Saul, and B. Sch¨olkopf, editors, NIPS 16, 2004. [9] G. E. Hinton. Learning translation invariant recognition in massively parallel networks. In Proceedings Conference on Parallel Architectures and Laguages Europe, pages 1–13. Springer, 1987. [10] T. Joachims. Training linear SVMs in linear time. In KDD, 2006. [11] Y. LeCun, L. D. Jackel, L. Bottou, A. Brunot, C. Cortes, J. S. Denker, H. Drucker, I. Guyon, U. A. M¨uller, E. S¨ackinger, P. Simard, and V. Vapnik. Comparison of learning algorithms for handwritten digit recognition. In F. Fogelman-Souli´e and P. Gallinari, editors, ICANN, 1995. [12] S. Shalev-Shwartz, Y. Singer, and N. Srebro. Pegasos: Primal estimated sub-gradient solver for SVM. In ICML, 2007. [13] P. Simard, Y. LeCun, and J. Denker. Efficient pattern recognition using a new transformation distance. In S. J. Hanson, J. D. Cowan, and C. L. Giles, editors, NIPS 5, 1993. [14] B. Taskar, C. Guestrin, and D. Koller. Max-margin Markov networks. In S. Thrun, L. 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The Noisy-Logical Distribution and its Application to Causal Inference Alan Yuille Department of Statistics University of California at Los Angeles Los Angeles, CA 90095 yuille@stat.ucla.edu Hongjing Lu Department of Psychology University of California at Los Angeles Los Angeles, CA 90095 hongjing@ucla.edu Abstract We describe a novel noisy-logical distribution for representing the distribution of a binary output variable conditioned on multiple binary input variables. The distribution is represented in terms of noisy-or’s and noisy-and-not’s of causal features which are conjunctions of the binary inputs. The standard noisy-or and noisy-andnot models, used in causal reasoning and artificial intelligence, are special cases of the noisy-logical distribution. We prove that the noisy-logical distribution is complete in the sense that it can represent all conditional distributions provided a sufficient number of causal factors are used. We illustrate the noisy-logical distribution by showing that it can account for new experimental findings on how humans perform causal reasoning in complex contexts. We speculate on the use of the noisy-logical distribution for causal reasoning and artificial intelligence. 1 Introduction The noisy-or and noisy-and-not conditional probability distributions are frequently studied in cognitive science for modeling causal reasoning [1], [2],[3] and are also used as probabilistic models for artificial intelligence [4]. It has been shown, for example, that human judgments of the power of causal cues in experiments involving two cues [1] can be interpreted in terms of maximum likelihood estimation and model selection using these types of models [3]. But the noisy-or and noisy-and-not distributions are limited in the sense that they can only represent a restricted set of all possible conditional distributions. This restriction is sometimes an advantage because there may not be sufficient data to determine the full conditional distribution. Nevertheless it would be better to have a representation that can expand to represent the full conditional distribution, if sufficient data is available, but can be reduced to simpler forms (e.g. standard noisy-or) if there is only limited data. This motivates us to define the noisy-logical distribution. This is defined in terms of noisy-or’s and noisy-and-not’s of causal features which are conjunctions of the basic input variables (inspired by the use of conjunctive features in [2] and the extensions in [5]). By restricting the choice of causal features we can obtain the standard noisy-or and noisy-and-not models. We prove that the noisy-logical distribution is complete in the sense that it can represent any conditional distribution provided we use all the causal features. Overall, it gives a distribution whose complexity can be adjusted by restricting the number of causal features. To illustrate the noisy-logical distribution we apply it to modeling some recent human experiments on causal reasoning in complex environments [6]. We show that noisy-logical distributions involving causal factors are able to account for human performance. By contrast, an alternative linear model gives predictions which are the opposite of the observed trends in human causal judgments. Section (2) presents the noisy-logical distribution for the case with two input causes (the case commonly studied in causal reasoning). In section (3) we specify the full noisy-logical distribution and 1 we prove its completeness in section (4). Section (5) illustrates the noisy-logical distribution by showing that it accounts for recent experimental findings in causal reasoning. 2 The Case with N = 2 causes In this section we study the simple case when the binary output effect E depends only on two binaryvalued causes C1, C2. This covers most of the work reported in the cognitive science literature [1],[3]. In this case, the probability distribution is specified by the four numbers P(E = 1|C1, C2), for C1 ∈{0, 1}, C2 ∈{0, 1}. To define the noisy-logical distribution over two variables P(E = 1|C1, C2), we introduce three concepts. Firstly, we define four binary-valued causal features Ψ0(.), Ψ1(.), Ψ2(.), Ψ3(.) which are functions of the input state ⃗C = (C1, C2). They are defined by Ψ0(⃗C) = 1, Ψ1(⃗C) = C1, Ψ2(⃗C) = C2, Ψ3(⃗C) = C1∧C2, where ∧denotes logical-and operation(i.e. C1∧C2 = 1 if C1 = C2 = 1 and C1 ∧C2 = 0 otherwise). Ψ3(⃗C) is the conjunction of C1 and C2. Secondly, we introduce binaryvalued hidden states E0, E1, E2, E3 which are caused by the corresponding features Ψ0, Ψ1, Ψ2, Ψ3. We define P(Ei = 1|Ψi; ωi) = ωiΨi with ωi ∈[0, 1], for i = 1, ..., 4 with ⃗ω = (ω1, ω2, ω3, ω4). Thirdly, we define the output effect E to be a logical combination of the states E0, E1, E2, E3 which we write in form δE,f(E0,E1,E2,E3), where f(., ., ., .) is a logic function which is formed by a combination of three logic operations AND, OR, NOT. This induces the noisy-logical distribution Pnl(E|⃗C; ⃗ω) = P E0,...,E3 δE,f(E0,E1,E2,E3) Q3 i=0 P(Ei|Ψi(⃗C); ωi). The noisy-logical distribution is characterized by the parameters ω0, ..., ω3 and the choice of the logic function f(., ., ., .). We can represent the distribution by a circuit diagram where the output E is a logical function of the hidden states E0, ..., E3 and each state is caused probabilistically by the corresponding causal features Ψ0, ..., Ψ3, as shown in Figure (1). Figure 1: Circuit diagram in the case with N = 2 causes. The noisy-logical distribution includes the commonly known distributions, noisy-or and noisy-andnot, as special cases. To obtain the noisy-or, we set E = E1 ∨E2 (i.e. E1 ∨E2 = 0 if E1 = E2 = 0 and E1 ∨E2 = 1 otherwise). A simple calculation shows that the noisy-logical distribution reduces to the noisy-or Pnor(E|C1, C2; ω1, ω2) [4], [1]: Pnl(E = 1|C1, C2; ω1, ω2) = X E1,E2 δ1,E1∨E2P(E1|Ψ1(⃗C); ω1)P(E2|Ψ2(⃗C); ω2) = ω1C1(1 −ω2C2) + (1 −ω1C1)ω2C2 + ω1ω2C1C2 = ω1C1 + ω2C2 −ω1ω2C1C2 = Pnor(E = 1|C1, C2; ω1, ω2)(1) To obtain the noisy-and-not, we set E = E1 ∧¬E2 (i.e. E1 ∧¬E2 = 1 if E1 = 1, E2 = 0 and E1 ∧¬E2 = 0 otherwise). The noisy-logical distribution reduces to the noisy-and-not Pn−and−not(E|C1, C2; ω1, ω2) [4],[?]: Pnl(E = 1|C1, C2; ω1, ω2) = X E1,E2 δ1,E1∧¬E2P(E1|Ψ1(⃗C); ω1)P(E2|Ψ2(⃗C); ω2) = ω1C1{1 −ω2C2} = Pn−and−not(E = 1|C1, C2; ω1, ω2) (2) 2 We claim that noisy-logical distributions of this form can represent any conditional distribution P(E|⃗C). The logical function f(E0, E1, E2, E3) will be expressed as a combination of logic operations AND-NOT, OR. The parameters of the distribution are given by ω0, ω1, ω2, ω3. The proof of this claim will be given for the general case in the next section. To get some insight, we consider the special case where we only know the values P(E|C1 = 1, C2 = 0) and P(E|C1 = 1, C2 = 1). This situation is studied in cognitive science where C1 is considered to be a background cause which always takes value 1, see [1] [3]. In this case, the only causal features are considered, Ψ1(⃗C) = C1 and Ψ2(⃗C) = C2. Result. The noisy-or and the noisy-and-not models, given by equations (1,2) are sufficient to fit any values of P(E = 1|1, 0) and P(E = 1|1, 1). (In this section we use P(E = 1|1, 0) to denote P(E = 1|C1 = 1, C2 = 0) and use P(E = 1|1, 1) to denote P(E = 1|C1 = 1, C2 = 1).) The noisy-or and noisy-and-not fit the cases when P(E = 1|1, 1) ≥P(E = 1|1, 0) and P(E = 1|1, 1) ≤P(E = 1|1, 0) respectively. In Cheng’s terminology [1] C2 is respectively a generative or preventative cause). Proof. We can fit both the noisy-or and noisy-and-not models to P(E|1, 0) by setting ω1 = P(E = 1|1, 0), so it remains to fit the models to P(E|1, 1). There are three cases to consider: (i) P(E = 1|1, 1) > P(E = 1|1, 0), (ii) P(E = 1|1, 1) < P(E = 1|1, 0), and (iii) P(E = 1|1, 1) = P(E = 1|1, 0). It follows directly from equations (1,2) that Pnor(E = 1|1, 1) ≥Pnor(E = 1|1, 0) and Pn−and−not(E = 1|1, 1) ≤Pn−and−not(E = 1|1, 0) with equality only if P(E = 1|1, 1) = P(E = 1|1, 0). Hence we must fit a noisy-or and a noisy-and-not model to cases (i) and (ii) respectively. For case (i), this requires solving P(E = 1|1, 1) = ω1 + ω2 −ω1ω2 to obtain ω2 = {P(E = 1|1, 1) −P(E = 1|1, 0)}/{1 −P(E = 1|1, 0)} (note that the condition P(E = 1|1, 1) > P(E = 1|1, 0) ensures that ω2 ∈[0, 1]). For case (ii), we must solve P(E = 1|1, 1) = ω1 −ω1ω2 which gives ω2 = {P(E = 1|1, 0) −P(E = 1|1, 1)}/P(E = 1|1, 0) (the condition P(E = 1|1, 1) < P(E = 1|1, 0) ensures that ω2 ∈[0, 1]). For case (iii), we can fit either model by setting ω2 = 0. 3 The Noisy-Logical Distribution for N causes We next consider representing probability distributions of form P(E|⃗C), where E ∈{0, 1} and ⃗C = (C1, ..., CN) where Ci ∈{0, 1}, ∀i = 1, .., N. These distributions can be characterized by the values of P(E = 1|⃗C) for all possible 2N values of ⃗C. We define the set of 2N binary-valued causal features {Ψi(⃗C) : i = 0, ..., 2N −1}. These features are ordered so that Ψ0(⃗C) = 1, Ψi(⃗C) = Ci : i = 1, .., N, ΨN+1(⃗C) = C1 ∧C2 is the conjunction of C1 and C2, and so on. The feature Ψ(⃗C) = Ca ∧Cb ∧... ∧Cg will take value 1 if Ca = Cb = ... = Cg = 1 and value 0 otherwise. We define binary variables {Ei : i = 0, ..., 2N −1} which are related to the causal features {Ψi : i = 0, ..., 2N −1} by distributions P(Ei = 1|Ψi; ωi) = ωiΨi, specified by parameters {ωi : i = 0, ..., 2N −1}. Then we define the output variable E to be a logical (i.e. deterministic) function of the {Ei : i = 0, ..., 2N −1}. This can be thought of as a circuit diagram. In particular, we define E = f(E0, ..., E2N−1) = (((((E1 ⊗E2) ⊗E3) ⊗E4....) where E1 ⊗E2 can be E1 ∨E2 or E1 ∧¬E2 (where ¬E means logical negation). This gives the general noisy-logical distribution, as shown in Figure (2). P(E = 1|⃗C; ⃗ω) = X ⃗E δE,f(E0,...,E2N −1) 2N−1 Y i=0 P(Ei = 1|Ψi; ωi). (3) 4 The Completeness Result This section proves that the noisy-logical distribution is capable of representing any conditional distribution. This is the main theoretical result of this paper. 3 Figure 2: Circuit diagram in the case with N causes. All conditional distributions can be represented in this form if we use all possible 2N causal features Ψ, choose the correct parameters ω, and select the correct logical combinations ⊗. Result We can represent any conditional distribution P(E|⃗C) defined on binary variables in terms of a noisy logical distribution given by equation (3). Proof. The proof is constructive. We show that any distribution P(E|⃗C) can be expressed as a noisy-logical distribution. We order the states ⃗C0, ..., ⃗C2N−1. This ordering must obey Ψi(⃗Ci) = 1 and Ψi(⃗Cj) = 0, ∀j < i. This ordering can be obtained by setting ⃗C0 = (0, ..., 0), then selecting the terms with a single conjunction (i.e. only one Ci is non-zero), then those with two conjunctions (i.e. two Ci’s are non-zero), then with three conjunctions, and so on. The strategy is to use induction to build a noisy-logical distribution which agrees with P(E|⃗C) for all values of ⃗C. We loop over the states and incrementally construct the logical function f(E0, ..., E2N−1) and estimate the parameters ω0, ..., ω2N−1. It is convenient to recursively define a variable Ei+1 = Ei ⊗Ei, so that f(E0, ..., E2N−1) = E2N−1. We start the induction using feature Ψ0(⃗C) = 1. Set E0 = E0 and ω0 = P(E|0, ..., 0). Then P(E0|⃗C0; ω0) = P(E|⃗C0), so the noisy-logical distribution fits the data for input ⃗C0. Now proceed by induction to determine EM+1 and ωM+1, assuming that we have determined EM and ω0, ..., ωM such that P(EM = 1|⃗Ci; ω0, ..., ωM) = P(E = 1|⃗Ci), for i = 0, ..., M. There are three cases to consider which are analogous to the cases considered in the section with two causes. Case 1. If P(E = 1|⃗CM+1) > P(EM = 1|⃗CM+1; ω0, ..., ωM) we need ΨM+1(⃗C) to be a generative feature. Set EM+1 = EM ∨EM+1 with P(EM+1 = 1|ΨM+1; ωM+1) = ωM+1ΨM+1. Then we obtain: P(EM+1 = 1|⃗CM+1; ω0, ., ωM+1) = P(EM = 1|⃗CM+1; ω0, ., ωM)+P(EM+1|ΨM+1(⃗C); ωM+1) −P(EM = 1|⃗CM+1; ω0, ., ωM)P(EM+1 = 1|ΨM+1(⃗C); ωM+1) = P(EM = 1|⃗CM+1; ω0, ., ωM)+ωM+1ΨM+1(⃗C)−P(EM = 1|⃗CM+1; ω0, ., ωM)ωM+1ΨM+1(⃗C) In particular, we see that P(EM+1 = 1|⃗Ci; ω0, ..., ωM+1) = P(EM = 1|⃗Ci; ω0, ..., ωM) = P(E = 1|⃗Ci) for i < M + 1 (using ΨM+1(⃗Ci) = 0, ∀i < M + 1). To determine the value of ωM+1, we must solve P(E = 1|⃗CM+1) = P(EM = 1|⃗CM+1; ω0, ..., ωM) + ωM+1 −P(EM = 1|⃗CM+1; ω0, ..., ωM)ωM+1 (using ΨM+1(⃗CM+1) = 1). This gives ωM+1 = {P(E = 1|⃗CM+1) − P(EM = 1|⃗CM+1; ω0, ..., ωM)}/{1 −P(EM = 1|⃗CM+1; ω0, ..., ωM+1)} (the conditions ensure that ωM+1 ∈[0, 1]). Case 2. If P(E = 1|⃗CM+1) < P(EM = 1|⃗CM+1; ω0, ..., ωM) we need ΨM+1(⃗C) to be a preventative feature. Set EM+1 = EM ∧¬EM+1 with P(EM+1 = 1|ΨM+1; ωM+1) = ωM+1ΨM+1. Then we obtain: P(EM+1 = 1|⃗CM+1; ω0, ..., ωM+1) = P(EM = 1|⃗CM+1; ω0, ..., ωM){1 −ωM+1ΨM+1(⃗C)}. (4) 4 As for the first case, P(EM+1 = 1|⃗Ci; ω0, ..., ωM+1) = P(EM = 1|⃗Ci; ω0, ..., ωM) = P(E = 1|⃗Ci) for i < M + 1 (because ΨM+1(⃗Ci) = 0, ∀i < M + 1). To determine the value of ωM+1 we must solve P(E = 1|⃗CM+1) = P(EM = 1|⃗CM+1; ω0, ..., ωM){1 −ωM+1} (using ΨM+1(⃗CM+1) = 1). This gives ωM+1 = {P(EM = 1|⃗CM+1; ω0, ..., ωM) −P(E = 1|⃗CM+1)}/P(EM = 1|⃗CM+1; ω0, ..., ωM) (the conditions ensure that ωM+1 ∈[0, 1]). Case 3. If P(E = 1|⃗CM+1) = P(EM = 1|⃗CM+1; ω0, ..., ωM), then we do nothing. 5 Cognitive Science Human Experiments We illustrate noisy-logical distributions by applying them to model two recent cognitive science experiments by Liljeholm and Cheng which involve causal reasoning in complex environments [6]. In these experiments, the participants are asked questions about the causal structure of the data. But the participants are not given enough data to determine the full distribution (i.e. not enough to determine the causal structure with certainty). Instead the experimental design forces them to choose between two different causal structures. We formulate this as a model selection problem [3]. Formally, we specify distributions P(D|⃗ω, Graph) for generating the data D from a causal model specified by Graph and parameterized by ⃗ω. These distributions will be of simple noisy-logical form. We set the prior distributions P(⃗ω|Graph) on the parameter values to be the uniform distribution. The evidence for the causal model is given by: P(D|Graph) = Z d⃗ωP(D|⃗ω, Graph)P(⃗ω|Graph). (5) We then evaluate the log-likelihood ratio log P (D|Graph1) P (D|Graph2) between two causal models Graph1 Graph2, called the causal support [3] and use this to predict the performance of the participants. This gives good fits to the experimental results. As an alternative theoretical model, we consider the possibility that the participants use the same causal structures, specified by Graph1 and Graph2, but use a linear model to combine cues. Formally, this corresponds to a model P(E = 1|C1, ..., CN) = ω1C1 + ... + ωNCN (with ωi ≥0, ∀i = 1, ..., N and ω1 + ... + ωN ≤1). This model corresponds [1, 3] to the classic Rescorla-Wagner learning model [8]. It cannot be expressed in simple noisy-logical form. Our simulations show that this model does not account for human participant performance . We note that previous attempts to model experiments with multiple causes and conjunctions by Novick and Cheng [2] can be interpreted as performing maximum likelihood estimation of the parameters of noisy-logical distributions (their paper helped inspire our work). Those experiments, however, were simpler than those described here and model selection was not used. The extensive literatures on two cases [1, 3] can also be interpreted in terms of noisy-logical models. 5.1 Experiment I: Multiple Causes In Experiment 1 of [6], the cover story involves a set of allergy patients who either did or did not have a headache, and either had or had not received allergy medicines A and B. The experimental participants were informed that two independent studies had been conducted in different labs using different patient groups. In the first study, patients were administered medicine A, whereas in the second study patients were administered both medicines A and B. A simultaneous presentation format [7] was used to display the specific contingency conditions used in both studies to the experimental subjects. The participants were then asked whether medicine B caused the headache. We represent this experiment as follows using binary-valued variables E, B1, B2, C1, C2. The variable E indicates whether a headache has occurred (E = 1) or not (E = 0). B1 = 1 and B2 = 1 notate background causes for the two studies (which are always present). C1 and C2 indicate whether medicine A and B are present respectively (e.g. C1 = 1 if A is present, C1 = 0 otherwise). The data D shown to the subjects can be expressed as D = (D1, D2) where D1 is the contingency table Pd(E = 1|B1 = 1, C1 = 0, C2 = 0), Pd(E = 1|B1 = 1, C1 = 1, C2 = 0) for the first study 5 and D2 is the contingency table Pd(E = 1|B2 = 1, C1 = 0, C2 = 0), Pd(E = 1|B2 = 1, C1 = 1, C2 = 1) for the second study. The experimental design forces the participants to choose between the two causal models shown on the left of figure (3). These causal models differ by whether C2 (i.e. medicine B) can have an effect or not. We set P(D|⃗ω, Graph) = P(D1|⃗ω1, Graph)P(D2|⃗ω2, Graph), where Di = {(Eµ, ⃗Cµ i )} (for i = 1, 2) is the contingency data. We express these distributions in form P(Di|⃗ωi, Graph) = Q µ Pi(Eµ|⃗Cµ i , ⃗ωµ i , Graph). For Graph1, P1(.) and P2(.) are P(E|B1, C1, ωB1, ωC1) and P(E|B2, C1, ωB2, ωC1). For Graph2, P1(.) and P2(.) are P(E|B1, C1, ωB1, ωC1) and P(E|B2, C1, C2, ωB2, ωC1, ωC2). All these P(E|.) are noisy-or distributions. For Experiment 1 there are two conditions [6], see table (1). In the first power-constant condition [6], the data is consistent with the causal structure for Graph1 (i.e. C2 has no effect) using noisy-or distributions. In the second ∆P-constant condition [6], the data is consistent with the causal structure for Graph1 but with noisy-or replaced by the linear distributions (e.g. P(E = 1|C1, ..., Cn) = ω1C1 + ... + ωnCn)). Table 1: Experimental conditions (1) and (2) for Experiment 1 (1) Pd(E = 1|B1 = 1, C1 = 0, C2 = 0), Pd(E = 1|B1 = 1, C1 = 1, C2 = 0) 16/24, 22/24 Pd(E = 1|B2 = 1, C1 = 0, C2 = 0), Pd(E = 1|B2 = 1, C1 = 1, C2 = 1) 0/24,18/24 (2) Pd(E = 1|B1 = 1, C1 = 0, C2 = 0), Pd(E = 1|B1 = 1, C1 = 1, C2 = 0) 0/24, 6/24 Pd(E = 1|B2 = 1, C1 = 0, C2 = 0), Pd(E = 1|B2 = 1, C1 = 1, C2 = 1) 16/24,22/24 5.2 Experiment I: Results We compare Liljeholm and Cheng’s experimental results with our theoretical simulations. These comparisons are shown on the right-hand-side of figure (3). The left panel shows the proportion of participants who decide that medicine B causes a headache for the two conditions. The right panel shows the predictions of our model (labeled ”noisy-logical”) together with predictions of a model that replaces the noisy-logical distributions by a linear model (labeled ”linear”). The simulations show that the noisy-logical model correctly predicts that participants (on average) judge that medicine B has no effect in the first experimental condition, but B does have an effect in the second condition. By contrast, the linear model makes the opposite (wrong) prediction. In summary, model selection comparing two noisy-logical models gives a good prediction of participant performance. Figure 3: Causal model and results for Experiment I. Left panel: two alternative causal models for the two studies. Right panel: the experimental results (proportion of patients who think medicine B causes headaches)) for the Power-constant and ∆P-constant conditions [6]. Far right, the causal support for the noisy-logic and linear models. 6 5.3 Experiment II: Causal Interaction Liljeholm and Cheng [6] also investigated causal interactions. The experimental design was identical to that used in Experiment 1, except that participants were presented with three studies in which only one medicine (A) was tested. Participants were asked to judge whether medicine A interacts with background causes that vary across the three studies. We define the background causes as B1,B2,B3 for the three studies, and C1 for medicine A. This experiment was also run under two different conditions, see table (2). The first power-constant condition [6] was consistent with a noisy-logical model, but the second power-varying condition [6] was not. Table 2: Experimental conditions (1) and (2) for Experiment 2 (1) P(E = 1|B1 = 1, C1 = 0), P(E = 1|B1 = 1, C1 = 1) 16/24, 22/24 P(E = 1|B2 = 1, C1 = 0), P(E = 1|B2 = 1, C1 = 1) 8/24,20/24 P(E = 1|B3 = 1, C1 = 0), P(E = 1|B3 = 1, C1 = 1) 0/24,18/24 (2) P(E = 1|B1 = 1, C1 = 0), P(E = 1|B1 = 1, C1 = 1) 0/24, 6/24 P(E = 1|B2 = 1, C1 = 0), P(E = 1|B2 = 1, C1 = 1) 0/24,12/24 P(E = 1|B3 = 1, C1 = 0), P(E = 1|B3 = 1, C1 = 1) 0/24,18/24 The experimental design caused participants to choose between two causal models shown on the left panel of figure (4). The probability of generating the data is given by P(D|⃗ω, Graph) = P(D1|⃗ω1, Graph)P(D2|⃗ω2, Graph)P(D3|⃗ω3, Graph). For Graph1, the P(Di|.) are noisyor distributions P(E|B1, C1, ωB1, ωC1), P(E|B2, C1, ωB2, ωC1), P(E|B3, C1, ωB3, ωC1). For Graph2, the P(Di|.) are P(E|B1, C1, ωB1, ωC1), P(E|B2, C1, B2C1, ωB2, ωC1, ωB2C1) and P(E|B3, C1, B3C1, ωB3, ωC1, ωB3C1). All the distributions are noisy-or on the unary causal features (e.g. B, C1), but the nature of the conjunctive cause B ∧C1 is unknown (i.e. not specified by the experimental design). Hence our theory considers the possibilities that it is a noisy-or (e.g. can produce headaches) or noisy-and-not (e.g. can prevent headaches), see graph 2 of Figure (4). 5.4 Results of Experiment II Figure (4) shows human and model performance for the two experimental conditions. Our noisylogical model is in agreement with human performance – i.e. there is no interaction between causes in the power-constant condition, but there is interaction in the power-varying condition. By contrast, the linear model predicts interaction in both conditions and hence fails to model human performance. Figure 4: Causal model and results for Experiment II. Left panel: two alternative causal models (one involving conjunctions) for the three studies . Right panel: the proportion of participants who think that there is an interaction (conjunction) between medicine A and the background for the powerconstant and power-varying conditions [6]. Far right, the causal support for the noisy-logical and linear models. 7 6 Summary The noisy-logical distribution gives a new way to represent conditional probability distributions defined over binary variables. The complexity of the distribution can be adjusted by restricting the set of causal factors. If all the causal factors are allowed, then the distribution can represent any conditional distribution. But by restricting the set of causal factors we can obtain standard distributions such as the noisy-or and noisy-and-not. We illustrated the noisy-logical distribution by modeling experimental findings on causal reasoning. Our results showed that this distribution fitted the experimental data and, in particular, accounted for the major trends (unlike the linear model). This is consistent with the success of noisy-or and noisyand-not models for accounting for experiments involving two causes [1], [2],[3]. This suggests that humans may make use of noisy-logical representations for causal reasoning. One attraction of the noisy-logical representation is that it helps clarify the relationship between logic and probabilities. Standard logical relationships between causes and effects arise in the limit as the ωi take values 0 or 1. We can, for example, bias the data towards a logical form by using a prior on the ⃗ω. This may be useful, for example, when modeling human cognition – evidence suggests that humans first learn logical relationships and, only later, move to probabilities. In summary, the noisy-logical distribution is a novel way to represent conditional probability distributions defined on binary variables. We hope this class of distributions will be useful for modeling cognitive phenomena and for applications to artificial intelligence. Acknowledgements We thank Mimi Liljeholm, Patricia Cheng, Adnan Darwiche, Keith Holyoak, Iasonas Kokkinos, and YingNian Wu for helpful discussions. Mimi and Patricia kindly gave us access to their experimental data. We acknowledge funding support from the W.M. Keck foundation and from NSF 0413214. References [1] P. W. Cheng. From covariation to causation: A causal power theory. Psychological Review, 104, 367405. 1997. [2] L.R. Novick and P.W. Cheng. Assessing interactive causal influence. Psychological Review, 111, 455-485. 2004. [3] T. L. Griffiths, and J. B. Tenenbaum. Structure and strength in causal induction. Cognitive Psychology, 51, 334-384, 2005. [4] J. Pearl, Probabilistic Reasoning in Intelligent Systems. Morgan-Kauffman, 1988. [5] C.N. Glymour. The Mind’s Arrow: Bayes Nets and Graphical Causal Models in Psychology. MIT Press. 2001. [6] M. Liljeholm and P. W. Cheng. When is a Cause the ”Same”? Coherent Generalization across Contexts. Psychological Science, in press. 2007. [7] M. J. Buehner, P. W. Cheng, and D. Clifford. From covariation to causation: A test of the assumption of causal power. Journal of Experimental Psychology: Learning, Memory, and Cognition, 29, 1119-1140, 2003. [8] R. A. Rescorla, and A. R. Wagner. A theory of Pavlovian conditioning: Variations in the effectiveness of reinforcement and nonreinforcement. In A. H. Black and W. F. Prokasy (Eds.), Classical conditioning II: Current theory and research (pp. 64-99). New York: Appleton-Century Crofts. 1972. 8
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Compressed Regression Shuheng Zhou∗ John Lafferty∗† Larry Wasserman‡† ∗Computer Science Department ‡Department of Statistics †Machine Learning Department Carnegie Mellon University Pittsburgh, PA 15213 Abstract Recent research has studied the role of sparsity in high dimensional regression and signal reconstruction, establishing theoretical limits for recovering sparse models from sparse data. In this paper we study a variant of this problem where the original n input variables are compressed by a random linear transformation to m ≪n examples in p dimensions, and establish conditions under which a sparse linear model can be successfully recovered from the compressed data. A primary motivation for this compression procedure is to anonymize the data and preserve privacy by revealing little information about the original data. We characterize the number of random projections that are required for ℓ1-regularized compressed regression to identify the nonzero coefficients in the true model with probability approaching one, a property called “sparsistence.” In addition, we show that ℓ1-regularized compressed regression asymptotically predicts as well as an oracle linear model, a property called “persistence.” Finally, we characterize the privacy properties of the compression procedure in information-theoretic terms, establishing upper bounds on the rate of information communicated between the compressed and uncompressed data that decay to zero. 1 Introduction Two issues facing the use of statistical learning methods in applications are scale and privacy. Scale is an issue in storing, manipulating and analyzing extremely large, high dimensional data. Privacy is, increasingly, a concern whenever large amounts of confidential data are manipulated within an organization. It is often important to allow researchers to analyze data without compromising the privacy of customers or leaking confidential information outside the organization. In this paper we show that sparse regression for high dimensional data can be carried out directly on a compressed form of the data, in a manner that can be shown to guard privacy in an information theoretic sense. The approach we develop here compresses the data by a random linear or affine transformation, reducing the number of data records exponentially, while preserving the number of original input variables. These compressed data can then be made available for statistical analyses; we focus on the problem of sparse linear regression for high dimensional data. Informally, our theory ensures that the relevant predictors can be learned from the compressed data as well as they could be from the original uncompressed data. Moreover, the actual predictions based on new examples are as accurate as they would be had the original data been made available. However, the original data are not recoverable from the compressed data, and the compressed data effectively reveal no more information than would be revealed by a completely new sample. At the same time, the inference algorithms run faster and require fewer resources than the much larger uncompressed data would require. The original data need not be stored; they can be transformed “on the fly” as they come in. 1 In more detail, the data are represented as a n × p matrix X. Each of the p columns is an attribute, and each of the n rows is the vector of attributes for an individual record. The data are compressed by a random linear transformation X 7→eX ≡ 8X, where 8 is a random m × n matrix with m ≪n. It is also natural to consider a random affine transformation X 7→eX ≡8X +1, where 1 is a random m × p matrix. Such transformations have been called “matrix masking” in the privacy literature [6]. The entries of 8 and 1 are taken to be independent Gaussian random variables, but other distributions are possible. We think of eX as “public,” while 8 and 1 are private and only needed at the time of compression. However, even with 1 = 0 and 8 known, recovering X from eX requires solving a highly under-determined linear system and comes with information theoretic privacy guarantees, as we demonstrate. In standard regression, a response variable Y = Xβ + ϵ ∈Rn is associated with the input variables, where ϵi are independent, mean zero additive noise variables. In compressed regression, we assume that the response is also compressed, resulting in the transformed response eY ∈Rm given by Y 7→ eY ≡8Y = 8Xβ + 8ϵ = eXβ + eϵ. Note that under compression, eϵi, i ∈{1, . . . , m}, in the transformed noise eϵ = 8ϵ are no longer independent. In the sparse setting, the parameter β ∈Rp is sparse, with a relatively small number s = ∥β∥0 of nonzero coefficients in β. The method we focus on is ℓ1-regularized least squares, also known as the lasso [17]. We study the ability of the compressed lasso estimator to identify the correct sparse set of relevant variables and to predict well. We omit details and technical assumptions in the following theorems for clarity. Our first result shows that the lasso is sparsistent under compression, meaning that the correct sparse set of relevant variables is identified asymptotically. Sparsistence (Theorem 3.3): If the number of compressed examples m satisfies C1s2 log nps ≤ m ≤√C2n/ log n, and the regularization parameter λm satisfies λm →0 and mλ2 m/ log p → ∞, then the compressed lasso estimator eβm = arg minβ 1 2m ∥eY −eXβ∥2 2 + λm∥β∥1 is sparsistent: P supp(eβm) = supp(β) →1 as m →∞, where supp(β) = {j : j ̸= 0}. Our second result shows that the lasso is persistent under compression. Roughly speaking, persistence [10] means that the procedure predicts well, as measured by the predictive risk R(β) = E Y −βT X 2, where X ∈Rp is a new input vector and Y is the associated response. Persistence is a weaker condition than sparsistency, and in particular does not assume that the true model is linear. Persistence (Theorem 4.1): Given a sequence of sets of estimators Bn,m ⊂Rp such that Bn,m = {β : ∥β∥1 ≤Ln,m} with log2(np) ≤m ≤n, the sequence of compressed lasso estimators eβn,m = arg min∥β∥1≤Ln,m ∥eY −eXβ∥2 2 is persistent with the predictive risk R(β) = E Y −βT X 2 over uncompressed data with respect to Bn,m, meaning that R(eβn,m) −inf∥β∥1≤Ln,m R(β) P −→ 0, as n →∞, in case Ln,m = o (m/ log(np))1/4. Our third result analyzes the privacy properties of compressed regression. We evaluate privacy in information theoretic terms by bounding the average mutual information I (eX; X)/np per matrix entry in the original data matrix X, which can be viewed as a communication rate. Bounding this mutual information is intimately connected with the problem of computing the channel capacity of certain multiple-antenna wireless communication systems [13]. Information Resistence (Propositions 5.1 and 5.2): The rate at which information about X is revealed by the compressed data eX satisfies rn,m = sup I (X; eX) np = O m n →0, where the supremum is over distributions on the original data X. As summarized by these results, compressed regression is a practical procedure for sparse learning in high dimensional data that has provably good properties. Connections with related literature are briefly reviewed in Section 2. Analyses of sparsistence, persistence and privacy properties appear in Section 3–5. Simulations for sparsistence and persistence of the compressed lasso are presented in Section 6. The proofs are included in the full version of the paper, available at http://arxiv. org/abs/0706.0534. 2 2 Background and Related Work In this section we briefly review related work in high dimensional statistical inference, compressed sensing, and privacy, to place our work in context. Sparse Regression. An estimator that has received much attention in the recent literature is the lasso bβn [17], defined as bβn = arg min 1 2n ∥Y −Xβ∥2 2 +λn∥β∥1, where λn is a regularization parameter. In [14] it was shown that the lasso is consistent in the high dimensional setting under certain assumptions. Sparsistency proofs for high dimensional problems have appeared recently in [20] and [19]. The results and method of analysis of Wainwright [19], where X comes from a Gaussian ensemble and ϵi is i.i.d. Gaussian, are particularly relevant to the current paper. We describe this Gaussian Ensemble result, and compare our results to it in Sections 3, 6.Given that under compression, the noise eϵ = 8ϵ is not i.i.d, one cannot simply apply this result to the compressed case. Persistence for the lasso was first defined and studied by Greenshtein and Ritov in [10]; we review their result in Section 4. Compressed Sensing. Compressed regression has close connections to, and draws motivation from compressed sensing [4, 2]. However, in a sense, our motivation is the opposite of compressed sensing. While compressed sensing of X allows a sparse X to be reconstructed from a small number of random measurements, our goal is to reconstruct a sparse function of X. Indeed, from the point of view of privacy, approximately reconstructing X, which compressed sensing shows is possible if X is sparse, should be viewed as undesirable; we return to this point in Section ??. Several authors have considered variations on compressed sensing for statistical signal processing tasks [5, 11]. They focus on certain hypothesis testing problems under sparse random measurements, and a generalization to classification of a signal into two or more classes. Here one observes y = 8x, where y ∈Rm, x ∈Rn and 8 is a known random measurement matrix. The problem is to select between the hypotheses e Hi : y = 8(si + ϵ). The proofs use concentration properties of random projection, which underlie the celebrated Johnson-Lindenstrauss lemma. The compressed regression problem we introduce can be considered as a more challenging statistical inference task, where the problem is to select from an exponentially large set of linear models, each with a certain set of relevant variables with unknown parameters, or to predict as well as the best linear model in some class. Privacy. Research on privacy in statistical data analysis has a long history, going back at least to [3]. We refer to [6] for discussion and further pointers into this literature; recent work includes [16]. The work of [12] is closely related to our work at a high level, in that it considers low rank random linear transformations of either the row space or column space of the data X. The authors note the JohnsonLindenstrauss lemma, and argue heuristically that data mining procedures that exploit correlations or pairwise distances in the data are just as effective under random projection. The privacy analysis is restricted to observing that recovering X from eX requires solving an under-determined linear system. We are not aware of previous work that analyzes the asymptotic properties of a statistical estimator under random projection in the high dimensional setting, giving information-theoretic guarantees, although an information-theoretic quantification of privacy was proposed in [1]. We cast privacy in terms of the rate of information communicated about X through eX, maximizing over all distributions on X, and identify this with the problem of bounding the Shannon capacity of a multi-antenna wireless channel, as modeled in [13]. Finally, it is important to mention the active area of cryptographic approaches to privacy from the theoretical computer science community, for instance [9, 7]; however, this line of work is quite different from our approach. 3 Compressed Regression is Sparsistent In the standard setting, X is a n × p matrix, Y = Xβ + ϵ is a vector of noisy observations under a linear model, and p is considered to be a constant. In the high-dimensional setting we allow p to grow with n. The lasso refers to the following: (P1) min ∥Y −Xβ∥2 2 such that ∥β∥1 ≤L. In Lagrangian form, this becomes: (P2) min 1 2n ∥Y −Xβ∥2 2 + λn∥β∥1. For an appropriate choice of the regularization parameter λ = λ(Y, L), the solutions of these two problems coincide. In compressed regression we project each column X j ∈Rn of X to a subspace of m dimensions, using an m × n random projection matrix 8. Let eX = 8X be the compressed design matrix, and 3 let eY = 8Y be the compressed response. Thus, the transformed noise eϵ is no longer i.i.d.. The compressed lasso is the following optimization problem, for eY = 8Xβ + 8ϵ = 8eX +eϵ, with em being the set of optimal solutions: (a) (eP2) min 1 2m ∥eY −eXβ∥2 2 + λm∥β∥1, (b) em = arg min β∈Rp 1 2m ∥eY −eXβ∥2 2 + λm∥β∥1. (1) Although sparsistency is the primary goal in selecting the correct variables, our analysis establishes conditions for the stronger property of sign consistency: Definition 3.1. (Sign Consistency) A set of estimators n is sign consistent with the true β if P ∃bβn ∈n s.t. sgn(bβn) = sgn(β) →1 as n →∞, where sgn(·) is given by sgn(x) = 1, 0, and −1 for x >, =, or < 0 respectively. As a shorthand, denote the event that a sign consistent solution exists with E sgn(bβn) = sgn(β∗) := ∃bβ ∈n such that sgn(bβ) = sgn(β∗) . Clearly, if a set of estimators is sign consistent then it is sparsistent. All recent work establishing results on sparsity recovery assumes some form of incoherence condition on the data matrix X. To formulate such a condition, it is convenient to introduce an additional piece of notation. Let S = {j : β j ̸= 0} be the set of relevant variables and let Sc = {1, . . . , p} \ S be the set of irrelevant variables. Then XS and XSc denote the corresponding sets of columns of the matrix X. We will impose the following incoherence condition; related conditions are used by [18] in a deterministic setting. Let ∥A∥∞= maxi Pp j=1 |Ai j| denote the matrix ∞-norm. Definition 3.2. (S-Incoherence) Let X be an n × p matrix and let S ⊂{1, . . . , p} be nonempty. We say that X is S-incoherent in case
1 n X T Sc XS
∞+
1 n X T S XS −I|S|
∞≤1 −η, for some η ∈(0, 1]. (2) Although not explicitly required, we only apply this definition to X such that columns of X satisfy
X j
2 2 = 2(n), ∀j ∈{1, . . . , p}. We can now state our main result on sparsistency. Theorem 3.3. Suppose that, before compression, Y = Xβ∗+ ϵ, where each column of X is normalized to have ℓ2-norm n, and ε ∼N(0, σ 2In). Assume that X is S-incoherent, where S = supp(β∗), and define s = |S| and ρm = mini∈S |β∗ i |. We observe, after compression, eY = eXβ∗+eϵ, where eY = 8Y, eX = 8X, andeϵ = 8ϵ, where 8i j ∼N(0, 1 n ). Let eβm ∈em as in (1b). If 16C1s2 η2 + 4C2s η (ln p + 2 log n + log 2(s + 1)) ≤m ≤ r n 16 log n (3) with C1 = 4e √ 6π ≈2.5044 and C2 = √ 8e ≈7.6885, and λm →0 satisfies (a) mη2λ2 m log(p −s) →∞, and (b) 1 ρm (r log s m + λm
( 1 n X T S XS)−1
∞ ) →0. (4) Then the compressed lasso is sparsistent: P supp(eβm) = supp(β) →1 as m →∞. 4 Compressed Regression is Persistent Persistence (Greenshtein and Ritov [10]) is a weaker condition than sparsistency. In particular, the assumption that E(Y|X) = βT X is dropped. Roughly speaking, persistence implies that a procedure predicts well. We review the arguments in [10] first; we then adapt it to the compressed case. Uncompressed Persistence. Consider a new pair (X, Y) and suppose we want to predict Y from X. The predictive risk using predictor βT X is R(β) = E(Y −βT X)2. Note that this is a well-defined quantity even though we do not assume that E(Y|X) = βT X. It is convenient to rewrite the risk in the following way: define Q = (Y, X1, . . . , X p) and γ = (−1, β1, . . . , βp)T , then R(β) = γ T 6γ, where 6 = E(QQT ). (5) 4 Let Q = (Q† 1 Q† 2 · · · Q† n)T , where Q† i = (Yi, X1i, . . . , X pi)T ∼Q, ∀i = 1, . . . , n are i.i.d. random vectors and the training error is bRn(β) = 1 n n X i=1 (Yi −X T i β)2 = γ T b6nγ, where b6n = 1 n QT Q. (6) Given Bn = {β : ∥β∥1 ≤Ln} for Ln = o (n/ log n)1/4 , we define the oracle predictor β∗,n = arg min∥β∥1≤Ln R(β), and the uncompressed lasso estimator bβn = arg min∥β∥1≤Ln bRn(β). Assumption 1. Suppose that, for each j and k, E |Z|q ≤q!Mq−2s/2, for every q ≥2 and some constants M and s, where Z = Q j Qk −E(Q j Qk), where Q j, Qk denote elements of Q. Following arguments in [10], it can be shown that under Assumption 1 and given a sequence of sets of estimators Bn = {β : ∥β∥1 ≤Ln} for Ln = o (n/ log n)1/4 , the sequence of uncompressed lasso estimators bβn = arg minβ∈Bn bRn(β) is persistent, i.e., R(bβn) −R(β∗,n) P→0. Compressed Persistence. For the compressed case, again we want to predict (X, Y), but now the estimator bβn,m is based on the lasso from the compressed data of size mn. Let γ = (−1, β1, . . . , βp)T as before and we replace bRn with bRn,m(β) = γ T b6n,mγ, where b6n,m = 1 mn QT 8T 8Q. (7) Given compressed sample size mn, let Bn,m = {β : ∥β∥1 ≤Ln,m}, where Ln,m = o mn log(npn) 1/4 . We define the compressed oracle predictor β∗,n,m = arg minβ : ∥β∥1≤Ln,m R(β) and the compressed lasso estimator bβn,m = arg minβ : ∥β∥1≤Ln,m bRn,m(β). Theorem 4.1. Under Assumption 1, we further assume that there exists a constant M1 > 0 such that E(Q2 j) < M1, ∀j, where Q j denotes the jth element of Q. For any sequence Bn,m ⊂Rp with log2(npn) ≤mn ≤n, where Bn,m consists of all coefficient vectors β such that ∥β∥1 ≤Ln,m = o (mn/ log(npn))1/4 , the sequence of compressed lasso procedures bβn,m = arg minβ∈Bn,m bRn,m(β) is persistent: R(bβn,m) −R(β∗,n,m) P→0, when pn = O enc for c < 1/2. The main difference between the sequence of compressed lasso estimators and the original uncompressed sequence is that n and mn together define the sequence of estimators for the compressed data. Here mn is allowed to grow from (log2(np)) to n; hence for each fixed n, bβn,m , ∀mn such that log2(np) < mn ≤n defines a subsequence of estimators. In Section 6 we illustrate the compressed lasso persistency via simulations to compare the empirical risks with the oracle risks on such a subsequence for a fixed n. 5 Information Theoretic Analysis of Privacy Next we derive bounds on the rate at which the compressed data eX reveal information about the uncompressed data X. Our general approach is to consider the mapping X 7→8X + 1 as a noisy communication channel, where the channel is characterized by multiplicative noise 8 and additive noise 1. Since the number of symbols in X is np we normalize by this effective block length to define the information rate rn,m per symbol as rn,m = supp(X) I (X; eX) np . Thus, we seek bounds on the capacity of this channel. A privacy guarantee is given in terms of bounds on the rate rn,m →0 decaying to zero. Intuitively, if the mutual information satisfies I (X; eX) = H(X) −H(X | eX) ≈0, then the compressed data eX reveal, on average, no more information about the original data X than could be obtained from an independent sample. The underlying channel is equivalent to the multiple antenna model for wireless communication [13], where there are n transmitter and m receiver antennas in a Raleigh flat-fading environment. The propagation coefficients between pairs of transmitter and receiver antennas are modeled by the matrix entries 8i j; they remain constant for a coherence interval of p time periods. Computing the 5 channel capacity over multiple intervals requires optimization of the joint density of pn transmitted signals, the problem studied in [13]. Formally, the channel is modeled as Z = 8X + γ 1, where γ > 0, 1i j ∼N(0, 1), 8i j ∼N(0, 1/n) and 1 n Pn i=1 E[X2 i j] ≤P, where the latter is a power constraint. Theorem 5.1. Suppose that E[X2 j] ≤P and the compressed data are formed by Z = 8X + γ 1, where 8 is m×n with independent entries 8i j ∼N(0, 1/n) and 1 is m× p with independent entries 1i j ∼N(0, 1). Then the information rate rn,m satisfies rn,m = supp(X) I (X; Z) np ≤ m n log 1 + P γ 2 . This result is implicitly contained in [13]. When 1 = 0, or equivalently γ = 0, which is the case assumed in our sparsistence and persistence results, the above analysis yields the trivial bound rn,m ≤∞. We thus derive a separate bound for this case; however, the resulting asymptotic order of the information rate is the same. Theorem 5.2. Suppose that E[X2 j] ≤P and the compressed data are formed by Z = 8X, where 8 is m × n with independent entries 8i j ∼N(0, 1/n). Then the information rate rn,m satisfies rn,m = supp(X) I (X; Z) np ≤ m 2n log (2πeP) . Under our sparsistency lower bound on m, the above upper bounds are rn,m = O(log(np)/n). We note that these bounds may not be the best possible since they are obtained assuming knowledge of the compression matrix 8, when in fact the privacy protocol requires that 8 and 1 are not public. 6 Experiments In this section, we report results of simulations designed to validate the theoretical analysis presented in previous sections. We first present results that show the compressed lasso is comparable to the uncompressed lasso in recovering the sparsity pattern of the true linear model. We then show results on persistence that are in close agreement with the theoretical results of Section 4. We only include Figures 1–2 here; additional plots are included in the full version. Sparsistency. Here we run simulations to compare the compressed lasso with the uncompressed lasso in terms of the probability of success in recovering the sparsity pattern of β∗. We use random matrices for both X and 8, and reproduce the experimental conditions of [19]. A design parameter is the compression factor f = n m , which indicates how much the original data are compressed. The results show that when the compression factor f is large enough, the thresholding behaviors as specified in (8) and (9) for the uncompressed lasso carry over to the compressed lasso, when X is drawn from a Gaussian ensemble. In general, the compression factor f is well below the requirement that we have in Theorem 3.3 in case X is deterministic. In more detail, we consider the Gaussian ensemble for the projection matrix 8, where 8i, j ∼N(0, 1/n) are independent. The noise is ϵ ∼N(0, σ 2), where σ 2 = 1. We consider Gaussian ensembles for the design matrix X with both diagonal and Toeplitz covariance. In the Toeplitz case, the covariance is given by T (ρ)i, j = ρ|i−j|; we use ρ = 0.1. [19] shows that when X comes from a Gaussian ensemble under these conditions, there exist fixed constants θℓand θu such that for any ν > 0 and s = supp(β), if n > 2(θu + ν)s log(p −s) + s + 1, (8) then the lasso identifies true variables with probability approaching one. Conversely, if n < 2(θℓ−ν)s log(p −s) + s + 1, (9) then the probability of recovering the true variables using the lasso approaches zero. In the following simulations, we carry out the lasso using procedure lars(Y, X) that implements the LARS algorithm of [8] to calculate the full regularization path. For the uncompressed case, we run lars(Y, X) such that Y = Xβ∗+ ϵ, and for the compressed case we run lars(8Y, 8X) such that 8Y = 8Xβ∗+8ϵ. The regularization parameter is λm = c p (log(p −s) log s)/m. The results show that the behavior under compression is close to the uncompressed case. Persistence. Here we solve the following ℓ1-constrained optimization problem eβ = arg min∥β∥1≤L ∥Y −Xβ∥2 directly, based on algorithms described by [15]. We constrain the solution to lie in the ball Bn = {∥β∥1 ≤Ln}, where Ln = n1/4/√log n. By [10], the uncompressed lasso 6 0 50 100 150 200 250 300 0.0 0.2 0.4 0.6 0.8 1.0 Toeplitz ρ=0.1; Fractional Power γ=0.5, α=0.2 Compressed dimension m Prob of success Uncompressed f = 120 1024 512 256 p=128 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 Identity; FP γ=0.5, α=0.2; p=1024 Control parameter θ Prob of success Uncomp. f = 5 f = 10 f = 20 f = 40 f = 80 f = 120 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 Toeplitz ρ=0.1; FP γ=0.5, α=0.2; p=1024 Control parameter θ Prob of success Uncomp. f = 5 f = 10 f = 20 f = 40 f = 80 f = 120 Figure 1: Plots of the number of samples versus the probability of success for recovering sgn(β∗). Each point on a curve for a particular θ or m, where m = 2θσ 2s log(p −s) + s + 1, is an average over 200 trials; for each trial, we randomly draw Xn×p, 8m×n, and ϵ ∈Rn. The covariance 6 = 1 n E X T X and model β∗are fixed across all curves in the plot. The sparsity level is s(p) = 0.2p1/2. The four sets of curves in the left plot are for p = 128, 256, 512 and 1024, with dashed lines marking m for θ = 1 and s = 2, 3, 5 and 6 respectively. In the plots on the right, each curve has a compression factor f ∈{5, 10, 20, 40, 80, 120} for the compressed lasso, thus n = f m; dashed lines mark θ = 1. For 6 = I, θu = θℓ= 1, while for 6 = T (0.1), θu ≈1.84 and θℓ≈0.46 [19], for the uncompressed lasso in (8) and in (9). 0 2000 4000 6000 8000 8 10 12 14 16 18 n=9000, p=128, s=9 Compressed dimension m Risk Uncompressed predictive Compressed predictive Compressed empirical Figure 2: Risk versus compressed dimension. We fix n = 9000 and p = 128, and set s(p) = 3 and Ln = 2.6874. The model is β∗= (−0.9, −1.7, 1.1, 1.3, −0.5, 2, −1.7, −1.3, −0.9, 0, . . . , 0)T so that
β∗ b
1 > Ln and β∗ b ̸∈Bn, and the uncompressed oracle predictive risk is R = 9.81. For each value of m, a data point corresponds to the mean empirical risk, which is defined in (7), over 100 trials, and each vertical bar shows one standard deviation. For each trial, we randomly draw Xn×p with i.i.d. row vectors xi ∼N(0, T (0.1)), and Y = Xβ∗+ ϵ. 7 estimator bβn is persistent over Bn. For the compressed lasso, given n and pn, and a varying compressed sample size m, we take the ball Bn,m = {β : ∥β∥1 ≤Ln,m} where Ln,m = m1/4/ p log(npn). The compressed lasso estimator bβn,m for log2(npn) ≤m ≤n, is persistent over Bn,m by Theorem 4.1. The simulations confirm this behavior. 7 Acknowlegments This work was supported in part by NSF grant CCF-0625879. References [1] D. Agrawal and C. C. Aggarwal. On the design and quantification of privacy preserving data mining algorithms. In In Proceedings of the 20th Symposium on Principles of Database Systems, May 2001. [2] E. Candès, J. Romberg, and T. Tao. Stable signal recovery from incomplete and inaccurate measurements. Communications in Pure and Applied Mathematics, 59(8):1207–1223, August 2006. [3] T. Dalenius. Towards a methodology for statistical disclosure control. Statistik Tidskrift, 15:429–444, 1977. [4] D. Donoho. Compressed sensing. IEEE Trans. Info. Theory, 52(4):1289–1306, April 2006. [5] M. Duarte, M. Davenport, M. Wakin, and R. Baraniuk. Sparse signal detection from incoherent projections. In Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, 2006. [6] G. Duncan and R. Pearson. Enhancing access to microdata while protecting confidentiality: Prospects for the future. Statistical Science, 6(3):219–232, August 1991. [7] C. Dwork. Differential privacy. In 33rd International Colloquium on Automata, Languages and Programming–ICALP 2006, pages 1–12, 2006. [8] B. Efron, T. Hastie, I. Johnstone, and R. Tibshirani. Least angle regression. Annals of Statistics, 32(2):407– 499, 2004. [9] J. Feigenbaum, Y. Ishai, T. Malkin, K. Nissim, M. J. Strauss, and R. N. Wright. Secure multiparty computation of approximations. ACM Trans. Algorithms, 2(3):435–472, 2006. [10] E. Greenshtein and Y. Ritov. Persistency in high dimensional linear predictor-selection and the virtue of over-parametrization. Journal of Bernoulli, 10:971–988, 2004. [11] J. Haupt, R. Castro, R. Nowak, G. Fudge, and A. Yeh. Compressive sampling for signal classification. In Proc. Asilomar Conference on Signals, Systems, and Computers, October 2006. [12] K. Liu, H. Kargupta, and J. Ryan. Random projection-based multiplicative data perturbation for privacy preserving distributed data mining. IEEE Trans. on Knowl. and Data Engin., 18(1), Jan. 2006. [13] T. L. Marzetta and B. M. Hochwald. Capacity of a mobile multiple-antenna communication link in Rayleigh flat fading. IEEE Trans. Info. Theory, 45(1):139–157, January 1999. [14] N. Meinshausen and B. Yu. Lasso-type recovery of sparse representations for high-dimensional data. Technical Report 720, Department of Statistics, UC Berkeley, 2006. [15] M. Osborne, B. Presnell, and B. Turlach. On the lasso and its dual. J. Comp. and Graph. Stat., 9(2):319– 337, 2000. [16] A. P. Sanil, A. Karr, X. Lin, and J. P. Reiter. Privacy preserving regression modelling via distributed computation. In Proceedings of Tenth ACM SIGKDD, 2004. [17] R. Tibshirani. Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Ser. B, 58(1):267–288, 1996. [18] J. Tropp. Greed is good: Algorithmic results for sparse approximation. IEEE Transactions on Information Theory, 50(10):2231–2242, 2004. [19] M. Wainwright. Sharp thresholds for high-dimensional and noisy recovery of sparsity. Technical Report 709, Department of Statistics, UC Berkeley, May 2006. [20] P. Zhao and B. Yu. On model selection consistency of lasso. J. Mach. Learn. Research, 7:2541–2567, 2007. 8
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DIFFRAC : a discriminative and flexible framework for clustering Francis R. Bach INRIA - Willow Project ´Ecole Normale Sup´erieure 45, rue d’Ulm, 75230 Paris, France francis.bach@mines.org Za¨ıd Harchaoui LTCI, TELECOM ParisTech and CNRS 46, rue Barrault 75634 Paris cedex 13, France zaid.harchaoui@enst.fr Abstract We present a novel linear clustering framework (DIFFRAC) which relies on a linear discriminative cost function and a convex relaxation of a combinatorial optimization problem. The large convex optimization problem is solved through a sequence of lower dimensional singular value decompositions. This framework has several attractive properties: (1) although apparently similar to K-means, it exhibits superior clustering performance than K-means, in particular in terms of robustness to noise. (2) It can be readily extended to non linear clustering if the discriminative cost function is based on positive definite kernels, and can then be seen as an alternative to spectral clustering. (3) Prior information on the partition is easily incorporated, leading to state-of-the-art performance for semi-supervised learning, for clustering or classification. We present empirical evaluations of our algorithms on synthetic and real medium-scale datasets. 1 Introduction Many clustering frameworks have already been proposed, with numerous applications in machine learning, exploratory data analysis, computer vision and speech processing. However, these unsupervised learning techniques have not reached the level of sophistication of supervised learning techniques, that is, for all methods, there are still a significant number of explicit or implicit parameters to tune for successful clustering, most generally, the number of clusters and the metric or the similarity structure over the space of configurations. In this paper, we present a discriminative and flexible framework for clustering (DIFFRAC), which is aimed at alleviating some of those practical annoyances. Our framework is based on a recent set of works [1, 2] that have used the support vector machine (SVM) cost function used for linear classification as a clustering criterion, with the intuitive goal of looking for clusters which are most linearly separable. This line of work has led to promising results; however, the large convex optimization problems that have to be solved prevent application to datasets larger than few hundreds data points.1 In this paper, we consider the maximum value of the regularized linear regression on indicator matrices. By choosing a square loss (instead of the hinge loss), we obtain a simple cost function which can be simply expressed in closed form and is amenable to specific efficient convex optimization algorithms, that can deal with large datasets of size 10,000 to 50,000 data points. Our cost function turns out to be a linear function of the “equivalence matrix” M, which is a square {0, 1}-matrix indexed by the data points, with value one for all pairs of data points that belong to the same clusters, and zero otherwise. In order to minimize this cost function with respect to M, we follow [1] and [2] by using convex outer approximations of the set of equivalence matrices, with a novel constraint on the minimum number of elements per cluster, which is based on the eigenvalues of M, and essential to the success of our approach. 1Recent work [3] has looked at more efficient formulations. In Section 2, we present a derivation of our cost function and of the convex relaxations. In Section 3, we show how the convex relaxed problem can be solved efficiently through a sequence of lower dimensional singular value decompositions, while in Section 4, we show how a priori knowledge can be incorporated into our framework. Finally, in Section 5, we present simulations comparing our new set of algorithms to other competing approaches. 2 Discriminative clustering framework In this section, we first assume that we are given n points x1, . . . , xn in Rd, represented in a matrix X ∈Rn×d. We represent the various partitions of {1, . . ., n} into k > 1 clusters by indicator matrices y ∈{0, 1}n×k such that y1k = 1n, where 1k and 1n denote the constant vectors of all ones, of dimensions k and n. We let denote Ik the set of k-class indicator matrices. 2.1 Discriminative clustering cost Given y, we consider the regularized linear regression problem of y given X, which takes the form: min w∈Rd×k, b∈R1×k 1 n∥y −Xw −1nb∥2 F + κ tr w⊤w, (1) where the Frobenius norm is defined for any vector or rectangular matrix as ∥A∥2 F = trAA⊤= trA⊤A. Denoting f(x) = w⊤x + b ∈Rk, this corresponds to a multi-label classification problem with square loss functions [4, 5]. The main advantage of this cost function is the possibility of (a) minimizing the regularized cost in closed form and (b) including a bias term by simply centering the data; namely, the global optimum is attained at w∗= (X⊤ΠnX + nκIn)−1X⊤Πny and b∗= 1 n1⊤ n (y −Xw∗), where Πn = In −1 n1n1⊤ n is the usual centering projection matrix. The optimal value is then equal to J(y, X, κ) = tr yy⊤A(X, κ), (2) where the n × n matrix A(X, κ) is defined as: A(X, κ)= 1 nΠn(In−X(X⊤ΠnX + nκI)−1X⊤)Πn. (3) The matrix A(X, κ) is positive semi-definite, i.e., for all u ∈Rn, u⊤A(X, κ)u ⩾0, and 1n is a singular vector of A(X, κ), i.e., A(X, κ)1n = 0. Following [1] and [2], we are thus looking for a k-class indicator matrix y such that tr yy⊤A(X, κ) is minimal, i.e., for a partition such that the clusters are most linearly separated, where the separability of clusters is measured through the minimum of the discriminative cost with respect to all linear classifiers. This combinatorial optimization is NP-hard in general [6], but efficient convex relaxations may be obtained, as presented in the next section. 2.2 Indicator and equivalence matrices The cost function defined in Eq. (2) only involves the matrix M = yy⊤∈Rn×n. We let denote Ek the set of “k-class equivalence matrices”, i.e., the set of matrices M such that there exists a k-class indicator matrix y with M = yy⊤. There are many outer convex approximations of the discrete sets Ek, based on different properties of matrices in Ek, that were used in different contexts, such as maximum cut problems [6] or correlation clustering [7]. We have the following usual properties of equivalence matrices (independent of k): if M ∈Ek, then (a) M is positive semidefinite (denoted as M ≽0), (b) M has nonnegative values (denoted as M ⩾0) , and (c) the diagonal of M is equal to 1n (denoted as diag(M) = 1n). Moreover, if M corresponds to at most k clusters, we have M ≽1 k1n1⊤ n , which is a consequence to the convex outer approximation of [6] for the maximum k-cut problem. We thus use the following convex outer approximation: Ck = {M ∈Rn×n, M = M ⊤, diag(M) = 1n, M ⩾0, M ≽1 k1n1⊤ n } ⊃Ek. Note that when k = 2, the constraints M ⩾0 (pointwise nonnegativity) is implied by the other constraints. 2.3 Minimum cluster sizes Given the discriminative nature of our cost function (and in particular that A(X, κ)1n = 0), the minimum value 0 is always obtained with M = 1n1⊤ n , a matrix of rank one, equivalent to a single cluster. Given the number of desired clusters, we thus need to add some prior knowledge regarding the size of those clusters. Following [1], we impose a minimum size λ0 for each cluster, through row sums and eigenvalues: Row sums If M ∈Ek, then M1n ⩾λ01n and M1n ⩽(n −(k −1)λ0)1n (the cluster must be smaller than (n −(k −1)λ0) if they are all larger than λ0)–this is the same constraint as in [1]. Eigenvalues When M ∈Ek, the sizes of the clusters are exactly the k largest eigenvalues of M. Thus, for a matrix in Ek, the minimum cluster size constraint is equivalent to Pn i=1 1λi(M)⩾λ0 ⩾ k, where λ1(M), . . . , λn(M) are the n eigenvalues of M. Functions of the form Φ(M) = Pn i=1 φ(λi(M)) are referred to as spectral functions and are particularly interesting in machine learning and optimization, since Φ inherits from φ many of its properties, such as differentiability and convexity [8]. The previous constraint can be seen as Φ(M) ⩾k, with φ(λ) = 1λ⩾λ0, which is not concave and thus does not lead to a convex constraint. In this paper we propose to use the concave upper envelope of this function, namely φλ0(λ) = min{λ/λ0, 1}, thus leading to a novel additional constraint. Our final convex relaxation is thus of minimizing trA(X, κ)M with respect to M ∈Ck and such that Φλ0(M) ⩾k, M1n ⩾λ01n and M1n ⩽(n −(k −1)λ0)1n, where Φλ0(M) = Pn i=1 min{λi(M)/λ0, 1}. The clustering results are empirically robust to the value of λ0. In all our simulations we use λ0 = ⌊n/2k⌋. 2.4 Comparison with K-means Our method bears some resemblance with the usual K-means algorithm. Indeed, in the unregularized case (κ = 0), we aim to minimize tr Πn(In −X(X⊤ΠnX)−1X⊤)Πnyy⊤. Results from [9] show that K-means aims at minimizing the following criterion with respect to y: min µ∈Rk×d ∥X −yµ∥2 F = tr(In −y(y⊤y)−1y⊤)(ΠnX)(ΠnX)⊤. The main differences between the two cost functions are that (1) we require an additional parameter, namely the minimum number of elements per cluster and (2) our cost function normalizes the data, while the K-means distortion measure normalizes the labels. This apparently little difference has a significant impact on the performance, as our method is invariant by affine scaling of the data, while K-means is only invariant by translation, isometries and isotropic scaling, and is very much dependent on how the data are presented (in particular the marginal scaling of the variables). In Figure 1, we compare the two algorithms on a simple synthetic task with noisy dimensions, showing that ours is more robust to noisy features. Note that using a discriminative criterion based on the square loss may lead to the masking problem [4], which can be dealt with in the usual way by using second-order polynomials or, equivalently, a polynomial kernel. 2.5 Kernels The matrix A(X, κ) in Eq. (3) can be expressed only in terms of the Gram matrix K = XX⊤. Indeed, using the matrix inversion lemma, we get: A(K, κ) = κΠn( eK + nκIn)−1Πn, (4) where eK = ΠnKΠn is the “centered Gram matrix” of the points X. We can thus apply our framework with any positive definite kernel [5]. 2.6 Additional relaxations Our convex optimization problem can be further relaxed. An interesting relaxation is obtained by (1) relaxing the constraints M ≽1 k1n1⊤ n into M ≽0, (2) relaxing diag(M) = 1n into trM = n, 0 10 20 30 −0.2 0 0.2 0.4 0.6 0.8 1 noise dimension clustering error K−means diffrac Figure 1: Comparison with K-means, on a two-dimensional dataset composed of two linearly separable bumps (100 data points, plotted in the left panel), with additional random independent noise dimensions (with normal distributions with same marginal variances as the 2D data). The clustering performance is plotted against the number of irrelevant dimensions, for regular K-means and our DIFFRAC approach (right panel, averaged over 50 replications with the standard deviation in dotted lines) . The clustering performance is measured by a metric between partitions defined in Section 5, which is always between 0 and 1. and (3) removing the constraint M ⩾0 and the constraints on the row sums. A short calculation shows that this relaxation leads to an eigenvalue problem: let A = Pn i=1 aiuiu⊤ i be an eigenvalue decomposition of A, where a1 ⩽· · · ⩽an are the sorted eigenvalues. The minimal value of the relaxed convex optimization problem is attained at M = Pj i=1 uiu⊤ i + (n −λ0j)uj+1u⊤ j+1, with j = ⌊n/λ0⌋. This additional relaxation into an eigenvalue problem is the basis of our efficient optimization algorithm in Section 3. In the kernel formulation, since the smallest eigenvectors of A = 1 nΠn( eK + nκIn)−1Πn are the same as the largest eigenvectors of eK, the relaxed problem is thus equivalent to kernel principal component analysis [10, 5] in the kernel setting, and in the linear setting to regular PCA (followed by our rounding procedure presented in Section 3.3). In the linear setting, since PCA has no clustering effects in general2, it is clear that the constraints that were removed are essential to the clustering performance. In the kernel setting, experiments have shown that the most important constraint to keep in order to achieve the best embedding and clustering is the constraint diag(M) = 1n. 3 Optimization Since φλ0(λ) = 1 2λ0 (λ + λ0 −|λ −λ0|), and the sum of singular values can be represented as a semidefinite program (SDP), our problem is an SDP. It can thus be solved to any given accuracy in polynomial time by general purpose interior-point methods [12]. However, the number of variables is O(n2) and thus the complexity of general purpose algorithms will be at least O(n7); this remains much too slow for medium scale problems, where the number of data points is between 1,000 and 10,000. We now present an efficient approximate method that uses the specificity of the problem to reduce the computational load. 3.1 Optimization by partial dualization We saw earlier that by relaxing some of the constraints, we get back an eigenvalue problem. Eigenvalue decompositions are among the most important tools in numerical algebra and algorithms and codes are heavily optimized for these, and it is thus advantageous to rely on a sequence of eigenvalue decompositions for large scale algorithms. We can dualize some constraints while keeping others; this leads to the following proposition: 2Recent results show however that it does have an effect when clusters are spherical Gaussians [11]. Proposition 1 The solution of the convex optimization problem defined in Section 2.3 can be obtained my maximizing F(β) = minM≽0,trM=n,Φλ0 (M)⩾k trB(β)M −b(β) with respect to β, where B(β) = A + Diag(β1) −1 2(β2 −β3)1⊤−1 21(β2 −β3)⊤−β4 + 1 2 β5β⊤ 5 β6 b(β) = β⊤ 1 1 −(n −(k −1)λ0)β⊤ 2 1 + λ0β⊤ 3 1 + kβ6/2 + β⊤ 5 1, and β1 ∈Rn, β2 ∈Rn +, β3 ∈Rn +, β4 ∈Rn×n + ,β5 ∈Rn, β6 ∈R+. The variables β1, β2, β3, β4, (β5, β6) correspond to the respective dualizations of the constraints diag(M) = 1n, M1n ⩽(n −(k −1)λ0)1n, M1n ⩾λ01n, M ⩾0, and M ≽1n1⊤ n k . The function J(B) = minM≽0,trM=n,Φλ0 (M)⩾k trBM is a spectral convex function and may be computed in closed form through an eigenvalue decomposition. Moreover, a subgradient may be easily computed, readily leading to a numerically efficient subgradient method in fewer dimensions than n2. Indeed, if we subsample the pointwise positivity constraint N ⩾0 (so that β4 has only a size smaller than n1/2 × n1/2), then the set of dual variables β we are trying to maximize has linear size in n (instead of the primal variable M being quadratic in n). More refined optimization schemes, based on smoothing of the spectral function J(B) by minM≽0,trM=n,Φλ0 (M)⩾k[trBM +εtrM 2] are also used to speed up convergence (steepest descent of a smoothed function is generally faster than subgradient iterations) [13]. 3.2 Computational complexity The running time complexity can be splitted into initialization procedures and per iteration complexity. The per iteration complexity depends directly on the cost of our eigenvalue problems, which themselves are linear in the matrix-vector operation with the matrix A (we only require a fixed small number of eigenvalues). In all situations, we manage to keep a linear complexity in the number n of data points. Note, however, that the number of descent iterations cannot be bounded a priori; in simulations we limit the number of those iterations to 200. For linear kernels with dimension d, the complexity of initialization is O(d2n), while the complexity of each iteration is proportional to the cost of performing a matrix-vector operation with A, that is, O(dn). For general kernels, the complexity of initialization is O(n3), while the complexity of each iteration is O(n2). However, using an incomplete Cholesky decomposition [5] makes all costs linear in n. 3.3 Rounding After the convex optimization, we obtain a low-rank matrix M ∈Ck which is pointwise nonnegative with unit diagonal, of the form UU ⊤where U ∈Rn×m. We need to project it back to the discrete Ek. We have explored several possibilities, all with similar results. We propose the following procedure: we first project M back to the set of matrices of rank k and unit diagonal, by computing an eigendecomposition, rescaling the first k eigenvectors to unit norms and then perform K-means, which is equivalent to performing the spectral clustering algorithm of [14] on the matrix M. 4 Semi-supervised learning Working with equivalence matrices M allows to easily include prior knowledge on the clusters [2, 15, 16], namely, “must-link” constraints (also referred to a positive constraints) for which we constrain an element of M to be one, and “must-not-link” constraints (also referred to as negative constraints), for which we constrain an element of M to be zero. Those two constraints are linear in M and can thus easily be included in our convex formulation. We assume throughout this section that we have a set of “must-link” pairs P+ and a set of “mustnot-link” pairs P−. Moreover, we assume that the set of positive constraints is closed, i.e., that if there is a path of positive constraints between two data points, then these two data points are already forming a pair in P+. If the set of positive pairs does not satisfy this assumption, a larger set of pairs can be obtained by transitive closure. 0 20 40 0 0.5 1 noise dimension clustering error 20 % x n 0 20 40 0 0.5 1 40 % x n noise dimension K−means diffrac Figure 2: Comparison with K-means in the semi-supervised setting, with data taken from Figure 1: clustering performance (averaged over 50 replications, with standard deviations in dotted) vs. number of irrelevant dimensions, with 20% × n and 40% × n random matching pairs used for semisupervision. Positive constraints Given our closure assumption on P+, we get a partition of {1, . . . , n} into p “chunks” of size greater or equal to 1. The singletons in this partition correspond to data points that are not involved in any positive constraints, while other subsets corresponds to chunks of data points that must occur together in the final partition. We let Cj, j = 1, . . . , p denote those groups, and let P denote the n × p {0, 1}-matrix defined such that each column (indexed by j) is equal to one for rows in Cj and zero otherwise. Forcing those groups is equivalent to considering M of the form M = PMP P ⊤, where MP is an equivalence matrix of size p. Note that the positive constraint Mij = 1 is in fact turned into the equality of columns (and thus rows by symmetry) i and j of M, which is equivalent when M ∈Ek, but much stronger for M ∈Ck. In our linear clustering framework, this is in fact equivalent to (a) replacing each chunk by its mean, (b) adding a weight equal to the number of elements in the group into the discriminative cost function and (c) modifying the regularization matrix to take into account the inner variance within each chunk. Positive constraints can be similarly included into K-means, to form a reduced weighted K-means problem, which is simpler than other approaches to deal with positive constraints [17]. In Figure 2, we compare constrained K-means and the DIFFRAC framework under the same setting as in Figure 1, with different numbers of randomly selected positive constraints. Negative constraints After the chunks corresponding to positive constraints have been collapsed to one point, we extend the set of negative constraints to those collapsed points (if the constraints were originally consistent, the negative constraints can be uniquely extended). In our optimization framework, we simply add a penalty function of the form 1 ε|P−| P (i,j)∈P−M 2 ij. The K-means rounding procedure also has to be constrained, e.g., using the procedure of [17]. 5 Simulations In this section, we apply the DIFFRAC framework to various clustering problems and situations. In all our simulations, we use the following distance between partitions B = B1 ∪ · · · ∪Bk and B′ = B′ 1 ∪· · · ∪B′ k′ into k and k′ disjoints subsets of {1, . . . , n}: d(B, B′) = k + k′ −2 P i,i′ Card(Bi∩B′ i′)2 Card(Bi)Card(B′ i′) 1/2 . d(B, B′) defines a distance over the set of partitions [9] which is always between 0 and (k + k′ −2)1/2. When comparing partitions, we use the squared distance 1 2d(B, B′)2, which is always between 0 and k+k′ 2 −1 (and between 0 and k −1, if the two partitions have the same number of clusters). 5.1 Clustering classification datasets We looked at the Isolet dataset (26 classes, 5,200 data points) from the UCI repository and the MNIST datasets of handwritten digits (10 classes, 5,000 data points). For each of those datasets, we compare the performances of K-means, RCA [18] and DIFFRAC, for linear and Gaussian kernels (referred to as “rbf”), for fixed value of the regularization parameter, with different levels of supervision. Results are presented in Table 1: on unsupervised problems, K-means and DIFFRAC Dataset K-means DIFFRAC RCA Mnist-linear 0% 5.6 ± 0.1 6.0 ± 0.4 Mnist-linear 20% 4.5 ± 0.3 3.6 ± 0.3 3.0 ± 0.2 Mnist-linear 40% 2.9 ± 0.3 2.2 ± 0.2 1.8 ± 0.4 Mnist-RBF 0% 5.6 ± 0.2 4.9 ± 0.2 Mnist-RBF 20% 4.6 ± 0.0 1.8 ± 0.4 4.1 ± 0.2 Mnist-RBF 40% 4.9 ± 0.0 0.9 ± 0.1 2.9 ± 0.1 Isolet-linear 0% 12.1 ± 0.6 12.3 ± 0.3 Isolet-linear 20% 10.5 ± 0.2 7.8 ± 0.8 9.5 ± 0.4 Isolet-linear 40% 9.2 ± 0.5 3.7 ± 0.2 7.0 ± 0.4 Isolet-RBF 0% 11.4 ± 0.4 11.0 ± 0.3 Isolet-RBF 20% 10.6 ± 0.0 7.5 ± 0.5 7.8 ± 0.5 Isolet-RBF 40% 10.0 ± 0.0 3.7 ± 1.0 6.9 ± 0.6 Table 1: Comparison of K-means, RCA and linear DIFFRAC, using the clustering metric defined in Section 5 (averaged over 10 replications), for linear and “rbf” kernels and various levels of supervision. have similar performance, while on semi-supervised problems, and in particular for nonlinear kernels, DIFFRAC outperforms both K-means and RCA. Note that all algorithms work on the same data representation (linear or kernelized) and that differences are due to the underlying clustering frameworks. 5.2 Semi-supervised classification To demonstrate the effectiveness of our method in a semi-supervised learning (SSL) context, we performed experiments on some benchmarks datasets for SSL described in [19]. We considered the following datasets: COIL, BCI and Text. We carried out the experiments in a transductive setting, i.e., the test set coincides with the set of unlabelled samples. This allowed us to conduct a fair comparison with the low density separation (LDS) algorithm of [19], which is an enhanced version of the so-called Transductive SVM. However, deriving “out-of-sample” extensions for our method is straightforward. A primary goal in semi-supervised learning is to take into account a large number of labelled points in order to dramatically reduce the number of labelled points required to achieve a competitive classification accuracy. Henceforth, our experimental setting consists in observing how fast the classification accuracy collapses as the number of labelled points increases. The less labelled points a method needs to achieve decent classification accuracy, the more it is relevant for semi-supervised learning tasks. As shown in Figure 3, our method yields competitive classification accuracy with very few labelled points on the three datasets. Moreover, DIFFRAC reaches unexpectedly good results on the Text dataset, where most semi-supervised learning methods usually show disappointing performance. One explanation might be that DIFFRAC acts as an “augmented”-clustering algorithm, whereas most semi-supervised learning algorithms are built as “augmented”-versions of traditional supervised learning algorithms such as LDS which is built on SVMs for instance. Hence, for datasets exhibiting multi-class structure such as Text, DIFFRAC is more able to utilize unlabelled points since it based on a multi-class clustering algorithm rather than algorithms based on binary SVMs, where multi-class extensions are currently unclear. Thus, our experiments support the fact that semisupervised learning algorithms built on clustering algorithms augmented with labelled data acting as hints on clusters are worth for investigation and further research. 6 Conclusion We have presented a discriminative framework for clustering based on the square loss and penalization through spectral functions of equivalence matrices. Our formulation enables the easy incorporation of semi-supervised constraints, which leads to state-of-the-art performance in semi-supervised learning. Moreover, our discriminative framework should allow to use existing methods for learning the kernel matrix from data [20]. Finally, we are currently investigating the use of DIFFRAC in semi-supervised image segmentation. In particular, early experiments on estimating the number of clusters using variation rates of our discriminative costs are very promising. 0 50 100 150 200 0 0.2 0.4 0.6 0.8 Number of labelled training points Test error Learning curve on Coil100 DIFFRAC LDS 0 50 100 150 0.25 0.3 0.35 0.4 0.45 0.5 Number of labelled training points Test error Learning curve on BCI DIFFRAC LDS 0 50 100 150 200 0.1 0.15 0.2 0.25 0.3 0.35 Number of labelled training points Test error Learning curve on Text DIFFRAC LDS Figure 3: Semi-supervised classification. References [1] L. Xu, J. Neufeld, B. Larson, and D. Schuurmans. Maximum margin clustering. In Adv. NIPS, 2004. [2] T. De Bie and N. Cristianini. Fast SDP relaxations of graph cut clustering, transduction, and other combinatorial problems. J. Mac. Learn. Res., 7:1409–1436, 2006. [3] K. Zhang, I. W. Tsang, and J. T. Kwok. Maximum margin clustering made practical. In Proc. ICML, 2007. [4] T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning. Springer-Verlag, 2001. [5] J. Shawe-Taylor and N. Cristianini. Kernel Methods for Pattern Analysis. Camb. Univ. Press, 2004. [6] A. Frieze and M. Jerrum. Improved approximation algorithms for MAX k-CUT and MAX BISECTION. In Integer Programming and Combinatorial Optimization, volume 920, pages 1–13. Springer, 1995. [7] C. Swamy. Correlation clustering: maximizing agreements via semidefinite programming. In ACM-SIAM Symp. Discrete algorithms, 2004. [8] A. S. Lewis and H. S. Sendov. Twice differentiable spectral functions. SIAM J. Mat. Anal. App., 23(2):368–386, 2002. [9] F R. Bach and M I. Jordan. Learning spectral clustering, with application to speech separation. J. Mac. Learn. Res., 7:1963–2001, 2006. [10] B. Sch¨olkopf, A. J. Smola, and K.-R. M¨uller. Nonlinear component analysis as a kernel eigenvalue problem. Neural Comp., 10(3):1299–1319, 1998. [11] N. Srebro, G. Shakhnarovich, and S. Roweis. An investigation of computational and informational limits in gaussian mixture clustering. In Proc. ICML, 2006. [12] S. Boyd and L. Vandenberghe. Convex Optimization. Camb. Univ. Press, 2003. [13] J. F. Bonnans, J. C. Gilbert, C. Lemar´echal, and C. A. Sagastizbal. Numerical Optimization Theoretical and Practical Aspects. Springer, 2003. [14] A. Y. Ng, M. I. Jordan, and Y. Weiss. On spectral clustering: analysis and an algorithm. In Adv. NIPS, 2002. [15] L. Xu and D. Schuurmans. Unsupervised and semi-supervised multi-class support vector machines. In Proc. AAAI, 2005. [16] M. Heiler, J. Keuchel, and C. Schn¨orr. Semidefinite clustering for image segmentation with a-priori knowledge. In Pattern Recognition, Proc. DAGM, 2005. [17] K. Wagstaff, C. Cardie, S. Rogers, and S. Schr¨odl. Constrained K-means clustering with background knowledge. In Proc. ICML, 2001. [18] A. Bar-Hillel, T. Hertz, N. Shental, and D. Weinshall. Learning distance functions using equivalence relations. In Proc. ICML, 2003. [19] O. Chapelle and A. Zien. Semi-supervised classification by low density separation. In Proc. AISTATS, 2004. [20] F. R. Bach, G. R. G. Lanckriet, and M. I. Jordan. Multiple kernel learning, conic duality, and the SMO algorithm. In Proc. ICML, 2004.
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Bundle Methods for Machine Learning Alexander J. Smola, S.V. N. Vishwanathan, Quoc V. Le NICTA and Australian National University, Canberra, Australia Alex.Smola@gmail.com, {SVN.Vishwanathan, Quoc.Le}@nicta.com.au Abstract We present a globally convergent method for regularized risk minimization problems. Our method applies to Support Vector estimation, regression, Gaussian Processes, and any other regularized risk minimization setting which leads to a convex optimization problem. SVMPerf can be shown to be a special case of our approach. In addition to the unified framework we present tight convergence bounds, which show that our algorithm converges in O(1/ϵ) steps to ϵ precision for general convex problems and in O(log(1/ϵ)) steps for continuously differentiable problems. We demonstrate in experiments the performance of our approach. 1 Introduction In recent years optimization methods for convex models have seen significant progress. Starting from the active set methods described by Vapnik [17] increasingly sophisticated algorithms for solving regularized risk minimization problems have been developed. Some of the most exciting recent developments are SVMPerf [5] and the Pegasos gradient descent solver [12]. The former computes gradients of the current solution at every step and adds those to the optimization problem. Joachims [5] prove an O(1/ϵ2) rate of convergence. For Pegasos Shalev-Shwartz et al. [12] prove an O(1/ϵ) rate of convergence, which suggests that Pegasos should be much more suitable for optimization. In this paper we extend the ideas of SVMPerf to general convex optimization problems and a much wider class of regularizers. In addition to this, we present a formulation which does not require the solution of a quadratic program whilst in practice enjoying the same rate of convergence as algorithms of the SVMPerf family. Our error analysis shows that the rates achieved by this algorithm are considerably better than what was previously known for SVMPerf, namely the algorithm enjoys O(1/ϵ) convergence and O(log(1/ϵ)) convergence, whenever the loss is sufficiently smooth. An important feature of our algorithm is that it automatically takes advantage of smoothness in the problem. Our work builds on [15], which describes the basic extension of SVMPerf to general convex problems. The current paper provides a) significantly improved performance bounds which match better what can be observed in practice and which apply to a wide range of regularization terms, b) a variant of the algorithm which does not require quadratic programming, yet enjoys the same fast rates of convergence, and c) experimental data comparing the speed of our solver to Pegasos and SVMPerf. Due to space constraints we relegate the proofs to an technical report [13]. 2 Problem Setting Denote by x ∈X and y ∈Y patterns and labels respectively and let l(x, y, w) be a loss function which is convex in w ∈W, where either W = Rd (linear classifier), or W is a Reproducing Kernel Hilbert Space for kernel methods. Given a set of m training patterns {xi, yi}m i=1 the regularized risk 1 functional which many estimation methods strive to minimize can be written as J(w) := Remp(w) + λΩ(w) where Remp(w) := 1 m m X i=1 l(xi, yi, w). (1) Ω(w) is a smooth convex regularizer such as 1 2 ∥w∥2, and λ > 0 is a regularization term. Typically Ωis cheap to compute and to minimize whereas the empirical risk term Remp(w) is computationally expensive to deal with. For instance, in the case of intractable graphical models it requires approximate inference methods such as sampling or semidefinite programming. To make matters worse the number of training observations m may be huge. We assume that the empirical risk Remp(w) is nonnegative. Figure 1: A lower bound on the convex empirical risk Remp(w) obtained by computing three tangents on the entire function. If J is differentiable we can use standard quasi-Newtons methods like LBFGS even for large values of m [8]. Unfortunately, it is not straightforward to extend these algorithms to optimize a non-smooth objective. In such cases one has to resort to bundle methods [3], which are based on the following elementary observation: for convex functions a first order Taylor approximation is a lower bound. So is the maximum over a set of Taylor approximations. Furthermore, the Taylor approximation is exact at the point of expansion. The idea is to replace Remp[w] by these lower bounds and to optimize the latter in conjunction with Ω(w). Figure 1 gives geometric intuition. In the remainder of the paper we will show that 1) This extends a number of existing algorithms; 2) This method enjoys good rates of convergence; and 3) It works well in practice. Note that there is no need for Remp[w] to decompose into individual losses in an additive fashion. For instance, scores, such as Precision@k [4], or SVM Ranking scores do not satisfy this property. Likewise, estimation problems which allow for an unregularized common constant offset or adaptive margin settings using the ν-trick fall into this category. The only difference is that in those cases the derivative of Remp[w] with respect to w no more decomposes trivially into a sum of gradients. 3 Bundle Methods 3.1 Subdifferential and Subgradient Before we describe the bundle method, it is necessary to clarify a key technical point. The subgradient is a generalization of gradients appropriate for convex functions, including those which are not necessarily smooth. Suppose w is a point where a convex function F is finite. Then a subgradient is the normal vector of any tangential supporting hyperplane of F at w. Formally µ is called a subgradient of F at w if, and only if, F(w′) ≥F(w) + ⟨w′ −w, µ⟩ ∀w′. (2) The set of all subgradients at a point is called the subdifferential, and is denoted by ∂wF(w). If this set is not empty then F is said to be subdifferentiable at w. On the other hand, if this set is a singleton then, the function is said to be differentiable at w. 3.2 The Algorithm Denote by wt ∈W the values of w which are obtained by successive steps of our method, Let at ∈W, bt ∈R, and set w0 = 0, a0 = 0, b0 = 0. Then, the Taylor expansion coefficients of Remp[wt] can be written as at+1 := ∂wRemp(wt) and bt+1 := Remp(wt) −⟨at+1, wt⟩. (3) Note that we do not require Remp to be differentiable: if Remp is not differentiable at wt we simply choose any element of the subdifferential as at+1. Since each Taylor approximation is a lower bound, we may take their maximum to obtain that Remp(w) ≥maxt ⟨at, w⟩+ bt. Moreover, by 2 Algorithm 1 Bundle Method(ϵ) Initialize t = 0, w0 = 0, a0 = 0, b0 = 0, and J0(w) = λΩ(w) repeat Find minimizer wt := argminw Jt(w) Compute gradient at+1 and offset bt+1. Increment t ←t + 1. until ϵt ≤ϵ virtue of the fact that Remp is a non-negative function we can write the following lower bounds on Remp and J respectively: Rt(w) := max t′≤t ⟨at′, w⟩+ bt′ and Jt(w) := λΩ(w) + Rt(w). (4) By construction Rt′ ≤Rt ≤Remp and Jt′ ≤Jt ≤J for all t′ ≤t. Define w∗:= argmin w J(w), γt := Jt+1(wt) −Jt(wt), wt := argmin w Jt(w), and ϵt := min t′≤t Jt′+1(wt′) −Jt(wt). The following lemma establishes some useful properties of γt and ϵt. Lemma 1 We have Jt′(wt′) ≤Jt(wt) ≤J(w∗) ≤J(wt) = Jt+1(wt) for all t′ ≤t. Furthermore, ϵt is monotonically decreasing with ϵt −ϵt+1 ≥Jt+1(wt+1) −Jt(wt) ≥0. Also, ϵt upper bounds the distance from optimality via γt ≥ϵt ≥mint′≤t J(wt′) −J(w∗). 3.3 Dual Problem Optimization is often considerably easier in the dual space. In fact, we will show that we need not know Ω(w) at all, instead it is sufficient to work with its Fenchel-Legendre dual Ω∗(µ) := supw ⟨w, µ⟩−Ω(w). If Ω∗is a so-called Legendre function [e.g. 10] the w at which the supremum is attained can be written as w = ∂µΩ∗(µ). In the sequel we will always assume that Ω∗is twice differentiable and Legendre. Examples include Ω∗(µ) = 1 2 ∥µ∥2 or Ω∗(µ) = P i exp[µ]i. Theorem 2 Let α ∈ Rt, denote by A = [a1, . . . , at] the matrix whose columns are the (sub)gradients, let b = [b1, . . . , bt]. The dual problem of minimize w Jt(w) := max t′≤t ⟨at′, w⟩+ bt′ + λΩ(w) is (5) maximize α J∗ t (α) := −λΩ∗(−λ−1Aα) + α⊤b subject to α ≥0 and ∥α∥1 = 1. (6) Furthermore, the optimal wt and αt are related by the dual connection wt = ∂Ω∗(−λ−1Aαt). Recall that for Ω(w) = 1 2 ∥w∥2 2 the Fenchel-Legendre dual is given by Ω∗(µ) = 1 2 ∥µ∥2 2. This is commonly used in SVMs and Gaussian Processes. The following corollary is immediate: Corollary 3 Define Q := A⊤A, i.e. Quv := ⟨au, av⟩. For quadratic regularization, i.e. minimizew max(0, maxt′≤t ⟨at′, w⟩+ bt′) + λ 2 ∥w∥2 2 the dual becomes maximize α − 1 2λα⊤Qα + α⊤b subject to α ≥0 and ∥α∥1 = 1. (7) This means that for quadratic regularization the dual optimization problem is a quadratic program where the number of variables equals the number of gradients computed previously. Since t is typically in the order of 10s to 100s, the resulting QP is very cheap to solve. In fact, we don’t even need to know the gradients explicitly. All that is required to define the QP are the inner products between gradient vectors ⟨au, av⟩. Later in this section we propose a variant which does away with the quadratic program altogether while preserving most of the appealing convergence properties of Algorithm 1. 3 3.4 Examples Structured Estimation Many estimation problems [14, 16] can be written in terms of a piecewise linear loss function l(x, y, w) = max y′∈Y ⟨φ(x, y′) −φ(x, y), w⟩+ ∆(y, y′) (8) for some suitable joint feature map φ, and a loss function ∆(y, y′). It follows from Section 3.1 that a subdifferential of (8) is given by ∂wl(x, y, w) = φ(x, y∗) −φ(x, y) where y∗:= argmax y′∈Y ⟨φ(x, y′) −φ(x, y), w⟩+ ∆(y, y′). (9) Since Remp is defined as a summation of loss terms, this allows us to apply Algorithm 1 directly for risk minimization: at every iteration t we find all maximal constraint violators for each (xi, yi) pair and compute the composite gradient vector. This vector is then added to the convex program we have so far. Joachims [5] pointed out this idea for the special case of φ(x, y) = yφ(x) and y ∈{±1}, that is, binary loss. Effectively, by defining a joint feature map as the sum over individual feature maps and by defining a joint loss ∆as the sum over individual losses SVMPerf performs exactly the same operations as we described above. Hence, for losses of type (8) our algorithm is a direct extension of SVMPerf to structured estimation. Exponential Families One of the advantages of our setting is that it applies to any convex loss function, as long as there is an efficient way of computing the gradient. That is, we can use it for cases where we are interested in modeling p(y|x; w) = exp(⟨φ(x, y), w⟩−g(w|x)) where g(w|x) = log Z Y exp ⟨φ(x, y′), w⟩dy′ (10) That is, g(w|x) is the conditional log-partition function. This type of losses includes settings such as Gaussian Process classification and Conditional Random Fields [1]. Such settings have been studied by Lee et al. [6] in conjunction with an ℓ1 regularizer Ω(w) = ∥w∥1 for structure discovery in graphical models. Choosing l to be the negative log-likelihood it follows that Remp(w) = m X i=1 g(w|xi) −⟨φ(xi, yi), w⟩and ∂wRemp(w) = m X i=1 Ey′∼p(y′|xi;w) [φ(xi, y′)] −φ(xi, yi). This means that column generation methods are therefore directly applicable to Gaussian Process estimation, a problem where large scale solvers were somewhat more difficult to find. It also shows that adding a new model becomes a matter of defining a new loss function and its corresponding gradient, rather than having to build a full solver from scratch. 4 Convergence Analysis While Algorithm 1 is intuitively plausible, it remains to be shown that it has good rates of convergence. In fact, past results, such as those by Tsochantaridis et al. [16] suggest an O(1/ϵ2) rate, which would make the application infeasible in practice. We use a duality argument similar to those put forward in [11, 16], both of which share key techniques with [18]. The crux of our proof argument lies in showing that ϵt −ϵt+1 ≥Jt+1(wt+1) − Jt(wt) (see Theorem 4) is sufficiently bounded away from 0. In other words, since ϵt bounds the distance from the optimality, at every step Algorithm 1 makes sufficient progress towards the optimum. Towards this end, we first observe that by strong duality the values of the primal and dual problems (5) and (6) are equal at optimality. Hence, any progress in Jt+1 can be computed in the dual. Next, we observe that the solution of the dual problem (6) at iteration t, denoted by αt, forms a feasible set of parameters for the dual problem (6) at iteration t+1 by means of the parameterization (αt, 0), i.e. by padding αt with a 0. The value of the objective function in this case equals Jt(wt). 4 To obtain a lower bound on the improvement due to Jt+1(wt+1) we perform a line search along ((1− η)αt, η) in (6). The constraint η ∈[0, 1] ensures dual feasibility. We will bound this improvement in terms of γt. Note that, in general, solving the dual problem (6) results in an increase which is larger than that obtained via the line search. The line search is employed in the analysis only for analytic tractability. We aim to lower-bound ϵt−ϵt+1 in terms of ϵt and solve the resultant difference equation. Depending on J(w) we will be able to prove two different convergence results. (a) For regularizers Ω(w) for which
∂2 µΩ∗(µ)
≤H∗we first experience a regime of progress linear in γt and a subsequent slowdown to improvements which are quadratic in γt. (b) Under the above conditions, if furthermore
∂2 wJ(w)
≤H, i.e. the Hessian of J is bounded, we have linear convergence throughout. We first derive lower bounds on the improvement Jt+1(wt+1)−Jt(wt), then the fact that for (b) the bounds are better. Finally we prove the convergence rates by solving the difference equation in ϵt. This reasoning leads to the following theorem: Theorem 4 Assume that ∥∂wRemp(w)∥≤G for all w ∈W, where W is some domain of interest containing all wt′ for t′ ≤t. Also assume that Ω∗has bounded curvature, i.e. let
∂2 µΩ∗(µ)
≤H∗ for all µ ∈ −λ−1 ¯A¯α where ¯α ≥0 and ∥¯α∥1 ≤1 . In this case we have ϵt −ϵt+1 ≥γt 2 min(1, λγt/4G2H∗) ≥ϵt 2 min(1, λϵt/4G2H∗). (11) Furthermore, if
∂2 wJ(w)
≤H, then we have ϵt −ϵt+1 ≥ γt/2 if γt > 4G2H∗/λ λ/8H∗ if 4G2H∗/λ ≥γt ≥H/2 λγt/4HH∗ otherwise (12) Note that the error keeps on halving initially and settles for a somewhat slower rate of convergence after that, whenever the Hessian of the overall risk is bounded from above. The reason for the difference in the convergence bound for differentiable and non-differentiable losses is that in the former case the gradient of the risk converges to 0 as we approach optimality, whereas in the former case, no such guarantees hold (e.g. when minimizing |x| the (sub)gradient does not vanish at the optimum). Two facts are worthwhile noting: a) The dual of many regularizers, e.g. squares norm, squared ℓp norm, and the entropic regularizer have bounded second derivative. See e.g. [11] for a discussion and details. Thus our condition
∂2 µΩ∗(µ)
≤H∗is not unreasonable. b) Since the improvements decrease with the size of γt we may replace γt by ϵt in both bounds and conditions without any ill effect (the bound only gets worse). Applying the previous result we obtain a convergence theorem for bundle methods. Theorem 5 Assume that J(w) ≥0 for all w. Under the assumptions of Theorem 4 we can give the following convergence guarantee for Algorithm 1. For any ϵ < 4G2H∗/λ the algorithm converges to the desired precision after n ≤log2 λJ(0) G2H∗+ 8G2H∗ λϵ −4 (13) steps. If furthermore the Hessian of J(w) is bounded, convergence to any ϵ ≤H/2 takes at most the following number of steps: n ≤log2 λJ(0) 4G2H∗+ 4H∗ λ max 0, H −8G2H∗/λ + 4HH∗ λ log(H/2ϵ) (14) Several observations are in order: firstly, note that the number of iterations only depends logarithmically on how far the initial value J(0) is away from the optimal solution. Compare this to the result of Tsochantaridis et al. [16], where the number of iterations is linear in J(0). 5 Secondly, we have an O(1/ϵ) dependence in the number of iterations in the non-differentiable case. This matches the rate of Shalev-Shwartz et al. [12]. In addition to that, the convergence is O(log(1/ϵ)) for continuously differentiable problems. Note that whenever Remp(w) is the average over many piecewise linear functions Remp(w) behaves essentially like a function with bounded Hessian as long as we are taking large enough steps not to “notice” the fact that the term is actually nonsmooth. Remark 6 For Ω(w) = 1 2 ∥w∥2 the dual Hessian is exactly H∗= 1. Moreover we know that H ≥λ since
∂2 wJ(w)
= λ +
∂2 wRemp(w)
. Effectively the rate of convergence of the algorithm is governed by upper bounds on the primal and dual curvature of the objective function. This acts like a condition number of the problem — for Ω(w) = 1 2w⊤Qw the dual is Ω∗(z) = 1 2z⊤Q−1z, hence the largest eigenvalues of Q and Q−1 would have a significant influence on the convergence. In terms of λ the number of iterations needed for convergence is O(λ−1). In practice the iteration count does increase with λ, albeit not as badly as predicted. This is likely due to the fact that the empirical risk Remp(w) is typically rather smooth and has a certain inherent curvature which acts as a natural regularizer in addition to the regularization afforded by λΩ[w]. 5 A Linesearch Variant The convergence analysis in Theorem 4 relied on a one-dimensional line search. Algorithm 1, however, uses a more complex quadratic program to solve the problem. Since even the simple updates promise good rates of convergence it is tempting to replace the corresponding step in the bundle update. This can lead to considerable savings in particular for smaller problems, where the time spent in the quadratic programming solver is a substantial fraction of the total runtime. To keep matters simple, we only consider quadratic regularization Ω(w) := 1 2 ∥w∥2. Note that Jt+1(η) := J∗ t+1((1 −η)αt, η) is a quadratic function in η, regardless of the choice of Remp[w]. Hence a line search only needs to determine first and second derivative as done in the proof of Theorem 4. It can be shown that ∂ηJt+1(0) = γt and ∂2 ηJt+1(0) = −1 λ ∥∂wJ(wt)∥2 = −1 λ ∥λwt + at+1∥2. Hence the optimal value of η is given by η = min(1, λγt/∥λwt + at+1∥2 2). (15) This means that we may update wt+1 = (1 −η)wt −η λat+1. In other words, we need not store past gradients for the update. To obtain γt note that we are computing Remp(wt) as part of the Taylor approximation step. Finally, Rt(wt) is given by w⊤A + b αt, hence it satisfies the same update relations. In particular, the fact that w⊤Aα = λ ∥w∥2 means that the only quantity we need to cache is b⊤αt as an auxiliary variable rt in order to compute γt efficiently. Experiments show that this simplified algorithm has essentially the same convergence properties. 6 Experiments In this section we show experimental results that demonstrate the merits of our algorithm and its analysis. Due to space constraints, we report results of experiments with two large datasets namely Astro-Physics (astro-ph) and Reuters-CCAT (reuters-ccat) [5, 12]. For a fair comparison with existing solvers we use the quadratic regularizer Ω(w) := λ 2 ∥w∥2, and the binary hinge loss. In our first experiment, we address the rate of convergence and its dependence on the value of λ. In Figure 2 we plot ϵt as a function of iterations for various values of λ using the QP solver at every iteration to solve the dual problem (6) to optimality. Initially, we observe super-linear convergence; this is consistent with our analysis. Surprisingly, even though theory predicts sub-linear speed of convergence for non-differentiable losses like the binary hinge loss (see (11)), our solver exhibits linear rates of convergence predicted only for differentiable functions (see (12)). We conjecture that the average over many piecewise linear functions, Remp(w), behaves essentially like a smooth function. As predicted, the convergence speed is inversely proportional to the value of λ. 6 Figure 2: We plot t as a function of the number of iterations. Note the logarithmic scale in t. Left: astro-ph; Right: reuters-ccat. Figure 3: Top: Objective function value as a function of time. Bottom: Objective function value as a function of iterations. Left: astro-ph, Right: reuters-ccat. The black line indicates the final value of the objective function + 0.001 . In our second experiment, we compare the convergence speed of two variants of the bundle method, namely, with a QP solver in the inner loop (which essentially boils down to SVMPerf) and the line search variant which we described in Section 5. We contrast these solvers with Pegasos [12] in the batch setting. Following [5] we set φ = 10 4 for reuters-ccat and φ = 2.10 4 for astro-ph. Figure 3 depicts the evolution of the primal objective function value as a function of both CPU time as well as the number of iterations. Following Shalev-Shwartz et al. [12] we investigate the time required by various solvers to reduce the objective value to within 0.001 of the optimum. This is depicted as a black horizontal line in our plots. As can be seen, Pegasos converges to this region quickly. Nevertheless, both variants of the bundle method converge to this value even faster (line search is slightly slower than Pegasos on astro-ph, but this is not always the case for many other large datasets we tested on). Note that both line search and Pegasos converge to within 0.001 precision rather quickly, but they require a large number of iterations to converge to the optimum. 7 Related Research Our work is closely related to Shalev-Shwartz and Singer [11] who prove mistake bounds for online algorithms by lower bounding the progress in the dual. Although not stated explicitly, essentially the same technique of lower bounding the dual improvement was used by Tsochantaridis et al. [16] to show polynomial time convergence of the SVMStruct algorithm. The main difference however is that Tsochantaridis et al. [16] only work with a quadratic objective function while the framework 7 proposed by Shalev-Shwartz and Singer [11] can handle arbitrary convex functions. In both cases, a weaker analysis led to O(1/ϵ2) rates of convergence for nonsmooth loss functions. On the other hand, our results establish a O(1/ϵ) rate for nonsmooth loss functions and O(log(1/ϵ)) rates for smooth loss functions under mild technical assumptions. Another related work is SVMPerf [5] which solves the SVM estimation problem in linear time. SVMPerf finds a solution with accuracy ϵ in O(md/(λϵ2)) time, where the m training patterns xi ∈Rd. This bound was improved by Shalev-Shwartz et al. [12] to ˜O(1/λδϵ) for obtaining an accuracy of ϵ with confidence 1 −δ. Their algorithm, Pegasos, essentially performs stochastic (sub)gradient descent but projects the solution back onto the L2 ball of radius 1/ √ λ. But, as our experiments show, performing an exact line search in the dual leads to a faster decrease in the value of primal objective. Note that Pegasos also can be used in an online setting. This, however, only applies whenever the empirical risk decomposes into individual loss terms (e.g. it is not applicable to multivariate performance scores). The third related strand of research considers gradient descent in the primal with a line search to choose the optimal step size, see e.g. [2, Section 9.3.1]. Under assumptions of smoothness and strong convexity – that is, the objective function can be upper and lower bounded by quadratic functions – it can be shown that gradient descent with line search will converge to an accuracy of ϵ in O(log(1/ϵ)) steps. The problem here is the line search in the primal, since evaluating the regularized risk functional might be as expensive as computing its gradient, thus rendering a line search in the primal unattractive. On the other hand, the dual objective is relatively simple to evaluate, thus making the line search in the dual, as performed by our algorithm, computationally feasible. Finally, we would like to point out connections to subgradient methods [7]. These algorithms are designed for nonsmooth functions, and essentially choose an arbitrary element of the subgradient set to perform a gradient descent like update. Let ∥Jw(w)∥≤G, and B(w∗, r) denote a ball of radius r centered around the minimizer of J(w). By applying the analysis of Nedich and Bertsekas [7] to the regularized risk minimization problem with Ω(w) := λ 2 ∥w∥2, Ratliff et al. [9] showed that subgradient descent with a fixed, but sufficiently small, stepsize will converge linearly to B(w∗, G/λ). References [1] Y. Altun, A. J. Smola, and T. Hofmann. Exponential families for conditional random fields. In UAI, pages 2–9, 2004. [2] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, 2004. [3] J. Hiriart-Urruty and C. Lemar´echal. Convex Analysis and Minimization Algorithms. 1993. [4] T. Joachims. A support vector method for multivariate performance measures. In ICML, pages 377–384, 2005. [5] T. Joachims. Training linear SVMs in linear time. In KDD, 2006. [6] S.-I. Lee, V. Ganapathi, and D. Koller. Efficient structure learning of Markov networks using L1regularization. In NIPS, pages 817–824, 2007. [7] A. Nedich and D. P. Bertsekas. Convergence rate of incremental subgradient algorithms. In Stochastic Optimization: Algorithms and Applications, pages 263–304. 2000. [8] J. Nocedal and S. J. Wright. Numerical Optimization. Springer, 1999. [9] N. Ratliff, J. Bagnell, and M. Zinkevich. (online) subgradient methods for structured prediction. In Proc. of AIStats, 2007. [10] R. T. Rockafellar. Convex Analysis. Princeton University Press, 1970. [11] S. Shalev-Shwartz and Y. Singer. Online learning meets optimization in the dual. In COLT, 2006. [12] S. Shalev-Shwartz, Y. Singer, and N. Srebro. Pegasos: Primal estimated sub-gradient solver for SVM. In ICML, 2007. [13] A. J. Smola, S. V. N. Vishwanathan, and Q. V. Le. Bundle methods for machine learning. JMLR, 2008. in preparation. [14] B. Taskar, C. Guestrin, and D. Koller. Max-margin Markov networks. In NIPS, pages 25–32, 2004. [15] C. H. Teo, Q. Le, A. Smola, and S. V. N. Vishwanathan. A scalable modular convex solver for regularized risk minimization. In KDD, 2007. [16] I. Tsochantaridis, T. Joachims, T. Hofmann, and Y. Altun. Large margin methods for structured and interdependent output variables. JMLR, 6:1453–1484, 2005. [17] V. Vapnik. The Nature of Statistical Learning Theory. Springer, 1995. [18] T. Zhang. Sequential greedy approximation for certain convex optimization problems. IEEE Trans. Information Theory, 49(3):682–691, 2003. 8
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Catching Up Faster in Bayesian Model Selection and Model Averaging Tim van Erven Peter Gr¨unwald Steven de Rooij Centrum voor Wiskunde en Informatica (CWI) Kruislaan 413, P.O. Box 94079 1090 GB Amsterdam, The Netherlands {Tim.van.Erven,Peter.Grunwald,Steven.de.Rooij}@cwi.nl Abstract Bayesian model averaging, model selection and their approximations such as BIC are generally statistically consistent, but sometimes achieve slower rates of convergence than other methods such as AIC and leave-one-out cross-validation. On the other hand, these other methods can be inconsistent. We identify the catch-up phenomenon as a novel explanation for the slow convergence of Bayesian methods. Based on this analysis we define the switch-distribution, a modification of the Bayesian model averaging distribution. We prove that in many situations model selection and prediction based on the switch-distribution is both consistent and achieves optimal convergence rates, thereby resolving the AIC-BIC dilemma. The method is practical; we give an efficient algorithm. 1 Introduction We consider inference based on a countable set of models (sets of probability distributions), focusing on two tasks: model selection and model averaging. In model selection tasks, the goal is to select the model that best explains the given data. In model averaging, the goal is to find the weighted combination of models that leads to the best prediction of future data from the same source. An attractive property of some criteria for model selection is that they are consistent under weak conditions, i.e. if the true distribution P ∗is in one of the models, then the P ∗-probability that this model is selected goes to one as the sample size increases. BIC [14], Bayes factor model selection [8], Minimum Description Length (MDL) model selection [3] and prequential model validation [5] are examples of widely used model selection criteria that are usually consistent. However, other model selection criteria such as AIC [1] and leave-one-out cross-validation (LOO) [16], while often inconsistent, do typically yield better predictions. This is especially the case in nonparametric settings, where P ∗can be arbitrarily well-approximated by a sequence of distributions in the (parametric) models under consideration, but is not itself contained in any of these. In many such cases, the predictive distribution converges to the true distribution at the optimal rate for AIC and LOO [15, 9], whereas in general BIC, the Bayes factor method and prequential validation only achieve the optimal rate to within an O(log n) factor [13, 20, 6]. In this paper we reconcile these seemingly conflicting approaches [19] by improving the rate of convergence achieved in Bayesian model selection without losing its convergence properties. First we provide an example to show why Bayes sometimes converges too slowly. Given priors on models M1, M2, . . . and parameters therein, Bayesian inference associates each model Mk with the marginal distribution pk, given in (1), obtained by averaging over the parameters according to the prior. In model selection the preferred model is the one with maximum a posteriori probability. By Bayes’ rule this is arg maxk pk(xn)w(k), where w(k) denotes the prior probability of Mk. We can further average over model indices, a process called Bayesian Model Averaging (BMA). The resulting distribution pbma(xn) = P k pk(xn)w(k) can be used for prediction. In a se1 quential setting, the probability of a data sequence xn := x1, . . . , xn under a distribution p typically decreases exponentially fast in n. It is therefore common to consider −log p(xn), which we call the codelength of xn achieved by p. We take all logarithms to base 2, allowing us to measure codelength in bits. The name codelength refers to the correspondence between codelength functions and probability distributions based on the Kraft inequality, but one may also think of the codelength as the accumulated log loss that is incurred if we sequentially predict the xi by conditioning on the past, i.e. using p(·|xi−1) [3, 6, 5, 11]. For BMA, we have −log pbma(xn) = Pn i=1 −log pbma(xi|xi−1). Here the ith term represents the loss incurred when predicting xi given xi−1 using pbma(·|xi−1), which turns out to be equal to the posterior average: pbma(xi|xi−1) = P k pk(xi|xi−1)w(k|xi−1). Prediction using pbma has the advantage that the codelength it achieves on xn is close to the codelength of pˆk, where ˆk is the index of best of the marginals p1, p2, . . . Namely, given a prior w on model indices, the difference between −log pbma(xn) = −log(P k pk(xn)w(k)) and −log pˆk(xn) must be in the range [0, −log w(ˆk)], whatever data xn are observed. Thus, using BMA for prediction is sensible if we are satisfied with doing essentially as well as the best model under consideration. However, it is often possible to combine p1, p2, . . . into a distribution that achieves smaller codelength than pˆk! This is possible if the index ˆk of the best distribution changes with the sample size in a predictable way. This is common in model selection, for example with nested models, say M1 ⊂M2. In this case p1 typically predicts better at small sample sizes (roughly, because M2 has more parameters that need to be learned than M1), while p2 predicts better eventually. Figure 1 illustrates this phenomenon. It shows the accumulated codelength difference −log p2(xn) −(−log p1(xn)) on “The Picture of Dorian Gray” by Oscar Wilde, where p1 and p2 are the Bayesian marginal distributions for the first-order and second-order Markov chains, respectively, and each character in the book is an outcome. Note that the example models M1 and M2 are very crude; for this particular application much better models are available. In more complicated, more realistic model selection scenarios, the models may still be wrong, but it may not be known how to improve them. Thus M1 and M2 serve as a simple illustration only. We used uniform priors on the model parameters, but for other common priors similar behaviour can be expected. Clearly p1 is better for about the first 100 000 outcomes, gaining a head start of approximately 40 000 bits. Ideally we should predict the initial 100 000 outcomes using p1 and the rest using p2. However, pbma only starts to behave like p2 when it catches up with p1 at a sample size of about 310 000, when the codelength of p2 drops below that of p1. Thus, in the shaded area pbma behaves like p1 while p2 is making better predictions of those outcomes: since at n = 100 000, p2 is 40 000 bits behind, and at n = 310 000, it has caught up, in between it must have outperformed p1 by 40 000 bits! The general pattern that first one model is −100000 −80000 −60000 −40000 −20000 0 20000 40000 60000 0 50000 100000 150000 200000 250000 300000 350000 400000 450000 Codelength difference with Markov order 1 (bits) Sample size Markov order 2 Bayesian Model Averaging Switch−Distribution Figure 1: The Catch-up Phenomenon better and then another occurs widely, both on real-world data and in theoretical settings. We argue that failure to take this effect into account leads to the suboptimal rate of convergence achieved by Bayes factor model selection and related methods. We have developed an alternative method to combine distributions p1 and p2 into a single distribution psw, which we call the switch-distribution, defined in Section 2. Figure 1 shows that psw behaves like p1 initially, but in contrast to pbma it starts to mimic p2 almost immediately after p2 starts making better predictions; it essentially does this no matter what sequence xn is actually observed. psw differs from pbma in that it is based on a prior distribution on sequences of models rather than simply a prior distribution on models. This allows us to avoid the implicit assumption that there is one model which is best at all sample sizes. After conditioning on past observations, the posterior we obtain gives a better indication of which model performs best at the current sample size, thereby achieving a faster rate of convergence. Indeed, the switch-distribution is related to earlier algorithms for tracking the best expert developed in the universal prediction literature [7, 18, 17, 10]; however, the applications we have in mind and the theorems we prove are completely different. In Sections 3 and 4 we show that model selection based on the switch-distribution is consistent (Theorem 1), but unlike standard 2 Bayes factor model selection achieves optimal rates of convergence (Theorem 2). Proofs of the theorems are in Appendix A. In Section 5 we give a practical algorithm that computes the switchdistribution for K (rather than 2) predictors in Θ(n · K) time. In the full paper, we will give further details of the proof of Theorem 1 and a more detailed discussion of Theorem 2 and the implications of both theorems. 2 The Switch-Distribution for Model Selection and Prediction Preliminaries Suppose X∞= (X1, X2, . . .) is a sequence of random variables that take values in sample space X ⊆Rd for some d ∈Z+ = {1, 2, . . .}. For n ∈N = {0, 1, 2, . . .}, let xn = (x1, . . ., xn) denote the first n outcomes of X∞, such that xn takes values in the product space X n = X1 × · · · × Xn. (We let x0 denote the empty sequence.) Let X ∗= S∞ n=0 X n. For m > n, we write Xm n+1 for (Xn+1, . . ., Xm), where m = ∞is allowed and we omit the subscript when n = 0. Any distribution P(X∞) may be defined by a sequential prediction strategy p that predicts the next outcome at any time n ∈N. To be precise: Given the previous outcomes xn at time n, this prediction strategy should issue a conditional density p(Xn+1|xn) with corresponding distribution P(Xn+1|xn) for the next outcome Xn+1. Such sequential prediction strategies are sometimes called prequential forecasting systems [5]. An instance is given in Example 1 below. We assume that the density p(Xn+1|xn) is taken relative to either the usual Lebesgue measure (if X is continuous) or the counting measure (if X is countable). In the latter case p(Xn+1|xn) is a probability mass function. It is natural to define the joint density p(xm|xn) = p(xn+1|xn) · · · p(xm|xm−1) and let P(X∞ n+1|xn) be the unique distribution such that, for all m > n, p(Xm n+1|xn) is the density of its marginal distribution for Xm n+1. To ensure that P(X∞ n+1|xn) is well-defined even if X is continuous, we impose the natural requirement that for any k ∈Z+ and any fixed event Ak+1 ⊆Xk+1 the probability P(Ak+1|xk) is a measurable function of xk, which holds automatically if X is countable. Model Selection and Prediction The goal in model selection is to choose an explanation for observed data xn from a potentially infinite list of candidate models M1, M2, . . . We consider parametric models, which are sets {pθ : θ ∈Θ} of prediction strategies pθ that are indexed by elements of Θ ⊆Rd, for some smallest possible d ∈N, the number of degrees of freedom. Examples of model selection are regression based on a set of basis functions such as polynomials (d is the number of coefficients of the polynomial), the variable selection problem in regression [15, 9, 20] (d is the number of variables), and histogram density estimation [13] (d is the number of bins). A model selection criterion is a function δ : X ∗→Z+ that, given any data sequence xn ∈X ∗, selects the model Mk with index k = δ(xn). We associate each model Mk with a single prediction strategy ¯pk. The bar emphasizes that ¯pk is a meta-strategy based on the prediction strategies in Mk. In many approaches to model selection, for example AIC and LOO, ¯pk is defined using some estimator ˆθk for each model Mk, which maps a sequence xn of previous observations to an estimated parameter value that represents a “best guess” of the true/best distribution in the model. Prediction is then based on this estimator: ¯pk(Xn+1 | xn) = pˆθk(xn)(Xn+1 | xn), which also defines a joint density ¯pk(xn) = ¯pk(x1) · · · ¯pk(xn|xn−1). The Bayesian approach to model selection or model averaging goes the other way around. We start out with a prior w on Θk, and define the Bayesian marginal density ¯pk(xn) = Z θ∈Θk pθ(xn)w(θ) dθ. (1) When ¯pk(xn) is non-zero this joint density induces a unique conditional density ¯pk(Xn+1 | xn) = ¯pk(Xn+1, xn)/¯pk(xn), which is equal to the mixture of pθ ∈Mk according to the posterior, w(θ|xn) = pθ(xn)w(θ)/ R pθ(xn)w(θ) dθ, based on xn. Thus the Bayesian approach also defines a prediction strategy ¯pk(Xn+1|xn), whose corresponding distribution may be thought of as an estimator. From now on we sometimes call the distributions induced by ¯p1, ¯p2, . . . “estimators”, even if they are Bayesian. This unified view is known as prequential or predictive MDL [11, 5]. Example 1. Suppose X = {0, 1}. Then a prediction strategy ¯p may be based on the Bernoulli model M = {pθ | θ ∈[0, 1]} that regards X∞as a sequence of independent, identically distributed Bernoulli random variables with Pθ(Xn+1 = 1) = θ. We may predict Xn+1 using the maximum likelihood (ML) estimator based on the past, i.e. using ˆθ(xn) = n−1 Pn i=1 xi. The prediction for x1 is then undefined. If we use a smoothed ML estimator such as the Laplace estimator, ˆθ′(xn) = 3 (n + 2)−1(Pn i=1 xi + 1), then all predictions are well-defined. Perhaps surprisingly, the predictor ¯p′ defined by ¯p′(Xn+1 | xn) = pˆθ′(xn)(Xn+1) equals the Bayesian predictive distribution based on a uniform prior. Thus in this case a Bayesian predictor and an estimation-based predictor coincide! The Switch-Distribution Suppose p1, p2, . . . is a list of prediction strategies for X∞. (Although here the list is infinitely long, the developments below can with little modification be adjusted to the case where the list is finite.) We first define a family Q = {qs : s ∈S} of combinator prediction strategies that switch between the original prediction strategies. Here the parameter space S is defined as S = {(t1, k1), . . . , (tm, km) ∈(N × Z+)m | m ∈Z+, 0 = t1 < . . . < tm}. (2) The parameter s ∈S specifies the identities of m constituent prediction strategies and the sample sizes, called switch-points, at which to switch between them. For s = ((t′ 1, k′ 1), . . . , (t′ m′, k′ m′)), we define ti(s) = t′ i, ki(s) = k′ i and m(s) = m′. We omit the argument when the parameter s is clear from context, e.g. we write t3 for t3(s). For each s ∈S the corresponding qs ∈Q is defined as: qs(Xn+1|xn) = pk1(Xn+1|xn) if n < t2, pk2(Xn+1|xn) if t2 ≤n < t3, ... ... pkm−1(Xn+1|xn) if tm−1 ≤n < tm, pkm(Xn+1|xn) if tm ≤n. (3) Switching to the same predictor multiple times is allowed. The extra switch-point t1 is included to simplify notation; we always take t1 = 0. Now the switch-distribution is defined as a Bayesian mixture of the elements of Q according to a prior π on S: Definition 1 (Switch-Distribution). Let π be a probability mass function on S. Then the switchdistribution Psw with prior π is the distribution for X∞such that, for any n ∈Z+, the density of its marginal distribution for Xn is given by psw(xn) = X s∈S qs(xn) · π(s). (4) Although the switch-distribution provides a general way to combine prediction strategies, in this paper it will only be applied to combine prediction strategies ¯p1, ¯p2, . . . that correspond to models. In this case we may define a corresponding model selection criterion δsw. To this end, let Kn+1 : S →Z+ be a random variable that denotes the strategy/model that is used to predict Xn+1 given past observations xn. Formally, Kn+1(s) = ki(s) iff ti(s) ≤n and i = m(s) ∨n < ti+1(s). Algorithm 1, given in Section 5, efficiently computes the posterior distribution on Kn+1 given xn: π(Kn+1 = k | xn) = P {s:Kn+1(s)=k} π s qs(xn) psw(xn) , (5) which is defined whenever psw(xn) is non-zero. We turn this into a model selection criterion δsw(xn) = arg maxk π(Kn+1 = k|xn) that selects the model with maximum posterior probability. 3 Consistency If one of the models, say with index k∗, is actually true, then it is natural to ask whether δsw is consistent, in the sense that it asymptotically selects k∗with probability 1. Theorem 1 below states that this is the case under certain conditions which are only slightly stronger than those required for the consistency of standard Bayes factor model selection. Bayes factor model selection is consistent if for all k, k′ ̸= k, ¯Pk(X∞) and ¯Pk′(X∞) are mutually singular, that is, if there exists a measurable set A ⊆X ∞such that ¯Pk(A) = 1 and ¯Pk′(A) = 0 [3]. For example, this can usually be shown to hold if the models are nested and for each k, Θk is a subset of Θk+1 of wk+1-measure 0 [6]. For consistency of δsw, we need to strengthen this to the requirement that, for all k′ ̸= k and all xn ∈X ∗, the distributions ¯Pk(X∞ n+1 | xn) and ¯Pk′(X∞ n+1 | xn) are mutually singular. For example, if X1, X2, . . . are i.i.d. according to each Pθ in all models, but also if X is countable and ¯pk(xn+1 | xn) > 0 for all k, all xn+1 ∈X n+1, then this conditional mutual singularity is automatically implied by ordinary mutual singularity of ¯Pk(X∞) and ¯Pk′(X∞). 4 Let Es = {s′ ∈S | m(s′) > m(s), (ti(s′), ki(s′)) = (ti(s), ki(s)) for i = 1, . . . , m(s)} denote the set of all possible extensions of s to more switch-points. Let ¯p1, ¯p2, . . . be Bayesian prediction strategies with respective parameter spaces Θ1, Θ2, . . . and priors w1, w2, . . ., and let π be the prior of the corresponding switch-distribution. Theorem 1 (Consistency of the Switch-Distribution). Suppose π is positive everywhere on {s ∈ S | m(s) = 1} and is such that there exists a positive constant c such that, for every s ∈S, c · π(s) ≥π(Es). Suppose further that ¯Pk(X∞ n+1 | xn) and ¯Pk′(X∞ n+1 | xn) are mutually singular for all k, k′ ∈Z+, k ̸= k′, xn ∈X ∗. Then, for all k∗∈Z+, for all θ∗∈Θk∗except for a subset of Θk∗of wk∗-measure 0, the posterior distribution on Kn+1 satisfies π(Kn+1 = k∗| Xn) n→∞ −→1 with Pθ∗-probability 1. (6) The requirement that c · π(s) ≥π(Es) is automatically satisfied if π is of the form: π(s) = πM(m)πK(k1) m Y i=2 πT(ti|ti > ti−1)πK(ki), (7) where πM, πK and πT are priors on Z+ with full support, and πM is geometric: πM(m) = θm−1(1−θ) for some 0 ≤θ < 1. In this case c = θ/(1 −θ). 4 Optimal Risk Convergence Rates Suppose X1, X2, . . . are distributed according to P ∗. We define the risk at sample size n ≥1 of the estimator ¯P relative to P ∗as Rn(P ∗, ¯P) = EXn−1∼P ∗[D(P ∗(Xn = · | Xn−1)∥¯P(Xn = · | Xn−1))], where D(·∥·) is the Kullback-Leibler (KL) divergence [4]. This is the standard definition of risk relative to KL divergence. The risk is always well-defined, and equal to 0 if ¯P(Xn+1 | Xn) is equal to P ∗(Xn+1 | Xn). The following identity connects information-theoretic redundancy and accumulated statistical risk (see [4] or [6, Chapter 15]): If P ∗admits a density p∗, then for all prediction strategies ¯p, EXn∼P ∗[−log ¯p(Xn) + log p∗(Xn)] = n X i=1 Ri(P ∗, ¯P). (8) For a union of parametric models M = S k≥1 Mk, we define the information closure ⟨M⟩= {P ∗| infP ∈M D(P ∗∥P) = 0}, i.e. the set of distributions for X∞that can be arbitrarily well approximated by elements of M. Theorem 2 below shows that, for a very large class of P ∗∈⟨M⟩, the switch-distribution defined relative to estimators ¯P1, ¯P2, . . . achieves the same risk as any other model selection criterion defined with respect to the same estimators, up to lower order terms; in other words, model averaging based on the switch-distribution achieves at least the same rate of convergence as model selection based on any model selection criterion whatsoever (the issue of averaging vs selection will be discussed at length in the full paper). The theorem requires that the prior π in (4) is of the form (7), and satisfies −log πM(m) = O(m) ; −log πK(k) = O(log k) ; −log πT(t) = O(log t). (9) Thus, πM, the prior on the total number of switch points, is allowed to decrease either polynomially or exponentially (as required for Theorem 1); πT and πK must decrease polynomially. For example, we could set πT(t) = πK(t) = 1/(t(t + 1)), or we could take the universal prior on the integers [12]. Let M∗⊂⟨M⟩be some subset of interest of the information closure of model M. M∗may consist of just a single, arbitrary distribution P ∗in ⟨M⟩\M – in that case Theorem 2 shows that the switchdistribution converges as fast as any other model selection criterion on any distribution in ⟨M⟩that cannot be expressed parametrically relative to M – or it may be a large, nonparametric family. In that case, Theorem 2 shows that the switch-distribution achieves the minimax convergence rate. For example, if the models Mk are k-bin histograms [13], then ⟨M⟩contains every distribution on [0, 1] with bounded continuous densities, and we may, for example, take M∗to be the set of all distributions on [0, 1] which have a differentiable density p∗such that p∗(x) and (d/dx)p∗(x) are bounded from below and above by some positive constants. We restrict ourselves to model selection criteria which, at sample size n, never select a model Mk with k > nτ for some arbitrarily large but fixed τ > 0; note that this condition will be met for most 5 practical model selection criteria. Let h : Z+ →R+ denote the minimax optimal achievable risk as a function of the sample size, i.e. h(n) = inf δ:X n→{1,2,...,⌈nτ ⌉} sup P ∗∈M∗sup n′≥n Rn′(P ∗, ¯Pδ), (10) where the infimum is over all model selection criteria restricted to sample size n, and ⌈·⌉denotes rounding up to the nearest integer. ¯pδ is the prediction strategy satisfying, for all n′ ≥n, all xn′ ∈X n′, ¯pδ(Xn′+1 | xn′) := ¯pδ(xn)(Xn′+1 | xn′), i.e. at sample size n it predicts xn+1 using ¯pk for the k = δ(Xn) chosen by δ, and it keeps predicting future xn′+1 by this k. We call h(n) the minimax optimal rate of convergence for model selection relative to data from M∗, model list M1, M2, . . ., and estimators ¯P1, ¯P2, . . . The definition is slightly nonstandard, in that we require a second supremum over n′ ≥n. This is needed because, as will be discussed in the full paper, it can sometimes happen that, for some P ∗, some k, some n′ > n, Rn′(P ∗, ¯Pk) > Rn(P ∗, ¯Pk) (see also [4, Section 7.1]). In cases where this cannot happen, such as regression with standard ML estimators, and in cases where, uniformly for all k, supn′≥n Rn′(P ∗, ¯Pk)−Rn(P ∗, ¯Pk) = o(Pn i=1 h(i)) (in the full paper we show that this holds for, for example, histogram density estimation), our Theorem 2 also implies minimax convergence in terms of the standard definition, without the supn′≥n. We expect that the supn′≥n can be safely ignored for most “reasonable” models and estimators. Theorem 2. Define Psw for some model class M = ∪k≥1Mk as in (4), where the prior π satisfies (9). Let M∗be a subset of ⟨M⟩with minimax rate h such that nh(n) is increasing, and nh(n)/(log n)2 →∞. Then lim sup n→∞ supP ∗∈M∗Pn i=1 Ri(P ∗, Psw) Pn i=1 h(i) ≤1. (11) The requirement that nh(n)/(log n)2 →∞will typically be satisfied whenever M∗\ M is nonempty. Then M∗contains P ∗that are “nonparametric” relative to the chosen sequence of models M1, M2, . . . Thus, the problem should not be “too simple”: we do not know whether (11) holds in the parametric setting where P ∗∈Mk for some k on the list. Theorem 2 expresses that the accumulated risk of the switch-distribution, as n increases, is not significantly larger than the accumulated risk of any other procedure. This “convergence in sum” has been considered before by, for example, [13, 4], and is compared to ordinary convergence in the full paper, where we will also give example applications of the theorem and further discuss (10). The proof works by bounding the redundancy of the switch-distribution, which, by (8), is identical to the accumulated risk. It is not clear whether similar techniques can be used to bound the individual risk. 5 Computing the Switch-Distribution Algorithm 1 sequentially computes the posterior probability on predictors p1, p2, . . .. It requires that π is a prior of the form in (7), and πM is geometric, as is also required for Theorem 1 and permitted in Theorem 2. The algorithm resembles FIXED-SHARE [7], but whereas FIXED-SHARE implicitly imposes a geometric distribution for πT, we allow general priors by varying the shared weight with n. We do require slightly more space to cope with πM. Algorithm 1 SWITCH(xN) ⊲K is the number of experts; θ is as in the definition of πM. for k=1, . . . , K do initialise wa k ←θ · πK(k); wb k ←(1 −θ) · πK(k) od Report prior π(K1) = wa K1 (a K-sized array) for n=1, . . . , N do for k=1, . . . , K do wa k ←wa k · pk(xn|xn−1); wb k ←wb k · pk(xn|xn−1) od (loss update) pool ←πT(Z = n | Z ≥n) · P k wa k (share update) for k=1, . . . , K do wa k ←wa k · πT(Z ̸= n | Z ≥n) + θ · pool · πK(k) wb k ←wb k + (1 −θ) · pool · πK(k) od Report posterior π(Kn+1 | xn) = (wa Kn+1+ wb Kn+1)/ P k(wa k + wb k) (a K-sized array) od This algorithm can be used to obtain fast convergence in the sense of Theorem 2, which can be extended to cope with a restriction to only the first K experts. Theorem 1 can be extended to show 6 consistency in this case as well. If πT(Z = n | Z ≥n) and πK(k) can be computed in constant time, then the running time is Θ(N · K), which is of the same order as that of fast model selection criteria like AIC and BIC. We will explain this algorithm in more detail in a forthcoming publication. Acknowledgements We thank Y. Mansour, whose remark over lunch at COLT 2005 sparked off all this research. We thank P. Harremo¨es and W. Koolen for mathematical support. This work was supported in part by the IST Programme of the European Community, under the PASCAL Network of Excellence, IST-2002-506778. This publication only reflects the authors’ views. A Proofs Proof of Theorem 1. Let Un = {s ∈S | Kn+1(s) ̸= k∗} denote the set of ‘bad’ parameters s that select an incorrect model. It is sufficient to show that lim n→∞ P s∈Un π s qs(Xn) P s∈S π s qs(Xn) = 0 with ¯Pk∗-probability 1. (12) To see this, suppose the theorem is false. Then there exists a Φ ⊆Θk∗with wk∗(Φ) > 0 such that (6) does not hold for any θ∗∈Φ. But then by definition of ¯Pk∗we have a contradiction with (12). Now let A = {s ∈S : km(s) ̸= k∗} denote the set of parameters that are bad for sufficiently large n. We observe that for each s′ ∈Un there exists at least one element s ∈A that uses the same sequence of switch-points and predictors on the first n + 1 outcomes (this implies that Ki(s) = Ki(s′) for i = 1, . . . , n + 1) and has no switch-points beyond n (i.e. tm(s) ≤n). Consequently, either s′ = s or s′ ∈Es. Therefore X s′∈Un π(s′)qs′(xn) ≤ X s∈A (π(s) + π(Es)) qs(xn) ≤(1 + c) X s∈A π(s)qs(xn). (13) Defining the mixture r(xn) = P s∈A π(s)qs(xn), we will show that lim n→∞ r(Xn) π(s = (0, k∗)) · ¯pk∗(Xn) = 0 with ¯Pk∗-probability 1. (14) Using (13) and the fact that P s∈S π(s)qs(xn) ≥π(s = (0, k∗)) · ¯pk∗(xn), this implies (12). For all s ∈A and xtm(s) ∈X tm(s), by definition Qs(X∞ tm+1|xtm) equals ¯Pkm(X∞ tm+1|xtm), which is mutually singular with ¯Pk∗(X∞ tm+1|xtm) by assumption. If X is a separable metric space, which holds because X ⊆Rd for some d ∈Z+, it can be shown that this conditional mutual singularity implies mutual singularity of Qs(X∞) and ¯Pk∗(X∞). To see this for countable X, let Bxtm be any event such that Qs(Bxtm |xtm) = 1 and ¯Pk∗(Bxtm |xtm) = 0. Then, for B = {y∞∈X ∞| y∞ tm+1 ∈ Bytm}, we have that Qs(B) = 1 and ¯Pk∗(B) = 0. In the uncountable case, however, B may not be measurable. We omit the full proof, which was shown to us by P. Harremo¨es. Any countable mixture of distributions that are mutually singular with Pk∗, in particular R, is mutually singular with Pk∗. This implies (14) by Lemma 3.1 of [2], which says that for any two mutually singular distributions R and P, the density ratio r(Xn)/p(Xn) goes to 0 as n →∞with P-probability 1. Proof of Theorem 2. We will show that for every α > 1, sup P ∗∈M∗ n X i=1 Ri(P ∗, Psw) ≤α n X i=1 h(i) + ǫα,n n X i=1 h(i), (15) where ǫα,n n→∞ −→0, and ǫα,1, ǫα,2, . . . are fixed constants that only depend on α, but not on the chosen subset M∗of ⟨M⟩. Theorem 2 is a consequence of (15), which we will proceed to prove. Let δn : X n →{1, . . . , ⌈nτ⌉} be a model selection criterion, restricted to samples of size n, that is minimax optimal, i.e. it achieves the infimum in (10). If such a δn does not exist, we take a δn that is almost minimax optimal in the sense that it achieves the infimum to within h(n)/n. For j ≥1, let tj = ⌈αj−1⌉−1. Fix an arbitrary n > 0 and let m be the unique integer such that tm < n ≤tm+1. We will first show that for arbitrary xn, psw achieves redundancy not much worse than qs with s = (t1, k1), . . . , (tm, km), where ki = δti(xti). Then we show that the redundancy of this qs is small enough for (15) to hold. Thus, to achieve this redundancy, it is sufficient to take only a logarithmic number m −1 of switch-points: m −1 < logα(n + 1). Formally, we have, for some c > 0, uniformly for all n, xn ∈X n, 7 −log psw(xn) = −log X s′∈S qs′(xn)π(s′) ≤−log qs(xn) −log πM(m) − m X j=1 log πT(tj)πK(kj) ≤−log qs(xn) + c log(n + 1) + cm(τ + 1) log n = −log qs(xn) + O((log n)2). (16) Here the second inequality follows because of (9), and the final equality follows because m ≤ logα(n + 1) + 1. Now fix any P ∗∈⟨M⟩. Since P ∗∈⟨M⟩, it must have some density p∗. Thus, applying (8), and then (16), and then (8) again, we find that n X i=1 Ri(P ∗, Psw) = EXn∼P ∗[−log psw(Xn) + log p∗(Xn)] ≤EXn∼P ∗[−log qs(Xn) + log p∗(Xn)] + O((log n)2) = n X i=1 Ri(P ∗, Qs) + O((log n)2) = m X j=1 min{tj+1,n} X i=tj+1 Ri(P ∗, ¯Pkj) + O((log n)2). (17) For i appearing in the second sum, with tj < i ≤ tj+1, we have Ri(P ∗, ¯Pkj) ≤ supi′≥tj+1 Ri′(P ∗, ¯Pkj) = supi′≥tj+1 Ri′(P ∗, ¯Pδtj (xtj )) ≤h(tj + 1), so that Ri(P ∗, ¯Pkj) ≤ 1 tj + 1 · (tj + 1)h(tj + 1) ≤ 1 tj + 1 · ih(i) ≤tj+1 tj + 1h(i) ≤αh(i), where the middle inequality follows because nh(n) is increasing (condition (b) of the theorem). Summing over i, we get Pm j=1 Pmin{tj+1,n} i=tj+1 Ri(P ∗, ¯Pkj) ≤α Pn i=1 h(i). Combining this with (17), it follows that Pn i=1 Ri(P ∗, Psw) ≤α Pn i=1 h(i) + O((log n)2). Because this holds for arbitrary P ∗∈M∗(with the constant in the O notation not depending on P ∗), (15) now follows by the requirement of Theorem 2 that nh(n)/(log n)2 →∞. References [1] H. Akaike. A new look at statistical model identification. IEEE T. Automat. Contr., 19(6):716–723, 1974. [2] A. Barron. Logically Smooth Density Estimation. PhD thesis, Stanford University, Stanford, CA, 1985. [3] A. Barron, J. Rissanen, and B. Yu. The minimum description length principle in coding and modeling. IEEE T. Inform. Theory, 44(6):2743–2760, 1998. [4] A. R. Barron. Information-theoretic characterization of Bayes performance and the choice of priors in parametric and nonparametric problems. In Bayesian Statistics 6, pages 27–52, 1998. [5] A. P. Dawid. Statistical theory: The prequential approach. J. Roy. Stat. Soc. A, 147, Part 2:278–292, 1984. [6] P. D. Gr¨unwald. The Minimum Description Length Principle. The MIT Press, 2007. [7] M. Herbster and M. K. Warmuth. Tracking the best expert. Machine Learning, 32:151–178, 1998. [8] R. E. Kass and A. E. Raftery. Bayes factors. J. Am. Stat. Assoc., 90(430):773–795, 1995. [9] K. Li. Asymptotic optimality of cp , cl, cross-validation and generalized cross-validation: Discrete index set. Ann. Stat., 15:958–975, 1987. [10] C. Monteleoni and T. Jaakkola. Online learning of non-stationary sequences. In Advances in Neural Information Processing Systems, volume 16, Cambridge, MA, 2004. MIT Press. [11] J. Rissanen. Universal coding, information, prediction, and estimation. IEEE T. Inform. Theory, IT-30(4): 629–636, 1984. [12] J. Rissanen. Stochastic Complexity in Statistical Inquiry. World Scientific, 1989. [13] J. Rissanen, T. P. Speed, and B. Yu. Density estimation by stochastic complexity. IEEE T. Inform. Theory, 38(2):315–323, 1992. [14] G. Schwarz. Estimating the dimension of a model. Ann. Stat., 6(2):461–464, 1978. [15] R. Shibata. Asymptotic mean efficiency of a selection of regression variables. Ann. I. Stat. Math., 35: 415–423, 1983. [16] M. Stone. An asymptotic equivalence of choice of model by cross-validation and Akaike’s criterion. J. Roy. Stat. Soc. B, 39:44–47, 1977. [17] P. Volf and F. Willems. Switching between two universal source coding algorithms. In Proceedings of the Data Compression Conference, Snowbird, Utah, pages 491–500, 1998. [18] V. Vovk. Derandomizing stochastic prediction strategies. Machine Learning, 35:247–282, 1999. [19] Y. Yang. Can the strengths of AIC and BIC be shared? Biometrica, 92(4):937–950, 2005. [20] Y. Yang. Model selection for nonparametric regression. Statistica Sinica, 9:475–499, 1999. 8
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Nearest-Neighbor-Based Active Learning for Rare Category Detection Jingrui He School of Computer Science Carnegie Mellon University jingruih@cs.cmu.edu Jaime Carbonell School of Computer Science Carnegie Mellon University jgc@cs.cmu.edu Abstract Rare category detection is an open challenge for active learning, especially in the de-novo case (no labeled examples), but of significant practical importance for data mining - e.g. detecting new financial transaction fraud patterns, where normal legitimate transactions dominate. This paper develops a new method for detecting an instance of each minority class via an unsupervised local-density-differential sampling strategy. Essentially a variable-scale nearest neighbor process is used to optimize the probability of sampling tightly-grouped minority classes, subject to a local smoothness assumption of the majority class. Results on both synthetic and real data sets are very positive, detecting each minority class with only a fraction of the actively sampled points required by random sampling and by Pelleg’s Interleave method, the prior best technique in the sparse literature on this topic. 1 Introduction In many real world problems, the proportion of data points in different classes is highly skewed: some classes dominate the data set (majority classes), and the remaining classes may have only a few examples (minority classes). However, it is very important to detect examples from the minority classes via active learning. For example, in fraud detection tasks, most of the records correspond to normal transactions, and yet once we identify a new type of fraud transaction, we are well on our way to stopping similar future fraud transactions [2]. Another example is in astronomy. Most of the objects in sky survey images are explainable by current theories and models. Only 0.001% of the objects are truly beyond the scope of current science and may lead to new discoveries [8]. Rare category detection is also a bottleneck in reducing the sampling complexity of active learning [1, 5]. The difference between rare category detection and outlier detection is that: in rare category detection, the examples from one or more minority classes are often self-similar, potentially forming compact clusters, while in outlier detection, the outliers are typically scattered. Currently, only a few methods have been proposed to address this challenge. For example, in [8], the authors assumed a mixture model to fit the data, and selected examples for labeling according to different criteria; in [6], the authors proposed a generic consistency algorithm, and proved upper bounds and lower bounds for this algorithm in some specific situations. Most of the existing methods require that the majority classes and the minority classes be separable or work best in the separable case. However, in real applications, the support regions of the majority and minority classes often overlap, which affects negatively the performance of these methods. In this paper, we propose a novel method for rare category detection in the context of active learning. We typically start de-novo, no category labels, though our algorithm makes no such assumption. Different from existing methods, we aim to solve the hard case, i.e. we do not assume separability or near-separability of the classes. Intuitively, the method makes use of nearest neighbors to measure local density around each example. In each iteration, the algorithm selects an example with the 1 maximum change in local density on a certain scale, and asks the oracle for its label. The method stops once it has found at least one example from each class (given the knowledge of the number of classes). When the minority classes form compact clusters and the majority class distribution is locally smooth, the method will select examples both on the boundary and in the interior of the minority classes, and is proved to be effective theoretically. Experimental results on both synthetic and real data sets show the superiority of our method over existing methods. The rest of the paper is organized as follows. In Section 2, we introduce our method and provide theoretical justification, first for binary classes and then for multiple classes. Section 3 gives experimental results. Finally, we conclude the paper in Section 4. 2 Rare category detection 2.1 Problem definition Given a set of unlabeled examples S = {x1, . . . , xn}, xi ∈Rd, which come from m distinct classes, i.e. yi ∈{1, . . . , m}, the goal is to find at least one example from each class by requesting as few total labels as possible. For the sake of simplicity, assume that there is only one majority class, which corresponds to yi = 1, and all the other classes are minority classes. 2.2 Rare category detection for the binary case First let us focus on the simplest case where m = 2, and Pr[yi = 1] ≫Pr[yi = 2] = p, i.e. p ≪1. Here, we assume that we have an estimate of the value of p a priori. Next, we introduce our method for rare category detection based on nearest neighbors, which is presented in Algorithm 1. The basic idea is to find maximum changes in local density, which might indicate the location of a rare category. The algorithm works as follows. Given the unlabeled set S and the prior of the minority class p, we first estimate the number K of minority class examples in S. Then, for each example, we record its distance from the Kth nearest neighbor, which could be realized by kd-trees [7]. The minimum distance over all the examples is assigned to r′. Next, we draw a hyper-ball centered at each example with radius r′, and count the number of examples enclosed by this hyper-ball, which is denoted as ni. ni is roughly in proportion to the local density. To measure the change of local density around a certain point xi, in each iteration of Step 3, we subtract nj of neighboring points from ni, and let the maximum value be the score of xi. The example with the maximum score is selected for labeling by the oracle. If the example is from the minority class, stop the iteration; otherwise, enlarge the neighborhood where the scores of the examples are re-calculated and continue. Before giving theoretical justification, here, we give an intuitive explanation of why the algorithm works. Assume that the minority class is concentrated in a small region and the probability distribution function (pdf) of the majority class is locally smooth. Firstly, since the support region of the minority class is very small, it is important to find its scale. The r′ value obtained in Step 1 will be used to calculate the local density ni. Since r′ is based on the minimum Kth nearest neighbor distance, it is never too large to smooth out changes of local density, and thus it is a good measure of the scale. Secondly, the score of a certain point, corresponding to the change in local density, is the maximum of the difference in local density between this point and all of its neighboring points. In this way, we are not only able to select points on the boundary of the minority class, but also points in the interior, given that the region is small. Finally, by gradually enlarging the neighborhood where the scores are calculated, we can further explore the interior of the support region, and increase our chance of finding a minority class example. 2.3 Correctness In this subsection, we prove that if the minority class is concentrated in a small region and the pdf of the majority class is locally smooth, the proposed algorithm will repeatedly sample in the region where minority class examples occur with high probability. Let f1(x) and f2(x) denote the pdf of the majority and minority classes respectively, where x ∈Rd. To be precise, we make the following assumptions. 2 Algorithm 1 Nearest-Neighbor-Based Rare Category Detection for the Binary Case (NNDB) Require: S, p 1: Let K = np. For each example, calculate the distance to its Kth nearest neighbor. Set r′ to be the minimum value among all the examples. 2: ∀xi ∈S, let NN(xi, r′) = {x|x ∈S, ∥x −xi∥≤r′}, and ni = |NN(xi, r′)|. 3: for t = 1 : n do 4: ∀xi ∈S, if xi has not been selected, then si = max xj∈NN(xi,tr′)(ni −nj); otherwise, si = −∞. 5: Query x = arg maxxi∈S si. 6: If the label of x is 2, break. 7: end for Assumptions 1. f2(x) is uniform within a hyper-ball B of radius r centered at b, i.e. f2(x) = 1 V (r), if x ∈B; and 0 otherwise, where V (r) ∝rd is the volume of B. 2. f1(x) is bounded and positive in B1, i.e. f1(x) ≥ c1p (1−p)V (r), ∀x ∈B and f1(x) ≤ c2p (1−p)V (r), ∀x ∈Rd, where c1, c2 > 0 are two constants. With the above assumptions, we have the following claim and theorem. Note that variants of the following proof apply if we assume a different minority class distribution, such as a tight Gaussian. Claim 1. ∀ϵ, δ > 0, if n ≥max{ 1 2c2 1p2 log 3 δ, 1 2(1−2−d)2p2 log 3 δ, 1 ϵ4V ( r2 2 )4 log 3 δ}, where r2 = r (1+c2) 1 d , and V ( r2 2 ) is the volume of a hyper-ball with radius r2 2 , then with probability at least 1−δ, r2 2 ≤r′ ≤r and | ni n −E( ni n )| ≤ϵV (r′), 1 ≤i ≤n, where V (r′) is the volume of a hyper-ball with radius r′. Proof. First, notice that the expected proportion of points falling inside B, E( |NN(b,r)| n ) ≥(c1+1)p, and that the maximum expected proportion of points falling inside any hyper-ball of radius r2 2 , max x∈Rd[E( |NN(x, r2 2 )| n )] ≤2−dp. Then Pr[r′ > r or r′ < r2 2 or ∃xi ∈S s.t. |ni n −E(ni n )| > ϵV (r′)] ≤Pr[r′ > r] + Pr[r′ < r2 2 ] + Pr[r′ ≥r2 2 and ∃xi ∈S s.t. |ni n −E(ni n )| > ϵV (r′)] ≤Pr[|NN(b, r)| < K] + Pr[max x∈Rd |NN(x, r2 2 )| > K] + n Pr[|ni n −E(ni n )| > ϵV (r′)|r′ ≥r2 2 ] = Pr[|NN(b, r) n | < p] + Pr[max x∈Rd |NN(x, r2 2 ) n | > p] + n Pr[|ni n −E(ni n )| > ϵV (r′)|r′ ≥r2 2 ] ≤e−2nc2 1p2 + e−2n(1−2−d)2p2 + 2ne−2nϵ2V (r′)2 where the last inequality is based on Hoeffding bound. Let e−2nc2 1p2 ≤ δ 3, e−2n(1−2−d)2p2 ≤ δ 3 and 2ne−2nϵ2V (r′) ≤2ne−2nϵ2V ( r2 2 )2 ≤ δ 3, we obtain n ≥ 1 2c2 1p2 log 3 δ, n ≥ 1 2(1−2−d)2p2 log 3 δ, and n ≥ 1 ϵ4V ( r2 2 )4 log 3 δ . ■ Based on Claim 1, we get the following theorem, which shows the effectiveness of the proposed method. Main Theorem. If 1. Let B2 be the hyper-ball centered at b with radius 2r. The minimum distance between the points inside B and the ones outside B2 is not too large, i.e. min{∥xi −xj∥|xi, xj ∈ S, ∥xi −b∥≤r, ∥xj −b∥> 2r} ≤α, where α is a positive parameter. 1Notice that here we are only dealing with the hard case where f1(x) is positive within B. In the separable case where the support regions of the two classes do not overlap, we can use other methods to detect the minority class, such as the one proposed in [8]. 3 2. f1(x) is locally smooth, i.e. ∀x, y ∈Rd, |f1(x)−f1(y)| ≤β∥x−y∥ α , where β ≤p2OV ( r2 2 ,r) 2d+1V (r)2 and OV ( r2 2 , r) is the volume of the overlapping region of two hyper-balls: one is of radius r, the other one is of radius r2 2 , and its center is on the sphere of the bigger one. 3. The number of examples is sufficiently large, i.e. n ≥max{ 1 2c2 1p2 log 3 δ, 1 2(1−2−d)2p2 log 3 δ, 1 (1−p)4β4V ( r2 2 )4 log 3 δ }. then with probability at least 1 −δ, after ⌈2α r2 ⌉iterations, NNDB will query at least one example whose probability of coming from the minority class is at least 1 3, and it will continue querying such examples until the ⌊( 2d p(1−p) −2) · α r ⌋th iteration. Proof. Based on Claim 1, using condition 3, if the number of examples is sufficiently large, then with probability at least 1−δ, r2 2 ≤r′ ≤r and | ni n −E( ni n )| ≤(1−p)βV (r′), 1 ≤i ≤n. According to condition 2, ∀xi, xj ∈S s.t. ∥xi −b∥> 2r, ∥xj −b∥> 2r and ∥xi −xj∥≤α, E( ni n ) and E( nj n ) will not be affected by the minority class, and |E( ni n ) −E( nj n )| ≤(1 −p)βV (r′) ≤(1 −p)βV (r). Note that α is always bigger than r. Based on the above inequalities, we have |ni n −nj n | ≤|ni n −E(ni n )| + |nj n −E(nj n )| + |E(ni n ) −E(nj n )| ≤3(1 −p)βV (r) (1) From inequality (1), it is not hard to see that ∀xi, xj ∈S, s.t. ∥xi −b∥> 2r and ∥xi −xj∥≤α, ni n −nj n ≤3(1 −p)βV (r), i.e. when tr′ = α, si n ≤3(1 −p)βV (r) (2) This is because if ∥xj −b∥≤2r, the minority class may also contribute to nj n , and thus the score may be even smaller. On the other hand, based on condition 1, there exist two points xk, xl ∈S, s.t. ∥xk −b∥≤r, ∥xl −b∥> 2r, and ∥xk −xl∥≤α. Since the contribution of the minority class to E( nk n ) is at least p·OV ( r2 2 ,r) V (r) , so E( nk n )−E( nl n ) ≥p·OV ( r2 2 ,r) V (r) −(1−p)βV (r′) ≥p·OV ( r2 2 ,r) V (r) −(1−p)βV (r). Since for any example xi ∈S, we have | ni n −E( ni n )| ≤(1 −p)βV (r′) ≤(1 −p)βV (r), therefore nk n −nl n ≥p · OV ( r2 2 , r) V (r) −3(1 −p)βV (r) ≥p · OV ( r2 2 , r) V (r) −3(1 −p)p2 · OV ( r2 2 , r) 2d+1V (r) Since p is very small, p ≫3(1−p)p2 2d+1 ; therefore, nk n −nl n > 3(1 −p)βV (r), i.e. when tr′ = α, sk n > 3(1 −p)βV (r) (3) In Step 4 of the proposed method, we gradually enlarge the neighborhood to calculate the change of local density. When tr′ = α, based on inequalities (2) and (3), ∀xi ∈S, ∥xi −b∥> 2r, we have sk > si. Therefore, in this round of iteration, we will pick an example from B2. In order for tr′ to be equal to α, the value of t would be ⌈α r′ ⌉≤⌈2α r2 ⌉. If we further increase t so that tr′ = cα, where c > 1, we have the following conclusion: ∀xi, xj ∈ S, s.t. ∥xi −b∥> 2r and ∥xi −xj∥≤cα, ni n −nj n ≤(c + 2)(1 −p)βV (r), i.e. si n ≤(c + 2)(1 − p)βV (r). As long as p ≥(c+2)(1−p)p2 2d , i.e. c ≤ 2d p(1−p) −2, then ∀xi ∈S, ∥xi −b∥> 2r, sk > si, and we will pick examples from B2. Since r′ ≤r, the method will continue querying examples in B2 until the ⌊( 2d p(1−p) −2) · α r ⌋th iteration. Finally, we show that the probability of picking a minority class example from B2 is at least 1 3. To this end, we need to calculate the maximum probability mass of the majority class within B2. Consider the case where the maximum value of f1(x) occurs at b, and this pdf decreases by β every time x moves away from b in the direction of the radius by α, i.e. the shape of f1(x) is a cone in (d + 1) dimensional space. Since f1(x) must integrate to 1, i.e. V ( αf1(b) β ) · f1(b) d+1 = 1, where V ( αf1(b) β ) is the volume of a hyper-ball with radius αf1(b) β , we have f1(b) = ( d+1 V (α)) 1 d+1 β d d+1 . 4 Therefore, the probability mass of the majority class within B2 is: V (2r)(f1(b) −2r α β) + 2r α β d + 1V (2r) < V (2r)f1(b) = V (2r)(d + 1 V (α)) 1 d+1 β d d+1 = 2d V (r) V (α) 1 d+1 (d + 1) 1 d+1 β d d+1 < (d + 1) 1 d+1 (2d+1V (r)β) d d+1 ≤(d + 1) 1 d+1 (p2 · OV ( r2 2 , r) V (r) ) d d+1 < 2p where V (2r) is the volume of a hyper-ball with radius 2r. Therefore, if we select a point at random from B2, the probability that this point is from the minority class is at least p p+(1−p)·2p ≥ p p+2p = 1 3. ■ 2.4 Rare category detection for multiple classes In subsection 2.2, we have discussed rare category detection for the binary case. In this subsection, we focus on the case where m > 2. To be specific, let p1, . . . , pm be the priors of the m classes, and p1 ≫pi, i ̸= 1. Our goal is to use as few label requests as possible to find at least one example from each class. The method proposed in subsection 2.2 can be easily generalized to multiple classes, which is presented in Algorithm 2. In this algorithm, we are given the priors of all the minority classes. Using each pi, we estimate the number Ki of examples from this class, and calculate the corresponding r′ i value in the same manner as NNDB. Then, we calculate the local density at each example based on different scales r′ i. In the outer loop of Step 9, we calculate the r′ value which is the minimum of all the r′ i whose corresponding classes have not been discovered yet and its index. In the inner loop of Step 11, we gradually enlarge the neighborhood to calculate the score of each example. This is the same as NNDB, except that we preclude the examples that are within a certain distance of any selected example from being selected. This heuristic is to avoid repeatedly selecting examples from the same discovered class. The inner loop stops when we find an example from an undiscovered class. Then we will update the r′ value and resume the inner loop. If the minority classes form compact clusters and are far apart from each other, NNDM is able to detect examples from each minority class with a small number of label requests. Algorithm 2 Nearest-Neighbor-Based Rare Category Detection for Multiple Classes (NNDM) Require: S, p2, . . . , pm 1: for i = 2 : m do 2: Let Ki = npi. 3: For each example, calculate the distance between this example and its Kth i nearest neighbor. Set r′ i to be the minimum value among all the examples. 4: end for 5: Let r′ 1 = maxm i=2 r′ i. 6: for i = 1 : m do 7: ∀xj ∈S, let NN(xj, r′ i) = {x|x ∈S, ∥x −xj∥≤r′ i}, and ni j = |NN(xj, r′ i)|. 8: end for 9: while not all the classes have been discovered do 10: Let r′ = min{r′ i|1 ≤i ≤m, and class i has not been discovered}, and s be the corresponding index, i.e. r′ = r′ s. 11: for t = 1 : n do 12: for each xi that has been selected and labeled yi, ∀x ∈S, s.t. ∥x −xi∥≤r′ yi, si = −∞; for all the other examples, si = max xj∈NN(xi,tr′)(ns i −ns j). 13: Query x = arg maxxi∈S si. 14: If x belongs to a class that has not been discovered, break. 15: end for 16: end while In NNDB and NNDM, we need the priors of the minority classes as the input. As we will see in the next section, our algorithms are robust against small perturbations in the priors. 5 3 Experimental results In this section, we compare our methods (NNDB and NNDM) with the best method proposed in [8] (Interleave) and random sampling (RS) on both synthetic and real data sets. In Interleave, we use the number of classes as the number of components in the mixture model. For both Interleave and RS, we run the experiment multiple times and report the average results. 3.1 Synthetic data sets Figure 1(a) shows a synthetic data set where the pdf of the majority class is Gaussian and the pdf of the minority class is uniform within a small hyper-ball. There are 1000 examples from the majority class and only 10 examples from the minority class. Using Interleave, we need to label 35 examples, using RS, we need to label 101 examples, and using NNDB, we only need to label 3 examples in order to sample one from the minority class, which are denoted as ‘x’ in Figure 1(b). Notice that the first 2 examples that NNDB selects are not from the correct region. This is because the number of examples from the minority class is very small, and the local density may be affected by the randomness in the data. −3 −2 −1 0 1 2 3 4 −1 0 1 2 3 4 5 (a) Data Set −3 −2 −1 0 1 2 3 4 −1 0 1 2 3 4 5 (b) Examples Selected by NNDB, denoted as ‘x’ Figure 1: Synthetic Data Set 1. In Figure 2(a), the X-shaped data consisting of 3000 examples correspond to the majority class, and the four characters ‘NIPS’ correspond to four minority classes, which consist of 138, 79, 118, and 206 examples respectively. Using Interleave, we need to label 1190 examples, using RS, we need to label 83 examples, and using NNDM, we only need to label 5 examples in order to get one from each of the minority classes, which are denoted as ‘x’ in Figure 2(b). Notice that in this example, Interleave is even worse than RS. This might be because some minority classes are located in the region where the density of the majority class is not negligible, and thus may be ‘explained’ by the majority-class mixture-model component. 3.2 Real data sets In this subsection, we compare different methods on two real data sets: Abalone [3] and Shuttle [4]. The first data set consists of 4177 examples, described by 7 dimensional features. The examples come from 20 classes: the proportion of the largest class is 16.50%, and the proportion of the smallest class is 0.34%. For the second data set, we sub-sample the original training set to produce a smaller data set with 4515 examples, described by 9 dimensional features. The examples come from 7 classes: the proportion of the largest class is 75.53%, and the proportion of the smallest class is 0.13%. The comparison results are shown in Figure 3(a) and Figure 3(b) respectively. From these figures, we can see that NNDM is significantly better than Interleave and RS: with Abalone data set, to find 6 0 50 100 150 200 250 0 20 40 60 80 100 120 140 160 180 200 (a) Data Set 0 50 100 150 200 250 0 20 40 60 80 100 120 140 160 180 200 (b) Examples Selected by NNDM, denoted as ‘x’ Figure 2: Synthetic Data Set 2. all the classes, Interleave needs 280 label requests, RS needs 483 label requests, and NNDM only needs 125 label requests; with Shuttle data set, to find all the classes, Interleave needs 140 label requests, RS needs 512 label requests, and NNDM only needs 87 label requests. This is because as the number of components becomes larger, the mixture model generated by Interleave is less reliable due to the lack of labeled examples, thus we need to select more examples. Furthermore, the majority and minority classes may not be near-separable, which is a disaster for Interleave. On the other hand, NNDM does not assume a generative model for the data, and only focuses on the change in local density, which is more effective on the two data sets. 0 100 200 300 400 500 0 5 10 15 20 Number of Selected Examples Classes Discovered NNDM Interleave RS (a) Abalone 0 100 200 300 400 500 600 1 2 3 4 5 6 7 Number of Selected Examples Classes Discovered NNDM Interleave RS (b) Shuttle Figure 3: Learning Curves for Real Data Sets 3.3 Imprecise priors The proposed algorithms need the priors of the minority classes as input. In this subsection, we test the robustness of NNDM against modest mis-estimations of the class priors. The performance of NNDB is similar to NNDM, so we omit the results here. In the experiments, we use the same data sets as in subsection 3.2, and add/subtract 5%, 10%, and 20% from the true priors of the minority classes. The results are shown in Figure 4. From these figures, we can see that NNDM is very robust to small perturbations in the priors. For example, with Abalone data set, if we subtract 10% from the true priors, only one more label request is needed in order to find all the classes. 7 0 50 100 150 200 250 0 5 10 15 20 Number of Selected Examples Classes Discovered −5% −10% −20% 0 +5% +10% +20% (a) Abalone 0 20 40 60 80 100 1 2 3 4 5 6 7 Number of Selected Examples Classes Discovered −5% −10% −20% 0 +5% +10% +20% (b) Shuttle Figure 4: Robustness Study 4 Conclusion In this paper, we have proposed a novel method for rare category detection, useful for de-novo active learning in serious applications. Different from existing methods, our method does not rely on the assumption that the data is near-separable. It works by selecting examples corresponding to regions with the maximum change in local density, and depending on scaling, it will select class-boundary or class-internal samples of minority classes. The method could be scaled up using kd-trees [7]. The effectiveness of the proposed method is guaranteed by theoretical justification, and its superiority over existing methods is demonstrated by extensive experimental results on both synthetic and real data sets. Moreover, it is very robust to modest perturbations in estimating true class priors. Acknowledgments This paper is based on work in part supported by the Defense Advanced Research Projects Agency (DARPA) under contract number NBCHD030010. References [1] M. Balcan, A. Beygelzimer, and J. Langford. Agnostic active learning. In Proc. of the 23rd Int. Conf. on Machine Learning, pages 65–72, 2006. [2] S. Bay, K. Kumaraswamy, M. Anderle, R. Kumar, and D. Steier. Large scale detection of irregularities in accounting data. In Proc. of the 6th Int. Conf. on Data Mining, pages 75–86, 2006. [3] C. Blake and C. Merz. Uci repository of machine learning databases. In http://www.ics.uci.edu/ machine/MLRepository.html, 1998. [4] P. Brazdil and J. Gama. Statlog repository. In http://www.niaad.liacc.up.pt/old/statlog/datasets/shuttle/shuttle.doc.html, 1991. [5] S. Dasgupta. Coarse sample complexity bounds for active learning. In Advances in Neural Information Processing Systems 19, 2005. [6] S. Fine and Y. Mansour. Active sampling for multiple output identification. In The 19th Annual Conf. on Learning Theory, pages 620–634, 2006. [7] A. Moore. A tutorial on kd-trees. Technical report, University of Cambridge Computer Laboratory, 1991. [8] D. Pelleg and A. Moore. Active learning for anomaly and rare-category detection. In Advances in Neural Information Processing Systems 18, 2004. 8
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Receptive Fields without Spike-Triggering Jakob H Macke j akob@ tuebi ngen. mpg. de Max Planck Institute for Biological Cybernetics Spemannstrasse 41 72076 T¨ubingen, Germany G ¨unther Zeck z eck@ neuro. mpg. de Max Planck Institute of Neurobiology Am Klopferspitze 1 8 821 52 Martinsried, Germany Matthias Bethge mbethge@ tuebi ngen. mpg. de Max Planck Institute for Biological Cybernetics Spemannstrasse 41 72076 T¨ubingen, Germany Abstract Stimulus selectivity of sensory neurons is often characterized by estimating their receptive field properties such as orientation selectivity. Receptive fields are usually derived from the mean (or covariance) of the spike-triggered stimulus ensemble. This approach treats each spike as an independent message but does not take into account that information might be conveyed through patterns of neural activity that are distributed across space or time. Can we find a concise description for the processing of a whole population of neurons analogous to the receptive field for single neurons? Here, we present a generalization of the linear receptive field which is not bound to be triggered on individual spikes but can be meaningfully linked to distributed response patterns. More precisely, we seek to identify those stimulus features and the corresponding patterns of neural activity that are most reliably coupled. We use an extension of reverse-correlation methods based on canonical correlation analysis. The resulting population receptive fields span the subspace of stimuli that is most informative about the population response. We evaluate our approach using both neuronal models and multi-electrode recordings from rabbit retinal ganglion cells. We show how the model can be extended to capture nonlinear stimulus-response relationships using kernel canonical correlation analysis, which makes it possible to test different coding mechanisms. Our technique can also be used to calculate receptive fields from multi-dimensional neural measurements such as those obtained from dynamic imaging methods. 1 Introduction Visual input to the retina consists of complex light intensity patterns. The interpretation of these patterns constitutes a challenging problem: for computational tasks like object recognition, it is not clear what information about the image should be extracted and in which format it should be represented. Similarly, it is difficult to assess what information is conveyed by the multitude of neurons in the visual pathway. Right from the first synapse, the information of an individual photoreceptor is signaled to many different cells with different temporal filtering properties, each of which is only a small unit within a complex neural network [20]. Even if we leave the difficulties imposed by nonlinearities and feedback aside, it is hard to judge what the contribution of any particular neuron is to the information transmitted. 1 The prevalent tool for characterizing the behavior of sensory neurons, the spike triggered average, is based on a quasi-linear model of neural responses [1 5]. For the sake of clarity, we consider an idealized model of the signaling channel y = Wx + ξ , (1 ) where y = ( y1 , . . . , yN) T denotes the vector of neural responses, x the stimulus parameters, W = ( w1 , . . . , wN) T the filter matrix with row ‘ k’ containing the receptive field wk of neuron k, and ξ is the noise. The spike-triggered average only allows description of the stimulus-response function (i.e. the wk) of one single neuron at a time. In order to understand the collective behavior of a neuronal population, we rather have to understand the behavior of the matrix W, and the structure of the noise correlations Σ ξ: Both of them influence the feature selectivity of the population. Can we find a compact description of the features that a neural ensemble is most sensitive to? In the case of a single cell, the receptive field provides such a description: It can be interpreted as the “favorite stimulus” of the neuron, in the sense that the more similar an input is to the receptive field, the higher is the spiking probability, and thus the firing rate of the neuron. In addition, the receptive field can easily be estimated using a spike-triggered average, which, under certain assumptions, yields the optimal estimate of the receptive field in a linear-nonlinear cascade model [1 1 ]. If we are considering an ensemble of neurons rather than a single neuron, it is not obvious what to trigger on: This requires assumptions about what patterns of spikes or modulations in firing rates across the population carry information about the stimulus. Rather than addressing the question “what features of the stimulus are correlated with the occurence of spikes”, the question now is: “What stimulus features are correlated with what patterns of spiking activity?” [1 4]. Phrased in the language of information theory, we are searching for the subspace that contains most of the mutual information between sensory inputs and neuronal responses. By this dimensionality reduction technique, we can find a compact description of the processing of the population. As an efficient implementation of this strategy, we present an extension of reverse-correlation methods based on canonical correlation analysis. The resulting population receptive fields (PRFs) are not bound to be triggered on individual spikes but are linked to response patterns that are simultaneously determined by the algorithm. We calculate the PRF for a population consisting of uniformly spaced cells with center-surround receptive fields and noise correlations, and estimate the PRF of a population of rabbit retinal ganglion cells from multi-electrode recordings. In addition, we show how our method can be extended to explore different hypotheses about the neural code, such as spike latencies or interval coding, which require nonlinear read out mechanisms. 2 From reverse correlation to canonical correlation We regard the stimulus at time t as a random variable Xt ∈Rn, and the neural response as Yt ∈Rm. For simplicity, we assume that the stimulus consists of Gaussian white noise, i.e. E( X) = 0 and Cov( X) = I. The spike-triggered average a of a neuron can be motivated by the fact that it is the direction in stimulus-space maximizing the correlation-coefficient ρ = Cov( aTX, Y1 ) Var( aTX) Var( Y1 ) . (2) between the filtered stimulus aTX and a univariate neural response Y1 . In the case of a neural population, we are not only looking for the stimulus feature a, but also need to determine what pattern of spiking activity b it is coupled with. The natural extension is to search for those vectors a1 and b1 that maximize ρ1 = Cov( aT 1 X, bT 1 Y) Var( aT 1 X) Var( bT 1 Y) . (3) We interpret a1 as the stimulus filter whose output is maximally correlated with the output of the “response filter” b1 . Thus, we are simultaneously searching for features of the stimulus that the neural system is selective for, and the patterns of activity that it uses to signal the presence or absence 2 of this feature. We refer to the vector a1 as the (first) population receptive field of the population, and b1 is the response feature corresponding to a1 . If a hypothetical neuron receives input from the population, and wants to decode the presence of the stimulus a1 , the weights of the optimal linear readout [1 6] could be derived from b1 . Canonical Correlation Analysis (CCA) [9] is an algorithm that finds the vectors a1 and b1 that maximize (3): We denote the covariances of X and Y by Σ x, Σ y, the cross-covariance by Σ xy, and the whitened cross-covariance by C = Σ ( −1 / 2) x Σ xyΣ ( −1 / 2) y . (4) Let C = UDVT denote the singular value decomposition of C, where the entries of the diagonal matrix D are non-negative and decreasing along the diagonal. Then, the k-th pair of canonical variables is given by ak = Σ ( −1 / 2) x uk and bk = Σ ( −1 / 2) y vk, where uk and vk are the k-th column vectors of U and V, respectively. Furthermore, the k-th singular value of C, i.e. the k-th diagonal entry of D is the correlation-coefficient ρk of aT k Xand bT k Y. The random variables aT i X and aT j X are uncorrelated for i = j. Importantly, the solution for the optimization problem in CCA is unique and can be computed efficiently via a single eigenvalue problem. The population receptive fields and the characteristic patterns are found by a joint optimization in stimulus and response space. Therefore, one does not need to know—or assume—a priori what features the population is sensitive to, or what spike patterns convey the information. The first K PRFs form a basis for the subspace of stimuli that the neural population is most sensitive to, and the individual basis vectors ak are sorted according to their “informativeness” [1 3, 1 7]. The mutual information between two one-dimensional Gaussian Variables with correlation ρ is given by MIGauss = −1 2 log( 1 −ρ2) , so maximizing correlation coefficients is equivalent to maximizing mutual information [3]. Assuming the neural response Y to be Gaussian, the subspace spanned by the first K vectors BK = ( b1 , . . . , bK) is also the K-subspace of stimuli that contains the maximal amount of mutual information between stimuli and neural response. That is BK = argmax B ∈Rn × k det ` B TΣ yB ´ det “ B T “ Σ y −Σ TxyΣ ( −1 ) x Σ xy ” B ” . (5) Thus, in terms of dimensionality reduction, CCA optimizes the same objective as oriented PCA [5]. In contrast to oriented PCA, however, CCA does not require one to know explicitly how the response covariance Σ y = Σ s + Σ ξ splits into signal Σ s and noise Σ ξ covariance. Instead, it uses the cross-covariance Σ xy which is directly available from reverse correlation experiments. In addition, CCA not only returns the most predictable response features b1 , . . . bK but also the most predictive stimulus components AK = ( a1 , . . . aK) . For general Y and for stimuli X with elliptically contoured distribution, MIGauss −J( ATX) provides a lower bound to the mutual information between ATX and BTY, where J( ATX) = 1 2 log( det( 2πeATΣ x A) ) −h( ATX) (6) is the Negentropy of ATX, and h( ATX) its differential entropy. Since for elliptically contoured distributions J( ATX) does not depend on A, the PRFs can be seen as the solution of a variational approach, maximizing a lower bound to the mutual information. Maximizing mutual information directly is hard, requires extensive amounts of data, and usually multiple repetitions of the same stimulus sequence. 3 The receptive field of a population of neurons 3.1 The effect of tuning functions and noise correlations To illustrate the relationship between the tuning-functions of individual neurons and the PRFs [22], we calculate the first PRF of a simple one-dimensional population model consisting of center3 surround neurons. Each tuning function is modeled by a “Difference of Gaussians” (DOG) f( x) = exp „ −1 2 “ x −c σ ” 2 « −A exp −1 2 „ x −c η « 2 ! (7) whose centers c are uniformly distributed over the real axis. The width η of the negative Gaussian is set to be twice as large as the width σ of the positive Gaussian. If the area of both Gaussians is the same ( A = 1 ) , the DC component of the DOG-fillter is zero, i.e. the neuron is not sensitive to the mean luminance of the stimulus. If the ratio between both areas becomes substantially unbalanced, the DC component will become the largest signal ( A ≈0) . In addition to the parameter A, we will study the length scale of noise correlations λ [1 8]. Specifically, we assume exponentially decaying noise correlation with Σ ξ( s) = exp( −| s| /λ) . As this model is invariant under spatial shifts, the first PRF can be calculated by finding the spatial frequency at which the SNR is maximal. That is, the first PRF can be used to estimate the passband of the population transfer function. The SNR is given by SNR( ω) = „ 1 + λ2 ω2 2λ „ e−ω 2 σ 2 + A2 e−η 2 ω 2 −2Ae−σ 2 + η 2 2 ω 2 « « 2 . (8) The passband of the first population filter moves as a function of both parameters A and λ. It equals the DC component for small A (i.e. large imbalance) and small λ (i.e. short correlation length). In this case, the mean intensity is the stimulus property that is most faithfully signaled by the ensemble. λ A 0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Figure 1 : Spatial frequency of the first PRF for the model described above. λ is the length-scale of the noise correlations, A is the weight of the negative Gaussian in the DOG-model. The region in the bottom left corner (bounded by the white line) is the part of the parameter-space in which the PRF equals the DC component. 3.2 The receptive field of an ensemble of retinal ganglion cells We mapped the population receptive fields of rabbit retinal ganglion cells recorded with a wholemount preparation. We are not primarily interested in prediction performance [1 2], but rather in dimensionality reduction: We want to characterize the filtering properties of the population. The neurons were stimulated with a 1 6 × 1 6 checkerboard consisting of binary white noise which was updated every 20ms. The experimental procedures are described in detail in [21 ]. After spikesorting, spike trains from 32 neurons were binned at 20ms resolution, and the response of a neuron to a stimulus at time t was defined to consist of the the spike-counts in the 1 0 bins between 40ms and 240ms after t. Thus, each population response Yt is a 320 dimensional vector. Figure 3.2 A) displays the first 6 PRFs, the corresponding patterns of neural activity (B) and their correlation coefficients ρk (which were calculated using a cross-validation procedure). It can be seen that the PRFs look very different to the usual center-surrond structure of retinal ganglion. However, one should keep in mind that it is really the space spanned by the PRFs that is relevant, and thus be careful when interpreting the actual filter shapes [1 5]. For comparison, we also plotted the single-cell receptive fields in Figure 3.2 C), and their projections into the spaced spanned by the first 6 PRFs. These plots suggest that a small number of PRFs might 4 be sufficient to approximate each of the receptive fields. To determine the dimensionality of the relevant subspace, we analyzed the correlation-coefficients ρk. The Gaussian Mutual Information MIGauss = −1 2 K k=1 log( 1 −ρ2 k) is an estimate of the information contained in the subspace spanned by the first K PRFs. Based on this measure, a 1 2 dimensional subspace accounts for 90% of the total information. In order to link the empirically estimated PRFs with the theoretical analysis in section 3.1 , we calculated the spectral properties of the first PRF. Our analysis revealed that most of the power is in the low frequencies, suggesting that the population is in the parameter-regime where the single-cell receptive fields have power in the DC-component and the noise-correlations have short range, which is certainly reasonable for retinal ganglion cells [4]. A) 0.51 0.44 0.38 0.35 0.29 0.27 B) Time → Neuron index 40 160 220 5 10 15 20 25 30 40 160 220 5 10 15 20 25 30 40 160 220 5 10 15 20 25 30 40 160 220 5 10 15 20 25 30 40 160 220 5 10 15 20 25 30 40 160 220 5 10 15 20 25 30 −0.2 0 0.2 C) Proj. RF RF Proj. RF RF Figure 2: The population receptive fields of a group of 32 retinal ganglion cells: A) the first 6 PRFs, as sorted by the correlation coefficient ρk B) the response features bk coupled with the PRFs. Each row of each image corresponds to one neuron, and each column to one time-bin. Blue color denotes enhanced activity, red suppressed. It can be seen that only a subset of neurons contributed to the first 6 PRFs. C) The single-cell receptive fields of 24 neurons from our population, and their projections into the space spanned by the 6 PRFs. 5 A) 1 5 10 15 20 30 40 50 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Correlations coefficients ρk PRF index B) 1 5 10 15 20 30 40 50 0 20 40 60 80 90 100 Percentage of MI Dimensionality of subspace Figure 3: A) Correlation coefficients ρk for the PRFs. Estimates and error-bars are calculated using a cross-validation procedure. B) Gaussian-MI of the subspace spanned by the first K PRFs. 4 Nonlinear extensions using Kernel Canonical Correlation Analysis Thus far, our model is completely linear: We assume that the stimulus is linearly related to the neural responses, and we also assume a linear readout of the response. In this section, we will explore generalizations of the CCA model using Kernel CCA: By embedding the stimulus-space nonlinearly in a feature space, nonlinear codes can be described. Kernel methods provide a framework for extending linear algorithms to the nonlinear case [8]. After projecting the data into a feature space via a feature maps φ and ψ, a solution is found using linear methods in the feature space. In the case of Kernel CCA [1 , 1 0, 2, 7] one seeks to find a linear relationship between the random variables ˆX = φ( X) and ˆY = ψ( Y) , rather than between X and Y. If an algorithm is purely defined in terms of dot-products, and if the dot-product in feature space k( s, t) = ψ( s) , ψ( t) can be computed efficiently, then the algorithm does not require explicit calculation of the feature maps φ and ψ. This “kernel-trick” makes it possible to work in high(or infinite)-dimensional feature spaces. It is worth mentioning that the space of patterns Y itself does not have to be a vector space. Given a data-set x1 . . . xn, it suffices to know the dot-products between any pair of training points, Kij : = ψ( yi) , ψ( yj) . The kernel function k( s, t) can be seen as a similiarity measure. It incorporates our assumptions about which spike-patterns should be regarded as similar “messages”. Therefore, the choice of the kernel-function is closely related to specifing what the search-space of potential neural codes is. A number of distance- and kernel-functions [6, 1 9] have been proposed to compute distances between spike-trains. They can be designed to take into account precisely timed pattern of spikes, or to be invariant to certain transformations such as temporal jitter. We illustrate the concept on simulated data: We will use a similarity measure based on the metric D interval [1 9] to estimate the receptive field of a neuron which does not use its firing rate, but rather the occurrence of specific interspike intervals to convey information about the stimulus. The metric D interval between two spike-trains is essentially the cost of matching their intervals by shifting, adding or deleting spikes. (We set k( s, t) = exp( −D( s, t) . In theory, this function is not guaranteed to be positive definite, which could lead to numerical problems, but we did not encounter any in our simulation.) If we consider coding-schemes that are based on patterns of spikes, the methods described here become useful even for the analysis of single neurons. We will here concentrate on a single neuron, but the analysis can be extended to patterns distributed across several neurons. Our hypothetical neuron encodes information in a pattern consisting of three spikes: The relative timing of the second spike is informative about the stimulus: The bigger the correlation between receptive field and stimulus r, st , the shorter is the interval. If the receptive field is very dissimilar to the stimulus, the interval is long. While the timing of the spikes relative to each other is precise, there is jitter in the timing of the pattern relative to the stimulus. Figure 4 A) is a raster plot of simulated spike-trains from this model, ordered by r, st . We also included noise spikes at random times. 6 A) C) D) B) 0 50 100 150 200 Time → Spike trains Figure 4: Coding by spike patterns: A) Receptive field of neuron described in Section 4. B) A subset of the simulated spike-trains, sorted with respect to the similarity between the shown stimulus and the receptive field of the model. The interval between the first two informative spikes in each trial is highlighted in red. C) Receptive field recovered by Kernel CCA, the correlation coefficient between real and estimated receptive field is 0.93. D) Receptive field derived using linear decoding, correlation coefficient is 0. 02. Using these spike-trains, we tried to recover the receptive field r without telling the algorithm what the indicating pattern was. Each stimulus was shown only once, and therefore, that every spikepattern occurred only once. We simulated 5000 stimulus presentations for this model, and applied Kernel CCA with a linear kernel on the stimuli, and the alignment-score on the spike-trains. By using incomplete Cholesky decompositions [2], one can compute Kernel CCA without having to calculate the full kernel matrix. As many kernels on spike trains are computationally expensive, this trick can result in substantial speed-ups of the computation. The receptive field was recovered (see Figure 4), despite the highly nonlinear encoding mechanism of the neuron. For comparison, we also show what receptive field would be obtained using linear decoding on the indicated bins. Although this neuron model may seem slightly contrived, it is a good proof of concept that, in principle, receptive fields can be estimated even if the firing rate gives no information at all about the stimulus, and the encoding is highly nonlinear. Our algorithm does not only look at patterns that occur more often than expected by chance, but also takes into account to what extent their occurrence is correlated to the sensory input. 5 Conclusions We set out to find a useful description of the stimulus-response relationship of an ensemble of neurons akin to the concept of receptive field for single neurons. The population receptive fields are found by a joint optimization over stimuli and spike-patterns, and are thus not bound to be triggered by single spikes. We estimated the PRFs of a group of retinal ganglion cells, and found that the first PRF had most spectral power in the low-frequency bands, consistent with our theoretical analysis. The stimulus we used was a white-noise sequence—it will be interesting to see how the informative subspace and its spectral properties change for different stimuli such as colored noise. The ganglion cell layer of the retina is a system that is relatively well understood at the level of single neurons. Therefore, our results can readily be compared and connected to those obtained using conventional analysis techniques. However, our approach has the potential to be especially useful in systems in which the functional significance of single cell receptive fields is difficult to interpret. 7 We usually assumed that each dimension of the response vector Y represents an electrode-recording from a single neuron. However, the vector Y could also represent any other multi-dimensional measurement of brain activity: For example, imaging modalities such as voltage-sensitive dye imaging yield measurements at multiple pixels simultaneously. Data from electro-physiological data, e.g. local field potentials, are often analyzed in frequency space, i.e. by looking at the energy of the signal in different frequency bands. This also results in a multi-dimensional representation of the signal. Using CCA, receptive fields can readily be estimated from these kinds of representations without limiting attention to single channels or extracting neural events. Acknowledgments We would like to thank A Gretton and J Eichhorn for useful discussions, and F J¨akel, J Butler and S Liebe for comments on the manuscript. References [1 ] S. Akaho. A kernel method for canonical correlation analysis. In International Meeting ofPsychometric Society, Osaka, 2001 . [2] F. R. Bach and M. I. Jordan. Kernel independent component analysis. Journal ofMachine Learning Research, 3: 1 : 48, 2002. [3] G. Chechik, A. Globerson, N. Tishby, and Y. Weiss. Information Bottleneck for Gaussian Variables. The Journal ofMachine Learning Research, 6: 1 65–1 88, 2005. [4] S. Devries and D. Baylor. Mosaic Arrangement of Ganglion Cell Receptive Fields in Rabbit Retina. Journal ofNeurophysiology, 78(4): 2048–2060, 1 997. [5] K. Diamantaras and S. 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Bialek. Analyzing neural responses to natural signals: maximally informative dimensions. Neural Comput, 1 6(2): 223–50, 2004. [1 8] H. Sompolinsky, H. Yoon, K. Kang, and M. Shamir. Population coding in neuronal systems with correlated noise. Phys Rev E Stat Nonlin Soft Matter Phys, 64(5 Pt 1 ): 051 904, 2001 . [1 9] J. Victor. Spike train metrics. Curr Opin Neurobiol, 1 5(5): 585–92, 2005. [20] H. W¨assle. Parallel processing in the mammalian retina. Nat Rev Neurosci, 5(1 0): 747–57, 2004. [21 ] G. M. Zeck, Q. Xiao, and R. H. Masland. The spatial filtering properties of local edge detectors and brisk-sustained retinal ganglion cells. Eur J Neurosci, 22(8): 201 6–26, 2005. [22] K. Zhang and T. Sejnowski. Neuronal Tuning: To Sharpen or Broaden?, 1 999. 8
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Robust Regression with Twinned Gaussian Processes Andrew Naish-Guzman & Sean Holden Computer Laboratory University of Cambridge Cambridge, CB3 0FD. United Kingdom {agpn2,sbh11}@cl.cam.ac.uk Abstract We propose a Gaussian process (GP) framework for robust inference in which a GP prior on the mixing weights of a two-component noise model augments the standard process over latent function values. This approach is a generalization of the mixture likelihood used in traditional robust GP regression, and a specialization of the GP mixture models suggested by Tresp [1] and Rasmussen and Ghahramani [2]. The value of this restriction is in its tractable expectation propagation updates, which allow for faster inference and model selection, and better convergence than the standard mixture. An additional benefit over the latter method lies in our ability to incorporate knowledge of the noise domain to influence predictions, and to recover with the predictive distribution information about the outlier distribution via the gating process. The model has asymptotic complexity equal to that of conventional robust methods, but yields more confident predictions on benchmark problems than classical heavy-tailed models and exhibits improved stability for data with clustered corruptions, for which they fail altogether. We show further how our approach can be used without adjustment for more smoothly heteroscedastic data, and suggest how it could be extended to more general noise models. We also address similarities with the work of Goldberg et al. [3]. 1 Introduction Regression data are often modelled as noisy observations of an underlying process. The simplest assumption is that all noise is independent and identically distributed (i.i.d.) zero-mean Gaussian, such that a typical set of samples appears as a cloud around the latent function. The Bayesian framework of Gaussian processes [4] is well-suited to these conditions, for which all computations remain tractable (see figure 1a). Furthermore, the Gaussian noise model enjoys the theoretical justification of the central limit theorem, which states that the sum of sufficiently many i.i.d. random variables of finite variance will be distributed normally. However, only rarely can perturbations affecting data in the real world be argued to have originated in the addition of many i.i.d. sources. The random component in the signal may be caused by human or measurement error, or it may be the manifestation of systematic variation invisible to a simplified model. In any case, if ever there is the possibility of encountering small quantities of highly implausible data, we require robustness, i.e. a model whose predictions are not greatly affected by outliers. Such demands render the standard GP inappropriate: the light tails of the Gaussian distribution cannot explain large non-Gaussian deviations, which either skew the mean interpolant away from the majority of the data, or force us to infer an unreasonably large (global) noise variance (see figure 1b). Robust methods use a heavy-tailed likelihood to allow the interpolant effectively to favour smoothness and ignore such erroneous data. Figure 1c shows how this can be achieved using a two-component noise model p(yn|fn) = (1 −ǫ)N yn ; fn , σ2 R + ǫN yn ; fn , σ2 O , (1) 1 (a) (b) (c) (d) Figure 1: Black dots show noisy samples from the sinc function. In panels (a) and (b), the behaviour of a GP with a Gaussian noise assumption is illustrated; the shaded region shows 95% confidence intervals. The presence of a single outlier is highly influential in this model, but the heavy-tailed likelihood (1) in panel (c) is more resilient. Unfortunately, even this model fails for the cluster of outliers in panel (d). Here, grey lines show ten repeated runs of the EP inference algorithm, while the black line and shaded region are their averaged mean and confidence intervals respectively—grossly at odds with those of the latent generative model. in which observations yn are Gaussian corruptions of fn, being drawn with probability ǫ from a large variance outlier distribution (σ2 O ≫σ2 R). Inference in this model is tractable, but impractical for all but the smallest problems due to the exponential explosion of terms in products of (1). In this paper, we address the more fundamental GP assumption of i.i.d. noise. Our research is motivated by observing how the predictive distribution suffers for heavy-tailed models when outliers appear in bursts: figure 1d replicates figure 1c, but introduces an additional three outliers. All parameters were taken from the optimal solution to (c), but even without the challenge of hyperparameter optimization there is now considerable uncertainty in the posterior since the competing interpretations of the cluster as signal or noise have similar posterior mass. Viewed another way, the tails of the effective log likelihood of four clustered observations have approximately one-quarter the weight of a single outlier, so the magnitude of the posterior peak associated with the robust solution is comparably reduced. One simple remedy is to make the tails of the likelihood heavier. However, since the noise model is global, this has ramifications across the entire data space, potentially causing underfitting elsewhere when real data are relegated to the tails. We can establish an optimal choice for the parameters by gradient ascent on the marginal likelihood, but it is entirely possible that no single setting will be universally satisfactory. The model introduced in this paper, which we call the twinned Gaussian process (TGP), generalizes the noise model (1) by using a GP gating function to choose between the “real” and “outlier distributions”: in regions of confidence, the tails can be made very light, encouraging the interpolant to hug the data points tightly; more dubious observations can be treated appropriately by broadening the noise distribution in their vicinity. Our model is also a specialization of the GP mixtures proposed by Tresp [1] and Rasmussen and Ghahramani [2]; indeed, the latter automatically infers the correct number of components to use. One may therefore wonder what can possibly be gained by restricting ourselves to a comparatively simple architecture. The answer is in the computational overhead required for the different approaches, since these more general models require inference by Monte Carlo methods. We argue that the two-component mixture is often a sensible distribution for modelling real data, with a natural interpretation and the heavy tails required for robustness; its weaknesses are exposed primarily when the noise distribution is not homoscedastic. The TGP largely solves this problem, and allows inference by an efficient expectation propagation (EP) [5] procedure (rather than resorting to more heavy duty Monte Carlo methods). Hence, provided a twocomponent mixture is likely to reflect adequately the noise on our data, the TGP will give similar results to the generalized mixtures mentioned above, but at a fraction of the cost. Goldberg et al. [3] suggest an approach to input-dependent noise in the spirit of the TGP, in which the log variance on observations is itself modelled as a GP (the logarithm since noise variance is a non-negative property). Inference is again analytically intractable, so Gibbs sampling is used to generate noise vectors from the posterior distribution by alternately fitting the signal process and fitting the noise process. A further stage of Gibbs sampling is required at each test point to estimate the predictive variance, making testing rather slow. Model selection is even slower, and the Metropolis-Hastings algorithm is suggested for updating hyperparameters. 2 2 Twinned Gaussian processes Given a domain X and covariance function K(·, ·) ∈X × X →R, a Gaussian process (GP) over the space of real-valued functions of X specifies the joint distribution at any finite set X ⊂X: p(f|X) = N(f ; 0 , Kf) , where the f = {fn}N n=1 are (latent) values associated with each xn ∈X, and Kf is the Gram matrix, the evaluation of the covariance function at all pairs (xi, xj). We apply Bayes’ rule to obtain the posterior distribution over the f, given the observed X and y, which with the assumption of i.i.d. Gaussian corrupted observations is also normally distributed. Predictions at X⋆are made by marginalizing over f in the (Gaussian) joint p(f, f⋆|X, y, X⋆). See [6] for a thorough introduction. Robust GP regression is achieved by using a leptokurtic likelihood distribution, i.e. one whose tails have more mass than the Gaussian. Common choices are the Laplace (or double exponential) distribution, Student’s t distribution, and the mixture model (1). In product with the prior, a heavy-tailed likelihood over an outlying observation does not exert the strong pull on the posterior witnessed with a light-tailed noise model. Kuss [7] describes how inference can be performed for all these likelihoods, and establishes that in many cases their performance is broadly comparable. Since it bears closest resemblance to the twinned GP, we are particularly interested in the mixture; however, in section 4, we include results for the Laplace model: it is the heaviest-tailed log concave distribution, which guarantees a unimodal posterior and allows more reliable EP convergence. In any case, all such methods make a global assumption about the noise distribution, and it is where this is inappropriate that our model is most beneficial. The graphical model for the TGP is shown in figure 2b. We augment the standard process over f with another GP over a set of variables u; this acts as a gating function, probabilistically dividing the domain between the real and outlier components of the noise model p(yn|fn) = σ(un)N yn ; fn , σ2 R + σ(−un)N yn ; fn , σ2 O , (2) where σ(un) .= Z un −∞ N(z ; 0 , 1) dz. In the TGP likelihood, we therefore mix two forms of Gaussian corruption, one strongly peaked at the observation, the other a broader distribution which provides the heavy tails, in proportion determined by u(x). This makes intuitive sense; crucially to us, it retains the advantage of tractability with respect to EP updates. The two priors may have quite different covariance structure, reflecting our different beliefs about correlations in the signal and in the noise domain. In addition, we accommodate prior beliefs about the prevalence of outliers with a non-zero mean process on u, p(u|X) = N(u ; mu , Ku) p(f|X) = N(f ; 0 , Kf) . Our model can be understood as lying between two extremes: observe that we recover the heavytailed (mixture of Gaussians) GP by forcing absolute correlation in u and adjusting the mean of the u-process to mu = σ−1(1 −e); conversely, if we remove all correlations in u, we return to a standard mixture model where independently we must decide to which component an input belongs. 3 Inference We begin with a very brief account of EP; for more details, see [5, 8]. Suppose we have an intractable distribution over f whose unnormalized form factorizes into a product of terms, such as a dense Gaussian prior t0(f, u) and a series of independent likelihoods {tn(yn|fn, un)}N n=1. EP constructs the approximate posterior as a product of scaled site functions ˜tn. For computational tractability, these sites are usually chosen from an exponential family with natural parameters θ, since in this case their product retains the same functional form as its components. The Gaussian (µ, Σ) has a natural parameterization (b, Π) = (Σ−1µ, −1 2Σ−1). If the prior is of this form, its site function is exact: p(f, u|y) = 1 Z t0(f, u) N Y n=1 tn(yn|fn, un) ≈q(f; θ) = t0(f, u) N Y n=1 zn˜tn(fn, un; θn), (3) 3 x1 x2 x3 xN f1 f2 f3 fN y1 y2 y3 yN (a) x1 x2 x3 xN f1 f2 f3 fN u1 u2 u3 uN y1 y2 y3 yN (b) Figure 2: In panel (a) we show a graphical model for the Gaussian process. The data ordinates are x, observations y, and the GP is over the latent f. The bold black lines indicate a fully-connected set. Panel (b) shows a graphical model for the twinned Gaussian process (TGP), in which an auxiliary set of hidden variables u describes the noisiness of the data. where Z is the marginal likelihood and zn are the scale parameters. Ideally, we would choose θ at the global minimum of some divergence measure d(p∥q), but the necessary optimization is usually intractable. EP is an iterative procedure that finds a minimizer of KL p(f, u|y)∥q(f, u; θ) on a pointwise basis: at each iteration, we select a new site n, and from the product of the cavity distribution formed by the current marginal with the omission of that site, and the true likelihood term tn, we obtain the so-called tilted distribution qn(fn, un; θ\n). A simpler optimization minθn KL qn(fn, un; θ\n)∥q(fn, un; θ) then fits only the parameters θn: this is equivalent to moment matching between the two distributions, with scale zn chosen to match the zeroth-order moments. After each site update, the moments at the remaining sites are liable to change, and several iterations may be required before convergence. The priors over u and f are independent, but we expect correlations in the posterior after conditioning on observations. To understand this, consider a single observation (xn, yn); in principle, it admits two explanations corresponding to its classification as either “outlier” or as “real” data: in general terms, either un > 0 and fn ≈yn, or un < 0 and fn respects the global structure of the signal. A diagram to assist the visualization of the behaviour of the posterior is provided in figure 3. Now, recall that the prior over u and f is p u f X ! = N u f ; mu 0 , Ku 0 0 Kf and the likelihood factorizes into a product of terms (2); our site approximations ˜tn are therefore Gaussian in (fn, un). Of importance for EP are the moments of the tilted distribution which we seek to match. These are most easily obtained by differentiation of the zeroth moments ZR and ZO of each component. We find ZR = ZZ f,u σ(u)N y ; f , σ2 R N u f ; µ , Σ dudf = Z ∞ 0 N z y ; µ , 1 0 0 σ2 R + Σ dz; writing the inner Gaussian as N zn yn ; µu µf , A C C BR , ZR = N(y ; µf , BR) σ(q), where q = µu + C BR (y −µf) q A −C2 BR . The integral for the outlier component is similar; ZO = N(y ; µf , BO) σ(−q). With partial derivatives ∂log Z ∂µ and ∂2 log Z ∂µµT we are equipped for EP; algorithmic details appear in Seeger’s note [8]. For efficiency, we make rank-two updates of the full approximate covariance on (f, u) during the EP loop, and refresh the posterior at the end of each cycle to avoid loss of precision. 4 f log p prior likelihood posterior EP -5 0 5 10 f u -5 0 5 10 -10 -5 0 5 10 f u -5 0 5 10 -10 -5 0 5 10 replacements f log p prior likelihood posterior EP -5 0 5 10 f u -5 0 5 10 -10 -5 0 5 10 f u -5 0 5 10 -10 -5 0 5 10 f log p prior likelihood posterior EP -5 0 5 10 f u -5 0 5 10 -10 -5 0 5 10 f u -5 0 5 10 -10 -5 0 5 10 f log p prior likelihood posterior EP -5 0 5 10 f u -5 0 5 10 -10 -5 0 5 10 f u -5 0 5 10 -10 -5 0 5 10 Figure 3: Using the twinned Gaussian process provides a natural resilience against clustered noisy data. The left-hand column illustrates the behaviour of a fixed heavy-tailed likelihood for one, two, four and five repeated observations at f = 5. (Outliers in real data are not necessarily so tightly packed, but the symmetry of this approximation allows us to treat them as a single unit: by “posterior”, for example, we mean the a posteriori belief in all the observations’ (identical) latent f.) The context is provided by the prior, which gives 95% confidence to data around f = 0 ±2. The top-left box illustrates how the influence of isolated outliers is mitigated by the standard mixture. However, a repeated observation (box two on the left) causes the EP solution to collapse onto the spike at the data (the log scale is deceptive: the second peak contributes only about 8% of the posterior mass). The twinned GP better preserves the marginal distribution of f by maintaining a joint distribution over both f and u: in the second and third columns respectively are contours of the true log joint (we use a broad zero-mean prior on u) and that inferred by EP, together with the marginal posterior over f. Only with a fifth observation—final box—is the context of f essentially overruled by the TGP approximation. The thick bar in the central column marks the cross-section corresponding to the unnormalized posterior from column one. 5 3.1 Predictions If the outlier component describes nuisance noise that should be eliminated, we require at test inputs x⋆only the marginal distribution p(f⋆|x⋆, X, y), obtained by marginalizing over u in the full (approximate) posterior N u f ; ˆµu ˆµf , ˆΣuu ˆΣuf ˆΣfu ˆΣff : p(f⋆|x⋆, X, y) = Z p(f⋆|x⋆, f)p(f|X, y)df ≈N f⋆; kT f⋆K−1 f ˆµf , kf ⋆⋆−kT f⋆K−1 f kf⋆+ kT f⋆K−1 f ˆΣffK−1 f kf⋆ . The noise process may itself be of interest, in which case we need to marginalize over both u⋆and f⋆in p(y⋆|x⋆, X, y) = ZZ p y⋆ x⋆, u f ! p u f X, y ! dudf ≈ ZZZZ p y⋆ x⋆, u⋆ f⋆ ! p u⋆ f⋆ u f ! N u f ; ˆµ , ˆΣ du⋆df⋆dudf. This distribution is no longer Gaussian, but its moments may be recovered easily by the same method used to obtain moments of the tilted distribution. EP provides in addition to the approximate moments of the posterior distribution an estimate of the marginal likelihood and its derivatives with respect to kernel hyperparameters. Again, we refer the interested reader to the algorithm presented in [8], adding here only that our implementation uses log noise values on (σ2 R, σ2 O) to allow for their unconstrained optimization. 3.2 Complexity The EP loop is dominated by the rank-two updates of the covariance. Each such update is O (2N)2 , making every N iterations O(4N 3). The posterior refresh is O(8N 3) since it requires the inverse of a 2N × 2N positive semi-definite matrix, most efficiently achieved through Cholesky factorization (this Cholesky factor can be retained for use in calculating the approximate log marginal likelihood). The total number of loops required for convergence of EP is typically independent of N, and can be upper bounded by a small constant, say 10, making the entire inference process O(8N 3) = O(N 3). Thus, our algorithm has the same limiting time complexity as i.i.d. robust regression by EP, which admittedly masks the larger coefficient that appears in approximating both u and f simultaneously. Additionally, the body of the EP loop is slightly slower, since the precision matrix in a standard GP can be obtained with a single division, whereas our model requires the inversion of a 2 × 2 matrix. 4 Experiments We identify two general noise characteristics for which our model may be suitable. The first is when the outlying observations can appear in clusters: we saw in figure 1d how these occurrences affect the standard mixture model. In fact the problem is quite severe, since the multimodality of the posterior impedes the convergence of EP, while the possibility of conflicting gradient information at the optima hampers procedures for evidence maximization. In figure 4 we illustrate how the TGP succeeds where the mixture and Laplace models fail; note how the mean process on u falls sharply in the contaminated regions. This is a stable solution, and hyperparameters can be fit reliably. A data set which exhibits the superior predictive modelling of the TGP in a domain where robust methods can also expect to perform well is provided by Kuss [7] in a variation on a set of Friedman [9]. The samples are drawn from a function of ten-dimensional vectors x which depend only on the first five components: f(x) = 10 sin(πx1x2) + 20(x3 −0.5)2 + 10x4 + 5x5. 6 -10 0 10 (a) Mixture noise -10 0 10 (b) Laplace noise -10 0 10 (c) TGP Figure 4: The corruptions are i.i.d. around x = −10, and highly correlated near x = 0. We generated ten sets of 90 training examples and 10000 test examples by sampling x uniformly in [0, 1]10, and adding to the training data noise N(0, 1). In our first experiment, we replicated the procedure of [7]: ten training points were added at random with outputs sampled from N(15, 9) (a value likely to lie in the same range as f). The results appear as Friedman (1) in figure 5. Observe that the r.m.s. error for the robust methods is similar, but the TGP is able to fit the variance far more accurately. In a second experiment, the training set was augmented with two Gaussian clusters each of five noisy observations. The cluster centres were drawn uniformly in [0, 1]10, with variance fixed at 10−3. Output values were then drawn from N(0, 1) for all ten points, to give highly correlated values distant from the underlying function (Friedman (2)). Now the TGP excels where the other methods offer no improvement on the standard GP; it also yields very confident predictions (cf. Friedman (1)), because once the outliers have been accounted for there are fewer corrupted regions; furthermore, estimates of where the data are corrupted can be recovered by considering the process on u. In both experiments, the training data were renormalized to zero mean and unit variance, and throughout, we used the anisotropic squared exponential for the f process (implementing so-called relevance determination), and an isotropic version for u. The approximate marginal likelihood was maximized on three to five randomly initialized models; we chose for testing the most favoured. The second domain of application is when the noise on the data is believed a priori to be a function of the input (i.e. heteroscedastic). The twinned GP can simulate this changing variance by modulating the u process, allocating varying weight to the two components. By way of example, the behaviour for the one-dimensional motorcycle set [10] is shown in fig. 5c. However, since the input-dependent noise is not modelled directly, there are two notable dangers associated with this approach: first, the predictive variance saturates when all weight has been apportioned to one or other component; second, the “outlier” component can dominate the variance estimates of the mixture. This is particularly problematic when variance on the data ranges over several orders of magnitude, such that the “outlier” width must be comparably broader than that of the “real” component. In such cases, only with extreme values of u can the smallest errors be predicted, but in consequence the process tends to sweep precipitately through the region of sensitivity where variance predictions can be made accurately. To circumvent these problems we might employ the warped GP [11] to rescale the process on u in a supervised manner, but we do not explore these ideas further here. test error neg. log probability GP Lap MixTGP GP Lap MixTGP 0 0.5 0.2 0.4 (a) Friedman (1) test error neg. log probability GP Lap MixTGP GP Lap MixTGP -1 0 0.2 0.4 0.6 (b) Friedman (2) (c) Motorcycle Figure 5: Results for the Friedman data, and the predictions of the TGP on the motorcycle set. 7 5 Extensions With prior knowledge of the nature of corruptions affecting the signal, we can seek to model the noise distribution more accurately, for example by introducing a compound likelihood for the outlier component pO(yn|fn) = P j αjN yn ; µj(fn) , σ2 j , P j αj = 1. This constrains the relative weight of outlier corruptions to be constant across the entire domain. A richer alternative is provided by extending the single u-process on noise to a series u(1), u(2), . . . , u(ν) of noise processes, and broadening the likelihood function appropriately. For example, with ν = 2, we may write p(yn|fn, u(1) n , u(2) n ) = σ(u(1) n )N yn ; fn , σ2 R + σ(−u(1) n )σ(u(2) n )N yn ; fn , σ2 O1 + σ(−u(1) n )σ(−u(2) n )N yn ; f0 , σ2 O2 . (4) In the former case, the preceding analysis applies with small changes: each component of the outlier distribution contributes moments independently. The second model introduces significant computational difficulty: firstly, we must maintain a posterior distribution over f and all ν us, yielding space requirements O(N(ν + 1)) and time complexity O(N 3(ν + 1)3). More importantly, the requisite moments needed in the EP loop are now intractable, although an inner EP loop can be used to approximate them, since the product of σs behaves in essence like the standard model for GP classification. We omit details, and defer experiments with such a model to future work. 6 Conclusions We have presented a method for robust GP regression that improves upon classical approaches by allowing the noise variance to vary in the input space. We found improved convergence on problems which upset the standard mixture model, and have shown how predictive certainty can be improved by adopting the TGP even for problems which do not. The model also allows an arbitrary process on u, such that specialized prior knowledge could be used to drive the inference over f to respecting regions which may otherwise be considered erroneous. A generalization of our ideas appears as the mixture of GPs [1], and the infinite mixture [2], but both involve a slow inference procedure. When faster solutions are required for robust inference, and a two-component mixture is an adequate model for the task, we believe the TGP is a very attractive option. References [1] Volker Tresp. Mixtures of Gaussian processes. In Advances in Neural Information Processing Systems, pages 654–660, 2000. [2] Carl Edward Rasmussen and Zoubin Ghahramani. Infinite mixtures of gaussian process experts. In Advances in Neural Information Processing Systems, 2002. [3] Paul Goldberg, Christopher Williams, and Christopher Bishop. Regression with input-dependent noise: a Gaussian process treatment. In Advances in Neural Information Processing Systems. MIT Press, 1998. [4] Edward Snelson and Zoubin Ghahramani. Sparse Gaussian processes using pseudo-inputs. In Advances in Neural Information Processing Systems 18. MIT Press, 2005. [5] Thomas Minka. A family of algorithms for approximate Bayesian inference. PhD thesis, Massachusetts Institute of Technology, 2001. [6] Carl Rasmussen and Christopher Williams. Gaussian processes for machine learning. MIT Press, 2006. [7] Malte Kuss. Gaussian process models for robust regression, classification and reinforcement learning. PhD thesis, Technische Universit¨at Darmstadt, 2006. [8] Matthias Seeger. Expectation propagation for exponential families, 2005. Available from http://www.cs.berkeley.edu/˜mseeger/papers/epexpfam.ps.gz. [9] J. H. Friedman. Multivariate adaptive regression splines. Annals of Statistics, 19(1):1–67, 1991. [10] B.W. Silverman. Some aspects of the spline smoothing approach to non-parametric regression curve fitting. Journal of the Royal Statistical Society B, 47:1–52, 1985. [11] Edward Snelson, Carl Edward Rasmussen, and Zoubin Ghahramani. Warped Gaussian processes. In Advances in Neural Information Processing Systems 16, 2003. 8
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New Outer Bounds on the Marginal Polytope David Sontag Tommi Jaakkola Computer Science and Artificial Intelligence Laboratory Massachusetts Institute of Technology Cambridge, MA 02139 dsontag,tommi@csail.mit.edu Abstract We give a new class of outer bounds on the marginal polytope, and propose a cutting-plane algorithm for efficiently optimizing over these constraints. When combined with a concave upper bound on the entropy, this gives a new variational inference algorithm for probabilistic inference in discrete Markov Random Fields (MRFs). Valid constraints on the marginal polytope are derived through a series of projections onto the cut polytope. As a result, we obtain tighter upper bounds on the log-partition function. We also show empirically that the approximations of the marginals are significantly more accurate when using the tighter outer bounds. Finally, we demonstrate the advantage of the new constraints for finding the MAP assignment in protein structure prediction. 1 Introduction Graphical models such as Markov Random Fields (MRFs) have been successfully applied to a wide variety of fields, from computer vision to computational biology. From the point of view of inference, we are generally interested in two questions: finding the marginal probabilities of specific subsets of the variables, and finding the Maximum a Posteriori (MAP) assignment. Both of these require approximate methods. We focus on a particular class of variational approximation methods that cast the inference problem as a non-linear optimization over the marginal polytope, the set of valid marginal probabilities. The selection of appropriate marginals from the marginal polytope is guided by the (non-linear) entropy function. Both the marginal polytope and the entropy are difficult to characterize in general, reflecting the hardness of exact inference calculations. Most message-passing algorithms for evaluating marginals, including belief propagation and tree-reweighted sum-product (TRW), operate instead within the local consistency polytope, characterized by pairwise consistent marginals. For general graphs, this is an outer bound of the marginal polytope. Various approximations have also been suggested for the entropy function. For example, in the TRW algorithm [10], the entropy is decomposed into a weighted combination of entropies of tree-structured distributions. Our goal here is to provide tighter outer bounds on the marginal polytope. We show how this can be achieved efficiently using a cutting-plane algorithm, iterating between solving a relaxed problem and adding additional constraints. Cutting-plane algorithms are a well-known technique for solving integer linear programs. The key to such approaches is to have an efficient separation algorithm which, given an infeasible solution, can quickly find a violated constraint, generally from a very large class of valid constraints on the set of integral solutions. The motivation for our approach comes from the cutting-plane literature for the maximum cut problem. Barahona et al. [3] showed that the MAP problem in pairwise binary MRFs is equivalent to a linear optimization over the cut polytope, which is the convex hull of all valid graph cuts. Tighter relaxations were obtained by using a separation algorithm together with the cutting-plane methodology. We extend this work by deriving a new class of outer bounds on the marginal polytope for 1 non-binary and non-pairwise MRFs. The key realization is that valid constraints can be constructed by a series of projections onto the cut polytope1. More broadly, we seek to highlight emerging connections between polyhedral combinatorics and probabilistic inference. 2 Background Markov Random Fields. Let x ∈χn denote a random vector on n variables, where, for simplicity, each variable xi takes on the values in χi = {0, 1, . . . , k −1}. The MRF is specified by a set of d real valued potentials or sufficient statistics φ(x) = {φk(x)} and a parameter vector θ ∈Rd: p(x; θ) = exp {⟨θ, φ(x)⟩−A(θ)}, A(θ) = log P x∈χnexp {⟨θ, φ(x)⟩} where ⟨θ, φ(x)⟩denotes the dot product of the parameters and the sufficient statistics. In pairwise MRFs, potentials are restricted to be at most over the edges in the graph. We assume that the potentials are indicator functions, i.e., φi,s(x) = δ(xi = s), and make use of the following notation: µi;s = Eθ[φi;s(x)] = p(xi = s; θ) and µij;st = Eθ[φij;st(x)] = p(xi = s, xj = t; θ). Variational inference. The inference task is to evaluate the mean vector µ = Eθ[φ(x)]. The log-partition function A(θ), a convex function of θ, plays a critical role in these calculations. In particular, we can write the log-partition function in terms of its Fenchel-Legendre conjugate [11]: A(θ) = supµ∈M {⟨θ, µ⟩−B(µ)} , (1) where B(µ) = −H(µ) is the negative entropy of the distribution parameterized by µ and is also convex. M is the set of realizable mean vectors µ known as the marginal polytope. More precisely, M := µ ∈Rd | ∃p(x) s.t. µ = Ep[φ(x)] . The value µ∗∈M that maximizes (1) is precisely the desired mean vector corresponding to p(x; θ). Both M and the entropy H(µ) are difficult to characterize in general and have to be approximated. We call the resulting approximate mean vectors pseudomarginals. Mean field algorithms optimize over an inner bound on the marginal polytope (which is not convex) by restricting the marginal vectors to those coming from simpler, e.g., fully factored, distributions. The entropy can be evaluated exactly in this case (the distribution is simple). Alternatively, we can relax the optimization to be over an outer bound on the marginal polytope and also bound the entropy function. Most message passing algorithms for evaluating marginal probabilities obtain locally consistent beliefs so that the pseudomarginals over the edges agree with the singleton pseudomarginals at the nodes. The solution is therefore sought within the local marginal polytope LOCAL(G) = { µ ≥0 | P s∈χiµi;s = 1, P t∈χjµij;st = µi;s } (2) Clearly, M ⊆LOCAL(G) since true marginals are also locally consistent. For trees, M = LOCAL(G). Both LOCAL(G) and M have the same integral vertices for general graphs [11, 6]. Belief propagation can be seen as optimizing pseudomarginals over LOCAL(G) with a (non-convex) Bethe approximation to the entropy [15]. The tree-reweighted sum-product algorithm [10], on the other hand, uses a concave upper bound on the entropy, expressed as a convex combination of entropies corresponding to the spanning trees of the original graph. The log-determinant relaxation [12] is instead based on a semi-definite outer bound on the marginal polytope combined with a Gaussian approximation to the entropy function. Since the moment matrix M1(µ) can be written as Eθ[(1 x)T (1 x)] for µ ∈M, the outer bound is obtained simply by requiring only that the pseudomarginals lie in SDEF1(Kn) = {µ ∈R+ | M1(µ) ⪰0}. Maximum a posteriori. The marginal polytope also plays a critical role in finding the MAP assignment. The problem is to find an assignment x ∈χn which maximizes p(x; θ), or equivalently: max x∈χn log p(x; θ) = max x∈χn⟨θ, φ(x)⟩−A(θ) = sup µ∈M ⟨θ, µ⟩−A(θ) (3) where the log-partition function A(θ) remains a constant and can be ignored. The last equality holds because the optimal value of the linear program is obtained at a vertex (integral solution). That is, when the MAP assignment x∗is unique, the maximizing µ∗is φ(x∗). 1For reasons of clarity, our results will be given in terms of the binary marginal polytope, also called the correlation polytope, which is equivalent to the cut polytope of the suspension graph of the MRF [6]. 2 Algorithm 1 Cutting-plane algorithm for probabilistic inference 1: OUTER ←LOCAL(G) 2: repeat 3: µ∗←argmaxµ∈OUTER {⟨θ, µ⟩−B∗(µ)} 4: Choose projection graph Gπ, e.g. single, k, or full 5: C ←Find Violated Inequalities(Gπ, Ψπ(µ∗)) 6: OUTER ←OUTER ∩C 7: until C = Rd (did not find any violated inequalities) Cycle inequalities. The marginal polytope can be defined by the intersection of a large number of linear inequalities. We focus on inequalities beyond those specifying LOCAL(G), in particular the cycle inequalities [4, 2, 6]. Assume the variables are binary. Given an assignment x ∈{0, 1}n, (i, j) ∈E is cut if xi ̸= xj. The cycle inequalities arise from the observation that a cycle must have an even (possibly zero) number of cut edges. Suppose we start at node i, where xi = 0. As we traverse the cycle, the assignment changes each time we cross a cut edge. Since we must return to xi = 0, the assignment can only change an even number of times. For a cycle C and any F ⊆C such that |F| is odd, this constraint can be written as P (i,j)∈C\F δ(xi ̸= xj)+P (i,j)∈F δ(xi = xj) ≥1. Since this constraint is valid for all assignments x ∈{0, 1}n, it holds also in expectation. Thus X (i,j)∈C\F (µij;10 + µij;01) + X (i,j)∈F (µij;00 + µij;11) ≥1 (4) is valid for any µ ∈M{0,1}, the marginal polytope of a binary pairwise MRF. For a chordless circuit C, the cycle inequalities are facets of M{0,1} [4]. They suffice to characterize M{0,1} for a graph G if and only if G has no K4-minor. Although there are exponentially many cycles and cycle inequalities for a graph, Barahona and Mahjoub [4, 6] give a simple algorithm to separate the whole class of cycle inequalities. To see whether any cycle inequality is violated, construct the undirected graph G′ = (V ′, E′) where V ′ contains nodes i1 and i2 for each i ∈V , and for each (i, j) ∈E, the edges in E′ are: (i1, j1) and (i2, j2) with weight µij;10 + µij;01, and (i1, j2) and (i2, j1) with weight µij;00 + µij;11. Then, for each node i ∈V we find the shortest path in G′ from i1 to i2. The shortest of all these paths will not use both copies of any node j (otherwise the path j1 to j2 would be shorter), and so defines a cycle in G and gives the minimum value of P (i,j)∈C\F (µij;10 + µij;01)+P (i,j)∈F (µij;00 + µij;11). If this is less than 1, we have found a violated cycle inequality; otherwise, µ satisfies all cycle inequalities. Using Dijkstra’s shortest paths algorithm with a Fibonacci heap [5], the separation problem can be solved in time O(n2 log n + n|E|). 3 Cutting-plane algorithm Our main result is the proposed Algorithm 1 given above. The algorithm alternates between solving for an upper bound of the log-partition function (see Eq. 1) and tightening the outer bound on the marginal polytope by incorporating valid constraints that are violated by the current pseudomarginals. The projection graph (line 4) is not needed for binary pairwise MRFs and will be described in the next section. We start the algorithm (line 1) with the loose outer bound on the marginal polytope given by the local consistency constraints. Tighter initial constraints, e.g., M1(µ) ⪰0, are possible as well. The separation algorithm returns a feasible set C given by the intersection of halfspaces, and we intersect this with OUTER to obtain a smaller feasible space, i.e. a tighter relaxation. The experiments in Section 5 use the separation algorithm for cycle inequalities. However, any class of valid constraints for the marginal polytope with an efficient separation algorithm may be used in line 5. Other examples besides the cycle inequalities include the odd-wheel and bicycle odd-wheel inequalities [6], and also linear inequalities that enforce positive semi-definiteness of M1(µ). The cutting-plane algorithm is in effect optimizing the variational objective (Eq. 1) over a relaxation of the marginal polytope defined by the intersection of all inequalities that can be returned in line 5. Any entropy approximation B∗(µ) can be used so long as we can efficiently solve the optimization problem in line 3. The log-determinant and TRW entropy approximations have two appealing fea3 Figure 1: Illustration of the projection Ψπ for one edge (i, j) ∈E where χi = {0, 1, 2} and χj = {0, 1, 2, 3}. The projection graph Gπ, shown on the right, has 3 partitions for i and 7 for j. tures. First, as upper bounds they permit the algorithm to be used for obtaining tighter upper bounds on the log-partition function. Second, the objective functions to be maximized are convex and can be solved efficiently using conditional gradient or other methods. When the algorithm terminates, we can use the last µ∗vector as an approximation to the single node and edge marginals. The results given in Section 5 use this method. The algorithm for MAP is the same, excluding the entropy function in line 3; the optimization is simply a linear program. Since all integral vectors in the relaxation OUTER are extreme points of the marginal polytope, any integral µ∗is the MAP assignment. 4 Generalization to non-binary MRFs In this section we give a new class of valid inequalities for the marginal polytope of non-binary and non-pairwise MRFs, and show how to efficiently separate this exponentially large set of inequalities. The key theoretical idea is to project the marginal polytope onto different binary marginal polytopes. Aggregation and projection are well-known techniques in polyhedral combinatorics for obtaining valid inequalities [6]. Given a linear projection Φ(x) = Ax, any valid inequality c′Φ(x) ≤b for Φ(x) also gives the valid inequality c′Ax ≤b for x. We obtain new inequalities by aggregating the values of each variable. For each variable i, let πq i be a partition of its values into two non-empty sets, i.e., the map πq i : χi →{0, 1} is surjective. Let πi = {π1 i , π2 i , . . .} be a collection of partitions of variable i. Define the projection graph Gπ = (Vπ, Eπ) so that there is a node for each πq i ∈πi, and nodes πq i and πr j are connected if (i, j) ∈E. We call the graph consisting of all possible variable partitions the full projection graph. In Figure 1 we show part of the full projection graph corresponding to one edge (i, j), where xi has three values and xj has four values. Intuitively, a partition for a variable splits its values into two clusters, resulting in a binary variable. For example, the (new) variable corresponding to the partition {0, 1}{2} of xi is 1 if xi = 2, and 0 otherwise. The following gives a projection of marginal vectors of non-binary MRFs onto the marginal polytope of the projection graph Gπ, which has binary variables for each partition. Definition 1. The linear map Ψπ takes µ ∈M and for each node v = πq i ∈Vπ assigns µ′ v;1 = P s∈χi s.t. πq i (s)=1 µi;s and for each edge e = (πq i , πr j) ∈Eπ assigns µ′ e;11 = P si∈χi,sj∈χj s.t. πq i (si)=πr j (sj)=1 µij;sisj. To construct valid inequalities for each projection we need to characterize the image space. Let M{0,1}(Gπ) denote the binary marginal polytope of the projection graph. Theorem 1. The image of the projection Ψπ is M{0,1}(Gπ), i.e. Ψπ : M →M{0,1}(Gπ). Proof. Since Ψπ is a linear map, it suffices to show that, for every extreme point µ ∈M, Ψπ(µ) ∈ M{0,1}(Gπ). The extreme points of M correspond one-to-one with assignments x ∈χn. Given an extreme point µ ∈M and variable v = πq i ∈Vπ, define x′(µ)v = P s∈χi s.t. πq i (s)=1 µi;s. Since µ is an extreme point, µi;s = 1 for exactly one value s, which implies that x′(µ) ∈{0, 1}|Vπ|. Then, Ψπ(µ) = E[φ(x′(µ))], showing that Ψπ(µ) ∈M{0,1}(Gπ). This result allows valid inequalities for M{0,1}(Gπ) to carry over to M. In general, the projection Ψπ will not be surjective. Suppose every variable has k values. The single projection graph, 4 where |πi| = 1 for all i, has one node per variable and is surjective. The full projection graph has O(2k) nodes per variable. A cutting-plane algorithm may begin by projecting onto a small graph, then expanding to larger graphs only after satisfying all inequalities given by the smaller one. The k−projection graph Gk = (Vk, Ek) has k partitions per variable corresponding to each value versus all the other values. These projections yield a new class of cycle inequalities for the marginal polytope. Consider a single projection graph Gπ, a cycle C in G, and any F ⊆C such that |F| is odd. Let πi be the partition for node i. We obtain the following valid inequality for µ ∈M by applying the projection Ψπ and the cycle inequality: X (i,j)∈C\F µπ ij(x′ i ̸= x′ j) + X (i,j)∈F µπ ij(x′ i = x′ j) ≥1, (5) where µπ ij(x′ i ̸= x′ j) = X si∈χi,sj∈χj s.t. πi(si)̸=πj(sj) µij;sisj (6) µπ ij(x′ i = x′ j) = X si∈χi,sj∈χj s.t. πi(si)=πj(sj) µij;sisj. (7) It is revealing to contrast (5) with P (i,j)∈C\F δ(xi ̸= xj) + P (i,j)∈F δ(xi = xj) ≥1. For x ∈χn, the latter holds only for |F| = 1. We can only obtain the more general inequality by fixing a partition of each node’s values. Theorem 2. For every single projection graph Gπ and every cycle inequality arising from a chordless circuit C on Gπ, ∃µ ∈LOCAL(G)\M such that µ violates that inequality. Proof. For each variable i ∈V , choose si, ti s.t. πi(si) = 1 and πi(ti) = 0. Assign µi;q = 0 for q ∈χi\{si, ti}. Similarly, for every (i, j) ∈E, assign µij;qr = 0 for q ∈χi\{si, ti} and r ∈χj\{sj, tj}. The polytope resulting from the projection of M onto the remaining values (e.g. µi;si) is isomorphic to M{0,1} for the graph Gπ. Barahona and Mahjoub [4] showed that the cycle inequality on the chordless circuit C is facet-defining for M{0,1}. Since C is over ≥3 variables from G, this cannot be a facet of LOCAL(G). Let LOCAL{0,1} be the projection of LOCAL(G) onto the remaining values. Thus, ∃µ′ ∈LOCAL{0,1}\M{0,1}, and we can assign µ accordingly. Note that the theorem implies that the projected cycle inequalities are strictly tighter than LOCAL(G), but it does not characterize how much is gained. If all n variables have k values, then there are O((2k)n) different single projection graphs. However, since for every cycle inequality in the single projection graphs there is an equivalent cycle inequality in the full projection graph, it suffices to consider just the full projection graph. Thus, even though the projection is not surjective, the full projection graph, which has O(n2k) nodes, allows us to efficiently obtain a tighter relaxation than any combination of projection graphs would give. In particular, the separation problem for all cycle inequalities (5) for all single projection graphs, when we allow some additional valid inequalities for M (arising from the cycle using more than one partition for some variables), can now be solved in time O(poly(n, 2k)). Related work. In earlier work, Althaus et al. [1] analyze the GMEC polyhedron, which is equivalent to the marginal polytope. They use a similar value-aggregation technique to derive valid constraints from the triangle inequalities. Koster et al. [8] investigate the Partial Constraint Satisfaction Problem polytope, which is also equivalent to the marginal polytope. They used value-aggregation to show that a class of cycle inequalities (corresponding to Eq. 5 for |F| = 1) are valid for this polytope, and give an algorithm to separate the inequalities for a single cycle. Interestingly, both papers showed that these constraints are facet-defining. Non-pairwise Markov random fields. These results could be applied to non-pairwise MRFs by first projecting the marginal vector onto the marginal polytope of a pairwise MRF. More generally, suppose we include additional variables corresponding to the joint probability of a cluster of variables. We need to add constraints enforcing that all variables in common between two clusters have the same marginals. For pairwise clusters these are simply the usual local consistency constraints. We can now apply the projections of the previous section, considering various partitions of each cluster variable, to obtain a tighter relaxation of the marginal polytope. 5 .5 2 4 6 8 0 0.1 0.2 0.3 0.4 0.5 Coupling, θ ∼ U[−x, x] Average Error TRW TRW + PSD TRW + Cycle TRW + Marg .5 2 4 6 8 0 0.1 0.2 0.3 0.4 0.5 Coupling, θ ∼ U[−x, x] Average Error Logdet Logdet + PSD Logdet + Cycle Logdet + Marg Figure 2: Accuracy of single node marginals on 10 node complete graph (100 trials). 5 Experiments Computing marginals. We experimented with Algorithm 1 using both the log-determinant [12] and the TRW [10] entropy approximations. These trials are on Ising models, which are pairwise MRFs with xi ∈{−1, 1} and potentials φi(x) = xi for i ∈V and φij(x) = xixj for (i, j) ∈E. Although TRW can efficiently optimize over the spanning tree polytope, for these experiments we simply use a weighted distribution over spanning trees, where each tree’s weight is the sum of the absolute value of its edge weights θij. The edge appearance probabilities for this distribution can be efficiently computed using the Matrix Tree Theorem [13]. We optimize the TRW objective with conditional gradient, using linear programming after each gradient step to project onto OUTER. We used the glpkmex and YALMIP optimization packages within Matlab, and wrote the separation algorithm for the cycle inequalities in Java. In Figure 2 we show results for 10 node complete graphs with θi ∼U[−1, 1] and θij ∼U[−x, x], where the coupling strength is varied along the x-axis of the figure. For each data point we averaged the results over 100 trials. The y-axis shows the average ℓ1 error of the single node marginals. These MRFs are highly coupled, and loopy belief propagation (not shown) with a .5 decay rate seldom converges. The TRW and log-determinant algorithms, optimizing over the local consistency polytope, give pseudomarginals only slightly better than loopy BP. Even adding the positive semi-definite constraint M1(µ) ⪰0, for which TRW must be optimized using conditional gradient and semidefinite programming for the projection step, does not improve the accuracy by much. However, both entropy approximations give significantly better pseudomarginals when used by our algorithm together with the cycle inequalities (see “TRW + Cycle” and “Logdet + Cycle” in the figures). For small MRFs, we can exactly represent the marginal polytope as the convex hull of its 2n vertices. We found that the cycle inequalities give nearly as good accuracy as the exact marginal polytope (see “TRW + Marg” and “Logdet + Marg”). Our work sheds some light on the relative value of the entropy approximation compared to the relaxation of the marginal polytope. When the MRF is weakly coupled, both entropy approximations do reasonably well using the local consistency polytope. This is not surprising: the limit of weak coupling is a fully disconnected graph, for which both the entropy approximation and the marginal polytope relaxation are exact. With the local consistency polytope, both entropy approximations get steadily worse as the coupling increases. In contrast, using the exact marginal polytope, we see a peak at θ = 2, then a steady improvement in accuracy as the coupling term grows. This occurs because the limit of strong coupling is the MAP problem, for which using the exact marginal polytope will give exact results. The interesting region is near the peak θ = 2, where the entropy term is neither exact nor outweighed by the coupling. Our algorithm seems to “solve” the part of the problem caused by the local consistency polytope relaxation: TRW’s accuracy goes from .33 to .15, and log-determinant’s accuracy from .17 to .076. The fact that neither entropy approximation can achieve accuracy below .07, even with the exact marginal polytope, motivates further research on improving this part of the approximation. 6 1 2 3 4 5 6 7 8 9 10 0.1 0.2 0.3 0.4 Average l1 Error 10×10 Grid Cutting Plane Iterations 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 Average Prediction Error 10×10 Grid Cutting Plane Iterations 3 9 15 21 27 33 39 45 0.1 0.2 0.3 0.4 0.5 Average l1 Error 20 Node Complete Cutting Plane Iterations 3 9 15 21 27 33 39 45 0 0.2 0.4 0.6 0.8 Average Prediction Error 20 Node Complete Cutting Plane Iterations Figure 3: Accuracy of single node marginals with TRW entropy, θi ∈U[−1, 1] and θij ∈U[−4, 4]. 0 500 1000 0 5 10 15 20 Amino Acids (Variables) Cutting Plane Iterations Figure 4: MAP for protein side-chain prediction with Rosetta energy function. Next, we looked at the number of iterations (in terms of the loop in Algorithm 1) the algorithm takes before all cycle inequalities are satisfied. In each iteration we add to OUTER at most2 n violated cycle inequalities, coming from the n shortest paths. In Figure 3 we show boxplots of the l1 error of the single node marginals for both 10x10 grid MRFs (40 trials) and 20 node complete MRFs (10 trials). We also show whether the pseudomarginals are on the correct side of .5, which is important if we were doing prediction based on the results from approximate inference. The middle line gives the median, the boxes show the upper and lower quartiles, and the whiskers show the extent of the data. Iteration 1 corresponds to TRW with only the local consistency constraints. For the grid MRFs, all of the cycle inequalities were satisfied within 10 iterations. We observed the same convergence results on a 30x30 grid, although we could not assess the accuracy due to the difficulty of exact marginals calculation. For the complete graph MRFs, the algorithm took many more iterations before all cycle inequalities were satisfied. Protein side-chain prediction. We next applied our algorithm to the problem of predicting protein side-chain configurations. Given the 3-dimensional structure of a protein’s backbone, the task is to predict the relative angle of each amino acid’s side-chain. The angles are discretized into at most 45 values. Yanover et al. [14] showed that minimization of the Rosetta energy function corresponds to finding the MAP assignment of a non-binary pairwise MRF. They also showed that the treereweighted max-product algorithm [9] can be used to solve the LP relaxation given by LOCAL(G), and that this succeeds in finding the MAP assignment for 339 of the 369 proteins in their data set. However, the optimal solution to the LP relaxation for the remaining 30 proteins, arguably the most difficult of the proteins, is fractional. Using the k-projection graph and projected cycle inequalities, we succeeded in finding the MAP assignment for all proteins except for the protein ‘1rl6’. We show in Figure 4 the number of cuttingplane iterations needed for each of the 30 proteins. In each iteration, we solve the LP relaxation, and, if the solution is not integral, run the separation algorithm to find violated inequalities. For the protein ‘1rl6’, after 12 cutting-plane iterations, the solution was not integral, and we could not find any violated cycle inequalities using the k-projection graph. We then tried using the full projection graph, and found the MAP after just one (additional) iteration. Figure 4 shows one of the cycle inequalities (5) in the full projection graph that was found to be violated. The cut edges indicate the 3 edges in F. The violating µ had µ36;s = .1667 for s ∈{0, 1, 2, 3, 4, 5}, µ38;6 = .3333, µ38;4 = .6667, µ43;s = .1667 for s ∈{1, 2, 4, 5}, µ43;3 = .3333, and zero for all other values of these variables. This example shows that the relaxation given by the full projection graph is strictly tighter than that of the k-projection graph. 2Many fewer inequalities were added, since not all cycles in G′ are simple cycles in G. 7 The commercial linear programming solver CPLEX 10.0 solves each LP relaxation in under 75 seconds. Using simple heuristics, the separation algorithm runs in seconds, and we find each protein’s MAP assignment in under 11.3 minutes. Kingsford et al. [7] found, and we also observed, that CPLEX’s branch-and-cut algorithm for solving integer linear programs also works well for these problems. One interesting future direction would be to combine the two approaches, using our new outer bounds within the branch-and-cut scheme. Our results show that the new outer bounds are powerful, allowing us to find the MAP solution for all of the MRFs, and suggesting that using them will also lead to significantly more accurate marginals for non-binary MRFs. 6 Conclusion The facial structure of the cut polytope, equivalently, the binary marginal polytope, has been wellstudied over the last twenty years. The cycle inequalities are just one of many large classes of valid inequalities for the cut polytope for which efficient separation algorithms are known. Our theoretical results can be used to derive outer bounds for the marginal polytope from any of the valid inequalities on the cut polytope. Our approach is particularly valuable because it takes advantage of the sparsity of the graph, and only uses additional constraints when they are guaranteed to affect the solution. An interesting open problem is to develop new message-passing algorithms which can incorporate cycle and other inequalities, to efficiently do the optimization within the cutting-plane algorithm. Acknowledgments The authors thank Amir Globerson and David Karger for helpful discussions. This work was supported in part by the DARPA Transfer Learning program. D.S. was also supported by a National Science Foundation Graduate Research Fellowship. References [1] E. Althaus, O. Kohlbacher, H.-P. Lenhof, and P. M¨uller. A combinatorial approach to protein docking with flexible side-chains. In RECOMB ’00, pages 15–24, 2000. [2] F. Barahona. On cuts and matchings in planar graphs. Mathematical Programming, 60:53–68, 1993. [3] F. Barahona, M. Gr¨otschel, M. Junger, and G. Reinelt. An application of combinatorial optimization to statistical physics and circuit layout design. Operations Research, 36(3):493–513, 1988. [4] F. Barahona and A. R. Mahjoub. On the cut polytope. Mathematical Programming, 36:157–173, 1986. [5] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein. Introduction to Algorithms. MIT Press, 2nd edition, 2001. [6] M. M. Deza and M. Laurent. Geometry of Cuts and Metrics, volume 15 of Algorithms and Combinatorics. Springer, 1997. [7] C. L. Kingsford, B. Chazelle, and M. Singh. Solving and analyzing side-chain positioning problems using linear and integer programming. Bioinformatics, 21(7):1028–1039, 2005. [8] A. Koster, S. van Hoesel, and A. Kolen. The partial constraint satisfaction problem: Facets and lifting theorems. Operations Research Letters, 23:89–97, 1998. [9] M. Wainwright, T. Jaakkola, and A. Willsky. MAP estimation via agreement on trees: message-passing and linear programming. IEEE Transactions on Information Theory, 51(11):3697–3717, November 2005. [10] M. Wainwright, T. Jaakkola, and A. Willsky. A new class of upper bounds on the log partition function. IEEE Transactions on Information Theory, 51:2313–2335, July 2005. [11] M. Wainwright and M. I. Jordan. Graphical models, exponential families and variational inference. Technical Report 649, UC Berkeley, Dept. of Statistics, 2003. [12] M. Wainwright and M. I. Jordan. Log-determinant relaxation for approximate inference in discrete Markov random fields. IEEE Transactions on Signal Processing, 54(6):2099–2109, June 2006. [13] D. B. West. Introduction to Graph Theory. Prentice Hall, 2001. [14] C. Yanover, T. Meltzer, and Y. Weiss. Linear programming relaxations and belief propagation – an empirical study. JMLR Special Issue on Machine Learning and Large Scale Optimization, 7:1887–1907, September 2006. [15] J. Yedidia, W. Freeman, and Y. Weiss. Bethe free energy, Kikuchi approximations, and belief propagation algorithms. Technical Report 16, Mitsubishi Electric Research Lab, 2001. 8
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Neural characterization in partially observed populations of spiking neurons Jonathan W. Pillow Peter Latham Gatsby Computational Neuroscience Unit, UCL 17 Queen Square, London WC1N 3AR, UK pillow@gatsby.ucl.ac.uk pel@gatsby.ucl.ac.uk Abstract Point process encoding models provide powerful statistical methods for understanding the responses of neurons to sensory stimuli. Although these models have been successfully applied to neurons in the early sensory pathway, they have fared less well capturing the response properties of neurons in deeper brain areas, owing in part to the fact that they do not take into account multiple stages of processing. Here we introduce a new twist on the point-process modeling approach: we include unobserved as well as observed spiking neurons in a joint encoding model. The resulting model exhibits richer dynamics and more highly nonlinear response properties, making it more powerful and more flexible for fitting neural data. More importantly, it allows us to estimate connectivity patterns among neurons (both observed and unobserved), and may provide insight into how networks process sensory input. We formulate the estimation procedure using variational EM and the wake-sleep algorithm, and illustrate the model’s performance using a simulated example network consisting of two coupled neurons. 1 Introduction A central goal of computational neuroscience is to understand how the brain transforms sensory input into spike trains, and considerable effort has focused on the development of statistical models that can describe this transformation. One of the most successful of these is the linear-nonlinearPoisson (LNP) cascade model, which describes a cell’s response in terms of a linear filter (or receptive field), an output nonlinearity, and an instantaneous spiking point process [1–5]. Recent efforts have generalized this model to incorporate spike-history and multi-neuronal dependencies, which greatly enhances the model’s flexibility, allowing it to capture non-Poisson spiking statistics and joint responses of an entire population of neurons [6–10]. Point process models accurately describe the spiking responses of neurons in the early visual pathway to light, and of cortical neurons to injected currents. However, they perform poorly both in higher visual areas and in auditory cortex, and often do not generalize well to stimuli whose statistics differ from those used for fitting. Such failings are in some ways not surprising: the cascade model’s stimulus sensitivity is described with a single linear filter, whereas responses in the brain reflect multiple stages of nonlinear processing, adaptation on multiple timescales, and recurrent feedback from higher-level areas. However, given its mathematical tractability and its accuracy in capturing the input-output properties of single neurons, the model provides a useful building block for constructing richer and more complex models of neural population responses. Here we extend the point-process modeling framework to incorporate a set of unobserved or “hidden” neurons, whose spike trains are unknown and treated as hidden or latent variables. The unobserved neurons respond to the stimulus and to synaptic inputs from other neurons, and their spiking 1 activity can in turn affect the responses of the observed neurons. Consequently, their functional properties and connectivity can be inferred from data [11–18]. However, the idea is not to simply build a more powerful statistical model, but to develop a model that can help us learn something about the underlying structure of networks deep in the brain. Although this expanded model offers considerably greater flexibility in describing an observed set of neural responses, it is more difficult to fit to data. Computing the likelihood of an observed set of spike trains requires integrating out the probability distribution over hidden activity, and we need sophisticated algorithms to find the maximum likelihood estimate of model parameters. Here we introduce a pair of estimation procedures based on variational EM (expectation maximization) and the wake-sleep algorithm. Both algorithms make use of a novel proposal density to capture the dependence of hidden spikes on the observed spike trains, which allows for fast sampling of hidden neurons’ activity. In the remainder of this paper we derive the basic formalism and demonstrate its utility on a toy problem consisting of two neurons, one of which is observed and one which is designated “hidden”. We show that a single-cell model used to characterize the observed neuron performs poorly, while a coupled two-cell model estimated using the wake-sleep algorithm performs much more accurately. 2 Multi-neuronal point-process encoding model We begin with a description of the encoding model, which generalizes the LNP model to incorporate non-Poisson spiking and coupling between neurons. We refer to this as a generalized linear pointprocess (glpp) model1 [8, 9]. For simplicity, we formulate the model for a pair of neurons, although it can be tractably applied to data from a moderate-sized populations (∼10-100 neurons). In this section we do not distinguish between observed and unobserved spikes, but will do so in the next. Let xt denote the stimulus at time t, and yt and zt denote the number of spikes elicited by two neurons at t, where t ∈[0, T ] is an index over time. Note that xt is a vector containing all elements of the stimulus that are causally related to the (scalar) responses yt and zt at time t. Furthermore, let us assume t takes on a discrete set of values, with bin size ∆, i.e., t ∈{0, ∆, 2∆, . . . , T}. Typically ∆is sufficiently small that we observe only zero or one spike in every bin: y t, zt ∈{0, 1}. The conditional intensity (or instantaneous spike rate) of each cell depends on both the stimulus and the recent spiking history via a bank of linear filters. Let y [t−τ,t) and z[t−τ,t) denote the (vector) spike train histories at time t. Here [t −τ, t) refers to times between t −τ and t −∆, so y[t−τ,t) ≡ (yt−τ, yt−τ+∆, ..., yt−2∆, yt−∆) and similarly for z[t−τ,t). The conditional intensities for the two cells are then given by λyt = f(ky · xt + hyy · y[t−τ,t) + hyz · z[t−τ,t)) λzt = f(kz · xt + hzz · z[t−τ,t) + hzy · y[t−τ,t)) (1) where ky and kz are linear filters representing each cell’s receptive field, hyy and hzz are filters operating on each cell’s own spike-train history (capturing effects like refractoriness and bursting), and hzy and hyz are a filters coupling the spike train history of each neuron to the other (allowing the model to capture statistical correlations and functional interactions between neurons). The “·” notation represents the standard dot product (performing a summation over either index or time): k · xt ≡ i kixit h · y[t−τ,t) ≡ t−∆ t′=t−τ ht′yt′, where the index i run over the components of the stimuli (which typically are time points extending into the past). The second expression generalizes to h · z[t−τ,t). The nonlinear function, f, maps the input to the instantaneous spike rate of each cell. We assume here that f is exponential, although any monotonic convex function that grows no faster than expo1We adapt this terminology from “generalized linear model” (glm), a much more general class of models from the statistics literature [19]; this model is a glm whose distribution function is Poisson. 2 x1 y1 x2 y2 x3 y3 x4 y4 z1 z2 z3 z4
hzz kz
hyy ky x y z x hzy
z[t-τ,t)
hyy y[t-τt) xt ky yt
hyz hzy f Figure 1: Schematic of generalized linear point-process (glpp) encoding model. a, Diagram of model parameters for a pair of coupled neurons. For each cell, the parameters consist of a stimulus filter (e.g., ky), a spike-train history filter (hyy), and a filter capturing coupling from the spike train history of the other cell (hzy). The filter outputs are summed, pass through an exponential nonlinearity, and drive spiking via an instantaneous point process. b, Equivalent diagram showing just the parameters of the neuron y, as used for drawing a sample yt. Gray boxes highlight the stimulus vector xt and spike train history vectors that form the input to the model on this time step. c, Simplified graphical model of the glpp causal structure, which allows us to visualize how the likelihood factorizes. Arrows between variables indicate conditional dependence. For visual clarity, temporal dependence is depicted as extending only two time bins, though in real data extends over many more. Red arrows highlight the dependency structure for a single time bin of the response y3. nentially is suitable [9]. Equation 1 is equivalent to f applied to a linear convolution of the stimulus and spike trains with their respective filters; a schematic is shown in figure 1. The probability of observing yt spikes in a bin of size ∆is given by a Poisson distribution with rate parameter λyt∆, P(yt|λyt) = (λyt∆)yt yt! e−λyt∆, (2) and likewise for P(zt|λzt). The likelihood of the full set of spike times is the product of conditionally independent terms, P(Y, Z|X, θ) = t P(yt|λyt)P(zt|λzt), (3) where Y and Z represent the full spike trains, X denotes the full set of stimuli, and θ ≡ {ky, kz, hyy, hzy, hzz, hyz} denotes the model parameters. This factorization is possible because λyt and λzt depend only on the process history up to time t, making y t and zt conditionally independent given the stimulus and spike histories up to t (see Fig. 1c). If the response at time t were to depend on both the past and future response, we would have a causal loop , preventing factorization and making both sampling and likelihood evaluation very difficult. The model parameters can be tractably fit to spike-train data using maximum likelihood. Although the parameter space may be high-dimensional (incorporating spike-history dependence over many time bins and stimulus dependence over a large region of time and space), the negative log-likelihood is convex with respect to the model parameters, making fast convex optimization methods feasible for finding the global maximum [9]. We can write the log-likelihood simply as log P(Y, Z|X, θ) = t (yt log λyt + zt log λzt −∆λyt −∆λzt) + c, (4) where c is a constant that does not depend on θ. 3 3 Generalized Expectation-Maximization and Wake-Sleep Maximizing log P(Y, Z|X, θ) is straightforward if both Y and Z are observed, but here we are interested in the case where Y is observed and Z is “hidden”. Consequently, we have to average over Z. The log-likelihood of the observed data is given by L(θ) ≡log P(Y |θ) = log Z P(Y, Z|θ), (5) where we have dropped X to simplify notation (all probabilities can henceforth be taken to also depend on X). This sum over Z is intractable in many settings, motivating the use of approximate methods for maximizing likelihood. Variational expectation-maximization (EM) [20,21] and the wake-sleep algorithm [22] are iterative algorithms for solving this problem by introducing a tractable approximation to the conditional probability over hidden variables, Q(Z|Y, φ) ≈P(Z|Y, θ), (6) where φ denotes the parameter vector determining Q. The idea behind variational EM can be described as follows. Concavity of the log implies a lower bound on the log-likelihood: L(θ) ≥ Z Q(Z|Y, φ) log P(Y, Z|θ) Q(Z|Y, φ) = log P(Y |θ) −DKL Q(Z|Y, φ), P(Z|Y, θ) , (7) where Q is any probability distribution over Z and D KL is the Kullback-Leibler (KL) divergence between Q and P (using P as shorthand for P(Z|Y, θ)), which is always ≥0. In standard EM, Q takes the same functional form as P, so that by setting φ = θ (the E-step), D KL is 0 and the bound is tight, since the right-hand-side of eq. 7 equals L(θ). Fixing φ, we then maximize the r.h.s. for θ (the M-step), which is equivalent to maximizing the expected complete-data log-likelihood (expectation taken w.r.t. Q), given by EQ(Z|Y,φ) log P(Y, Z|θ) ≡ Z Q(Z|Y, φ) log P(Y, Z|θ). (8) Each step increases a lower bound on the log-likelihood, which can always be made tight, so the algorithm converges to a fixed point that is a maximum of L(θ). The variational formulation differs in allowing Q to take a different functional form than P (i.e., one for which eq. 8 is easier to maximize). The variational E-step involves minimizing D KL(Q, P) with respect to φ, which remains positive if Q does not approximate P exactly; the variational M-step is unchanged from the standard algorithm. In certain cases, it is easier to minimize the KL divergence DKL(P, Q) than DKL(Q, P), and doing so in place of the variational E-step above results in the wake-sleep algorithm [22]. In this algorithm, we fit φ by minimizing DKL(P, Q) averaged over Y , which is equivalent to maximizing the expectation EP (Y,Z|θ) log Q(Z|Y, φ) ≡ Y,Z P(Y, Z|θ) log Q(Z|Y, φ), (9) which bears an obvious symmetry to eq. 8. Thus, both steps of the wake-sleep algorithm involve maximizing an expected log-probability. In the “wake” step (identical to the M-step), we fit the true model parameters θ by maximizing (an approximation to) the log-probability of the observed data Y . In the “sleep” step, we fit φ by trying to find a distribution Q that best approximates the conditional dependence of Z on Y , averaged over the joint distribution P(Y, Z|θ). We can therefore think of the wake phase as learning a model of the data (parametrized by θ), and the sleep phase as learning a consistent internal description of that model (parametrized by φ). Both variational-EM and the wake-sleep algorithm work well when Q closely approximates P, but may fail to converge to a maximum of the likelihood if there is a significant mismatch. Therefore, the efficiency of these methods depends on choosing a good approximating distribution Q(Z|Y, φ) — one that closely matches P(Z|Y, θ). In the next section we show that considerations of the spike generation process can provide us with a good choice for Q. 4 hyz
y[t-τ , t+τ]
hzz z[t-t,t zt xt kz x1 z1 x2 z2 x3 z3 x4 z4 y1 y2 y3 y4 Figure 2: Schematic diagram of the (acausal) model for the proposal density Q(Z|Y, φ), the conditional density on hidden spikes given the observed spike data. a, Conditional model schematic, which allows zt to depend on the observed response both before and after t. b, Graphical model showing causal structure of the acausal model, with arrows indicating dependency. The observed spike responses (gray circles) are no longer dependent variables, but regarded as fixed, external data, which is necessary for computing Q(zt|Y, φ). Red arrows illustrate the dependency structure for a single bin of the hidden response, z3. 4 Estimating the model with partially observed data To understand intuitively why the true P(Z|Y, θ) is difficult to sample, and to motivate a reasonable choice for Q(Z|Y, φ), let us consider a simple example: suppose a single hidden neuron (whose full response is Z) makes a strong excitatory connection to an observed neuron (whose response is Y ), so that if zt = 1 (i.e., the hidden neuron spikes at time t), it is highly likely that y t+1 = 1 (i.e., the observed neuron spikes at time t + 1). Consequently, under the true P(Z|Y, θ), which is the probability over Z in all time bins given Y in all time bins, if y t+1 = 1 there is a high probability that zt = 1. In other words, zt exhibits an acausal dependence on yt+1. But this acausal dependence is not captured in Equation 3, which expresses the probability over z t as depending only on past events at time t, ignoring the future event yt+1 = 1. Based on this observation — essentially, that the effect of future observed spikes on the probability of unobserved spikes depends on the connection strength between the two neurons — we approximate P(Z|Y, θ) using a separate point-process model Q(Z|Y, φ), which contains set of acausal linear filters from Y to Z. Thus we have ˜λzt = exp(˜kz · xt + ˜hzz · z[t−τ,t) + ˜hzy · y[t−τ,t+τ)). (10) As above, ˜kz, ˜hzz and ˜hzy are linear filters; the important difference is that ˜hzy · y[t−τ,t+τ) is a sum over past and future time: from t −τ to t + τ −∆. For this model, the parameters are φ = (˜kz, ˜hzz, ˜hzy). Figure 2 illustrates the model architecture. We now have a straightforward way to implement the wake-sleep algorithm, using samples from Q to perform the wake phase (estimating θ), and samples from P(Y, Z|θ) to perform the sleep phase (estimating φ). The algorithm works as follows: • Wake: Draw samples {Zi} ∼Q(Z|Y, φ), where Y are the observed spike trains and φ is the current set of parameters for the acausal point-process model Q. Evaluate the expected complete-data log-likelihood (eq. 8) using Monte Carlo integration: EQ log P(Y, Z|θ) = lim N→∞ 1 N N i=1 log P(Y, Zi|θ). (11) This is log-concave in θ, meaning that we can efficiently find its global maximum to fit θ. 5 • Sleep: Draw samples {Yj, Zj} ∼P(Y, Z|θ), the true encoding distribution with current parameters θ. (Note these samples are pure “fantasy” data, drawn without reference to the observed Y ). As above, compute the expected log-probability (eq. 9) using these samples: EP (Y,Z|θ) log Q(Z|Y, φ) = lim N→∞ 1 N N i=1 log Q(Zj|Yj, φ), (12) which is also log-concave and thus efficiently maximized for φ. One advantage of the wake-sleep algorithm is that each complete iteration can be performed using only a single set of samples drawn from Q and P. A theoretical drawback to wake-sleep, however, is that the sleep step is not guaranteed to increase a lower-bound on the log-likelihood, as in variationalEM (wake-sleep minimizes the “wrong” KL divergence). We can implement variational-EM using the same approximating point-process model Q, but we now require multiple steps of sampling for a complete E-step. To perform a variational E-step, we draw samples (as above) from Q and use them to evaluate both the KL divergence DKL Q(Z|Y, φ)||P(Z|Y, θ) and its gradient with respect to φ. We can then perform noisy gradient descent to find a minimum, drawing a new set of samples for each evaluation of DKL(Q, P). The M-step is equivalent to the wake phase of wake-sleep, achievable with a single set of samples. One additional use for the approximating point-process model Q is as a “proposal” distribution for Metropolis-Hastings sampling of the true P(Z|Y, θ). Such samples can be used to evaluate the true log-likelihood, for comparison with the variational lower bound, and for noisy gradient ascent of the likelihood to examine how closely these approximate methods converge to the true ML estimate. For fully observed data, such samples also provide a useful means for measuring how much the entropy of one neuron’s response is reduced by knowing the responses of its neighbors. 5 Simulations: a two-neuron example To verify the method, we applied it to a pair of neurons (as depicted in fig. 1), simulated using a stimulus consisting of a long presentation of white noise. We denoted one of the neurons ”observed” and the other ”hidden”. The parameters used for the simulation are depicted in fig. 3. The cells have similarly-shaped biphasic stimulus filters with opposite sign, like those commonly observed in ON and OFF retinal ganglion cells. We assume that the ON-like cell is observed, while the OFF-like cell is hidden. Both cells have spike-history filters that induce a refractory period following a spike, with a small peak during the relative refractory period that elicits burst-like responses. The hidden cell has a strong positive coupling filter hzy onto the observed cell, which allows spiking activity in the hidden cell to excite the observed cell (despite the fact that the two cells receive opposite-sign stimulus input). For simplicity, we assume no coupling from the observed to the hidden cell 2. Both types of filters were defined in a linear basis consisting of four raised cosines, meaning that each filter is specified by four parameters, and the full model contains 20 parameters (i.e., 2 stimulus filters and 3 spike-train filters). Fig. 3b shows rasters of the two cells’ responses to a repeated presentations of a 1s Gaussian whitenoise stimulus with a framerate of 100Hz. Note that the temporal structure of the observed cell’s response is strongly correlated with that of the hidden cell due to the strong coupling from hidden to observed (and the fact that the hidden cell receives slightly stronger stimulus drive). Our first task is to examine whether a standard, single-cell glpp model can capture the mapping from stimuli to spike responses. Fig. 3c shows the parameters obtained from such a fit to the observed data, using 10s of the response to a non-repeating white noise stimulus (1000 samples, 251 spikes). Note that the estimated stimulus filter (red) has much lower amplitude than the stimulus filter of the true model (gray). Fig. 3d shows the parameters obtained for an observed and a hidden neuron, estimated using wake-sleep as described in section 4. Fig. 3e-f shows a comparison of the performance of the two models, indicating that the coupled model estimated with wake-sleep does a much better job of capturing the temporal structure of the observed neuron’s response (accounting for 60% vs. 15% of 2Although the stimulus and spike-history filters bear a rough similarity to those observed in retinal ganglion cells, the coupling used here is unlike coupling filters observed (to our knowledge) between ON and OFF cells in retinal data; it is assumed purely for demonstration purposes. 6 ! ! ! ! ! "! ! ! ! !# ! "! ! ! ! # "
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kz ky hzz hyy hzy '()
Figure 3: Simulation results. a, Parameters used for generating simulated responses. The top row shows the filters determining the input to the observed cell, while the bottom row shows those influencing the hidden cell. b, Raster of spike responses of observed and hidden cells to a repeated, 1s Gaussian white noise stimulus (top). c, Parameter estimates for a single-cell glpp model fit to the observed cell’s response, using just the stimulus and observed data (estimates in red; true observedcell filters in gray). d, Parameters obtained using wake-sleep to estimate a coupled glpp model, again using only the stimulus and observed spike times. e, Response raster of true observed cell (obtained by simulating the true two-cell model), estimated single-cell model and estimated coupled model. f, Peri-stimulus time histogram (PSTH) of the above rasters showing that the coupled model gives much higher accuracy predicting the true response. the PSTH variance). The single-cell model, by contrast, exhibits much worse performance, which is unsurprising given that the standard glpp encoding model can capture only quasi-linear stimulus dependencies. 6 Discussion Although most statistical models of spike trains posit a direct pathway from sensory stimuli to neuronal responses, neurons are in fact embedded in highly recurrent networks that exhibit dynamics on a broad range of time-scales. To take into account the fact that neural responses are driven by both stimuli and network activity, and to understand the role of network interactions, we proposed a model incorporating both hidden and observed spikes. We regard the observed spike responses as those recorded during a typical experiment, while the responses of unobserved neurons are modeled as latent variables (unrecorded, but exerting influence on the observed responses). The resulting model is tractable, as the latent variables can be integrated out using approximate sampling methods, and optimization using variational EM or wake-sleep provides an approximate maximum likelihood estimate of the model parameters. As shown by a simple example, certain settings of model parameters necessitate the incorporation unobserved spikes, as the standard single-stage encoding model does not accurately describe the data. In future work, we plan to examine the quantitative performance of the variational-EM and wakesleep algorithms, to explore their tractability in scaling to larger populations, and to apply them to real neural data. The model offers a promising tool for analyzing network structure and networkbased computations carried out in higher sensory areas, particularly in the context where data are only available from a restricted set of neurons recorded within a larger population. 7 References [1] I. Hunter and M. Korenberg. The identification of nonlinear biological systems: Wiener and hammerstein cascade models. Biological Cybernetics, 55:135–144, 1986. [2] N. Brenner, W. Bialek, and R. de Ruyter van Steveninck. Adaptive rescaling optimizes information transmission. Neuron, 26:695–702, 2000. [3] H. Plesser and W. Gerstner. Noise in integrate-and-fire neurons: From stochastic input to escape rates. Neural Computation, 12:367–384, 2000. [4] E. J. Chichilnisky. A simple white noise analysis of neuronal light responses. Network: Computation in Neural Systems, 12:199–213, 2001. [5] E. P. Simoncelli, L. Paninski, J. Pillow, and O. Schwartz. Characterization of neural responses with stochastic stimuli. In M. Gazzaniga, editor, The Cognitive Neurosciences, pages 327–338. MIT Press, 3rd edition, 2004. [6] M. Berry and M. Meister. Refractoriness and neural precision. Journal of Neuroscience, 18:2200–2211, 1998. [7] K. Harris, J. Csicsvari, H. Hirase, G. Dragoi, and G. Buzsaki. 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Bayesian Agglomerative Clustering with Coalescents Yee Whye Teh Gatsby Unit University College London ywteh@gatsby.ucl.ac.uk Hal Daum´e III School of Computing University of Utah me@hal3.name Daniel Roy CSAIL MIT droy@mit.edu Abstract We introduce a new Bayesian model for hierarchical clustering based on a prior over trees called Kingman’s coalescent. We develop novel greedy and sequential Monte Carlo inferences which operate in a bottom-up agglomerative fashion. We show experimentally the superiority of our algorithms over the state-of-the-art, and demonstrate our approach in document clustering and phylolinguistics. 1 Introduction Hierarchically structured data abound across a wide variety of domains. It is thus not surprising that hierarchical clustering is a traditional mainstay of machine learning [1]. The dominant approach to hierarchical clustering is agglomerative: start with one cluster per datum, and greedily merge pairs until a single cluster remains. Such algorithms are efficient and easy to implement. Their primary limitations—a lack of predictive semantics and a coherent mechanism to deal with missing data— can be addressed by probabilistic models that handle partially observed data, quantify goodness-offit, predict on new data, and integrate within more complex models, all in a principled fashion. Currently there are two main approaches to probabilistic models for hierarchical clustering. The first takes a direct Bayesian approach by defining a prior over trees followed by a distribution over data points conditioned on a tree [2, 3, 4, 5]. MCMC sampling is then used to obtain trees from their posterior distribution given observations. This approach has the advantages and disadvantages of most Bayesian models: averaging over sampled trees can improve predictive capabilities, give confidence estimates for conclusions drawn from the hierarchy, and share statistical strength across the model; but it is also computationally demanding and complex to implement. As a result such models have not found widespread use. [2] has the additional advantage that the distribution induced on the data points is exchangeable, so the model can be coherently extended to new data. The second approach uses a flat mixture model as the underlying probabilistic model and structures the posterior hierarchically [6, 7]. This approach uses an agglomerative procedure to find the tree giving the best posterior approximation, mirroring traditional agglomerative clustering techniques closely and giving efficient and easy to implement algorithms. However because the underlying model has no hierarchical structure, there is no sharing of information across the tree. We propose a novel class of Bayesian hierarchical clustering models and associated inference algorithms combining the advantages of both probabilistic approaches above. 1) We define a prior and compute the posterior over trees, thus reaping the benefits of a fully Bayesian approach; 2) the distribution over data is hierarchically structured allowing for sharing of statistical strength; 3) we have efficient and easy to implement inference algorithms that construct trees agglomeratively; and 4) the induced distribution over data points is exchangeable. Our model is based on an exchangeable distribution over trees called Kingman’s coalescent [8, 9]. Kingman’s coalescent is a standard model from population genetics for the genealogy of a set of individuals. It is obtained by tracing the genealogy backwards in time, noting when lineages coalesce together. We review Kingman’s coalescent in Section 2. Our own contribution is in using it as a prior over trees in a hierarchical clustering model (Section 3) and in developing novel inference procedures for this model (Section 4). !!"# !!"$ !!"% !! !&"' !&"# !&"$ !&"% & !( !%") !% !!") !! !&") & &") ! !") t1 t2 t3 −∞ t0 = 0 δ1 δ2 δ3 x1 x2 x3 x4 y{1,2} y{3,4} y{1,2,3,4} z {{1, 2, 3, 4}} {{1, 2}, {3, 4}} {{1}, {2}, {3}, {4}} {{1}, {2}, {3, 4}} π(t) = !! !" !# !$ % $ !# !$&' !$ !%&' % %&' $ $&' # #&' (a) (b) (c) t Figure 1: (a) Variables describing the n-coalescent. (b) Sample path from a Brownian diffusion coalescent process in 1D, circles are coalescent points. (c) Sample observed points from same in 2D, notice the hierarchically clustered nature of the points. 2 Kingman’s coalescent Kingman’s coalescent is a standard model in population genetics describing the common genealogy (ancestral tree) of a set of individuals [8, 9]. In its full form it is a distribution over the genealogy of a countably infinite set of individuals. Like other nonparametric models (e.g. Gaussian and Dirichlet processes), Kingman’s coalescent is most easily described and understood in terms of its finite dimensional marginal distributions over the genealogies of n individuals, called n-coalescents. We obtain Kingman’s coalescent as n→∞. Consider the genealogy of n individuals alive at the present time t = 0. We can trace their ancestry backwards in time to the distant past t = −∞. Assume each individual has one parent (in genetics, haploid organisms), and therefore genealogies of [n] = {1, ..., n} form a directed forest. In general, at time t≤0, there are m (1≤m≤n) ancestors alive. Identify these ancestors with their corresponding sets ρ1, ..., ρm of descendants (we will make this identification throughout the paper). Note that π(t) = {ρ1, ..., ρm} form a partition of [n], and interpret t7→π(t) as a function from (−∞, 0] to the set of partitions of [n]. This function is piecewise constant, left-continuous, monotonic (s≤t implies that π(t) is a refinement of π(s)), and π(0)={{1}, ..., {n}} (see Figure 1a). Further, π completely and succinctly characterizes the genealogy; we shall henceforth refer to π as the genealogy of [n]. Kingman’s n-coalescent is simply a distribution over genealogies of [n], or equivalently, over the space of partition-valued functions like π. More specifically, the n-coalescent is a continuous-time, partition-valued, Markov process, which starts at {{1}, ..., {n}} at present time t = 0, and evolves backwards in time, merging (coalescing) lineages until only one is left. To describe the Markov process in its entirety, it is sufficient to describe the jump process (i.e. the embedded, discrete-time, Markov chain over partitions) and the distribution over coalescent times. Both are straightforward and their simplicity is part of the appeal of Kingman’s coalescent. Let ρli, ρri be the ith pair of lineages to coalesce, tn−1 < ···< t1 < t0 = 0 be the coalescent times and δi = ti−1−ti > 0 be the duration between adjacent events (see Figure 1a). Under the n-coalescent, every pair of lineages merges independently with exponential rate 1. Thus the first pair amongst m lineages merge with rate m 2 = m(m−1) 2 . Therefore δi ∼Exp n−i+1 2 independently, the pair ρli, ρri is chosen from among those right after time ti, and with probability one a random draw from the n-coalescent is a binary tree with a single root at t=−∞and the n individuals at time t=0. The genealogy is: π(t) = {{1}, ..., {n}} if t = 0; πti−1 −ρli −ρri + (ρli ∪ρri) if t = ti; πti if ti+1 < t < ti. (1) Combining the probabilities of the durations and choices of lineages, the probability of π is simply: p(π) = Qn−1 i=1 n−i+1 2 exp − n−i+1 2 δi / n−i+1 2 = Qn−1 i=1 exp − n−i+1 2 δi (2) The n-coalescent has some interesting statistical properties [8, 9]. The marginal distribution over tree topologies is uniform and independent of the coalescent times. Secondly, it is infinitely exchangeable: given a genealogy drawn from an n-coalescent, the genealogy of any m contemporary individuals alive at time t ≤0 embedded within the genealogy is a draw from the m-coalescent. Thus, taking n →∞, there is a distribution over genealogies of a countably infinite population for which the marginal distribution of the genealogy of any n individuals gives the n-coalescent. Kingman called this the coalescent. 3 Hierarchical clustering with coalescents We take a Bayesian approach to hierarchical clustering, placing a coalescent prior on the latent tree and modeling the observed data with a tree structured Markov process evolving forward in time. We will alter our terminology from genealogy to tree, from n individuals at present time to n observed data points, and from individuals on the genealogy to latent variables on the tree-structured distribution. Let x = {x1, ..., xn} be n observed data points at the leaves of a tree π drawn from the n-coalescent. π has n −1 coalescent points, the ith occuring when ρli and ρri merge at time ti to form ρi = ρli ∪ρri. Let tli and tri be the times at which ρli and ρri are themselves formed. We use a continuous-time Markov process to define the distribution over the n data points x given the tree π. The Markov process starts in the distant past, evolves forward in time, splits at each coalescent point, and evolves independently down both branches until we reach time 0, when n data points are observations of the process at the n leaves of the tree. The joint distribution described by this process respects the conditional independences implied by the structure of the directed tree π. Let yρi be a latent variable that takes on the value of the Markov process at ρi just before it splitsLet y{i} = xi at leaf i. See Figure 1a. To complete the description of the likelihood model, let q(z) be the initial distribution of the Markov process at time t = −∞, and kst(x, y) be the transition probability from state x at time s to state y at time t. This Markov process need be neither stationary nor ergodic. Marginalizing over paths of the Markov process, the joint probability over the latent variables and the observations is: p(x, y, z|π) = q(z)k−∞tn−1(z, yρn−1) Qn−1 i=1 ktitli(yρi, yρli)ktitri(yρi, yρri) (3) Notice that the marginal distributions for each observation p(xi|π) are identical and given by the Markov process at time 0. However the observations are not independent as they share the same sample path down the Markov process until it splits. In fact the amount of dependence between two observations is a function of the time at which the observations coalesce. A more recent coalescent time implies larger dependence. The overall distribution induced on the observations p(x) inherits the infinite exchangeability of the n-coalescent. We consider in Section 4.3 a brownian diffusion (Figures 1(b,c)) and a simple independent sites mutation process on multinomial vectors. 4 Agglomerative sequential Monte Carlo and greedy inference We develop two classes of efficient and easily implementable inference algorithms for our hierarchical clustering model based on sequential Monte Carlo (SMC) and greedy schemes respectively. In both classes, the latent variables are integrated out, and the trees are constructed in a bottom-up fashion. The full tree π can be expressed as a series of n −1 coalescent events, ordered backwards in time. The ith coalescent event involves the merging of the two subtrees with leaves ρli and ρri and occurs at a time δi before the previous coalescent event. Let θi = {δj, ρlj, ρrj for j ≤i} denote the first i coalescent events. θn−1 is equivalent to π and we shall use them interchangeably. We assume that the form of the Markov process is such that the latent variables {yρi}n−1 i=1 and z can be efficiently integrated out using an upward pass of belief propagation on the tree. Let Mρi(y) be the message passed from yρi to its parent; M{i}(y) = δxi(y) is point mass at xi for leaf i. Mρi(y) is proportional to the likelihood of the observations at the leaves below coalescent event i, given that yρi = y. Belief propagation computes the messages recursively up the tree; for i = 1, ..., n −1: Mρi(y) = Z−1 ρi (x, θi) Q b=l,r R ktitbi(y, yb)Mρbi(yb) dyb (4) where Zρi(x, θi) is a normalization constant. The choice of Z does not affect the computed probability of x, but does impact the accuracy and efficiency of our inference algorithms. We found that Zρi(x, θi) = RR q(z)k−∞ti(z, y)Mρi(y) dy dz worked well. At the root, we have: Z−∞(x, θn−1) = RR q(z)k−∞tn−1(z, y)Mρn−1(y) dy dz (5) The marginal probability p(x|π) is now given by the product of normalization constants: p(x|π) = Z−∞(x, θn−1) Qn−1 i=1 Zρi(x, θi) (6) Multiplying in the prior (2) over π, we get the joint probability for the tree π and observations x: p(x, π) = Z−∞(x, θn−1) Qn−1 i=1 exp − n−i+1 2 δi Zρi(x, θi) (7) Our inference algorithms are based upon (7). The sequential Monte Carlo (SMC) algorithms approximate the posterior over the tree θn−1 using a weighted sum of samples, while the greedy algorithms construct θn−1 by maximizing local terms in (7). Both proceeds by iterating over i = 1, ..., n −1, choosing a duration δi and a pair of subtrees ρli, ρri to coalesce at each iteration. This choice is based upon the ith term in (7), interpreted as the product of a local prior exp − n−i+1 2 δi and a local likelihood Zρi(x, θi) for choosing δi, ρli and ρri given θi−1. 4.1 Sequential Monte Carlo algorithms SMC algorithms approximate the posterior by iteratively constructing a weighted sum of point masses. At iteration i −1, particle s consists of θs i−1 = {δs j, ρs lj, ρs rj for j < i}, and has weight ws i−1. At iteration i, s is extended by sampling δs i , ρs li and ρs ri from a proposal distribution fi(δs i , ρs li, ρs ri|θs i−1), and the weight is updated by: ws i = ws i−1exp − n−i+1 2 δs i Zρi(x, θs i )/fi(δs i , ρs li, ρs ri|θs i−1) (8) After n −1 iterations, we obtain a set of trees θs n−1 and weights ws n−1. The joint distribution is approximated by: p(π, x) ≈P s ws n−1δθs n−1(π), while the posterior is approximated with the weights normalized. An important aspect of SMC is resampling, which places more particles in high probability regions and prunes particles stuck in low probability regions. We resample as in Algorithm 5.1 of [10] when the effective sample size ratio as estimated in [11] falls below one half. SMC-PriorPrior. The simplest proposal distribution is to sample δs i , ρs li and ρs ri from the local prior. δs i is drawn from an exponential with rate n−i+1 2 and ρs li, ρs ri are drawn uniformly from all available pairs. The weight updates (8) reduce to multiplying by Zρi(x, θs i ). This approach is computationally very efficient, but performs badly with many objects due to the uniform draws over pairs. SMC-PriorPost. The second approach addresses the suboptimal choice of pairs to coalesce. We first draw δs i from its local prior, then draw ρs li, ρs ri from the local posterior: fi(ρs li, ρs ri|δs i , θs i−1) ∝Zρi(x, θs i−1, δs i , ρs li, ρs ri); ws i = ws i−1 P ρ′ l,ρ′r Zρi(x, θs i−1, δs i , ρ′ l, ρ′ r) (9) This approach is more computationally demanding since we need to evaluate the local likelihood of every pair. It also performs significantly better than SMC-PriorPrior. We have found that it works reasonably well for small data sets but fails in larger ones for which the local posterior for δi is highly peaked. SMC-PostPost. The third approach is to draw all of δs i , ρs li and ρs ri from their posterior: fi(δs i , ρs li, ρs ri|θs i−1) ∝exp − n−i+1 2 δs i Zρi(x, θs i−1, δs i , ρs li, ρs ri) ws i = ws i−1 P ρ′ l,ρ′r R exp − n−i+1 2 δ′ Zρi(x, θs i−1, δ′, ρ′ l, ρ′ r) dδ′ (10) This approach requires the fewest particles, but is the most computationally expensive due to the integral for each pair. Fortunately, for the case of Brownian diffusion process described below, these integrals are tractable and related to generalized inverse Gaussian distributions. 4.2 Greedy algorithms SMC algorithms are attractive because they can produce an arbitrarily accurate approximation to the full posterior as the number of samples grow. However in many applications a single good tree is often sufficient. We describe a few greedy algorithms to construct a good tree. Greedy-MaxProb: the obvious greedy algorithm is to pick δi, ρli and ρri maximizing the ith term in (7). We do so by computing the optimal δi for each pair of ρli, ρri, and then picking the pair maximizing the ith term at its optimal δi. Greedy-MinDuration: pick the pair to coalesce whose optimal duration is minimum. Both algorithms require recomputing the optimal duration for each pair at each iteration, since the prior rate n−i+1 2 on the duration varies with the iteration i. The total computational cost is thus O(n3). We can avoid this by using the alternative view of the n-coalesent as a Markov process where each pair of lineages coalesces at rate 1. Greedy-Rate1: for each pair ρli and ρri we determine the optimal δi, replacing the n−i+1 2 prior rate with 1. We coalesce the pair with most recent time (as in Greedy-MinDuration). This reduces the complexity to O(n2). We found that all three performed similarly, and use Greedy-Rate1 in our experiments as it is faster. 4.3 Examples Brownian diffusion. Consider the case of continuous data evolving via Brownian diffusion. The transition kernel kst(y, ·) is a Gaussian centred at y with variance (t −s)Λ, where Λ is a symmetric positive definite covariance matrix. Because the joint distribution (3) over x, y and z is Gaussian, we can express each message Mρi(y) as a Gaussian with mean byρi and covariance Λvρi. The local likelihood is: Zρi(x, θi) = |2πbΛi|−1 2 exp −1 2 ||byρli −byρri||2 bΛi ; bΛi = Λ(vρli +vρri +tli+tri−2ti) (11) where ∥x∥Ψ = x⊤Ψ−1x is the Mahanalobis norm. The optimal duration δi can also be solved for, δi = 1 4 n−i+1 2 −1q 4 n−i+1 2 ||byρli −byρri||2 Λ+D2 −D −1 2(vρli +vρri +tli+tri−2ti−1) (12) where D is the dimensionality. The message at the newly coalesced point has parameters: vρi = (vρli + tli −ti)−1 + (vρri + tri −ti)−1−1; byρi = byρli vρli+tli−ti + byρri vρri+tri−ti vρi (13) Multinomial vectors. Consider a Markov process acting on multinomial vectors with each entry taking one of K values and evolving independently. Entry d evolves at rate λd and has equilibrium distribution vector qd. The transition rate matrix is Qd = λd(q⊤ h 111K −Ik) where 111K is a vector of K ones and IK is identity matrix of size K, while the transition probability matrix for entry d in a time interval of length t is eQdt = e−λdtIK + (1 −e−λdt)q⊤ d 111K. Representing the message for entry d from ρi to its parent as a vector M d ρi = [M d1 ρi , ..., M dK ρi ]⊤, normalized so that qd · M d ρi = 1, the local likelihood terms and messages are computed as, Zd ρi(x, θi) = 1 −eλh(2ti−tli−tri) 1 −PK k=1 qdkM dk ρliM dk ρri (14) M d ρi = (1 −eλd(ti−tli)(1 −M d ρli))(1 −eλd(ti−tri)(1 −M d ρri))/Zd ρi(x, θi) (15) Unfortunately the optimal δi cannot be solved analytically and we use Newton steps to compute it. 4.4 Hyperparameter estimation We perform hyperparameter estimation by iterating between estimating a tree, and estimating the hyperparameters. In the Brownian case, we place an inverse Wishart prior on Λ and the MAP posterior ˆΛ is available in a standard closed form. In the multinomial case, the updates are not available analytically and are solved iteratively. Further information on hyperparameter estimation, as well predictive densities and more experiments are available in a longer technical report. 5 Experiments Synthetic Data Sets. In Figure 2 we compare the various SMC algorithms and Greedy-Rate1 on a range of synthetic data sets drawn from the Brownian diffusion coalescent process itself (Λ = ID) to investigate the effects of various parameters on the efficacy of the algorithms1. Generally SMCPostPost performed best, followed by SMC-PriorPost, SMC-PriorPrior and Greedy-Rate1. With increasing D the amount of data given to the algorithms increases and all algorithms do better, especially Greedy-Rate1. This is because the posterior becomes concentrated and the Greedy-Rate1 approximation corresponds well with the posterior. As n increases, the amount of data increases as well and all algorithms perform better. However, the posterior space also increases and SMCPriorPrior which simply samples from the prior over genealogies does not improve as much. We see this effect as well when S is small. As S increases all SMC algorithms improve. Finally, the algorithms were surprisingly robust when there is mismatch between the generated data sets’ λ and the λ used by the model. We expected all models to perform worse with SMC-PostPost best able to maintain its performance (though this is possibly due to our experimental setup). MNIST and SPAMBASE. We compare the performance of Greedy-Rate1 to two other hierarchical clustering algorithms: average-linkage and Bayesian hierarchical clustering (BHC) [6]. In MNIST, 1Each panel was generated from independent runs. Data set variance affected all algorithms, varying overall performance across panels. However, trends in each panel are still valid, as they are based on the same data. 4 6 8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 (a) average log predictive D : dimensions 4 6 8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 (b) n : observations 0.5 1 2 −1.6 −1.4 −1.2 −1 −0.8 −0.6 (c) λ: mutation rate 10 30 50 70 −1.6 −1.4 −1.2 −1 −0.8 −0.6 (d) S : particles SMC−PostPost SMC−PriorPost SMC−PriorPrior Greedy−Rate1 Figure 2: Predictive performance of algorithms as we vary (a) the numbers of dimensions D, (b) observations n, (c) the mutation rate λ (Λ = λID), and (d) number of samples S. In each panel other parameters are fixed to their middle values (we used S = 50) in other panels, and we report log predictive probabilities on one unobserved entry, averaged over 100 runs. MNIST SPAMBASE Avg-link BHC Coalescent Avg-link BHC Coalescent Purity .363±.004 .392±.006 .412±.006 .616±.007 .711±.010 .689±.008 Subtree .581±.005 .579±.005 .610±.005 .607±.011 .549±.015 .661±.012 LOO-acc .755±.005 .763±.005 .773±.005 .846±.010 .832±.010 .861±.008 Table 1: Comparative results. Numbers are averages and standard errors over 50 and 20 repeats. we use 20 exemplars from each of 10 digits from the MNIST data set, reduced via PCA to 20 dimensions, repeating the experiment 50 times. In SPAMBASE, we use 100 examples of 57 binary attributes from each of 2 classes, repeating 20 times. We present purity scores [6], subtree scores (#{interior nodes with all leaves of same class}/(n −#classes)) and leave-one-out accuracies (all scores between 0 and 1, higher better). The results are in Table 1; except for purity on SPAMBASE, ours gives the best performance. Experiments not presented here show that all greedy algorithms perform about the same and that performance improves with hyperparameter updates. Phylolinguistics. We apply Greedy-Rate1 to a phylolinguistic problem: language evolution. Unlike previous research [12] which studies only phonological data, we use a full typological database of 139 binary features over 2150 languages: the World Atlas of Language Structures (WALS) [13]. The data is sparse: about 84% of the entries are unknown. We use the same version of the database as extracted by [14]. Based on the Indo-European subset of this data for which at most 30 features are unknown (48 languages total), we recover the coalescent tree shown in Figure 3(a). Each language is shown with its genus, allowing us to observe that it teases apart Germanic and Romance languages, but makes a few errors with respect to Iranian and Greek. Indo-European Data Avg-link BHC Coalescent Purity 0.510 0.491 0.813 Subtree 0.414 0.414 0.690 LOO-acc 0.538 0.590 0.769 Whole World Data Avg-link BHC Coalescent Purity 0.162 0.160 0.269 Subtree 0.227 0.099 0.177 LOO-acc 0.080 0.248 0.369 Table 2: Comparative performance of various algorithms on phylolinguistics data. Next we compare predictive abilities to other algorithms. We take a subset of WALS and tested on 5% of withheld entries, restoring these with various techniques: Greedy-Rate1; nearest neighbors (use value from nearest observed neighbor); averagelinkage (nearest neighbor in the tree); and probabilistic PCA (latent dimensions in 5, 10, 20, 40, chosen optimistically). We use five subsets of the WALS database, obtained by sorting both the languages and features of the database according to sparsity and using a varying percentage (10% −50%) of the densest portion. The results are in Figure 3(b). Our approach performed reasonably well. Finally, we compare the trees generated by Greedy-Rate1 with trees generated by either averagelinkage or BHC, using the same evaluation criteria as for MNIST and SPAMBASE, using language genus as classes. The results are in Table 5, where we can see that the coalescent significantly outperforms the other methods. 0 0.1 0.2 [Armenian] Armenian (Eastern) [Armenian] Armenian (Western) [Indic] Bengali [Indic] Marathi [Indic] Maithili [Iranian] Ossetic [Indic] Nepali [Indic] Sinhala [Indic] Kashmiri [Indic] Hindi [Indic] Panjabi [Iranian] Pashto [Slavic] Czech [Baltic] Latvian [Baltic] Lithuanian [Slavic] Russian [Slavic] Ukrainian [Slavic] Serbian−Croatian [Slavic] Slovene [Slavic] Polish [Albanian] Albanian [Romance] Catalan [Romance] Italian [Romance] Portuguese [Romance] Romanian [Slavic] Bulgarian [Greek] Greek (Modern) [Romance] Spanish [Germanic] Danish [Germanic] Norwegian [Germanic] Swedish [Germanic] Icelandic [Germanic] English [Germanic] Dutch [Germanic] German [Romance] French [Iranian] Kurdish (Central) [Iranian] Persian [Iranian] Tajik [Celtic] Breton [Celtic] Cornish [Celtic] Welsh [Celtic] Gaelic (Scots) [Celtic] Irish (a) Coalescent for a subset of Indo-European languages from WALS. 0.1 0.2 0.3 0.4 0.5 72 74 76 78 80 82 Coalescent Neighbor Agglomerative PPCA (b) Data restoration on WALS. Y-axis is accuracy; X-axis is percentage of data set used in experiments. At 10%, there are N = 215 languages, H = 14 features and p = 94% observed data; at 20%, N = 430, H = 28 and p = 80%; at 30%: N = 645, H = 42 and p = 66%; at 40%: N = 860, H = 56 and p = 53%; at 50%: N = 1075, H = 70 and p = 43%. Results are averaged over five folds with a different 5% hidden each time. (We also tried a “mode” prediction, but its performance is in the 60% range in all cases, and is not depicted.) Figure 3: Results of the phylolinguistics experiments. LLR (t) Top Words Top Authors (# papers) 32.7 (-2.71) bifurcation attractors hopfield network saddle Mjolsness (9) Saad (9) Ruppin (8) Coolen (7) 0.106 (-3.77) voltage model cells neurons neuron Koch (30) Sejnowski (22) Bower (11) Dayan (10) 83.8 (-2.02) chip circuit voltage vlsi transistor Koch (12) Alspector (6) Lazzaro (6) Murray (6) 140.0 (-2.43) spike ocular cells firing stimulus Sejnowski (22) Koch (18) Bower (11) Dayan (10) 2.48 (-3.66) data model learning algorithm training Jordan (17) Hinton (16) Williams (14) Tresp (13) 31.3 (-2.76) infomax image ica images kurtosis Hinton (12) Sejnowski (10) Amari (7) Zemel (7) 31.6 (-2.83) data training regression learning model Jordan (16) Tresp (13) Smola (11) Moody (10) 39.5 (-2.46) critic policy reinforcement agent controller Singh (15) Barto (10) Sutton (8) Sanger (7) 23.0 (-3.03) network training units hidden input Mozer (14) Lippmann (11) Giles (10) Bengio (9) Table 3: Nine clusters discovered in NIPS abstracts data. NIPS. We applied Greedy-Rate1 to all NIPS abstracts through NIPS12 (1740, total). The data was preprocessed so that only words occuring in at least 100 abstracts were retained. The word counts were then converted to binary. We performed one iteration of hyperparameter re-estimation. In the supplemental material, we depict the top levels of the coalescent tree. Here, we use the tree to generate a flat clustering. To do so, we use the log likelihood ratio at each branch in the coalescent to determine if a split should occur. If the log likelihood ratio is greater than zero, we break the branch; otherwise, we recurse down. On the NIPS abstracts, this leads to nine clusters, depicted in Table 3. Note that clusters two and three are quite similar—had we used a slighly higher log likelihood ratio, they would have been merged (the LLR for cluster 2 was only 0.105). Note that the clustering is able to tease apart Bayesian learning (cluster 5) and non-bayesian learning (cluster 7)—both of which have Mike Jordan as their top author! 6 Discussion We described a new model for Bayesian agglomerative clustering. We used Kingman’s coalescent as our prior over trees, and derived efficient and easily implementable greedy and SMC inference algorithms for the model. We showed empirically that our model gives better performance than other agglomerative clustering algorithms, and gives good results on applications to document modeling and phylolinguistics. Our model is most similar in spirit to the Dirichlet diffusion tree of [2]. Both use infinitely exchangeable priors over trees. While [2] uses a fragmentation process for trees, our prior uses the reverse—a coalescent process instead. This allows us to develop simpler inference algorithms than those in [2] (we have not compared our model against the Dirichlet diffusion tree due to the complexity of implementing it). It will be interesting to consider the possibility of developing similar agglomerative style algorithms for [2]. [3] also describes a hierarchical clustering model involving a prior over trees, but his prior is not infinitely exchangeable. [5] uses tree-consistent partitions to model relational data; it would be interesting to apply our approach to their setting. Another related work is the Bayesian hierarchical clustering of [6], which uses an agglomerative procedure returning a tree structured approximate posterior for a Dirichlet process mixture model. As opposed to our work [6] uses a flat mixture model and does not have a notion of distributions over trees. There are a number of unresolved issues with our work. Firstly, our algorithms take O(n3) computation time, except for Greedy-Rate1 which takes O(n2) time. Among the greedy algorithms we see that there are no discernible differences in quality of approximation thus we recommend GreedyRate1. It would be interesting to develop SMC algorithms with O(n2) runtime, and compare these against Greedy-Rate1 on real world problems. Secondly, there are unanswered statistical questions. For example, since our prior is infinitely exchangeable, by de Finetti’s theorem there is an underlying random distribution for which our observations are i.i.d. draws. What is this underlying random distribution, and how do samples from this distribution look like? We know the answer for at least a simple case: if the Markov process is a mutation process with mutation rate α/2 and new states are drawn i.i.d. from a base distribution H, then the induced distribution is a Dirichlet process DP(α, H) [8]. Another issue is that of consistency—does the posterior over random distributions converge to the true distribution as the number of observations grows? Finally, it would be interesting to generalize our approach to varying mutation rates, and to non-binary trees by using generalizations to Kingman’s coalescent called Λ-coalescents [15]. References [1] R. O. Duda and P. E. Hart. Pattern Classification And Scene Analysis. Wiley and Sons, New York, 1973. [2] R. M. Neal. Defining priors for distributions using Dirichlet diffusion trees. Technical Report 0104, Department of Statistics, University of Toronto, 2001. [3] C. K. I. Williams. A MCMC approach to hierarchical mixture modelling. In Advances in Neural Information Processing Systems, volume 12, 2000. [4] C. Kemp, T. L. Griffiths, S. Stromsten, and J. B. Tenenbaum. Semi-supervised learning with trees. In Advances in Neural Information Processing Systems, volume 16, 2004. [5] D. M. Roy, C. Kemp, V. Mansinghka, and J. B. Tenenbaum. Learning annotated hierarchies from relational data. In Advances in Neural Information Processing Systems, volume 19, 2007. [6] K. A. Heller and Z. Ghahramani. Bayesian hierarchical clustering. In Proceedings of the International Conference on Machine Learning, volume 22, 2005. [7] N. Friedman. Pcluster: Probabilistic agglomerative clustering of gene expression profiles. Technical Report Technical Report 2003-80, Hebrew University, 2003. [8] J. F. C. Kingman. On the genealogy of large populations. Journal of Applied Probability, 19:27–43, 1982. Essays in Statistical Science. [9] J. F. C. Kingman. The coalescent. Stochastic Processes and their Applications, 13:235–248, 1982. [10] P. Fearnhead. Sequential Monte Carlo Method in Filter Theory. PhD thesis, Merton College, University of Oxford, 1998. [11] R. M. Neal. Annealed importance sampling. Technical Report 9805, Department of Statistics, University of Toronto, 1998. [12] A. McMahon and R. McMahon. Language Classification by Numbers. Oxford University Press, 2005. [13] M. Haspelmath, M. Dryer, D. Gil, and B. Comrie, editors. The World Atlas of Language Structures. Oxford University Press, 2005. [14] H. Daum´e III and L. Campbell. A Bayesian model for discovering typological implications. In Proceedings of the Annual Meeting of the Association for Computational Linguistics, 2007. [15] J. Pitman. Coalescents with multiple collisions. Annals of Probability, 27:1870–1902, 1999.
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Distributed Inference for Latent Dirichlet Allocation David Newman, Arthur Asuncion, Padhraic Smyth, Max Welling Department of Computer Science University of California, Irvine newman,asuncion,smyth,welling @ics.uci.edu Abstract We investigate the problem of learning a widely-used latent-variable model – the Latent Dirichlet Allocation (LDA) or “topic” model – using distributed computation, where each of processors only sees of the total data set. We propose two distributed inference schemes that are motivated from different perspectives. The first scheme uses local Gibbs sampling on each processor with periodic updates—it is simple to implement and can be viewed as an approximation to a single processor implementation of Gibbs sampling. The second scheme relies on a hierarchical Bayesian extension of the standard LDA model to directly account for the fact that data are distributed across processors—it has a theoretical guarantee of convergence but is more complex to implement than the approximate method. Using five real-world text corpora we show that distributed learning works very well for LDA models, i.e., perplexity and precision-recall scores for distributed learning are indistinguishable from those obtained with single-processor learning. Our extensive experimental results include large-scale distributed computation on 1000 virtual processors; and speedup experiments of learning topics in a 100-million word corpus using 16 processors. 1 Introduction Very large data sets, such as collections of images, text, and related data, are becoming increasingly common, with examples ranging from digitized collections of books by companies such as Google and Amazon, to large collections of images at Web sites such as Flickr, to the recent Netflix customer recommendation data set. These data sets present major opportunities for machine learning, such as the ability to explore much richer and more expressive models, as well as providing new and interesting domains for the application of learning algorithms. However, the scale of these data sets also brings significant challenges for machine learning, particularly in terms of computation time and memory requirements. For example, a text corpus with 1 million documents, each containing 1000 words on average, will require approximately 12 Gbytes of memory to store the words, which is beyond the main memory capacity for most single processor machines. Similarly, if one were to assume that a simple operation (such as computing a probability vector over categories using Bayes rule) would take on the order of
sec per word, then a full pass through words will take 1000 seconds. Thus, algorithms that make multiple passes over this sized corpus (such as occurs in many clustering and classification algorithms) will have run times in days. An obvious approach for addressing these time and memory issues is to distribute the learning algorithm over multiple processors [1, 2, 3]. In particular, with processors, it is somewhat trivial to get around the memory problem by distributing of the total data to each processor. However, the computation problem remains non-trivial for a fairly large class of learning algorithms, namely how to combine local processing on each of the processors to arrive at a useful global solution. 1 In this general context we investigate distributed learning algorithms for the LDA model [4]. LDA models are arguably among the most successful recent learning algorithms for analyzing count data such as text. However, they can take days to learn for large corpora, and thus, distributed learning would be particularly useful for this type of model. The novel contributions of this paper are as follows: We introduce two algorithms that perform distributed inference for LDA models, one of which is simple to implement but does not necessarily sample from the correct posterior distribution, and the other which optimizes the correct posterior quantity but is more complex to implement and slower to run. We demonstrate that both distributed algorithms produce models that are statistically indistinguishable (in terms of predictive power) from models obtained on a single-processor, and they can learn these models much faster than using a single processor and only requiring storage of th of the data on each processor. 2 Latent Dirichlet Allocation Before introducing our distributed algorithms for LDA, we briefly review the standard LDA model. LDA models each of documents as a mixture over latent topics, each being a multinomial distribution over a word vocabulary. For document , we first draw a mixing proportion from a Dirichlet with parameter . For the
word in the document, a topic is drawn with topic chosen with probability , then word is drawn from the
topic, with taking on value with probability . Finally, a Dirichlet prior with parameter is placed on the topics . Thus, the generative process is given by "!$# % & ' "!$# (% ) * & + ,.-0/ (1) Given the observed words 1+2 3 , the task of Bayesian inference is to compute the posterior distribution over the latent topic indices 4*2 , the mixing proportions , and the topics . An efficient procedure is to use collapsed Gibbs sampling [5], where and are marginalized out, and the latent variables 4 are sampled. Given the current state of all but one variable , the conditional probability of is 576 2 8 49 : 1 : : 7;=< 6 ?>A@ 9 ; 6 $>A@ 9 B -C/ ; 6 D $>A@ 9 ; (2) where the superscript E7 F means the corresponding data-item is excluded in the count values, and where @ GH 2JI KL& 2 : M 2 . We use the convention that missing indices are summed out: @N O2+P @ GQ and @ R2SP @ GQ . 3 Distributed Inference Algorithms for LDA We now present two versions of LDA where the data and the parameters are distributed over distinct processors. We distribute the documents over processors, with 2UT documents on each processor. We partition the data 1 (words from the documents) into 1A2 1 :WVXVWVX: 1Y :WVWVXVW: 1 and the corresponding topic assignments into 4Z2 4 :WVWVXVW: 4 Y :WVWVXVW: 4 , where 1 Y and 4 Y only exist on processor 5 . Document-specific counts @[ are likewise distributed, however every processor maintains its own copy of word-topic and topic counts, @7 and @ . We denote processor-specific counts as @N Y : @ Y and @ Y . 3.1 Approximate Distributed Inference In our Approximate Distributed LDA model (AD-LDA), we simply implement LDA on each processor, and simultaneous Gibbs sampling is performed independently on each of the processors, as if each processor thinks it is the only processor. On processor 5 , given the current state of all but one variable ) Y , the topic assignment to the
word in document , Y]\ 4 Y is sampled from: 576 ) Y 2 8 4 9 Y Y : 1 : : ^;_< 6 `>a@ 9 Y Y ; 6 $>a@ 9 Y B -C/ Y ; 6 D ?>A@ 9 Y Y ; (3) 2 β α ij Z ij X k w| φ j k| θ K D Nj jp k| θ γ k β p α ijp Z ijp X k w| Φ kp w| ϕ K P Njp Dp P Figure 1: (Left) Graphical model for LDA. (Right) Graphical model for HD-LDA. Variables are repeated over the indices of the random variables. Square boxes indicate parameters. Note that @ Y is not the result of separate LDA models running on separate data. In particular P @ Y 2 , where is the total number of words across all processors, as opposed to the number of words on processor 5 . After processor 5 has reassigned 4 Y , we have modified counts @N Y , @ Y , and @ Y . To merge back to a single set of counts, after a number of Gibbs sampling steps (e.g., after a single pass through the data on each processor) we perform the global update, using a reduce-scatter operation, @ @ > Y 6 @ Y
@ ; : @ Y @ (4) where @N are the counts that all processors started with before the sweep of the Gibbs sampler. The counts @ are computed by @ 2 P @ . Note that this global update correctly reflects the topic assignments 4 (i.e., @N can also be regenerated using 4 ). We can consider this algorithm to be an approximation to the single-processor Gibbs sampler in the following sense: at the start of each iteration, all of the processors have the same set of counts. However, as each processor starts sampling, the global count matrix is changing in a way that is unknown to each processor. Thus, in Equation 3, the sampling is not being done according to the true current global count (or true posterior distribution), but to an approximation. We have experimented with “repairing” reversibility of the sampler by adding a phase which re-traces the Gibbs moves starting at the (global) end-state, but we found that, due to the curse-of-dimensionality, virtually all steps ended up being rejected. 3.2 Hierarchical Distributed Inference A more principled way to model parallel processes is to build them directly into the probabilistic model. Imagine a parent collection of topics . This parent has children Y which represent the topic distributions on the various processors. We assume is sampled from according to a Dirichlet distribution with topic-dependent strength parameter . The model that lives on each processor is simply an LDA model. Hence, the generative process is given by, _ "!$# &% _# : % Y _# : % [ Y !$# _ M% Y "!$# Y % YZ " Y Y 7 , -C/ (5) The graphical model corresponding to this Hierarchical Distributed LDA (HD-LDA) is shown on the right of Figure 1, with standard LDA shown on the left for comparison. This model is different than the two other topic hierarchies we found in the literature, namely 1) the deeper version of the hierarchical Dirichlet process mentioned in [6] and 2) Pachinko allocation [7]. The first places a deeper hierarchical prior on (instead of on ) while the second deals with a document-specific hierarchy of topic-assignments. These types of hierarchies do not suit our need to facilitate parallel computation. 3 As is the case for LDA, inference for HD-LDA is most efficient if we marginalize out and . We derive the following conditional probabilities necessary for the Gibbs sampler, 576 ) Y 2 8 4 9 Y Y : 1 : : : ; < 6 Y >A@ 9 Y Y ; 6 B -C/ >A@ 9 Y B -C/W Y ; 6 >A@ 9 Y Y ; (6) In our experiments we learn MAP estimates for the global variables , and . Alternatively, one can derive Gibbs sampling equations using the auxiliary variable method explained in [6], but we leave exploration of this inference technique for future research. Inference is thus based on integrating out and , sampling 4 and learning the MAP value of , and . The entire algorithm can be understood as expectation maximization on a collapsed space where the M-step corresponds to MAP-updates and the E-step corresponds to sampling. As such, the proposed Monte Carlo EM (MCEM) algorithm is guaranteed to converge in expectation (e.g., [8]). The MAP learning rules are derived by using the bounds derived in [9]. They are given by Y
=>A Y P G 6 Y >A@N Y ;
6 Y; >AJP # 6 Y >A@ Y;
6 Y;%
=>A P Y _ 6 _ >a@ Y ;
6 _ ; > P Y # 6 >A@ Y;
6 ;% _
=> P Y _ 6 _ >a@ Y ;
6 _ ; (
> P Y _ 6 _ >a@ Y ;
6 _ ; (7) where is the digamma function. Careful selection of hyper-parameters is critical to making HDLDA work well, and we used our experience with AD-LDA to guide these choices. For AD-LDA P @ Y 2 , but for HD-LDA P @ Y , so we choose and to make the mode of 2 . We set 2 > . Finally we choose and to make the mode of ^Y 2 V , matching the value of used in our LDA and AD-LDA experiments. We can view HD-LDA as a mixture model with LDA mixture components, where the data have been hard-assigned to their respective clusters (processors). The parameters of the clusters are generated from a shared prior distribution. This view clarifies the procedure we have adopted for testing: First we sample assignment variables Y for the first half of the test document (analogous to folding-in). Given these samples we compute the likelihood of the test document under the model for each processor. Assuming equal prior weights for each processor we then compute responsibilities, which are given by the likelihoods, normalized over processors. The probability of the remainder of the test document is then given by the responsibility-weighted average over the processors. 4 Experiments The two distributed algorithms are initialized by first randomly assigning topics to 4 , then from this counting topics in documents, @^ Y , and words in topics, @N Y , for each processor. Recall for AD-LDA that the count arrays @N Y 2J@ are the same on every processor (initially, and after every global update). For each run of each algorithm, a sample was taken after 500 iterations of the Gibbs sampler, well after the typical burn-in period of 200-300 iterations. Multiple processors were simulated in software (by separating data, running sequentially through each processor, and simulating the global update step), except for the speedup experiments which were run on a 16processor computer. It is not obvious a priori that the AD-LDA algorithm will in general converge to a useful result. Later in this section we describe a set of systematic empirical results with AD-LDA, but we first use an illustrative toy example to provide some insight as to how AD-LDA learns a model. The toy example has 2
words, 2 topics. The left panel of Figure 2 shows the distance between the model’s estimate of a particular topic-word distribution and the true distribution, as a function of Gibbs iterations, for both single-processor LDA and AD-LDA with 2 . LDA and AD-LDA have qualitatively the same 3-phase learning dynamics1. The first 4 or so iterations (“early burnin”) correspond to somewhat random movement close to the randomly initialized starting point. In 1For clarity, the results in this figure are plotted for a single run, single data set, etc.—we observed qualitatively similar results over a large variety of such simulations 4 0 5 10 15 20 25 30 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Iteration L1 norm early burn−in burn−in equilibrium LDA AD−LDA proc1 AD−LDA proc2 0.55 0.6 0.65 0.7 0.2 0.25 0.3 0.35 0.4 start topic mode LDA AD−LDA proc1 AD−LDA proc2 0.55 0.6 0.65 0.7 0.2 0.25 0.3 0.35 start topic mode LDA AD−LDA proc1 AD−LDA proc2 AD−LDA proc3 ...etc... AD−LDA proc10 Figure 2: (Left) distance to the mode for LDA and for P=2 AD-LDA. (Center) Projection of topics onto simplex, showing convergence to mode. (Right) Same setup as center panel, but with 2 processors. the next phase (“burn-in”) both algorithms rapidly move in parameter space towards the posterior mode. And finally at equilibrium, both are sampling around the mode. The center panel of Figure 2 plots the same run, in the 2-d planar simplex corresponding to the 3-word topic distribution. This panel shows the paths in parameter space of each model, taking a few small steps near the starting point (top right corner), moving down to the true solution (bottom left), and then sampling near the posterior mode for the rest of the iterations. For each Gibbs iteration, the parameters corresponding to each of the two individual processors, and those parameters after merging, are shown (for ADLDA). We observed that after the initial few iterations, the individual processor steps and the merge step each resulted in a move closer to the mode. The right panel in Figure 2 illustrates the same qualitative behavior as in the center panel, but now for 10 processors. One might worry that the AD-LDA algorithm would get “trapped” close to the initial starting point, e.g., due to repeated label mismatching of the topics across processors. In practice we have consistently observed that the algorithm quickly discards such configurations (due to the stochastic nature of the moves) and “latches” onto a consistent labeling that then rapidly moves it towards the posterior mode. It is useful to think of LDA as an approximation to stochastic descent in the space of assignment variables 4 . On a single processor, one can view Gibbs sampling during burn-in as a stochastic algorithm to move up the likelihood surface. With multiple processors, each processor computes an upward direction in its own subspace, keeping all other directions fixed. The global update step then recombines these directions by vector-addition, in the same way as one would compute a gradient using finite differences. This is expected to be accurate as long as the surface is locally convex or concave, but will break down at saddle-points. We conjecture AD-LDA works reliably because saddle points are 1) unstable and 2) rare due to the fact that the posterior appears often to be highly peaked for LDA models and high-dimensional count data sets. To evaluate AD-LDA and HD-LDA systematically, we measured performance using test set perplexity, computed as Perp 6 1 test ; 2 6
test 5^6 1 test ;.; . For every test document, half the words (at random) are put in a fold-in part, and the remaining words are put in a test part. The document mix is learned using the fold-in part, and log probability is computed using this mix and words from the test part, ensuring that the test words are never seen before being used. For AD-LDA, the perplexity computation exactly follows that of LDA, since a single set of topic counts @[ are saved when a sample is taken. In contrast, all copies of @^ Y are required to compute perplexity for HD-LDA, as described in the previous section. Except where stated, perplexities are computed for all algorithms using a2 samples from the posterior (from 10 different chains) using
5^6 1 test ; 2 2 ?>A@ $>A@ 2 > @ D $>a@ (8) with the analogous expression being used for HD-LDA. We compared LDA (Gibbs sampling on a single processor) and our two distributed algorithms, ADLDA and HD-LDA, using three data sets: KOS (from dailykos.com), NIPS (from books.nips.cc) and NYTIMES (from ldc.upenn.edu). Each data set was split into a training set and a test set. Size parameters for these data sets are shown in Table 1. For each corpus is the vocabulary size and is the total number of words. Using the three data sets and the three models we computed test set 5 KOS NIPS NYTIMES train 3000 1500 300,000 6906 12,419 102,660 410,000 1,900,000 100,000,000 test 430 184 34,658 Table 1: Size parameters for the three data sets used in perplexity and speedup experiments. perplexities for a range of topics , and for number of processors, , ranging from 10 to 1000 for our distributed models. P=1 P=10 P=100 1350 1400 1450 1500 1550 1600 1650 1700 1750 Perplexity T=8 T=16 T=32 T=64 LDA AD−LDA HD−LDA P=1 P=10 P=100 1400 1500 1600 1700 1800 1900 2000 Perplexity T=10 T=20 T=40 T=80 LDA AD−LDA HD−LDA Figure 3: Test perplexity of models versus number of processors P for KOS (left) and NIPS (right). P=1 corresponds to LDA (circles), and AD-LDA (crosses), and HD-LDA (squares) are shown at P=10 and 100 . Figure 3 clearly shows that, for a fixed number of topics, the perplexity results are essentially the same whether we use single-processor LDA or either of the two algorithms with data distributed across multiple processors (either 10 or 100). The figure shows the test set perplexity for KOS (left) and NIPS (right), versus number of processors, . The 2 perplexity is computed by LDA (circles), and we use our distributed models – AD-LDA (crosses), and HD-LDA (squares) – to compute the 2 and 2 perplexities. Though not shown, perplexities for AD-LDA remained approximately constant as the number of processors was further increased to 2 for KOS and 2 for NIPS, demonstrating effective distributed learning with only 3 documents on each processor. It is worth emphasizing that, despite no formal convergence guarantees, the approximate distributed algorithm converged to good solutions in every single one of the more than one thousand experiments we did using five real-world data sets, plus synthesized data sets designed to be “hard” to learn (i.e., topics mutually exclusively distributed over processors)—page limitations preclude a full description of all these results in this paper. 50 100 150 200 250 300 350 400 1700 1750 1800 1850 1900 1950 2000 Iteration Perplexity LDA AD−LDA P=10 AD−LDA P=100 HD−LDA P=10 HD−LDA P=100 0 100 200 300 400 500 600 700 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 Number of Topics Perplexity LDA AD−LDA P=10 HD−LDA P=10 Figure 4: (Left) Test perplexity versus iteration. (Right) Test perplexity versus number of topics. To properly determine the utility of the distributed algorithms, it is necessary to check whether the parallelized samplers are systematically converging more slowly than single processor sampling. If 6 AP FR 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Precision TF−IDF LDA AD−LDA HD−LDA 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 Number of Processors, P Speedup Perfect AD−LDA Figure 5: (Left) Precision/recall results. (Right). Parallel speedup results. this were the case, it would mitigate the computational gains of parallelization. In fact our experiments consistently showed (somewhat surprisingly) that the convergence rate for the distributed algorithms is just as rapid as for the single processor case. As an example, Figure 4 (left) shows test perplexity versus iteration number of the Gibbs sampler (NIPS, 2 ). During burn-in, up to iteration 200, the distributed models are actually converging slightly faster than single processor LDA. Also note that 1 iteration of AD-LDA (or HD-LDA) on a parallel computer takes a fraction of the wall-clock time of 1 iteration of LDA. We also investigated whether the results were sensitive to the number of topics used in the models, e.g., perhaps the distributed algorithms’ performance diverges when the number of topics becomes very large. Figure 4 (right) shows the test set perplexity computed on the NIPS data set using 2 samples, as a function of the number of topics, for the different algorithms and a fixed number of processors 2 (not shown here are the results for the KOS data set which were quite similar). The perplexities of the different algorithms closely track each other as varies. Sometimes the distributed algorithms produce slightly lower perplexities than those of single processor LDA. This lower perplexity may be due to: for AD-LDA, parameters constantly splitting and merging producing an internal averaging effect; and for HD-LDA, test perplexity being computed using copies of saved parameters. Finally, to demonstrate that the low perplexities obtained from the distributed algorithms with 2 processors are not just due to averaging effects, we split the NIPS corpus into one hundred 15document collections, and ran LDA separately on each of these hundred collections. Test perplexity ( 2 ) computed by averaging 100-separate LDA models was 2117, versus the P=100 test perplexity of 1575 for AD-LDA and HD-LDA. This shows that simple averaging of results from separate processors does not perform nearly as well as the distributed coordinated learning. Our distributed algorithms also perform well under other performance metrics. We performed precision/recall calculations using TREC’s AP and FR collections and measured performance using the well-known mean average precision (MAP) metric used in IR research. Figure 5 (left) again shows that AD-LDA and HD-LDA (both using P=10) perform similarly to LDA. All three LDA models have significantly higher precision than TF-IDF on the AP and FR collections (significance was computed using a t-test at the 0.05 level). These calculations were run with 2 . The per-processor per-iteration time and space complexity of LDA and AD-LDA are shown in Table 2. AD-LDA’s memory requirement scales well as collections grow, because while and can get arbitrarily large (which can be offset by increasing ), the vocabulary size asymptotes. Similarly the time complexity scales well since the leading order term is divided by . The term accounts for the communication cost of the reduce-scatter operation on the count difference 6 @ Y
@ ; , which is executed in
stages. Because of the additional term, parallel efficiency will depend on , with increasing efficiency as this ratio increases. Space and time complexity of HD-LDA are similar to that of AD-LDA, but HD-LDA has bigger constants. Using our large NYTIMES data set, we performed speedup experiments on a 16-processor SMP shared memory computer using 2 1, 2, 4, 8 and 16 processors (since we did not have access to a distributed memory computer). The single processor LDA run with 1000 iterations for this data set involves flops, and takes more than 10 days on a 3GHz workstation, so it is an ideal 7 LDA AD-LDA Space >A 6 > ; 6 >A ; >A Time > > Table 2: Space and time complexity of LDA and AD-LDA. computation to speed up. The speedup results, shown in Figure 5 (right), show reasonable parallel efficiency, with a V speedup using 2 processors. This speedup reduces our NYTIMES 10day run (880 sec/iteration on 1 processor) to the order of 1 day (105 sec/iteration on 16 processors). Note, however, that while the implementation on an SMP machine captures some distributed effects (e.g. time to synchronize), it does not accurately reflect the extra time for communication. However, we do expect that for problems with large , parallel efficiency will be high. 5 Discussion and Conclusions Prior work on parallelizing probabilistic learning algorithms has focused largely on EMoptimization algorithms, e.g., parallel updates of expected sufficient statistics for mixture models [2, 1]. In the statistical literature, the idea of running multiple MCMC chains in parallel is one approach to parallelization (e.g., the method of parallel tempering), but requires that each processor store a copy of the full data set. Since MCMC is inherently sequential, parallel sampling using distributed subsets of the data will not in general yield a proper MCMC sampler except in special cases [10]. Mimno and McCallum [11] recently proposed the DCM-LDA model, where processorspecific sets of topics are learned independently on each processor for local subsets of data, without any communication between processors, followed by a global clustering of the topics from the different processors. While this method is highly scalable, it does not lead to single global set of topics that represent individual documents, nor is it defined by a generative process. We proposed two different approaches to distributing MCMC sampling across different processors for an LDA model. With AD-LDA we sample from an approximation to the posterior density by allowing different processors to concurrently sample latent topic assignments on their local subsets of the data. Despite having no formal convergence guarantees, AD-LDA works very well empirically and is easy to implement. With HD-LDA we adapt the underlying LDA model to map to the distributed computational infrastructure. While this model is more complicated than AD-LDA, and slower to run (because of digamma evaluations), it inherits the usual convergence properties of MCEM. Careful selection of hyper-parameters was critical to making HD-LDA work well. In conclusion, both of our proposed algorithms learn models with predictive performance that is no different than single-processor LDA. On each processor they burn-in and converge at the same rate as LDA, yielding significant speedups in practice. The space and time complexity of both models make them scalable to run on enormous problems, for example, collections with billions to trillions of words. There are several potentially interesting research directions that can be pursued using the algorithms proposed here as a starting point, e.g., using asynchronous local communication (as opposed to the environment of synchronous global communications covered in this paper) and more complex schemes that allow data to adaptively move from one processor to another. The distributed scheme of AD-LDA can also be used to parallelize other machine learning algorithms. Using the same principles, we have implemented distributed versions of NMF and PLSA, and initial results suggest that these distributed algorithms also work well in practice. 6 Acknowledgements This material is based upon work supported by the National Science Foundation: DN and PS were supported by NSF grants SCI-0225642, CNS-0551510, and IIS-0083489, AA was supported by an NSF graduate fellowship, and MW was supported by grants IIS-0535278 and IIS-0447903. 8 References [1] C. Chu, S. Kim, Y. Lin, Y. Yu, G. Bradski, A. Ng, and K. Olukotun. Map-Reduce for machine learning on multicore. In NIPS 19, pages 281–288. MIT Press, Cambridge, MA, 2007. [2] W. Kowalczyk and N. Vlassis. Newscast EM. In NIPS 17, pages 713–720. MIT Press, Cambridge, MA, 2005. [3] A. Das, M. Datar, A. Garg, and S. Rajaram. Google news personalization: Scalable online collaborative filtering. In 16th International World Wide Web Conference, 2007. [4] D. Blei, A. Ng, and M. Jordan. Latent Dirichlet allocation. JMLR, 3:993–1022, 2003. [5] T. Griffiths and M. Steyvers. Finding scientific topics. In Proceedings of the National Academy of Sciences, volume 101, pages 5228–5235, 2004. [6] Y.W. Teh, M. Jordan, M. Beal, and A. Blei. Sharing clusters among related groups: Hierarchical Dirichlet processes. In NIPS 17, pages 1385–1392. MIT Press, Cambridge, MA, 2005. [7] W. Li and A. McCallum. Pachinko allocation: DAG-structured mixture models of topic correlations. In ICML, pages 577–584, 2006. [8] G. Wei and M. Tanner. A Monte Carlo implementation of the EM algorithm and the poor man’s data augmentation algorithms. Journal of the American Statistical Association, 85(411):699– 704, 1990. [9] T. Minka. Estimating a Dirichlet distribution. http://research.microsoft.com/minka/papers/dirichlet/, 2003. [10] A. Brockwell. Parallel markov chain monte carlo simulation by pre-fetching. In J.Comp.Graph.Stats, volume 15, pages 246–261, 2006. [11] A. McCallum D. Mimno. Organizing the oca: Learning faceted subjects from a library of digital books. In Joint Conference in Digital Libraries, pages 376–385, 2007. 9
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Simulated Annealing: Rigorous finite-time guarantees for optimization on continuous domains Andrea Lecchini-Visintini Department of Engineering University of Leicester, UK alv1@leicester.ac.uk John Lygeros Automatic Control Laboratory ETH Zurich, Switzerland. lygeros@control.ee.ethz.ch Jan Maciejowski Department of Engineering University of Cambridge, UK jmm@eng.cam.ac.uk Abstract Simulated annealing is a popular method for approaching the solution of a global optimization problem. Existing results on its performance apply to discrete combinatorial optimization where the optimization variables can assume only a finite set of possible values. We introduce a new general formulation of simulated annealing which allows one to guarantee finite-time performance in the optimization of functions of continuous variables. The results hold universally for any optimization problem on a bounded domain and establish a connection between simulated annealing and up-to-date theory of convergence of Markov chain Monte Carlo methods on continuous domains. This work is inspired by the concept of finite-time learning with known accuracy and confidence developed in statistical learning theory. Optimization is the general problem of finding a value of a vector of variables θ that maximizes (or minimizes) some scalar criterion U(θ). The set of all possible values of the vector θ is called the optimization domain. The elements of θ can be discrete or continuous variables. In the first case the optimization domain is usually finite, such as in the well-known traveling salesman problem; in the second case the optimization domain is a continuous set. An important example of a continuous optimization domain is the set of 3-D configurations of a sequence of amino-acids in the problem of finding the minimum energy folding of the corresponding protein [1]. In principle, any optimization problem on a finite domain can be solved by an exhaustive search. However, this is often beyond computational capacity: the optimization domain of the traveling salesman problem with 100 cities contains more than 10155 possible tours. An efficient algorithm to solve the traveling salesman and many similar problems has not yet been found and such problems remain reliably solvable only in principle [2]. Statistical mechanics has inspired widely used methods for finding good approximate solutions in hard discrete optimization problems which defy efficient exact solutions [3, 4, 5, 6]. Here a key idea has been that of simulated annealing [3]: a random search based on the Metropolis-Hastings algorithm, such that the distribution of the elements of the domain visited during the search converges to an equilibrium distribution concentrated around the global optimizers. Convergence and finite-time performance of simulated annealing on finite domains has been evaluated in many works, e.g. [7, 8, 9, 10]. On continuous domains, most popular optimization methods perform a local gradient-based search and in general converge to local optimizers; with the notable exception of convex criteria where convergence to the unique global optimizer occurs [11]. Simulated annealing performs a global search and can be easily implemented on continuous domains. Hence it can be considered a powerful complement to local methods. In this paper, we introduce for the first time rigorous guarantees on the finite-time performance of simulated annealing on continuous domains. We will show that it is possible to derive simulated annealing algorithms which, with an arbitrarily high level of confidence, find an approximate solution to the problem of optimizing a function of continuous variables, within a specified tolerance to the global optimal solution after a known finite number of steps. Rigorous guarantees on the finite-time performance of simulated annealing in the optimization of functions of continuous variables have never been obtained before; the only results available state that simulated annealing converges to a global optimizer as the number of steps grows to infinity, e.g. [12, 13, 14, 15]. The background of our work is twofold. On the one hand, our notion of approximate solution to a global optimization problem is inspired by the concept of finite-time learning with known accuracy and confidence developed in statistical learning theory [16, 17]. We actually maintain an important aspect of statistical learning theory which is that we do not introduce any particular assumption on the optimization criterion, i.e. our results hold regardless of what U is. On the other hand, we ground our results on the theory of convergence, with quantitative bounds on the distance to the target distribution, of the Metropolis-Hastings algorithm and Markov Chain Monte Carlo (MCMC) methods, which has been one of the main achievements of recent research in statistics [18, 19, 20, 21]. In this paper, we will not develop any ready-to-use optimization algorithm. We will instead introduce a general formulation of the simulated annealing method which allows one to derive new simulated annealing algorithms with rigorous finite-time guarantees on the basis of existing theory. The Metropolis-Hastings algorithm and the general family of MCMC methods have many degrees of freedom. The choice and comparison of specific algorithms goes beyond the scope of the paper. The paper is organized in the following sections. In Simulated annealing we introduce the method and fix the notation. In Convergence we recall the reasons why finite-time guarantees for simulated annealing on continuous domains have not been obtained before. In Finite-time guarantees we present the main result of the paper. In Conclusions we state our findings and conclude the paper. 1 Simulated annealing The original formulation of simulated annealing was inspired by the analogy between the stochastic evolution of the thermodynamic state of an annealing material towards the configurations of minimal energy and the search for the global minimum of an optimization criterion [3]. In the procedure, the optimization criterion plays the role of the energy and the state of the annealed material is simulated by the evolution of the state of an inhomogeneous Markov chain. The state of the chain evolves according to the Metropolis-Hastings algorithm in order to simulate the Boltzmann distribution of thermodynamic equilibrium. The Boltzmann distribution is simulated for a decreasing sequence of temperatures (“cooling”). The target distribution of the cooling procedure is the limiting Boltzmann distribution, for the temperature that tends to zero, which takes non-zero values only on the set of global minimizers [7]. The original formulation of the method was for a finite domain. However, simulated annealing can be generalized straightforwardly to a continuous domain because the Metropolis-Hastings algorithm can be used with almost no differences on discrete and continuous domains The main difference is that on a continuous domain the equilibrium distributions are specified by probability densities. On a continuous domain, Markov transition kernels in which the distribution of the elements visited by the chain converges to an equilibrium distribution with the desired density can be constructed using the Metropolis-Hastings algorithm and the general family of MCMC methods [22]. We point out that Boltzmann distributions are not the only distributions which can be adopted as equilibrium distributions in simulated annealing [7]. In this paper it is convenient for us to adopt a different type of equilibrium distribution in place of Boltzmann distributions. 1.1 Our setting The optimization criterion is U : Θ →[0, 1], with Θ ⊂RN. The assumption that U takes values in the interval [0, 1] is a technical one. It does not imply any serious loss of generality. In general, any bounded optimization criterion can be scaled to take values in [0, 1]. We assume that the optimization task is to find a global maximizer; this can be done without loss of generality. We also assume that Θ is a bounded set. We consider equilibrium distributions defined by probability density functions proportional to [U(θ) + δ]J where J and δ are two strictly positive parameters. We use π(J) to denote an equilibrium distribution, i.e. π(J)(dθ) ∝[U(θ) + δ]JπLeb(dθ) where πLeb is the standard Lebesgue measure. Here, J−1 plays the role of the temperature: if the function U(θ) plus δ is taken to a positive power J then as J increases (i.e. as J−1 decreases) [U(θ) + δ]J becomes increasingly peaked around the global maximizers. The parameter δ is an offset which guarantees that the equilibrium densities are always strictly positive, even if U takes zero values on some elements of the domain. The offset δ is chosen by the user and we show later that our results allow one to make an optimal selection of δ. The zero-temperature distribution is the limiting distribution, for J →∞, which takes non-zero values only on the set of global maximizers. It is denoted by π(∞). In the generic formulation of the method, the Markov transition kernel of the k-th step of the inhomogeneous chain has equilibrium distribution π(Jk) where {Jk}k=1,2,... is the “cooling schedule”. The cooling schedule is a non-decreasing sequence of positive numbers according to which the equilibrium distribution become increasingly sharpened during the evolution of the chain. We use θk to denote the state of the chain and Pθk to denote its probability distribution. The distribution Pθk obviously depends on the initial condition θ0. However, in this work, we don’t need to make this dependence explicit in the notation. Remark 1: If, given an element θ in Θ, the value U(θ) can be computed directly, we say that U is a deterministic criterion, e.g. the energy landscape in protein structure prediction [1]. In problems involving random variables, the value U(θ) may be the expected value U(θ) = R g(x, θ)px(x; θ)dx of some function g which depends on both the optimization variable θ, and on some random variable x which has probability density px(x; θ) (which may itself depend on θ). In such problems it is usually not possible to compute U(θ) directly, either because evaluation of the integral requires too much computation, or because no analytical expression for px(x; θ) is available. Typically one must perform stochastic simulations in order to obtain samples of x for a given θ, hence obtain sample values of g(x, θ), and thus construct a Monte Carlo estimate of U(θ). The Bayesian design of clinical trials is an important application area where such expected-value criteria arise [23]. The authors of this paper investigate the optimization of expected-value criteria motivated by problems of aircraft routing [24]. In the particular case that px(x; θ) does not depend on θ, the optimization task is often called “empirical risk minimization”, and is studied extensively in statistical learning theory [16, 17]. The results of this paper apply in the same way to the optimization of both deterministic and expected-value criteria. The MCMC method developed by M¨uller [25, 26] allows one to construct simulated annealing algorithms for the optimization of expected-value criteria. M¨uller [25, 26] employs the same equilibrium distributions as those described in our setting; in his context J is restricted to integer values. 2 Convergence The rationale of simulated annealing is as follows: if the temperature is kept constant, say Jk = J, then the distribution of the state of the chain Pθk tends to the equilibrium distribution π(J); if J →∞ then the equilibrium distribution π(J) tends to the zero-temperature distribution π(∞); as a result, if the cooling schedule Jk tends to infinity, one obtains that Pθk “follows” π(Jk) and that π(Jk) tends to π(∞) and eventually that the distribution of the state of the chain Pθk tends to π(∞). The theory shows that, under conditions on the cooling schedule and the Markov transition kernels, the distribution of the state of the chain Pθk actually converges to the target zero-temperature distribution π(∞) as k →∞[12, 13, 14, 15]. Convergence to the zero-temperature distribution implies that asymptotically the state of the chain eventually coincides with a global optimizer with probability one. The difficulty which must be overcome in order to obtain finite step results on simulated annealing algorithms on a continuous domain is that usually, in an optimization problem defined over continuous variables, the set of global optimizers has zero Lebesgue measure (e.g. a set of isolated points). If the set of global optimizers has zero measure then the set of global optimizers has null probability according to the equilibrium distributions π(J) for any finite J and, as a consequence, according to the distributions Pθk for any finite k. Put another way, the probability that the state of the chain visits the set of global optimizers is constantly zero after any finite number of steps. Hence the confidence of the fact that the solution provided by the algorithm in finite time coincides with a global optimizer is also constantly zero. Notice that this is not the case for a finite domain, where the set of global optimizers is of non-null measure with respect to the reference counting measure [7, 8, 9, 10]. It is instructive to look at the issue also in terms of the rate of convergence to the target zerotemperature distribution. On a discrete domain, the distribution of the state of the chain at each step and the zero-temperature distribution are both standard discrete distributions. It is then possible to define a distance between them and study the rate of convergence of this distance to zero. This analysis allows one to obtain results on the finite-time behavior of simulated annealing [7, 8]. On a continuous domain and for a set of global optimizers of measure zero, the target zero-temperature distribution π(∞) ends up being a mixture of probability masses on the set of global optimizers. In this situation, although the distribution of the state of the chain Pθk still converges asymptotically to π(∞), it is not possible to introduce a sensible distance between the two distributions and a rate of convergence to the target distribution cannot even be defined (weak convergence), see [12, Theorem 3.3]. This is the reason that until now there have been no guarantees on the performance of simulated annealing on a continuous domain after a finite number of computations: by adopting the zero-temperature distribution π(∞) as the target distribution it is only possible to prove asymptotic convergence in infinite time to a global optimizer. Remark 2: The standard distance between two distributions, say µ1 and µ2, on a continuous support is the total variation norm ∥µ1 −µ2∥T V = supA |µ1(A) −µ2(A)|, see e.g. [21]. In simulated annealing on a continuous domain the distribution of the state of the chain Pθk is absolutely continuous with respect to the Lebesgue measure (i.e. πLeb(A) = 0 ⇒Pθk(A) = 0), by construction for any finite k. Hence if the set of global optimizers has zero Lebesgue measure then it has zero measure also according to Pθk. The set of global optimizers has however measure 1 according to π(∞). The distance ∥Pθk −π(∞)∥T V is then constantly 1 for any finite k. It is also worth mentioning that if the set of global optimizers has zero measure then asymptotic convergence to the zero-temperature distribution π(∞) can be proven only under the additional assumptions of continuity and differentiability of U [12, 13, 14, 15]. 3 Finite-time guarantees In general, optimization algorithms for problems defined on continuous variables can only find approximate solutions in finite time [27]. Given an element θ of a continuous domain how can we assess how good it is as an approximate solution to an optimization problem? Here we introduce the concept of approximate global optimizer to answer this question. The definition is given for a maximization problem in a continuous but bounded domain. We use two parameters: the value imprecision ǫ (greater than or equal to 0) and the residual domain α (between 0 and 1) which together determine the level of approximation. We say that θ is an approximate global optimizer of U with value imprecision ǫ and residual domain α if the function U takes values strictly greater than U(θ)+ǫ only on a subset of values of θ no larger than an α portion of the optimization domain. The formal definition is as follows. Definition 1 Let U : Θ →R be an optimization criterion where Θ ⊂RN is bounded. Let πLeb denote the standard Lebesgue measure. Let ǫ ≥0 and α ∈[0, 1] be given numbers. Then θ is an approximate global optimizer of U with value imprecision ǫ and residual domain α if πLeb{θ′ ∈Θ : U(θ′) > U(θ) + ǫ} ≤α πLeb(Θ) . In other words, the value U(θ) is within ǫ of a value which is greater than the values that U takes on at least a 1 −α portion of the domain. The smaller ǫ and α are, the better is the approximation of a true global optimizer. If both α and ǫ are equal to zero then U(θ) coincides with the essential supremum of U. Our definition of approximate global optimizer carries an important property, which holds regardless of what the criterion U is: if ǫ and α have non-zero values then the set of approximate global optimizers always has non-zero Lebesgue measure. It follows that the probability that the chain visits the set of approximate global optimizers can be non-zero. Hence, it is sensible to study the confidence of the fact that the solution found by simulated annealing in finite time is an approximate global optimizer. Remark 3: The intuition that our notion of approximate global optimizer can be used to obtain formal guarantees on the finite-time performance of optimization methods based on a stochastic search of the domain is already apparent in the work of Vidyasagar [17, 28]. Vidyasagar [17, 28] introduces a similar definition and obtains rigorous finite-time guarantees in the optimization of expected value criteria based on uniform independent sampling of the domain. Notably, the number of independent samples required to guarantee some desired accuracy and confidence turns out to be polynomial in the values of the desired imprecision, residual domain and confidence. Although the method of Vidyasagar is not highly sophisticated, it has had considerable success in solving difficult control system design applications [28, 29]. Its appeal stems from its rigorous finite-time guarantees which exist without the need for any particular assumption on the optimization criterion. Here we show that finite-time guarantees for simulated annealing can be obtained by selecting a distribution π(J) with a finite J as the target distribution in place of the zero-temperature distribution π(∞). The fundamental result is the following theorem which allows one to select in a rigorous way δ and J in the target distribution π(J). It is important to stress that the result holds universally for any optimization criterion U on a bounded domain. The only minor requirement is that U takes values in [0, 1]. Theorem 1 Let U : Θ →[0, 1] be an optimization criterion where Θ ⊂RN is bounded. Let J ≥1 and δ > 0 be given numbers. Let θ be a multivariate random variable with distribution π(J)(dθ) ∝[U(θ) + δ]JπLeb(dθ). Let α ∈(0, 1] and ǫ ∈[0, 1] be given numbers and define σ = 1 1 + 1 + δ ǫ + 1 + δ J 1 α 1 + δ ǫ + δ −1 1 + δ δ . (1) Then the statement “θ is an approximate global optimizer of U with value imprecision ǫ and residual domain α” holds with probability at least σ. Proof. See Appendix A. The importance of the choice of a target distribution π(J) with a finite J is that π(J) is absolutely continuous with respect to the Lebesgue measure. Hence, the distance ∥Pθk −π(J)∥TV between the distribution of the state of the chain Pθk and the target distribution π(J) is a meaningful quantity. Convergence of the Metropolis-Hastings algorithm and MCMC methods in total variation norm is a well studied problem. The theory provides simple conditions under which one derives upper bounds on the distance to the target distribution which are known at each step of the chain and decrease monotonically to zero as the number of steps of the chain grows. The theory has been developed mainly for homogeneous chains [18, 19, 20, 21]. In the case of simulated annealing, the factor that enables us to employ these results is the absolute continuity of the target distribution π(J) with respect to the Lebesgue measure. However, simulated annealing involves the simulation of inhomogeneous chains. In this respect, another important fact is that the choice of a target distribution π(J) with a finite J implies that the inhomogeneous Markov chain can in fact be formed by a finite sequence of homogeneous chains (i.e. the cooling schedule {Jk}k=1,2,... can be chosen to be a sequence that takes only a finite set of values). In turn, this allows one to apply the theory of homogeneous MCMC methods to study the convergence of Pθk to π(J) in total variation norm. On a bounded domain, simple conditions on the ‘proposal distribution’ in the iteration of the simulated annealing algorithm allows one to obtain upper bounds on ∥Pθk −π(J)∥TV that decrease geometrically to zero as k →∞, without the need for any additional assumption on U [18, 19, 20, 21]. It is then appropriate to introduce the following finite-time result. Theorem 2 Let the notation and assumptions of Theorem 1 hold. Let θk, with distribution Pθk, be the state of the inhomogeneous chain of a simulated annealing algorithm with target distribution π(J). Then the statement “θk is an approximate global optimizer of U with value imprecision ǫ and residual domain α” holds with probability at least σ −∥Pθk −π(J)∥TV. The proof of the theorem follows directly from the definition of the total variation norm. It follows that if simulated annealing is implemented with an algorithm which converges in total variation distance to a target distribution π(J) with a finite J, then one can state with confidence arbitrarily close to 1 that the solution found by the algorithm after the known appropriate finite number of steps is an approximate global optimizer with the desired approximation level. For given non-zero values of ǫ, α the value of σ given by (1) can be made arbitrarily close to 1 by choice of J; while the distance ∥Pθk −π(J)∥TV can be made arbitrarily small by taking the known sufficient number of steps. It can be shown that there exists the possibility of making an optimal choice of δ and J in the target distribution π(J). In fact, for given ǫ and α and a given value of J there exists an optimal choice of δ which maximizes the value of σ given by (1). Hence, it is possible to obtain a desired σ with the smallest possible J. The advantage of choosing the smallest J, consistent with the required approximation and confidence, is that it will decrease the number of steps required to achieve the desired reduction of ∥Pθk −π(J)∥TV. 4 Conclusions We have introduced a new formulation of simulated annealing which admits rigorous finite-time guarantees in the optimization of functions of continuous variables. First, we have introduced the notion of approximate global optimizer. Then, we have shown that simulated annealing is guaranteed to find approximate global optimizers, with the desired confidence and the desired level of accuracy, in a known finite number of steps, if a proper choice of the target distribution is made and conditions for convergence in total variation norm are met. The results hold for any optimization criterion on a bounded domain with the only minor requirement that it takes values between 0 and 1. In this framework, simulated annealing algorithms with rigorous finite-time guarantees can be derived by studying the choice of the proposal distribution and of the cooling schedule, in the generic iteration of simulated annealing, in order to ensure convergence to the target distribution in total variation norm. To do this, existing theory of convergence of the Metropolis-Hastings algorithm and MCMC methods on continuous domains can be used [18, 19, 20, 21]. Vidyasagar [17, 28] has introduced a similar definition of approximate global optimizer and has shown that approximate optimizers with desired accuracy and confidence can be obtained with a number of uniform independent samples of the domain which is polynomial in the accuracy and confidence parameters. In general, algorithms developed with the MCMC methodology can be expected to be equally or more efficient than uniform independent sampling. Acknowledgments Work supported by EPSRC, Grant EP/C014006/1, and by the European Commission under projects HYGEIA FP6-NEST-4995 and iFly FP6-TREN-037180. We thank S. Brooks, M. Vidyasagar and D. M. Wolpert for discussions and useful comments on the paper. A Proof of Theorem 1 Let ¯α ∈(0, 1] and ρ ∈(0, 1] be given numbers. Let Uδ(θ) := U(θ) + δ. Let πδ be a normalized measure such that πδ(dθ) ∝Uδ(θ)πLeb(dθ). In the first part of the proof we find a lower bound on the probability that θ belongs to the set {θ ∈Θ : πδ{θ′ ∈Θ : ρ Uδ(θ′) > Uδ(θ)} ≤¯α} . Let y¯α := inf{y : πδ{θ ∈Θ : Uδ(θ) ≤y} ≥1 −¯α}. To start with we show that the set {θ ∈Θ : πδ{θ′ ∈Θ : ρ Uδ(θ′) > Uδ(θ)} ≤¯α} coincides with {θ ∈Θ : Uδ(θ) ≥ρ y¯α}. Notice that the quantity πδ{θ ∈Θ : Uδ(θ) ≤y} is a right-continuous non-decreasing function of y because it has the form of a distribution function (see e.g. [30, p.162] and [17, Lemma 11.1]). Therefore we have πδ{θ ∈Θ : Uδ(θ) ≤y¯α} ≥1 −¯α and y ≥ρ y¯α ⇒ πδ{θ′ ∈Θ : ρ Uδ(θ′) ≤y} ≥1 −¯α ⇒ πδ{θ′ ∈Θ : ρ Uδ(θ′) > y} ≤¯α . Moreover, y < ρ y¯α ⇒ πδ{θ′ ∈Θ : ρ Uδ(θ′) ≤y} < 1 −¯α ⇒ πδ{θ′ ∈Θ : ρ Uδ(θ′) > y} > ¯α and taking the contrapositive one obtains πδ{θ′ ∈Θ : ρ Uδ(θ′) > y} ≤¯α ⇒ y ≥ρ y¯α. Therefore {θ ∈Θ : Uδ(θ) ≥ρ y¯α} ≡{θ ∈Θ : πδ{θ′ ∈Θ : ρ Uδ(θ′) > Uδ(θ)} ≤¯α}. We now derive a lower bound on π(J){θ ∈Θ : Uδ(θ) ≥ρ y¯α}. Let us introduce the notation A¯α := {θ ∈Θ : Uδ(θ) < y¯α}, ¯A¯α := {θ ∈Θ : Uδ(θ) ≥y¯α}, B¯α,ρ := {θ ∈Θ : Uδ(θ) < ρ y¯α} and ¯B¯α,ρ := {θ ∈Θ : Uδ(θ) ≥ρ y¯α}. Notice that B¯α,ρ ⊆A¯α and ¯A¯α ⊆¯B¯α,ρ. The quantity πδ{θ ∈Θ : Uδ(θ) < y} as a function of y is the left-continuous version of πδ{θ ∈Θ : Uδ(θ) ≤ y}[30, p.162]. Hence, the definition of y¯α implies πδ(A¯α) ≤1 −¯α and πδ( ¯A¯α) ≥¯α. Notice that πδ(A¯α) ≤1 −¯α ⇒ δπLeb(A¯α) R Θ Uδ(θ)πLeb(dθ) ≤1 −¯α , πδ( ¯A¯α) ≥¯α ⇒ (1 + δ)πLeb( ¯A¯α) R Θ Uδ(θ)πLeb(dθ) ≥¯α . Hence, πLeb( ¯A¯α) > 0 and πLeb(A¯α) πLeb( ¯A¯α) ≤1 −¯α ¯α 1 + δ δ . Notice that πLeb( ¯A¯α) > 0 implies πLeb( ¯B¯α,ρ) > 0. We obtain π(J){θ ∈Θ : Uδ(θ) ≥ρ y¯α} = 1 1 + R B ¯ α,ρ Uδ(θ)JπLeb(dθ) R ¯ B ¯ α,ρ Uδ(θ)JπLeb(dθ) ≥ 1 1 + R B ¯ α,ρ Uδ(θ)JπLeb(dθ) R ¯ A ¯ α Uδ(θ)JπLeb(dθ) ≥ 1 1 + ρ JyJ ¯α yJ ¯α πLeb(B¯α,ρ) πLeb( ¯A¯α) ≥ 1 1 + ρ J πLeb(A¯α) πLeb( ¯A¯α) ≥ 1 1 + ρ J 1 −¯α ¯α 1 + δ δ . Since {θ ∈Θ : Uδ(θ) ≥ρ y¯α} ≡{θ ∈Θ : πδ{θ′ ∈Θ : ρ Uδ(θ′) > Uδ(θ)} ≤¯α} the first part of the proof is complete. In the second part of the proof we show that the set {θ ∈Θ : πδ{θ′ ∈Θ : ρ Uδ(θ′) > Uδ(θ)} ≤¯α} is contained in the set of approximate global optimizers of U with value imprecision ˜ǫ := (ρ−1 −1)(1 + δ) and residual domain ˜α := 1+δ ˜ǫ+δ ¯α. Hence, we show that {θ ∈Θ : πδ{θ′ ∈ Θ : ρ Uδ(θ′) > Uδ(θ)} ≤¯α} ⊆{θ ∈Θ : πLeb{θ′ ∈Θ : U(θ′) > U(θ) + ˜ǫ} ≤˜α πLeb(Θ)}. We have U(θ′) > U(θ) + ˜ǫ ⇔ ρ Uδ(θ′) > ρ [Uδ(θ) + ˜ǫ] ⇒ ρ Uδ(θ′) > Uδ(θ) which is proven by noticing that ρ [Uδ(θ) + ˜ǫ] ≥ Uδ(θ) ⇔ 1 −ρ ≥ U(θ)(1 −ρ) and U(θ) ∈[0, 1]. Hence {θ′ ∈Θ : ρ Uδ(θ′) > Uδ(θ)} ⊇{θ′ ∈Θ : U(θ′) > U(θ) + ˜ǫ} . Therefore πδ{θ′ ∈Θ : ρ Uδ(θ′) > Uδ(θ)} ≤¯α ⇒πδ{θ′ ∈Θ : U(θ′) > U(θ) + ˜ǫ} ≤¯α . Let Qθ,˜ǫ := {θ′ ∈Θ : U(θ′) > U(θ) + ˜ǫ} and notice that πδ{θ′ ∈Θ : U(θ′) > U(θ) + ˜ǫ} = Z Qθ,˜ǫ U(θ′)πLeb(dθ′) + δπLeb(Qθ,˜ǫ) Z Θ U(θ′)πLeb(dθ′) + δπLeb(Θ) . We obtain πδ{θ′ ∈Θ : U(θ′) > U(θ) + ˜ǫ} ≤¯α ⇒˜ǫ πLeb(Qθ,˜ǫ) + δπLeb(Qθ,˜ǫ) ≤¯α(1 + δ)πLeb(Θ) ⇒πLeb{θ′ ∈Θ : U(θ′) > U(θ) + ˜ǫ} ≤˜α πLeb(Θ) . Hence we can conclude that πδ{θ′ ∈Θ : ρ Uδ(θ′) > Uδ(θ)} ≤¯α ⇒πLeb{θ′ ∈Θ : U(θ′) > U(θ) + ˜ǫ} ≤˜α πLeb(Θ) and the second part of the proof is complete. We have shown that given ¯α ∈(0, 1], ρ ∈(0, 1], ˜ǫ := (ρ−1 −1)(1 + δ), ˜α := 1+δ ˜ǫ+δ ¯α and σ := 1 1 + ρ J 1 −¯α ¯α 1 + δ δ = 1 1 + 1 + δ ˜ǫ + 1 + δ J 1 ˜α 1 + δ ˜ǫ + δ −1 1 + δ δ , the statement “θ is an approximate global optimizer of U with value imprecision ˜ǫ and residual domain ˜α” holds with probability at least σ. Notice that ˜ǫ ∈[0, 1] and ˜α ∈(0, 1] are linked through a bijective relation to ρ ∈[ 1+δ 2+δ, 1] and ¯α ∈(0, ˜ǫ+δ 1+δ]. The statement of the theorem is eventually obtained by expressing σ as a function of desired ˜ǫ = ǫ and ˜α = α. □ References [1] D. J. Wales. Energy Landscapes. Cambridge University Press, Cambridge, UK, 2003. [2] D. Achlioptas, A. Naor, and Y. Peres. Rigorous location of phase transitions in hard optimization problems. Nature, 435:759–764, 2005. [3] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi. Optimization by Simulated Annealing. Science, 220(4598):671–680, 1983. [4] E. Bonomi and J. Lutton. 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Comparison of objective functions for estimating linear-nonlinear models Tatyana O. Sharpee Computational Neurobiology Laboratory, the Salk Institute for Biological Studies, La Jolla, CA 92037 sharpee@salk.edu Abstract This paper compares a family of methods for characterizing neural feature selectivity with natural stimuli in the framework of the linear-nonlinear model. In this model, the neural firing rate is a nonlinear function of a small number of relevant stimulus components. The relevant stimulus dimensions can be found by maximizing one of the family of objective functions, R´enyi divergences of different orders [1, 2]. We show that maximizing one of them, R´enyi divergence of order 2, is equivalent to least-square fitting of the linear-nonlinear model to neural data. Next, we derive reconstruction errors in relevant dimensions found by maximizing R´enyi divergences of arbitrary order in the asymptotic limit of large spike numbers. We find that the smallest errors are obtained with R´enyi divergence of order 1, also known as Kullback-Leibler divergence. This corresponds to finding relevant dimensions by maximizing mutual information [2]. We numerically test how these optimization schemes perform in the regime of low signal-to-noise ratio (small number of spikes and increasing neural noise) for model visual neurons. We find that optimization schemes based on either least square fitting or information maximization perform well even when number of spikes is small. Information maximization provides slightly, but significantly, better reconstructions than least square fitting. This makes the problem of finding relevant dimensions, together with the problem of lossy compression [3], one of examples where informationtheoretic measures are no more data limited than those derived from least squares. 1 Introduction The application of system identification techniques to the study of sensory neural systems has a long history. One family of approaches employs the dimensionality reduction idea: while inputs are typically very high-dimensional, not all dimensions are equally important for eliciting a neural response [4, 5, 6, 7, 8]. The aim is then to find a small set of dimensions {ˆe1, ˆe2, . . .} in the stimulus space that are relevant for neural response, without imposing, however, a particular functional dependence between the neural response and the stimulus components {s1, s2, . . .} along the relevant dimensions: P(spike|s) = P(spike)g(s1, s2, ..., sK), (1) If the inputs are Gaussian, the last requirement is not important, because relevant dimensions can be found without knowing a correct functional form for the nonlinear function g in Eq. (1). However, for non-Gaussian inputs a wrong assumption for the form of the nonlinearity g will lead to systematic errors in the estimate of the relevant dimensions themselves [9, 5, 1, 2]. The larger the deviations of the stimulus distribution from a Gaussian, the larger will be the effect of errors in the presumed form of the nonlinearity function g on estimating the relevant dimensions. Because inputs derived from a natural environment, either visual or auditory, have been shown to be strongly non-Gaussian [10], we will concentrate here on system identification methods suitable for either Gaussian or non-Gaussian stimuli. To find the relevant dimensions for neural responses probed with non-Gaussian inputs, Hunter and Korenberg proposed an iterative scheme [5] where the relevant dimensions are first found by assuming that the input–output function g is linear. Its functional form is then updated given the current estimate of the relevant dimensions. The inverse of g is then used to improve the estimate of the relevant dimensions. This procedure can be improved not to rely on inverting the nonlinear function g by formulating optimization problem exclusively with respect to relevant dimensions [1, 2], where the nonlinear function g is taken into account in the objective function to be optimized. A family of objective functions suitable for finding relevant dimensions with natural stimuli have been proposed based on R´enyi divergences [1] between the the probability distributions of stimulus components along the candidate relevant dimensions computed with respect to all inputs and those associated with spikes. Here we show that the optimization problem based on the R´enyi divergence of order 2 corresponds to least square fitting of the linear-nonlinear model to neural spike trains. The KullbackLeibler divergence also belongs to this family and is the R´enyi divergence of order 1. It quantifies the amount of mutual information between the neural response and the stimulus components along the relevant dimension [2]. The optimization scheme based on information maximization has been previously proposed and implemented on model [2] and real cells [11]. Here we derive asymptotic errors for optimization strategies based on R´enyi divergences of arbitrary order, and show that relevant dimensions found by maximizing Kullback-Leibler divergence have the smallest errors in the limit of large spike numbers compared to maximizing other R´enyi divergences, including the one which implements least squares. We then show in numerical simulations on model cells that this trend persists even for very low spike numbers. 2 Variance as an Objective Function One way of selecting a low-dimensional model of neural response is to minimize a χ2-difference between spike probabilities measured and predicted by the model after averaging across all inputs s: χ2[v] = Z dsP(s) ·P(spike|s) P(spike) −P(spike|s · v) P(spike) ¸2 , (2) where dimension v is the relevant dimension for a given model described by Eq. (1) [multiple dimensions could also be used, see below]. Using the Bayes’ rule and rearranging terms, we get: χ2[v] = Z dsP(s) ·P(s|spike) P(s) −P(s · v|spike) P(s · v) ¸2 = Z ds[P(s|spike)]2 P(s) − Z dx[Pv(x|spike)]2 Pv(x) . (3) In the last integral averaging has been carried out with respect to all stimulus components except for those along the trial direction v, so that integration variable x = s · v. Probability distributions Pv(x) and Pv(x|spike) represent the result of this averaging across all presented stimuli and those that lead to a spike, respectively: Pv(x) = Z dsP(s)δ(x −s · v), Pv(x|spike) = Z dsP(s|spike)δ(x −s · v), (4) where δ(x) is a delta-function. In practice, both of the averages (4) are calculated by bining the range of projections values x and computing histograms normalized to unity. Note that if there multiple spikes are sometimes elicited, the probability distribution P(x|spike) can be constructed by weighting the contribution from each stimulus according to the number of spikes it elicited. If neural spikes are indeed based on one relevant dimension, then this dimension will explain all of the variance, leading to χ2 = 0. For all other dimensions v, χ2[v] > 0. Based on Eq. (3), in order to minimize χ2 we need to maximize F[v] = Z dxPv(x) ·Pv(x|spike) Pv(x) ¸2 , (5) which is a R´enyi divergence of order 2 between probability distribution Pv(x|spike) and Pv(x), and are part of a family of f-divergences measures that are based on a convex function of the ratio of the two probability distributions (instead of a power α in a R´enyi divergence of order α) [12, 13, 1]. For optimization strategy based on R´enyi divergences of order α, the relevant dimensions are found by maximizing: F (α)[v] = 1 α −1 Z dxPv(x) ·Pv(x|spike) Pv(x) ¸α . (6) By comparison, when the relevant dimension(s) are found by maximizing information [2], the goal is to maximize Kullback-Leibler divergence, which can be obtained by taking a formal limit α →1: I[v] = Z dxPv(x)Pv(x|spike) Pv(x) ln Pv(x|spike) Pv(x) = Z dxPv(x|spike) ln Pv(x|spike) Pv(x) . (7) Returning to the variance optimization, the maximal value for F[v] that can be achieved by any dimension v is: Fmax = Z ds[P(s|spike)]2 P(s) . (8) It corresponds to the variance in the firing rate averaged across different inputs (see Eq. (9) below). Computation of the mutual information carried by the individual spike about the stimulus relies on similar integrals. Following the procedure outlined for computing mutual information [14], one can use the Bayes’ rule and the ergodic assumption to compute Fmax as a time-average: Fmax = 1 T Z dt ·r(t) ¯r ¸2 , (9) where the firing rate r(t) = P(spike|s)/∆t is measured in time bins of width ∆t using multiple repetitions of the same stimulus sequence . The stimulus ensemble should be diverse enough to justify the ergodic assumption [this could be checked by computing Fmax for increasing fractions of the overall dataset size]. The average firing rate ¯r = P(spike)/∆t is obtained by averaging r(t) in time. The fact that F[v] < Fmax can be seen either by simply noting that χ2[v] ≥0, or from the data processing inequality, which applies not only to Kullback-Leibler divergence, but also to R´enyi divergences [12, 13, 1]. In other words, the variance in the firing rate explained by a given dimension F[v] cannot be greater than the overall variance in the firing rate Fmax. This is because we have averaged over all of the variations in the firing rate that correspond to inputs with the same projection value on the dimension v and differ only in projections onto other dimensions. Optimization scheme based on R´enyi divergences of different orders have very similar structure. In particular, gradient could be evaluated in a similar way: ∇vF (α) = α α −1 Z dxPv(x|spike) [⟨s|x, spike⟩−⟨s|x⟩] d dx "µPv(x|spike) Pv(x) ¶α−1# , (10) where ⟨s|x, spike⟩= R ds sδ(x−s·v)P(s|spike)/P(x|spike), and similarly for ⟨s|x⟩. The gradient is thus given by a weighted sum of spike-triggered averages ⟨s|x, spike⟩−⟨s|x⟩conditional upon projection values of stimuli onto the dimension v for which the gradient of information is being evaluated. The similarity of the structure of both the objective functions and their gradients for different R´enyi divergences means that numeric algorithms can be used for optimization of R´enyi divergences of different orders. Examples of possible algorithms have been described [1, 2, 11] and include a combination of gradient ascent and simulated annealing. Here are a few facts common to this family of optimization schemes. First, as was proved in the case of information maximization based on Kullback-Leibler divergence [2], the merit function F (α)[v] does not change with the length of the vector v. Therefore v ·∇vF = 0, as can also be seen directly from Eq. (10), because v · ⟨s|x, spike⟩= x and v · ⟨s|x⟩= x. Second, the gradient is 0 when evaluated along the true receptive field. This is because for the true relevant dimension according to which spikes were generated, ⟨s|s1, spike⟩= ⟨s|s1⟩, a consequence of the fact that relevant projections completely determine the spike probability. Third, merit functions, including variance and information, can be computed with respect to multiple dimensions by keeping track of stimulus projections on all the relevant dimensions when forming probability distributions (4). For example, in the case of two dimensions v1 and v2, we would use Pv1,v2(x1, x2|spike) = Z ds δ(x1 −s · v1)δ(x2 −s · v2)P(s|spike), Pv1,v2(x1, x2) = Z ds δ(x1 −s · v1)δ(x2 −s · v2)P(s), (11) to compute the variance with respect to the two dimensions as F[v1, v2] = R dx1dx2 [P(x1, x2|spike)]2 /P(x1, x2). If multiple stimulus dimensions are relevant for eliciting the neural response, they can always be found (provided sufficient number of responses have been recorded) by optimizing the variance according to Eq. (11) with the correct number of dimensions. In practice this involves finding a single relevant dimension first, and then iteratively increasing the number of relevant dimensions considered while adjusting the previously found relevant dimensions. The amount by which relevant dimensions need to be adjusted is proportional to the contribution of subsequent relevant dimensions to neural spiking (the corresponding expression has the same functional form as that for relevant dimensions found by maximizing information, cf. Appendix B [2]). If stimuli are either uncorrelated or correlated but Gaussian, then the previously found dimensions do not need to be adjusted when additional dimensions are introduced. All of the relevant dimensions can be found one by one, by always searching only for a single relevant dimension in the subspace orthogonal to the relevant dimensions already found. 3 Illustration for a model simple cell Here we illustrate how relevant dimensions can be found by maximizing variance (equivalent to least square fitting), and compare this scheme with that of finding relevant dimensions by maximizing information, as well as with those that are based upon computing the spike-triggered average. Our goal is to reconstruct relevant dimensions of neurons probed with inputs of arbitrary statistics. We used stimuli derived from a natural visual environment [11] that are known to strongly deviate from a Gaussian distribution. All of the studies have been carried out with respect to model neurons. Advantage of doing so is that the relevant dimensions are known. The example model neuron is taken to mimic properties of simple cells found in the primary visual cortex. It has a single relevant dimension, which we will denote as ˆe1. As can be seen in Fig. 1(a), it is phase and orientation sensitive. In this model, a given stimulus s leads to a spike if the projection s1 = s · ˆe1 reaches a threshold value θ in the presence of noise: P(spike|s)/P(spike) ≡g(s1) = ⟨H(s1 −θ+ξ)⟩, where a Gaussian random variable ξ with variance σ2 models additive noise, and the function H(x) = 1 for x > 0, and zero otherwise. The parameters θ for threshold and the noise variance σ2 determine the input–output function. In what follows we will measure these parameters in units of the standard deviation of stimulus projections along the relevant dimension. In these units, the signal-to-noise ratio is given by σ. Figure 1 shows that it is possible to obtain a good estimate of the relevant dimension ˆe1 by maximizing either information, as shown in panel (b), or variance, as shown in panel(c). The final value of the projection depends on the size of the dataset, as will be discussed below. In the example shown in Fig. 1 there were ≈50, 000 spikes with average probability of spike ≈0.05 per frame, and the reconstructed vector has a projection ˆvmax · ˆe1 = 0.98 when maximizing either information or variance. Having estimated the relevant dimension, one can proceed to sample the nonlinear input– output function. This is done by constructing histograms for P(s · ˆvmax) and P(s · ˆvmax|spike) of projections onto vector ˆvmax found by maximizing either information or variance, and taking their ratio. Because of the Bayes’ rule, this yields the nonlinear input–output function g of Eq. (1). In Fig. 1(d) the spike probability of the reconstructed neuron P(spike|s · ˆvmax) (crosses) is compared with the probability P(spike|s1) used in the model (solid line). A good match is obtained. In actuality, reconstructing even just one relevant dimension from neural responses to correlated non-Gaussian inputs, such as those derived from real-world, is not an easy problem. This fact can be appreciated by considering the estimates of relevant dimension obtained from the spike-triggered average (STA) shown in panel (e). Correcting the STA by second-order correlations of the input ensemble through a multiplication by the inverse covariance matrix results in a very noisy estimate, 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 -6 -4 -2 0 2 4 6 0.0 0.2 0.4 0.6 0.8 1.0 -6 -4 -2 0 2 4 6 0.0 0.2 0.4 0.6 0.8 1.0 STA decorrelated STA decorrelated STA (a) (b) (c) (d) (e) (f) (g) (h) truth regularized filtered stimulus (sd=1) filtered stimulus (sd=1) spike probability spike probability maximally informative dimension dimension of maximal variance maximizing information (x) variance (x) decorrelated STA (x) regularized decorrelated STA (x) truth Figure 1: Analysis of a model visual neuron with one relevant dimension shown in (a). Panels (b) and (c) show normalized vectors ˆvmax found by maximizing information and variance, respectively; (d) The probability of a spike P(spike|s · ˆvmax) (blue crosses – information maximization, red crosses – variance maximization) is compared to P(spike|s1) used in generating spikes (solid line). Parameters of the model are σ = 0.5 and θ = 2, both given in units of standard deviation of s1, which is also the units for the x-axis in panels (d and h). The spike–triggered average (STA) is shown in (e). An attempt to remove correlations according to the reverse correlation method, C−1 a priorivsta (decorrelated STA), is shown in panel (f) and in panel (g) with regularization (see text). In panel (h), the spike probabilities as a function of stimulus projections onto the dimensions obtained as decorrelated STA (blue crosses) and regularized decorrelated STA (red crosses) are compared to a spike probability used to generate spikes (solid line). shown in panel (f). It has a projection value of 0.25. Attempt to regularize the inverse of covariance matrix results in a closer match to the true relevant dimension [15, 16, 17, 18, 19] and has a projection value of 0.8, as shown in panel (g). While it appears to be less noisy, the regularized decorrelated STA can have systematic deviations from the true relevant dimensions [9, 20, 2, 11]. Preferred orientation is less susceptible to distortions than the preferred spatial frequency [19]. In this case regularization was performed by setting aside 1/4 of the data as a test dataset, and choosing a cutoff on the eigenvalues of the input covariances matrix that would give the maximal information value on the test dataset [16, 19]. 4 Comparison of Performance with Finite Data In the limit of infinite data the relevant dimensions can be found by maximizing variance, information, or other objective functions [1]. In a real experiment, with a dataset of finite size, the optimal vector found by any of the R´enyi divergences ˆv will deviate from the true relevant dimension ˆe1. In this section we compare the robustness of optimization strategies based on R´enyi divergences of various orders, including least squares fitting (α = 2) and information maximization (α = 1), as the dataset size decreases and/or neural noise increases. The deviation from the true relevant dimension δv = ˆv −ˆe1 arises because the probability distributions (4) are estimated from experimental histograms and differ from the distributions found in the limit of infinite data size. The effects of noise on the reconstruction can be characterized by taking the dot product between the relevant dimension and the optimal vector for a particular data sample: ˆv · ˆe1 = 1 −1 2δv2, where both ˆv and ˆe1 are normalized, and δv is by definition orthogonal to ˆe1. Assuming that the deviation δv is small, we can use quadratic approximation to expand the objective function (obtained with finite data) near its maximum. This leads to an expression δv = −[H(α)]−1∇F (α), which relates deviation δv to the gradient and Hessian of the objective function evaluated at the vector ˆe1. Subscript (α) denotes the order of the R´enyi divergence used as an objective function. Similarly to the case of optimizing information [2], the Hessian of R´enyi divergence of arbitrary order when evaluated along the optimal dimension ˆe1 is given by H(α) ij = −α Z dxP(x|spike)Cij(x) ·P(x|spike) P(x) ¸α−3 · d dx µP(x|spike) P(x) ¶¸2 , (12) where Cij(x) = (⟨sisj|x⟩−⟨si|x⟩⟨sj|x⟩) are covariance matrices of inputs sorted by their projection x along the optimal dimension. When averaged over possible outcomes of N trials, the gradient is zero for the optimal direction. In other words, there is no specific direction towards which the deviations δv are biased. Next, in order to measure the expected spread of optimal dimensions around the true one ˆe1, we need to evaluate ⟨δv2⟩= Tr h ⟨∇F (α)∇F (α)T ⟩ £ H(α)¤−2i , and therefore need to know the variance of the gradient of F averaged across different equivalent datasets. Assuming that the probability of generating a spike is independent for different bins, we find that ⟨∇F (α) i ∇F (α) j ⟩= B(α) ij /Nspike, where B(α) ij = α2 Z dxP(x|spike)Cij(x) ·P(x|spike) P(x) ¸2α−4 · d dx P(x|spike) P(x) ¸2 . (13) Therefore an expected error in the reconstruction of the optimal filter by maximizing variance is inversely proportional to the number of spikes: ˆv · ˆe1 ≈1 −1 2⟨δv2⟩= 1 −Tr′[BH−2] 2Nspike , (14) where we omitted superscripts (α) for clarity. Tr′ denotes the trace taken in the subspace orthogonal to the relevant dimension (deviations along the relevant dimension have no meaning [2], which mathematically manifests itself in dimension ˆe1 being an eigenvector of matrices H and B with the zero eigenvalue). Note that when α = 1, which corresponds to Kullback-Leibler divergence and information maximization, A ≡Hα=1 = Bα=1. The asymptotic errors in this case are completely determined by the trace of the Hessian of information, ⟨δv2⟩∝Tr′ £ A−1¤ , reproducing the previously published result for maximally informative dimensions [2]. Qualitatively, the expected error ∼D/(2Nspike) increases in proportion to the dimensionality D of inputs and decreases as more spikes are collected. This dependence is in common with expected errors of relevant dimensions found by maximizing information [2], as well as methods based on computing the spike-triggered average both for white noise [1, 21, 22] and correlated Gaussian inputs [2]. Next we examine which of the R´enyi divergences provides the smallest asymptotic error (14) for estimating relevant dimensions. Representing the covariance matrix as Cij(x) = γik(x)γjk(x) (exact expression for matrices γ will not be needed), we can express the Hessian matrix H and covariance matrix for the gradient B as averages with respect to probability distribution P(x|spike): B = Z dxP(x|spike)b(x)bT (x), H = Z dxP(x|spike)a(x)bT (x), (15) where the gain function g(x) = P(x|spike)/P(x), and matrices bij(x) = αγij(x)g′(x) [g(x)]α−2 and aij(x) = γij(x)g′(x)/g(x). Cauchy-Schwarz identity for scalar quantities states that, ⟨b2⟩/⟨ab⟩2 ≥1/⟨a2⟩, where the average is taken with respect to some probability distribution. A similar result can also be proven for matrices under a Tr operation as in Eq. (14). Applying the matrix-version of the Cauchy-Schwarz identity to Eq. (14), we find that the smallest error is obtained when Tr′[BH−2] = Tr′[A−1], with A = Z dxP(x|spike)a(x)aT (x), (16) Matrix A corresponds to the Hessian of the merit function for α = 1: A = H(α=1). Thus, among the various optimization strategies based on R´enyi divergences, Kullback-Leibler divergence (α = 1) has the smallest asymptotic errors. The least square fitting corresponds to optimization based on R´enyi divergence with α = 2, and is expected to have larger errors than optimization based on Kullback-Leibler divergence (α = 1) implementing information maximization. This result agrees with recent findings that Kullback-Leibler divergence is the best distortion measure for performing lossy compression [3]. Below we use numerical simulations with model cells to compare the performance of information (α = 1) and variance (α = 2) maximization strategies in the regime of relatively small numbers of spikes. We are interested in the range 0.1 ∼ < D/Nspike ∼ < 1, where the asymptotic results do not necessarily apply. The results of simulations are shown in Fig. 2 as a function of D/Nspike, as well as with varying neural noise levels. To estimate sharper (less noisy) input/output functions with σ = 1.5, 1.0, 0.5, 0.25, we used larger number of bins (16, 21, 32, 64), respectively. Identical numerical algorithms, including the number of bins, were used for maximizing variance and information. The relevant dimension for each simulated spike train was obtained as an average of 4 jackknife estimates computed by setting aside 1/4 of the data as a test set. Results are shown after 1000 line optimizations (D = 900), and performance on the test set was checked after every line optimization. As can be seen, generally good reconstructions with projection values ∼ > 0.7 can be obtained by maximizing either information or variance, even in the severely undersampled regime D < Nspike. We find that reconstruction errors are comparable for both information and variance maximization strategies, and are better or equal (at very low spike numbers) than STA-based methods. Information maximization achieves significantly smaller errors than the least-square fitting, when we analyze results for all simulations for four different models cells and spike numbers (p < 10−4, paired t-test). 0 0 0.5 1.0 1.5 2.0 2.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 STA decorrelated STA maximizing information maximizing variance A B C D A B C D D / N spike projection on true dimension 0 0.5 1.0 1.5 2.0 2.5 0.5 0.6 0.7 0.8 0.9 1.0 maximizing information maximizing variance regularized decorrelated STA D / Nspike A B C D Figure 2: Projection of vector ˆvmax obtained by maximizing information (red filled symbols) or variance (blue open symbols) on the true relevant dimension ˆe1 is plotted as a function of ratio between stimulus dimensionality D and the number of spikes Nspike, with D = 900. Simulations were carried out for model visual neurons with one relevant dimension from Fig. 1(a) and the input-output function Eq.(1) described by threshold θ = 2.0 and noise standard deviation σ = 1.5, 1.0, 0.5, 0.25 for groups labeled A (△), B (▽), C (⃝), and D (2), respectively. The left panel also shows results obtained using spike-triggered average (STA, gray) and decorrelated STA (dSTA, black). In the right panel, we replot results for information and variance optimization together with those for regularized decorrelated STA (RdSTA, green open symbols). All error bars show standard deviations. 5 Conclusions In this paper we compared accuracy of a family of optimization strategies for analyzing neural responses to natural stimuli based on R´enyi divergences. Finding relevant dimensions by maximizing one of the merit functions, R´enyi divergence of order 2, corresponds to fitting the linear-nonlinear model in the least-square sense to neural spike trains. Advantage of this approach over standard least square fitting procedure is that it does not require the nonlinear gain function to be invertible. We derived errors expected for relevant dimensions computed by maximizing R´enyi divergences of arbitrary order in the asymptotic regime of large spike numbers. The smallest errors were achieved not in the case of (nonlinear) least square fitting of the linear-nonlinear model to the neural spike trains (R´enyi divergence of order 2), but with information maximization (based on Kullback-Leibler divergence). Numeric simulations on the performance of both information and variance maximization strategies showed that both algorithms performed well even when the number of spikes is very small. With small numbers of spikes, reconstructions based on information maximization had also slightly, but significantly, smaller errors those of least-square fitting. 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Structured Learning with Approximate Inference Alex Kulesza and Fernando Pereira∗ Department of Computer and Information Science University of Pennsylvania {kulesza, pereira}@cis.upenn.edu Abstract In many structured prediction problems, the highest-scoring labeling is hard to compute exactly, leading to the use of approximate inference methods. However, when inference is used in a learning algorithm, a good approximation of the score may not be sufficient. We show in particular that learning can fail even with an approximate inference method with rigorous approximation guarantees. There are two reasons for this. First, approximate methods can effectively reduce the expressivity of an underlying model by making it impossible to choose parameters that reliably give good predictions. Second, approximations can respond to parameter changes in such a way that standard learning algorithms are misled. In contrast, we give two positive results in the form of learning bounds for the use of LP-relaxed inference in structured perceptron and empirical risk minimization settings. We argue that without understanding combinations of inference and learning, such as these, that are appropriately compatible, learning performance under approximate inference cannot be guaranteed. 1 Introduction Structured prediction models commonly involve complex inference problems for which finding exact solutions is intractable [1]. There are two ways to address this difficulty. Directly, models used in practice can be restricted to those for which inference is feasible, such as conditional random fields on trees [2] or associative Markov networks with binary labels [3]. More generally, however, efficient but approximate inference procedures have been devised that apply to a wide range of models, including loopy belief propagation [4, 5], tree-reweighted message passing [6], and linear programming relaxations [7, 3], all of which give efficient approximate predictions for graphical models of arbitrary structure. Since some form of inference is the dominant subroutine for all structured learning algorithms, it is natural to see good approximate inference techniques as solutions to the problem of tractable learning as well. A number of authors have taken this approach, using inference approximations as drop-in replacements during training, often with empirical success [3, 8]. And yet there has been little theoretical analysis of the relationship between approximate inference and reliable learning. We demonstrate with two counterexamples that the characteristics of approximate inference algorithms relevant for learning can be distinct from those, such as approximation guarantees, that make them appropriate for prediction. First, we show that approximations can reduce the expressivity of a model, making previously simple concepts impossible to implement and hence to learn, even though inference meets an approximation guarantee. Second, we show that standard learning algorithms can be led astray by inexact inference, failing to find valid model parameters. It is therefore crucial to choose compatible inference and learning procedures. ∗This work is based on research supported by NSF ITR IIS 0428193. 1 With these considerations in mind, we prove that LP-relaxation-based approximate inference procedures are compatible with the structured perceptron [9] as well as empirical risk minimization with a margin criterion using the PAC-Bayes framework [10, 11]. 2 Setting Given a scoring model S(y|x) over candidate labelings y for input x, exact Viterbi inference is the computation of the optimal labeling h(x) = arg max y S(y|x) . (1) In a prediction setting, the goal of approximate inference is to compute efficiently a prediction with the highest possible score. However, in learning a tight relationship between the scoring model and true utility cannot be assumed; after all, learning seeks to find such a relationship. Instead, we assume a fixed loss function L(y|x) that measures the true cost of predicting y given x, a distribution D over inputs x, and a parameterized scoring model Sθ(y|x) with associated optimal labeling function hθ and inference algorithm Aθ. Exact inference implies Aθ = hθ. Learning seeks the risk minimizer: θ∗= arg min θ Ex∼D [L(Aθ(x)|x)] . (2) Successful learning, then, requires two things: the existence of θ for which risk is suitably low, and the ability to find such θ efficiently. In this work we consider the impact of approximate inference on both criteria. We model our examples as pairwise Markov random fields (MRFs) defined over a graph G = (V, E) with probabilistic scoring model P(y|x) ∝ Y i∈V ψi(yi|x) Y ij∈E ψij(yi, yj|x) , (3) where ψi(yi|x) and ψij(yi, yj|x) are positive potentials. For learning, we use log-linear potentials ψi(yi|x) = exp(w·f(x, yi)) assuming a feature function f(·) and parameter vector w. Since MRFs are probabilistic, we also refer to Viterbi inference as maximum a posteriori (MAP) inference. 3 Algorithmic separability The existence of suitable model parameters θ is captured by the standard notion of separability. Definition 1. A distribution D (which can be empirical) is separable with respect to a model Sθ(y|x) and loss L(y|x) if there exists θ such that Ex∼D [L(hθ(x), x)] = 01. However, approximate inference may not be able to match exactly the separating hypothesis hθ. We need a notion of separability that takes into account the (approximate) inference algorithm. Definition 2. A distribution D is algorithmically separable with respect to parameterized inference algorithm Aθ and loss L(y|x) if there exists θ such that Ex∼D [L(Aθ(x), x)] = 0. While separability characterizes data distributions with respect to models, algorithmic separability characterizes data distributions with respect to inference algorithms. Note that algorithmic separability is more general than standard separability for any decidable model, since we can design an (inefficient) algorithm Aθ(x) = hθ(x)2. However, we show by counterexample that even algorithms with provable approximation guarantees can make separable problems algorithmically inseparable. 3.1 LP-relaxed inference Consider the simple Markov random field pictured in Figure 1, a triangle in which each node has as its set of allowed labels a different pair of the three possible labels A, B, and C. Let the node potentials ψi(yi) be fixed to 1 so that labeling preferences derive only from edge potentials. For positive 1Separability can be weakened to allow nonzero risk, but for simplicity we focus on the strict case. 2Note further that algorithmic separability supports inference algorithms that are not based on any abstract model at all; such algorithms can describe arbitrary “black box” functions from parameters to predictions. It seems unlikely, however, that such algorithms are of much use since their parameters cannot be easily learned. 2 constants λij, define edge potentials ψij(yi, yj) = exp(λij) whenever yi = yj and ψij(yi, yj) = 1 otherwise. Then the joint probability of a configuration y = (y1, y2, y3) is given by P(y) ∝ Y ij:yi=yj exp(λij) = exp X i,j I(yi = yj)λij (4) and the MAP labeling is arg maxy hP i,j I(yi = yj)λij i . Note that this example is associative; that is, neighboring nodes are enFigure 1: A simple MRF. Each node is annotated with its allowed labels. couraged to take identical labels (λij > 0). We can therefore perform approximate inference using a linear programming (LP) relaxation and get a multiplicative approximation guarantee [3]. We begin by writing an integer program for computing the MAP labeling; below, µi(yi) indicates node i taking label yi (which ranges over the two allowed labels for node i) and µij(yi, yj) indicates nodes i and j taking labels yi and yj, respectively. max µ λ12µ12(B, B) + λ23µ23(C, C) + λ31µ31(A, A) s.t. X yi µi(yi) ≤1 ∀i µij(yi, yj) ≤µi(yi) ∀ij, yi, yj µ ∈{0, 1}dim(µ) Integer programming is NP-hard, so we use an LP-relaxation by replacing the integrality constraint with µ ≥0. Letting i∗j∗= arg maxij λij, it is easy to see that the correct MAP configuration assigns matching labels to nodes i∗and j∗and an arbitrary label to the third. The score for this configuration is λi∗j∗. However, the LP-relaxation may generate fractional solutions. In particular, whenever (λ12 + λ23 + λ31)/2 > λi∗j∗the configuration that assigns to every node both of its allowed labels in equal proportion—µ = 1/2—is optimal. The fractional labeling µ = 1/2 is the most uninformative possible; it suggests that all labelings are equally valid. Even so, (λ12 + λ23 + λ31)/2 ≤3λi∗j∗/2 by the definition of i∗j∗, so LP-relaxed inference for this MRF has a relatively good approximation ratio of 3/2. 3.2 Learning with LP-relaxed inference Suppose now that we wish to learn to predict labelings y from instances of the MRF in Figure 1 with positive features given by x = (x12, x23, x31). We will parameterize the model using a positive weight vector w = (w12, w23, w31), letting λij = wijxij. Suppose the data distribution gives equal probability to inputs x = (4, 3, 3), (3, 4, 3), and (3, 3, 4), and that the loss function is defined as follows. Given x, let i∗j∗= arg maxij xij. Then assigning matching labels to nodes i∗and j∗and an arbitrary label to the third node yields a 0-loss configuration. All other configurations have positive loss. It is clear, first of all, that this problem is separable; if w = (1, 1, 1), λij = xij and the solution to the integer program above coincides with the labeling rule. Furthermore, there is margin: any weight vector in a neighborhood of (1, 1, 1) assigns the highest probability to the correct labeling. Using LP-relaxed inference, however, the problem is impossible to learn. In order to correctly label the instance x = (4, 3, 3) we must have, at a minimum, λ12 > λ23, λ31 (equivalently 4w12 > 3w23, 3w31) since the 0-loss labeling must have higher objective score than any other labeling. Reasoning similarly for the remaining instances, any separating weight vector must satisfy 4wij > 3wkl for each pair of edges (ij, kl). Without loss of generality, assume an instance to be labeled has feature vector x = (4, 3, 3). Then, 1 2(λ12 + λ23 + λ31) = 1 2(4w12 + 3w23 + 3w31) > 1 2(4w12 + 33 4w12 + 33 4w12) > 4w12 = λ12 . 3 As a result, LP-relaxed inference predicts µ = 1/2. The data cannot be correctly labeled using an LP-relaxation with any choice of weight vector, and the example is therefore algorithmically inseparable. 4 Insufficiency of algorithmic separability We cannot expect to learn without algorithmic separability; no amount of training can hope to be successful when there simply do not exist acceptable model parameters. Nevertheless, we could draw upon the usual techniques for dealing with (geometric) inseparability in this case. Approximate inference introduces another complication, however. Learning techniques exploit assumptions about the underlying model to search parameter space; the perceptron, for example, assumes that increasing weights for features present in correct labelings but not incorrect labelings will lead to better predictions. While this is formally true with respect to an underlying linear model, inexact inference methods can disturb and even invert such assumptions. 4.1 Loopy inference Loopy belief propagation (LBP) is a common approximate inference procedure in which maxproduct message passing, known to be exact for trees, is applied to arbitrary, cyclic graphical models [5]. While LBP is, of course, inexact, its behavior can be even more problematic for learning. Because LBP does not respond to model parameters in the usual way, its predictions can lead a learner away from appropriate parameters even for algorithmically separable problems. Consider the simple MRF shown in Figure 2 and discussed previously in Figure 2: An MRF on which LBP is inexact. [6]. All nodes are binary and take labels from the set {−1, 1}. Suppose that node potentials are assigned by type, where each node is of type A or B as indicated and α and β are real-valued parameters: ψA(−1) = 1 ψA(1) = eα ψB(−1) = 1 ψB(1) = eβ Also let edge potentials ψij(yi, yj) be equal to the constant λ when yi = yj and 1 otherwise. Define λ to be sufficiently positive that the MAP configuration is either (−1, −1, −1, −1) or (1, 1, 1, 1), abbreviated by −1 and 1, respectively. In particular, the solution is −1 when α + β < 0 and 1 otherwise. With slight abuse of notation we can write yMAP = sign(α + β). We now investigate the behavior of LBP on this example. In general, max-product LBP on pairwise MRFs requires iterating the following rule to update messages mij(yj) from node i to node j, where yj ranges over the possible labels for node j and N(i) is the neighbor set of node i. mij(yj) = max yi ψij(yi, yj)ψi(yi) Y k∈N(i)\{j} mki(yi) (5) Since we take λ to be suitably positive in our example, we can eliminate the max, letting yi = yj, and then divide to remove the edge potentials ψij(yj, yj) = λ. When messages are initialized uniformly to 1 and passed in parallel, symmetry also implies that messages are completely determined by the the types of the relevant nodes. The updates are then as follows. mAB(−1) = mBA(−1) mAB(1) = eαmBA(1) mBA(−1) = mAB(−1)mBB(−1) mBA(1) = eβmAB(1)mBB(1) mBB(−1) = m2 AB(−1) mBB(1) = eβm2 AB(1) Note that messages mij(−1) remain fixed at 1 after any number of updates. Messages mAB(1), mBA(1), and mBB(1) always take the form exp(pα + qβ) for appropriate values of p and q, and it is easy to show by iterating the updates that, for all three messages, p and q go to ∞while the ratio q/p converges to γ ≈1.089339. The label 1 messages, therefore, approach 0 when α + γβ < 0 and ∞when α + γβ > 0. Note that after message normalization (mij(−1) + mij(1) = 1 for all ij) the algorithm converges in either case. 4 (a) y = −1 (b) y = 1 Figure 3: A two-instance training set. Within each instance, nodes of the same shading share a feature vector, as annotated. Below each instance is its correct labeling. Beliefs are computed from the converged messages as bi(yi) ∝Q j∈N(i) mji(yi), so we can express the prediction of LBP as yLBP = sign(α + γβ). Intuitively, then, LBP gives a slight preference to the B-type nodes because of their shared edge. If α and β are both positive or both negative, or if α and β differ in sign but |β| > |α| or |α| > γ|β|, LBP finds the correct MAP solution. However, when the strength of the A nodes only slightly exceeds that of the B nodes (γ|β| > |α| > |β|), the preference exerted by LBP is significant enough to flip the labels. For example, if α = 1 and β = −0.95, the true MAP configuration is 1 but LBP converges to −1. 4.2 Learning with LBP Suppose now that we wish to use the perceptron algorithm with LBP inference to learn the twoinstance data set shown in Figure 3. For each instance the unshaded nodes are annotated with a feature vector xα = (xα1, xα2) and the shaded nodes are annotated with a feature vector xβ = (xβ1, xβ2). We wish to learn weights w = (w1, w2), modeling node potentials as before with α = w · xα and β = w · xβ. Assume that edge potentials remain fixed using a suitably positive λ. By the previous analysis, the data are algorithmically separated by w∗= (1, −1). On instance (a), α = 1, β = −0.95, and LBP correctly predicts −1. Instance (b) is symmetric. Note that although the predicted configurations are not the true MAP labelings, they correctly match the training labels. The weight vector (1, −1) is therefore an ideal choice in the context of learning. The problem is also separated in the usual sense by the weight vector (−1, 1). Since we can think of the MAP decision problem as computFigure 4: Perceptron learning path. ing sign(α + β) = sign (w · (xα + xβ)), we can apply the perceptron algorithm with update w ←w −ˆy(xα + xβ), where ˆy is the sign of the proposed labeling. The standard perceptron mistake bound guarantees that separable problems require only a finite number of iterations with exact inference to find a separating weight vector. Here, however, LBP causes the perceptron to diverge even though the problem is not only separable but also algorithmically separable. Figure 4 shows the path of the weight vector as it progresses from the origin over the first 20 iterations of the algorithm. During each pass through the data the weight vector is updated twice: once after mislabeling instance (a) (w ←w −(1, 0.95)), and again after mislabeling instance (b) (w ←w + (0.95, 1)). The net effect is w ←w + (−0.05, 0.05). The weight vector continually moves in the opposite direction of w∗= (1, −1), and learning diverges. 4.3 Discussion To understand why perceptron learning fails with LBP, it is instructive to visualize the feasible regions of weight space. Exact inference correctly labels instance (a) whenever w1 + 0.95w2 < 0, and, similarly, instance (b) requires a weight vector with 0.95w1 + w2 > 0. Weights that satisfy both constraints are feasible, as depicted in Figure 5(a). For LBP, the preference given to nodes 2 and 3 is effectively a scaling of xβ by γ ≈1.089339, so a feasible weight vector must satisfy 5 (a) Exact inference (b) LBP Figure 5: The feasible regions of weight space for exact inference and LBP. Each numbered gray halfspace indicates the region in which the corresponding instance is correctly labeled; their intersection is the feasible region, colored black. w1 + 0.95γw2 < 0 and 0.95γw1 + w2 > 0. Since 0.95γ > 1, these constraints define a completely different feasible region of weight space, shown in Figure 5(b). It is clear from the figures why perceptron does not succeed; it assumes that pushing weights into the feasible region of Figure 5(a) will produce correct labelings, while under LBP the exact opposite is required. Algorithmic separability, then, is necessary for learning but may not be sufficient. This does not imply that no algorithm can learn using LBP; a grid search on weight space, for example, will be slow but successful. Instead, care must be taken to ensure that learning and inference are appropriately matched. In particular, it is generally invalid to assume that an arbitrary choice of approximate inference will lead to useful results when the learning method expects exact feedback. 5 Learning bounds for approximate inference In contrast to the failure of LBP in Section 4, appropriate pairs of inference and learning algorithms do exist. We give two bounds using LP-relaxed inference for MRFs with log-linear potentials. First, under the assumption of algorithmic separability, we show that the structured perceptron of Collins [9] makes only a finite number of mistakes. Second, we show using the PAC-Bayesian framework [11] that choosing model parameters to minimize a margin-based empirical risk function (assuming “soft” algorithmic separability) gives rise to a bound on the true risk. In both cases, the proofs are directly adapted from known results using the following characterization of LP-relaxation. Claim 1. Let z = (z1, . . . , zk) be the vector of 0/1 optimization variables for an integer program P. Let Z ⊆{0, 1}dim(z) be the feasible set of P. Then replacing integrality constraints in P with box constraints 0 ≤zi ≤1 yields an LP with a feasible polytope having vertices Z′ ⊇Z. Proof. Each z ∈Z is integral and thus a vertex of the polytope defined by box constraints alone. The remaining constraints appear in P and by definition do not exclude any element of Z. The addition of constraints cannot eliminate a vertex without rendering it infeasible. Thus, Z ⊆Z′. □ We can encode the MAP inference problem for MRFs as an integer program over indicators z with objective w · Φ(x, z) for some Φ linear in z (see, for example, [6]). By Claim 1 and the fact that an optimal vertex always exists, LP-relaxed inference given an input x computes LPw(x) = arg max z∈Z′(x) w · Φ(x, z) . (6) We can think of this as exact inference over an expanded set of labelings Z′(x), some of which may not be valid (i.e., z ∈Z′(x) may be fractional). To simplify notation, we will assume that labelings y are always translated into corresponding indicator values z. 5.1 Perceptron Theorem 1 (adapted from Theorem 1 in [9]). Given a sequence of input/labeling pairs {(xi, zi)}, suppose that there exists a weight vector w∗with unit norm and γ > 0 such that, for all i, w∗· (Φ(xi, zi) −Φ(xi, z)) ≥γ for all z ∈Z′(xi) \ {zi}. (The instances are algorithmically separable with margin γ.) Suppose that there also exists R such that ∥Φ(xi, zi) −Φ(xi, z)∥≤R for all z ∈Z′(xi). Then the structured perceptron makes at most R2/γ2 mistakes. 6 Proof sketch. Let wk be the weight vector before the kth mistake; w1 = 0. Following the proof of Collins without modification, we can show that ∥wk+1∥≥kγ. We now bound ∥wk+1∥in the other direction. If (xk, zk) is the instance on which the kth update occurs and zLP(k) = LPwk(xk), then by the update rule, ∥wk+1∥2 = ∥wk∥2 + 2wk · (Φ(xk, zk) −Φ(xk, zLP(k))) + ∥Φ(xk, zk) −Φ(xk, zLP(k))∥2 ≤∥wk∥2 + R2 . (7) The inequality follows from the fact that LP-relaxed inference maximizes w · Φ(xk, z) over all z ∈Z′(xk), so the middle term is nonpositive. Hence, by induction, ∥wk+1∥2 ≤kR2. Combining the two bounds, k2γ2 ≤∥wk+1∥2 ≤kR2, hence k ≤R2/γ2. □ 5.2 PAC-Bayes The perceptron bound applies when data are perfectly algorithmically separable, but we might also hope to use LP-relaxed inference in the presence of noisy or otherwise almost-separable data. The following theorem adapts an empirical risk minimization bound using the PAC-Bayes framework to show that LP-relaxed inference can also be used to learn successfully in these cases. The measure of empirical risk for a weight vector w over a sample S = (x1, . . . , xm) is defined as follows. ˆR(w, S) = 1 m m X i=1 max z∈Hw(xi) L(z|xi) Hw(x) = {z′ ∈Z′(x) | w · (Φ(x, LPw(x)) −Φ(x, z′)) ≤|LPw(x) −z′|} (8) Intuitively, ˆR accounts for the maximum loss of any z that is closer in score than in 1-norm to the LP prediction. Such z are considered “confusable” at test time. The PAC-Bayesian setting requires that, after training, weight vectors are drawn from some distribution Q(w); however, a deterministic version of the bound can also be proved. Theorem 2 (adapted from Theorem 3 in [11]). Suppose that loss function L(y|x) is bounded between 0 and 1 and can be expanded to L(z|x) for all z ∈Z′(x); that is, loss can be defined for every potential value of LP(x). Let ℓ= dim(z) be the number of indicator variables in the LP, and let R bound the 2-norm of a feature vector for a single clique. Let Q(w) be a symmetric Gaussian centered at w as defined in [11]. Then with probability at least 1 −δ over the choice of a sample S of size m from distribution D over inputs x, the following holds for all w. Ex∼D,w′∼Q(w) [L(LPw′(x)|x)] ≤ˆR(w, S) + s R2∥w∥2 ln( 2ℓm R2∥w∥2 ) + ln( m δ ) 2(m −1) + R2∥w∥2 m (9) The proof in [11] can be directly adapted; the only significant changes are the use of Z′ in place of the set Y of possible labelings and reasoning as above using the definition of LP-relaxed inference. 6 Related work A number of authors have applied inference approximations to a wide range of learning problems, sometimes with theoretical analysis of approximation quality and often with good empirical results [8, 12, 3]. However, none to our knowledge has investigated the theoretical relationship between approximation and learning performance. Daume et al. [13] developed a method for using a linear model to make decisions during a search-based approximate inference process. They showed that perceptron updates give rise to a mistake bound under the assumption that parameters leading to correct decisions exist. Such results are analogous to those presented in Section 5 in that performance bounds follow from an (implicit) assumption of algorithmic separability. Wainright [14] proved that when approximate inference is required at test time due to computational constraints, using an inconsistent (approximate) estimator for learning can be beneficial. His result suggests that optimal performance is obtained when the methods used for training and testing are appropriately aligned, even if those methods are not independently optimal. In contrast, we consider learning algorithms that use identical inference for both training and testing, minimizing a general measure of empirical risk rather than maximizing data likelihood, and argue for compatibility between the learning method and inference process. 7 Roth et al. [15] consider learning independent classifiers for single labels, essentially using a trivial form of approximate inference. They show that this method can outperform exact inference learning when algorithmic separability holds precisely because approximation reduces expressivity; i.e., less complex models require fewer samples to train accurately. When the data are not algorithmically separable, exact inference provides better performance if a large enough sample is available. It is interesting to note that both of our counterexamples involve strong edge potentials. These are precisely the kinds of examples that are difficult to learn using independent classifiers. 7 Conclusion Effective use of approximate inference for learning depends on two considerations that are irrelevant for prediction. First, the expressivity of approximate inference, and consequently the bias for learning, can vary significantly from that of exact inference. Second, learning algorithms can misinterpret feedback received from approximate inference methods, leading to poor results or even divergence. However, when algorithmic separability holds, the use of LP-relaxed inference with standard learning frameworks yields provably good results. Future work includes the investigation of alternate inference methods that, while potentially less suitable for prediction alone, give better feedback for learning. Conversely, learning methods that are tailored specifically to particular inference algorithms might show improved performance over those that assume exact inference. Finally, the notion of algorithmic separability and the ways in which it might relate (through approximation) to traditional separability deserve further study. References [1] Gregory F. Cooper. The computational complexity of probabilistic inference using Bayesian belief networks (research note). Artif. Intell., 42(2-3):393–405, 1990. [2] John D. Lafferty, Andrew McCallum, and Fernando C. N. Pereira. Conditional random fields: Probabilistic models for segmenting and labeling sequence data. In ICML ’01: Proceedings of the Eighteenth International Conference on Machine Learning, pages 282–289, 2001. [3] Ben Taskar, Vassil Chatalbashev, and Daphne Koller. Learning associative Markov networks. In ICML ’04: Proceedings of the twenty-first international conference on Machine learning, page 102, 2004. [4] Judea Pearl. Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 1988. [5] Kevin Murphy, Yair Weiss, and Michael Jordan. Loopy belief propagation for approximate inference: An empirical study. In Proceedings of the 15th Annual Conference on Uncertainty in Artificial Intelligence (UAI-99), pages 467–47, 1999. [6] M.J. Wainwright, T.S. Jaakkola, and A.S. Willsky. MAP estimation via agreement on trees: messagepassing and linear programming. IEEE Transactions on Information Theory, 51(11):3697–3717, 2005. [7] D. Roth and W. Yih. A linear programming formulation for global inference in natural language tasks. In Proc. of the Conference on Computational Natural Language Learning (CoNLL), pages 1–8, 2004. [8] Charles Sutton and Andrew McCallum. Collective segmentation and labeling of distant entities in information extraction. Technical Report TR # 04-49, University of Massachusetts, 2004. [9] Michael Collins. Discriminative training methods for hidden Markov models: theory and experiments with perceptron algorithms. In EMNLP ’02: Proceedings of the ACL-02 conference on Empirical methods in natural language processing, pages 1–8, 2002. [10] David A. McAllester. PAC-bayesian stochastic model selection. Machine Learning, 51(1):5–21, 2003. [11] David McAllester. Generalization bounds and consistency for structured labeling. In Predicting Structured Data. MIT Press, To Appear. [12] Charles Sutton and Andrew McCallum. Piecewise training of undirected models. In 21st Conference on Uncertainty in Artificial Intelligence, 2005. [13] Hal Daum´e III and Daniel Marcu. Learning as search optimization: Approximate large margin methods for structured prediction. In International Conference on Machine Learning (ICML), 2005. [14] Martin J. Wainwright. Estimating the ”wrong” graphical model: Benefits in the computation-limited setting. Journal of Machine Learning Research, 7:1829–1859, 2006. [15] V. Punyakanok, D. Roth, W. Yih, and D. Zimak. Learning and inference over constrained output. In Proc. of the International Joint Conference on Artificial Intelligence (IJCAI), pages 1124–1129, 2005. 8
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On Ranking in Survival Analysis: Bounds on the Concordance Index Vikas C. Raykar, Harald Steck, Balaji Krishnapuram CAD and Knowledge Solutions (IKM CKS), Siemens Medical Solutions Inc., Malvern, USA {vikas.raykar,harald.steck,balaji.krishnapuram}@siemens.com Cary Dehing-Oberije, Philippe Lambin Maastro Clinic, University Hospital Maastricht, University Maastricht, GROW, The Netherlands {cary.dehing,philippe.lambin}@maastro.nl Abstract In this paper, we show that classical survival analysis involving censored data can naturally be cast as a ranking problem. The concordance index (CI), which quantifies the quality of rankings, is the standard performance measure for model assessment in survival analysis. In contrast, the standard approach to learning the popular proportional hazard (PH) model is based on Cox’s partial likelihood. We devise two bounds on CI–one of which emerges directly from the properties of PH models–and optimize them directly. Our experimental results suggest that all three methods perform about equally well, with our new approach giving slightly better results. We also explain why a method designed to maximize the Cox’s partial likelihood also ends up (approximately) maximizing the CI. 1 Introduction Survival analysis is a well-established field in medical statistics concerned with analyzing/predicting the time until the occurrence of an event of interest–e.g., death, onset of a disease, or failure of a machine. It is applied not only in clinical research, but also in epidemiology, reliability engineering, marketing, insurance, etc. The time between a well-defined starting point and the occurrence of the event is called the survival time or failure time, measured in clock time or in another appropriate scale, e.g., mileage of a car. Survival time data are not amenable to standard statistical methods because of its two special features–(1) the continuous survival time often follows a skewed distribution, far from normal, and (2) a large portion of the data is censored (see Sec. 2). In this paper we take a machine learning perspective and cast survival analysis as a ranking problem–where the task is to rank the data points based on their survival times rather than to predict the actual survival times. One of the most popular performance measures for assessing learned models in survival analysis is the Concordance Index (CI), which is similar to the Wilcoxon-Mann-Whitney statistic [13, 10] used in bi-partite ranking problems. Given the CI as a performance measure, we develop approaches that learn models by directly optimizing the CI. As optimization of the CI is computationally expensive, we focus on maximizing two lower bounds on the CI, namely the log-sigmoid and the exponential bounds, which are described in Sec. 4, 5, and 6. Interestingly, the log-sigmoid bound arises in a natural way from the Proportional Hazard (PH) model, which is the standard model used in classical survival analysis, see Sec. 5.2. Moreover, as the PH models are learned by optimizing Cox’s partial likelihood in classical survival analysis, we show in Sec. 8 that maximizing this likelihood also ends up (approximately) maximizing the CI. Our experiments in Sec. 9 show that optimizing our two lower bounds and Cox’s likelihood yields very similar results with respect to the CI, with the proposed lower bounds being slightly better. 1 2 Survival analysis Survival analysis has been extensively studied in the statistics community for decades, e.g., [4, 8]. A primary focus is to build statistical models for survival time T ∗ i of individual i of a population. 2.1 Censored data A major problem is the fact that the period of observation C∗ i can be censored for many individuals i. For instance, a patient may move to a different town and thus be no longer available for a clinical trial. Also at the end of the trial a lot of patients may actually survive. For such cases the exact survival time may be longer than the observation period. Such data are referred to as right-censored, and C∗ i is also called the censoring time. For such individuals, we only know that they survived for at least C∗ i , i.e., our actual observation is Ti = min(T ∗ i , C∗ i ). Let xi ∈Rd be the associated d-dimensional vector of covariates (explanatory variables) for the ith individual. In clinical studies, the covariates typically include demographic variables, such as age, gender, or race; diagnosis information like lab tests; or treatment information, e.g., dosage. An important assumption generally made is that T ∗ i and C∗ i are independent conditional on xi, i.e., the cause for censoring is independent of the survival time. With the indicator function δi, which equals 1 if failure is observed (T ∗ i ≤C∗ i ) and 0 if data is censored (T ∗ i > C∗ i ), the available training data can be summarized as D = {Ti, xi, δi}N i=1 for N patients. The objective is to learn a predictive model for the survival time as a function of the covariates. 2.2 Failure time distributions The failures times are typically modeled to follow a distribution, which absorbs both truly random effects and causes unexplained by the (available) covariates. This distribution is characterized by the survival function S(t) = Pr[T > t] for t > 0, which is the probability that the individual is still alive at time t. A related function commonly used is the hazard function. If T has density function p, then the hazard function is defined by λ(t) = lim∆t→0 Pr[t < T ≤t + ∆t|T > t]/∆t = p(t)/S(t). The hazard function measures the instantaneous rate of failure, and provides more insight into the failure mechanisms. The function Λ(t) = R t 0 λ(u)du is called the cumulative hazard function, and it holds that S(t) = e−Λ(t) [4]. 2.3 Proportional hazard model Proportional hazard (PH) models have become the standard for studying the effect of the covariates on the survival time distributions, e.g., [8]. Specifically, the PH model assumes a multiplicative effect of the covariates on the hazard function, i.e., λ(t|x) = λ0(t)ew⊤x, (1) where λ(t|x) is the hazard function of a person with covariates x; λ0(t) is the so-called baseline hazard function (i.e., when x = 0), which is typically based on the exponential or the Weibull distributions; w is a set of unknown regression parameters, and ew⊤x is the relative hazard function. Equivalent formulations for the cumulative hazard function and the survival function include Λ(t|x) = Λ0(t)ew⊤x, and S(t|x) = e−Λ0(t)ew⊤x = e − h ew⊤x R λ0(t)dt i . (2) 2.4 Cox’s partial likelihood Cox noticed that a semi-parametric approach is sufficient for estimating the weights w in PH models [2, 3], i.e., the baseline hazard function can remain completely unspecified. Only a parametric assumption concerning the effect of the covariates on the hazard function is required. Parameter estimates in the PH model are obtained by maximizing Cox’s partial likelihood (of the weights) [2, 3]: L(w) = Y Ti uncensored ew⊤xi P Tj≥Ti ew⊤xj . (3) 2 (a) (b) −10 −8 −6 −4 −2 0 2 4 6 8 10 −10 −8 −6 −4 −2 0 2 z Indicator function Log−sigmoid lower bound Exponential lower bound (c) Figure 1: Order graphs representing the ranking constraints. (a) No censored data and (b) with censored data. The empty circle represents a censored point. The points are arranged in the increasing value of their survival times with the lowest being at the bottom. (c) Two concave lower bounds on the 0-1 indicator function. Each term in the product is the probability that the ith individual failed at time Ti given that exactly one failure has occurred at time Ti and all individuals for which Tj ≥Ti are at risk of failing. Cox and others have shown that this partial log-likelihood can be treated as an ordinary log-likelihood to derive valid (partial) maximum likelihood estimates of w [2, 3]. The interesting properties of the Cox’s partial likelihood include: (1) due to its parametric form, it can be optimized in a computationally efficient way; (2) it depends only on the ranks of the observed survival times, cf. the inequality Tj ≥Ti in Eq. 3, rather than on their actual numerical values. We outline this connection to the ranking of the times Ti–and hence the concordance index–in Sec. 8. 3 Ordering of Survival times Casting survival analysis as ranking problem is an elegant way of dealing not only with the typically skewed distributions of survival times, but also with the censoring of the data: Two subjects’ survival times can be ordered not only if (1) both of them are uncensored but also if (2) the uncensored time of one is smaller than the censored survival time of the other. This can be visualized by means of an order graph G = (V, E), cf. also Fig. 1. The set of vertices V represents all the individuals, where each filled vertex indicates an observed/uncensored survival time, while an empty circle denotes a censored observation. Existence of an edge Eij implies that Ti < Tj. An edge cannot originate from a censored point. 3.1 Concordance index For these reasons, the concordance index (CI) or c-index is one of the most commonly used performance measures of survival models, e.g., [6]. It can be interpreted as the fraction of all pairs of subjects whose predicted survival times are correctly ordered among all subjects that can actually be ordered. In other words, it is the probability of concordance between the predicted and the observed survival. It can be written as c(D, G, f) = 1 |E| X Eij 1f(xi)<f(xj) (4) with the indicator function 1a<b = 1 if a < b, and 0 otherwise; |E| denotes the number of edges in the order graph. f(xi) is the predicted survival time for subject i by the model f. Equivalently, the concordance index can also be written explicitly as c = 1 |E| X Ti uncensored X Tj>Ti 1f(xi)<f(xj). (5) This index is a generalization of the Wilcoxon-Mann-Whitney statistics [13, 10] and thus of the area under the ROC curve (AUC) to regression problems in that it can (1) be applied to continuous 3 output variables and (2) account for censoring of the data. Like for the AUC, c = 1 indicates perfect prediction accuracy and c = 0.5 is as good as a random predictor. 3.2 Maximizing the CI—The Ranking Problem Since we evaluate the predictive accuracy of a survival model in terms of the concordance index, it is natural to formulate the learning problem to directly maximize the concordance index. Note that, while the concordance index has been used widely to evaluate a learnt model, it is not generally used as an objective function during training. As the concordance index is invariant to any monotone transformation of the survival times, the model learnt by maximizing the c-index is actually a ranking/scoring function. Our goal is to predict whether the survival time of one individual is larger than the one of another individual. Very often the doctor would like to know whether a particular kind of treatment results in an increase in the survival time and the exact absolute value of the survival time is not important. In terms of ranking problems studied in machine learning this is an N-partite ranking problem, where every data point is a class in itself. Formulating it as a ranking problem allows us to naturally incorporate the censored data. Once we have formulated it as a ranking problem we can use various ranking algorithms proposed in the machine learning literature [5, 7, 1, 12]. In this paper we use the algorithm proposed by [12]. More formally, we would like to learn a ranking function f from a suitable function class F, such that f(xi) > f(xj) implies that the survival time of patient i is larger than the one of patient j. Given the data D and the order graph G, the optimal ranking function is bf = arg maxf∈F c(D, G, f). As to prevent overfitting on the training data, regularization can be added to this equation, see Secs. 5 and 6. In many cases, sufficient regularization is also achieved by restricting the function class F, e.g., it may contain only linear functions. For ease of exposition we will consider the family of linear ranking functions 1 in this paper: F = {fw}, where for any x, w ∈Rd, fw(x) = w⊤x. 4 Lower bounds on the CI Maximizing the CI is a discrete optimization problem, which is computationally expensive. For this reason, we resort to maximizing a differentiable and concave lower bound on the 0-1 indicator function in the concordance index, cf. Eqs. 4 and 5. In this paper we focus on the log-sigmoid lower bound [12], cf. Sec. 5, and exponential lower bound, cf. Sec. 6, which are suitably scaled as to be tight at the origin and also in the asymptotic limit of large positive values, see also Fig. 1(c). We will also show how these bounds relate to the classical approaches in survival analysis: as it turns out, for the family of linear ranking functions, these two approaches are closely related to the PH model commonly used in survival analysis, cf. Sec. 5.2. 5 Log-sigmoid lower bound The first subsection discusses the lower bound on the concordance index based on the log-sigmoid function. The second subsection shows that this bound arises naturally when using proportional hazard models. 5.1 Lower bound The sigmoid function is defined as σ(z) = 1/(1+e−z), While it is an approximation to the indicator function, it is not a lower bound. In contrast, the scaled version of the log of the sigmoid function, log [2σ(z)]/ log 2, is a lower bound on the indicator function (Fig. 1(c)), i.e., 1z>0 ≥1 + (log σ(z)/log 2). (6) The log-sigmoid function is concave and asymptotically linear for large negative values, and may hence be considered a differentiable approximation to the hinge loss, which is commonly used for 1Generalization to non-linear functions can be achieved easily by using kernels: the linear ranking function class F is replaced by H, a reproducing kernel Hilbert space (RKHS). The ranking function then is of the form f(x) = PN i=1 αik(x, xi) where k is the kernel of the RHKS H. 4 training support vector machines. The lower bound on the concordance index (cf. Eq. 4) follows immediately: c = 1 |E| X Eij 1f(xj)−f(xi)>0 ≥1 |E| X Eij 1 + (log σ[f(xj) −f(xi)]/log 2) ≡bcLS, (7) which can efficiently be maximized by gradient-based methods (cf. Sec 7). Given the linear ranking function fw(x) = w⊤x, the bound bcLS becomes bcLS(w) = 1 |E| X Eij 1 + (log σ[w⊤(xj −xi)]/log 2). (8) As to avoid overfitting, we penalize functions with a large norm w in the standard way, and obtain the regularized version bcLSreg(w) = −λ 2 ∥w∥2 + bcLS(w). (9) 5.2 Connection to the PH model The concordance index can be interpreted as the probability of correct ranking (as defined by the given order graph) given a function f. Its probabilistic version can thus be cast as a likelihood. Under the assumption that each pair (j, i) is independent of any other pair, the log-likelihood reads L(fw, D, G) = log Y Eij Pr [fw(xi) < fw(xj)|w] . (10) As this independence assumption obviously does not hold among all pairs due to transitivity (even though the individual samples i are assumed i.i.d.), it provides a lower bound on the concordance index. While the probability of correct pairwise ordering, Pr [fw(xi) < fw(xj)|w], is often chosen to be sigmoid in the ranking literature [1], we show in the following that the sigmoid function arises naturally in the context of PH models. Let T(w⊤x) denote the survival time for the patient with covariates x or relative log-hazard w⊤x. A larger hazard corresponds to a smaller survival time, cf. Sec. 2. Hence Pr [fw(xi) < fw(xj)|w] = Pr[T(w⊤xj) > T(w⊤xi)|w] = Z ∞ 0 Pr[T(w⊤xj) > t]p(t|xi)dt = Z ∞ 0 S(t|xj)p(t|xi)dt = Z ∞ 0 −S(t|xj)S ′(t|xi)dt, where p(t|xi) is the density function of T for patient i with covariate xi, and S(t|xi) is the corresponding survival function; S ′(t) = dS(t)/dt = −p(t). Using Eq. 2 of the PH model, we continue the manipulations: Pr [fw(xi) < fw(xj)|w] = −ew⊤xi Z ∞ 0 e −Λ0(t) ew⊤xj +ew⊤xi Λ ′ 0(t)dt = ew⊤xi ew⊤xj + ew⊤xi = σ[w⊤(xi −xj)]. (11) This derivation shows that the probability of correct pairwise ordering indeed follows the sigmoid function. Assuming a prior Pr[w] = N(w|0, λ−1) for regularization, the optimal maximum aposteriori (MAP) estimator is of the form bwMAP = arg max L(w), where the posterior L(w) takes the form of a penalized log-likelihood: L(w) = −λ 2 ∥w∥2 + X Eij log σ wT (xj −xi) . (12) This expression is equivalent to (8) except for a few constants that are irrelevant for optimization problem, which justifies our choice of regularization in Eq. 8. 5 6 Exponential lower bound The exponential 1 −e−z can serve as an alternative lower bound on the step indicator function (see Fig. 1(c)). The concordance index can then be lower-bounded by c ≥ 1 |E| X Eij 1 −e−[f(xj)−f(xi)] ≡bcE. (13) Analogous to the log-sigmoid bound, for the linear ranking function fw(x) = w⊤x, the lower bound bcE simplifies to bcE(w) = 1 |E| X Eij 1 −e−w⊤(xj−xi), (14) and, penalizing functions with large norm w, the regularized version reads bcEreg(w) = −λ 2 ∥w∥2 + 1 |E| X Eij 1 −e−w⊤(xj−xi). (15) 7 Gradient based learning In order to maximize the regularized concave surrogate we can use any gradient-based learning technique. We use the Polak-Ribi`ere variant of nonlinear conjugate gradients (CG) algorithm [11]. The CG method only needs the gradient g(w) and does not require evaluation of the function. It also avoids the need for computing the second derivatives. The convergence of CG is much faster than that of the steepest descent. Using the fact that dσ(z)/dz = σ(z)[1 −σ(z)] and 1 −σ(z) = σ(−z), the gradient of Eq. 9 (log-sigmoid bound) is given by ∇wbcLSreg(w) = −λw − 1 |E| log 2 P Eij(xi − xj)σ wT (xi −xj) , and the gradient of Eq. 15 (exponential bound) by ∇wbcEreg(w) = −λw − 1 |E| P Eij(xi −xj)e−w⊤(xj−xi). 8 Is Cox’s partial likelihood a lower bound on the CI ? Our experimental results (Sec. 9) indicate that the Coxs method and our proposed methods showed similar performance when assessed using the CI. While our proposed method was formulated to explicitly maximize a lower bound on the concordance index, the Coxs method maximized the partial likelihood. One suspects whether Coxs partial likelihood itself is a lower bound on the concordance index. The argument presented below could give an indication as to why a method which maximizes the partial likelihood also ends up (approximately) maximizing the concordance index. We re-write the exponential bound on the CI for proportional hazard models from Sec. 6 bcE(w) = 1 |E| X Ti uncensored X Tj≥Ti 1 −e−w⊤(xi−xj) = 1 −1 |E| X Ti uncensored e−w⊤xi[ X Tj≥Ti ew⊤xj] = 1 −No |E| 1 No X Ti uncensored 1/zi ! , where zi = ew⊤xi P Tj≥Ti ew⊤xj ∈[0, 1]. (16) Note that we have replaced Tj > Ti by Tj ≥Ti, assuming that there are no ties in the data, i.e., no two survival times are identical, analogous to Cox’s partial likelihood approach (cf. Sec. 2.4). The number of uncensored observations is denoted by No. The Cox’s partial likelihood can be written in terms of zi as L(w) = Q Ti uncensored zi = ⟨zi⟩No geom, where ⟨zi⟩geom denotes the geometric mean of the zi with uncensored Ti. Using the inequality zi ≥min zi the concordance index can be bounded as c ≥1 −No |E| 1 min zi . (17) This says maximizing min zi maximizes a lower bound on the concordance index. While this does not say anything about the Cox’s partial likelihood it still gives a useful insight. Since max zi = 1 (because zi = 1 for the largest uncensored Ti), maximizing min zi can be expected to approximately maximize the geometric mean of zi, and hence the Cox’s partial likelihood. 6 Table 1: Summary of the five data sets used. N is the number of patients. d is the number of covariates used. Dataset N d Missing Censored MAASTRO 285 19 3.6% 30.5% SUPPORT-1 477 26 14.9% 36.4% SUPPORT-2 314 26 16.6% 43.0% SUPPORT-4 149 26 22.0% 10.7% MELANOMA 191 4 0.0% 70.2% 9 Experiments In this section we compare the performance of the two different lower bounds on the CI—the logsigmoid, exponential, and Cox’s partial likelihood—on five medical data sets. 9.1 Medical datasets Table 1 summarizes the five data sets we used in our experiments. A substantial amount of data is censored and also missing. The MAASTRO dataset concerns the survival time of non-small cell lung cancer patients, which we analyzed as part of our collaboration. The other medical data sets are publicly available: The SUPPORT dataset 2 is a random sample from Phases I and II of the SUPPORT [9](Study to Understand Prognoses Preferences Outcomes and Risks of Treatment) study. As suggested in [6] we split the dataset into three different datasets, each corresponding to a different cause of death. The MELANOMA data 3 is from a clinical study of skin cancer. 9.2 Evaluation procedure For each data set, 70% of the examples were used for training and the remaining 30% as the hold-out set for testing. We chose the optimal value of regularization parameter λ (cf. Eqs. 9 and 15) based on five-fold cross validation on the training set. The tolerance for the conjugate gradient procedure was set to 10−3. The conjugate-gradient optimization procedure was initialized to the zero vector. All the covariates were normalized to have zero mean and unit variance. As missing values were not the focus of this paper, we used a simple imputation technique. For each missing value, we imputed a sample drawn from a Gaussian distribution with its mean and variance estimated from the available values of the other patients. 9.3 Results The performance was evaluated in terms of the concordance index and the results are tabulated in Table 2. We compare the following methods–(1) Cox’s partial likelihood method, and (2) the proposed ranking methods with log-sigmoid and exponential lower bounds. The following observations can be made–(1) The proposed linear ranking method performs slightly better than the Cox’s partial likelihood method, but the difference does not appear significant. This agrees with our insights that Cox’s partial likelihood may also end up maximizing the CI. (2) The exponential bound shows slightly better performance than the log-sigmoid bound, which may indicate that the tightness of the bound for positive z in Fig. 1(c) is more important than for negative z in our data sets. However the difference is not significant. 10 Conclusions In this paper, we outlined several approaches for maximizing the concordance index, the standard performance measure in survival analysis when cast as a ranking problem. We showed that, for the widely-used proportional hazard models, the log-sigmoid function arises as a natural lower bound on the concordance index. We presented an approach for directly optimizing this lower bound in a computationally efficient way. This optimization procedure can also be applied to other lower bounds, like the exponential one. Apart from that, we showed that maximizing Cox’s partial likelihood can be understood as (approximately) maximizing a lower bound on the concordance index, which explains the high CI-scores of proportional hazard models observed in practice. Optimization of each of these three lower bounds results in about the same CI-score in our experiments, with our new approach giving tentatively better results. 2http://biostat.mc.vanderbilt.edu/twiki/bin/view/Main/DataSets. 3www.stat.uni-muenchen.de/service/datenarchiv/melanoma/melanoma_e.html 7 Table 2: Concordance indices for the different methods and datasets. The mean and the standard deviation are computed over a five fold cross-validation. The results are also shown for a fixed holdout set. CI for CI for CI for training set test set holdout mean [± std] mean [± std] set MAASTRO Cox PH 0.65 [±0.02] 0.57 [±0.09] 0.64 log-sigmoid 0.69 [±0.02] 0.60 [±0.06] 0.64 exponential 0.69 [±0.02] 0.64 [±0.08] 0.65 SUPPORT-1 Cox PH 0.76 [±0.01] 0.74 [±0.05] 0.79 log-sigmoid 0.83 [±0.01] 0.77 [±0.04] 0.79 exponential 0.83 [±0.01] 0.79 [±0.02] 0.82 SUPPORT-2 Cox PH 0.70 [±0.02] 0.63 [±0.06] 0.69 log-sigmoid 0.79 [±0.01] 0.68 [±0.06] 0.65 exponential 0.78 [±0.02] 0.68 [±0.09] 0.70 SUPPORT-4 Cox PH 0.78 [±0.01] 0.68 [±0.09] 0.64 log-sigmoid 0.80 [±0.01] 0.74 [±0.12] 0.71 exponential 0.79 [±0.01] 0.73 [±0.03] 0.71 MELANOMA Cox PH 0.63 [±0.03] 0.62 [±0.09] 0.54 log-sigmoid 0.76 [±0.02] 0.70 [±0.10] 0.55 exponential 0.76 [±0.01] 0.65 [±0.11] 0.55 Acknowledgements We are grateful to R. Bharat Rao for encouragement and support of this work, and to the anonymous reviewers for their valuable comments. References [1] C.J.C. Burges, T. Shaked, E. Renshaw, A. Lazier, M. Deeds, N. Hamilton, and G. Hullender. Learning to rank using gradient descent. In Proceeding of the 22th International Conference on Machine Learning, 2005. [2] D. R. Cox. Regression models and life-tables (with discussion). Journal of the Royal Statistical Society, Series B, 34(2):187–220, 1972. [3] D. R. Cox. Partial likelihood. Biometrika, 62(2):269–276, 1975. [4] D. R. Cox and D. Oakes. Analysis of survival data. Chapman and Hall, 1984. [5] Y. Freund, R. Iyer, and R. Schapire. An efficient boosting algorithm for combining preferences. Journal of Machine Learning Research, 4:933–969, 2003. [6] F. E. Harrell Jr. Regression Modeling Strategies, With Applications to Linear Models, Logistic Regression, and Survival Analysis. Springer, 2001. [7] R. Herbrich, T. Graepel, P. Bollmann-Sdorra, and K. Obermayer. Learning preference relations for information retrieval. ICML-98 Workshop: Text Categorization and Machine Learning, pages 80–84, 1998. [8] J. D. Kalbfleisch and R. L. Prentice. The statistical analysis of failure time data. Wiley-Interscience, 2002. [9] W.A. Knaus, F. E. Harrell, J. Lynn, et al. The support prognostic model: Objective estimates of survival for seriously ill hospitalized adults. Annals of Internal Medicine, 122:191–203, 1995. [10] H. B. Mann and D. R. Whitney. On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other. The Annals of Mathematical Statistics, 18(1):50–60, 1947. [11] J. Nocedal and S. J. Wright. Numerical Optimization. Springer, 1999. [12] V. C. Raykar, R. Duraiswami, and B. Krishnapuram. A fast algorithm for learning large scale preference relations. In M. Meila and X. Shen, editors, Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics, pages 385–392, 2007. [13] F. Wilcoxon. 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Competition adds complexity Judy Goldsmith Department of Computer Science University of Kentucky Lexington, KY goldsmit@cs.uky.edu Martin Mundhenk Friedrich-Schiller-Universit¨at Jena Jena, Germany mundhenk@cs.uni-jena.de Abstract It is known that determinining whether a DEC-POMDP, namely, a cooperative partially observable stochastic game (POSG), has a cooperative strategy with positive expected reward is complete for NEXP. It was not known until now how cooperation affected that complexity. We show that, for competitive POSGs, the complexity of determining whether one team has a positive-expected-rewardstrategy is complete for NEXPNP. 1 Introduction From online auctions to Texas Hold’em, AI is captivated by multi-agent interactions based on competition. The problem of finding a winning strategy harks back to the first days of chess programs. Now, we are starting to have the capacity to handle issues like stochastic games, partial information, and real-time video inputs for human player modeling. This paper looks at the complexity of computations involving the first two factors: partially observable stochastic games (POSGs). There are many factors that could affect the complexity of different POSG models: Do the players, collectively, have sufficient information to reconstruct a state? Do they communicate or cooperate? Is the game zero sum, or do the players’ individual utilities depend on other players’ utilities? Do the players even have models for other players’ utilities? The ultimate question is, what is the complexity of finding a winning strategy for a particular player, with no assumptions about joint observations or knowledge of other players’ utilities. Since a special case of this is the DEC-POMDP, where finding an optimal (joint, cooperative) policy is known to be NEXP-hard [1], this problem cannot be any easier than in NEXP. We show that one variant of this problem is hard for the class NEXPNP. 2 Definitions and Preliminaries 2.1 Partially observable stochastic games A partially observable stochastic game (POSG) describes multi-player stochastic game with imperfect information by its states and the consequences of the players actions on the system. We follow the definition from [2] and denote it as a tuple M = (I,S,s0,A,O,t,o,r), where • I is the finite set {1,2,...,k} of agents (or players), S is a finite set of states, with distinguished initial state s0 ∈S, A is a finite set of actions, and O is a finite set of observations • t : S × Ak × S →[0,1] is the transition probability function, where t(s,a1,...,ak,s′) is the probability that state s′ is reached from state s when each agent i chooses action ai • o : S × I →O is the observation function , where o(s,i) is the observation made in state s by agent i, and 1 • r : S × Ak × I →Z is the reward function, where r(s,a1,...,ak,i) is the reward gained by agent i in state s, when the agents take actions a1,...,ak. (Z is the set of integers.) A POSG where all agents have the same reward function is called a decentralized partiallyobservable Markov decision process (see [1]). Let M = (I,S,s0,A,O,t,o,r) be a POSG. A step of M is a transition from one state to another according to the transition probability function t. A run of M is a sequence of steps that starts in the initial state s0. The outcome of each step is probabilistic and depends on the actions chosen. For each agent, a policy describes how to choose actions depending on observations made during the run of the process. A (history-dependent) policy π chooses an action dependent on all observations made by the agent during the run of the process. This is described as a function π : O∗→A, mapping each finite sequence of observations to an action. A trajectory θ of length |θ| = m for M is a sequence of states θ = σ1,σ2,...,σm (m ≥1, σi ∈S) which starts with the initial state of M , i.e. σ1 = s0. Given policies π1,...,πk, each trajectory θ has a probability prob(θ,π1,...,πk). We will use some abbreviations in the sequel. For π1,...,πk we will write πk 1, and for π1(o(σ1,1)···o(σj,1)),...,πk(o(σ1,k)···o(σj,k)) we will write πk 1(θ j 1) accordingly. Then prob(θ,π1,...,πk) is defined by prob(θ,πk 1) = |θ|−1 ∏ i=1 t(σi,πk 1(θ i 1),σi+1) . We use Tl(s) to denote all length l trajectories which start in the initial state s0 and end in state s. The expected reward Ri(s,l,πk 1) obtained by agent i in state s after exactly l steps under policies πk 1 is the reward obtained in s by the actions according to πk 1 weighted by the probability that s is reached after l steps, Ri(s,l,πk 1) = ∑ θ∈Tl(s),θ=(σ1,...,σl) r(s,πk 1(θ l 1),i)·prob(θ,πk 1) . A POSG may behave differently under different policies. The quality of a policy is determined by its performance, i.e. by the sum of expected rewards received on it. We use |M | to denote the size of the representation of M .1 The short-term performance for policies πk 1 for agent i with POSG M is the expected sum of rewards received by agent i during the next |M | steps by following the policy πk 1, i.e. perfi(M ,πk 1) = ∑ s∈S Ri(s,|M |,πk 1) . The performance is also called the expected reward. Agents may cooperate or compete in a stochastic game. We want to know whether a stochastic game can be won by some agents. This is formally expressed in the following decision problems. The cooperative agents problem for k agents: instance: a POSG M for k agents query: are there policies π1,...,πk under which every agent has positive performance ? (I.e. ∃π1,...,πk : Vk i=1 perfi(M ,πk 1) > 0 ?) The competing agents problem for 2k agents: instance: a POSG M for 2k agents query: are there policies π1,...,πk under which all agents 1,2,...,k have positive performance independent of which policies agents k +1,k +2,...,2k choose? (I.e. ∃π1,...,πk∀πk+1,...,π2k : Vk i=1 perfi(M ,π2k 1 ) > 0 ?) It was shown by Bernstein et al. [1] that the cooperative agents problem for two or more agents is complete for NEXP. 1The size of the representation of M is the number of bits to encode the entire model, where the function t, o, and r are encoded by tables. We do not consider smaller representations. In fact, smaller representations may increase the complexity. 2 2.2 NEXPNP A Turing machine M has exponential running time, if there is a polynomial p such that for every input x, the machine M on input x halts after at most 2p(|x|) steps. NEXP is the class of sets that can be decided by a nondeterministic Turing machine within exponential time. NEXPNP is the class of sets that can be decided by a nondeterministic oracle Turing machine within exponential time, when a set in NP is used as an oracle. Similar as for the class NPNP, it turns out that a NEXPNP computation can be performed by an NEXP oracle machine that asks exactly one query to a co NP oracle and accepts if and only if the oracle accepts. 2.3 Domino tilings Domino tiling problems are useful for reductions between different kinds of computations. They have been proposed by Wang [3], and we will use it according to the following definition. Definition 2.1 We use [m] to denote the set {0,1,2,...,m −1}. A tile type T = (V,H) consists of two finite sets V,H ⊆N×N. A T-tiling of an m-square (m ∈N) is a mapping τ : [m]×[m] →N that satisfies both the following conditions. 1. Every pair of two neighboured tiles in the same row is in H. I.e. for all r ∈[m] and c ∈[m−1], (τ(r,c),τ(r,c+1)) ∈H. 2. Every pair of two neighboured tiles in the same column is in V. I.e. for all r ∈[m−1] and c ∈[m], (τ(r,c),τ(r +1,c)) ∈V. The exponential square tiling problem is the set of all pairs (T,1k), where T is a tile type and 1k is a string consisting of k 1s (k ∈N), such that there exists a T-tiling of the 2k-square. It was shown by Savelsbergh and van Emde Boas [4] that the exponential square tiling problem is complete for NEXP. We will consider the following variant, which we call the exponential Σ2 square tiling problem: given a pair (T,1k), does there exist a row w of tiles and a T-tiling of the 2k-square with final row w, such that there exists no T-tiling of the 2k-square with initial row w? The proof technique of Theorem 2.29 in [4], which translates Turing machine computations into tilings, is very robust in the sense that simple variants of the square tiling problem can analogously be shown to be complete for different complexity classes. Together with the above characterization of NEXPNP it can be used to prove the following. Theorem 2.2 The exponential Σ2 square tiling problem is complete for NEXPNP. 3 Results POSGs can be seen as a generalization of partially-observable Markov decision processes (POMDPs) in that POMDPs have only one agent and POSGs allow for many agents. Papadimitriou and Tsitsiklis [5] proved that it is PSPACE-complete to decide the cooperative agents problem for POMDPs. The result of Bernstein et al. [1] shows that in case of history-dependent policies, the complexity of POSGs is greater than the complexity of POMDPs. We show that this difference does not appear when stationary policies are considered instead of history-dependent policies. For POMDPs, the problem appears to be NP-complete [6]. A stationary policy is a mapping O →A from observations to actions. Whenever the same observation is made, the same action is chosen by a stationary policy. Theorem 3.1 For any k ≥2, the cooperative agents problem for k agents for stationary policies is NP-complete. Proof We start with proving NP-hardness. A POSG with only one agent is a POMDP. The problem of deciding, for a given POMDP M , whether there exists a stationary policy such that the short-term performance of M is greater than 0, is NP-complete [6]. Hence, the cooperative agents problem for stationary policies is NP-hard. 3 It remains to show containment in NP. Let M = (I,S,s0,A,O,t,o,r) be a POSG. We assume that t is represented in a straightforward way as a table. Let π1,...,πk be a sequence of stationary policies for the k agents. This sequence can be straightforwardly represented using not more space than the representation of t takes. Under a fixed sequence of policies, the performance of the POSG for all of the agents can be calculated in polynomial time. Using a guess and check approach (guess the stationary policies and evaluate the POSG), this shows that the cooperative agents problem for stationary policies is in NP. 2 In the same way we can characterize the complexity of a problem that we will need in the proof of Lemma 3.3. Corollary 3.2 The following problem is coNP-complete. instance: a POSG M for k agents query: do all agents under every stationary policy have positive performance? (I.e. ∀stationary π1 ...πk : Vk i=1 perfi(M ,πk 1) > 0 ?) The cooperative agents problem was shown to be NEXP-complete by Bernstein et al. [1]. Not surprisingly, if the agents compete, the problem becomes harder. Lemma 3.3 For every k ≥1, the competing agents problem for 2k agents is in NEXPNP. Proof The basic idea is as follows. We guess policies π1,π2,...,πk for agents 1,2,...,k, and construct a POSG that “implements” these policies and leaves open the actions chosen by agents k +1,...,2k. This new POSG has states for all short-term trajectories through the origin POSG. Therefore, its size is exponential in the size of the origin POSG. Because the history is stored in every state, and the POSG is loop-free, it turns out that the new POSG can be taken as a POMDP for which a (joint) policy with positive reward is searched. This problem is known to be NP-complete. Let M = (I,S,s0,A,O,t,o,r) be a POSG with 2k agents, and let π1,...,πk be short-term policies for M . We define a k-agent POSG M ′ = (I′,S′,s′ 0,A,O′,t′,o′,r′) as follows2. In M ′, we have as agents those of M , whose policies are not fixed, i.e. I′ = {k + 1,...,2k}. The set of states of M ′ is the cross product of states from M and all trajectories up to length |M | over S, i.e. S′ = S ×S≤|M |+1. The meaning of state (s,u) ∈S′ is, that state s can be reached on a trajectory u (that ends with s) through M with the fixed policies. The initial state s′ 0 is s′ 0 = (s0,s0). The state (s0,ε) is taken as a special sink state. After |M |+2 steps, the sink state is entered in M ′ and it is not left thereafter. All rewards gained in the sink state are 0. Now for the transition probabilities. If s is reached on trajectory u in M and the actions a1,...,ak are according to the fixed policies π1,...,πk, then the probabiliy of reaching state s′ on trajectory us′ according to t in M is the same as to reach (s′,us′) in M ′ from (s,u). In the formal description, the sink state has to be considered, too. t′((s,u),ak,...,a2k,(ˆs, ˆu)) = 0, if u ̸= ε and uˆs ̸= ˆu t(s,π1(o(us,1)),··· ,πk(o(us,k)),ak+1,...,a2k, ˆs), if ˆu = uˆs, | ˆu| ≤|M |, u ̸= ε 1, if |u| = |M |+1 or u = ε, and ˆu = ε The observation in M ′ is the sequence of observations made in the trajectory that is contained in each state, i.e. o′((s,w)) = o(w), where o(ε) is any element of O. Finally, the rewards. Essentially, we are interested in the rewards obtained by the agents 1,2,...,k. The rewards obtained by the other agents have no impact on this, only the actions the other agents choose. Therefore, agent i obtains the rewards in M ′ that are obtained by agent i−k in M . In this way, the agents k +1,...,2k obtain in M ′ the same rewards that are obtained by agents 1,2,...,k in M , and this is what we are interested in. This results in r′((s,u),ak,...,a2k,i) = r(s,π1(o(u,1)),··· ,πk(o(u,k)),ak+1,...,a2k,i −k) for i = k +1,...,2k. 2S≤|M | denotes the set of sequences up to |M | elements from S. The empty sequence is denoted by ε. For w ∈S≤|M | we use o(w,i) to describe the sequence of observations made by agent i on trajectory w. The concatenation of sequences u and w is denoted uw. We do not distinguish between elements of sets and sequences of one element. 4 Notice that the size of M ′ is exponential in the size of M . The sink state in M ′ is the only state that lies on a loop. This means, that on all trajectories through M ′, the sink state is the only state that may appear more than once. All states other than the sink state contain the full history of how they are reached. Therefore, there is a one-to-one correspondence between history-dependent policies for M and stationary policies for M ′ (with regard to horizon |M |). Moreover, the corresponding policies have the same performances. Claim 1 Let π1,...,π2k be short-term policies for M , and let ˆπk+1,..., ˆπ2k be their corresponding stationary policies for M ′. For |M | steps and i = 1,2,...,k, perfi(M ,π2k 1 ) = perfi+k(M ′, ˆπ2k k+1). Thus, this yields an NEXPNP algorithm to decide the competitive agents problem. The input is a POSG M for 2k agents. In the first step, the policies for the agents 1,2,...,k are guessed. This takes nondeterministic exponential time. In the second step, the POSG M ′ is constructed from the input M and the guessed policies. This takes exponential time (in the length of the input M ). Finally, the oracle is queried whether M ′ has positive performance for all agents under all stationary policies. This problem belongs to coNP (Corollary 3.2). Henceforth, the algorithm shows the competing agents problem to be in NEXPNP. 2 Lemma 3.4 For every k ≥2, the competing agents problem for 2k agents is hard for NEXPNP. Proof We give a reduction from the exponential Σ2 square tiling problem to the competing agents problem. Let T = (T,1k) be an instance of the exponential Σ2 square tiling problem, where T = (V,H) is a tile type. We will show how to construct a POSG M with 4 agents from it, such that T is a positive instance of the exponential Σ2 square tiling problem if and only if (1) agents 1 and 2 have a tiling for the 2k square with final row w such that (2) agents 3 and 4 have no tiling for the 2k square with initial row w. The basic idea for checking of tilings with POSGs for two agents stems from Bernstein et al. [1], but we give a slight simplification of their proof technique, and in fact have to extend it for four agents later on. The POSG is constructed so that on every trajectory each agent sees a position in the square. This position is chosen by the process. The only action of the agent that has impact on the process is putting a tile on the given position. In fact, the same position is observed by the agents in different states of the POSG. From a global point of view, the process splits into two parts. The first part checks whether both agents know the same tiling, without checking that it is a correct tiling. In the state where the agents are asked to put their tiles on the given position, a high negative reward is obtained if the agents put different tiles on that position. ”High negative” means that, if there is at least one trajectory on which such a reward is obtained, then the performance of the whole process will be negative. The second part checks whether the tiling is correct. The idea is to give both the agents neighboured positions in the square and to ask each which tile she puts on that position. Notice that the agents do not know in which part of the process they are. This means, that they do not know whether the other agent is asked for the same position, or for its upper or right neighbour. This is why the agents cannot cheat the process. A high negative reward will be obtained if the agents’ tiles do not fit together. For the first part, we need to construct is a POSG Pk for two agents, that allows both agents to make the same sequence of observations consisting of 2k bits. This sequence is randomly chosen, and encodes a position in a 2k ×2k grid. At the end, state same is reached, at which no observation is made. At this state, it will be checked whether both agents put the same tile at this position (see later on). The task of Pk is to provide both agents with the same position. Figure 1 shows an example for a 24 × 24-square. The initial state is s4. Dashed arrows indicate transitions with probability 1 2 independent of the actions. The observation of agent 1 is written on the left hand side of the states, and the observations of agent 2 at the right hand side. In s4, the agents make no observation. In Pk both agents always make the same observations. The second part is more involved. The goal is to provide both agents with neighboured positions in the square. Eventually, it is checked whether the tiles they put on the neighboured positions are according to the tile type T. Because the positions are encoded in binary, we can make use 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 s4 same Figure 1: P4 s hori 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 Figure 2: C3,4 row column check tiles s same 1 0 1 0 1 0 1 0 1 1 1 1 0 0 0 0 0 1 1 0 Figure 3: L3,4 of the following fact of subsequent binary numbers. Let u = u1...uk and w = w1 ...wk be bitwise representation of strings. if nw = nu + 1, then for some index l it holds that (1) ui = wi for i = 1,2,...,l −1, (2) wl = 1 and ul = 0, and (3) wj = 0 and u j = 1 for j = l +1,...,k. The POSG Cl,k is intended to provide the agents with two neighboured positions in the same row, where the index of the leftmost bit of the column encoding where both positions distinguish is l. (The C stands for column.) Figure 2 shows an example for the 24-square. The “final state” of Cl,k is the state hori, from which it is checked whether the agents put horizontically fitting tiles together. In the same way, a POSG Rl,k can be constructed (R stands for row), whose task is, to check whether two tiles in neighboured rows correspond to a correct tiling. This POSG has the final state vert, from which on it is checked whether two tiles fit vertically. Finally, we have to construct the last part of the POSG. It consists of the states same, hori, vert (as mentioned above), good, bad, and sink. All transitions between these states are deterministic (i.e. with probability 1). From state same the state good is reached, if both agents take the same action – otherwise bad is reached. From state hori the state good is reached, if action a1 by agent 1 and a2 by agent 2 make a pair (a1,a2) in H, i.e. in the set of horizontically correct pairs of tiles – otherwise bad is reached. Similarly, from state vert the state good is reached, if action a1 by agent 1 and a2 by agent 2 make a pair (a1,a2) in V. All these transitions are with reward 0. From state good the state sink is reached on every action with reward 1, and from state bad the state sink is reached on every action with reward −(22k+2). When the state sink is reached, the process stays there on any action, and all agents obtain reward 0. All rewards are the same for both agents. (This part can be seen in the overall picture in Figure 4). From these POSGs we construct a POSG T2,k that checks whether two agents know the same correct tiling for a 2k ×2k square, as described above. There are 2k+1 parts of T2,k. The initial state of each part can be reached with one step from the initial state s0 of T2,k. The parts of T2,k are as follows. • P2k with initial state s (checks whether two agents have the same tiling) • For each l = 1,2,...,k, we take Cl,k. Let cl be the initial state of Cl,k. 6 Pk C1,k Ck,k R1,k Rk,k s0 sk c1 ck r1 rk same hori vert good bad sink Figure 4: T2,k • For each l = 1,2,...,k, we take Rl,k. Let rl be the initial state of Rl,k. There are 22k + 2 · ∑k l=1 2k · 2l−1 =: tr(k) trajectories with probability > 0 through T2,k. Notice that tr(k) < 22k+2. From the initial state s0 of T2,k, each of the initial states of the parts is reachable independent on the action chosen by the agents. We will give transition probabilities to the transition from s0 to each of the initial states of the parts in a way, that eventually each trajectory has the same probability. t(s0,a1,a2,s′) = ( 22k tr(k), if s′ = s, i.e. the initial state of Pk 2k+l−1 tr(k) if s ∈{rl,cl | l = 1,2,...,k} In the initial state s0 and in the initial states of all parts, the observation ε is made. When a state same, hori, vert is reached, each agent has made 2k + 3 observations, where the first and last are ε and the remaining 2k are each in {0,1}. Such a state is the only one where the actions of the agents have impact on the process. Because of the partial observability, they cannot know in which part of T2,k they are. The agents can win, if they both know the same correct tiling and interpret the sequence of observations as the position in the grid they are asked to put a tile on. On the other hand, if both agents know different tilings or the tiling they share is not correct, then at least one trajectory will end in a bad state and has reward −(22k+2). The structure of the POSG is given in Figure 4. Claim 2 Let (T,1k) be an instance of the exponential square tiling problem. (1) There exists a polynomial time algorithm, that on input (T,1k) outputs T2,k. (2) There exists a T-tiling of the 2k square if and only if there exist policies for the agents under which T2,k has performance > 0. Part (1) is straightforward. Part (2) is not much harder. If there exists a T-tiling of the 2k square, both agents use the same policy according to this tiling. Under these policies, state bad will not be reached. This guarantess performance > 0 for both agents. For the other direction: if there exist policies for the agents under which T2,k has performance > 0, then state bad is not reached. Hence, both agents use the same policy. It can be shown inductively that this policy “is” a T-tiling of the 2k square. 7 The POSG for the competing agents problem with 4 agents consists of three parts. The first part is a copy of T2,k. It is used to check whether the first square can be tiled correctly (by agents 1 and 2). In this part, the negative rewards are increased in a way that guarantees the performance of the POSG to be negative whenever agents 1 and 2 do not correctly tile their square. The second part is a modified copy of T2,k. It is used to check whether the second square can be tiled correctly (by agents 3 and 4). Whenever state bad is left in this copy, reward 0 is obtained, and whenever state good is left, reward −1 is obtained. The third part checks whether agent 1 puts the same tiles into the last row of its square as agent 3 puts into the first row of its square. (See L3,4 in Figure 3 as an example.) If this succeeds, the performance of the third part equals 0, otherwise it has performance 1. These three parts run in parallel. If agents 1 and 2 have a tiling for the first square, the performance of the first part equals 1. • If agents 3 and 4 are able to continue this tiling through their square, the performance of the second part equals −1 and the performance of the third part equals 0. At all, the performance of the POSG under these policies equals 0. • If agents 3 and 4 are not able to continue this tiling through their square, then the performance of part 2 and part 3 is strictly greater −1. At all, the performance of the POSG under these policies is > 0. 2 Lemmas 3.3 and 3.4 together yield completeness of the competing agents problem. Theorem 3.5 For every k ≥2, the competing agents problem for 2k agents is complete for NEXPNP. 4 Conclusion We have shown that competition makes life—and computation—more complex. However, in order to do so, we needed teamwork. It is not yet clear what the complexity is of determining the existence of a good strategy for Player I in a 2-person POSG, or a 1-against-many POSG. There are other variations that can be shown to be complete for NEXPNP, a complexity class that, shockingly, has not been well explored. We look forward to further results about the complexity of POSGs, and to additional NEXPNP-completeness results for familiar AI and ML problems. References [1] Daniel S. Bernstein, Robert Givan, Neil Immerman, and Shlomo Zilberstein. The complexity of decentralized control of Markov decision processes. Math. Oper. Res., 27(4):819–840, 2002. [2] E. Hansen, D. Bernstein, and S. Zilberstein. Dynamic programming for partially observable stochastic games. In Proceedings of the Nineteenth National Conference on Artificial Intelligence (AAAI-04), pages 709–715, 2004. [3] Hao Wang. Proving theorems by pattern recognition II. Bell Systems Technical Journal, 40:1– 42, 1961. [4] M. Savelsbergh and P. van Emde Boas. Bounded tiling, an alternative to satisfiability. In Gerd Wechsung, editor, 2nd Frege Conference, volume 20 of Mathematische Forschung, pages 354– 363. Akademie Verlag, Berlin, 1984. [5] C.H. Papadimitriou and J.N. Tsitsiklis. The complexity of Markov decision processes. Mathematics of Operations Research, 12(3):441–450, 1987. [6] Martin Mundhenk, Judy Goldsmith, Christopher Lusena, and Eric Allender. Complexity results for finite-horizon Markov decision process problems. Journal of the ACM, 47(4):681–720,2000. 8
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Classification via Minimum Incremental Coding Length (MICL) John Wright∗, Yi Ma Coordinated Science Laboratory University of Illinois at Urbana-Champaign {jnwright,yima}@uiuc.edu Yangyu Tao, Zhouchen Lin, Heung-Yeung Shum Visual Computing Group Microsoft Research Asia {v-yatao,zhoulin,hshum}@microsoft.com Abstract We present a simple new criterion for classification, based on principles from lossy data compression. The criterion assigns a test sample to the class that uses the minimum number of additional bits to code the test sample, subject to an allowable distortion. We prove asymptotic optimality of this criterion for Gaussian data and analyze its relationships to classical classifiers. Theoretical results provide new insights into relationships among popular classifiers such as MAP and RDA, as well as unsupervised clustering methods based on lossy compression [13]. Minimizing the lossy coding length induces a regularization effect which stabilizes the (implicit) density estimate in a small-sample setting. Compression also provides a uniform means of handling classes of varying dimension. This simple classification criterion and its kernel and local versions perform competitively against existing classifiers on both synthetic examples and real imagery data such as handwritten digits and human faces, without requiring domain-specific information. 1 Introduction One quintessential problem in statistical learning [9, 20] is to construct a classifier from labeled training data (xi, yi) ∼iid pX,Y (x, y). Here, xi ∈Rn is the observation, and yi ∈{1, . . . , K} its associated class label. The goal is to construct a classifier g : Rn →{1, . . . , K} which minimizes the expected risk (or probability of error): g∗= arg min E[Ig(X)̸=Y ], where the expectation is taken with respect to pX,Y . When the conditional class distributions pX|Y (x|y) and the class priors pY (y) are given, the maximum a posterior (MAP) assignment ˆy(x) = arg miny∈{1,...,K} −ln pX|Y (x|y) −ln pY (y) (1) gives the optimal classifier. This amounts to a minimum coding length principle: the optimal classifier minimizes the Shannon optimal (lossless) coding length of the test data x with respect to the distribution of the true class. The first term is the number of bits needed to code x w.r.t. the distribution of class y, and the second term is the number of bits needed to code the label y for x. Issues with Learning the Distributions from Training Samples. In the typical classification setting, the distributions pX|Y (x|y) and pY (y) need to be learned from a set of labeled training ∗The authors gratefully acknowledge support from grants NSF Career IIS-0347456, NSF CRS-EHS0509151, NSF CCF-TF-0514955, and ONR YIP N00014-05-1-0633. 1 data. Conventional approaches to model estimation (implicitly) assume that the distributions are nondegenerate and the samples are sufficiently dense. However, these assumptions fail in many classification problems which are vital for applications in computer vision [10,11]. For instance, the set of images of a human face taken from different angles and under different lighting conditions often lie in a low-dimensional subspace or submanifold of the ambient space [2]. As a result, the associated distributions are degenerate or nearly degenerate. Moreover, due to the high dimensionality of imagery data, the set of training images is typically sparse. Inferring the generating probability distribution pX,Y from a sparse set of samples is an inherently ill-conditioned problem [20]. Furthermore, in the case of degenerate distributions, the classical likelihood function (1) does not have a well-defined maximum [20]. Thus, to infer the distribution from the training data or to use it to classify new observations, the distribution or its likelihood function needs to be properly “regularized.” Typically, this is accomplished either explicitly via smoothness constraints, or implicitly via parametric assumptions on the distribution [3]. However, even if the distributions are assumed to be generic Gaussians, explicit regularization is still necessary to achieve good small-sample performance [6]. In many real problems in computer vision, the distributions associated with different classes of data have different model complexity. For instance, when detecting a face in an image, features associated with the face often have a low-dimensional structure which is “embedded” as a submanifold in a cloud of essentially random features from the background. Model selection criteria such as minimum description length (MDL) [12, 16] serve as important modifications to MAP for model estimation across classes of different complexity. It selects the model that minimizes the overall coding length of the given (training) data, hence the name “minimum description length” [1]. Notice, however, that MDL does not specify how the model complexity should be properly accounted for when classifying new test data among models that have different dimensions. Solution from Lossy Data Coding. Given the difficulty of learning the (potentially degenerate) distributions pX|Y (x|y) from a few samples in a high-dimensional space, it makes more sense to seek good “surrogates” for implementing the minimum coding length principle (1). Our idea is to measure how efficiently a new observation can be encoded by each class of the training data subject to an allowable distortion, and to assign the new observation to the class that requires the minimum number of additional bits. We dub this criterion “minimum incremental coding length” (MICL) for classification. It provides a counterpart of the MDL principle for model estimation and as a surrogate for the minimum coding length principle for classification. The proposed MICL criterion naturally addresses the issues of regularization and model complexity. Regularization is introduced through the use of lossy coding, i.e. coding the test data x upto an allowable distortion1 (placing our approach along the lines of lossy MDL [15]). This contrasts with Shannon’s optimal lossless coding length, which requires precise knowledge of the true distributions. Lossy coding length also accounts for model complexity by directly measuring the difference in the volume (hence dimension) of the training data with and without the new observation. Relationships to Existing Classifiers. While MICL and MDL both minimize a coding-theoretic objective, MICL differs strongly from traditional MDL approaches to classification such as those proven inconsistent in [8]. Those methods chose a decision boundary that minimizes the total number of bits needed to code the boundary and the samples it incorrectly classifies. In contrast, MICL uses coding length directly as a measure of how well the training data represent the new sample. The inconsistency result of [8] does not apply in this modified context. Within the lossy data coding framework, we establish that the MICL criterion leads to a family of classifiers that generalize the conventional MAP classifier (1). We prove that for Gaussian distributions, the MICL criterion asymptotically converges to a regularized version of MAP2 (see Theorem 1) and give a precise estimate of the convergence rate (see Theorem 2). Thus, lossy coding induces a regularization effect similar to Regularized Discriminant Analysis (RDA) [6], with similar gains in finite sample performance with respect to MAP/QDA. The fully Bayesian approach to model estimation, in which posterior distributions over model parameters are estimated also provides finite sample gains over 1Information Bottleneck also uses lossy coding, but in an unsupervised manner, for clustering, feature selection and dimensionality reduction [19]. We apply lossy coding in the supervised (classification) setting. 2MAP subject to a Gaussian assumption is often referred to as Quadratic Discriminant Analysis (QDA) [9]. 2 ML/MAP [14]. However, that method is sensitive to the choice of prior when the number of samples is less than the dimension of the space, a situation that poses no difficulty to our proposed classifier. When the distributions involved are not Gaussian, the MICL criterion can still be applied locally, similar to the popular k-Nearest Neighbor (k-NN) classifier. However, the local MICL classifier significantly improves the k-NN classifier as it accounts for both the number of samples and the distribution of the samples within the neighborhood. MICL can also be kernelized to handle nonlinear/non-Gaussian data, an extension similar to the generalization of Support Vector Machines (SVM) to nonlinear decision boundaries. The kernelized version of MICL provides a simple alternative to the SVM approach of constructing a linear decision boundary in the embedded (kernel) space, and better exploits the covariance structure of the embedded data. 2 Classification Criterion and Analysis 2.1 Minimum Incremental Coding Length. A lossy coding scheme [5] maps vectors X = (x1, . . . , xm) ∈Rn×m to a sequence of binary bits, from which the original vectors can be recovered upto an allowable distortion E[∥ˆx −x∥2] ≤ε2. The length of the bit sequence is then a function Lε(X) : Rn×m →Z+. If we encode each class of training data Xj .= {xi : yi = j} separately using Lε(Xj) bits, the entire training dataset can be represented by a two-part code using PK j=1 Lε(Xj) −|Xj| log2 pY (j) bits. Here, the second term is the minimum number of bits needed to (losslessly) code the class labels yi. Now, suppose we are given a test observation x ∈Rn, whose associated class label y(x) = j is unknown. If we code x jointly with the training data Xj of the jth class, the number of additional bits needed to code the pair (x, y) is δLε(x, j) = Lε(Xj ∪{x})−Lε(Xj)+L(j). Here, the first two terms measure the excess bits needed to code (x, Xj) upto distortion ε2, while the last term L(j) is the cost of losslessly coding the label y(x) = j. One may view these as “finite-sample lossy” surrogates for the Shannon coding lengths in the ideal classifier (1). This interpretation naturally leads to the following classifier: Criterion 1 (Minimum Incremental Coding Length). Assign x to the class which minimizes the number of additional bits needed to code (x, ˆy), subject to the distortion ε: ˆy(x) .= arg minj∈{1,...,K} δLε(x, j). (2) The above criterion (2) can be taken as a general principle for classification, in the sense that it can be applied using any lossy coding scheme. Nevertheless, effective classification demands that the chosen coding scheme be approximately optimal for the given data. From a finite sample perspective, Lε should approximate the Kolmogorov complexity of X, while in an asymptotic, statistical setting it should approach the lower bound given by the rate-distortion of the generating distribution [5]. Lossy Coding of Gaussian Data. We will first consider a coding length function Lε introduced and rigorously justified in [13], which is (asymptotically) optimal for Gaussians. The (implicit) use of a coding scheme which is optimal for Gaussian sources is equivalent to assuming that the conditional class distributions pX|Y can be well-approximated by Gaussians. After rigorously analyzing this admittedly restrictive scenario, we will extend the MICL classifier (with this same Lε function) to arbitrary, multimodal distributions via an effective local Gaussian approximation. For a multivariate Gaussian source N(µ, Σ), the average number of bits needed to code a vector subject to a distortion ε2 is approximately Rε(Σ) .= 1 2 log2 det I+ n ε2 Σ (bits/vector). Observations X = (x1, . . . , xm) with sample mean ˆµ = 1 m P i xi and covariance ˆΣ(X) = 1 m−1 P i(xi − ˆµ)(xi −ˆµ)T can be represented upto expected distortion ε2 using ≈mRε(ˆΣ) bits. The optimal codebook is adaptive to the data, and can be encoded by representing the principal axes of the covariance using an additional nRε(ˆΣ) bits. Encoding the mean vector µ requires an additional n 2 log2 1 + ˆµT ˆµ ε2 bits. The total number of bits required to code X is therefore Lε(X) .= m + n 2 log2 det I + n ε2 ˆΣ(X) + n 2 log2 1 + ˆµT ˆµ ε2 . (3) 3 MICL k-NN SVM-RBF Figure 1: MICL harnesses linear structure in the data to interpolate (left) and extrapolate (center) in sparsely sampled regions. Popular classifiers such as k-NN and SVM-RBF do not (right). The first term gives the number of bits needed to represent the distribution of the xi about their mean, and the second gives the cost of representing the mean. The above function well-approximates the optimal coding length for Gaussian data, and has also been shown to give a good upper bound on the number of bits needed to code finitely many samples lying on a linear subspace (e.g., a degenerate Gaussian distribution) [13]. Coding the Class Label. Since the label Y is discrete, it can be coded losslessly. If the test class labels Y are known to have the marginal distribution P[Y = j] = πj, then the optimal coding lengths are (within one bit): L(j) = −log2 πj. In practice, we may replace πj with the estimate ˆπj = |Xj|/m. Notice that as in the MAP classifier, the πj essentially form a prior on class labels. Combining this coding length the class label with the coding length function (3) for the observations, we summarize the MICL criterion (2) as Algorithm 1 below: Algorithm 1 (MICL Classifier). 1: Input: m training samples partitioned into K classes X1, X2, . . . , XK and a test sample x. 2: Compute prior distribution of class labels ˆπj = |Xj|/m. 3: Compute incremental coding length of x for each class: δLε(x, j) = Lε(Xj ∪{x}) −Lε(Xj) −log2 ˆπj, where Lε(X) .= m+n 2 log2 det I + n ε2ˆΣ(X) + n 2 log2 1 + ˆµT ˆµ ε2 . 4: Output: ˆy(x) = arg minj=1,...,K δLε(x, j). The Lε(Xj ∪{x}) can be computed in O(min(m, n)2) time (see [21]), allowing the MICL classifier to be directly applied to high-dimensional data. Figure 1 shows the performance of Algorithm 1 on two toy problems. In both cases, the MICL criterion harnesses the covariance structure of the data to achieve good classification in sparsely sampled regions. In the left example, the criterion interpolates the data structure to achieve correct classification, even near the origin where the samples are sparse. In the right example, the criterion extrapolates the horizontal line to the other side of the plane. Methods such as k-NN and SVM do not achieve the same effect. Notice, however, that these decision boundaries are similar to what MAP/QDA would give. This raises an important question: what is the precise relationship between MICL and MAP, and when is MICL superior? 2.2 Asymptotic Behavior and Relationship to MAP In this section, we analyze the asymptotic behavior of Algorithm 1 as the number of training samples goes to infinity. The following result, whose proof is given in [21], indicates that MICL converges to a regularized version of ML/MAP, subject to a reward on the dimension of the classes: Theorem 1 (Asymptotic MICL [21]). Let the training samples {(xi, yi)}m i=1 ∼iid pX,Y (x, y), with µj .= E[X|Y = j], Σj .= Cov(X|Y = j). Then as m →∞, the MICL criterion coincides (asymptotically, with probability one) with the decision rule ˆy(x) = argmax j=1,...,K LG x µj, Σj + ε2 n I + ln πj + 1 2Dε(Σj), (4) where LG(·| µ, Σ) is the log-likelihood function for a N(µ, Σ) distribution , and Dε(Σj) .= tr(Σj(Σj + ε2 n I)−1) is the effective dimension of the j-th model, relative to the distortion ε2. 4 −2 −1 0 1 2 3 15 27 39 51 63 75 log ε Number of training samples RMAP − RMICL −0.04 −0.02 0 0.02 0.04 0.06 −2 −1 0 1 3 15 27 39 51 63 log ε Number of training samples RMAP − RMICL 0 0.02 0.04 0.06 10 22 34 10 20 30 40 50 Ambient dimension Number of training samples RMAP − RMICL 0 0.2 0.4 0.6 0.8 10 22 34 10 20 30 40 50 Ambient dimension Number of training samples RRDA − RMICL −0.05 0 0.05 0.1 0.15 0.2 Figure 2: Left: Excess risk incurred by using MAP rather than MICL, as a function of ε and m. (a) isotropic Gaussians. (b) anisotropic Gaussians. Right: Excess risk for nested classes, as a function of n and m. (c) MICL vs. MAP. (d) MICL vs. RDA. In all examples, MICL is superior for n ≫m. This result shows that asymptotically, MICL generates a family of MAP-like classifiers parametrized by the distortion ε2. If all of the distributions are nondegenerate (i.e. their covariance matrices Σj are nonsingular), then limε→0(Σj + ε2 n I) = Σj and limε→0 Dε(Σj) = n, a constant across the various classes. Thus, for nondegenerate data, the family of classifiers induced by MICL contains the conventional MAP classifier (1) at ε = 0. Given a finite number, m, of samples, any reasonable rule for choosing the distortion ε2 should therefore ensure that ε →0 as m →∞. This guarantees that for non-degenerate distributions, MICL converges to the asymptotically optimal MAP criterion. Simulations (e.g., Figure 1) suggest that the limiting behavior provides useful information even for finite training data. The following result, proven in [21], verifies that the MICL discriminant functions δLε(x, j) converge quickly to their limiting form δL∞ ε (x, j): Theorem 2 (MICL Convergence Rate [21]). As the number of samples, m →∞, the MICL criterion (2) converges to its asymptotic form, (4) at a rate of m−1 2 . More specifically, with probability at least 1 −α, δLε(z, j) −δL∞ ε (z, j) ≤c(α) · m−1 2 for some constant c(α) > 0. 2.3 Improvements over MAP In the above, we have established the fact that asymptotically, the MICL criterion (4) is just as good as the MAP criterion. Nevertheless, the MICL criterion makes several important modifications to MAP, which significantly improve its performance on sparsely sampled or degenerate data. Regularization and Finite-Sample Behavior. Notice that the first two terms of the asymptotic MICL criterion (4) have the form of a MAP criterion, based on an N(µ, Σ + ε2 n I) distribution. This is somewhat equivalent to softening the distribution by ε2 n along each dimension, and has two important effects. First, it renders the associated MAP decision rule well-defined, even if the true data distribution is (almost) degenerate. Even for non-degenerate distributions, there is empirical evidence that for appropriately chosen ε, ˆΣ + ε2 n I gives more stable finite-sample classification [6]. Figure 2 demonstrates this effect on two simple examples. The generating distributions are parameterized as (a) µ1 = [−1 2, 0], µ2 = [ 1 2, 0], Σ1 = Σ2 = I, and (b) µ1 = [−3 4, 0], µ2 = [ 3 4, 0], Σ1 = diag(1, 4), Σ2 = diag(4, 1). In each example, we vary the number of training samples, m, and the distortion ε. For each (m, ε) combination, we draw m training samples from two Gaussian distributions N(µi, Σi), i = 1, 2, and estimate the Bayes risk of the resulting MICL and MAP classifiers. This procedure is repeated 500 times, to estimate the overall Bayes risk with respect to variations in the training data. Figure 2 visualizes the difference in risks, RMAP −RMICL. Positive values indicate that MICL is outperforming MAP. The red line approximates the zero level-set, where the two methods perform equally well. In the isotropic case (a), MICL outperforms MAP for all sufficiently large ε. with a larger performance gain when the number of samples is small. In the anisotropic case (b), for most ε, MICL dramatically outperforms MAP for small sample sizes. We will see in the next example that this effect becomes more pronounced as the dimension increases. Dimension Reward. The effective dimension term Dε(Σj) in the large-n MICL criterion (4) can be rewritten as Dε(Σj) = Pn i=1 λi/( ε2 n + λi), where λi is the ith eigenvalue of Σj. If the data lie near a d-dimensional subspace (λ1 . . . λd ≫ε2/n and λd+1 . . . λn ≪ε2/n), Dε ≈d. In general, 5 Dε can be viewed as “softened” estimate of the dimension3, relative to the distortion ε2. MICL therefore rewards distributions that have relatively higher dimension.4 However, this effect is somewhat countered by the regularization induced by ε, which rewards lower dimensional distributions. Figure 2(right) empirically compares MICL to the conventional MAP and the regularized MAP (or RDA [6]). We draw m samples from three nested Gaussian distributions: one of full rank n, one of rank n/2, and one of rank 1. All samples are corrupted by 4% Gaussian noise. We estimate the Bayes risk for each (m, n) combination as in the previous example. The regularization parameter in RDA and the distortion ε for MICL are chosen independently for each trial by cross validation. Plotted are the (estimated) differences in risk, RMAP −RMICL (Fig. 2 (c)) and RRDA −RMICL (Fig. 2 (d)). The red lines again correspond to the zero level-set of the difference. Unsurprisingly, MICL outperforms MAP for most (m, n), and that the effect is most pronounced when n is large and m is small. When m is much smaller than n (e.g. the bottom row of Figure 2 right), MICL demonstrates a significant performance gain with respect to RDA. As the number of samples increases, there is a region where RDA is slightly better. For most (m, n), MICL and RDA are close in performance. 2.4 Extensions to Non-Gaussian Data In practice, the data distribution(s) of interest may not be Gaussian. If the rate-distortion function is known, one could, in principle, carry out similar analysis as for the Gaussian case. Nevertheless, in this subsection, we discuss two practical modifications to the MICL criterion that are applicable to arbitrary distributions and preserve the desirable properties discussed in the previous subsections. Kernel MICL Criterion. Since XX T and X T X have the same non-zero eigenvalues, log2 det I+α XX T = log2 det I+α X T X . (5) This identity shows that Lε(X) can also be computed from the inner products between the xi. If the data x (of each class) are not Gaussian but there exists a nonlinear map ψ : Rn →H such that the transformed data ψ(x) are (approximately) Gaussian, we can replace the inner product xT 1 x2 with a symmetric positive definite kernel function k(x1, x2) .= ψ(x1)T ψ(x2). Choosing a proper kernel function will improve classification performance for non-Gaussian distributions. In practice, popular choices include the polynomial kernel k(x1, x2) = (xT 1 x2 + 1)d, the radial basis function (RBF) kernel k(x1, x2) = exp(−γ∥x1 −x2∥2) and their variants. Implementation details, including how to properly account for the mean and dimension of the embedded data, are given in [21]. A similar transformation is used to generate nonlinear decision boundaries with SVM. Notice, however, that whereas SVM constructs a linear decision boundary in the lifted space H, kernel MICL exploits the covariance structure of the lifted data, generating decision boundaries that are (asymptotically) quadratic. In Section 3 we will see that even for real data whose statistical nature is unclear, kernel MICL outperforms SVM when applied with the same kernel function. Local MICL Criterion. For real data whose distribution is unknown, it may be difficult to find an appropriate kernel function. In this case, MICL can still be applied locally, in a neighborhood of the test sample x. Let N k(x) denote the k nearest neighbors of x in the training set X. Training data in this neighborhood that belong to each class are N k j (x) .= Xj ∩N k(x), j = 1, . . . , K. In the MICL classifier (Algorithm 1), we replace the incremental coding length δLε(x, j) by its local version: δLε(x, j) = Lε(N k j (x) ∪{x}) −Lε(N k j (x)) + L(j), (6) with L(j) = −log2(|N k j (x)|/|N k(x)|). Theorem 1 implies that this gives a universal classifier: Corollary 3. Suppose the conditional density pj(x) = p(x|y = j) of each class is nondegenerate. Then if k = o(m) and k, m →∞, the local MICL criterion converges to the MAP criterion (1). This follows, since as the radius of the neighborhood shrinks, the cost of coding the class label, −log2(|N k j (x)|/|N k(x)|) →−log2 pj(x), dominates the coding length, (6). In this asymptotic setting the local MICL criterion behaves like k-Nearest Neighbor (k-NN). However, the finitesample behavior of the local MICL criterion can differ drastically from that of k-NN, especially 3This quantity has been dubbed the effective number of parameters in the context of ridge regression [9]. 4This contrasts with the dimension penalties typical in model selection/estimation. 6 (a) KMICL-RBF (b) SVM-RBF (c) LMICL (d) 5-NN Figure 3: Nonlinear extensions to MICL, compared to SVM and k-NN. Local MICL produces a smoother and more intuitive decision boundary than k-NN. Kernel MICL and SVM produce similar boundaries, that are smoother and better respect the data structure than those given by local methods. Method Error Method Error LMICL 1.6% SVM-Poly [20] 1.4% k-NN 3.1% Best [18] 0.4% Method Error Method Error LMICL 4.9% KMICL-Poly 4.7% k-NN 5.3% SVM-Poly [4] 5.3% Table 1: Results for handwritten digit recognition. Left: MNIST dataset. Right: USPS dataset, with identical preprocessing and kernel function. Here, kernel-MICL slightly outperforms SVM. when the samples are sparse and the distributions involved are almost degenerate. In this case, from (4), local MICL effectively approximates the local shape of the distribution pj(x) by a (regularized) Gaussian, exploiting structure in the distribution of the nearest neighbors (see figure 3). 3 Experiments with Real Imagery Data Using experiments on real data, we demonstrate that MICL and its nonlinear variants approach the best results from more sophisticated systems, without relying on domain-specific information. Handwritten Digit Recognition. We first test the MICL classifier on two standard datasets for handwritten digit recognition (Table 1 top). The MNIST handwritten digit dataset [10] consists of 60,000 training images and 10,000 test images. We achieved better results using the local version of MICL, due to non-Gaussian distribution of the data. With k = 20 and ε = 150, local MICL achieves a test error 1.59%, outperforming simple methods such as k-NN as well as many more complicated neural network approaches (e.g. LeNet-1 [10]). MICL’s error rate approaches the best result for a generic learning machine (1.4% error for SVM with a degree-4 polynomial kernel). Problem-specific approaches have resulted in lower error rates, however, with the best reported result achieved using a specially engineered neural network [18]. We also test on the challenging USPS digits database (Table 1 bottom). Here, even humans have considerable difficulties (≈2.5% error). With k = 35 and ε = 0.03, local MICL achieves an error rate of 4.88%, again outperforming k-NN. We further compare the performance of kernel MICL to SVM (using [4]) on this dataset with the same homogeneous, degree 3 polynomial kernel, and identical preprocessing (normalization and centering), allowing us to compare pure classification performace. Here, SVM achieves a 5.3% error, while kernel-MICL achieves an error rate of 4.7% with distortion ε = 0.0067 (chosen automatically by cross-validation). Using domain-specific information, one can achieve better results. For instance [17] achieves 2.7% error using tangent distance to a large number of prototypes. Other preprocessing steps, synthetic training images, or more advanced skew-correction and normalization techniques have been applied to lower the error rate for SVM (e.g. 4.1% in [20]). While we have avoided extensive preprocessing here, so as to isolate the effect of the classifier, such preprocessing can be readily incorporated into our framework. Face Recognition. We further verify MICL’s effectiveness on sparsely sampled high-dimensional data using the Yale Face Database B [7], which tests illumination sensitivity of face recognition algorithms. Following [7,11], we use subsets 1 and 2 for training, and report the average test error across the four subsets. We apply Algorithm 1, not the local or kernel version, with ε = 75. MICL significantly outperforms classical subspace techniques on this problem (see Table 2), with error 0.9% near the best reported results in [7, 11] that were obtained using a domain-specific model of 7 Method Error Method Error MICL 0.9% Eigenface [7] 25.8% Subspace [7] 4.6% Best [11] 0% Subsets 1,2 (training) Subsets 1-4 (testing) Table 2: Face recognition under widely varying illumination. MICL outperforms classical face recognition methods such as Eigenfaces on Yale Face Database B [7]. illumination for face images. We suggest that the source of this improved performance is precisely the regularization induced by lossy coding. In this problem the number of training vectors per class, 19, is small compared to the dimension, n = 32, 256 (for raw 168 × 192 images). Simulations (e.g. Figure 2) show that this is exactly the circumstance in which MICL is superior to MAP and even RDA. Interestingly, this suggests that directly exploiting degenerate or low-dimensional structures via MICL renders dimensionality reduction before classifying unnecessary or even undesirable. 4 Conclusion We have proposed and studied a new information theoretic classification criterion, Minimum Incremental Coding Length (MICL), establishing its optimality for Gaussian data. MICL generates a family of classifiers that inherit many of the good properties of MAP, RDA, and k-NN, while extending their working conditions to sparsely sampled or degenerate high-dimensional observations. MICL and its kernel and local versions approach best reported performance on high-dimensional visual recognition problems without domain-specific engineering. Due to its simplicity and flexibility, we believe MICL can be successfully applied to a wide range of real-world classification problems. References [1] A. Barron, J. Rissanen, and B. Yu. The minimum description length principle in coding and modeling. IEEE Transactions on Information Theory, 44(6):2743–2760, 1998. [2] R. Basri and D. Jacobs. Lambertian reflection and linear subspaces. PAMI, 25(2):218– 233, 2003. [3] P. Bickel and B. Li. Regularization in statistics. TEST, 15(2):271–344, 2006. [4] C. Chang and C. Lin. LIBSVM: a library for support vector machines, 2001. [5] T. Cover and J. Thomas. Elements of Information Theory. Wiley Series in Telecommunications, 1991. [6] J. Friedman. Regularized discriminant analysis. JASA, 84:165–175, 1989. [7] A. Georghiades, P. Belhumeur, and D. Kriegman. From few to many: Illumination cone models for face recognition under variable lighting and pose. PAMI, 23(6):643–660, 2001. [8] P. Grunwald and J. Langford. Suboptimal behaviour of Bayes and MDL in classification under misspecification. In Proceedings of Conference on Learning Theory, 2004. [9] T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning. Springer, 2001. [10] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998. [11] K. Lee, J. Ho, and D. Kriegman. Acquiring linear subspaces for face recognition under variable lighting. PAMI, 27(5):684–698, 2005. [12] J. Li. A source coding approach to classification by vector quantization and the principle of minimum description length. In IEEE DCC, pages 382–391, 2002. [13] Y. Ma, H. Derksen, W. Hong, and J. Wright. Segmentation of multivariate mixed data via lossy data coding and compression. PAMI, 29(9):1546–1562, 2007. [14] D. MacKay. Developments in probabilistic modelling with neural networks – ensemble learning. In Proc. 3rd Annual Symposium on Neural Networks, pages 191–198, 1995. [15] M. Madiman, M. Harrison, and I. Kontoyiannis. Minimum description length vs. maximum likelihood in lossy data compression. In IEEE International Symposium on Information Theory, 2004. [16] J. Rissanen. Modeling by shortest data description. Automatica, 14:465–471, 1978. [17] P. Simard, Y. LeCun, and J. Denker. Efficient pattern recognition using a new transformation distance. In Proceedings of NIPS, volume 5, 1993. [18] P. Simard, D. Steinkraus, and J. Platt. Best practice for convolutional neural networks applied to visual document analysis. In ICDAR, pages 958–962, 2003. [19] N. Tishby, F. Pereira, and W. Bialek. The information bottleneck method. In Allerton, 1999. [20] V. Vapnik. The Nature of Statistical Learning Theory. Springer, 2000. [21] J. Wright, Y. Tao, Z. Lin, Y. Ma, and H. Shum. Classification via minimum incremental coding length (MICL). Technical report, UILU-ENG-07-2201, http://perception.csl.uiuc.edu/coding, 2007. 8
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Kernel Measures of Conditional Dependence Kenji Fukumizu Institute of Statistical Mathematics 4-6-7 Minami-Azabu, Minato-ku Tokyo 106-8569 Japan fukumizu@ism.ac.jp Arthur Gretton Max-Planck Institute for Biological Cybernetics Spemannstraße 38, 72076 T¨ubingen, Germany arthur.gretton@tuebingen.mpg.de Xiaohai Sun Max-Planck Institute for Biological Cybernetics Spemannstraße 38, 72076 T¨ubingen, Germany xiaohi@tuebingen.mpg.de Bernhard Sch¨olkopf Max-Planck Institute for Biological Cybernetics Spemannstraße 38, 72076 T¨ubingen, Germany bernhard.schoelkopf@tuebingen.mpg.de Abstract We propose a new measure of conditional dependence of random variables, based on normalized cross-covariance operators on reproducing kernel Hilbert spaces. Unlike previous kernel dependence measures, the proposed criterion does not depend on the choice of kernel in the limit of infinite data, for a wide class of kernels. At the same time, it has a straightforward empirical estimate with good convergence behaviour. We discuss the theoretical properties of the measure, and demonstrate its application in experiments. 1 Introduction Measuring dependence of random variables is one of the main concerns of statistical inference. A typical example is the inference of a graphical model, which expresses the relations among variables in terms of independence and conditional independence. Independent component analysis employs a measure of independence as the objective function, and feature selection in supervised learning looks for a set of features on which the response variable most depends. Kernel methods have been successfully used for capturing (conditional) dependence of variables [1, 5, 8, 9, 16]. With the ability to represent high order moments, mapping of variables into reproducing kernel Hilbert spaces (RKHSs) allows us to infer properties of the distributions, such as independence and homogeneity [7]. A drawback of previous kernel dependence measures, however, is that their value depends not only on the distribution of the variables, but also on the kernel, in contrast to measures such as mutual information. In this paper, we propose to use the Hilbert-Schmidt norm of the normalized conditional crosscovariance operator, and show that this operator encodes the dependence structure of random variables. Our criterion includes a measure of unconditional dependence as a special case. We prove in the limit of infinite data, under assumptions on the richness of the RKHS, that this measure has an explicit integral expression which depends only on the probability densities of the variables, despite being defined in terms of kernels. We also prove that its empirical estimate converges to the kernelindependent value as the sample size increases. Furthermore, we provide a general formulation for 1 the “richness” of an RKHS, and a theoretically motivated kernel selection method. We successfully apply our measure in experiments on synthetic and real data. 2 Measuring conditional dependence with kernels The probability law of a random variable X is denoted by PX, and the space of the square integrable functions with probability P by L2(P). The symbol X⊥⊥Y | Z indicates the conditional independence of X and Y given Z. The null space and the range of an operator T are written N(T) and R(T), respectively. 2.1 Dependence measures with normalized cross-covariance operators Covariance operators on RKHSs have been successfully used for capturing dependence and conditional dependence of random variables, by incorporating high order moments [5, 8, 16]. We give a brief review here; see [5, 6, 2] for further detail. Suppose we have a random variable (X, Y ) on X ×Y, and RKHSs HX and HY on X and Y, respectively, with measurable positive definite kernels kX and kY. Throughout this paper, we assume the integrability (A-1) E[kX (X, X)] < ∞, E[kY(Y, Y )] < ∞. This assumption ensures HX ⊂L2(PX) and HY ⊂L2(PY ). The cross-covariance operator ΣY X : HX →HY is defined by the unique bounded operator that satisfies ⟨g, ΣY Xf⟩HY = Cov[f(X), g(Y )] ( = E[f(X)g(Y )] −E[f(X)]E[g(Y )]) (1) for all f ∈HX and g ∈HY. If Y = X, ΣXX is called the covariance operator, which is self-adjoint and positive. The operator ΣY X naturally extends the covariance matrix CY X on Euclidean spaces, and represents higher order correlations of X and Y through f(X) and g(Y ) with nonlinear kernels. It is known [2] that the cross-covariance operator can be decomposed into the covariance of the marginals and the correlation; that is, there exists a unique bounded operator VY X such that ΣY X = Σ1/2 Y Y VY XΣ1/2 XX, (2) R(VY X) ⊂R(ΣY Y ), and N(VY X)⊥⊂R(ΣXX). The operator norm of VY X is less than or equal to 1. We call VY X the normalized cross-covariance operator (NOCCO, see also [4]). While the operator VY X encodes the same information regarding the dependence of X and Y as ΣY X, the former rather expresses the information more directly than ΣY X, with less influence of the marginals. This relation can be understood as an analogue to the difference between the covariance Cov[X, Y ] and the correlation Cov[X, Y ]/(Var(X)Var(Y ))1/2. Note also that kernel canonical correlation analysis [1] uses the largest eigenvalue of VY X and its corresponding eigenfunctions [4]. Suppose we have another random variable Z on Z and RKHS (HZ, kZ), which satisfy the analog to (A-1). We then define the normalized conditional cross-covariance operator, VY X|Z = VY X −VY ZVZX, (3) for measuring the conditional dependence of X and Y given Z, where VY Z and VZX are defined similarly to Eq. (2). The operator VY X|Z may be better understood by expressing it as VY X|Z = Σ−1/2 Y Y ΣY X −ΣY ZΣ−1 ZZΣZX Σ−1/2 XX , where ΣY X|Z = ΣY X −ΣY ZΣ−1 ZZΣZX can be interpreted as a nonlinear extension of the conditional covariance matrix CY X −CY ZC−1 ZZCZX of Gaussian random variables. The operator ΣY X can be used to determine the independence of X and Y : roughly speaking, ΣY X = O if and only if X⊥⊥Y . Similarly, a relation between ΣY X|Z and conditional independence, X⊥⊥Y | Z, has been established in [5]: if the extended variables ¨X = (X, Z) and ¨Y = (Y, Z) are used, X⊥⊥Y | Z is equivalent to Σ ¨ X ¨Y |Z = O. We will give a rigorous treatment in Section 2.2 Noting that the conditions ΣY X = O and ΣY X|Z = O are equivalent to VY X = O and VY X|Z = O, respectively, we propose to use the Hilbert-Schmidt norms of the latter operators as dependence 2 measures. Recall that an operator A : H1 →H2 is called Hilbert-Schmidt if for complete orthonormal systems (CONSs) {φi} of H1 and {ψj} of H2, the sum P i,j⟨ψj, Aφi⟩2 H2 is finite (see [13]). For a Hilbert-Schmidt operator A, the Hilbert-Schmidt (HS) norm ∥A∥HS is defined by ∥A∥2 HS = P i,j⟨ψj, Aφi⟩2 H2. It is easy to see that this sum is independent of the choice of CONSs. Provided that VY X and VY X|Z are Hilbert-Schmidt, we propose the following measures: ICOND(X, Y |Z) = ∥V ¨Y ¨ X|Z∥2 HS, (4) INOCCO(X, Y ) = ∥VY X∥2 HS. (5) A sufficient condition that these operators are Hilbert-Schmidt will be discussed in Section 2.3. It is easy to provide empirical estimates of the measures. Let (X1, Y1, Z1), . . . , (Xn, Yn, Zn) be an i.i.d. sample from the joint distribution. Using the empirical mean elements bm(n) X = 1 n Pn i=1 kX ( · , Xi) and bm(n) Y = 1 n Pn i=1 kY( · , Yi), an estimator of ΣY X is bΣ(n) Y X = 1 n Pn i=1(kY( · , Yi) −bm(n) Y ) kX ( · , Xi) −bm(n) X , · HX . bΣ(n) XX and bΣ(n) Y Y are defined similarly. The estimators of VY X and VY X|Z are respectively bV (n) Y X = bΣ(n) Y Y + εnI −1/2bΣ(n) Y X bΣ(n) XX + εnI −1/2, where εn > 0 is a regularization constant used in the same way as [1, 5], and bV (n) Y X|Z = bV (n) Y X −bV (n) Y Z bV (n) ZX, (6) from Eq. (3). The HS norm of the finite rank operator bV (n) Y X|Z is easy to calculate. Let GX, GY , and GZ be the centered Gram matrices, such that GX,ij = ⟨kX ( · , Xi) −bm(n) X , kX ( · , Xj) −bm(n) X ⟩HX and so on, and define RX, RY , and RZ as RX = GX(GX +nεnIn)−1, RY = GY (GY +nεnIn)−1, and RZ = GZ(GZ + nεnIn)−1. The empirical dependence measures are then ˆICOND n ≡
bV (n) ¨Y ¨ X|Z
2 HS = Tr R ¨Y R ¨ X −2R ¨Y R ¨ XRZ + R ¨Y RZR ¨ XRZ , (7) ˆINOCCO n (X, Y ) ≡
bV (n) Y X
2 HS = Tr RY RX , (8) where the extended variables are used for ˆICOND n . These empirical estimators, and use of εn, will be justified in Section 2.4 by showing the convergence to INOCCO and ICOND. With the incomplete Cholesky decomposition [17] of rank r, the complexity to compute ˆICOND n is O(r2n). 2.2 Inference on probabilities by characteristic kernels To relate INOCCO and ICOND with independence and conditional independence, respectively, the RKHS should contain a sufficiently rich class of functions to represent all higher order moments. Similar notions have already appeared in the literature: universal kernel on compact domains [15] and Gaussian kernels on the entire Rm characterize independence via the cross-covariance operator [8, 1]. We now discuss a unified class of kernels for inference on probabilities. Let (X, B) be a measurable space, X a random variable on X, and (H, k) an RKHS on X satisfying assumption (A-1). The mean element of X on H is defined by the unique element mX ∈H such that ⟨mX, f⟩H = E[f(X)] for all f ∈H (see [7]). If the distribution of X is P, we also use mP to denote mX. Letting P be the family of all probabilities on (X, B), we define the map Mk by Mk : P →H, P 7→mP . The kernel k is said to be characteristic1 if the map Mk is injective, or equivalently, if the condition EX∼P [f(X)] = EX∼Q[f(X)] (∀f ∈H) implies P = Q. The notion of a characteristic kernel is an analogy to the characteristic function EP [e √−1uT X], which is the expectation of the Fourier kernel kF (x, u) = e √−1uT x. Noting that mP = mQ iff EP [k(u, X)] = EQ[k(u, X)] for all u ∈X, the definition of a characteristic kernel generalizes the well-known property of the characteristic function that EP [kF (u, X)] uniquely determines a Borel probability P on Rm. The next lemma is useful to show that a kernel is characteristic. 1Although the same notion was called probability-determining in [5], we call it ”characteristic” by analogy with the characteristic function. 3 Lemma 1. Let q ≥1. Suppose that (H, k) is an RKHS on a measurable space (X, B) with k measurable and bounded. If H + R (the direct sum of the two RKHSs) is dense in Lq(X, P) for any probability P on (X, B), the kernel k is characteristic. Proof. Assume mP = mQ. By the assumption, for any ε > 0 and a measurable set A, there is a function f ∈H and c ∈R such that |EP [f(X)] + c −P(A)| < ε and |EQ[f(Y )] + c −Q(A)| < ε, from which we have |P(A) −Q(A)| < 2ε. Since ε > 0 is arbitrary, this means P(A) = Q(A). Many popular kernels are characteristic. For a compact metric space, it is easy to see that the RKHS given by a universal kernel [15] is dense in L2(P) for any P, and thus characteristic (see also [7] Theorem 3). It is also important to consider kernels on non-compact spaces, since many standard random variables, such as Gaussian variables, are defined on non-compact spaces. The next theorem implies that many kernels on the entire Rm, including Gaussian and Laplacian, are characteristic. The proof is an extension of Theorem 2 in [1], and is given in the supplementary material. Theorem 2. Let φ(z) be a continuous positive function on Rm with the Fourier transform ˜φ(u), and k be a kernel of the form k(x, y) = φ(x −y). If for any ξ ∈Rm there exists τ0 such that R ˜φ(τ(u+ξ))2 ˜φ(u) du < ∞for all τ > τ0, then the RKHS associated with k is dense in L2(P) for any Borel probability P on Rm. Hence k is characteristic with respect to the Borel σ-field. The assumptions to relate the operators with independence are well described by using characteristic kernels and denseness. The next result generalizes Corollary 9 in [5] (we omit the proof: see [5, 6]). Theorem 3. (i) Assume (A-1) for the kernels. If the product kX kY is characteristic, then we have VY X = O ⇐⇒ X⊥⊥Y. (ii) Denote ¨X = (X, Z) and k ¨ X = kX kZ. In addition to (A-1), assume that the product k ¨ X kY is a characteristic kernel on (X × Z) × Y, and HZ + R is dense in L2(PZ). Then, VY ¨ X|Z = O ⇐⇒ X⊥⊥Y | Z. From the above results, we can guarantee that VY X and VY ¨ X|Z will detect independence and conditional independence, if we use a Gaussian or Laplacian kernel either on a compact set or the whole of Rm. Note also that we can substitute V ¨Y ¨ X|Z for VY ¨ X|Z in Theorem 3 (ii). 2.3 Kernel-free integral expression of the measures A remarkable property of INOCCO and ICOND is that they do not depend on the kernels under some assumptions, having integral expressions containing only the probability density functions. The probability EZ[PX|Z ⊗PY |Z] on X × Y is defined by EZ[PY |Z ⊗PX|Z](B × A) R E[χB(Y )|Z = z]E[χA(X)|Z = z]dPZ(z) for A ∈BX and B ∈BY. Theorem 4. Let µX and µY be measures on X and Y, respectively, and assume that the probabilities PXY and EZ[PX|Z ⊗PY |Z] are absolutely continuous with respect to µX × µY with probability density functions pXY and pX⊥⊥Y |Z, respectively. If HZ + R and (HX ⊗HY) + R are dense in L2(PZ) and L2(PX ⊗PY ), respectively, and VY X and VY ZVZX are Hilbert-Schmidt, then we have ICOND = ∥VY X|Z∥2 HS = Z Z X×Y pXY (x, y) pX(x)pY (y) −pX⊥⊥Y |Z(x, y) pX(x)pY (y) 2 pX(x)pY (y)dµX dµY, where pX and pY are the density functions of the marginal distributions PX and PY , respectively. As a special case of Z = ∅, we have INOCCO = ∥VY X∥2 HS = Z Z X×Y pXY (x, y) pX(x)pY (y) −1 2 pX(x)pY (y)dµX dµY. (9) Sketch of the proof (see the supplement for the complete proof). Since it is known [8] that ΣZZ is Hilbert-Schmidt under (A-1), there exist CONSs {φi}∞ i=1 ⊂HX and {ψj}∞ j=1 ⊂HY consisting of the eigenfunctions of ΣXX and ΣY Y , respectively, with ΣXXφi = λiφi (λi ≥0) and ΣY Y ψj = 4 νjψj (νj ≥0). Then, ∥VY X|Z∥2 HS admits the expansion P∞ i,j=1 n ⟨ψj, VY Xφi⟩2 HY −2⟨ψj, VY Xφi⟩HY⟨ψj, VY ZVZXφi⟩HY + ⟨ψj, VY ZVZXφi⟩2 HY o . Let IX + = {i ∈N | λi > 0} and IY + = {i ∈N | νi > 0}, and define ˜φi = (φi −E[φi(X)])/√λi and ˜ψj = (ψj −E[ψj(Y )])/√νj for i ∈IX + and j ∈IY + . For simplicity, L2 denotes L2(PX ⊗PY ). With the notations ˜φ0 = 1 and ˜ψ0 = 1, it is easy to see that the class {˜φi ˜ψj}i∈IX + ∪{0},j∈IY + ∪{0} is a CONS of L2. From Parseval’s equality, the first term of the above expansion is rewritten as P i∈IX + ,j∈IY + ⟨˜ψj, ΣY X ˜φi⟩2 HY = P i∈IX + ,j∈IY + EY X ˜ψj(Y )˜φi(X) 2 = P i∈IX + ,j∈IY + ˜φi ˜ψj, pXY pXpY L2 =
pXY (x,y) pX(x)pY (y)
2 L2 −P i∈IX + E ˜φi(X)] −P j∈IY + E ˜ψj(Y )] −1 =
pXY (x,y) pX(x)pY (y)
2 L2 −1. By a similar argument, the second and third term of the expansion are rewritten as −2 pXY pXpY , pX⊥ ⊥Y |Z pXpY L2 + 2 and
pX⊥ ⊥Y |Z pXpY
2 L2 −1, respectively. This completes the proof. Many practical kernels, such as the Gaussian and Laplacian, satisfy the assumptions in the above theorem, as we saw in Theorems 2 and the remark after Lemma 1. While the empirical estimate from finite samples depends on the choice of kernels, it is a desirable property for the empirical dependence measure to converge to a value that depends only on the distributions of the variables. Eq. (9) shows that, under the assumptions, INOCCO is equal to the mean square contingency, a well-known dependence measure[14] commonly used for discrete variables. As we show in Section 2.4, ˆINOCCO n works as a consistent kernel estimator of the mean square contingency. The expression of Eq. (9) can be compared with the mutual information, MI(X, Y ) = Z Z X×Y pXY (x, y) log pXY (x, y) pX(x)pY (y)dµX dµY. Both the mutual information and the mean square contingency are nonnegative, and equal to zero if and only if X and Y are independent. Note also that from log z ≤z −1, the inequality MI(X, Y ) ≤INOCCO(X, Y ) holds under the assumptions of Theorem 4. While the mutual information is the best known dependence measure, its finite sample empirical estimate is not straightforward, especially for continuous variables. The direct estimation of a probability density function is infeasible if the joint space has even a moderate number of dimensions. 2.4 Consistency of the measures It is important to ask whether the empirical measures converge to the population value I COND and INOCCO, since this provides a theoretical justification for the empirical measures. It is known [4] that bV (n) Y X converges in probability to VY X in operator norm. The next theorem asserts convergence in HS norm, provided that VY X is Hilbert-Schmidt. Although the proof is analogous to the case of operator norm, it is more involved to discuss the HS norm. We give it in the supplementary material. Theorem 5. Assume that VY X, VY Z, and VZX are Hilbert-Schmidt, and that the regularization constant εn satisfies εn →0 and ε3 nn →∞. Then, we have the convergence in probability ∥bV (n) Y X −VY X∥HS →0 and ∥bV (n) Y X|Z −VY X|Z∥HS →0 (n →∞). (10) In particular, ˆINOCCO n →INOCCO and ˆICOND n →ICOND (n →∞) in probability. 2.5 Choice of kernels As with all empirical measures, the sample estimates ˆINOCCO n and ˆICOND n are dependent on the kernel, and the problem of choosing a kernel has yet to be solved. Unlike supervised learning, there are no easy criteria to choose a kernel for dependence measures. We propose a method of choosing a kernel by considering the large sample behavior. We explain the method only briefly in this paper. The basic idea is that a kernel should be chosen so that the covariance operator detects independence of variables as effectively as possible. It has been recently shown [10], under the independence of 5 −4 −2 0 2 4 −4 −2 0 2 4 −4 −2 0 2 4 −4 −2 0 2 4 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Angle INOCCO Figure 1: Left and Middle: Examples of data (θ = 0 and θ = π/4). Right: The marks ”o” and ”+” show ˆINOCCO n for each angle and the 95th percentile of the permutation test, respectively. X and Y , that the measure HSIC = ∥bΣ(n) Y X∥2 HS ([8]) multiplied by n converges to an infinite mixture of χ2 distributions with variance Varlim[nHSIC] = 2∥ΣXX∥2 HS∥ΣY Y ∥2 HS. We choose a kernel so that the bootstrapped variance VarB[nHSIC] of nHSIC is close to this theoretical limit variance. More precisely, we compare the ratio T = VarB[nHSIC]/Varlim[nHSIC] for various candidate kernels. In preliminary experiments for choosing the variance parameter σ of Gaussian kernels, we often observed the ratio decays and saturates below 1, as σ increases. Therefore, we use σ starting the saturation by choosing the minimum of σ among all candidates that satisfy |Tσ −α| ≤ (1 + δ) minσ |Tσ −α| for δ > 0, α ∈(0, 1]. We always use δ = 0.1 and α = 0.5. We can expect that the chosen kernel uses the data effectively. While there is no rigorous theoretical guarantee, in the next section we see that the method gives a reasonable result for ˆINOCCO n and ˆICOND n . 3 Experiments To evaluate the dependence measures, we use a permutation test of independence for data sets with various degrees of dependence. The test randomly permutes the order of Y1, . . . , Yn to make many samples independent of (X1, . . . , Xn), thus simulating the null distribution under independence. For the evaluation of ˆICOND n , the range of Z is partitioned into Z1, . . . , ZL with the same number of data, and the sample {(Xi, Yi) | Zi ∈Zℓ} within the ℓ-th bin is randomly permuted. The significance level is always set to 5%. In the following experiments, we always use Gaussian kernels e− 1 2σ2 ∥x1−x2∥2 and choose σ by the method proposed in Section 2.5. Synthetic data for dependence. The random variables X(0) and Y (0) are independent and uniformly distributed on [−2, 2] and [a, b] ∪[−b, −a], respectively, so that (X (0), Y (0)) has a scalar covariance matrix. (X(θ), Y (θ)) is the rotation of (X(0), Y (0)) by θ ∈[0, π/4] (see Figure 1). X(θ) and Y (θ) are always uncorrelated, but dependent for θ ̸= 0. We generate 100 sets of 200 data. We perform permutation tests with ˆINOCCO n , HSIC = ∥bΣ(n) Y X∥2 HS, and the mutual information (MI). For the empirical estimates of MI, we use the advanced method from [11], with no need for explicit estimation of the densities. Since ˆINOCCO n is an estimate of the mean square contingency, we also apply a relevant contingency-table-based independence test ([12]), partitioning the variables into bins. Figure 1 shows the values of ˆINOCCO n for a sample. In Table 1, we see that the results of ˆINOCCO n are stable w.r.t. the choice of εn, provided it is sufficiently small. We fix εn = 10−6 for all remaining experiments. While all the methods are able to detect the dependence, ˆINOCCO n with the asymptotic choice of σ is the most sensitive to very small dependence. We also observe the chosen parameters σY for Y increase from 0.58 to 2.0 as θ increases. The small σY for small θ seems reasonable, because the range of Y is split into two small regions. Chaotic time series. We evaluate a chaotic time series derived from the coupled H´enon map. The variables X and Y are four dimensional: the components X1, X2, Y1, and Y2 follow the dynamics (X1(t + 1), X2(t + 1)) = (1.4 −X1(t)2 + 0.3X2(t), X1(t)), (Y1(t + 1), Y2(t + 1)) = (1.4 − {γX1(t)Y1(t) + (1 −γ)Y2(t)2} + 0.1Y2(t), Y1(t)), and X3, X4, Y3, Y4 are independent noise with N(0, (0.5)2). X and Y are independent for γ = 0, while they are synchronized chaos for γ > 0 (see Figure 2 for examples). A sample consists of 100 data generated from this system. Table 2 6 Angle (degree) 0 4.5 9 13.5 18 22.5 27 31.5 36 40.5 45 ˆINOCCO n (ε = 10−4, Median) 94 23 0 0 0 0 0 0 0 0 0 ˆINOCCO n (ε = 10−6, Median) 92 20 1 0 0 0 0 0 0 0 0 ˆINOCCO n (ε = 10−8, Median) 93 15 0 0 0 0 0 0 0 0 0 ˆINOCCO n (Asymp. Var.) 94 11 0 0 0 0 0 0 0 0 0 HSIC (Median) 93 92 63 5 0 0 0 0 0 0 0 HSIC (Asymp. Var.) 93 44 1 0 0 0 0 0 0 0 0 MI (#Nearest Neighbors = 1) 93 62 11 0 0 0 0 0 0 0 0 MI (#Nearest Neighbors = 3) 96 43 0 0 0 0 0 0 0 0 0 MI (#Nearest Neighbors = 5) 97 49 0 0 0 0 0 0 0 0 0 Conting. Table (#Bins= 3) 100 96 46 9 1 0 0 0 0 0 0 Conting. Table (#Bins= 4) 98 29 0 0 0 0 0 0 0 0 0 Conting. Table (#Bins= 5) 98 82 5 0 0 0 0 0 0 0 0 Table 1: Comparison of dependence measures. The number of times independence is accepted out of 100 permutation tests is shown. ”Asymp. Var.” is the method in Section 2.5. ”Median” is a heuristic method [8] which chooses σ as the median of pairwise distances of the data. −2 −1 0 1 2 −2 −1 0 1 2 X1(t) X2(t) −2 −1 0 1 2 −1 0 1 2 X1(t) Y1(t) 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 1 I(Xt+1,Yt|Xt) Thresh (α=0.05) 0 0.2 0.4 0.6 0 1 2 3 4 5 I(Yt+1,Xt|Yt) Thresh (α=0.05) (a) Plot of H´enon map (b) Xt,1-Yt,1 (γ = 0.25) (c) I(Xt+1, Yt|Xt) (d) I(Yt+1, Xt|Yt) Figure 2: Chaotic time series. (a,b): examples of data. (c,d) examples of ˆICOND n (colored ”o”) and the threshholds of the permutation test with significance level 5% (black ”+”). shows the results of permutation tests of independence for the instantaneous pairs (X(t), Y (t))100 t=1. The proposed ˆINOCCO n outperforms the other methods in capturing small dependence. Next, we apply ˆICOND n to detect the causal structure of the same time series. Note that the series X is a cause of Y for γ > 0, but there is no opposite causality, i.e., Xt+1⊥⊥Yt | Xt and Yt+1 ̸⊥⊥Xt | Yt. In Table 3, it is remarkable that ˆICOND n detects the small causal influence from Xt to Yt+1 for γ ≥0.1, while for γ = 0 the result is close to the theoretical value of 95%. Graphical modeling from medical data. This is the inference of a graphical model from data with no time structure. The data consist of three variables, creatinine clearance (C), digoxin clearance (D), urine flow (U). These were taken from 35 patients, and analyzed with graphical models in [3, Section 3.1.4.]. From medical knowledge, D should be independent of U when controlling C. Table 4 shows the results of the permutation tests and a comparison with the linear method. The relation D⊥⊥U | C is strongly affirmed by ˆICOND n , while the partial correlation does not find it. γ (strength of coupling) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ˆINOCCO n 97 66 21 1 0 1 0 HSIC 75 70 58 52 13 1 0 MI (k = 3) 87 91 83 73 23 6 0 MI (k = 5) 87 88 75 67 23 5 0 MI (k = 7) 87 86 75 64 21 5 0 Table 2: Results for the independence tests for the chaotic time series. The number of times independence was accepted out of 100 permutation tests is shown. γ = 0 implies independence. 7 H0: Yt is not a cause of Xt+1 H0: Xt is not a cause of Yt+1 γ (coupling) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ˆINOCCO n 97 96 93 85 81 68 75 96 0 0 0 0 0 0 HSIC 94 94 92 81 60 73 66 93 95 85 56 1 1 1 Table 3: Results of the permutation test of non-causality for the chaotic time series. The number of times non-causality was accepted out of 100 tests is shown. Kernel measure Linear method ˆICOND n P-value (partial) correl. P-value D⊥⊥U | C 1.458 0.924 Parcorr(D, U|C) 0.4847 0.0037 C⊥⊥D 0.776 <0.001 Corr(C, D) 0.7754 0.0000 C⊥⊥U 0.194 0.117 Corr(C, U) 0.3092 0.0707 D⊥⊥U 0.343 0.023 Corr(D, U) 0.5309 0.0010 Table 4: Graphical modeling from the medical data. Higher P-values indicate (conditional) independence more strongly. 4 Concluding remarks There are many dependence measures, and further theoretical and experimental comparison is important. That said, one unambiguous strength of the kernel measure we propose is its kernel-free population expression. It is interesting to ask if other classical dependence measures, such as the mutual information, can be estimated by kernels (in a broader sense than the expansion about independence of [9]). A relevant measure is the kernel generalized variance (KGV [1]), which is based on a sum of the logarithm of the eigenvalues of VY X, while INOCCO is their squared sum. It is also interesting to investigate whether the KGV has a kernel-free expression. Another topic for further study is causal inference with the proposed measure, both with and without time information ([16]). References [1] F. Bach and M. Jordan. Kernel independent component analysis. J. Machine Learning Res., 3:1–48, 2002. [2] C. Baker. Joint measures and cross-covariance operators. Trans. Amer. Math. Soc., 186:273–289, 1973. [3] D. Edwards. Introduction to graphical modelling. Springer verlag, New York, 2000. [4] K. Fukumizu, F. Bach, and A. Gretton. Statistical consistency of kernel canonical correlation analysis. J. Machine Learning Res., 8:361–383, 2007. [5] K. Fukumizu, F. Bach, and M. Jordan. Dimensionality reduction for supervised learning with reproducing kernel Hilbert spaces. J. Machine Learning Res., 5:73–99, 2004. [6] K. Fukumizu, F. Bach, and M. Jordan. Kernel dimension reduction in regression. Tech Report 715, Dept. Statistics, University of California, Berkeley, 2006. [7] A. Gretton, K. Borgwardt, M. Rasch, B. Sch¨olkopf, and A. Smola. 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Trans-dimensional MCMC for Bayesian Policy Learning Matt Hoffman Dept. of Computer Science University of British Columbia hoffmanm@cs.ubc.ca Arnaud Doucet Depts. of Statistics and Computer Science University of British Columbia arnaud@cs.ubc.ca Nando de Freitas Dept. of Computer Science University of British Columbia nando@cs.ubc.ca Ajay Jasra Dept. of Mathematics Imperial College London ajay.jasra@imperial.ac.uk Abstract A recently proposed formulation of the stochastic planning and control problem as one of parameter estimation for suitable artificial statistical models has led to the adoption of inference algorithms for this notoriously hard problem. At the algorithmic level, the focus has been on developing Expectation-Maximization (EM) algorithms. In this paper, we begin by making the crucial observation that the stochastic control problem can be reinterpreted as one of trans-dimensional inference. With this new interpretation, we are able to propose a novel reversible jump Markov chain Monte Carlo (MCMC) algorithm that is more efficient than its EM counterparts. Moreover, it enables us to implement full Bayesian policy search, without the need for gradients and with one single Markov chain. The new approach involves sampling directly from a distribution that is proportional to the reward and, consequently, performs better than classic simulations methods in situations where the reward is a rare event. 1 Introduction Continuous state-space Markov Decision Processes (MDPs) are notoriously difficult to solve. Except for a few rare cases, including linear Gaussian models with quadratic cost, there is no closed-form solution and approximations are required [4]. A large number of methods have been proposed in the literature relying on value function approximation and policy search; including [3, 10, 14, 16, 18]. In this paper, we follow the policy learning approach because of its promise and remarkable success in complex domains; see for example [13, 15]. Our work is strongly motivated by a recent formulation of stochastic planning and control problems as inference problems. This line of work appears to have been initiated in [5], where the authors used EM as an alternative to standard stochastic gradient algorithms to maximize an expected cost. In [2], a planning problem under uncertainty was solved using a Viterbi algorithm. This was later extended in [21]. In these works, the number of time steps to reach the goal was fixed and the plans were not optimal in expected reward. An important step toward surmounting these limitations was taken in [20, 19]. In these works, the standard discounted reward control problem was expressed in terms of an infinite mixture of MDPs. To make the problem tractable, the authors proposed to truncate the infinite horizon time. Here, we make the observation that, in this probabilistic interpretation of stochastic control, the objective function can be written as the expectation of a positive function with respect to a transdimensional probability distribution, i.e. a probability distribution defined on a union of subspaces 1 of different dimensions. By reinterpreting this function as a (artificial) marginal likelihood, it is easy to see that it can also be maximized using an EM-type algorithm in the spirit of [5]. However, the observation that we are dealing with a trans-dimensional distribution enables us to go beyond EM. We believe it creates many opportunities for exploiting a large body of sophisticated inference algorithms in the decision-making context. In this paper, we propose a full Bayesian policy search alternative to the EM algorithm. In this approach, we set a prior distribution on the set of policy parameters and derive an artificial posterior distribution which is proportional to the prior times the expected reward. In the simpler context of myopic Bayesian experimental design, a similar method was developed in [11] and applied successfully to high-dimensional problems [12]. Our method can be interpreted as a trans-dimensional extension of [11]. We sample from the resulting artificial posterior distribution using a single transdimensional MCMC algorithm, which only involves a simple modification of the MCMC algorithm developed to implement the EM. Although the Bayesian policy search approach can benefit from gradient information, it does not require gradients. Moreover, since the target is proportional to the expected reward, the simulation is guided to areas of high reward automatically. In the fixed policy case, the value function is often computed using importance sampling. In this context, our algorithm could be reinterpreted as an MCMC algorithm sampling from the optimal importance distribution. 2 Model formulation We consider the following class of discrete-time Markov decision processes (MDPs): X1 ∼µ(·) Xn| (Xn−1 = x, An−1 = a) ∼fa (·| x) Rn| (Xn = x, An = a) ∼ga (·| x) An| (Xn = x, θ) ∼πθ (·| x) , (1) where n = 1, 2, . . . is a discrete-time index, µ(·) is the initial state distribution, {Xn} is the X−valued state process, {An} is the A−valued action process, {Rn} is a positive real-valued reward process, fa denotes the transition density, ga the reward density and πθ is a randomized policy. If we have a deterministic policy then πθ (a| x) = δϕθ(x) (a). In this case, the transition model fa (·| x) assumes the parametrization fθ (·| x). The reward model could also be parameterized as gθ (·| x). It should be noted that for this work we will be working within a model-based framework and as a result will require knowledge of the transition model (although it could be learned). We are here interested in maximizing with respect to the parameters of the policy θ the expected future reward V π µ (θ) = E " ∞ X n=1 γn−1Rn # , where 0 < γ < 1 is a discount factor and the expectation is with respect to the probabilistic model defined in (1). As shown in [20], it is possible to re-write this objective of optimizing an infinite horizon discounted reward MDP (where the reward happens at each step) as one of optimizing an infinite mixture of finite horizon MDPs (where the reward only happens at the last time step). In particular, we note that by introducing the trans-dimensional probability distribution on ] {k}× X k × Ak × R+ given by pθ (k, x1:k, a1:k, rk) = (1 −γ) γk−1µ (x1) gak (rk| xk) k Y n=2 fan−1 (xn| xn−1) k Y n=1 πθ (an| xn) , (2) we can easily rewrite V π µ (θ) as an infinite mixture model of finite horizon MDPs, with the reward happening at the last horizon step; namely at k. Specifically we have: V π µ (θ) = (1 −γ)−1 Epθ [RK] = (1 −γ)−1 ∞ X k=1 Z rkpθ (k, x1:k, a1:k, rk) dx1:kda1:kdrk (3) 2 for a randomized policy. Similarly, for a deterministic policy, the representation (3) also holds for the trans-dimensional probability distribution defined on ] {k} × X k × R+ given by pθ (k, x1:k, rk) = (1 −γ) γk−1µ (x1) gθ (rk| xk) k Y n=2 fθ (xn| xn−1) . (4) The representation (3) was also used in [6] to compute the value function through MCMC for a fixed θ. In [20], this representation is exploited to maximize V π µ (θ) using the EM algorithm which, applied to this problem, proceeds as follows at iteration i θi = arg max θ∈Θ Q (θi−1, θ) where Q (θi−1, θ) = Eepθi−1 [log (RK.pθ (K, X1:K, A1:K, RK))] , epθ (k, x1:k, a1:k, rk) = rkpθ (k, x1:k, a1:k, rk) Epθ [RK] . Unlike [20], we are interested in problems with potentially nonlinear and non-Gaussian properties. In these situations, the Q function cannot be calculated exactly. The standard Monte Carlo EM approach consists of sampling from epθ (k, x1:k, a1:k, rk) using MCMC to obtain a Monte Carlo estimate of the Q function. As epθ (k, x1:k, a1:k, rk) is proportional to the reward, the samples will consequently be drawn in regions of high reward. This is a particularly interesting feature in situations where the reward function is concentrated in a region of low probability mass under pθ (k, x1:k, rk), which is often the case in high-dimensional control settings. Note that if we wanted to estimate V π µ (θ) using importance sampling, then the distribution epθ (k, x1:k, a1:k, rk) corresponds to the optimal zero-variance importance distribution. Alternatively, instead of sampling from epθ (k, x1:k, a1:k, rk) using MCMC, we could proceed as in [20] to derive forward-backward algorithms to implement the E-step which can be implemented here using Sequential Monte Carlo (SMC) techniques. We have in fact done this using the smoothing algorithms proposed in [9]. However, we will focus the discussion on a different MCMC approach based on trans-dimensional simulation. As shown in the experiments, the latter does considerably better. Finally, we remark that for a deterministic policy, we can introduce the trans-dimensional distribution: epθ (k, x1:k, rk) = rkpθ (k, x1:k, rk) Epθ [RK] . In addition, and for ease of presentation only, we focus the discussion on deterministic policies and reward functions gθ (rn| xn) = δr(xn) (rn) ; the extension of our algorithms to the randomized case is straightforward. 3 Bayesian policy exploration The EM algorithm is particularly sensitive to initialization and might get trapped in a severe local maximum of V π µ (θ). Moreover, in the general state-space setting that we are considering, the particle smoothers in the E-step can be very expensive computationally. To address these concerns, we propose an alternative full Bayesian approach. In the simpler context of experimental design, this approach was successfully developed in [11], [12]. The idea consists of introducing a vague prior distribution p (θ) on the parameters of the policy θ. We then define the new artificial probability distribution defined on Θ × ] {k} × X k by p (θ, k, x1:k) ∝r (xk) pθ (k, x1:k) p (θ) . By construction, this target distribution admits the following marginal in θ p (θ) ∝V π µ (θ) p (θ) and we can select an improper prior distribution p (θ) ∝1 if R Θ V π µ (θ) dθ < ∞. 3 If we could sample from p (θ), then the generated samples θ(i) would concentrate themselves in regions where V π µ (θ) is large. We cannot sample from p (θ) directly but we can developed a trans-dimensional MCMC algorithm which will generate asymptotically samples from p (θ, k, x1:k), hence samples from p (θ). Our algorithm proceeds as follows. Assume the current state of the Markov chain targeting p (θ, k, x1:k) is (θ, k, x1:k). We propose first to update the components (k, x1:k) conditional upon θ using a combination of birth, death and update moves using the reversible jump MCMC algorithm [7, 8, 17]. Then we propose to update θ conditional upon the current value of (k, x1:k). This can be achieved using a simple Metropolis-Hastings algorithm or a more sophisticated dynamic Monte Carlo schemes. For example, if gradient information is available, one could adopt Langevin diffusions and the hybrid Monte Carlo algorithm [1]. The overall algorithm is depicted in Figure 1. The details of the reversible jump algorithm are presented in the following section. 1. Initialization: set (k(0), x(0) 1:k(0), θ(0)). 2. For i = 0 to N −1 • Sample u ∼U[0,1]. • If (u ≤bk) – then carry out a “birth” move: Increase the horizon length of the MDP, say k(i) = k(i−1) + 1 and insert a new state. – else if (u ≤bk + dk) then carry out a “death” move: decrease the horizon length of the MDP, say k(i) = k(i−1) −1 and an existing state. – else let k(i) = k(i−1) and generate samples x(i) 1:k(i) of the MDP states. End If. • Sample the policy parameters θ(i) conditional on the samples (x(i) 1:k(i), k(i)). Figure 1: Generic reversible jump MCMC for Bayesian policy learning. We note that for a given θ the samples of the states and horizon generated by this Markov chain will also be distributed (asymptotically) according to the trans-dimensional distribution epθ (k, x1:k). Hence, they can be easily adapted to generate a Monte Carlo estimate of Q (θi−1, θ). This allows us to side-step the need for expensive smoothing algorithms in the E-step. The trans-dimensional simulation approach has the advantage that the samples will concentrate themselves automatically in regions where epθ (k) has high probability masses. Moreover, unlike in the EM framework, it is no longer necessary to truncate the time domain. 4 Trans-Dimensional Markov chain Monte Carlo We present a simple reversible jump method composed of two reversible moves (birth and death) and several update moves. Assume the current state of the Markov chain targeting epθ (k, x1:k) is (k, x1:k). With probability1 bk, we propose a birth move; that is we sample a location uniformly in the interval {1, ..., k + 1}, i.e. J ∼U {1, ..., k + 1}, and propose the candidate (k + 1, x1:j−1, x∗, xj:k) where X∗∼qθ (·| xj−1:j). This candidate is accepted with probability Abirth = min{1, αbirth} where we have for j ∈{2, ..., k −1} αbirth = epθ (k + 1, x1:j−1, x∗, xj:k) dk+1 epθ (k, x1:k) bkqθ (x∗| xj−1:j) = γfθ (x∗| xj−1) fθ (xj| x∗) dk+1 fθ (xj| xj−1) bkqθ (x∗| xj−1:j) , for j = 1 αbirth = γµ (x∗) fθ (x1| x∗) dk+1 µ (x1) bkqθ (x∗| x1) , 1In practice we can set the birth and death probabilities such that bk = dk = uk = 1/3. 4 and j = k + 1 αbirth = γr (x∗) fθ (x∗| xk) dk+1 r (xk) bkqθ (x∗| xk) . With probability dk, we propose a death move; that is J ∼U {1, ..., k} and we propose the candidate (k −1, x1:j−1, xj+1:k) which is accepted with probability Adeath = min{1, αdeath} where for j ∈{2, ..., k −1} αdeath = epθ (k −1, x1:j−1, xj+1:k) bk+1qθ (xj| xj−1:j+1) epθ (k, x1:k) dk = fθ (xj+1| xj−1) bk+1qθ (xj| xj−1:j+1) γfθ (xj+1| xj) fθ (xj| xj−1) dk , for j = 1 αdeath = µ (x2) qθ (x1| x2) bk+1 γµ (x1) fθ (x2| x1) dk , and for j = k αdeath = r (xk−1) qθ (xk| xk−1) bk+1 γr (xk) fθ (xk| xk−1) dk . The αbirth and αdeath terms derived above can be thought of as ratios between the distribution over the newly proposed state of the chain (i.e. after the birth/death) and the current state. These terms must also ensure reversibility and the dimension-matching requirement for reversible jump MCMC. For more information see [7, 8]. Finally with probability uk = 1 −bk −dk, we propose a standard (fixed dimensional) move where we update all or a subset of the components x1:k using say Metropolis-Hastings or Gibbs moves. There are many design possibilities for these moves. In general, one should block some of the variables so as to improve the mixing time of the Markov chain. If one adopts a simple one-at-a time Metropolis-Hastings scheme with proposals qθ (x∗| xj−1:j+1) to update the j-th term, then the candidate is accepted with probability Aupdate = min{1, αupdate} where for j ∈{2, ..., k −1} αupdate = epθ (k, x1:j−1, x∗, xj+1:k) qθ (xj| xj−1, x∗, xj+1) epθ (k, x1:k) qθ (x∗| xj−1:j+1) = fθ (x∗| xj−1) fθ (xj+1| x∗) qθ (xj| xj−1, x∗, xj+1) fθ (xj| xj−1) fθ (xj+1| xj) qθ (x∗| xj−1:j+1) , for j = 1 αupdate = µ (x∗) fθ (x2| x∗) qθ (x1| x∗, x2) µ (x1) fθ (x2| x1) qθ (x∗| x1:2) , and for j = k αupdate = r (x∗) fθ (x∗| xk−1) qθ (xk| x∗, xk−1) r (xk) fθ (xk| xk−1) qθ (x∗| xk−1:k) . Under weak assumptions on the model, the Markov chain {K(i), X(i) 1:K} generated by this transition kernel will be irreducible and aperiodic and hence will generate asymptotically samples from the target distribution epθ (k, x1:k). We emphasize that the structure of the distributions epθ (x1:k| k) will not in many applications vary significantly with k and we often have epθ (x1:k| k) ≈epθ (x1:k| k + 1). Hence the probability of having the reversible moves accepted will be reasonable. Standard Bayesian applications of reversible jump MCMC usually do not enjoy this property and it makes it more difficult to design fast mixing algorithms. In this respect, our problem is easier. 5 Experiments It should be noted from the outset that the results presented in this paper are preliminary, and serve mainly as an illustration of the Monte Carlo algorithms presented earlier. With that note aside, even these simple examples will give us some intuition about the algorithms’ performance and behavior. 5 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Figure 2: This figure shows an illustration of the 2d state-space described in section 5. Ten sample points are shown distributed according to µ, the initial distribution, and the contour plot corresponds to the reward function r. The red line denotes the policy parameterized by some angle θ, while a path is drawn in blue sampled from this policy. We are also very optimistic as to the possible applications of analytic expressions for linear Gaussian models, but space has not allowed us to present simulations for this class of models here. We will consider state- and action-spaces X = A = R2 such that each state x ∈X is a 2d position and each action a ∈A is a vector corresponding to a change in position. A new state at time n is given by Xn = Xn−1 + An−1 + νn−1 where νn−1 denotes zero-mean Gaussian noise. Finally we will let µ be a normal distribution about the origin, and consider a reward (as in [20]) given by an unnormalized Gaussian about some point m, i.e. r(x) = exp(−1 2(x −m)T Σ−1(x −m)). An illustration of this space can be seen in Figure 2 where m = (1, 1). For these experiments we chose a simple, stochastic policy parameterized by θ ∈[0, 2π]. Under this policy, an action An = (w + δ) · (cos(θ + ω), sin(θ + ω)) is taken where δ and ω are normally distributed random variables and w is some (small) constant step-length. Intuitively, this policy corresponds to choosing a direction θ in which the agent will walk. While unrealistic from a real-world perspective, this allows us a method to easily evaluate and plot the convergence of our algorithm. For a state-space with initial distribution and reward function defined as in Figure 2 the optimal policy corresponds to θ = π/4. We first implemented a simple SMC-based extension of the EM algorithm described in [20], wherein a particle filter was used for the forwards/backwards filters. The plots in Figure 3 compare the SMC-based and trans-dimensional approaches performing on this synthetic example. Here the inferred value of θ is shown against CPU time, averaged over 5 runs. The first thing of note is the terrible performance of the SMC-based algorithm—in fact we had to make the reward broader and closer to the initial position in order to ensure that the algorithm converges in a reasonable amount of time. This comes as no surprise considering the O(N 2k2 max) time complexity necessary for computing the importance weights. While there do exist methods [9] for reducing this complexity to O(N log Nk2 max), the discrepancy between this and the reversible jump MCMC method suggests that the MCMC approach may be more adapted to this class of problems. In the finite/discrete case it is also possible, as shown by Toussaint et al (2006), to reduce the k2 max term to kmax by calculating updates only using messages from the backwards recursion. The SMC method might further be improved by better choices for the artificial distribution ηn(xn) in the backwards filter. In this problem we used a vague Gaussian centered on the relevant state-space. It is however possible that any added benefit from a more informative η distribution is counterbalanced by the time required to calculate this η, for example by simulating particles forward in order to find the invariant distribution, etc. Also shown in figure 3 is the performance of a Monte Carlo EM algorithm using reversible jump MCMC in the E-step. Both this and the fully Bayesian approach perform comparably, although the fully Bayesian approach shows less in-run variance, as well as less variance between runs. The EM algorithm was also more sensitive, and we were forced to increase the number of samples N used 6 0 500 1000 1500 2000 2500 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Convergence of θ as a function of time θ (in radians) cpu time (in seconds) Two−filter EM Monte Carlo EM Bayes. policy search Optimal (baseline) 0 200 400 600 800 1000 0.7 0.75 0.8 0.85 Convergence of θ as a function of time θ (in radians) cpu time (in seconds) Monte Carlo EM Bayes. policy search Optimal (baseline) Figure 3: The left figure shows estimates for the policy parameter θ as a function of the CPU time used to calculate that value. This data is shown for the three discussed Monte Carlo algorithms as applied to a synthetic example and has been averaged over five runs; error bars are shown for the SMC-based EM algorithm. Because of the poor performance of the SMC-based algorithm it is difficult to compare the performance of the other two algorithms using only this plot. The right figure shows a smoothed and “zoomed” version of the right plot in order to show the reversible-jump EM algorithm and the fully Bayesian algorithm in more detail. In both plots a red line denotes the known optimal policy parameter of π/4. by the E-step as the algorithm progressed, as well as controlling the learning rate with a smoothing parameter. For higher dimensional and/or larger models it is not inconceivable that this could have an adverse affect on the algorithms performance. Finally, we also compared the proposed Bayesian policy exploration method to the PEGASUS [14] approach using a local search method. We initially tried using a policy-gradient approach, but because of the very highly-peaked rewards the gradients become very poorly scaled and would have required more tuning. As shown in Figure 4, the Bayesian strategy is more efficient in this rare event setting. As the dimension of the state-space increases, we expect this difference to become even more pronounced. 6 Discussion We believe that formulating stochastic control as a trans-dimensional inference problem is fruitful. This formulation relies on minimal assumptions and allows us to apply modern inference algorithms to solve control problems. We have focused here on Monte Carlo methods and have presented— to the best of our knowledge—the first application of reversible jump MCMC to policy search. Our results, on an illustrative example, showed that this trans-dimensional MCMC algorithm is more effective that standard policy search methods and alternative Monte Carlo methods relying on particle filters. However, this methodology remains to be tested on high-dimensional problems. For such scenarios, we expect that it will be necessary to develop more efficient MCMC strategies to explore the policy space efficiently. References [1] C. Andrieu, N. de Freitas, A. Doucet, and M. I. Jordan. An introduction to MCMC for machine learning. Machine Learning, 50:5–43, 2003. [2] H. Attias. Planning by probabilistic inference. In Uncertainty in Artificial Intelligence, 2003. [3] J. Baxter and P. L. Bartlett. Infinite-horizon policy-gradient estimation. Journal of Artificial Intelligence Research, 15:319–350, 2001. [4] D. P. Bertsekas. Dynamic Programming and Optimal Control. Athena Scientific, 1995. [5] P. Dayan and G. E. Hinton. Using EM for reinforcement learning. Neural Computation, 9:271–278, 1997. 7 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 number of samples taken from transition-model 0.2 0.3 0.4 0.5 0.6 0.7 0.8 policy parameter (theta) Evolution of policy parameters against transition-model samples rjmdp pegasus optimal Figure 4: Convergence of PEGASUS and our Bayesian policy search algorithm when started from θ = 0 and converging to the optimum of θ∗= π/4. The plots are averaged over 10 runs. For our algorithm we plot samples taken directly from the MCMC algorithm itself: plotting the empirical average would produce an estimate whose convergence is almost immediate, but we also wanted to show the “burn-in” period. For both algorithms lines denoting one standard deviation are shown and performance is plotted against the number of samples taken from the transition model. [6] A. Doucet and V. B. Tadic. On solving integral equations using Markov chain Monte Carlo methods. Technical Report CUED-F-INFENG 444, Cambridge University Engineering Department, 2004. [7] P. J. Green. Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82:711–732, 1995. [8] P. J. Green. Trans-dimensional Markov chain Monte Carlo. In Highly Structured Stochastic Systems, 2003. [9] M. Klaas, M. Briers, N. de Freitas, A. Doucet, and S. Maskell. Fast particle smoothing: If i had a million particles. In International Conference on Machine Learning, 2006. [10] G. Lawrence, N. Cowan, and S. Russell. Efficient gradient estimation for motor control learning. In Uncertainty in Artificial Intelligence, pages 354–36, 2003. [11] P. M¨uller. Simulation based optimal design. Bayesian Statistics, 6, 1999. [12] P. M¨uller, B. Sans´o, and M. De Iorio. Optimal Bayesian design by inhomogeneous Markov chain simulation. J. American Stat. Assoc., 99:788–798, 2004. [13] A. Ng, A. Coates, M. Diel, V. Ganapathi, J. Schulte, B. Tse, E. Berger, and E. Liang. Inverted autonomous helicopter flight via reinforcement learning. In International Symposium on Experimental Robotics, 2004. [14] A. Y. Ng and M. I. Jordan. PEGASUS: A policy search method for large MDPs and POMDPs. In Uncertainty in Artificial Intelligence, 2000. [15] J. Peters and S. Schaal. Policy gradient methods for robotics. In IEEE International Conference on Intelligent Robotics Systems, 2006. [16] M. Porta, N. Vlassis, M. T. J. Spaan, and P. Poupart. Point-based value iteration for continuous POMDPs. Journal of Machine Learning Research, 7:2329–2367, 2006. [17] S. Richardson and P. J. Green. On Bayesian analysis of mixtures with an unknown number of components. Journal of the Royal Statistical Society B, 59(4):731–792, 1997. [18] S. Thrun. Monte Carlo POMDPs. In S. Solla, T. Leen, and K.-R. M¨uller, editors, Neural Information Processing Systems, pages 1064–1070. MIT Press, 2000. [19] M. Toussaint, S. Harmeling, and A. Storkey. Probabilistic inference for solving (PO)MDPs. Technical Report EDI-INF-RR-0934, University of Edinburgh, School of Informatics, 2006. [20] M. Toussaint and A. Storkey. Probabilistic inference for solving discrete and continuous state Markov decision processes. In International Conference on Machine Learning, 2006. [21] D. Verma and R. P. N. Rao. Planning and acting in uncertain environments using probabilistic inference. In IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, 2006. 8
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Temporal Difference Updating without a Learning Rate Marcus Hutter RSISE@ANU and SML@NICTA Canberra, ACT, 0200, Australia marcus@hutter1.net www.hutter1.net Shane Legg IDSIA, Galleria 2, Manno-Lugano CH-6928, Switzerland shane@vetta.org www.vetta.org/shane Abstract We derive an equation for temporal difference learning from statistical principles. Specifically, we start with the variational principle and then bootstrap to produce an updating rule for discounted state value estimates. The resulting equation is similar to the standard equation for temporal difference learning with eligibility traces, so called TD(λ), however it lacks the parameter α that specifies the learning rate. In the place of this free parameter there is now an equation for the learning rate that is specific to each state transition. We experimentally test this new learning rule against TD(λ) and find that it offers superior performance in various settings. Finally, we make some preliminary investigations into how to extend our new temporal difference algorithm to reinforcement learning. To do this we combine our update equation with both Watkins’ Q(λ) and Sarsa(λ) and find that it again offers superior performance without a learning rate parameter. 1 Introduction In the field of reinforcement learning, perhaps the most popular way to estimate the future discounted reward of states is the method of temporal difference learning. It is unclear who exactly introduced this first, however the first explicit version of temporal difference as a learning rule appears to be Witten [9]. The idea is as follows: The expected future discounted reward of a state s is, V s := E rk + γrk+1 + γ2rk+2 + · · · |sk = s , where the rewards rk, rk+1, . . . are geometrically discounted into the future by γ < 1. From this definition it follows that, V s = E rk + γV sk+1|sk = s . (1) Our task, at time t, is to compute an estimate V t s of V s for each state s. The only information we have to base this estimate on is the current history of state transitions, s1, s2, . . . , st, and the current history of observed rewards, r1, r2, . . . , rt. Equation (1) suggests that at time t + 1 the value of rt +γVst+1 provides us with information on what V t s should be: If it is higher than V t st then perhaps this estimate should be increased, and vice versa. This intuition gives us the following estimation heuristic for state st, V t+1 st := V t st + α rt + γV t st+1 −V t st , where α is a parameter that controls the rate of learning. This type of temporal difference learning is known as TD(0). 1 One shortcoming of this method is that at each time step the value of only the last state st is updated. States before the last state are also affected by changes in the last state’s value and thus these could be updated too. This is what happens with so called temporal difference learning with eligibility traces, where a history, or trace, is kept of which states have been recently visited. Under this method, when we update the value of a state we also go back through the trace updating the earlier states as well. Formally, for any state s its eligibility trace is computed by, Et s := γλEt−1 s if s ̸= st, γλEt−1 s + 1 if s = st, where λ is used to control the rate at which the eligibility trace is discounted. The temporal difference update is then, for all states s, V t+1 s := V t s + αEt s r + γV t st+1 −V t st . (2) This more powerful version of temporal different learning is known as TD(λ) [7]. The main idea of this paper is to derive a temporal difference rule from statistical principles and compare it to the standard heuristic described above. Superficially, our work has some similarities to LSTD(λ) ([2] and references therein). However LSTD is concerned with finding a least-squares linear function approximation, it has not yet been developed for general γ and λ, and has update time quadratic in the number of features/states. On the other hand, our algorithm “exactly” coincides with TD/Q/Sarsa(λ) for finite state spaces, but with a novel learning rate derived from statistical principles. We therefore focus our comparison on TD/Q/Sarsa. For a recent survey of methods to set the learning rate see [1]. In Section 2 we derive a least squares estimate for the value function. By expressing the estimate as an incremental update rule we obtain a new form of TD(λ), which we call HL(λ). In Section 3 we compare HL(λ) to TD(λ) on a simple Markov chain. We then test it on a random Markov chain in Section 4 and a non-stationary environment in Section 5. In Section 6 we derive two new methods for policy learning based on HL(λ), and compare them to Sarsa(λ) and Watkins’ Q(λ) on a simple reinforcement learning problem. Section 7 ends the paper with a summary and some thoughts on future research directions. 2 Derivation The empirical future discounted reward of a state sk is the sum of actual rewards following from state sk in time steps k, k + 1, . . ., where the rewards are discounted as they go into the future. Formally, the empirical value of state sk at time k for k = 1, ..., t is, vk := ∞ X u=k γu−kru, (3) where the future rewards ru are geometrically discounted by γ < 1. In practice the exact value of vk is always unknown to us as it depends not only on rewards that have been already observed, but also on unknown future rewards. Note that if sm = sn for m ̸= n, that is, we have visited the same state twice at different times m and n, this does not imply that vn = vm as the observed rewards following the state visit may be different each time. Our goal is that for each state s the estimate V t s should be as close as possible to the true expected future discounted reward V s. Thus, for each state s we would like Vs to be close to vk for all k such that s = sk. Furthermore, in non-stationary environments we would like to discount old evidence by some parameter λ ∈(0, 1]. Formally, we want to minimise the loss function, L := 1 2 t X k=1 λt−k vk −V t sk 2. (4) For stationary environments we may simply set λ = 1 a priori. As we wish to minimise this loss, we take the partial derivative with respect to the value estimate of each state and set to zero, ∂L ∂V ts = − t X k=1 λt−k vk −V t sk δsks = V t s t X k=1 λt−kδsks − t X k=1 λt−kδsksvk = 0, 2 where we could change V t sk into V t s due to the presence of the Kronecker δsks, defined δxy := 1 if x = y, and 0 otherwise. By defining a discounted state visit counter N t s := Pt k=1 λt−kδsks we get V t s N t s = t X k=1 λt−kδsksvk. (5) Since vk depends on future rewards rk, Equation (5) can not be used in its current form. Next we note that vk has a self-consistency property with respect to the rewards. Specifically, the tail of the future discounted reward sum for each state depends on the empirical value at time t in the following way, vk = t−1 X u=k γu−kru + γt−kvt. Substituting this into Equation (5) and exchanging the order of the double sum, V t s N t s = t−1 X u=1 u X k=1 λt−kδsksγu−kru + t X k=1 λt−kδsksγt−kvt = t−1 X u=1 λt−u u X k=1 (λγ)u−kδsksru + t X k=1 (λγ)t−kδsksvt = Rt s + Et svt, where Et s := Pt k=1(λγ)t−kδsks is the eligibility trace of state s, and Rt s := Pt−1 u=1 λt−uEu s ru is the discounted reward with eligibility. Et s and Rt s depend only on quantities known at time t. The only unknown quantity is vt, which we have to replace with our current estimate of this value at time t, which is V t st. In other words, we bootstrap our estimates. This gives us, V t s N t s = Rt s + Et sV t st. (6) For state s = st, this simplifies to V t st = Rt st/(N t st −Et st). Substituting this back into Equation (6) we obtain, V t s N t s = Rt s + Et s Rt st N tst −Etst . (7) This gives us an explicit expression for our V estimates. However, from an algorithmic perspective an incremental update rule is more convenient. To derive this we make use of the relations, N t+1 s = λN t s + δst+1s, Et+1 s = λγEt s + δst+1s, Rt+1 s = λRt s + λEt srt, with N 0 s = E0 s = R0 s = 0. Inserting these into Equation (7) with t replaced by t + 1, V t+1 s N t+1 s = Rt+1 s + Et+1 s Rt+1 st+1 N t+1 st+1 −Et+1 st+1 = λRt s + λEt srt + Et+1 s Rt st+1 + Et st+1rt N tst+1 −γEtst+1 . By solving Equation (6) for Rt s and substituting back in, V t+1 s N t+1 s = λ V t s N t s −Et sV t st + λEt srt + Et+1 s N t st+1V t st+1 −Et st+1V t st + Et st+1rt N tst+1 −γEtst+1 = λN t s + δst+1s V t s −δst+1sV t s −λEt sV t st + λEt srt + Et+1 s N t st+1V t st+1 −Et st+1V t st + Et st+1rt N tst+1 −γEtst+1 . Dividing through by N t+1 s (= λN t s+ δst+1s), V t+1 s = V t s + −δst+1sV t s −λEt sV t st + λEt srt λN ts+ δst+1s 3 + (λγEt s + δst+1s)(N t st+1V t st+1 −Et st+1V t st + Et st+1rt) (N tst+1 −γEtst+1)(λN ts+ δst+1s) . Making the first denominator the same as the second, then expanding the numerator, V t+1 s = V t s + λEt srtN t st+1 −λEt sV t stN t st+1 −δst+1sV t s N t st+1 −λγEt st+1Et srt (N tst+1 −γEtst+1)(λN ts+ δst+1s) + λγEt st+1Et sV t st + γEt st+1V t s δst+1s + λγEt sN t st+1V t st+1 −λγEt sEt st+1V t st (N tst+1 −γEtst+1)(λN ts+ δst+1s) + λγEt sEt st+1rt + δst+1sN t st+1V t st+1 −δst+1sEt st+1V t st + δst+1sEt st+1rt (N tst+1 −γEtst+1)(λN ts+ δst+1s) . After cancelling equal terms (keeping in mind that in every term with a Kronecker δxy factor we may assume that x = y as the term is always zero otherwise), and factoring out Et s we obtain, V t+1 s = V t s + Et s λrtN t st+1 −λV t stN t st+1 + γV t s δst+1s + λγN t st+1V t st+1 −δst+1sV t st + δst+1srt (N tst+1 −γEtst+1)(λN ts+ δst+1s) Finally, by factoring out λN t st+1 + δst+1s we obtain our update rule, V t+1 s = V t s + Et s βt(s, st+1) rt + γV t st+1 −V t st , (8) where the learning rate is given by, βt(s, st+1) := 1 N tst+1 −γEtst+1 N t st+1 N ts . (9) Examining Equation (8), we find the usual update equation for temporal difference learning with eligibility traces (see Equation (2)), however the learning rate α has now been replaced by βt(s, st+1). This learning rate was derived from statistical principles by minimising the squared loss between the estimated and true state value. In the derivation we have exploited the fact that the latter must be self-consistent and then bootstrapped to get Equation (6). This gives us an equation for the learning rate for each state transition at time t, as opposed to the standard temporal difference learning where the learning rate α is either a fixed free parameter for all transitions, or is decreased over time by some monotonically decreasing function. In either case, the learning rate is not automatic and must be experimentally tuned for good performance. The above derivation appears to theoretically solve this problem. The first term in βt seems to provide some type of normalisation to the learning rate, though the intuition behind this is not clear to us. The meaning of second term however can be understood as follows: N t s measures how often we have visited state s in the recent past. Therefore, if N t s ≪ N t st+1 then state s has a value estimate based on relatively few samples, while state st+1 has a value estimate based on relatively many samples. In such a situation, the second term in βt boosts the learning rate so that V t+1 s moves more aggressively towards the presumably more accurate rt +γV t st+1. In the opposite situation when st+1 is a less visited state, we see that the reverse occurs and the learning rate is reduced in order to maintain the existing value of Vs. 3 A simple Markov process For our first test we consider a simple Markov process with 51 states. In each step the state number is either incremented or decremented by one with equal probability, unless the system is in state 0 or 50 in which case it always transitions to state 25 in the following step. When the state transitions from 0 to 25 a reward of 1.0 is generated, and for a transition from 50 to 25 a reward of -1.0 is generated. All other transitions have a reward of 0. We set the discount value γ = 0.99 and then computed the true discounted value of each state by running a brute force Monte Carlo simulation. We ran our algorithm 10 times on the above Markov chain and computed the root mean squared error in the value estimate across the states at each time step averaged across each run. The optimal 4 0.0 0.5 1.0 1.5 2.0 Time x1e+4 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 RMSE HL(1.0) TD(0.9) a = 0.1 TD(0.9) a = 0.2 Figure 1: 51 state Markov process averaged over 10 runs. The parameter a is the learning rate α. 0.0 0.5 1.0 1.5 2.0 Time x1e+4 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 RMSE HL(1.0) TD(0.9) a = 8.0/sqrt(t) TD(0.9) a = 2.0/cbrt(t) Figure 2: 51 state Markov process averaged over 300 runs. value of λ for HL(λ) was 1.0, which was to be expected given that the environment is stationary and thus discounting old experience is not helpful. For TD(λ) we tried various different learning rates and values of λ. We could find no settings where TD(λ) was competitive with HL(λ). If the learning rate α was set too high the system would learn as fast as HL(λ) briefly before becoming stuck. With a lower learning rate the final performance was improved, however the initial performance was now much worse than HL(λ). The results of these tests appear in Figure 1. Similar tests were performed with larger and smaller Markov chains, and with different values of γ. HL(λ) was consistently superior to TD(λ) across these tests. One wonders whether this may be due to the fact that the implicit learning rate that HL(λ) uses is not fixed. To test this we explored the performance of a number of different learning rate functions on the 51 state Markov chain described above. We found that functions of the form κ t always performed poorly, however good performance was possible by setting κ correctly for functions of the form κ √ t and κ 3√ t. As the results were much closer, we averaged over 300 runs. These results appear in Figure 2. With a variable learning rate TD(λ) is performing much better, however we were still unable to find an equation that reduced the learning rate in such a way that TD(λ) would outperform HL(λ). This is evidence that HL(λ) is adapting the learning rate optimally without the need for manual equation tuning. 4 Random Markov process To test on a Markov process with a more complex transition structure, we created a random 50 state Markov process. We did this by creating a 50 by 50 transition matrix where each element was set to 0 with probability 0.9, and a uniformly random number in the interval [0, 1] otherwise. We then scaled each row to sum to 1. Then to transition between states we interpreted the ith row as a probability distribution over which state follows state i. To compute the reward associated with each transition we created a random matrix as above, but without normalising. We set γ = 0.9 and then ran a brute force Monte Carlo simulation to compute the true discounted value of each state. The λ parameter for HL(λ) was simply set to 1.0 as the environment is stationary. For TD we experimented with a range of parameter settings and learning rate decrease functions. We found that a fixed learning rate of α = 0.2, and a decreasing rate of 1.5 3√ t performed reasonable well, but never as well as HL(λ). The results were generated by averaging over 10 runs, and are shown in Figure 3. Although the structure of this Markov process is quite different to that used in the previous experiment, the results are again similar: HL(λ) preforms as well or better than TD(λ) from the beginning to the end of the run. Furthermore, stability in the error towards the end of the run is better with HL(λ) and no manual learning tuning was required for these performance gains. 5 0 1000 2000 3000 4000 5000 Time 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 RMSE HL(1.0) TD(0.9) a = 0.2 TD(0.9) a = 1.5/cbrt(t) Figure 3: Random 50 state Markov process. The parameter a is the learning rate α. 0.0 0.5 1.0 1.5 2.0 Time x1e+4 0.00 0.05 0.10 0.15 0.20 0.25 0.30 RMSE HL(0.9995) TD(0.8) a = 0.05 TD(0.9) a = 0.05 Figure 4: 21 state non-stationary Markov process. 5 Non-stationary Markov process The λ parameter in HL(λ), introduced in Equation (4), reduces the importance of old observations when computing the state value estimates. When the environment is stationary this is not useful and so we can set λ = 1.0, however in a non-stationary environment we need to reduce this value so that the state values adapt properly to changes in the environment. The more rapidly the environment is changing, the lower we need to make λ in order to more rapidly forget old observations. To test HL(λ) in such a setting, we used the Markov chain from Section 3, but reduced its size to 21 states to speed up convergence. We used this Markov chain for the first 5,000 time steps. At that point, we changed the reward when transitioning from the last state to middle state to from -1.0 to be 0.5. At time 10,000 we then switched back to the original Markov chain, and so on alternating between the models of the environment every 5,000 steps. At each switch, we also changed the target state values that the algorithm was trying to estimate to match the current configuration of the environment. For this experiment we set γ = 0.9. As expected, the optimal value of λ for HL(λ) fell from 1 down to about 0.9995. This is about what we would expect given that each phase is 5,000 steps long. For TD(λ) the optimal value of λ was around 0.8 and the optimum learning rate was around 0.05. As we would expect, for both algorithms when we pushed λ above its optimal value this caused poor performance in the periods following each switch in the environment (these bad parameter settings are not shown in the results). On the other hand, setting λ too low produced initially fast adaption to each environment switch, but poor performance after that until the next environment change. To get accurate statistics we averaged over 200 runs. The results of these tests appear in Figure 4. For some reason HL(0.9995) learns faster than TD(0.8) in the first half of the first cycle, but only equally fast at the start of each following cycle. We are not sure why this is happening. We could improve the initial speed at which HL(λ) learnt in the last three cycles by reducing λ, however that comes at a performance cost in terms of the lowest mean squared error attained at the end of each cycle. In any case, in this non-stationary situation HL(λ) again performed well. 6 Windy Gridworld Reinforcement learning algorithms such as Watkins’ Q(λ) [8] and Sarsa(λ) [5, 4] are based on temporal difference updates. This suggests that new reinforcement learning algorithms based on HL(λ) should be possible. For our first experiment we took the standard Sarsa(λ) algorithm and modified it in the obvious way to use an HL temporal difference update. In the presentation of this algorithm we have changed notation slightly to make things more consistent with that typical in reinforcement learning. Specifically, we have dropped the t super script as this is implicit in the algorithm specification, and have 6 Algorithm 1 HLS(λ) Initialise Q(s, a) = 0, N(s, a) = 1 and E(s, a) = 0 for all s, a Initialise s and a repeat Take action a, observed r, s′ Choose a′ by using ǫ-greedy selection on Q(s′, ·) ∆←r + γQ(s′, a′) −Q(s, a) E(s, a) ←E(s, a) + 1 N(s, a) ←N(s, a) + 1 for all s, a do β((s, a), (s′, a′)) ← 1 N(s′,a′)−γE(s′,a′) N(s′,a′) N(s,a) end for for all s, a do Q(s, a) ←Q(s, a) + β (s, a), (s′, a′) E(s, a)∆ E(s, a) ←γλE(s, a) N(s, a) ←λN(s, a) end for s ←s′; a ←a′ until end of run defined Q(s, a) := V(s,a), E(s, a) := E(s,a) and N(s, a) := N(s,a). Our new reinforcement learning algorithm, which we call HLS(λ) is given in Algorithm 1. Essentially the only changes to the standard Sarsa(λ) algorithm have been to add code to compute the visit counter N(s, a), add a loop to compute the β values, and replace α with β in the temporal difference update. To test HLS(λ) against standard Sarsa(λ) we used the Windy Gridworld environment described on page 146 of [6]. This world is a grid of 7 by 10 squares that the agent can move through by going either up, down, left of right. If the agent attempts to move off the grid it simply stays where it is. The agent starts in the 4th row of the 1st column and receives a reward of 1 when it finds its way to the 4th row of the 8th column. To make things more difficult, there is a “wind” blowing the agent up 1 row in columns 4, 5, 6, and 9, and a strong wind of 2 in columns 7 and 8. This is illustrated in Figure 5. Unlike in the original version, we have set up this problem to be a continuing discounted task with an automatic transition from the goal state back to the start state. We set γ = 0.99 and in each run computed the empirical future discounted reward at each point in time. As this value oscillated we also ran a moving average through these values with a window length of 50. Each run lasted for 50,000 time steps as this allowed us to see at what level each learning algorithm topped out. These results appear in Figure 6 and were averaged over 500 runs to get accurate statistics. Despite putting considerable effort into tuning the parameters of Sarsa(λ), we were unable to achieve a final future discounted reward above 5.0. The settings shown on the graph represent the best final value we could achieve. In comparison HLS(λ) easily beat this result at the end of the run, while being slightly slower than Sarsa(λ) at the start. By setting λ = 0.99 we were able to achieve the same performance as Sarsa(λ) at the start of the run, however the performance at the end of the run was then only slightly better than Sarsa(λ). This combination of superior performance and fewer parameters to tune suggest that the benefits of HL(λ) carry over into the reinforcement learning setting. Another popular reinforcement learning algorithm is Watkins’ Q(λ). Similar to Sarsa(λ) above, we simply inserted the HL(λ) temporal difference update into the usual Q(λ) algorithm in the obvious way. We call this new algorithm HLQ(λ)(not shown). The test environment was exactly the same as we used with Sarsa(λ) above. The results this time were more competitive (these results are not shown). Nevertheless, despite spending a considerable amount of time fine tuning the parameters of Q(λ), we were unable to beat HLQ(λ). As the performance advantage was relatively modest, the main benefit of HLQ(λ) was that it achieved this level of performance without having to tune a learning rate. 7 Figure 5: [Windy Gridworld] S marks the start state and G the goal state, at which the agent jumps back to S with a reward of 1. 0 1 2 3 4 5 Time x1e+4 0 1 2 3 4 5 6 Future Discounted Reward HLS(0.995) e = 0.003 Sarsa(0.5) a = 0.4 e = 0.005 Figure 6: Sarsa(λ) vs. HLS(λ) in the Windy Gridworld. Performance averaged over 500 runs. On the graph, e represents the exploration parameter ǫ, and a the learning rate α. 7 Conclusions We have derived a new equation for setting the learning rate in temporal difference learning with eligibility traces. The equation replaces the free learning rate parameter α, which is normally experimentally tuned by hand. In every setting tested, be it stationary Markov chains, non-stationary Markov chains or reinforcement learning, our new method produced superior results. To further our theoretical understanding, the next step would be to try to prove that the method converges to correct estimates. This can be done for TD(λ) under certain assumptions on how the learning rate decreases over time. Hopefully, something similar can be proven for our new method. In terms of experimental results, it would be interesting to try different types of reinforcement learning problems and to more clearly identify where the ability to set the learning rate differently for different state transition pairs helps performance. It would also be good to generalise the result to episodic tasks. Finally, just as we have successfully merged HL(λ) with Sarsa(λ) and Watkins’ Q(λ), we would also like to see if the same can be done with Peng’s Q(λ) [3], and perhaps other reinforcement learning algorithms. Acknowledgements This research was funded by the Swiss NSF grant 200020-107616. References [1] A. P. George and W. B. Powell. Adaptive stepsizes for recursive estimation with applications in approximate dynamic programming. Journal of Machine Learning, 65(1):167–198, 2006. [2] M. G. Lagoudakis and R. Parr. Least-squares policy iteration. Journal of Machine Learning Research, 4:1107–1149, 2003. [3] J. Peng and R. J. Williams. Increamental multi-step Q-learning. Machine Learning, 22:283–290, 1996. [4] G. A. Rummery. Problem solving with reinforcement learning. PhD thesis, Cambridge University, 1995. [5] G. A. Rummery and M. Niranjan. On-line Q-learning using connectionist systems. Technial Report CUED/F-INFENG/TR 166, Engineering Department, Cambridge University, 1994. [6] R. Sutton and A. Barto. Reinforcement learning: An introduction. Cambridge, MA, MIT Press, 1998. [7] R. S. Sutton. Learning to predict by the methods of temporal differences. Machine Learning, 3:9–44, 1988. [8] C.J.C.H Watkins. Learning from Delayed Rewards. PhD thesis, King’s College, Oxford, 1989. [9] I. H. Witten. An adaptive optimal controller for discrete-time markov environments. Information and Control, 34:286–295, 1977. 8
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Bayes-Adaptive POMDPs St´ephane Ross McGill University Montr´eal, Qc, Canada sross12@cs.mcgill.ca Brahim Chaib-draa Laval University Qu´ebec, Qc, Canada chaib@ift.ulaval.ca Joelle Pineau McGill University Montr´eal, Qc, Canada jpineau@cs.mcgill.ca Abstract Bayesian Reinforcement Learning has generated substantial interest recently, as it provides an elegant solution to the exploration-exploitation trade-off in reinforcement learning. However most investigations of Bayesian reinforcement learning to date focus on the standard Markov Decision Processes (MDPs). Our goal is to extend these ideas to the more general Partially Observable MDP (POMDP) framework, where the state is a hidden variable. To address this problem, we introduce a new mathematical model, the Bayes-Adaptive POMDP. This new model allows us to (1) improve knowledge of the POMDP domain through interaction with the environment, and (2) plan optimal sequences of actions which can tradeoff between improving the model, identifying the state, and gathering reward. We show how the model can be finitely approximated while preserving the value function. We describe approximations for belief tracking and planning in this model. Empirical results on two domains show that the model estimate and agent’s return improve over time, as the agent learns better model estimates. 1 Introduction In many real world systems, uncertainty can arise in both the prediction of the system’s behavior, and the observability of the system’s state. Partially Observable Markov Decision Processes (POMDPs) take both kinds of uncertainty into account and provide a powerful model for sequential decision making under these conditions. However most solving methods for POMDPs assume that the model is known a priori, which is rarely the case in practice. For instance in robotics, the POMDP must reflect exactly the uncertainty on the robot’s sensors and actuators. These parameters are rarely known exactly and therefore must often be approximated by a human designer, such that even if this approximate POMDP could be solved exactly, the resulting policy may not be optimal. Thus we seek a decision-theoretic planner which can take into account the uncertainty over model parameters during the planning process, as well as being able to learn from experience the values of these unknown parameters. Bayesian Reinforcement Learning has investigated this problem in the context of fully observable MDPs [1, 2, 3]. An extension to POMDP has recently been proposed [4], yet this method relies on heuristics to select actions that will improve the model, thus forgoing any theoretical guarantee on the quality of the approximation, and on an oracle that can be queried to provide the current state. In this paper, we draw inspiration from the Bayes-Adaptive MDP framework [2], which is formulated to provide an optimal solution to the exploration-exploitation trade-off. To extend these ideas to POMDPs, we face two challenges: (1) how to update Dirichlet parameters when the state is a hidden variable? (2) how to approximate the infinite dimensional belief space to perform belief monitoring and compute the optimal policy. This paper tackles both problem jointly. The first problem is solved by including the Dirichlet parameters in the state space and maintaining belief states over these parameters. We address the second by bounding the space of Dirichlet parameters to a finite subspace necessary for ǫ-optimal solutions. We provide theoretical results for bounding the state space while preserving the value function and we use these results to derive approximate solving and belief monitoring algorithms. We compare several belief approximations in two problem domains. Empirical results show that the agent is able to learn good POMDP models and improve its return as it learns better model estimate. 2 POMDP A POMDP is defined by finite sets of states S, actions A and observations Z. It has transition probabilities {T sas′}s,s′∈S,a∈A where T sas′ = Pr(st+1 = s′|st = s, at = a) and observation probabilities {Osaz}s∈S,a∈A,z∈Z where Osaz = Pr(zt = z|st = s, at−1 = a). The reward function R : S × A →R specifies the immediate reward obtained by the agent. In a POMDP, the state is never observed. Instead the agent perceives an observation z ∈Z at each time step, which (along with the action sequence) allows it to maintain a belief state b ∈∆S. The belief state specifies the probability of being in each state given the history of action and observation experienced so far, starting from an initial belief b0. It can be updated at each time step using Baye’s rule: bt+1(s′) = Os′atzt+1 P s∈S T sats′bt(s) P s′′∈s Os′′atzt+1 P s∈S T sats′′bt(s). A policy π : ∆S → A indicates how the agent should select actions as a function of the current belief. Solving a POMDP involves finding the optimal policy π∗ that maximizes the expected discounted return over the infinite horizon. The return obtained by following π∗from a belief b is defined by Bellman’s equation: V ∗(b) = maxa∈A P s∈S b(s)R(s, a) + γ P z∈Z Pr(z|b, a)V ∗(τ(b, a, z)) , where τ(b, a, z) is the new belief after performing action a and observation z and γ ∈[0, 1) is the discount factor. Exact solving algorithms [5] are usually intractable, except on small domains with only a few states, actions and observations. Various approximate algorithms, both offline [6, 7, 8] and online [9], have been proposed to tackle increasingly large domains. However, all these methods requires full knowledge of the POMDP model, which is a strong assumption in practice. Some approaches do not require knowledge of the model, as in [10], but these approaches generally require a lot of data and do not address the exploration-exploitation tradeoff. 3 Bayes-Adaptive POMDP In this section, we introduce the Bayes-Adaptive POMDP (BAPOMDP) model, an optimal decisiontheoretic algorithm for learning and planning in POMDPs under parameter uncertainty. Throughout we assume that the state, action, and observation spaces are finite and known, but that the transition and observation probabilities are unknown or partially known. We also assume that the reward function is known as it is generally specified by the user for the specific task he wants to accomplish, but the model can easily be generalised to learn the reward function as well. To model the uncertainty on the transition T sas′ and observation Osaz parameters, we use Dirichlet distributions, which are probability distributions over the parameters of multinomial distributions. Given φi, the number of times event ei has occurred over n trials, the probabilities pi of each event follow a Dirichlet distribution, i.e. (p1, . . . , pk) ∼Dir(φ1, . . . , φk). This distribution represents the probability that a discrete random variable behaves according to some probability distribution (p1, . . . , pk), given that the counts (φ1, . . . , φk) have been observed over n trials (n = Pk i=1 φi). Its probability density function is defined by: f(p, φ) = 1 B(φ) Qk i=1 pφi−1 i , where B is the multinomial beta function. The expected value of pi is E(pi) = φi Pk j=1 φj . 3.1 The BAPOMDP Model The BAPOMDP is constructed from the model of the POMDP with unknown parameters. Let (S, A, Z, T, O, R, γ) be that model. The uncertainty on the distributions T sa· and Os′a· can be represented by experience counts: φa ss′∀s′ represents the number of times the transition (s, a, s′) occurred, similarly ψa s′z∀z is the number of times observation z was made in state s′ after doing action a. Let φ be the vector of all transition counts and ψ be the vector of all observation counts. Given the count vectors φ and ψ, the expected transition probability for T sas′ is: T sas′ φ = φa ss′ P s′′∈S φa ss′′ , and similarly for Os′az: Os′az ψ = ψa s′z P z′∈Z ψa s′z′ . The objective of the BAPOMDP is to learn an optimal policy, such that actions are chosen to maximize reward taking into account both state and parameter uncertainty. To model this, we follow the Bayes-Adaptive MDP framework, and include the φ and ψ vectors in the state of the BAPOMDP. Thus, the state space S′ of the BAPOMDP is defined as S′ = S × T × O, where T = {φ ∈N|S|2|A||∀(s, a), P s′∈S φa ss′ > 0} represents the space in which φ lies and O = {ψ ∈N|S||A||Z||∀(s, a), P z∈Z ψa sz > 0} represents the space in which ψ lies. The action and observation sets of the BAPOMDP are the same as in the original POMDP. Transition and observation functions of the BAPOMDP must capture how the state and count vectors φ, ψ evolve after every time step. Consider an agent in a given state s with count vectors φ and ψ, which performs action a, causing it to move to state s′ and observe z. Then the vector φ′ after the transition is defined as φ′ = φ + δa ss′, where δa ss′ is a vector full of zeroes, with a 1 for the count φa ss′, and the vector ψ′ after the observation is defined as ψ′ = ψ + δa s′z, where δa s′z is a vector full of zeroes, with a 1 for the count ψa s′z. Note that the probabilities of such transitions and observations occurring must be defined by considering all models and their probabilities as specified by the current Dirichlet distributions, which turn out to be their expectations. Hence, we define T ′ and O′ to be: T ′((s, φ, ψ), a, (s′, φ′, ψ′)) = T sas′ φ Os′az ψ , if φ′ = φ + δa ss′ and ψ′ = ψ + δa s′z 0, otherwise. (1) O′((s, φ, ψ), a, (s′, φ′, ψ′), z) = 1, if φ′ = φ + δa ss′ and ψ′ = ψ + δa s′z 0, otherwise. (2) Note here that the observation probabilities are folded into the transition function, and that the observation function becomes deterministic. This happens because a state transition in the BAPOMDP automatically specifies which observation is acquired after transition, via the way the counts are incremented. Since the counts do not affect the reward, the reward function of the BAPOMDP is defined as R′((s, φ, ψ), a) = R(s, a); the discount factor of the BAPOMDP remains the same. Using these definitions, the BAPOMDP has a known model specified by the tuple (S′, A, Z, T ′, O′, R′, γ). The belief state of the BAPOMDP represents a distribution over both states and count values. The model is learned by simply maintaining this belief state, as the distribution will concentrate over most likely models, given the prior and experience so far. If b0 is the initial belief state of the unknown POMDP, and the count vectors φ0 ∈T and ψ0 ∈O represent the prior knowledge on this POMDP, then the initial belief of the BAPOMDP is: b′ 0(s, φ0, ψ0) = {b0(s), if (φ, ψ) = (φ0, ψ0); 0, otherwise}. After actions are taken, the uncertainty on the POMDP model is represented by mixtures of Dirichlet distributions (i.e. mixtures of count vectors). Note that the BAPOMDP is in fact a POMDP with a countably infinite state space. Hence the belief update function and optimal value function are still defined as in Section 2. However these functions now require summations over S′ = S × T × O. Maintaining the belief state is practical only if the number of states with non-zero probabilities is finite. We prove this in the following theorem: Theorem 3.1. Let (S′, A, Z, T ′, O′, R′, γ) be a BAPOMDP constructed from the POMDP (S, A, Z, T, O, R, γ). If S is finite, then at any time t, the set S′ b′ t = {σ ∈S′|b′ t(σ) > 0} has size |S′ b′ t| ≤|S|t+1. Proof. Proof available in [11]. Proceeds by induction from b′ 0. The proof of this theorem suggests that it is sufficient to iterate over S and S′ b′ t−1 in order to compute the belief state b′ t when an action and observation are taken in the environment. Hence, Algorithm 3.1 can be used to update the belief state. 3.2 Exact Solution for BAPOMDP in Finite Horizons The value function of a BAPOMDP for finite horizons can be represented by a finite set Γ of functions α : S′ →R, as in standard POMDP. For example, an exact solution can be computed using function τ(b, a, z) Initialize b′ as a 0 vector. for all (s, φ, ψ, s′) ∈S′ b × S do b′(s′, φ + δa ss′, ψ + δa s′z) ←b′(s′, φ + δa ss′, ψ + δa s′z) + b(s, φ, ψ)T sas′ φ Os′az ψ end for return normalized b′ Algorithm 3.1: Exact Belief Update in BAPOMDP. dynamic programming (see [5] for more details): Γa 1 = {αa|αa(s, φ, ψ) = R(s, a)}, Γa,z t = {αa,z i |αa,z i (s, φ, ψ) = γ P s′∈S T sas′ φ Os′az ψ α′ i(s′, φ + δa ss′, ψ + δa s′z), α′ i ∈Γt−1}, Γa t = Γa 1 ⊕Γa,z1 t ⊕Γa,z2 t ⊕· · · ⊕Γ a,z|Z| t , (where ⊕is the cross sum operator), Γt = S a∈A Γa t . (3) Note here that the definition of αa,z i (s, φ, ψ) is obtained from the fact that T ′((s, φ, ψ), a, (s′, φ′, ψ′))O′((s, φ, ψ), a, (s′, φ′, ψ′), z) = 0 except when φ′ = φ + δa ss′ and ψ′ = ψ + δa s′z. The optimal policy is extracted as usual: πΓ(b) = argmaxα∈Γ P σ∈S′ b α(σ)b(σ). In practice, it will be impossible to compute αa,z i (s, φ, ψ) for all (s, φ, ψ) ∈S′. In order to compute these more efficiently, we show in the next section that the infinite state space can be reduced to a finite state space, while still preserving the value function to arbitrary precision for any horizon t. 4 Approximating the BAPOMDP: Theory and Algorithms Solving a BAPOMDP exactly for all belief states is impossible in practice due to the dimensionnality of the state space (in particular to the fact that the count vectors can grow unbounded). We now show how we can reduce this infinite state space to a finite state space. This allows us to compute an ǫoptimal value function over the resulting finite-dimensionnal belief space using standard POMDP techniques. Various methods for belief tracking in the infinite model are also presented. 4.1 Approximate Finite Model We first present an upper bound on the value difference between two states that differ only by their model estimate φ and ψ. This bound uses the following definitions: given φ, φ′ ∈T , and ψ, ψ′ ∈O, define Dsa S (φ, φ′) = P s′∈S T sas′ φ −T sas′ φ′ and Dsa Z (ψ, ψ′) = P z∈Z Osaz ψ −Osaz ψ′ , and N sa φ = P s′∈S φa ss′ and N sa ψ = P z∈Z ψa sz. Theorem 4.1. Given any φ, φ′ ∈T , ψ, ψ′ ∈O, and γ ∈(0, 1), then for all t: sup αt∈Γt,s∈S |αt(s, φ, ψ) −αt(s, φ′, ψ′)| ≤2γ||R||∞ (1−γ)2 sup s,s′∈S,a∈A h Dsa S (φ, φ′) + Ds′a Z (ψ, ψ′) + 4 ln(γ−e) P s′′∈S|φa ss′′−φ′a ss′′| (N sa φ +1)(N sa φ′ +1) + P z∈Z|ψa s′z−ψ′a s′z| (N s′a ψ +1)(N s′a ψ′ +1) Proof. Proof available in [11] finds a bound on a 1-step backup and solves the recurrence. We now use this bound on the α-vector values to approximate the space of Dirichlet parameters within a finite subspace. We use the following definitions: given any ǫ > 0, define ǫ′ = ǫ(1−γ)2 8γ||R||∞, ǫ′′ = ǫ(1−γ)2 ln(γ−e) 32γ||R||∞ , N ǫ S = max |S|(1+ǫ′) ǫ′ , 1 ǫ′′ −1 and N ǫ Z = max |Z|(1+ǫ′) ǫ′ , 1 ǫ′′ −1 . Theorem 4.2. Given any ǫ > 0 and (s, φ, ψ) ∈S′ such that ∃a ∈A, s′ ∈S, N s′a φ > N ǫ S or N s′a ψ > N ǫ Z, then ∃(s, φ′, ψ′) ∈S′ such that ∀a ∈A, s′ ∈S, N s′a φ′ ≤N ǫ S and N s′a ψ′ ≤N ǫ Z where |αt(s, φ, ψ) −αt(s, φ′, ψ′)| < ǫ holds for all t and αt ∈Γt. Proof. Proof available in [11]. Theorem 4.2 suggests that if we want a precision of ǫ on the value function, we just need to restrict the space of Dirichlet parameters to count vectors φ ∈˜Tǫ = {φ ∈N|S|2|A||∀a ∈A, s ∈S, 0 < N sa φ ≤N ǫ S} and ψ ∈˜Oǫ = {ψ ∈N|S||A||Z||∀a ∈A, s ∈S, 0 < N sa ψ ≤N ǫ Z}. Since ˜Tǫ and ˜Oǫ are finite, we can define a finite approximate BAPOMDP as the tuple ( ˜Sǫ, A, Z, ˜Tǫ, ˜Oǫ, ˜Rǫ, γ) where ˜Sǫ = S × ˜Tǫ × ˜Oǫ is the finite state space. To define the transition and observation functions over that finite state space, we need to make sure that when the count vectors are incremented, they stay within the finite space. To achieve, this we define a projection operator Pǫ : S′ →˜Sǫ that simply projects every state in S′ to their closest state in ˜Sǫ. Definition 4.1. Let d : S′ × S′ →R be defined such that: d(s, φ, ψ, s′, φ′, ψ′) = 2γ||R||∞ (1−γ)2 sup s,s′∈S,a∈A h Dsa S (φ, φ′) + Ds′a Z (ψ, ψ′) + 4 ln(γ−e) P s′′∈S |φa ss′′−φ′a ss′′| (N as φ +1)(N as φ′ +1) + P z∈Z |ψa s′z−ψ′a s′z| (N as′ ψ +1)(N as′ ψ′ +1) , if s = s′ 8γ||R||∞ (1−γ)2 1 + 4 ln(γ−e) + 2||R||∞ (1−γ) , otherwise. Definition 4.2. Let Pǫ : S′ →˜Sǫ be defined as Pǫ(s) = arg min s′∈˜Sǫ d(s, s′) The function d uses the bound defined in Theorem 4.1 as a distance between states that only differs by their φ and ψ vectors, and uses an upper bound on that value when the states differ. Thus Pǫ always maps states (s, φ, ψ) ∈S′ to some state (s, φ′, ψ′) ∈˜Sǫ. Note that if σ ∈˜Sǫ, then Pǫ(σ) = σ. Using Pǫ, the transition and observation function are defined as follows: ˜Tǫ((s, φ, ψ), a, (s′, φ′, ψ′)) = T sas′ φ Os′az ψ , if (s′, φ′, ψ′) = Pǫ(s′, φ + δa ss′, ψ + δa s′z) 0, otherwise. (4) ˜Oǫ((s, φ, ψ), a, (s′, φ′, ψ′), z) = 1, if (s′, φ′, ψ′) = Pǫ(s′, φ + δa ss′, ψ + δa s′z) 0, otherwise. (5) These definitions are the same as the one in the infinite BAPOMDP, except that now we add an extra projection to make sure that the incremented count vectors stays in ˜Sǫ. Finally, the reward function ˜Rǫ : ˜Sǫ × A →R is defined as ˜Rǫ((s, φ, ψ), a) = R(s, a). Theorem 4.3 bounds the value difference between α-vectors computed with this finite model and the α-vector computed with the original model. Theorem 4.3. Given any ǫ > 0, (s, φ, ψ) ∈S′ and αt ∈Γt computed from the infinite BAPOMDP. Let ˜αt be the α-vector representing the same conditionnal plan as αt but computed with the finite BAPOMDP ( ˜Sǫ, A, Z, ˜Tǫ, ˜Oǫ, ˜Rǫ, γ), then |˜αt(Pǫ(s, φ, ψ)) −αt(s, φ, ψ)| < ǫ 1−γ . Proof. Proof available in [11]. Solves a recurrence over the 1-step approximation in Thm. 4.2. Because the state space is now finite, solution methods from the literature on finite POMDPs could theoretically be applied. This includes en particular the equations for τ(b, a, z) and V ∗(b) that were presented in Section 2. In practice however, even though the state space is finite, it will generally be very large for small ǫ, such that it may still be intractable, even for small domains. We therefore favor a faster online solution approach, as described below. 4.2 Approximate Belief Monitoring As shown in Theorem 3.1, the number of states with non-zero probability grows exponentially in the planning horizon, thus exact belief monitoring can quickly become intractable. We now discuss different particle-based approximations that allow polynomial-time belief tracking. Monte Carlo sampling: Monte Carlo sampling algorithms have been widely used for sequential state estimation [12]. Given a prior belief b, followed by action a and observation z, the new belief b′ is obtained by first sampling K states from the distribution b, then for each sampled s a new state s′ is sampled from T(s, a, ·). Finally, the probability O(s′, a, z) is added to b′(s′) and the belief b′ is re-normalized. This will capture at most K states with non-zero probabilities. In the context of BAPOMDPs, we use a slight variation of this method, where (s, φ, ψ) are first sampled from b, and then a next state s′ ∈S is sampled from the normalized distribution T sa· φ O·az ψ . The probability 1/K is added directly to b′(s′, φ + δa ss′, ψ + δa s′z). Most Probable: Alternately, we can do the exact belief update at a given time step, but then only keep the K most probable states in the new belief b′ and renormalize b′. Weighted Distance Minimization: The two previous methods only try to approximate the distribution τ(b, a, z). However, in practice, we only care most about the agent’s expected reward. Hence, instead of keeping the K most likely states, we can keep K states which best approximate the belief’s value. As in the Most Probable method, we do an exact belief update, however in this case we fit the posterior distribution using a greedy K-means procedure, where distance is defined as in Definition 4.1, weighted by the probability of the state to remove. See [11] for algorithmic details. 4.3 Online planning While the finite model presented in Section 4.1 can be used to find provably near-optimal policies offline, this will likely be intractable in practice due to the very large state space required to ensure good precision. Instead, we turn to online lookahead search algorithms, which have been proposed for solving standard POMDPs [9]. Our approach simply performs dynamic programming over all the beliefs reachable within some fixed finite planning horizon from the current belief. The action with highest return over that finite horizon is executed and then planning is conducted again on the next belief. To further limit the complexity of the online planning algorithm, we used the approximate belief monitoring methods detailed above. Its overall complexity is in O((|A||Z|)DCb) where D is the planning horizon and Cb is the complexity of updating the belief. 5 Empirical Results We begin by evaluating the different belief approximations introduced above. To do so, we use a simple online d-step lookahead search, and compare the overall expected return and model accuracy in two different problems: the well-known Tiger [5] and a new domain called Follow. Given T sas′ and Os′az the exact probabilities of the (unknown) POMDP, the model accuracy is measured in terms of the weighted sum of L1-distance, denoted WL1, between the exact model and the probable models in a belief state b: WL1(b) = P (s,φ,ψ)∈S′ b b(s, φ, ψ)L1(φ, ψ) L1(φ, ψ) = P a∈A P s′∈S hP s∈S |T sas′ φ −T sas′| + P z∈Z |Os′az ψ −Os′az| i (6) 5.1 Tiger In the Tiger problem [5], we consider the case where the transition and reward parameters are known, but the observation probabilities are not. Hence, there are four unknown parameters: OLl, OLr, ORl, ORr (OLr stands for Pr(z = hear right|s = tiger Left, a = Listen)). We define the observation count vector ψ = (ψLl, ψLr, ψRl, ψRr). We consider a prior of ψ0 = (5, 3, 3, 5), which specifies an expected sensor accuracy of 62.5% (instead of the correct 85%) in both states. Each simulation consists of 100 episodes. Episodes terminate when the agent opens a door, at which point the POMDP state (i.e. tiger’s position) is reset, but the distribution over count vector is carried over to the next episode. Figures 1 and 2 show how the average return and model accuracy evolve over the 100 episodes (results are averaged over 1000 simulations), using an online 3-step lookahead search with varying belief approximations and parameters. Returns obtained by planning directly with the prior and exact model (without learning) are shown for comparison. Model accuracy is measured on the initial belief of each episode. Figure 3 compares the average planning time per action taken by each approach. We observe from these figures that the results for the Most Probable and Weighted Distance approximations are very similar and perform well even with few particles (lines are overlapping in many places, making Weighted Distance results hard to see). On the other hand, the performance of Monte Carlo is significantly affected by the number of particles and had to use much more par0 20 40 60 80 100 −4 −3 −2 −1 0 1 2 Episode Return Most Probable (2) Monte Carlo (64) Weighted Distance (2) Prior model Exact model Figure 1: Return with different belief approximations. 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 Episode WL1 Most Probable (2) Monte Carlo (64) Weighted Distance (2) Figure 2: Model accuracy with different belief approximations. MP (2) MC (64) WD (2) 0 5 10 15 20 Planning Time/Action (ms) Figure 3: Planning Time with different belief approximations. ticles (64) to obtain an improvement over the prior. This may be due to the sampling error that is introduced when using fewer samples. 5.2 Follow We propose a new POMDP domain, called Follow, inspired by an interactive human-robot task. It is often the case that such domains are particularly subject to parameter uncertainty (due to the difficulty in modelling human behavior), thus this environment motivates the utility of Bayes-Adaptive POMDP in a very practical way. The goal of the Follow task is for a robot to continuously follow one of two individuals in a 2D open area. The two subjects have different motion behavior, requiring the robot to use a different policy for each. At every episode, the target person is selected randomly with Pr = 0.5 (and the other is not present). The person’s identity is not observable (except through their motion). The state space has two features: a binary variable indicating which person is being followed, and a position variable indicating the person’s position relative to the robot (5×5 square grid with the robot always at the center). Initially, the robot and person are at the same position. Both the robot and the person can perform five motion actions {NoAction, North, East, South, West}. The person follows a fixed stochastic policy (stationary over space and time), but the parameters of this behavior are unknown. The robot perceives observations indicating the person’s position relative to the robot: {Same, North, East, South, West, Unseen}. The robot perceives the correct observation Pr = 0.8 and Unseen with Pr = 0.2. The reward R = +1 if the robot and person are at the same position (central grid cell), R = 0 if the person is one cell away from the robot, and R = −1 if the person is two cells away. The task terminates if the person reaches a distance of 3 cells away from the robot, also causing a reward of -20. We use a discount factor of 0.9. When formulating the BAPOMDP, the robot’s motion model (deterministic), the observation probabilities and the rewards are assumed to be known. We maintain a separate count vector for each person, representing the number of times they move in each direction, i.e. φ1 = (φ1 NA, φ1 N, φ1 E, φ1 S, φ1 W ), φ2 = (φ2 NA, φ2 N, φ2 E, φ2 S, φ2 W ). We assume a prior φ1 0 = (2, 3, 1, 2, 2) for person 1 and φ2 0 = (2, 1, 3, 2, 2) for person 2, while in reality person 1 moves with probabilities Pr = (0.3, 0.4, 0.2, 0.05, 0.05) and person 2 with Pr = (0.1, 0.05, 0.8, 0.03, 0.02). We run 200 simulations, each consisting of 100 episodes (of at most 10 time steps). The count vectors’ distributions are reset after every simulation, and the target person is reset after every episode. We use a 2-step lookahead search for planning in the BAPOMDP. Figures 4 and 5 show how the average return and model accuracy evolve over the 100 episodes (averaged over the 200 simulations) with different belief approximations. Figure 6 compares the planning time taken by each approach. We observe from these figures that the results for the Weighted Distance approximations are much better both in terms of return and model accuracy, even with fewer particles (16). Monte Carlo fails at providing any improvement over the prior model, which indicates it would require much more particles. Running Weighted Distance with 16 particles require less time than both Monte Carlo and Most Probable with 64 particles, showing that it can be more time efficient for the performance it provides in complex environment. 0 20 40 60 80 100 −8 −6 −4 −2 0 2 Episode Return Most Probable (64) Monte Carlo (64) Weighted Distance (16) Prior model Exact model Figure 4: Return with different belief approximations. 0 20 40 60 80 100 0 0.5 1 1.5 2 Episode WL1 Most Probable (64) Monte Carlo (64) Weighted Distance (16) Figure 5: Model accuracy with different belief approximations. MP (64) MC (64) WD (16) 0 50 100 150 200 Planning Time/Action (ms) Figure 6: Planning Time with different belief approximations. 6 Conclusion The objective of this paper was to propose a preliminary decision-theoretic framework for learning and acting in POMDPs under parameter uncertainty. This raises a number of interesting challenges, including (1) defining the appropriate model for POMDP parameter uncertainty, (2) approximating this model while maintaining performance guarantees, (3) performing tractable belief updating, and (4) planning action sequences which optimally trade-off exploration and exploitation. We proposed a new model, the Bayes-Adaptive POMDP, and showed that it can be approximated to ǫ-precision by a finite POMDP. We provided practical approaches for belief tracking and online planning in this model, and validated these using two experimental domains. Results in the Follow problem, showed that our approach is able to learn the motion patterns of two (simulated) individuals. This suggests interesting applications in human-robot interaction, where it is often essential that we be able to reason and plan under parameter uncertainty. Acknowledgments This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds Qu´eb´ecois de la Recherche sur la Nature et les Technologies (FQRNT). References [1] R. Dearden, N. Friedman, and N. Andre. Model based bayesian exploration. In UAI, 1999. [2] M. Duff. Optimal Learning: Computational Procedure for Bayes-Adaptive Markov Decision Processes. PhD thesis, University of Massachusetts, Amherst, USA, 2002. [3] P. Poupart, N. Vlassis, J. Hoey, and K. Regan. An analytic solution to discrete bayesian reinforcement learning. In Proc. ICML, 2006. [4] R. Jaulmes, J. Pineau, and D. Precup. Active learning in partially observable markov decision processes. In ECML, 2005. [5] L. P. Kaelbling, M. L. Littman, and A. R. Cassandra. Planning and acting in partially observable stochastic domains. Artificial Intelligence, 101:99–134, 1998. [6] J. Pineau, G. Gordon, and S. Thrun. Point-based value iteration: an anytime algorithm for POMDPs. In IJCAI, pages 1025–1032, Acapulco, Mexico, 2003. [7] M. Spaan and N. Vlassis. Perseus: randomized point-based value iteration for POMDPs. JAIR, 24:195– 220, 2005. [8] T. Smith and R. Simmons. Heuristic search value iteration for POMDPs. In UAI, Banff, Canada, 2004. [9] S. Paquet, L. Tobin, and B. Chaib-draa. An online POMDP algorithm for complex multiagent environments. In AAMAS, 2005. [10] Jonathan Baxter and Peter L. Bartlett. Infinite-horizon policy-gradient estimation. Journal of Artificial Intelligence Research (JAIR), 15:319–350, 2001. [11] St´ephane Ross, Brahim Chaib-draa, and Joelle Pineau. Bayes-adaptive pomdps. Technical Report SOCSTR-2007.6, McGill University, 2007. [12] A. Doucet, N. de Freitas, and N. Gordon. Sequential Monte Carlo Methods In Practice. Springer, 2001.
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Regulator Discovery from Gene Expression Time Series of Malaria Parasites: a Hierarchical Approach Jos´e Miguel Hern´andez-Lobato Escuela Polit´ecnica Superior Universidad Aut´onoma de Madrid, Madrid, Spain Josemiguel.hernandez@uam.es Tjeerd Dijkstra Leiden Malaria Research Group LUMC, Leiden, The Netherlands t.dijkstra@lumc.nl Tom Heskes Institute for Computing and Information Sciences Radboud University Nijmegen, Nijmegen, The Netherlands t.heskes@science.ru.nl Abstract We introduce a hierarchical Bayesian model for the discovery of putative regulators from gene expression data only. The hierarchy incorporates the knowledge that there are just a few regulators that by themselves only regulate a handful of genes. This is implemented through a so-called spike-and-slab prior, a mixture of Gaussians with different widths, with mixing weights from a hierarchical Bernoulli model. For efficient inference we implemented expectation propagation. Running the model on a malaria parasite data set, we found four genes with significant homology to transcription factors in an amoebe, one RNA regulator and three genes of unknown function (out of the top ten genes considered). 1 Introduction Bioinformatics provides a rich source for the application of techniques from machine learning. Especially the elucidation of regulatory networks underlying gene expression has lead to a cornucopia of approaches: see [1] for review. Here we focus on one aspect of network elucidation, the identification of the regulators of the causative agent of severe malaria, Plasmodium falciparum. Several properties of the parasite necessitate a tailored algorithm for regulator identification: • In most species gene regulation takes place at the first stage of gene expression when a DNA template is transcribed into mRNA. This transcriptional control is mediated by specific transcription factors. Few specific transcription factors have been identified in Plasmodium based on sequence homology with other species [2, 3]. This could be due to Plasmodium possessing a unique set of transcription factors or due to other mechanisms of gene regulation, e.g. at the level of mRNA stability or post-transcritional regulation. • Compared with yeast, gene expression in Plasmodium is hardly changed by perturbations e.g. by adding chemicals or changing temperature [4]. The biological interpretation of this finding is that the parasite is so narrowly adapted to its environment inside a red blood cell that it follows a stereotyped gene expression program. From a machine learning point of view, this finding means that network elucidation techniques relying on perturbations of gene expression cannot be used. • Similar to yeast [5], data for three different strains of the parasite with time series of gene expression are publicly available [6]. These assay all of Plasmodium’s 5,600 genes for about 50 time points. In contrast to yeast, there are no ChIP-chip data available and fewer then ten transcription factor binding motifs are known. 1 Together, these properties point to a vector autoregressive model making use of the gene expression time series. The model should not rely on sequence homology information but it should be flexible enough to integrate sequence information in the future. This points to a Bayesian model as favored approach. 2 The model We start with a semi-realistic model of transcription based on Michaelis-Menten kinetics [1] and subsequently simplify to obtain a linear model. Denoting the concentration of a certain mRNA transcript at time t by z(t) we write: dz(t) dt = V1a1(t)M1 K1 + a1(t)M1 · · · VNaN(t)MN KN + aN(t)MN p(t) −1 τz z(t), (1) with aj(t) the concentration of the j-th activator (positive regulator), p(t) the concentration of RNA polymerase and Vj, Kj, Mj and τz reaction constants. N denotes the number of potential activators. The activator is thought to bind to DNA motifs upstream of the transcription start site and binds RNA polymerase which reads the DNA template to produce an mRNA transcript. Mj can be thought of as the multiplicity of the motif, τz captures the characteristic life time of the transcript. While reasonably realistic, this equation harbors too many unknowns for reliable inference: 3N + 1 with N ≈1000. We proceed with several simplifications: • aj(t) ≪Kj: activator concentration is low; • p(t) = p0 is constant; • dz(t) dt ≈z(t+Δ)−z(t) Δ with Δ the sampling period; • Δ ≈τz: sampling period roughly equal to transcript life time. Counting time in units of Δ and taking logarithms on both sides, Equation (1) then simplifies to log z(t + 1) = C + M1 log a1(t) + · · · + MN log aN(t), with C = log(T V1 · · · VNp0/(K1 · · · KN)). This is a linear model for gene expression level given the expression levels of a set of activators. With a similar derivation one can include repressors [1]. 2.1 A Bayesian model for sparse linear regression Let y be a vector with the log expression of the target gene and X = (x1, . . . , xN) a matrix whose columns contain the log expression of the candidate regulators. Assuming that the measurements are corrupted with additive Gaussian noise, we get y ∼N(Xβ, σ2I) where β = (β1, . . . , βN)T is a vector of regression coefficients and σ2 is the variance of the noise. Such a linear model is commonly used [7, 8, 9]. Both y and x1, . . . , xN are mean-centered vectors with T measurements. We specify an inverse gamma (IG) prior for σ2 so that P(σ2) = IG(σ2, ν/2, νλ/2), where λ is a prior estimate of σ2 and ν is the sample size associated with that estimate. We assume that a priori all components βi are independent and take a so-called “spike and slab prior” [10] for each of them. That is, we introduce binary latent variables γi, with γi = 1 if xi takes part in the regression of y and γi = 0 otherwise. Given γ, the prior on β then reads P(β|γ) = N i=1 P(βi|γi) = N i=1 N(βi, 0, v1)γi N(βi, 0, v0)1−γi , where N(x, μ, σ2) denotes a Gaussian density with mean μ and variance σ2 evaluated at x. In order to enforce sparsity, the variance v1 of the slab should be larger than the variance v0 of the spike. Instead of picking the hyperparameters v1 and v0 directly, it is convenient to pick a threshold of practical significance δ so that P(γi = 1) gets more weight when |βi| > δ and P(γi = 0) gets more weight when |βi| < δ [10]. In this way, given δ and one of v1 or v0, we pick the other one such that δ2 = log(v1/v0) v−1 0 −v−1 1 . (2) 2 Finally, we assign independent Bernoulli priors to the components of the latent vector γ: P(γ) = N i=1 Bern(γi, w) = N i=1 wγi(1 −w)1−γi , so that each of the x1, . . . , xN can independently take part in the regression with probability w. We can identify the candidate genes whose expression is more likely to be correlated with the target gene by means of the posterior distribution of γ: P(γ|y, X) = β,σ2 P(γ, β, σ2|y, X) dβ dσ2 ∝ β,σ2 P(γ, β, σ2, y|X) dβ dσ2 , where P(γ, β, σ2, y|X) = N(y, Xβ, σ2I)P(β|γ)P(γ)P(σ2) = T t=1 N(yt, N i=1 xi,tβi, σ2) N i=1 N(βi, 0, v1)γi N(βi, 0, v0)1−γi N i=1 Bern(γi, w) IG(σ2, ν/2, νλ/2) . (3) Unfortunately, this posterior distribution cannot be computed exactly if the number N of candidate genes is larger than 25. An approximation based on Markov Chain Monte Carlo (MCMC) methods has been proposed in [11]. 2.2 A hierarchical model for gene regulation In the section above we made use of the prior information that a target gene is typically regulated by a small number of regulators. We have not yet made use of the prior information that a regulator typically regulates more than one gene. We incorporate this information by a hierarchical extension of our previous model. We introduce a vector τ of binary latent variables where τi = 1 if gene i is a regulator and τi = 0 otherwise. The following joint distribution captures this idea: P(τ, γ, β, σ2|X) = ⎡ ⎣ N j=1 T −1 t=1 N(xj,t+1, N i=1, i̸=j xi,tβj,i, σ2 j ) ⎤ ⎦ ⎡ ⎣ N j=1 N i=1,i̸=j N(βj,i, 0, v1)γj,i N(βj,i, 0, v0)1−γj,i ⎤ ⎦ ⎡ ⎣ N j=1 IG(σ2 j , νj/2, νjλj/2) ⎤ ⎦ ⎡ ⎣ N j=1 N i=1,i̸=j Bern(γj,i, w1)τi Bern(γj,i, w0)1−τi ⎤ ⎦ N i=1 Bern(τi, w) . (4) In this hierarchical model, γ is a matrix of binary latent variables where γj,i = 1 if gene i takes part in the regression of gene j and γj,i = 0 otherwise. The relationship between regulators and regulatees suggests that P(γj,i = 1|τi = 1) should be bigger than P(γj,i = 1|τi = 0) and thus w1 > w0. Matrix β contains regression coefficients where βj,i is the regression coefficient between the expression of gene i and the delayed expression of gene j. Hyperparameter w represents the prior probability of any gene being a regulator and the elements σ2 j of the vector σ2 contain the variance of the noise in each of the N regressions. Hyperparameters λj and νj have the same meaning as in the model for sparse linear regression. The corresponding plate model is illustrated in Figure 1. We can identify the genes more likely to be regulators by means of the posterior distribution P(τ|X). Compared with the sparse linear regression model we expanded the number of latent variables from O(N) to O(N 2). In order to keep inference feasible we turn to an approximate inference technique. 3 Figure 1: The hierarchical model for gene regulation. β γ τ σ λ ν 3 Expectation propagation The Expectation Propagation (EP) algorithm [12] allows to perform approximate Bayesian inference. In all Bayesian problems, the joint distribution of the model parameters θ and a data set D = {(xi, yi) : i = 1, . . . , n} with i.i.d. elements can be expressed as a product of terms P(θ, D) = n i=1 P(yi|xi, θ)P(θ) = n+1 i=1 ti(θ) , (5) where tn+1(θ) = P(θ) is the prior distribution for θ and ti(θ) = P(yi|xi, θ) for i = 1, . . . , n. Expectation propagation proceeds to approximate (5) with a product of simpler terms n+1 i=1 ti(θ) ≈ n+1 i=1 ˜ti(θ) = Q(θ) , (6) where all the term approximations ˜ti are restricted to belong to the same family F of exponential distributions, but they do not have to integrate 1. Note that Q will also be in F because F is closed under multiplication. Each term approximation ˜ti is chosen so that Q(θ) = ˜ti(θ) j̸=i ˜tj(θ) = ˜ti(θ)Q\i(θ) is as close as possible to ti(θ) j̸=i ˜tj(θ) = ti(θ)Q\i(θ) , in terms of the direct Kullback-Leibler (K-L) divergence. The pseudocode of the EP algorithm is: 1. Initialize the term approximations ˜ti and Q to be uniform. 2. Repeat until all ˜ti converge: (a) Choose a ˜ti to refine and remove it from Q to get Q\i (e.g. dividing Q by ˜ti). (b) Update the term ˜ti so that it minimizes the K-L divergence between tiQ\i and ˜tiQ\i. (c) Re-compute Q so that Q = ˜tiQ\i. The optimization problem in step (b) is solved by matching sufficient statistics between a distribution Q′ within the F family and tiQ\i, the new ˜ti is then equal to Q′/Q\i. Because Q belongs to the exponential family it is generally trivial to calculate its normalization constant. Once Q is normalized it can approximate P(θ|D). Finally, EP is not guaranteed to converge, although convergence can be improved by means of damped updates or double-loop algorithms [13]. 3.1 EP for sparse linear regression The application of EP to the models of Section 2 introduces some nontrivial technicalities. Furthermore, we describe several techniques to speed up the EP algorithm. We approximate P(γ, β, σ2, y|X) for sparse linear regression by means of a factorized exponential distribution: P(γ, β, σ2, y|X) ≈ N i=1 Bern(γi, qi)N(βi, μi, si) IG(σ2, a, b) ≡Q(γ, β, σ2) , (7) 4 where {qi, μi, si : i = 1, . . . , N}, a and b are free parameters. Note that in the approximation Q(γ, β, σ2) all the components of the vectors γ and β and the variable σ2 are considered to be independent; this allows the approximation of P(γ|y, X) by
n i=1 Bern(γi, qi). We tune the parameters of Q(γ, β, σ2) by means of EP over the unnormalized density P(γ, β, σ2, y|X). Such density appears in (3) as a product of T +N terms (not counting the priors) which correspond to the ti terms in (5). This way, we have T + N term approximations with the same form as (7) and which correspond to the term approximations ˜ti in (6). The complexity is O(T N) per iteration, because updating any of the first T term approximations requires N operations. However, some of the EP update operations require to compute integrals which do not have a closed form expression. To avoid that, we employ the following simplifications when we update the first T term approximations: 1. When updating the parameters {μi, si : i = 1, . . . , N} of the Gaussians in the term approximations, we approximate a Student’s t-distribution by means of a Gaussian distribution with the same mean and variance. This approximation becomes more accurate as the degrees of freedom of the t-distribution increase. 2. When updating the parameters {a, b} of the IG in the term approximations, instead of propagating the sufficient statistics of an IG distribution we propagate the expectations of 1/σ2 and 1/σ4. To achieve this, we have to perform two approximations like the one stated above. Note that in this case we are not minimizing the direct K-L divergence. However, at convergence, we expect the resulting IG in (7) to be sufficiently accurate. In order to improve convergence, we re-update all the N last term approximations each time one of the first T term approximations is updated. Computational complexity does not get worse than O(T N) and the resulting algorithm turns out to be faster. By comparison, the MCMC method in [11] takes O(N 2) steps to generate a single sample from P(γ|y, X). On problems of much smaller size than we will consider in our experiments, one typically requires on the order of 10000 samples to obtain reasonably accurate estimates [10]. 3.2 EP for gene regulation We approximate P(τ, γ, β, σ2|X) by the factorized exponential distribution Q(τ, γ, β, σ2) = ⎡ ⎣ N j=1 N i=1,i̸=j Bern(γj,i, wj,i) ⎤ ⎦ N i=1 Bern(τi, ti) ⎡ ⎣ N j=1 N i=1,i̸=j N(βj,i, μj,i, sj,i) ⎤ ⎦ ⎡ ⎣ N j=1 IG(σ2 j , aj, bj) ⎤ ⎦, where {aj, bj, ti, wj,i, μj,i, sj,i : i = 1, . . . , N ; j = 1, . . . , N ; i ̸= j} are free parameters. The posterior probability P(τ|X) that indicates which genes are more likely to be regulators can then be approximated by
N i=1 Bern(τi, ti). Again, we fix the parameters in Q(τ, γ, β, σ2) by means of EP over the joint density P(τ, γ, β, σ2|X). It is trivial to adapt the EP algorithm used in the sparse linear regression model to this new case: the terms to be approximated are the same as before except for the new N(N −1) terms for the prior on γ. As in the previous section and in order to improve convergence, we re-update all the N(N −1) term approximations corresponding to the prior on β each time N of the N(T −1) term approximations corresponding to regressions are updated. In order to reduce memory requirements, we associate all the N(N −1) terms for the prior on β into a single term, which we can do because they are independent so that we only store in memory one term approximation instead of N(N −1). We also group the N(N −1) terms for the prior on γ into N independent terms and the N(T −1) terms for the regressions into T −1 independent terms. Assuming a constant number of iterations (in our experiments, we need at most 20 iterations for EP to converge), the computational complexity and the memory requirements of the resulting algorithm are O(T N 2). This indicates that it is feasible to analyze data sets which contain the expression pattern of thousands of genes. An MCMC algorithm would require O(N 3) to generate just a single sample. 5 4 Experiments with artificial data We carried out experiments with artificially generated data in order to validate the EP algorithms. In the experiments for sparse linear regression we fixed the hyperparameters in (3) so that ν = 3, λ is the sample variance of the target vector y, v1 = 1, δ = N −1, v0 is chosen according to (2) and w = N −1. In the experiment for gene regulation we fixed the hyperparameters in (4) so that w = (N −1)−1, νi = 3 and λi is the sample variance of the vector xi, w1 = 10−1(N − 1)−1, w0 = 10−2(N −1)−1, v1 = 1, δ = 0.2 and v0 is chosen according to (2). Although the posterior probabilities are sensitive to some of the choices, the orderings of these probabilities, e.g., to determine the most likely regulators, are robust to even large changes. 4.1 Sparse linear regression In the first experiment we set T = 50 and generated x1, . . . , x6000 ∼N(0, 32I) candidate vectors and a target vector y = x1 −x2 + 0.5 x3 −0.5 x4 + ε, where ε ∼N(0, I). The EP algorithm assigned values close to 1 to w1 and w2, the parameters w3 and w4 obtained values 5.2 · 10−3 and 0.5 respectively and w5, . . . , w6000 were smaller than 3 · 10−4. We repeated the experiment several times (each time using new data) and obtained similar results on each run. In the second experiment we set T = 50 and generated a target vector y ∼N(0, 32I) and x1, . . . , x500 candidate vectors so that xi = y + εi for i = 2, . . . , 500, where εi ∼N(0, I). The candidate vector x1 is generated as x1 = y + 0.5 ε1 where ε1 ∼N(0, I). This way, the noise in x1 is twice as small as the noise in the other candidate vectors. Note that all the candidate vectors are highly correlated with each other and with the target vector. This is what happens in gene expression data sets where many genes show similar expression patterns. We ran the EP algorithm 100 times (each time using new data) and it always assigned to all the w1, . . . , w500 more or less the same value of 6 · 10−4. However, w1 obtained the highest value on 54 of the runs and it was among the three ws with highest value on 87 of the runs. Finally, we repeated these experiments setting N = 100, using the MCMC method of [11] and the EP algorithm for sparse linear regression. Both techniques produced results that are statistically indistinguishable (the approximations obtained through EP fall within the variation of the MCMC method), for EP within a fraction of the time of MCMC. 4.2 Gene regulation In this experiment we set T = 50 and generated a vector z with T + 1 values from a sinusoid. We then generated 49 more vectors x2, ..., x50 where xi,t = zt+εi,t for i = 2, . . . , 50 and t = 1, . . . , T, where εi,t ∼N(0, σ2) and σ is one fourth of the sample standard deviation of z. We also generated a vector x1 so that x1,t = zt+1 + εt where t = 1, . . . , T and εt ∼N(0, σ2). In this way, x1 acts as a regulator for x2, ..., x50. A single realization of the vectors x1, . . . , x50 is displayed on the left of Figure 2. We ran the EP algorithm for gene regulation over 100 different realizations of x1, . . . , x50. The algorithm assigned t1 the highest value on 33 of the runs and x1 was ranked among the top five on 74 of the runs. This indicates that the EP algorithm can successfully detect small differences in correlations and should be able to find new regulators in real microarray data. 5 Experiments with real microarray data We applied our algorithm to four data sets. The first is a yeast cell-cycle data set from [5] which is commonly used as a benchmark for regulator discovery. Data sets two through four are from three different Plasmodium strains [6]. Missing values were imputed by nearest neighbors [14] and the hyperparameters were fixed at the same values as in Section 4. The yeast cdc15 data set contains 23 measurements of 6178 genes. We singled out 751 genes which met a minimum criterion for cell cycle regulation [5]. The top ten genes with the highest values for τ along with their annotation from the Saccharomyces Genome database are listed in table 5: the top two genes are specific transcription factors and IOC2 is associated with transcription regulation. As 4% of the yeast genome is associated with transcription the probability of this occurring by chance is 0.0062. However, although the result is statistically significant, we were disappointed to find none of the known cell-cycle regulators (like ACE2, FKH* or SWI*) among the top ten. 6 0 10 20 30 40 50 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 Measurement Expression Regulator Regulatees 0 10 20 30 40 50 −2 −1 0 1 2 3 Measurement Expression Gene PF11_321 Genes positively regulated Genes negatively regulated Figure 2: Left: Plot of the vectors x2, ..., x50 in grey and the vector x1 in black. The vector x1 contains the expression of a regulator which would determine the expressions in x2, ..., x50. Right: Expressions of gene PF11 321 (black) and the 100 genes which are more likely to be regulated by it (light and dark grey). Two clusters of positively and negatively regulated genes can be appreciated. rank standard name common name annotation 1 YLR098c CHA4 DNA binding transcriptional activator 2 YOR315w SFG1 putative transcription factor for growth of superficial pseudohyphae 3 YJL073w JEM1 DNAJ-like chaperone 4 YOR023c AHC1 subunit of the ADA histone acetyl transferase complex 5 YOR105w dubious open reading frame 6 YLR095w IOC2 transcription elongation 7 YOR321w PMT3 protein O-mannosyl transferase 8 YLR231c BNA5 kynureninase 9 YOR248w dubious open reading frame 10 YOR247w SRL1 mannoprotein The three data sets for the malaria parasite [6] contain 53 measurements (3D7), 50 measurements (Dd2) and 48 measurements (HB3). We focus on 3D7 as this is the sequenced reference strain. We singled out 751 genes who showed the highest variation as quantified by the interquartile range of the expression measurements. The top ten genes with the highest values for τ along with their annotation from PlasmoDB are listed in table 5. Recalling the motivation for our approach, the paucity of known transcription factors, we cannot expect to find many annotated regulators in PlasmoDB version 5.4. Thus, we list the BLASTP hits provided by PlasmoDB instead of the absent annotation. These hits were the highest scoring ones outside of the genus Plasmodium. We find four genes with a large identity to transcription factors in Dictyostelium (a recently sequenced social amoebe) and one annotated helicase which typically functions in post-transcriptional regulation. Interestingly three genes have no known function and could be regulators. rank standard name annotation or selected BLASTP hits 1 PFC0950c 25% identity to GATA binding TF in Dictyostelium 2 PF11 0321 25% identity to putative WRKY TF in Dictyostelium 3 PFI1210w no BLASTP matches outside Plasmodium genus 4 MAL6P1.233 no BLASTP matches outside Plasmodium genus 5 PFD0175c 32% identity to GATA binding TF in Dictyostelium 6 MAL7P1.34 35% identity to GATA binding TF in Dictyostelium 7 MAL6P1.182 N-acetylglucosaminyl-phosphatidylinositol de-n-acetylase 8 PF13 0140 dihydrofolate synthase/folylpolyglutamate synthase 9 PF13 0138 no BLASTP matches outside Plasmodium genus 10 MAL13P1.14 DEAD box helicase Results for the HB3 strain were similar in that five putative regulators were found. Somewhat disappointing, we found only one putative regulator (a helicase) among the top ten genes for Dd2. 7 6 Conclusion and discussion Our approach enters a field full of methods enforcing sparsity ([15, 8, 7, 16, 9]). Our main contributions are: a hierarchical model to discover regulators, a tractable algorithm for fast approximate inference in models with many interacting variables, and the application to malaria. Arguably most related is the hierarchical model in [15]. The covariates in this model are a dozen external variables, coding experimental conditions, instead of the hundreds of expression levels of other genes as in our model. Furthermore, the prior in [15] enforces sparsity on the “columns” of β to implement the idea that some genes are not influenced by any of the experimental conditions. Our prior, on the other hand, enforces sparsity on the “rows” in order to find regulators. Future work could include more involved priors, e.g., enforcing sparsity on both “rows” and “columns” or incorporating information from DNA sequence data. The approximate inference techniques described in this paper make it feasible to evaluate such extensions in a fraction of the time required by MCMC methods. References [1] T.S. Gardner and J.J. Faith. Reverse-engineering transcription control networks. Physics of Life Reviews, 2:65–88, 2005. [2] R. Coulson, N. Hall, and C. Ouzounis. Comparative genomics of transcriptional control in the human malaria parasite Plasmodium falciparum. Genome Res., 14:1548–1554, 2004. [3] S. Balaji, M.M. Babu, L.M. Iyer, and L. Aravind. Discovery of the principal specific transcription factors of apicomplexa and their implication for the evolution of the ap2-integrase dna binding domains. Nucleic Acids Research, 33(13):3994–4006, 2005. [4] T. Sakata and E.A. Winzeler. Genomics, systems biology and drug development for infectuous diseases. Molecular BioSystems, 3:841–848, 2007. [5] P.T. Spellman, G. Sherlock, V.R. Iyer, K. Anders, M.B. Eisen, P.O. Brown, and D. Botstein. Comprehensive identification of cell cycle-regulated genes of the yeast Saccharomyces cerevisiae by microarray hybridization. Molecular Biology of the Cell, 9(12):3273–3297, 1998. [6] M. LLinas, Z. Bozdech, E. D. Wong, A.T. Adai, and J. L. DeRisi. Comparative whole genome transcriptome analysis of three Plasmodium falciparum strains. Nucleic Acids Research, 34(4):1166–1173, 2006. [7] M. Beal. Variational Algorithms for Approximate Bayesian Inference. PhD thesis, UCL, 2003. [8] C. Sabatti and G.M. James. Bayesian sparse hidden components analysis for transcription regulation networks. Bioinformatics, 22(6):739–746, 2006. [9] S.T. Jensen, G. Chen, and C.J. Stoeckert. Bayesian variable selection and data integration for biological regulatory networks. The Annals of Applied Statistics, 1:612–633, 2007. [10] E.I. George and R.E. McCulloch. Approaches for Bayesian variable selection. Statistica Sinica, 7:339–374, 1997. [11] E.I. George and R.E. McCulloch. Variable selection via Gibbs sampling. Journal of the American Statistical Association, 88(423):881–889, 1993. [12] T. Minka. A family of algorithms for approximate Bayesian inference. PhD thesis, MIT, 2001. [13] T. Heskes and O. Zoeter. Expectation propagation for approximate inference in dynamic Bayesian networks. In UAI-2002, pages 216–223, 2002. [14] O. Troyanskaya, M. Cantor, P. Brown, T. Hastie, R. Tibshirani, and D. Botstein. Missing value estimation methods for dna microarrays. Bioinformatics, 17(6):520–525, 2001. [15] J. Lucas, C. Carvalho, Q. Wang, A. Bild, J. Nevins, and M. West. Sparse statistical modelling in gene expression genomics. In K.A. Do, P. M¨uller, and M. Vannucci, editors, Bayesian inference for gene expression and proteomics. Springer, 2006. [16] M.Y. Park, T. Hastie, and R. Tibshirani. Averaged gene expressions for regression. 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Predictive Matrix-Variate t Models Shenghuo Zhu Kai Yu Yihong Gong NEC Labs America, Inc. 10080 N. Wolfe Rd. SW3-350 Cupertino, CA 95014 {zsh,kyu,ygong}@sv.nec-labs.com Abstract It is becoming increasingly important to learn from a partially-observed random matrix and predict its missing elements. We assume that the entire matrix is a single sample drawn from a matrix-variate t distribution and suggest a matrixvariate t model (MVTM) to predict those missing elements. We show that MVTM generalizes a range of known probabilistic models, and automatically performs model selection to encourage sparse predictive models. Due to the non-conjugacy of its prior, it is difficult to make predictions by computing the mode or mean of the posterior distribution. We suggest an optimization method that sequentially minimizes a convex upper-bound of the log-likelihood, which is very efficient and scalable. The experiments on a toy data and EachMovie dataset show a good predictive accuracy of the model. 1 Introduction Matrix analysis techniques, e.g., singular value decomposition (SVD), have been widely used in various data analysis applications. An important class of applications is to predict missing elements given a partially observed random matrix. For example, putting ratings of users into a matrix form, the goal of collaborative filtering is to predict those unseen ratings in the matrix. To predict unobserved elements in matrices, the structures of the matrices play an importance role, for example, the similarity between columns and between rows. Such structures imply that elements in a random matrix are no longer independent and identically-distributed (i.i.d.). Without the i.i.d. assumption, many machine learning models are not applicable. In this paper, we model the random matrix of interest as a single sample drawn from a matrixvariate t distribution, which is a generalization of Student-t distribution. We call the predictive model under such a prior by matrix-variate t model (MVTM). Our study shows several interesting properties of the model. First, it continues the line of gradual generalizations across several known probabilistic models on random matrices, namely, from probabilistic principle component analysis (PPCA) [11], to Gaussian process latent-variable models (GPLVMs)[7], and to multi-task Gaussian processes (MTGPs) [13]. MVTMs can be further derived by analytically marginalizing out the hyper-parameters of these models. From a Bayesian modeling point of view, the marginalization of hyper-parameters means an automatic model selection and usually leads to a better generalization performance [8]; Second, the model selection by MVTMs explicitly encourages simpler predictive models that have lower ranks. Unlike the direct rank minimization, the log-determinant terms in the form of matrix-variate t prior offers a continuous optimization surface (though non-convex) for rank constraint; Third, like multivariate Gaussian distributions, a matrix-variate t prior is consistent under marginalization, that means, if a matrix follows a matrix-variate t distribution, its any sub-matrix follows a matrix-variate t distribution as well. This property allows to generalize distributions for finite matrices to infinite stochastic processes. Σ Ω T Y (a) Σ Ω S T Y (b) Σ Ω R T Y (c) Σ I S T Y (d) Figure 1: Models for matrix prediction. (a) MVTM. (b) and (c) are two normal-inverse-Wishart models, equivalent to MVTM when the covariance variable S (or R) is marginalized. (d) MTGP, which requires to optimize the covariance variable S. Circle nodes represent for random variables, shaded nodes for (partially) observable variables, text nodes for given parameters. Under a Gaussian noise model, the matrix-variate t distribution is not a conjugate prior. It is thus difficult to make predictions by computing the mode or mean of the posterior distribution. We suggest an optimization method that sequentially minimizes a convex upper-bound of the log-likelihood, which is highly efficient and scalable. In the experiments, the algorithm shows very good efficiency and excellent prediction accuracy. This paper is organized as follows. We review three existing models and introduce the matrix-variate t models in Section 2. The prediction methods are proposed in Section 3. In Section 4, the MVTM is compared with some other models. We illustrate the MVTM with the experiments on a toy example and on the movie-rating data in Section 5. We conclude in Section 6. 2 Predictive Matrix-Variate t Models 2.1 A Family of Probabilistic Models for Matrix Data In this section we introduce three probabilistic models in the literature. Let Y be a p × m observational matrix and T be the underlying p × m noise-free random matrix. We assume Yi,j = Ti,j + ϵi,j, ϵi,j ∼N(0, σ2), where Yi,j denotes the (i, j)-th element of Y. If Y is partially observed, then YI denotes the set of observed elements and I is the corresponding index set. Probabilistic Principal Component Analysis (PPCA) [11] assumes that yj, the j-th column vector of Y, can be generated from a latent vector vj in a k-dimensional linear space (k < p). The model is defined as yj = Wvj + µ + ϵj and vj ∼Nk(vj; 0, Ik), where ϵj ∼Np(ϵj; 0, σ2Ip), and W is a p × k loading matrix. By integrating out vj, we obtain the marginal distribution yj ∼ Np(yj; µ, WW⊤+ σ2Ip). Since the columns of Y are conditionally independent, letting S take the place of WW⊤, PPCA is similar1 to Yi,j = Ti,j + ϵi,j, T ∼Np,m(T; 0, S, Im), where Np,m(·; 0, S, Im) is a matrix-variate normal prior with zero mean, covariance S between rows, and identity covariance Im between columns. PPCA aims to estimate the parameter W by maximum likelihood. Gaussian Process Latent-Variable Model (GPLVM) [7] formulates a latent-variable model in a slightly unconventional way. It considers the same linear relationship from latent representation vj to observations yj. Instead of treating vj as random variables, GPLVM assigns a prior on W and see {vj} as parameters yj = Wvj + ϵj, and W ∼Np,k(W; 0, Ip, Ik), where the elements of W are independent Gaussian random variables. By marginalizing out W, we obtain a distribution that each row of Y is an i.i.d. sample from a Gaussian process prior with the covariance VV⊤+ σ2Im and V = [v1, . . . , vm]⊤. Letting R take the place of VV⊤, we rewrite a similar model as Yi,j = Ti,j + ϵi,j, T ∼Np,m(T; 0, Ip, R). 1Because it requires S to be positive definite and W is usually low rank, they are not equivalent. From a matrix modeling point of view, GPLVM estimates the covariance between the rows and assume the columns to be conditionally independent. Multi-task Gaussian Process (MTGP) [13] is a multi-task learning model where each column of Y is a predictive function of one task, sampled from a Gaussian process prior, yj = tj + ϵj, and tj ∼Np(0, S), where ϵj ∼Np(0, σ2Ip). It introduces a hierarchical model where an inverseWishart prior is added for the covariance, Yi,j = Ti,j + ϵi,j, T ∼Np,m(T; 0, S, Im), S ∼IWp(S; ν, Ip) MTGP utilizes the inverse-Wishart prior as the regularization and obtains a maximum a posteriori (MAP) estimate of S. 2.2 Matrix-Variate t Models The models introduced in the previous section are closely related to each other. PPCA models the row covariance of Y, GPLVM models the column covariance, and MTGP assigns a hyper prior to prevent over-fitting when estimating the (row) covariance. From a matrix modeling point of view, capturing the dependence structure of Y by its row or column covariance is a matter of choices, which are not fundamentally different.2 There is no reason to favor one choice over the other. By introducing the matrix-variate t models (MVTMs), they can be unified to be the same model. From a Bayesian modeling viewpoint, one should marginalize out as many variables as possible [8]. We thus extend the MTGP model in two directions: (1) assume T ∼Np,m(T; 0, S, Im) that have covariances on both sides of the matrix; (2) marginalize the covariance S on one side (see Figure 1(b)). Then we have a marginal distribution of T Pr(T) = Z Np,m(T; 0, S, Im)IWp(S; ν, Ip)dS = tp,m(T; ν, 0, Ip, Im), (1) which is a matrix-variate t distribution. Because the inverse-Wishart distribution may have different degree-of-freedom definition in literature, we use the definition in [5]. Following the definition in [6], the matrix-variate t distribution of p × m matrix T is given by tp,m(T; ν, M, Σ, Ω) def = 1 Z |Σ|−m 2 |Ω|−p 2 Ip + Σ−1(T −M)Ω−1(T −M)⊤ −ν+m+p−1 2 , where ν is the degree of freedom; M is a p × m matrix; Σ and Ωare positive definite matrices of size p×p and m×m, respectively; Z = (νπ) mp 2 Γp( ν+p−1 2 )/Γp( ν+m+p−1 2 ); Γp(·) is a multivariate gamma function, and | · | stands for determinant. The model can be depicted as Figure 1(a). One important property of matrix-variate t distribution is that the marginal distribution of its sub-matrix still follows a matrix-variate t distribution with the same degree of freedom (see Section 3.1). Therefore, we can expand it to the infinite dimensional stochastic process. By Eq. (1), we can see that Figure 1(a) and Figure 1(b) describe two equivalent models. Comparing them with the MTGP model represented in Figure 1(d), we can see that the difference lies in whether S is point estimated or integrated out. Interestingly, the same matrix-variate t distribution can be equivalently derived by putting another hierarchical generative process on the covariance R, as described in Figure 1(c), where R follows an inverse-Wishart distribution. In other words, integrating the covariance on either side, we obtain the same model. This implies that the model controls the complexity of the covariances on both sides of the matrix. Neither PPCA nor GPLVM has such a property. The matrix-variate t distribution involves a determinant term of T, which becomes a log-determinant term in log-likelihood or KL-divergence. The log-determinant term encourages the sparsity of matrix T with lower rank. This property has been used as the heuristic for minimizing the rank of the matrix in [3]. Student’s t priors were applied to enforce sparse kernel machine [10]. Here we say a few words about the given parameters. Though we can use evidence framework[8] or other methods to estimate ν, the results are not good in many cases(see [4]). Usually we just set 2GPLVM offers an advantage of using nonlinear covariance function based on attributes. ν to a small number. Similar to ν, the estimated σ2 does not give us a good result either, but crossvalidation is a good choice. For the mean matrix M, in our experiments, we just use sample average for all observed elements. For some tasks, when we have prior knowledge about the covariance between columns or between rows, we can use the covariance matrices in the places of Im or Ip. 3 Prediction Methods When the evaluation of the prediction is the sum of individual losses, the optimal prediction is to find the individual mode of the marginal posterior distribution, i.e., arg maxTij Pr(Tij|YI). However, there is no exact solution for the marginal posterior. We have two ways to approximate the optimal prediction. One way to make prediction is to compute the mode of the joint posterior distribution of T, i.e. the prediction problem is bT = arg max T {ln Pr(YI|T) + ln Pr(T)} . (2) The computation of this estimation is usually easy. We discuss it in Section 3.3. An alternative way is to use the individual mean of the posterior distribution to approximate the individual mode. Since the joint of individual mean happens to be the mean of the joint distribution, we only need to compute the joint posterior distribution. The problem of prediction by means is written as T = E(T|YI). (3) However, it is usually difficult to compute the exact mean. One estimation method is the Monte Carlo method, which is computationally intensive. In Section 3.4, we discuss an approximation to compute the mean. From our experiments, the prediction by means usually outperforms the prediction by modes. Before discussing the prediction methods, we introduce a few useful properties in Section 3.1 and suggest an optimization method as the efficient tool for prediction in Section 3.2. 3.1 Properties The MVTM has a rich set of properties. We list a few in the following Theorem. Theorem 1. If n m q Θ Φ p Ψ T ∼tp+q,m+n(·; ν, 0, q p Iq 0 0 Ip , n m In 0 0 Im ), (4) then Pr(T) =tp,m(T; ν, 0, Ip, Im), (5) Pr(T|Θ, Φ, Ψ) =tp,m(T; ν + q + n, M, (Ip + ΨBΨ⊤), (Im + Φ⊤AΦ)), (6) Pr(Θ) =tq,n(Θ; ν, 0, Iq, In), (7) Pr(Φ|Θ) =tq,m(Φ; ν + n, 0, A−1, Im), (8) Pr(Ψ|Θ, Φ) =tp,n(Ψ; ν + q, 0, Ip, B−1) = Pr(Ψ|Θ), (9) E(T|Θ, Φ, Ψ) =M, (10) Cov vec T⊤ |Θ, Φ, Ψ =(ν + q + n −2)−1(Ip + ΨBΨ⊤) ⊗(Im + Φ⊤AΦ), (11) where A def = (ΘΘ⊤+ Iq)−1, B def = (Θ⊤Θ + In)−1, and M def = ΨΘ⊤AΦ = ΨBΘ⊤Φ. This theorem can be directly derived from Theorem 4.3.1 and 4.3.9 in [6] with a little calculus. It provides some insights about MVTMs. The marginal distribution in Eq. (5) has the same form as the joint distribution, therefore the matrix-variate t distribution is extensible to an infinite dimensional stochastic process. As conditional distribution in Eq. (6) is still a matrix-variate t distribution, we can use it to approximate the posterior distribution, which we use in Section 3.4. We encounter log-determinant terms in computation of the mode or mean estimation. The following theorem provides a quadratic upper bounds for the log-determinant terms, which makes it possible to apply the optimization method in Section 3.2. Lemma 1. If X is a p × p positive definite matrices, it holds that ln |X| ≤tr (X) −p. The equality holds when X is an orthonormal matrix. Proof. Let {λ1, · · · , λp} be the eigenvalues of X. We have ln |X| = P i ln λi and tr (X) = P i λi. Since ln λi ≤λi −1, we have the inequality. The equality holds when λi = 1. Therefore, when X is an orthonormal matrix (especially X = Ip), the equality holds. Theorem 2. If Σ is a p × p positive definite matrix, Ωis an m × m positive definite matrix, and T and T0 are p × m matrices, it holds that ln |Σ + TΩ−1T⊤| ≤h(T; T0, Σ, Ω) + h0(T0, Σ, Ω), where h(T; T0, Σ, Ω) def =tr (Σ + T0Ω−1T⊤ 0 )−1TΩ−1T⊤ , h0(T0, Σ, Ω) def = ln |Σ + T0Ω−1T⊤ 0 | + tr (Σ + T0Ω−1T⊤ 0 )−1Σ −p The equality holds when T = T0. Also it holds that ∂ ∂Th(T; T0, Σ, Ω) T=T0 = 2(Σ + T0Ω−1T⊤ 0 )−1T0Ω−1 = ∂ ∂T ln |Σ + TΩ−1T⊤| T=T0 . Applying Lemma 1 with X = (Σ + T0Ω−1T⊤ 0 )−1(Σ + TΩ−1T⊤), we obtain the inequality. By some calculus we have the equality of the first-order derivative. Actually h(·) is a quadratic convex function with respect to T, as (Σ + T0Ω−1T⊤ 0 )−1 and Ω−1 are positive definite matrices. 3.2 Optimization Method Once the objective is given, the prediction becomes an optimization problem. We use an EMstyle optimization method to make the prediction. Suppose J (T) be the objective function to be minimized. If we can find an auxiliary function, Q(T; T′), having the following properties, we can apply this method. 1. J (T) ≤Q(T; T′) and J (T′) = Q(T′; T′), 2. ∂J (T)/∂T|T=T′ = ∂Q(T; T′)/∂T T=T′ , 3. For a fixed T′, Q(T; T′) is quadratic and convex with respect to T. Starting from any T0, as long as we can find a T1 such that Q(T1, T0) < Q(T0, T0), we have J (T0) = Q(T0, T0) > Q(T1, T0) ≥J (T1). If there exists a global minimum point of J (T), there exists a global minimum point of Q(T; T0) as well, because Q(T; T0) is upper bound of J (T). Since Q(T; T0) is quadratic with the respect to T, we can apply the Newton-Raphson method to minimize Q(T; T0). As long as T0 is not a local minimum, maximum or saddle point of J , we can find a T to reduce Q(T; T0), because Q(T; T0) has the same derivative as J (T) at T0. Usually, a random starting point, T0 is unlikely to be a local maximum, then T1 can not be a local maximum. If T0 is a local maximum, we can reselect a point, which is not. After we find a Ti, we repeat the procedure to find a Ti+1 so that J (Ti+1) < J (Ti), unless Ti is a local minimum or saddle point of J . Repeating this procedure, Ti converges a local minimum or saddle point of J , as long as T0 is not a local maximum. 3.3 Mode Prediction Following Eq. (2), the goal is to minimize the objective function b J (T) def = ℓ(T) + ν+m+p−1 2 ln Ip + TT⊤ , (12) where ℓ(T) def = −ln Pr(YI) = 1 2σ2 P (i,j)∈I(Tij −Yij)2 + const. As b J contains a log-determinant term, minimizing b J by nonlinear optimization is slow. Here, we introduce an auxiliary function, Q(T; T′) def = ℓ(T) + h(T; T′, Ip, Im) + h0(T′, Ip, Im). (13) By Corollary 2, we have that b J (T) ≤Q(T; T′), b J (T′) = Q(T′, T′), and Q(T, T′) has the same first-order derivative as b J (T) at T′. Because l and h are quadratic and convex, Q is quadratic and convex as well. Therefore, we can apply the optimization method in Section 3.2 to minimize b J . However, when the size of T is large, to find bT is still time consuming and requires a very large space. In many tasks, we only need to infer a small portion of bT. Therefore, we consider a low rank approximation, using UV⊤to approximate T, where U is a p × k matrix and V is an m × k matrix. The problem of Eq. (2) is approximated by arg minU,V b J (UV⊤). We can minimize J1 by alternatively optimizing U and V. We can put the final result in a canonical format as bT ≈USV⊤, where U and V are semi-orthonormal and S is a k × k diagonal matrix. This result can be consider as the SVD of an incomplete matrix using matrix-variate t regularization. The details are skipped because of the limit space. 3.4 Variational Mean Prediction As the difficulty in explicitly computing the posterior distribution of T, we take a variational approach to approximate its posterior distribution by a matrix-variate t distribution via an expanded model. We expand the model by adding matrix variate Θ, Φ and Ψ with distribution as Eq. (4). Since the marginal distribution, Eq. (5), is the same as the prior of T, we can derive the original model by marginalizing out Θ, Φ and Ψ. However, instead of integrating out Θ, Φ and Ψ, we use them as the parameters to approximate T’s posterior distribution. Therefore, the estimation of the parameters is to minimize −ln Pr(YI, Θ, Φ, Ψ) = −ln Pr(Θ, Φ, Ψ) −ln Z Pr(T|Θ, Φ, Ψ) Pr(YI|T)dT (14) over Θ, Φ and Ψ. The first term in the RHS of Eq. (14) can be written as −ln Pr(Θ, Φ, Ψ) = −ln Pr(Θ) −ln Pr(Φ|Θ) −ln Pr(Ψ|Θ, Φ) = ν+q+n+p+m−1 2 ln |Iq + ΘΘ⊤| + ν+q+n+m−1 2 ln |Im + Φ⊤AΦ| + ν+q+n+p−1 2 ln |Ip + ΨBΨ⊤| + const. (15) Due to the convexity of negative logarithm, the second term in the RHS of Eq. (14) is bounded by ℓ(ΨB 1 2 Θ⊤A 1 2 Φ) + 1 2σ2(ν+q+n−2) X (i,j)∈I (1 + [ΨBΨ⊤]ii)(1 + [Φ⊤AΦ]jj) + const. (16) because −ln Pr(YI|T) is quadratic respective to T, thus we only need integration using the mean and variance of Tij of Pr(T|Θ, Φ, Ψ), which is given by Eq. (10) and (11). The parameter estimation not only reduce the loss (the term of ℓ(·)), but also reduce the variance. Because of this, the prediction by means usually outperforms the prediction by modes. Let J be the sum of the right-hand-side of Eq. (15) and (16), which can be considered as the upper bound of Eq. (14) (ignoring constants). Here, we estimate the parameters by minimizing J . Because A and B involve the inverse of quadratic term of Θ, it is awkward to directly optimize Θ, Φ, Ψ. We reparameterize J by U def = ΨB1/2, V def = Φ⊤A1/2, and S def = Θ. We can easily apply the optimization method in Section 3.2 to find optimal U, V and S. After estimation U, V and S, by Theorem 1, we can compute T = M = USV⊤. The details are skipped because of the limit space. 4 Related work Maximum Margin Matrix Factorization (MMMF) [9] is not in the framework of stochastic matrix analysis, but there are some similarities between MMMF and our mode estimation in Section 3.3. Using trace norm on the matrix as regularization, MMMF overcomes the over-fitting problem in factorizing matrix with missing values. From the regularization viewpoint, the prediction by mode of MVTM uses log-determinants as the regularization term in Eq. (12). The log-determinants encourage sparsity predictive models. Stochastic Relational Models (SRMs) [12] extend MTGPs by estimating the covariance matrices for each side. The covariance functions are required to be estimated from observation. By maximizing marginalized likelihood, the estimated S and R reflect the information of the dependency structure. Then the relationship can be predicted with S and R. During estimating S and R, inverseWishart priors with parameter Σ and Ωare imposed to S and R respectively. MVTM differs from SRM in integrating out the hyper-parameters or maximizing out. As MacKay suggests [8], “one should integrate over as many variables as possible”. Robust Probabilistic Projections (RPP)[1] uses Student-t distribution to extends PPCA by scaling each feature vector by an independent random variable. Written in a matrix format, RPP is T ∼Np,m(T; µ1⊤, WW⊤, U), U = diag {ui} , ui ∼IG(ui|ν 2, ν 2), where IG is inverse Gamma distribution. Though RPP unties the scale factors between feature vectors, which could make the estimation more robust, it does not integrate out the covariance matrix, which we did in MVTM. Moreover inherited from PPCA, RPP implicitly uses independence assumption of feature vectors. Also RPP results different models depending on which side we assume to be independent, therefore it is not suitable for matrix prediction. 5 Experiments 2 4 6 8 10 12 14 16 18 20 5 10 15 20 25 30 (a) Original Matrix 2 4 6 8 10 12 14 16 18 20 5 10 15 20 25 30 (b) With Noise (0.32) 2 4 6 8 10 12 14 16 18 20 5 10 15 20 25 30 (c) MMMF (0.27) 2 4 6 8 10 12 14 16 18 20 5 10 15 20 25 30 (d) PPCA (0.26) 2 4 6 8 10 12 14 16 18 20 5 10 15 20 25 30 (e) SRM (0.22) 2 4 6 8 10 12 14 16 18 20 5 10 15 20 25 30 (f) MVTM mode (0.20) 2 4 6 8 10 12 14 16 18 20 5 10 15 20 25 30 (g) MVTM mean (0.192) 2 4 6 8 10 12 14 16 18 20 5 10 15 20 25 30 (h) MCMC (0.185) Figure 2: Experiments on synthetic data. RMSEs are shown in parentheses. 0 0.5 1 1.5 2 2.5 3 3.5 4 1 2 3 4 5 6 7 8 9 10 singular values index MMMF MVTM-mode MVTM-mean Figure 3: Singular values of recovered matrices in descent order. Synthetic data: We generate a 30 × 20 matrix (Figure 2(a)), then add noise with σ2 = 0.1 (Figure 2(b)). The root mean squared noise is 0.32. We select 70% elements as the observed data and the rest elements are for prediction. We apply MMMF [9], PPCA[11], MTGP[13], SRM [12], our MVTM prediction-by-means and predictionby-modes methods. The number of dimensions for low rank approximation is 10. We also apply MCMC method to infer the matrix. The reconstruction matrix and root mean squared errors of prediction on the unobserved elements (comparing to the original matrix) are shown in Figure 2(c)-2(g), respectively. MTGP has the similar result as PPCA, we do not show the result. MVTM is in favor of sparse predictive models. To verify this, we depict the singular values of the MMMF method and two MVTM prediction methods in Figure 3. There are only two singular user mean movie mean MMMF PPCA MVTM (mode) MVTM (mean) RMSE 1.425 1.387 1.186 1.165 1.162 1.151 MAE 1.141 1.103 0.943 0.915 0.898 0.887 Table 1: RMSE (root mean squred error) and MAE (mean absolute error) of experiments on Eachmovie data. All standard errors are 0.001 or less. values of the MVTM prediction-by-means method are non-zeros. The singular values of the mode estimation decrease faster than the MMMF ones at beginning, but decrease slower after a threshold. This confirms that the log-determinants automatically determine the intrinsic rank of the matrices. Eachmovie data: We test our algorithms on Eachmovie from [2]. The dataset contains 74, 424 users’ 2, 811, 718 ratings on 1, 648 movies, i.e. about 2.29% are rated by zero-to-five stars. We put all ratings into a matrix, and randomly select 80% as observed data to predict the remaining ratings. The random selection was carried out 10 times independently. We compare our approach with other three approaches: 1) USER MEAN predicting rating by the sample mean of the same user’ ratings; 2) MOVIE MEAN, predicting rating by the sample mean of users’ ratings of the same movie; 3) MMMF[9]; 4) PPCA[11]. We do not have a scalable implementation for other approaches compared in the previous experiment. The number of dimensions is 10. The results are shown in Table 1. Two MVTM prediction methods outperform the other methods. 6 Conclusions In this paper we introduce matrix-variate t models for matrix prediction. The entire matrix is modeled as a sample drawn from a matrix-variate t distribution. An MVTM does not require the independence assumption over elements. The implicit model selection of the MVTM encourages sparse models with lower ranks. To minimize the log-likelihood with log-determinant terms, we propose an optimization method by sequentially minimizing its convex quadratic upper bound. The experiments show that the approach is accurate, efficient and scalable. References [1] C. Archambeau, N. Delannay, and M. Verleysen. Robust probabilistic projections. In ICML, 2006. [2] J. Breese, D. Heckerman, and C. Kadie. Empirical analysis of predictive algorithms for collaborative filtering. In UAI-98, pages 43–52, 1998. [3] M. Fazel, H. Haitham, and S. P. Boyd. Log-det heuristic for matrix rank minimization with applications to hankel and euclidean distance matrices. In Proceedings of the American Control Conference, 2003. [4] C. Fernandez and M. F. J. Steel. Multivariate Student-t regression models: Pitfalls and inference. Biometrika, 86(1):153–167, 1999. [5] A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin. Bayesian Data Analysis. Chapman & Hall/CRC, New York, 2nd edition, 2004. [6] A. K. Gupta and D. K. Nagar. Matrix Variate Distributions. Chapman & Hall/CRC, 2000. [7] N. Lawrence. Probabilistic non-linear principal component analysis with gaussian process latent variable models. J. Mach. Learn. Res., 6:1783–1816, 2005. [8] D. J. C. MacKay. Comparison of approximate methods for handling hyperparameters. Neural Comput., 11(5):1035–1068, 1999. [9] J. D. M. Rennie and N. Srebro. Fast maximum margin matrix factorization for collaborative prediction. In ICML, 2005. [10] M. E. Tipping. Sparse bayesian learning and the relevance vector machine. Journal of Machine Learning Research, 1:211–244, 2001. [11] M. E. Tipping and C. M. Bishop. Probabilistic principal component analysis. Journal of the Royal Statisitical Scoiety, B(61):611–622, 1999. [12] K. Yu, W. Chu, S. Yu, V. Tresp, and Z. Xu. Stochastic relational models for discriminative link prediction. In Advances in Neural Information Processing Systems 19 (NIPS), 2006. [13] K. Yu, V. Tresp, and A. Schwaighofer. Learning Gaussian processes from multiple tasks. In ICML, 2005.
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Convex Clustering with Exemplar-Based Models Danial Lashkari Polina Golland Computer Science and Artificial Intelligence Laboratory Massachusetts Institute of Technology Cambridge, MA 02139 {danial, polina}@csail.mit.edu Abstract Clustering is often formulated as the maximum likelihood estimation of a mixture model that explains the data. The EM algorithm widely used to solve the resulting optimization problem is inherently a gradient-descent method and is sensitive to initialization. The resulting solution is a local optimum in the neighborhood of the initial guess. This sensitivity to initialization presents a significant challenge in clustering large data sets into many clusters. In this paper, we present a different approach to approximate mixture fitting for clustering. We introduce an exemplar-based likelihood function that approximates the exact likelihood. This formulation leads to a convex minimization problem and an efficient algorithm with guaranteed convergence to the globally optimal solution. The resulting clustering can be thought of as a probabilistic mapping of the data points to the set of exemplars that minimizes the average distance and the information-theoretic cost of mapping. We present experimental results illustrating the performance of our algorithm and its comparison with the conventional approach to mixture model clustering. 1 Introduction Clustering is one of the most basic problems of unsupervised learning with applications in a wide variety of fields. The input is either vectorial data, that is, vectors of data points in the feature space, or proximity data, the pairwise similarity or dissimilarity values between the data points. The choice of the clustering cost function and the optimization algorithm employed to solve the problem determines the resulting clustering [1]. Intuitively, most methods seek compact clusters of data points, namely, clusters with relatively small intra-cluster and high inter-cluster distances. Other approaches, such as Spectral Clustering [2], look for clusters of more complex shapes lying on some low dimensional manifolds in the feature space. These methods typically transform the data such that the manifold structures get mapped to compact point clouds in a different space. Hence, they do not remove the need for efficient compact-cluster-finding techniques such as k-means. The widely used Soft k-means method is an instance of maximum likelihood fitting of a mixture model through the EM algorithm. Although this approach yields satisfactory results for problems with a small number of clusters and is relatively fast, its use of a gradient-descent algorithm for minimization of a cost function with many local optima makes it sensitive to initialization. As the search space grows, that is, the number of data points or clusters increases, it becomes harder to find a good initialization. This problem often arises in emerging applications of clustering for large biological data sets such as gene-expression. Typically, one runs the algorithm many times with different random initializations and selects the best solution. More sophisticated initialization methods have been proposed to improve the results but the challenge of finding good initialization for EM algorithm remains [4]. We aim to circumvent the initialization procedure by designing a convex problem whose global optimum can be found with a simple algorithm. It has been shown that mixture modeling can 1 be formulated as an instance of iterative distance minimization between two sets of probability distributions [3]. This formulation shows that the non-convexity of mixture modeling cost function comes from the parametrization of the model components . More precisely, any mixture model is, by definition, a convex combination of some set of distributions. However, for a fixed number of mixture components, the set of all such mixture models is usually not convex when the distributions have, say, free mean parameters in the case of normal distributions. Inspired by combinatorial, non-parametric methods such as k-medoids [5] and affinity propagation [6], our main idea is to employ the notion of exemplar finding, namely, finding the data points which could best describe the data set. We assume that the clusters are dense enough such that there is always a data point very close to the real cluster centroid and, thus, restrict the set of possible cluster means to the set of data points. Further, by taking all data points as exemplar candidates, the modeling cost function becomes convex. A variant of EM algorithm finds the globally optimal solution. Convexity of the cost function means that the algorithm will unconditionally converge to the global minimum. Moreover, since the number of clusters is not specified a priori, the algorithm automatically finds the number of clusters depending only on one temperature-like parameter. This parameter, which is equivalent to a common fixed variance in case of Gaussian models, defines the width scale of the desired clusters in the feature space. Our method works exactly in the same way with both proximity and vectorial data, unifying their treatment and providing insights into the modeling assumptions underlying the conversion of feature vectors into pairwise proximity data. In the next section, we introduce our maximum likelihood function and the algorithm that maximizes it. In Section 3, we make a connection to the Rate-Distortion theory as a way to build intuition about our objective function. Section 4 presents implementation details of our algorithm. Experimental results comparing our method with a similar mixture model fitting method are presented in Section 5, followed by a discussion of the algorithm and the related work in Section 6. 2 Convex Cost Function Given a set of data points X = {x1, · · · , xn} ⊂IRd, mixture model clustering seeks to maximize the scaled log-likelihood function l({qj}k j=1, {mj}k j=1; X) = 1 n n X i=1 log k X j=1 qjf(xi; mj) , (1) where f(x; m) is an exponential family distribution on random variable X. It has been shown that there is a bijection between regular exponential families and a broad family of divergences called Bregman divergence [7]. Most of the well-known distance measures, such as Euclidean distance or Kullback-Leibler divergence (KL-divergence) are included in this family. We employ this relationship and let our model be an exponential family distribution on X of the form f(x; m) = C(x) exp(−dφ(x, m)) where dφ is some Bregman divergence and C(x) is independent of m. Note that with this representation, m is the expected value of X under the distribution f(x; m). For instance, taking Euclidean distance as the divergence, we obtain normal distribution as our model f. In this work, we take models of the above form whose parameters m lie in the same space as data vectors. Thus, we can restrict the set of mixture components to the distributions centered at the data points, i.e., mj ∈X. Yet, for a specified number of clusters k, the problem still has a combinatorial nature of choosing the right k cluster centers among n data points. To avoid this problem, we increase the number of possible components to n and represent all data points as cluster-center candidates. The new log-likelihood function is l({qj}n j=1; X) = 1 n n X i=1 log n X j=1 qjfj(xi) = 1 n n X i=1 log n X j=1 qje−βdφ(xi,xj) + const. , (2) where fj(x) is an exponential family member with its expectation parameter equal to the jth data vector and the constant denotes a term that does not depend on the unknown variables {qj}n j=1. The constant scaling factor β in the exponent controls the sharpness of mixture components. We maximize l(·; X) over the set of all mixture distributions Q = n Q|Q(·) = Pn j=1 qjfj(·) o . 2 The log-likelihood function (2) can be expressed in terms of the KL-divergence by defining ˆP(x) = 1/n, x ∈X, to be the empirical distribution of the data on IRd and by noting that D( ˆP∥Q) = − X x∈X ˆP(x) log Q(x) −H( ˆP) = −l({qj}n j=1; X) + const. (3) where H( ˆP) is the entropy of the empirical distribution and does not depend on the unknown mixture coefficients {qj}n j=1. Consequently, the maximum likelihood problem can be equivalently stated as the minimization of the KL-divergence between ˆP and the set of mixture distributions Q. It is easy to see that unlike the unconstrained set of mixture densities considered by the likelihood function (1), set Q is convex. Our formulation therefore leads to a convex minimization problem. Furthermore, it is proved in [3] that for such a problem, the sequence of distributions Q(t) with corresponding weights {q(t) j }n j=1 defined iteratively via q(t+1) j = q(t) j X x∈X ˆP(x)fj(x) Pn j′=1 q(t) j′ fj′(x) (4) is guaranteed to converge to the global optimum solution Q∗if the support of the initial distribution is the entire index set, i.e., q(0) j > 0 for all j. 3 Connection to Rate-Distortion Problems Now, we present an equivalent statement of our problem on the product set of exemplars and data points. This alternative formulation views our method as an instance of lossy data compression and directly implies the optimality of the algorithm (4). The following proposition is introduced and proved in [3]: Proposition 1. Let Q′ be the set of distributions of the complete data random variable (J, X) ∈ {1, · · · , n} × IRd with elements Q′(j, x) = qjfj(x). Let P′ be the set of all distributions on the same random variable (J, X) which have ˆP as their marginal on X. Then, min Q∈Q D( ˆP∥Q) = min P ′∈P′,Q′∈Q′D(P ′∥Q′) (5) where Q is the set of all marginal distributions of elements of Q′ on X. Furthermore, if Q∗and (P ′∗, Q′∗) are the corresponding optimal arguments, Q∗is the marginal of Q′∗. This proposition implies that we can express our problem of minimizing (3) as minimization of D(P ′∥Q′) where P ′ and Q′ are distributions of the random variable (J, X). Specifically, we define: Q′(j, x) = qjC(x)e−βdφ(x,xj) P ′(j, x) = ˆP(x)P ′(j|x) = 1 nrij, x = xi ∈X; 0, otherwise (6) where qj and rij = P ′(j|x = xi) are probability distributions over the set {j}n j=1. This formulation ensures that P ′ ∈P′, Q′ ∈Q′ and the objective function is expressed only in terms of variables qj and P ′(j|x) for x ∈X. Our goal is then to solve the minimization problem in the space of distributions of random variable (J, I) ∈{j}n j=1×{j}n j=1, namely, in the product space of exemplar × data point indices. Substituting expressions (6) into the KL-divergence D(P ′∥Q′), we obtain the equivalent cost function: D(P ′∥Q′) = 1 n n X i,j=1 rij log rij qj + βdφ(xi, xj) + const. (7) It is straightforward to show that for any set of values rij, setting qj = 1 n P i rij minimizes (7). Substituting this expression into the cost function, we obtain the final expression D(P ′∥Q′∗(P ′)) = 1 n n X i,j=1 rij log rij 1 n P i′ ri′j + βdφ(xi, xj) + const. , = I(I; J) + βEI,Jdφ(xi, xj) + const. (8) 3 where the first term is the mutual information between the random variables I (data points) and J (exemplars) under the distribution P ′ and the second term is the expected value of the pairwise distances with the same distribution on indices. The n2 unknown values of rij lie on n separate n-dimensional simplices. These parameters have the same role as cluster responsibilities in soft k-means: they stand for the probability of data point xi choosing data point xj as its cluster-center. The algorithm described in (4) is in fact the same as the standard Arimoto-Blahut algorithm [10] commonly used for solving problems of the form (8). We established that the problem of maximizing log-likelihood function (2) is equivalent to the minimization of objective function (8). This helps us to interpret this problem in the framework of Rate-Distortion theory. The data set can be thought of as an information source with a uniform distribution on the alphabet X. Such a source has entropy log n, which means that any scheme for encoding an infinitely long i.i.d. sequence generated by this source requires on average this number of bits per symbol, i.e., has a rate of at least log n. We cannot compress the information source beyond this rate without tolerating some distortion, when the original data points are encoded into other points with nonzero distances between them. We can then consider rij’s as a probabilistic encoding of our data set onto itself with the corresponding average distortion D = EI,Jdφ(xi, xj) and the rate I(I; J). A solution r∗ ij that minimizes (8) for some β yields the least rate that can be achieved having no more than the corresponding average distortion D. This rate is usually denoted by R(D), a function of average distortion, and is called the rate-distortion function [8]. Note that we have ∂R/∂D = −β, 0 < β < ∞at any point on the rate-distortion function graph. The weight qj for the data point xj is a measure of how likely this point is to appear in the compressed representation of the data set, i.e., to be an exemplar. Here, we can rigorously quantify our intuitive idea that higher number of clusters (corresponding to higher rates) is the inherent cost of attaining lower average distortion. We will see an instance of this rate-distortion trade-off in Section 5. 4 Implementation The implementation of our algorithm costs two matrix-vector multiplications per iteration, that is, has a complexity of order n2 per iteration, if solved with no approximations. Letting sij = exp(−βdφ(xi, xj)) and using two auxiliary vectors z and η, we obtain the simple update rules z(t) i = n X j=1 sijq(t) j η(t) j = 1 n n X i=1 sij z(t) i q(t+1) j = η(t) j q(t) j (9) where the initialization q(0) j is nonzero for all the data points we want to consider as possible exemplars. At the fixed point, the values of ηj are equal to 1 for all data points in the support of qj and are less than 1 otherwise [10]. In practice, we compute the gap between maxj (log ηj) and P j qj log ηj in each iteration and stop the algorithm when this gap becomes less than a small threshold. Note that the soft assignments r(t) ij = q(t) j sij/nz(t) i need to be computed only once after the algorithm has converged. Any value of β ∈[0, ∞) yields a different solution to (8) with different number of nonzero qj values. Smaller values of β correspond to having wider clusters and greater values correspond to narrower clusters. Neither extreme, one assigning all data points to the central exemplar and the other taking all data points as exemplars, is interesting. For reasonable ranges of β, the solution is sparse and the resulting number of nonzero components of qj determines the final number of clusters. Similar to other interior-point methods, the convergence of our algorithm becomes slow as we move close to the vertices of the probability simplex where some qj’s are very small. In order to improve the convergence rate, after each iteration, we identify all qj’s that are below a certain threshold (10−3/n in our experiments,) set them to zero and re-normalize the entire distribution over the remaining indices. This effectively excludes the corresponding points as possible exemplars and reduces the cost of the following iterations. In order to further speed up the algorithm for very large data sets, we can search over values of sij for any i and keep only the largest no values in any row turning the proximity matrix into a sparse one. The reasoning is simply that we expect any point to be represented in the final solution with exemplars relatively close to it. We observed that as long as no values are a few times greater than the expected number of data points in each cluster, the final results remain almost the same 4 0 100 200 300 400 500 600 700 800 900 1000 0 1 2 3 4 5 6 Average Distortion Rate (bits) 0 0.5 1 1.5 2 2.5 0 2 4 6 8 10 12 β/βo Figure 1: Left: rate-distortion function for the example described in the text. The line with slope −βo is also illustrated for comparison (dotted line) as well as the point corresponding to β = βo (cross) and the line tangent to the graph at that point. Right: the exponential of rate (dotted line) and number of hard clusters for different values of beta (solid line.) The rate is bounded above by logarithm of number of clusters. with or without this preprocessing. However, this approximation decreases the running time of the algorithm by a factor n/no. 5 Experimental Results To illustrate some general properties of our method, we apply it to the set of 400 random data points in IR2 shown in Figure 2. We use Euclidean distance and run the algorithm for different values of β. Figure 1 (left) shows the resulting rate-distortion function for this example. As we expect, the estimated rate-distortion function is smooth, monotonically decreasing and convex. To visualize the clustering results, we turn the soft responsibilities into hard assignments. Here, we first choose the set of exemplars to be the set of all indices j that are MAP estimate exemplars for some data point i under P ′(j|xi). Then, any point is assigned to its closest exemplar. Figure 2 illustrates the shapes of the resulting hard clusters for different values of β. Since β has dimensions of inverse variance in the case of Gaussian models, we chose an empirical value βo = n2 log n/ P i,j ∥xi −xj∥2 so that values β around βo give reasonable results. We can see how clusters split when we increase β. Such cluster splitting behavior also occurs in the case of a Gaussian mixture model with unconstrained cluster centers and has been studied as the phase transitions of a corresponding statistical system [9]. The nature of this connection remains to be further investigated. The resulting number of hard clusters for different values of β are shown in Figure 1 (right). The figure indicates two regions of β with relatively stable number of clusters, namely 4 and 10, while other cluster numbers have a more transitory nature with varying β. The distribution of data points in Figure 2 shows that this is a reasonable choice of number of clusters for this data set. However, we also observe some fluctuations in the number of clusters even in the more stable regime of values of β. Comparing this behavior with the monotonicity of our rate shows how, by turning the soft assignments into the hard ones, we lose the strong optimality guarantees we have for the original soft solution. Nevertheless, since our global optimum is minimum to a well justified cost function, we expect to obtain relatively good hard assignments. We further discuss this aspect of the formulation in Section 6. The main motivation for developing a convex formulation of clustering is to avoid the well-known problem of local optima and sensitivity to initialization. We compare our method with a regular mixture model of the form (1) where f(x; m) is a Gaussian distribution and the problem is solved using the EM algorithm. We will refer to this regular mixture model as the soft k-means. The kmeans algorithm is a limiting case of this mixture-model problem when β →∞, hence the name soft k-means. The comparison will illustrate how employing convexity helps us better explore the search space as the problem grows in complexity. We use synthetic data sets by drawing points from unit variance Gaussian distributions centered around a set of vectors. There is an important distinction between the soft k-means and our algorithm: although the results of both algorithms depend on the choice of β, only the soft k-means needs the number of clusters k as an input. We run the two algorithms for five different values of β which were empirically found 5 −40 −30 −20 −10 0 10 20 30 40 −40 −30 −20 −10 0 10 20 30 40 (a) −40 −30 −20 −10 0 10 20 30 40 −40 −30 −20 −10 0 10 20 30 40 (b) −40 −30 −20 −10 0 10 20 30 40 −40 −30 −20 −10 0 10 20 30 40 (c) −40 −30 −20 −10 0 10 20 30 40 −40 −30 −20 −10 0 10 20 30 40 (d) −40 −30 −20 −10 0 10 20 30 40 −40 −30 −20 −10 0 10 20 30 40 (e) −40 −30 −20 −10 0 10 20 30 40 −40 −30 −20 −10 0 10 20 30 40 (f) Figure 2: The clusters found for different values of β, (a) 0.1βo (b) 0.5βo (c) βo (d) 1.2βo (e) 1.6βo (f) 1.7βo. The exemplar data point of each cluster is denoted by a cross. The range of normal distributions for any mixture model is illustrated here by circles around these exemplar points with radius equal to the square root of the variance corresponding to the value of β used by the algorithm (σ = (2β)−1/2). Shapes and colors denote cluster labels. to yield reasonable results for the problems presented here. As a measure of clustering quality, we use micro-averaged precision. We form the contingency tables for the cluster assignments found by the algorithm and the true cluster labels. The percentage of the total number of data points assigned to the right cluster is taken as the precision value of the clustering result. Out of the five runs with different values of β, we take the result with the best precision value for any of the two algorithms. In the first experiment, we look at the performance of the two algorithms as the number of clusters increases. Different data sets are generated by drawing 3000 data points around some number of cluster centers in IR20 with all clusters having the same number of data points. Each component of any data-point vector comes from an independent Gaussian distribution with unit variance around the value of the corresponding component of its cluster center. Further, we randomly generate components of the cluster-center vectors from a Gaussian distribution with variance 25 around zero. In this experiment, for any value of β, we repeat soft k-means 1000 times with random initialization and pick the solution with the highest likelihood value. Figure 3 (left) presents the precision values as a function of the number of clusters in the mixture distribution that generates the 3000 data points. The error bars summarize the standard deviation of precision over 200 independently generated data sets. We can see that performance of soft k-means drops as the number of clusters increases while our performance remains relatively stable. Consequently, as illustrated in Figure 3 (right), 6 5 6 8 10 12 15 20 25 30 75 80 85 90 95 100 105 Number of Clusters Average Precision Convex Clustering Soft k‐means 5 6 8 10 12 15 20 25 30 -5 0 5 10 15 20 25 Number of Clusters Average Precision Gain Figure 3: Left: average precision values of Convex Clustering and Soft k-means for different numbers of clusters in 200 data sets of 3000 data points. Right: precision gain of using Convex Clustering in the same experiment. the average precision difference of the two algorithms increases with increasing number of clusters. Since the total number of data points remains the same, increasing the number of clusters results in increasing complexity of the problem with presumably more local minima to the cost function. This trend agrees with our expectation that the results of the convex algorithm improves relative to the original one with a larger search space. As another way of exploring the complexity of the problem, in our second experiment, we generate data sets with different dimensionality. We draw 100 random vectors, with unit variance Gaussian distribution in each component, around any of the 40 cluster centers to make data sets of total 4000 data points. The cluster centers are chosen to be of the form (0, · · · , 0, √ 50, 0, · · · , 0) where we change the position of the nonzero component to make different cluster centers. In this way, the pairwise distance between all cluster centers is 50 by formation. Figure 4 (left) presents the precision values found for the two algorithms when 4000 points lie in spaces with different dimensionality. Soft k-means was repeated 100 times with random initialization for any value of β. Again, the relative performance of Convex Clustering when compared to soft k-means improves with the increasing problem complexity. This is another evidence that for larger data sets the less precise nature of our constrained search, as compared to the full mixture models, is well compensated by its ability to always find its global optimum. In general the value of β should be tuned to find the desired solution. We plan to develop a more systematic way for choosing β. 6 Discussion and Related Work Since only the distances take part in our formulation and the values of data point vectors are not required, we can extend this method to any proximity data. Given a matrix Dn×n = [dij] that describes the pairwise symmetric or asymmetric dissimilarities between data points, we can replace dφ(xi, xj)’s in (8) with dij’s and solve the same minimization problem whose convexity can be directly verified. The algorithm works in exactly the same way and all the aforementioned properties carry over to this case as well. A previous application of rate-distortion theoretic ideas in clustering led to the deterministic annealing (DA). In order to avoid local optima, DA gradually decreases an annealing parameter, tightening the bound on the average distortion [9]. However, at each temperature the same standard EM updates are used. Consequently, the method does not provide strong guarantees on the global optimality of the resulting solution. Affinity propagation is another recent exemplar-based clustering algorithm. It finds the exemplars by forming a factor graph and running a message passing algorithm on the graph as a way to minimize the clustering cost function [6]. If the data point i is represented by the data point ci, assuming a common preference parameter value λ for all data points, the objective function of affinity propagation can be stated as P i dici + λk where k is the number of found clusters. The second term is needed to put some cost on picking any point as an exemplar to prevent the trivial case of sending any point to itself. Outstanding results have been reported for the affinity propagation [6] but theoretical guarantees on its convergence or optimality are yet to be established. 7 50 75 100 125 150 85 90 95 100 Number of Dimensions Average Precision Convex Clustering Soft k‐means 50 75 100 125 150 8 10 12 14 16 18 Number of Dimensions Average Precision Gain Figure 4: Left: average precision values of Convex Clustering and Soft k-means for different data dimensionality in 100 data sets of 4000 data points with 40 clusters. Right: precision gain of using Convex Clustering in the same experiment. We can interpret our algorithm as a relaxation of this combinatorial problem to the soft assignment case by introducing probabilities P(ci = j) = rij of associating point i with an exemplar j. The marginal distribution qj = 1 n P i rij is the probability that point j is an exemplar. In order to use analytical tools for solving this problem, we have to turn the regularization term k into a continuous function of assignments. A possible choice might be H(q), entropy of distribution qj, which is bounded above by log k. However, the entropy function is concave and any local or global minimum of a concave minimization problem over a simplex occurs in an extreme point of the feasible domain which in our case corresponds to the original combinatorial hard assignments [11]. In contrast, using mutual information I(I, J) induced by rij as the regularizing term turns the problem into a convex problem. Mutual information is convex and serves as a lower bound on H(q) since it is always less than the entropy of both of its random variables. Now, by letting λ = 1/β we arrive to our cost function in (8). We can therefore see that our formulation is a convex relaxation of the original combinatorial problem. In conclusion, we proposed a framework for constraining the search space of general mixture models to achieve global optimality of the solution. In particular, our method promises to be useful in problems with large data sets where regular mixture models fail to yield consistent results due to their sensitivity to initialization. We also plan to further investigate generalization of this idea to the models with more elaborate parameterizations. Acknowledgements. This research was supported in part by the NIH NIBIB NAMIC U54EB005149, NCRR NAC P41-RR13218 grants and by the NSF CAREER grant 0642971. References [1] J. Puzicha, T. Hofmann, and J. M. Buhmann, “Theory of proximity based clustering: Structure detection by optimization,” Pattern Recognition, Vol. 33, No. 4, pp. 617–634, 2000. [2] A. Y. Ng, M. I. Jordan, and Y. Weiss, “On Spectral Clustering: Analysis and an Algorithml,” Advances in Neural Information Processing Systems, Vol. 14, pp. 849–856, 2001. [3] I. Csisz´ar and P. Shields, “Information Theory and Statistics: A Tutorial,” Foundations and Trends in Communications and Information Theory, Vol. 1, No. 4, pp. 417–528, 2004. [4] M. Meil˘a, and D. Heckerman, “An Experimental Comparison of Model-Based Clustering Methods,” Machine Learning, Vol. 42, No. 1-2, pp. 9–29, 2001. [5] J. Han, and M. Kamber, Data Mining: Concepts and Techniques, Morgan Kaufmann, 2001. [6] B. J. Frey, and D. Dueck, “Clustering by Passing Messages Between Data Points,” Science, Vol. 315, No. 5814, pp. 972–976, 2007. [7] A. Banerjee, S. Merugu, I. S.Dhillon, and J. Ghosh, “Clustering with Bregman Divergences,” Journal of Machine Learning Research, Vol. 6, No. 6, pp. 1705-1749, 2005. [8] T. M. Cover, and J. A. Thomas, Elements of information theory, New York, Wiley, 1991. [9] K. Rose, “Deterministic Annealing for Clustering, Compression, Classification, Regression, and Related Optimization Problems,” Proceedings of the IEEE, Vol. 86, No. 11, pp. 2210–2239, 1998. [10] R. E. .Blahut, “Computation of Channel Capacity and Rate-Distortion Functions,” IEEE Transactions on Information Theory, Vol. IT-18, No. 4, pp. 460–473, 1974. [11] M. Pardalos, and J. B. Rosen, “Methods for Global Concave Minimization: A Bibliographic Survey,” SIAM Review, Vol. 28, No. 3., pp. 367–379, 1986. 8
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Learning Bounds for Domain Adaptation John Blitzer, Koby Crammer, Alex Kulesza, Fernando Pereira, and Jennifer Wortman Department of Computer and Information Science University of Pennsylvania, Philadelphia, PA 19146 {blitzer,crammer,kulesza,pereira,wortmanj}@cis.upenn.edu Abstract Empirical risk minimization offers well-known learning guarantees when training and test data come from the same domain. In the real world, though, we often wish to adapt a classifier from a source domain with a large amount of training data to different target domain with very little training data. In this work we give uniform convergence bounds for algorithms that minimize a convex combination of source and target empirical risk. The bounds explicitly model the inherent trade-off between training on a large but inaccurate source data set and a small but accurate target training set. Our theory also gives results when we have multiple source domains, each of which may have a different number of instances, and we exhibit cases in which minimizing a non-uniform combination of source risks can achieve much lower target error than standard empirical risk minimization. 1 Introduction Domain adaptation addresses a common situation that arises when applying machine learning to diverse data. We have ample data drawn from a source domain to train a model, but little or no training data from the target domain where we wish to use the model [17, 3, 10, 5, 9]. Domain adaptation questions arise in nearly every application of machine learning. In face recognition systems, training images are obtained under one set of lighting or occlusion conditions while the recognizer will be used under different conditions [14]. In speech recognition, acoustic models trained by one speaker need to be used by another [12]. In natural language processing, part-of-speech taggers, parsers, and document classifiers are trained on carefully annotated training sets, but applied to texts from different genres or styles [7, 6]. While many domain-adaptation algorithms have been proposed, there are only a few theoretical studies of the problem [3, 10]. Those studies focus on the case where training data is drawn from a source domain and test data is drawn from a different target domain. We generalize this approach to the case where we have some labeled data from the target domain in addition to a large amount of labeled source data. Our main result is a uniform convergence bound on the true target risk of a model trained to minimize a convex combination of empirical source and target risks. The bound describes an intuitive tradeoff between the quantity of the source data and the accuracy of the target data, and under relatively weak assumptions we can compute it from finite labeled and unlabeled samples of the source and target distributions. We use the task of sentiment classification to demonstrate that our bound makes correct predictions about model error with respect to a distance measure between source and target domains and the number of training instances. Finally, we extend our theory to the case in which we have multiple sources of training data, each of which may be drawn according to a different distribution and may contain a different number of instances. Several authors have empirically studied a special case of this in which each instance is weighted separately in the loss function, and instance weights are set to approximate the target domain distribution [10, 5, 9, 11]. We give a uniform convergence bound for algorithms that min1 imize a convex combination of multiple empirical source risks and we show that these algorithms can outperform standard empirical risk minimization. 2 A Rigorous Model of Domain Adaptation We formalize domain adaptation for binary classification as follows. A domain is a pair consisting of a distribution D on X and a labeling function f : X →[0, 1].1 Initially we consider two domains, a source domain ⟨DS, fS⟩and a target domain ⟨DT , fT ⟩. A hypothesis is a function h : X →{0, 1}. The probability according the distribution DS that a hypothesis h disagrees with a labeling function f (which can also be a hypothesis) is defined as ǫS(h, f) = Ex∼DS [ |h(x) −f(x)| ] . When we want to refer to the risk of a hypothesis, we use the shorthand ǫS(h) = ǫS(h, fS). We write the empirical risk of a hypothesis on the source domain as ˆǫS(h). We use the parallel notation ǫT (h, f), ǫT (h), and ˆǫT (h) for the target domain. We measure the distance between two distributions D and D′ using a hypothesis class-specific distance measure. Let H be a hypothesis class for instance space X, and AH be the set of subsets of X that are the support of some hypothesis in H. In other words, for every hypothesis h ∈H, {x : x ∈X, h(x) = 1} ∈AH. We define the distance between two distributions as: dH(D, D′) = 2 sup A∈AH |PrD [A] −PrD′ [A]| . For our purposes, the distance dH has an important advantage over more common means for comparing distributions such as L1 distance or the KL divergence: we can compute dH from finite unlabeled samples of the distributions D and D′ when H has finite VC dimension [4]. Furthermore, we can compute a finite-sample approximation to dH by finding a classifier h ∈H that maximally discriminates between (unlabeled) instances from D and D′ [3]. For a hypothesis space H, we define the symmetric difference hypothesis space H∆H as H∆H = {h(x) ⊕h′(x) : h, h′ ∈H} , where ⊕is the XOR operator. Each hypothesis g ∈H∆H labels as positive all points x on which a given pair of hypotheses in H disagree. We can then define AH∆H in the natural way as the set of all sets A such that A = {x : x ∈X, h(x) ̸= h′(x)} for some h, h′ ∈H. This allows us to define as above a distance dH∆H that satisfies the following useful inequality for any hypotheses h, h′ ∈H, which is straight-forward to prove: |ǫS(h, h′) −ǫT (h, h′)| ≤1 2dH∆H(DS, DT ) . We formalize the difference between labeling functions by measuring error relative to other hypotheses in our class. The ideal hypothesis minimizes combined source and target risk: h∗= argmin h∈H ǫS(h) + ǫT (h) . We denote the combined risk of the ideal hypothesis by λ = ǫS(h∗)+ǫT (h∗) . The ideal hypothesis explicitly embodies our notion of adaptability. When the ideal hypothesis performs poorly, we cannot expect to learn a good target classifier by minimizing source error.2 On the other hand, for the kinds of tasks mentioned in Section 1, we expect λ to be small. If this is the case, we can reasonably approximate target risk using source risk and the distance between DS and DT . We illustrate the kind of result available in this setting with the following bound on the target risk in terms of the source risk, the difference between labeling functions fS and fT , and the distance between the distributions DS and DT . This bound is essentially a restatement of the main theorem of Ben-David et al. [3], with a small correction to the statement of their theorem. 1This notion of domain is not the domain of a function. To avoid confusion, we will always mean a specific distribution and function pair when we say domain. 2Of course it is still possible that the source data contains relevant information about the target function even when the ideal hypothesis performs poorly — suppose, for example, that fS(x) = 1 if and only if fT (x) = 0 — but a classifier trained using source data will perform poorly on data from the target domain in this case. 2 Theorem 1 Let H be a hypothesis space of VC-dimension d and US, UT be unlabeled samples of size m′ each, drawn from DS and DT , respectively. Let ˆdH∆H be the empirical distance on US, UT , induced by the symmetric difference hypothesis space. With probability at least 1 −δ (over the choice of the samples), for every h ∈H, ǫT (h) ≤ǫS(h) + 1 2 ˆdH∆H(US, UT ) + 4 s 2d log(2m′) + log( 4 δ ) m′ + λ . The corrected proof of this result can be found Appendix A.3 The main step in the proof is a variant of the triangle inequality in which the sides of the triangle represent errors between different decision rules [3, 8]. The bound is relative to λ. When the combined error of the ideal hypothesis is large, there is no classifier that performs well on both the source and target domains, so we cannot hope to find a good target hypothesis by training only on the source domain. On the other hand, for small λ (the most relevant case for domain adaptation), Theorem 1 shows that source error and unlabeled H∆H-distance are important quantities for computing target error. 3 A Learning Bound Combining Source and Target Data Theorem 1 shows how to relate source and target risk. We now proceed to give a learning bound for empirical risk minimization using combined source and target training data. In order to simplify the presentation of the trade-offs that arise in this scenario, we state the bound in terms of VC dimension. Similar, tighter bounds could be derived using more sophisticated measures of complexity such as PAC-Bayes [15] or Rademacher complexity [2] in an analogous way. At train time a learner receives a sample S = (ST , SS) of m instances, where ST consists of βm instances drawn independently from DT and SS consists of (1−β)m instances drawn independently from DS. The goal of a learner is to find a hypothesis that minimizes target risk ǫT (h). When β is small, as in domain adaptation, minimizing empirical target risk may not be the best choice. We analyze learners that instead minimize a convex combination of empirical source and target risk: ˆǫα(h) = αˆǫT (h) + (1 −α)ˆǫS(h) We denote as ǫα(h) the corresponding weighted combination of true source and target risks, measured with respect to DS and DT . We bound the target risk of a domain adaptation algorithm that minimizes ˆǫα(h). The proof of the bound has two main components, which we state as lemmas below. First we bound the difference between the target risk ǫT (h) and weighted risk ǫα(h). Then we bound the difference between the true and empirical weighted risks ǫα(h) and ˆǫα(h). The proofs of these lemmas, as well as the proof of Theorem 2, are in Appendix B. Lemma 1 Let h be a hypothesis in class H. Then |ǫα(h) −ǫT (h)| ≤(1 −α) 1 2dH∆H(DS, DT ) + λ . The lemma shows that as α approaches 1, we rely increasingly on the target data, and the distance between domains matters less and less. The proof uses a similar technique to that of Theorem 1. Lemma 2 Let H be a hypothesis space of VC-dimension d. If a random labeled sample of size m is generated by drawing βm points from DT and (1 −β)m points from DS, and labeling them according to fS and fT respectively, then with probability at least 1 −δ (over the choice of the samples), for every h ∈H |ˆǫα(h) −ǫα(h)| < s α2 β + (1 −α)2 1 −β r d log(2m) −log δ 2m . 3A longer version of this paper that includes the omitted appendix can be found on the authors’ websites. 3 The proof is similar to standard uniform convergence proofs [16, 1], but it uses Hoeffding’s inequality in a different way because the bound on the range of the random variables underlying the inequality varies with α and β. The lemma shows that as α moves away from β (where each instance is weighted equally), our finite sample approximation to ǫα(h) becomes less reliable. Theorem 2 Let H be a hypothesis space of VC-dimension d. Let US and UT be unlabeled samples of size m′ each, drawn from DS and DT respectively. Let S be a labeled sample of size m generated by drawing βm points from DT and (1 −β)m points from DS, labeling them according to fS and fT , respectively. If ˆh ∈H is the empirical minimizer of ˆǫα(h) on S and h∗ T = minh∈H ǫT (h) is the target risk minimizer, then with probability at least 1 −δ (over the choice of the samples), ǫT (ˆh) ≤ǫT (h∗ T ) + 2 s α2 β + (1 −α)2 1 −β r d log(2m) −log δ 2m + 2(1 −α) 1 2 ˆdH∆H(US, UT ) + 4 s 2d log(2m′) + log( 4 δ ) m′ + λ . When α = 0 (that is, we ignore target data), the bound is identical to that of Theorem 1, but with an empirical estimate for the source error. Similarly when α = 1 (that is, we use only target data), the bound is the standard learning bound using only target data. At the optimal α (which minimizes the right hand side), the bound is always at least as tight as either of these two settings. Finally note that by choosing different values of α, the bound allows us to effectively trade off the small amount of target data against the large amount of less relevant source data. We remark that when it is known that λ = 0, the dependence on m in Theorem 2 can be improved; this corresponds to the restricted or realizable setting. 4 Experimental Results We evaluate our theory by comparing its predictions to empirical results. While ideally Theorem 2 could be directly compared with test error, this is not practical because λ is unknown, dH∆H is computationally intractable [3], and the VC dimension d is too large to be a useful measure of complexity. Instead, we develop a simple approximation of Theorem 2 that we can compute from unlabeled data. For many adaptation tasks, λ is small (there exists a classifier which is simultaneously good for both domains), so we ignore it here. We approximate dH∆H by training a linear classifier to discriminate between the two domains. We use a standard hinge loss (normalized by dividing by the number of instances) and apply the quantity 1 − hinge loss in place of the actual dH∆H. Let ζ(US, UT ) be our approximation to dH∆H, computed from source and target unlabeled data. For domains that can be perfectly separated with margin, ζ(US, UT ) = 1. For domains that are indistinguishable, ζ(US, UT )=0. Finally we replace the VC dimension sample complexity term with a tighter constant C. The resulting approximation to the bound of Theorem 2 is f(α) = s C m α2 β + (1 −α)2 1 −β + (1 −α)ζ(US, UT ) . (1) Our experimental results are for the task of sentiment classification. Sentiment classification systems have recently gained popularity because of their potential applicability to a wide range of documents in many genres, from congressional records to financial news. Because of the large number of potential genres, sentiment classification is an ideal area for domain adaptation. We use the data provided by Blitzer et al. [6], which consists of reviews of eight types of products from Amazon.com: apparel, books, DVDs, electronics, kitchen appliances, music, video, and a catchall category “other”. The task is binary classification: given a review, predict whether it is positive (4 or 5 out of 5 stars) or negative (1 or 2 stars). We chose the “apparel” domain as our target domain, and all of the plots on the right-hand side of Figure 1 are for this domain. We obtain empirical curves for the error as a function of α by training a classifier using a weighted hinge loss. Suppose the target domain has weight α and there are βm target training instances. Then we scale the loss of target training instance by α/β and the loss of a source training instance by (1 −α)/(1 −β). 4 (a) vary distance, mS = 2500, (c) ζ(US, UT ) = 0.715, (e) ζ(US, UT ) = 0.715, mT = 1000 mS = 2500, vary mT vary mS, mT = 2500 0 0.2 0.4 0.6 0.8 1 Dist: 0.780 Dist: 0.715 Dist: 0.447 Dist: 0.336 0 0.2 0.4 0.6 0.8 1 mT: 250 mT: 500 mT: 1000 mT: 2000 0 0.2 0.4 0.6 0.8 1 mS: 250 mS: 500 mS: 1000 mS: 2500 (b) vary sources, mS = 2500, (d) source = dvd, mS = 2500, (f) source = dvd, mT = 1000 vary mT vary mS, mT = 2500 0 0.1 0.4 0.6 0.8 1 books: 0.78 dvd: 0.715 electronics: 0.447 kitchen: 0.336 0 0.2 0.4 0.6 0.8 1 mT: 250 mT: 500 mT: 1000 mT: 2000 0 0.2 0.4 0.6 0.8 1 mS: 250 mS: 500 mS: 1000 mS: 2500 Figure 1: Comparing the bound with test error for sentiment classification. The x-axis of each figure shows α. The y-axis shows the value of the bound or test set error. (a), (c), and (e) depict the bound, (b), (d), and (f) the test error. Each curve in (a) and (b) represents a different distance. Curves in (c) and (d) represent different numbers of target instances. Curves in (e) and (f) represent different numbers of source instances. Figure 1 shows a series of plots of equation 1 (on the top) coupled with corresponding plots of test error (on the bottom) as a function of α for different amounts of source and target data and different distances between domains. In each pair of plots, a single parameter (distance, number of target instances mT , or number of source instances mS) is varied while the other two are held constant. Note that β = mT /(mT + mS). The plots on the top part of Figure 1 are not meant to be numerical proxies for the true error (For the source domains “books” and “dvd”, the distance alone is well above 1 2). Instead, they are scaled to illustrate that the bound is similar in shape to the true error curve and that relative relationships are preserved. By choosing a different C in equation 1 for each curve, one can achieve complete control over their minima. In order to avoid this, we only use a single value of C =1600 for all 12 curves on the top part of Figure 1. First note that in every pair of plots, the empirical error curves have a roughly convex shape that mimics the shape of the bounds. Furthermore the value of α which minimizes the bound also has a low empirical error for each corresponding curve. This suggests that choosing α to minimize the bound of Theorem 2 and subsequently training a classifier to minimize the empirical error ˆǫα(h) can work well in practice, provided we have a reasonable measure of complexity.4 Figures 1a and 1b show that more distant source domains result in higher target error. Figures 1c and 1d illustrate that for more target data, we have not only lower error in general, but also a higher minimizing α. Finally, figures 1e and 1f depict the limitation of distant source data. With enough target data, no matter how much source data we include, we always prefer to use only the target data. This is reflected in our bound as a phase transition in the value of the optimal α (governing the tradeoff between source and target data). The phase transition occurs when mT = C/ζ(US, UT )2 (See Figure 2). 4Although Theorem 2 does not hold uniformly for all α as stated, this is easily remedied via an application of the union bound. The resulting bound will contain an additional logarithmic factor in the complexity term. 5 Source Target ×102 5,000 50,000 722,000 11 million 167 million 32 30 28 26 24 0 0.5 1 Figure 2: An example of the phase transition in the optimal α. The value of α which minimizes the bound is indicated by the intensity, where black means α = 1 (corresponding to ignoring source and learning only from target data). We fix C = 1600 and ζ(US, UT ) = 0.715, as in our sentiment results. The x-axis shows the number of source instances (log-scale). The y-axis shows the number of target instances. A phase transition occurs at 3,130 target instances. With more target instances than this, it is more effective to ignore even an infinite amount of source data. 5 Learning from Multiple Sources We now explore an extension of our theory to the case of multiple source domains. We are presented with data from N distinct sources. Each source Sj is associated with an unknown underlying distribution Dj over input points and an unknown labeling function fj. From each source Sj, we are given mj labeled training instances, and our goal is to use these instances to train a model to perform well on a target domain ⟨DT , fT ⟩, which may or may not be one of the sources. This setting is motivated by several new domain adaptation algorithms [10, 5, 11, 9] that weigh the loss from training instances depending on how “far” they are from the target domain. That is, each training instance is its own source domain. As in the previous sections, we will examine algorithms that minimize convex combinations of training errors over the labeled examples from each source domain. As before, we let mj = βjm with PN j=1 βj = 1. Given a vector α = (α1, · · · , αN) of domain weights with P j αj = 1, we define the empirical α-weighted error of function h as ˆǫα(h) = N X j=1 αjˆǫj(h) = N X j=1 αj mj X x∈Sj |h(x) −fj(x)| . The true α-weighted error ǫα(h) is defined analogously. Let Dα be a mixture of the N source distributions with mixing weights equal to the components of α. Finally, analogous to λ in the single-source setting, we define the error of the multi-source ideal hypothesis for a weighting α as γα = min h {ǫT (h) + ǫα(h)} = min h {ǫT (h) + N X j=1 αjǫj(h)} . The following theorem gives a learning bound for empirical risk minimization using the empirical α-weighted error. Theorem 3 Suppose we are given mj labeled instances from source Sj for j = 1 . . . N. For a fixed vector of weights α, let ˆh = argminh∈H ˆǫα(h), and let h∗ T = argminh∈H ǫT (h). Then for any δ ∈(0, 1), with probability at least 1 −δ (over the choice of samples from each source), ǫT (ˆh) ≤ǫT (h∗ T ) + 2 v u u t N X j=1 α2 j βj r d log 2m −log δ 2m + 2 γα + 1 2dH∆H(Dα, DT ) . 6 (a) Source. More girls than boys (b) Target. Separator from (c) Weighting sources to match uniform mixture is suboptimal target is optimal Females Males learned separator optimal separator learned separator errors Females Males Target optimal & learned separator Figure 3: A 1-dimensional example illustrating how non-uniform mixture weighting can result in optimal error. We observe one feature, which we use to predict gender. (a) At train time we observe more females than males. (b) Learning by uniformly weighting the training data causes us to learn a suboptimal decision boundary, (c) but by weighting the males more highly, we can match the target data and learn an optimal classifier. The full proof is in appendix C. Like the proof of Theorem 2, it is split into two parts. The first part bounds the difference between the α-weighted error and the target error similar to lemma 1. The second is a uniform convergence bound for ˆǫα(h) similar to lemma 2. Theorem 3 reduces to Theorem 2 when we have only two sources, one of which is the target domain (that is, we have some small number of target instances). It is more general, though, because by manipulating α we can effectively change the source domain. This has two consequences. First, we demand that there exists a hypothesis h∗which has low error on both the α-weighted convex combination of sources and the target domain. Second, we measure distance between the target and a mixture of sources, rather than between the target and a single source. One question we might ask is whether there exist settings where a non-uniform weighting can lead to a significantly lower value of the bound than a uniform weighting. This can happen if some non-uniform weighting of sources accurately approximates the target domain. As a hypothetical example, suppose we are trying to predict gender from height (Figure 3). Each instance is drawn from a gender-specific Gaussian. In this example, we can find the optimal classifier by weighting the “males” and “females” components of the source to match the target. 6 Related Work Domain adaptation is a widely-studied area, and we cannot hope to cover every aspect and application of it here5. Instead, in this section we focus on other theoretical approaches to domain adaptation. While we do not explicitly address the relationship in this paper, we note that domain adaptation is closely related to the setting of covariate shift, which has been studied in statistics. In addition to the work of Huang et al. [10], several other authors have considered learning by assigning separate weights to the components of the loss function corresponding to separate instances. Bickel at al. [5] and Jiang and Zhai [11] suggest promising empirical algorithms that in part inspire our Theorem 3. We hope that our work can help to explain when these algorithms are effective. Dai et al. [9] considered weighting instances using a transfer-aware variant of boosting, but the learning bounds they give are no stronger than bounds which completely ignore the source data. Crammer et al. [8] consider learning when the marginal distribution on instances is the same across sources but the labeling function may change. This corresponds in our theory to cases where dH∆H = 0 but λ is large. Like us they consider multiple sources, but their notion of weighting is less general. They consider only including or discarding a source entirely. Li and Bilmes [13] give PAC-Bayesian learning bounds for adaptation using “divergence priors”. They place source-centered prior on the parameters of a model learned in the target domain. Like 5The NIPS 2006 Workshop on Learning When Test and Training Inputs have Different Distributions (http://ida.first.fraunhofer.de/projects/different06/) contains a good set of references on domain adaptation and related topics. 7 our model, the divergence prior also emphasizes the tradeoff between source and target. In our model, though, we measure the divergence (and consequently the bias) of the source domain from unlabeled data. This allows us to choose the best tradeoff between source and target labeled data. 7 Conclusion In this work we investigate the task of domain adaptation when we have a large amount of training data from a source domain but wish to apply a model in a target domain with a much smaller amount of training data. Our main result is a uniform convergence learning bound for algorithms which minimize convex combinations of source and target empirical risk. Our bound reflects the trade-off between the size of the source data and the accuracy of the target data, and we give a simple approximation to it that is computable from finite labeled and unlabeled samples. This approximation makes correct predictions about model test error for a sentiment classification task. Our theory also extends in a straightforward manner to a multi-source setting, which we believe helps to explain the success of recent empirical work in domain adaptation. Our future work has two related directions. First, we wish to tighten our bounds, both by considering more sophisticated measures of complexity [15, 2] and by focusing our distance measure on the most relevant features, rather than all the features. We also plan to investigate algorithms that choose a convex combination of multiple sources to minimize the bound in Theorem 3. 8 Acknowledgements This material is based upon work partially supported by the Defense Advanced Research Projects Agency (DARPA) under Contract No. NBCHD030010. 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 DARPA or Department of Interior-National Business Center (DOI-NBC). References [1] M. Anthony and P. Bartlett. Neural Network Learning: Theoretical Foundations. Cambridge University Press, Cambridge, 1999. [2] P. Barlett and S. Mendelson. Rademacher and gaussian complexities: Risk bounds and structural results. JMLR, 3:463–482, 2002. [3] S. Ben-David, J. Blitzer, K. Crammer, and F. Pereira. Analysis of representations for domain adaptation. In NIPS, 2007. [4] S. Ben-David, J. Gehrke, and D. Kifer. Detecting change in data streams. In VLDB, 2004. [5] S. Bickel, M. Br¨uckner, and T. Scheffer. Discriminative learning for differing training and test distributions. In ICML, 2007. [6] J. Blitzer, M. Dredze, and F. Pereira. Biographies, bollywood, boomboxes and blenders: Domain adaptation for sentiment classification. In ACL, 2007. [7] C. Chelba and A. Acero. Empirical methods in natural language processing. In EMNLP, 2004. [8] K. Crammer, M. Kearns, and J. Wortman. Learning from multiple sources. In NIPS, 2007. [9] W. Dai, Q. Yang, G. Xue, and Y. Yu. Boosting for transfer learning. In ICML, 2007. [10] J. Huang, A. Smola, A. Gretton, K. Borgwardt, and B. Schoelkopf. Correcting sample selection bias by unlabeled data. In NIPS, 2007. [11] J. Jiang and C. Zhai. Instance weighting for domain adaptation. In ACL, 2007. [12] C. Legetter and P. Woodland. Maximum likelihood linear regression for speaker adaptation of continuous density hidden markov models. Computer Speech and Language, 9:171–185, 1995. [13] X. Li and J. Bilmes. A bayesian divergence prior for classification adaptation. In AISTATS, 2007. [14] A. Martinez. Recognition of partially occluded and/or imprecisely localized faces using a probabilistic approach. In CVPR, 2007. [15] D. McAllester. Simplified PAC-Bayesian margin bounds. In COLT, 2003. [16] V. Vapnik. Statistical Learning Theory. John Wiley, New York, 1998. [17] P. Wu and T. Dietterich. Improving svm accuracy by training on auxiliary data sources. In ICML, 2004. 8
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SpAM: Sparse Additive Models Pradeep Ravikumar† Han Liu†‡ John Lafferty∗† Larry Wasserman‡† †Machine Learning Department ‡Department of Statistics ∗Computer Science Department Carnegie Mellon University Pittsburgh, PA 15213 Abstract We present a new class of models for high-dimensional nonparametric regression and classification called sparse additive models (SpAM). Our methods combine ideas from sparse linear modeling and additive nonparametric regression. We derive a method for fitting the models that is effective even when the number of covariates is larger than the sample size. A statistical analysis of the properties of SpAM is given together with empirical results on synthetic and real data, showing that SpAM can be effective in fitting sparse nonparametric models in high dimensional data. 1 Introduction Substantial progress has been made recently on the problem of fitting high dimensional linear regression models of the form Yi = X T i β + ϵi, for i = 1, . . . , n. Here Yi is a real-valued response, Xi is a p-dimensional predictor and ϵi is a mean zero error term. Finding an estimate of β when p > n that is both statistically well-behaved and computationally efficient has proved challenging; however, the lasso estimator (Tibshirani (1996)) has been remarkably successful. The lasso estimator bβ minimizes the ℓ1-penalized sums of squares X i (Yi −X T i β) + λ p X j=1 |β j| (1) with the ℓ1 penalty ∥β∥1 encouraging sparse solutions, where many components bβ j are zero. The good empirical success of this estimator has been recently backed up by results confirming that it has strong theoretical properties; see (Greenshtein and Ritov, 2004; Zhao and Yu, 2007; Meinshausen and Yu, 2006; Wainwright, 2006). The nonparametric regression model Yi = m(Xi)+ϵi, where m is a general smooth function, relaxes the strong assumptions made by a linear model, but is much more challenging in high dimensions. Hastie and Tibshirani (1999) introduced the class of additive models of the form Yi = p X j=1 m j(Xi j) + ϵi (2) which is less general, but can be more interpretable and easier to fit; in particular, an additive model can be estimated using a coordinate descent Gauss-Seidel procedure called backfitting. An extension of the additive model is the functional ANOVA model Yi = X 1≤j≤p m j(Xi j) + X j<k m j,k(Xi j, Xik) + X j<k<ℓ m j,k,ℓ(Xi j, Xik, Xiℓ) + · · · + ϵi (3) 1 which allows interactions among the variables. Unfortunately, additive models only have good statistical and computational behavior when the number of variables p is not large relative to the sample size n. In this paper we introduce sparse additive models (SpAM) that extend the advantages of sparse linear models to the additive, nonparametric setting. The underlying model is the same as in (2), but constraints are placed on the component functions {m j}1≤j≤p to simultaneously encourage smoothness of each component and sparsity across components; the penalty is similar to that used by the COSSO of Lin and Zhang (2006). The SpAM estimation procedure we introduce allows the use of arbitrary nonparametric smoothing techniques, and in the case where the underlying component functions are linear, it reduces to the lasso. It naturally extends to classification problems using generalized additive models. The main results of the paper are (i) the formulation of a convex optimization problem for estimating a sparse additive model, (ii) an efficient backfitting algorithm for constructing the estimator, (iii) simulations showing the estimator has excellent behavior on some simulated and real data, even when p is large, and (iv) a statistical analysis of the theoretical properties of the estimator that support its good empirical performance. 2 The SpAM Optimization Problem In this section we describe the key idea underlying SpAM. We first present a population version of the procedure that intuitively suggests how sparsity is achieved. We then present an equivalent convex optimization problem. In the following section we derive a backfitting procedure for solving this optimization problem in the finite sample setting. To motivate our approach, we first consider a formulation that scales each component function g j by a scalar β j, and then imposes an ℓ1 constraint on β = (β1, . . . , βp)T . For j ∈{1, . . . , p}, let H j denote the Hilbert space of measurable functions f j(x j) of the single scalar variable x j, such that E( f j(X j)) = 0 and E( f j(X j)2) < ∞, furnished with the inner product D f j, f ′ j E = E f j(X j) f ′ j(X j) . (4) Let Hadd = H1 + H2 + . . . , Hp denote the Hilbert space of functions of (x1, . . . , x p) that have an additive form: f (x) = P j f j(x j). The standard additive model optimization problem, in the population setting, is min f j∈H j, 1≤j≤p E Y −Pp j=1 f j(X j) 2 (5) and m(x) = E(Y | X = x) is the unknown regression function. Now consider the following modification of this problem that imposes additional constraints: (P) min β∈Rp,g j∈H j E Y −Pp j=1 β jg j(X j) 2 (6a) subject to p X j=1 |β j| ≤L (6b) E g2 j = 1, j = 1, . . . , p (6c) E g j = 0, j = 1, . . . , p (6d) noting that g j is a function while β is a vector. Intuitively, the constraint that β lies in the ℓ1-ball {β : ∥β∥1 ≤L} encourages sparsity of the estimated β, just as for the parametric lasso. When β is sparse, the estimated additive function f (x) = Pp j=1 f j(x j) = Pp j=1 β jg j(x j) will also be sparse, meaning that many of the component functions f j(·) = β jg j(·) are identically zero. The constraints (6c) and (6c) are imposed for identifiability; without (6c), for example, one could always satisfy (6a) by rescaling. While this optimization problem makes plain the role ℓ1 regularization of β to achieve sparsity, it has the unfortunate drawback of not being convex. More specifically, while the optimization problem is convex in β and {g j} separately, it is not convex in β and {g j} jointly. 2 However, consider the following related optimization problem: (Q) min f j∈H j E Y −Pp j=1 f j(X j) 2 (7a) subject to p X j=1 q E( f 2 j (X j)) ≤L (7b) E( f j) = 0, j = 1, . . . , p. (7c) This problem is convex in {f j}. Moreover, the solutions to problems (P) and (Q) are equivalent: n β∗ j o , n g∗ j o optimizes (P) implies n f ∗ j = β∗ j g∗ j o optimizes (Q); n f ∗ j o optimizes (Q) implies n β∗ j = (∥f j∥2)T o , n g∗ j = f ∗ j /∥f ∗ j ∥2 o optimizes (P). While optimization problem (Q) has the important virtue of being convex, the way it encourages sparsity is not intuitive; the following observation provides some insight. Consider the set C ⊂R4 defined by C = ( f11, f12, f21, f22)T ∈R4 : q f 2 11 + f 2 12 + q f 2 21 + f 2 22 ≤L . Then the projection π12C onto the first two components is an ℓ2 ball. However, the projection π13C onto the first and third components is an ℓ1 ball. In this way, it can be seen that the constraint P j
f j
2 ≤L acts as an ℓ1 constraint across components to encourage sparsity, while it acts as an ℓ2 constraint within components to encourage smoothness, as in a ridge regression penalty. It is thus crucial that the norm
f j
2 appears in the constraint, and not its square
f j
2 2. For the purposes of sparsity, this constraint could be replaced by P j
f j
q ≤L for any q ≥1. In case each f j is linear, ( f j(x1 j), . . . , f (xnj)) = β j(x1 j, . . . , xnj), the optimization problem reduces to the lasso. The use of scaling coefficients together with a nonnegative garrote penalty, similar to our problem (P), is considered by Yuan (2007). However, the component functions g j are fixed, so that the procedure is not asymptotically consistent. The form of the optimization problem (Q) is similar to that of the COSSO for smoothing spline ANOVA models (Lin and Zhang, 2006); however, our method differs significantly from the COSSO, as discussed below. In particular, our method is scalable and easy to implement even when p is much larger than n. 3 A Backfitting Algorithm for SpAM We now derive a coordinate descent algorithm for fitting a sparse additive model. We assume that we observe Y = m(X) + ϵ, where ϵ is mean zero Gaussian noise. We write the Lagrangian for the optimization problem (Q) as L( f, λ, µ) = 1 2E Y −Pp j=1 f j(X j) 2 + λ p X j=1 q E( f 2 j (X j)) + X j µ jE( f j). (8) Let R j = Y −P k̸= j fk(Xk) be the jth residual. The stationary condition for minimizing L as a function of f j, holding the other components fk fixed for k ̸= j, is expressed in terms of the Frechet derivative δL as δL( f, λ, µ; δf j) = E ( f j −R j + λv j)δf j = 0 (9) for any δf j ∈H j satisfying E(δf j) = 0, where v j ∈∂ q E( f 2 j ) is an element of the subgradient, satisfying q Ev2 j ≤1 and v j = f j .q E( f 2 j ) if E( f 2 j ) ̸= 0. Therefore, conditioning on X j, the stationary condition (9) implies f j + λv j = E(R j | X j). (10) Letting Pj = E[R j | X j] denote the projection of the residual onto H j, the solution satisfies 1 + λ q E( f 2 j ) f j = Pj if E(P2 j ) > λ (11) 3 Input: Data (Xi, Yi), regularization parameter λ. Initialize f j = f (0) j , for j = 1, . . . , p. Iterate until convergence: For each j = 1, . . . , p: Compute the residual: R j = Y −P k̸= j fk(Xk); Estimate the projection Pj = E[R j | X j] by smoothing: bPj = S j R j; Estimate the norm s j = q E[Pj]2 using, for example, (15) or (35); Soft-threshold: f j = 1 −λ bs j + bPj; Center: f j ←f j −mean( f j). Output: Component functions f j and estimator b m(Xi) = P j f j(Xi j). Figure 1: THE SPAM BACKFITTING ALGORITHM and f j = 0 otherwise. Condition (11), in turn, implies 1 + λ q E( f 2 j ) q E( f 2 j ) = q E(P2 j ) or q E( f 2 j ) = q E(P2 j ) −λ. (12) Thus, we arrive at the following multiplicative soft-thresholding update for f j: f j = 1 − λ q E(P2 j ) + Pj (13) where [·]+ denotes the positive part. In the finite sample case, as in standard backfitting (Hastie and Tibshirani, 1999), we estimate the projection E[R j | X j] by a smooth of the residuals: bPj = S j R j (14) where S j is a linear smoother, such as a local linear or kernel smoother. Let bs j be an estimate of q E[P2 j ]. A simple but biased estimate is bs j = 1 √n ∥bPj∥2 = q mean(bP2 j ). (15) More accurate estimators are possible; an example is given in the appendix. We have thus derived the SpAM backfitting algorithm given in Figure 1. While the motivating optimization problem (Q) is similar to that considered in the COSSO (Lin and Zhang, 2006) for smoothing splines, the SpAM backfitting algorithm decouples smoothing and sparsity, through a combination of soft-thresholding and smoothing. In particular, SpAM backfitting can be carried out with any nonparametric smoother; it is not restricted to splines. Moreover, by iteratively estimating over the components and using soft thresholding, our procedure is simple to implement and scales to high dimensions. 3.1 SpAM for Nonparametric Logistic Regression The SpAM backfitting procedure can be extended to nonparametric logistic regression for classification. The additive logistic model is P(Y = 1 | X) ≡p(X; f ) = exp Pp j=1 f j(X j) 1 + exp Pp j=1 f j(X j) (16) 4 where Y ∈{0, 1}, and the population log-likelihood is ℓ( f ) = E Y f (X) −log (1 + exp f (X)) . Recall that in the local scoring algorithm for generalized additive models (Hastie and Tibshirani, 1999) in the logistic case, one runs the backfitting procedure within Newton’s method. Here one iteratively computes the transformed response for the current estimate f0 Zi = f0(Xi) + Yi −p(Xi; f0) p(Xi; f0)(1 −p(Xi; f0)) (17) and weights w(Xi) = p(Xi; f0)(1 −p(Xi; f0), and carries out a weighted backfitting of (Z, X) with weights w. The weighted smooth is given by bPj = S j(wR j) S jw . (18) To incorporate the sparsity penalty, we first note that the Lagrangian is given by L( f, λ, µ) = E log (1 + exp f (X)) −Y f (X) + λ p X j=1 q E( f 2 j (X j)) + X j µ jE( f j) (19) and the stationary condition for component function f j is E p −Y | X j + λv j = 0 where v j is an element of the subgradient ∂ q E( f 2 j ). As in the unregularized case, this condition is nonlinear in f , and so we linearize the gradient of the log-likelihood around f0. This yields the linearized condition E w(X)( f (X) −Z) | X j + λv j = 0. When E( f 2 j ) ̸= 0, this implies the condition E w | X j + λ q E( f j)2 f j(X j) = E(wR j | X j). (20) In the finite sample case, in terms of the smoothing matrix S j, this becomes f j = S j(wR j) S jw + λ .q E( f 2 j ) . (21) If ∥S j(wR j)∥2 < λ, then f j = 0. Otherwise, this implicit, nonlinear equation for f j cannot be solved explicitly, so we propose to iterate until convergence: f j ← S j(wR j) S jw + λ√n ∥f j∥2 . (22) When λ = 0, this yields the standard local scoring update (18). An example of logistic SpAM is given in Section 5. 4 Properties of SpAM 4.1 SpAM is Persistent The notion of risk consistency, or persistence, was studied by Juditsky and Nemirovski (2000) and Greenshtein and Ritov (2004) in the context of linear models. Let (X, Y) denote a new pair (independent of the observed data) and define the predictive risk when predicting Y with f (X) by R( f ) = E(Y −f (X))2. (23) Since we consider predictors of the form f (x) = P j β jg j(x j) we also write the risk as R(β, g) where β = (β1, . . . , βp) and g = (g1, . . . , gp). Following Greenshtein and Ritov (2004), we say that an estimator b mn is persistent relative to a class of functions Mn if R(b mn) −R(m∗ n) P→0 (24) where m∗ n = argmin f ∈Mn R( f ) is the predictive oracle. Greenshtein and Ritov (2004) showed that the lasso is persistent for the class of linear models Mn = {f (x) = xT β : ∥β∥1 ≤Ln} if Ln = o((n/ log n)1/4). We show a similar result for SpAM. Theorem 4.1. Suppose that pn ≤enξ for some ξ < 1. Then SpAM is persistent relative to the class of additive models Mn = n f (x) = Pp j=1 β jg j(x j) : ∥β∥1 ≤Ln o if Ln = o n(1−ξ)/4 . 5 4.2 SpAM is Sparsistent In the case of linear regression, with m j(X j) = βT j X j, Wainwright (2006) shows that under certain conditions on n, p, s = |supp(β)|, and the design matrix X, the lasso recovers the sparsity pattern asymptotically; that is, the lasso estimator bβn is sparsistent: P supp(β) = supp(bβn) →1. We show a similar result for SpAM with the sparse backfitting procedure. For the purpose of analysis, we use orthogonal function regression as the smoothing procedure. For each j = 1, . . . , p let ψ j be an orthogonal basis for H j. We truncate the basis to finite dimension dn, and let dn →∞such that dn/n →0. Let 9 j denote the n × d matrix 9 j(i, k) = ψ jk(Xi j). If A ⊂{1, . . . , p}, we denote by 9A the n × d|A| matrix where for each i ∈A, 9i appears as a submatrix in the natural way. The SpAM optimization problem can then be written as min β 1 2n Y −Pp j=1 9 jβ j 2 + λn p X j=1 r 1 n βT j 9T j 9 jβ j (25) where each β j is a d-dimensional vector. Let S denote the true set of variables {j : m j ̸= 0}, with s = |S|, and let Sc denote its complement. Let bSn = {j : bβ j ̸= 0} denote the estimated set of variables from the minimizer bβn of (25). Theorem 4.2. Suppose that 9 satisfies the conditions 3max 1 n 9T S 9S ≤Cmax < ∞ and 3min 1 n 9T S 9S ≥Cmin > 0 (26)
1 n 9T Sc9S 1 n 9T S 9S −1
2 2 ≤ s Cmin Cmax 1 −δ √s , for some 0 < δ ≤1 (27) Let the regularization parameter λn →0 be chosen to satisfy λn p sdn →0, s dnλn →0, and dn(log dn + log(p −s)) nλ2n →0. (28) Then SpAM is sparsistent: P bSn = S −→1. 5 Experiments In this section we present experimental results for SpAM applied to both synthetic and real data, including regression and classification examples that illustrate the behavior of the algorithm in various conditions. We first use simulated data to investigate the performance of the SpAM backfitting algorithm, where the true sparsity pattern is known. We then apply SpAM to some real data. If not explicitly stated otherwise, the data are always rescaled to lie in a d-dimensional cube [0, 1]d, and a kernel smoother with Gaussian kernel is used. To tune the penalization parameter λ, we use a C p statistic, which is defined as C p( bf ) = 1 n n X i=1 Yi −Pp j=1 bf j(X j) 2 + 2bσ 2 n p X j=1 trace(S j) 1[ bf j ̸= 0] (29) where S j is the smoothing matrix for the j-th dimension and bσ 2 is the estimated variance. 5.1 Simulations We first apply SpAM to an example from (Härdle et al., 2004). A dataset with sample size n = 150 is generated from the following 200-dimensional additive model: Yi = f1(xi1) + f2(xi2) + f3(xi3) + f4(xi4) + ϵi (30) f1(x) = −2 sin(2x), f2(x) = x2 −1 3, f3(x) = x −1 2, f4(x) = e−x + e−1 −1 (31) and f j(x) = 0 for j ≥5 with noise ϵi ∼N(0, 1). These data therefore have 196 irrelevant dimensions. The results of applying SpAM with the plug-in bandwidths are summarized in Figure 2. 6 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Component Norms 194 9 94 2 4 3 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 12 14 Cp 0.0 0.2 0.4 0.6 0.8 1.0 sample size prob. of correct recovery 0 10 20 30 40 50 60 70 80 90 110 130 150 p=128 p=256 0.0 0.2 0.4 0.6 0.8 1.0 −4 −2 2 4 l1=97.05 x1 m1 0.0 0.2 0.4 0.6 0.8 1.0 −4 −2 2 4 6 l1=88.36 x2 m2 0.0 0.2 0.4 0.6 0.8 1.0 −6 −4 −2 2 4 6 l1=90.65 x3 m3 0.0 0.2 0.4 0.6 0.8 1.0 −4 −2 2 4 6 l1=79.26 x4 m4 0.0 0.2 0.4 0.6 0.8 1.0 −6 −4 −2 2 4 6 zero x5 m5 0.0 0.2 0.4 0.6 0.8 1.0 −6 −4 −2 2 4 6 zero x6 m6 Figure 2: (Simulated data) Upper left: The empirical ℓ2 norm of the estimated components as plotted against the tuning parameter λ; the value on the x-axis is proportional to P j ∥bf j∥2. Upper center: The C p scores against the tuning parameter λ; the dashed vertical line corresponds to the value of λ which has the smallest C p score. Upper right: The proportion of 200 trials where the correct relevant variables are selected, as a function of sample size n. Lower (from left to right): Estimated (solid lines) versus true additive component functions (dashed lines) for the first 6 dimensions; the remaining components are zero. 5.2 Boston Housing The Boston housing data was collected to study house values in the suburbs of Boston; there are altogether 506 observations with 10 covariates. The dataset has been studied by many other authors (Härdle et al., 2004; Lin and Zhang, 2006), with various transformations proposed for different covariates. To explore the sparsistency properties of our method, we add 20 irrelevant variables. Ten of them are randomly drawn from Uniform(0, 1), the remaining ten are a random permutation of the original ten covariates, so that they have the same empirical densities. The full model (containing all 10 chosen covariates) for the Boston Housing data is: medv = α + f1(crim) + f2(indus) + f3(nox) + f4(rm) + f5(age) + f6(dis) + f7(tax) + f8(ptratio) + f9(b) + f10(lstat) (32) The result of applying SpAM to this 30 dimensional dataset is shown in Figure 3. SpAM identifies 6 nonzero components. It correctly zeros out both types of irrelevant variables. From the full solution path, the important variables are seen to be rm, lstat, ptratio, and crim. The importance of variables nox and b are borderline. These results are basically consistent with those obtained by other authors (Härdle et al., 2004). However, using C p as the selection criterion, the variables indux, age, dist, and tax are estimated to be irrelevant, a result not seen in other studies. 5.3 SpAM for Spam Here we consider an email spam classification problem, using the logistic SpAM backfitting algorithm from Section 3.1. This dataset has been studied by Hastie et al. (2001), using a set of 3,065 emails as a training set, and conducting hypothesis tests to choose significant variables; there are a total of 4,601 observations with p = 57 attributes, all numeric. The attributes measure the percentage of specific words or characters in the email, the average and maximum run lengths of upper case letters, and the total number of such letters. To demonstrate how SpAM performs well with sparse data, we only sample n = 300 emails as the training set, with the remaining 4301 data points used as the test set. We also use the test data as the hold-out set to tune the penalization parameter λ. The results of a typical run of logistic SpAM are summarized in Figure 4, using plug-in bandwidths. 7 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 Component Norms 17 7 5 6 3 8 10 4 0.0 0.2 0.4 0.6 0.8 1.0 20 30 40 50 60 70 80 Cp 0.0 0.2 0.4 0.6 0.8 1.0 −10 10 20 l1=177.14 x1 m1 0.0 0.2 0.4 0.6 0.8 1.0 −10 10 20 l1=1173.64 x4 m4 0.0 0.2 0.4 0.6 0.8 1.0 −10 10 20 l1=478.29 x8 m8 0.0 0.2 0.4 0.6 0.8 1.0 −10 10 20 l1=1221.11 x10 m10 Figure 3: (Boston housing) Left: The empirical ℓ2 norm of the estimated components versus the regularization parameter λ. Center: The C p scores against λ; the dashed vertical line corresponds to best C p score. Right: Additive fits for four relevant variables. λ(×10−3) ERROR # ZEROS SELECTED VARIABLES 5.5 0.2009 55 { 8,54} 5.0 0.1725 51 { 8, 9, 27, 53, 54, 57} 4.5 0.1354 46 {7, 8, 9, 17, 18, 27, 53, 54, 57, 58} 4.0 0.1083 (√) 20 {4, 6–10, 14–22, 26, 27, 38, 53–58} 3.5 0.1117 0 ALL 3.0 0.1174 0 ALL 2.5 0.1251 0 ALL 2.0 0.1259 0 ALL 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 0.12 0.14 0.16 0.18 0.20 penalization parameter Empirical prediction error Figure 4: (Email spam) Classification accuracies and variable selection for logistic SpAM. 6 Acknowlegments This research was supported in part by NSF grant CCF-0625879 and a Siebel Scholarship to PR. References GREENSHTEIN, E. and RITOV, Y. (2004). Persistency in high dimensional linear predictor-selection and the virtue of over-parametrization. Journal of Bernoulli 10 971–988. HÄRDLE, W., MÜLLER, M., SPERLICH, S. and WERWATZ, A. (2004). Nonparametric and Semiparametric Models. Springer-Verlag Inc. HASTIE, T. and TIBSHIRANI, R. (1999). Generalized additive models. Chapman & Hall Ltd. HASTIE, T., TIBSHIRANI, R. and FRIEDMAN, J. H. (2001). The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer-Verlag. JUDITSKY, A. and NEMIROVSKI, A. (2000). Functional aggregation for nonparametric regression. Ann. Statist. 28 681–712. LIN, Y. and ZHANG, H. H. (2006). Component selection and smoothing in multivariate nonparametric regression. Ann. Statist. 34 2272–2297. MEINSHAUSEN, N. and YU, B. (2006). Lasso-type recovery of sparse representations for high-dimensional data. Tech. Rep. 720, Department of Statistics, UC Berkeley. TIBSHIRANI, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B, Methodological 58 267–288. WAINWRIGHT, M. (2006). Sharp thresholds for high-dimensional and noisy recovery of sparsity. Tech. Rep. 709, Department of Statistics, UC Berkeley. YUAN, M. (2007). Nonnegative garrote component selection in functional ANOVA models. In Proceedings of AI and Statistics, AISTATS. ZHAO, P. and YU, B. (2007). On model selection consistency of lasso. J. of Mach. Learn. Res. 7 2541–2567. 8
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Bayesian Inference for Spiking Neuron Models with a Sparsity Prior Sebastian Gerwinn Jakob H Macke Matthias Seeger Matthias Bethge Max Planck Institute for Biological Cybernetics Spemannstrasse 41 72076 Tuebingen, Germany {firstname.surname}@tuebingen.mpg.de Abstract Generalized linear models are the most commonly used tools to describe the stimulus selectivity of sensory neurons. Here we present a Bayesian treatment of such models. Using the expectation propagation algorithm, we are able to approximate the full posterior distribution over all weights. In addition, we use a Laplacian prior to favor sparse solutions. Therefore, stimulus features that do not critically influence neural activity will be assigned zero weights and thus be effectively excluded by the model. This feature selection mechanism facilitates both the interpretation of the neuron model as well as its predictive abilities. The posterior distribution can be used to obtain confidence intervals which makes it possible to assess the statistical significance of the solution. In neural data analysis, the available amount of experimental measurements is often limited whereas the parameter space is large. In such a situation, both regularization by a sparsity prior and uncertainty estimates for the model parameters are essential. We apply our method to multi-electrode recordings of retinal ganglion cells and use our uncertainty estimate to test the statistical significance of functional couplings between neurons. Furthermore we used the sparsity of the Laplace prior to select those filters from a spike-triggered covariance analysis that are most informative about the neural response. 1 Introduction A central goal of systems neuroscience is to identify the functional relationship between environmental stimuli and a neural response. Given an arbitrary stimulus we would like to predict the neural response as well as possible. In order to achieve this goal with limited amount of data, it is essential to combine the information in the data with prior knowledge about neural function. To this end, generalized linear models (GLMs) have proven to be particularly useful as they allow for flexible model architectures while still being tractable for estimation. The GLM neuron model consists of a linear filter, a static nonlinear transfer function and a Poisson spike generating mechanism. To determine the neural response to a given stimulus, the stimulus is first convolved with the linear filter (i.e. the receptive field of the neuron). Subsequently, the filter output is converted into an instantaneous firing rate via a static nonlinear transfer function, and finally spikes are generated from an inhomogeneous Poisson-process according to this firing rate. Note, however, that the GLM neuron model is not limited to describe neurons with Poisson firing statistics. Rather, it is possible to incorporate influences of its own spiking history on the neural response. That is, the firing rate is then determined by a combination of both the external 1 stimulus and the spiking-history of the neuron. Thus, the model can account for typical effects such as refractory periods, bursting behavior or spike-frequency adaptation. Last but not least, the GLM neuron model can also be applied for populations of coupled neurons by making each neuron dependent not only on its own spiking activity but also on the spiking history of all the other neurons. In previous work (Pillow et al., 2005; Chornoboy et al., 1988; Okatan et al., 2005) it has been shown how point-estimates of the GLM-parameters can be obtained using maximum-likelihood (or maximum a posteriori (MAP)) techniques. Here, we extend this approach one step further by using Bayesian inference methods in order to obtain an approximation to the full posterior distribution, rather than point estimates. In particular, the posterior determines confidence intervals for every linear weight, which facilitates the interpretation of the model and its parameters. For example, if a weight describes the strength of coupling between two neurons, then we can use these confidence intervals to test whether this weight is significantly different from zero. In this way, we can readily distinguish statistical significant interactions between neurons from spurious couplings. Another application of the Bayesian GLM neuron model arises in the context of spike-triggered covariance analysis. Spike-triggered covariance basically employs a quadratic expansion of the external stimulus parameter space and is often used in order to determine the most informative subspace. By combining spike-triggered covariance analysis with the Bayesian GLM framework, we will present a new method for selecting the filters of this subspace. Feature selection in the GLM neuron model can be done by the assumption of a Laplace prior over the linear weights, which naturally leads to sparse posterior solutions. Consequently, all weights are equally strongly pushed to zero. This contrasts the Gaussian prior which pushes weights to zero proportional to their absolute value. In this sense, the Laplace prior can also be seen as an efficient regularizer, which is well suited for the situation when a large range of alternative explanations for the neural response shall be compared on the basis of limited data. As we do not perform gradient descent on the posterior, differentiability of the posterior is not required. The paper is organized as follows: In section 2, we describe the model, and the “expectation propagation” algorithm (Minka, 2001; Opper & Winther, 2000) used to find the approximate posterior distribution. In section 3, we estimate the receptive fields, spike-history effects and functional couplings of a small population of retinal ganglion cells. We demonstrate that for small training sets, the Laplace-prior leads to superior performance compared to a Gaussian-prior, which does not lead to sparse solutions. We use the confidence intervals to test whether the functional couplings between the neurons are significant. In section 4, we use the GLM neuron model to describe a complex cell response recorded in macaque primary visual cortex: After computing the spike-triggered covariance (STC) we determine the relevant stimulus subspace via feature selection in our model. In contrast to the usual approach, the selection of the subspace in our case becomes directly linked to an explicit neuron model which also takes into account the spike-history dependence of the spike generation. 2 Generalized Linear Models and Expectation Propagation 2.1 Generalized Linear Models Let Xt ∈Rd, t ∈[0, T] denote a time-varying stimulus and Di = {ti,j} the spike-times of i = 1, . . . , n neurons. Here Xt consists of the sensory input at time t and can include preceeding input frames as well. We assume that the stimulus can only change at distinct time points, but can be evaluated at continous time t. We would like to incorporate spike-history effects, couplings between neurons and dependence on nonlinear features of the stimulus. Therefore, we describe the effective input to a neuron via the following feature-map: ψ(t) = ψst (Xt) M i ψsp({ti,j ∈Di : ti,j < t}), where ψsp represents the spike time history and ψst the possibly nonlinear feature map for the stimulus. That is, the complete feature vector ψ contains possibly nonlinear features of the stimulus and the spike history of every neuron. Any feature which is causal in the sense that it does not depend on future events can be used. We model the spike history dependence by a set of small time 2 windows [t −τl, t −τ ′ l) in which occuring spikes are counted. (ψsp,i({ti,j ∈Di : ti,j < t}))l = X j:ti,j<t 1[t−τl,t−τ ′ l )(ti,j) , where 1[a,b)(t) denotes the indicator function which is one if t ∈[a, b) and zero otherwise. In other words, for each neuron there is a set of windows l = 1, . . . , L with time-lags τl and width τl −τ ′ l describing its spiking history. More precisely, the rate can only change if the stimulus changes or a spike leaves or enters one of these windows. Thus, we obtain a sequence of changepoints 0 = ˜t0 < ˜t1 < · · · < ˜tj < · · · < T, where each feature ψi(t) is constant in [˜tj−1, ˜tj), attaining the value ψi,j. In the GLM neuron model setting the instantanious firing rate of neuron i is obtained by a linear filter of the feature map: p(spike|Xt, {ti,j ∈D : ti,j < t}) = λ(wT i ψ(t)), (1) where λ is the nonlinear transfer function. Following general point process theory (Snyder & Miller, 1991) and using the fact that the features stay constant between two changepoints we can write down the likelihood P(D|{w}) = Qn i=1 Li(wi), where each Li(wi) has the form Li(wi) ∝ Y j φi,j(ui,j), ui,j = wT i ψj, φi,j(ui,j) = λi(ui,j) P t∈Di δ(t−˜tj) exp(−λi(ui,j)(˜tj −˜tj−1)) . The function δ(.) in the second equation is defined to be one if and only if its argument equals zero. The sum therfore is 1 iff a spike of neuron i occurs at changepoint ˜tj. Note that the changepoints ˜tj depend on the spikes and therefore, the process is not Poissonian, as it might be suggested by the functional form of the likelihood. As it has been shown in (Paninski, 2004), the likelihood is log-concave in wi if λi(·) is both convex and log-concave. We are using the transfer function λi(u) = eu which, in particular, gives rise to a log-linear point process model. Alternatively, one could also use λi(u) = eu1u<0 + (1 + u)1u≥0, which grows only linearly (cf. Harris et al. (2003); Pillow et al. (2005)). While we require all rates λi(t) to be piecewise constant, it should be noted that we do not restrict ourselves to a uniform quantization of the time axis. In this way, we achieve an efficient architecture for which the density of change points automatically adapts to the speed with which the input signal is changing. The choice of the prior distribution can play a central role when coping with limited amount of data. We use a Laplace prior distribution over the weights in order to favor sparse solutions over those which explain the data equally well but require more weights different from zero (c.f. Tibshirani (1996)): P(wi) ∝ Y k e−ρk|wk,i|. (2) Thus, prior factors have the form φi,k(ui,k) = ρk 2 exp(−ρk|ui,k|) with ψk = (1l=k)l and ui,k = wT i ψk as above. In our applications, we allowed the prior variance 2 ρ2 k of the stimulus-dependent features to be different from the variance of the spike-history features. The posterior takes the form: P(w|D) ∝ Y i,j φi,j(ui,j), where each φi,j individually instantiates a Generalized Linear Model (either corresponding to a likelihood factor or to a prior factor). As the posterior factorizes over neurons, we can perform our analysis for each neuron seperately. Therefore, for simplicity we drop the subscript i in the following. Our model does not assume or require any specific stimulus distribution. In particular, it is not limited to white noise stimuli or elliptically contoured distributions but it can be used without modification for other stimulus distributions such as natural image sequences. Finally, this framework allows exact sampling of spike trains due to the piecewise constant rate. 3 2.2 Expectation Propagation As exact Bayesian inference is intractable in our model, we seek to find a good approximation to the full posterior. In our case all likelihood and prior factors are log-concave. Therefore, the posterior is unimodal and a Gaussian approximation is well suited. A frequently used technique for this purpose is the Laplace-approximation which computes a quadratic approximation to the log-posterior based on the Hessian around the maximum. For the Laplacian prior, however, this approach falls short since the distribution is not differentiable at zero. Instead, we employ the Expectation Propagation (EP) algorithm (Minka, 2001; Opper & Winther, 2000). In this approximation technique, each factor (also called site) φj of the posterior is replaced by an unnormalised Gaussian: N U(uj|bj, πj) = exp(−1 2πju2 j + bjuj) =: ˆφ(uj), πj ≥0 where the bj, πj are called the site parameters. The approximation aims at minimizing the KullbackLeibler divergence between the full posterior P(w|D) and the approximation, Q(w) ≈Q j ˆφ(uj). The log-concavity of the model implies that all πj ≥0, which supports the numerical stability of the EP algorithm. Some of the πj may even be 0, as long as Q(w) is a (normalizable) Gaussian. An EP update at j consists of computing the Gaussian cavity distribution Q\j ∝Qˆφ−1 j and the nonGaussian tilted distribution ˆP ∝Q\jφj, then updating bj, πj such that the new Q′ has the same mean and covariance as ˆP (moment matching). This is iterated in random order over the sites until convergence. We omit the detailed update schemes here and refer to (Seeger et al., 2007; Seeger, 2005). Convergence guarantees for EP applied to non-Gaussian log-concave models have not been shown so far. Nevertheless it is reported that at least in the log-concave case EP behaves stable (e.g., Rasmussen & Williams (2006)), and we observe quick convergence in our case ( ≤20 iterations over all sites are required). The model still contains hyperparameters, namely the prior variances 2 ρ2 k . In each experiment, these were determined via a standard crossvalidation procedure (80% training data, 10% validation, 10% test). 3 Modeling retinal ganglion cells: Which cells are functionally coupled? We applied the GLM neuron model to multi-electrode recordings of three rabbit retinal ganglion cells. The stimulus consisted of 32767 frames each of which showing a random 16×16 checkerboard pattern with a refresh rate of 50 Hz (data provided by G. Zeck, see (Zeck et al., 2005)). First, in order to investigate the role of the Laplace prior, we trained a single cell GLM neuron model on datasets of different sizes with either a Laplace prior or a Gaussian prior. The models, which have the same number of parameters, were compared by evaluating their negative log-likelihood on an independent test set. As can be seen on the right the choice of prior becomes less important for large training sets as the weights are sufficiently constrained by the data. For each training set size a separate crossvalidation was carried out. Errorbars were obtained by drawing 100 samples from the posterior. training data-set size (% of complete dataset) neg. log likelihood score on test-set Fig. 1 shows the spatiotemporal receptive field of each neuron, as well as the filters describing the influence of spiking history and input from other cells. For conciseness, we only plot the filters for 80 and 120 ms time lags, but the fitted model included 60 and 140 ms time lags as well. The strongly positive weights on the diagonal of figure 1(c) for the spiking history can be interpreted as “self-excitation”. In this way, it is possible to model the bursting behavior exhibited by the cells in our recordings (see also Fig. 2). The strongly negative weights at small time lags represent refractory periods. The red lines correspond to 3 standard deviations of the posterior. The first neuron seems to elicit ”bursts” at lower frequencies. Note the different scaling of the y-axis for diagonal and off-diagonal terms. By analyzing the coupling terms, we can see that there is significant interaction 4 between cells 2 and 3, but not between any other pair of cells. As our prior assumption is that the couplings are 0, this interaction-term is not merely a consequence of our choice of prior. As a result of our crossvalidation it turns out that the prior variance for spike history weights should be set to very large values (ρ= 0.1, variance = 2 1 ρ2 ) meaning that these are well determinated by the data. In contrast, prior variances for the stimulus weights should be more strongly biased towards zero (ρ = 150). (a) GLM (b) STA (c) Figure 1: (a): Stimulus dependence inferred by the GLM for the three neurons (columns) at different time lags (rows). 2 of 4 time lags are plotted (60, 140 ms not shown). (b): Spike-triggered average for the same neurons and time lags as in (a). (c): Causal dependencies between the three neurons. Each plot shows the value of the linear weight as a function of increasing time lag τl (in ms). Shown are posterior mean and three std. dev. (indicated in red). Different scaling of the y-axis is used for diagonal and off-diagonal plots. STA GLM GLM with couplings Neuron 1 0.2199 0.2442 0.3576 Neuron 2 0.1746 0.2348 0.3320 Neuron 3 0.1828 0.3319 0.4202 Mean 0.1924 0.2703 0.3699 Table 1: Predictions performance of different models. Entries correspond to the correlation coefficient between the predicted rate of each model and spikes on a test set. Both rate and spikes are binned in 5 ms bins. The first GLM models neither connections nor self-feedback. Because of the regularization by the prior the spatio-temporal receptive fields are much smoother than the spike-triggered average ones, see Fig. 1(a). The receptive fields of the STA seems to be 5 Figure 2: Predicted rate for the GLM neuron model with and without any spike history and the predicted rate for the STA for the same neurons as in the other plots. For the STA the linear response is rectified. Rate for the GLM with spike dependence is obtained by averaging over 1000 sampled spike-trains. Rates are rescaled to have the same standard deviation. more smeared out which might be due to the fact that it cannot model bursting behavior. The more conservative estimate of the sparse neuron model should increase the prediction performance. To verify this, we calculated the linear response from the spike-triggered average and the rate of our GLM neuron model. In order to have the same number of parameters we neglected all connections. As a model free performance measure we used the correlation coefficient between the spike trains and the rates (each are binned in 5 ms bins). For the GLM with couplings, rates were estimated by sampling 1000 spike trains with the posterior mean as linear weights. As our model explicitly includes the nonlinearity during fitting, the rate is more sharply peaked around the spikes, see Fig. 2. The prediction performance can be increased even further by modeling couplings between neurons as summarized in Tab. 1. 4 Modeling complex cells: How many filters do we need? Complex cells in primary visual cortex exhibit strongly nonlinear response properties which cannot be well described by a single linear filter, but rather requires a set of filters. A common approach for finding these filters is based on the covariance of the spike-triggered ensemble: Eigenvectors of eigenvalues that are much bigger (or smaller) than the eigenvalues of the whole stimulus ensemble indicate directions in stimulus space to which the cell is sensitive to. Usually, a statistical hypothesis test on the eigenvalue-spectrum is used to decide how many of the eigenvectors ei are needed to model the cells (Simoncelli et al., 2004; Touryan et al., 2002; Rust et al., 2005; Steveninck & Bialek, 1988). Here, we take a different approach: We use the confidence intervals of our GLM neuron model to determine the relevant dimensions within the subspace revealed by STC. We first apply STC to find the space spanned by a set of eigenvectors that is substantially larger than the expected dimensionality of the relevant subspace. Next, we fit a nonlinear function ni to the filter-outputs fi(Xt) = ⟨Xt, ei⟩. Finally, we linearly combine the ni(t), resulting in a model of the same form as equation (1) with (ψst)i(Xt) = ni(fi(Xt)) 6 (a) (b) Figure 3: (a): 24 out of 40 Filters estimated by STC. The filters are ordered according to their logratio of their eigenvalue to the corresponding eigenvalue of the complete stimulus ensemble (from left to right). Highlighted filter are those with significant non-zero weights, red indicating excitatory and blue inhibitory filters. (b) Upper: Posterior mean +/- 3 std. dev. Filter indices are ordered in the same way as in (a). Lower: Predicted rate on a test set for STC and for the GLM neuron model with spike history dependence on a test set. As the model is linear in the weights wi, we can use the GLM neuron model to fit these weights and obtain confidence intervals. If a filter fi(t) is not needed for explaining the cells response, its corresponding weight wi will automatically be set to zero by the model due to the sparsity prior. This provides an alternative, model-based method of determining the number of filters required to model the cell. The significance of each filter is not determined by a separate hypothesis test on the spectrum of the spike-triggered covariance, but rather by assessing its influence on the neural activity within the full model. As in the previous application, we can model the spike history effects with an additional feature vector ψsp to take into account temporal dynamics of single neurons or couplings. Before applying our method to real data, we tested it on data generated from an artificial complex cell similar to the one in (Rust et al., 2005). On this simulated data we were able to recover the original filters. We then fitted this GLM neuron model to data recorded from a complex cell in primary visual cortex of an anesthetized macaque monkey (same data as in (Rust et al., 2005)). We first extracted 40 filters which eigenvalues were most different to their corresponding eigenvalues of the complete stimulus ensemble. Any nonlinear regression procedure could be used to fit a nonlinearity to each filter output. We used a simple quadratic regression technique. Having fixed the first nonlinearity we approximated the posterior as above. The resulting confidence intervals for the linear weights are plotted in Fig. 3(b). The filters with significant non-zero weights are highlighted in Fig. 3(a). Red indicates exitatory and blue inhibitory effects on the firing rate. Using 3 std. dev. confidence intervals 9 excitatory and 8 inhibitory filters turned out to be significant in our model. The number of filters is similar to that reportet in Rust et al., who regarded 7 excitatory and 7 inhibitory filters as significant (Rust et al., 2005). The rank order of the linear weights is closely related but not identical to the order of eigenvalues, as can be seen in Fig. 3(b), top. 5 Summary and Conclusions We have shown how approximate Bayesian inference within the framework of generalized linear models can be used to address the problem of identifying relevant features of neural data. More precisely, the use of a sparsity prior favors sparse posterior solutions: non-zero weights are assigned only to those features which which are critical for explaining the data. Furthermore, the explicit 7 uncertainty information obtained from the posterior distribution enables us to identify ranges of statistical significance and therefore facilitates the interpretation of the solution. We used this technique to determine couplings between neurons in a multi-cell recording and demonstrated an increase in prediction performance due to regularization by the sparsity prior. Also, in the context of spiketriggered covariance analysis, we used our method to determine the relevant stimulus subspace within the space spanned by the eigenvectors. Our subspace selection method is directly linked to an explicit neuron model which also takes into account the spike-history dependence of the spike generation. Acknowledgements We would like to thank G¨unther Zeck and Nicole Rust for generously providing their data and for useful discussions. References Chornoboy, E., Schramm, L., & Karr, A.(1988). Maximum likelihood identification of neural point process systems. Biological Cybernetics, 59, 265-275. Harris, K., Csicsvari, J., Hirase, H., Dragoi, G., & Buzsaki, G. (2003). Organization of cell assemblies in the hippocampus. Nature, 424(6948), 552–6. Minka, T. (2001). Expectation propagation for approximate Bayesian inference. Uncertainty in Artificial Intelligence, 17, 362–369. Okatan, M., Wilson, M. A., & Brown, E. N. (2005). Analyzing functional connectivity using a network likelihood model of ensemble neural spiking activity. Neural Computation, 17, 19271961. Opper, M., & Winther, O. (2000). Gaussian Processes for Classification: Mean-Field Algorithms. Neural Computation, 12(11), 2655-2684. Paninski, L. (2004). Maximum likelihood estimation of cascade point-process neural encoding models. Network, 15(4), 243–262. Pillow, J. W., Paninski, L., Uzzell, V. J., Simoncelli, E. P., & Chichilnisky, E. J. (2005). Prediction and decoding of retinal ganglion cell responses with a probabilistic spiking model. J Neurosci, 25(47), 11003–11013. Rasmussen, C., & Williams, C.(2006). Gaussian processes for machine learning. Springer. Rust, N., Schwartz, O., Movshon, J., & Simoncelli, E.(2005). Spatiotemporal Elements of Macaque V1 Receptive Fields. Neuron, 46(6), 945–956. Seeger, M. (2005). Expectation propagation for exponential families (Tech. Rep.). University of California at Berkeley. (See www.kyb.tuebingen.mpg.de/bs/people/seeger.) Seeger, M., Steinke, F., & Tsuda, K. (2007). Bayesian inference and optimal design in the sparse linear model. AI and Statistics. Simoncelli, E., Paninski, L., Pillow, J., & Schwartz, O.(2004). Characterization of neural responses with stochastic stimuli. In M. Gazzaniga (Ed.), (Vol. 3, pp. 327–338). MIT Press. Snyder, D., & Miller, M. (1991). Random point processes in time and space. Springer Texts in Electrical Engineering. Steveninck, R., & Bialek, W. (1988). Real-Time Performance of a Movement-Sensitive Neuron in the Blowfly Visual System: Coding and Information Transfer in Short Spike Sequences. Proceedings of the Royal Society of London. Series B, Biological Sciences, 234(1277), 379–414. Tibshirani, R. (1996). Regression Shrinkage and Selection via the Lasso. Journal of the Royal Statistical Society. Series B (Methodological), 58(1), 267–288. Touryan, J., Lau, B., & Dan, Y.(2002). Isolation of Relevant Visual Features from Random Stimuli for Cortical Complex Cells. Journal of Neuroscience, 22(24), 10811. Zeck, G. M., Xiao, Q., & Masland, R. H. (2005). The spatial filtering properties of local edge detectors and brisk-sustained retinal ganglion cells. Eur J Neurosci, 22(8), 2016-26. 8
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Unconstrained Online Handwriting Recognition with Recurrent Neural Networks Alex Graves TUM, Germany alex@idsia.ch Santiago Fern´andez IDSIA, Switzerland santiago@idsia.ch Marcus Liwicki University of Bern, Switzerland liwicki@iam.unibe.ch Horst Bunke University of Bern, Switzerland bunke@iam.unibe.ch J¨urgen Schmidhuber IDSIA, Switzerland and TUM, Germany juergen@idsia.ch Abstract In online handwriting recognition the trajectory of the pen is recorded during writing. Although the trajectory provides a compact and complete representation of the written output, it is hard to transcribe directly, because each letter is spread over many pen locations. Most recognition systems therefore employ sophisticated preprocessing techniques to put the inputs into a more localised form. However these techniques require considerable human effort, and are specific to particular languages and alphabets. This paper describes a system capable of directly transcribing raw online handwriting data. The system consists of an advanced recurrent neural network with an output layer designed for sequence labelling, combined with a probabilistic language model. In experiments on an unconstrained online database, we record excellent results using either raw or preprocessed data, well outperforming a state-of-the-art HMM based system in both cases. 1 Introduction Handwriting recognition is traditionally divided into offline and online recognition. Offline recognition is performed on images of handwritten text. In online handwriting the location of the pen-tip on a surface is recorded at regular intervals, and the task is to map from the sequence of pen positions to the sequence of words. At first sight, it would seem straightforward to label raw online inputs directly. However, the fact that each letter or word is distributed over many pen positions poses a problem for conventional sequence labelling algorithms, which have difficulty processing data with long-range interdependencies. The problem is especially acute for unconstrained handwriting, where the writing style may be cursive, printed or a mix of the two, and the degree of interdependency is therefore difficult to determine in advance. The standard solution is to preprocess the data into a set of localised features. These features typically include geometric properties of the trajectory in the vicinity of every data point, pseudo-offline information from a generated image, and character level shape characteristics [6, 7]. Delayed strokes (such as the crossing of a ‘t’ or the dot of an ‘i’) require special treatment because they split up the characters and therefore interfere with localisation. HMMs [6] and hybrid systems incorporating time-delay neural networks and HMMs [7] are commonly trained with such features. The issue of classifying preprocessed versus raw data has broad relevance to machine learning, and merits further discussion. Using hand crafted features often yields superior results, and in some cases can render classification essentially trivial. However, there are three points to consider in favour of raw data. Firstly, designing an effective preprocessor requires considerable time and expertise. Secondly, hand coded features tend to be more task specific. For example, features designed 1 for English handwriting could not be applied to languages with substantially different alphabets, such as Arabic or Chinese. In contrast, a system trained directly on pen movements could be applied to any alphabet. Thirdly, using raw data allows feature extraction to be built into the classifier, and the whole system to be trained together. For example, convolutional neural networks [10], in which a globally trained hierarchy of network layers is used to extract progressively higher level features, have proved effective at classifying raw images, such as objects in cluttered scenes or isolated handwritten characters [15, 11]. (Note than convolution nets are less suitable for unconstrained handwriting, because they require the text images to be presegmented into characters [10]). In this paper, we apply a recurrent neural network (RNN) to online handwriting recognition. The RNN architecture is bidirectional Long Short-Term Memory [3], chosen for its ability to process data with long time dependencies. The RNN uses the recently introduced connectionist temporal classification output layer [2], which was specifically designed for labelling unsegmented sequence data. An algorithm is introduced for applying grammatical constraints to the network outputs, thereby providing word level transcriptions. Experiments are carried out on the IAM online database [12] which contains forms of unconstrained English text acquired from a whiteboard. The performance of the RNN system using both raw and preprocessed input data is compared to that of an HMM based system using preprocessed data only [13]. To the best of our knowledge, this is the first time whole sentences of unconstrained handwriting have been directly transcribed from raw online data. Section 2 describes the network architecture, the output layer and the algorithm for applying grammatical constraints. Section 3 provides experimental results, and conclusions are given in Section 4. 2 Method 2.1 Bidirectional Long Short-Term Memory One of the key benefits of RNNs is their ability to make use of previous context. However, for standard RNN architectures, the range of context that can in practice be accessed is limited. The problem is that the influence of a given input on the hidden layer, and therefore on the network output, either decays or blows up exponentially as it cycles around the recurrent connections. This is often referred to as the vanishing gradient problem [4]. Long Short-Term Memory (LSTM; [5]) is an RNN architecture designed to address the vanishing gradient problem. An LSTM layer consists of multiple recurrently connected subnets, known as memory blocks. Each block contains a set of internal units, known as cells, whose activation is controlled by three multiplicative ‘gate’ units. The effect of the gates is to allow the cells to store and access information over long periods of time. For many tasks it is useful to have access to future as well past context. Bidirectional RNNs [14] achieve this by presenting the input data forwards and backwards to two separate hidden layers, both of which are connected to the same output layer. Bidirectional LSTM (BLSTM) [3] combines the above architectures to provide access to long-range, bidirectional context. 2.2 Connectionist Temporal Classification Connectionist temporal classification (CTC) [2] is an objective function designed for sequence labelling with RNNs. Unlike previous objective functions it does not require pre-segmented training data, or postprocessing to transform the network outputs into labellings. Instead, it trains the network to map directly from input sequences to the conditional probabilities of the possible labellings. A CTC output layer contains one more unit than there are elements in the alphabet L of labels for the task. The output activations are normalised with the softmax activation function [1]. At each time step, the first |L| outputs are used to estimate the probabilities of observing the corresponding labels. The extra output estimates the probability of observing a ‘blank’, or no label. The combined output sequence estimates the joint probability of all possible alignments of the input sequence with all possible labellings. The probability of a particular labelling can then be estimated by summing over the probabilities of all the alignments that correspond to it. More precisely, for an input sequence x of length T, choosing a label (or blank) at every time step according to the probabilities implied by the network outputs defines a probability distribution 2 over the set of length T sequences of labels and blanks. We denote this set L′T , where L′ = L ∪ {blank}. To distinguish them from labellings, we refer to the elements of L′T as paths. Assuming that the label probabilities at each time step are conditionally independent given x, the conditional probability of a path π ∈L′T is given by p(π|x) = T Y t=1 yt πt, (1) where yt k is the activation of output unit k at time t. Denote the set of sequences of length less than or equal to T on the alphabet L as L≤T . Then Paths are mapped onto labellings l ∈L≤T by an operator B that removes first the repeated labels, then the blanks. For example, both B(a, −, a, b, −) and B(−, a, a, −, −, a, b, b) yield the labelling (a,a,b). Since the paths are mutually exclusive, the conditional probability of a given labelling l ∈L≤T is the sum of the probabilities of all paths corresponding to it: p(l|x) = X π∈B−1(l) p(π|x). (2) Although a naive calculation of the above sum would be unfeasible, it can be efficiently evaluated with a graph-based algorithm [2], similar to the forward-backward algorithm for HMMs. To allow for blanks in the output paths, for each label sequence l ∈L≤T consider a modified label sequence l′ ∈L′≤T , with blanks added to the beginning and the end and inserted between every pair of labels. The length of l′ is therefore |l′| = 2|l| + 1. For a labelling l, define the forward variable αt(s) as the summed probability of all paths whose length t prefixes are mapped by B onto the length s/2 prefix of l, i.e. αt(s) = P(π1:t : B(π1:t) = l1:s/2, πt = l′ s|x) = X π: B(π1:t)=l1:s/2 tY t′=1 yt′ πt′, (3) where, for some sequence s, sa:b is the subsequence (sa, sa+1, ..., sb−1, sb), and s/2 is rounded down to an integer value. The backward variables βt(s) are defined as the summed probability of all paths whose suffixes starting at t map onto the suffix of l starting at label s/2 βt(s) = P(πt+1:T : B(πt:T ) = ls/2:|l|, πt = l′ s|x) = X π: B(πt:T )=ls/2:|l| T Y t′=t+1 yt′ πt′ (4) Both the forward and backward variables are calculated recursively [2]. The label sequence probability is given by the sum of the products of the forward and backward variables at any time step: p(l|x) = |l′| X s=1 αt(s)βt(s). (5) The objective function for CTC is the negative log probability of the network correctly labelling the entire training set. Let S be a training set, consisting of pairs of input and target sequences (x, z), where target sequence z is at most as long as input sequence x. Then the objective function is: OCT C = − X (x,z)∈S ln (p(z|x)). (6) The network can be trained with gradient descent by differentiating OCT C with respect to the outputs, then using backpropagation through time to differentiate with respect to the network weights. Noting that the same label (or blank) may be repeated several times for a single labelling l, we define the set of positions where label k occurs as lab(l, k) = {s : l′ s = k}, which may be empty. We then set l = z and differentiate (5) with respect to the unnormalised network outputs at k to obtain: ∂OCT C ∂at k = −∂ln (p(z|x)) ∂at k = yt k − 1 p(z|x) X s∈lab(z,k) αt(s)βt(s). (7) 3 Once the network is trained, we would ideally label some unknown input sequence x by choosing the most probable labelling l∗: l∗= arg max l p(l|x). (8) Using the terminology of HMMs, we refer to the task of finding this labelling as decoding. Unfortunately, we do not know of a tractable decoding algorithm that is guaranteed to give optimal results. However a simple and effective approximation is given by assuming that the most probable path corresponds to the most probable labelling, i.e. l∗≈B arg max π p(π|x) . (9) 2.3 Integration with an External Grammar For some tasks we want to constrain the output labellings according to a predefined grammar. For example, in speech and handwriting recognition, the final transcriptions are usually required to form sequences of dictionary words. In addition it is common practice to use a language model to weight the probabilities of particular sequences of words. We can express these constraints by altering the probabilities in (8) to be conditioned on some probabilistic grammar G, as well as the input sequence x: l∗= arg max l p(l|x, G). (10) Absolute requirements, for example that l contains only dictionary words, can be incorporated by setting the probability of all sequences that fail to meet them to 0. At first sight, conditioning on G seems to contradict a basic assumption of CTC: that the labels are conditionally independent given the input sequences (see Eqn. (1)). Since the network attempts to model the probability of the whole labelling at once, there is nothing to stop it from learning inter-label transitions direct from the data, which would then be skewed by the external grammar. However, CTC networks are typically only able to learn local relationships such as commonly occurring pairs or triples of labels. Therefore as long as G focuses on long range label interactions (such as the probability of one word following another when the outputs are letters) it doesn’t interfere with the dependencies modelled by CTC. The basic rules of probability tell us that p(l|x, G) = p(l|x)p(l|G)p(x) p(x|G)p(l) , where we have used the fact that x is conditionally independent of G given l. If we assume x is independent of G, this reduces to p(l|x, G) = p(l|x)p(l|G) p(l) . That assumption is in general false, since both the input sequences and the grammar depend on the underlying generator of the data, for example the language being spoken. However it is a reasonable first approximation, and is particularly justifiable in cases where the grammar is created using data other than that from which x was drawn (as is common practice in speech and handwriting recognition, where independent textual corpora are used to generate language models). Finally, if we assume that all label sequences are equally probable prior to any knowledge about the input or the grammar, we can drop the p(l) term in the denominator to get l∗= arg max l p(l|x)p(l|G). (11) Note that, since the number of possible label sequences is finite (because both L and |l| are finite), assigning equal prior probabilities does not lead to an improper prior. We now describe an algorithm, based on the token passing algorithm for HMMs [16], that allows us to find an approximate solution to (11) for a simple grammar. Let G consist of a dictionary D containing W words, and a set of W 2 bigrams p(w| ˆw) that define the probability of making a transition from word ˆw to word w. The probability of any labelling that does not form a sequence of dictionary words is 0. For each word w, define the modified word w′ as w with blanks added at the beginning and end and between each pair of labels. Therefore |w′| = 2|w| + 1. Define a token tok = (score, history) to be a pair consisting of a real valued score and a history of previously visited words. In fact, 4 each token corresponds to a particular path through the network outputs, and its score is the log probability of that path. The basic idea of the token passing algorithm is to pass along the highest scoring tokens at every word state, then maximise over these to find the highest scoring tokens at the next state. The transition probabilities are used when a token is passed from the last state in one word to the first state in another. The output word sequence is given by the history of the highest scoring end-of-word token at the final time step. At every time step t of the length T output sequence, each segment s of each modified word w′ holds a single token tok(w, s, t). This is the highest scoring token reaching that segment at that time. In addition we define the input token tok(w, 0, t) to be the highest scoring token arriving at word w at time t, and the output token tok(w, −1, t) to be the highest scoring token leaving word w at time t. 1: Initialisation: 2: for all words w ∈D do 3: tok(w, 1, 1) = (ln(y1 b), (w)) 4: tok(w, 2, 1) = (ln(y1 w1), (w)) 5: if |w| = 1 then 6: tok(w, −1, 1) = tok(w, 2, 1) 7: else 8: tok(w, −1, 1) = (−∞, ()) 9: tok(w, s, 1) = (−∞, ()) for all s ̸= −1 10: Algorithm: 11: for t = 2 to T do 12: sort output tokens tok(w, −1, t −1) by ascending score 13: for all words w ∈D do 14: w∗= arg max ˆ w∈D tok( ˆw, −1, t −1).score + ln (p(w| ˆw)) 15: tok(w, 0, t).score = tok(w∗, −1, t −1).score + ln (p(w|w∗)) 16: tok(w, 0, t).history = tok(w∗, −1, t −1).history + w 17: for segment s = 1 to |w′| do 18: P = {tok(w, s, t −1), tok(w, s −1, t −1)} 19: if w′ s ̸= blank and s > 2 and w′ s−2 ̸= w′ s then 20: add tok(w, s −2, t −1) to P 21: tok(w, s, t) = token in P with highest score 22: tok(w, s, t).score += ln(yt w′ s) 23: tok(w, −1, t) = highest scoring of {tok(w, |w′|, t), tok(w, |w′| −1, t)} 24: Termination: 25: find output token tok∗(w, −1, T) with highest score at time T 26: output tok∗(w, −1, T).history Algorithm 1: CTC Token Passing Algorithm The algorithm’s worst case complexity is O(TW 2), since line 14 requires a potential search through all W words. However, because the output tokens tok(w, −1, T) are sorted in order of score, the search can be terminated when a token is reached whose score is less than the current best score with the transition included. The typical complexity is therefore considerably lower, with a lower bound of O(TWlogW) to account for the sort. If no bigrams are used, lines 14-16 can be replaced by a simple search for the highest scoring output token, and the complexity reduces to O(TW). Note that this is the same as the complexity of HMM decoding, if the search through bigrams is exhaustive. Much work has gone into developing more efficient decoding techniques (see e.g. [9]), typically by pruning improbable branches from the tree of labellings. Such methods are essential for applications where a rapid response is required, such as real time transcription. In addition, many decoders use more sophisticated language models than simple bigrams. Any HMM decoding algorithm could be applied to CTC outputs in the same way as token passing. However, we have stuck with a relatively basic algorithm since our focus here is on recognition rather than decoding. 5 3 Experiments The experimental task was online handwriting recognition, using the IAM-OnDB handwriting database [12], which is available for public download from http://www.iam.unibe.ch/ fki/iamondb/ For CTC, we record both the character error rate, and the word error rate using Algorithm 1 with a language model and a dictionary. For the HMM system, the word error rate is quoted from the literature [13]. Both the character and word error rate are defined as the total number of insertions, deletions and substitutions in the algorithm’s transcription of test set, divided by the combined length of the target transcriptions in the test set. We compare results using both raw inputs direct from the pen sensor, and a preprocessed input representation designed for HMMs. 3.1 Data and Preprocessing IAM-OnDB consists of pen trajectories collected from 221 different writers using a ‘smart whiteboard’ [12]. The writers were asked to write forms from the LOB text corpus [8], and the position of their pen was tracked using an infra-red device in the corner of the board. The input data consisted of the x and y pen coordinates, the points in the sequence when individual strokes (i.e. periods when the pen is pressed against the board) end, and the times when successive position measurements were made. Recording errors in the x, y data were corrected by interpolating to fill in for missing readings, and removing steps whose length exceeded a certain threshold. IAM-OnDB is divided into a training set, two validation sets, and a test set, containing respectively 5364, 1438, 1518 and 3859 written lines taken from 775, 192, 216 and 544 forms. The data sets contained a total of 3,298,424, 885,964, 1,036,803 and 2,425,5242 pen coordinates respectively. For our experiments, each line was used as a separate sequence (meaning that possible dependencies between successive lines were ignored). The character level transcriptions contain 80 distinct target labels (capital letters, lower case letters, numbers, and punctuation). A dictionary consisting of the 20, 000 most frequently occurring words in the LOB corpus was used for decoding, along with a bigram language model optimised on the training and validation sets [13]. 5.6% of the words in the test set were not in the dictionary. Two input representations were used. The first contained only the offset of the x, y coordinates from the top left of the line, the time from the beginning of the line, and the marker for the ends of strokes. We refer to this as the raw input representation. The second representation used state-of-theart preprocessing and feature extraction techniques [13]. We refer to this as the preprocessed input representation. Briefly, in order to account for the variance in writing styles, the pen trajectories were normalised with respect to such properties as the slant, skew and width of the letters, and the slope of the line as a whole. Two sets of input features were then extracted, the first consisting of ‘online’ features, such as pen position, pen speed, line curvature etc., and the second consisting of ‘offline’ features created from a two dimensional window of the image created by the pen. 3.2 Experimental Setup The CTC network used the BLSTM architecture, as described in Section 2.1. The forward and backward hidden layers each contained 100 single cell memory blocks. The input layer was fully connected to the hidden layers, which were fully connected to themselves and the output layer. The output layer contained 81 units (80 characters plus the blank label). For the raw input representation, there were 4 input units and a total of 100,881 weights. For the preprocessed representation, there were 25 inputs and 117,681 weights. tanh was used for the cell activation functions and logistic sigmoid in the range [0, 1] was used for the gates. For both input representations, the data was normalised so that each input had mean 0 and standard deviation 1 on the training set. The network was trained with online gradient descent, using a learning rate of 10−4 and a momentum of 0.9. Training was stopped after no improvement was recorded on the validation set for 50 training epochs. The HMM setup [13] contained a separate, left-to-right HMM with 8 states for each character (8 ∗ 81 = 648 states in total). Diagonal mixtures of 32 Gaussians were used to estimate the observation 6 Table 1: Word Error Rate (WER) on IAM-OnDB. LM = language model. CTC results are a mean over 4 runs, ± standard error. All differences were significant (p < 0.01) System Input LM WER HMM preprocessed ✓ 35.5% [13] CTC raw 30.1 ± 0.5% CTC preprocessed 26.0 ± 0.3% CTC raw ✓ 22.8 ± 0.2% CTC preprocessed ✓ 20.4 ± 0.3% probabilities. All parameters, including the word insertion penalty and the grammar scale factor, were optimised on the validation set. 3.3 Results The character error rate for the CTC network with the preprocessed inputs was 11.5 ± 0.05%. From Table 1 we can see that with a dictionary and a language model this translates into a mean word error rate of 20.4%, which is a relative error reduction of 42.5% compared to the HMM. Without the language model, the error reduction was 26.8%. With the raw input data CTC achieved a character error rate of 13.9 ± 0.1%, and word error rates that were close to those recorded with the preprocessed data, particularly when the language model was present. The key difference between the input representations is that the raw data is less localised, and therefore requires more use of context. A useful indication of the network’s sensitivity to context is provided by the derivatives of the output yt k at a particular point t in the data sequence with respect to the inputs xt′ k at all points 1 ≤t′ ≤T. We refer to these derivatives as the sequential Jacobian. Looking at the relative magnitude of the sequential Jacobian over time gives an idea of the range of context used, as illustrated in Figure 1. 4 Conclusion We have combined a BLSTM CTC network with a probabilistic language model. We have applied this system to an online handwriting database and obtained results that substantially improve on a state-of-the-art HMM based system. We have also shown that the network’s performance with raw sensor inputs is comparable to that with sophisticated preprocessing. As far as we are aware, our system is the first to successfully recognise unconstrained online handwriting using raw inputs only. Acknowledgments This research was funded by EC Sixth Framework project “NanoBioTact”, SNF grant 200021111968/1, and the SNF program “Interactive Multimodal Information Management (IM)2”. References [1] J. S. Bridle. Probabilistic interpretation of feedforward classification network outputs, with relationships to statistical pattern recognition. In F. Fogleman-Soulie and J.Herault, editors, Neurocomputing: Algorithms, Architectures and Applications, pages 227–236. Springer-Verlag, 1990. [2] A. Graves, S. Fern´andez, F. Gomez, and J. Schmidhuber. Connectionist temporal classification: Labelling unsegmented sequence data with recurrent neural networks. In Proc. 23rd Int. Conf. on Machine Learning, Pittsburgh, USA, 2006. [3] A. Graves and J. Schmidhuber. Framewise phoneme classification with bidirectional LSTM and other neural network architectures. Neural Networks, 18(5-6):602–610, June/July 2005. [4] S. Hochreiter, Y. Bengio, P. Frasconi, and J. Schmidhuber. Gradient flow in recurrent nets: the difficulty of learning long-term dependencies. In S. C. Kremer and J. F. Kolen, editors, A Field Guide to Dynamical Recurrent Neural Networks. IEEE Press, 2001. [5] S. Hochreiter and J. Schmidhuber. Long Short-Term Memory. Neural Comp., 9(8):1735–1780, 1997. [6] J. Hu, S. G. Lim, and M. K. Brown. Writer independent on-line handwriting recognition using an HMM approach. Pattern Recognition, 33:133–147, 2000. 7 Figure 1: Sequential Jacobian for an excerpt from the IAM-OnDB, with raw inputs (left) and preprocessed inputs (right). For ease of visualisation, only the derivative with highest absolute value is plotted at each time step. The reconstructed image was created by plotting the pen coordinates recorded by the sensor. The individual strokes are alternately coloured red and black. For both representations, the Jacobian is plotted for the output corresponding to the label ‘i’ at the point when ‘i’ is emitted (indicated by the vertical dashed lines). Because bidirectional networks were used, the range of sensitivity extends in both directions from the dashed line. For the preprocessed data, the Jacobian is sharply peaked around the time when the output is emitted. For the raw data it is more spread out, suggesting that the network makes more use of long-range context. Note the spike in sensitivity to the very end of the raw input sequence: this corresponds to the delayed dot of the ‘i’. [7] S. Jaeger, S. Manke, J. Reichert, and A. Waibel. On-line handwriting recognition: the NPen++ recognizer. Int. Journal on Document Analysis and Recognition, 3:169–180, 2001. [8] S. Johansson, R. Atwell, R. Garside, and G. Leech. The tagged LOB corpus user’s manual; Norwegian Computing Centre for the Humanities, 1986. [9] P. Lamere, P. Kwok, W. Walker, E. Gouvea, R. Singh, B. Raj, and P. Wolf. Design of the CMU Sphinx-4 decoder. In Proc. 8th European Conf. on Speech Communication and Technology, Aug. 2003. [10] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. Proc. IEEE, 86(11):2278–2324, Nov. 1998. [11] Y. LeCun, F. Huang, and L. Bottou. Learning methods for generic object recognition with invariance to pose and lighting. In Proc. of CVPR’04. IEEE Press, 2004. [12] M. Liwicki and H. Bunke. IAM-OnDB - an on-line English sentence database acquired from handwritten text on a whiteboard. In Proc. 8th Int. Conf. on Document Analysis and Recognition, volume 2, pages 956–961, 2005. [13] M. Liwicki, A. Graves, S. Fern´andez, H. Bunke, and J. Schmidhuber. A novel approach to on-line handwriting recognition based on bidirectional long short-term memory networks. In Proc. 9th Int. Conf. on Document Analysis and Recognition, Curitiba, Brazil, Sep. 2007. [14] M. Schuster and K. K. Paliwal. Bidirectional recurrent neural networks. IEEE Transactions on Signal Processing, 45:2673–2681, Nov. 1997. [15] P. Y. Simard, D. Steinkraus, and J. C. Platt. Best practices for convolutional neural networks applied to visual document analysis. In Proc. 7th Int. Conf. on Document Analysis and Recognition, page 958, Washington, DC, USA, 2003. IEEE Computer Society. [16] S. Young, N. Russell, and J. Thornton. Token passing: A simple conceptual model for connected speech recognition systems. Technical Report CUED/F-INFENG/TR38, Cambridge University Eng. Dept., 1989. 8
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The Epoch-Greedy Algorithm for Contextual Multi-armed Bandits John Langford Yahoo! Research jl@yahoo-inc.com Tong Zhang Department of Statistics Rutgers University tongz@rci.rutgers.edu Abstract We present Epoch-Greedy, an algorithm for contextual multi-armed bandits (also known as bandits with side information). Epoch-Greedy has the following properties: 1. No knowledge of a time horizon T is necessary. 2. The regret incurred by Epoch-Greedy is controlled by a sample complexity bound for a hypothesis class. 3. The regret scales as O(T 2/3S1/3) or better (sometimes, much better). Here S is the complexity term in a sample complexity bound for standard supervised learning. 1 Introduction The standard k-armed bandits problem has been well-studied in the literature (Lai & Robbins, 1985; Auer et al., 2002; Even-dar et al., 2006, for example). It can be regarded as a repeated game between two players, with every stage consisting of the following: The world chooses k rewards r1, ..., rk ∈[0, 1]; the player chooses an arm i ∈{1, k} without knowledge of the world’s chosen rewards, and then observes the reward ri. The contextual bandits setting considered in this paper is the same except for a modification of the first step, in which the player also observes context information x which can be used to determine which arm to pull. The contextual bandits problem has many applications and is often more suitable than the standard bandits problem, because settings with no context information are rare in practice. The setting considered in this paper is directly motivated by the problem of matching ads to web-page contents on the internet. In this problem, a number of ads (arms) are available to be placed on a number of web-pages (context information). Each page visit can be regarded as a random draw of the context information (one may also include the visitor’s online profile as context information if available) from an underlying distribution that is not controlled by the player. A certain amount of revenue is generated when the visitor clicks on an ad. The goal is to put the most relevant ad on each page to maximize the expected revenue. Although one may potentially put multiple ads on each web-page, we focus on the problem that only one ad is placed on each page (which is like pulling an arm given context information). The more precise definition is given in Section 2. Prior Work. The problem of bandits with context has been analyzed previously (Pandey et al., 2007; Wang et al., 2005), typically under additional assumptions such as a correct prior or knowledge of the relationship between the arms. This problem is also known as associative reinforcement learning (Strehl et al., 2006, for example) or bandits with side information. A few results under as weak or weaker assumptions are directly comparable. 1. The Exp4 algorithm (Auer et al., 1995) notably makes no assumptions about the world. Epoch-Greedy has a worse regret bound in T (O(T 2/3) rather than O(T 1/2)) and is only 1 analyzed under an IID assumption. An important advantage of Epoch-Greedy is a much better dependence on the size of the set of predictors. In the situation where the number of predictors is infinite but with finite VC-Dimension d, Exp4 has a vacuous regret bound while Epoch-Greedy has a regret bound no worse than O(T 2/3(ln m)1/3). Sometimes we can achieve much better dependence on T, depending on the structure of the hypothesis space. For example, we will show that it is possible to achieve O(ln T) regret bound using Epoch-Greedy, while this is not possible with Exp4 or any simple modification of it. Another substantial advantage is reduced computational complexity. The ERM step in Epoch-Greedy can be replaced with any standard learning algorithm that achieves approximate loss minimization, making guarantees that degrade gracefully with the approximation factor. Exp4 on the other hand requires computation proportional to the explicit count of hypotheses in a hypothesis space. 2. The random trajectories method (Kearns et al., 2000) for learning policies in reinforcement learning with hard horizon T = 1 is essentially the same setting. In this paper, bounds are stated for a batch oriented setting where examples are formed and then used for choosing a hypothesis. Epoch-Greedy takes advantage of this idea, but it also has analysis which states that it trades off the number of exploration and exploitation steps so as to maximize the sum of rewards incurred during both exploration and exploitation. What we do. We present and analyze the Epoch-Greedy algorithm for multiarmed bandits with context. This has all the nice properties stated in the abstract, resulting in a practical algorithm for solving this problem. The paper is broken up into the following sections. 1. In Section 2 we present basic definitions and background. 2. Section 3 presents the Epoch-Greedy algorithm along with a regret bound analysis which holds without knowledge of T. 3. Section 4 analyzes the instantiation of the Epoch-Greedy algorithm in several settings. 2 Contextual bandits We first formally define contextual bandit problems and algorithms to solve them. Definition 2.1 (Contextual bandit problem) In a contextual bandits problem, there is a distribution P over (x, r1, ..., rk), where x is context, a ∈{1, . . . , k} is one of the k arms to be pulled, and ra ∈[0, 1] is the reward for arm a. The problem is a repeated game: on each round, a sample (x, r1, ..., rk) is drawn from P, the context x is announced, and then for precisely one arm a chosen by the player, its reward ra is revealed. Definition 2.2 (Contextual bandit algorithm) A contextual bandits algorithm B determines an arm a ∈{1, . . . , k} to pull at each time step t, based on the previous observation sequence (x1, a1, ra,1), . . . , (xt−1, at−1, ra,t−1), and the current context xt. Our goal is to maximize the expected total reward PT t=1 E(xt,⃗rt)∼P [ra,t]. Note that we use the notation ra,t = rat to improve readability. Similar to supervised learning, we assume that we are given a set H consisting of hypotheses h : X →{1, . . . , k}. Each hypothesis maps side information x to an arm a. A natural goal is to choose arms to compete with the best hypothesis in H. We introduce the following definition. Definition 2.3 (Regret) The expected reward of a hypothesis h is R(h) = E(x,⃗r)∼D rh(x) . Consider any contextual bandits algorithm B. Let ZT = {(x1,⃗r1), . . . , (xT ,⃗rT )}, and the expected regret of B with respect to a hypothesis h be: ∆R(B, h, T) = TR(h) −EZT ∼P T T X t=1 rB(x),t. 2 The expected regret of B up to time T with respect to hypothesis space H is defined as ∆R(B, H, T) = sup h∈H ∆R(B, h, T). The main challenge of the contextual bandits problem is that when we pull an arm, rewards of other arms are not observed. Therefore it is necessary to try all arms (explore) in order to form an accurate estimation. In this context, methods we investigate in the paper make explicit distinctions between exploration and exploitation steps. In an exploration step, the goal is to form unbiased samples by randomly pulling all arms to improve the accuracy of learning. Because it does not focus on the best arm, this step leads to large immediate regret but can potentially reduce regret for the future exploitation steps. In an exploitation step, the learning algorithm suggests the best hypothesis learned from the samples formed in the exploration steps, and the arm given by the hypothesis is pulled: the goal is to maximize immediate reward (or minimize immediate regret). Since the samples in the exploitation steps are biased (toward the arm suggested by the learning algorithm using previous exploration samples), we do not use them to learn the hypothesis for the future steps. That is, in methods we consider, exploitation does not help us to improve learning accuracy for the future. More specifically, in an exploration step, in order to form unbiased samples, we pull an arm a ∈ {1, . . . , k} uniformly at random. Therefore the expected regret comparing to the best hypothesis in H can be as large as O(1). In an exploitation step, the expected regret can be much smaller. Therefore a central theme we examine in this paper is to balance the trade-off between exploration and exploitation, so as to achieve a small overall expected regret up to some time horizon T. Note that if we decide to pull a specific arm a with side information x, we do not observe rewards ra′ for a′ ̸= a. In order to apply standard sample complexity analysis, we first show that exploration samples, where a is picked uniformly at random, can create a standard learning problem without missing observations. This is simply achieved by setting fully observed rewards r′ such that r′ a′(ra) = kI(a′ = a)ra, (1) where I(·) is the indicator function. The basic idea behind this transformation from partially observed to fully observed data dates back to the analysis of “Sample Selection Bias” (Heckman, 1979). The above rule is easily generalized to other distribution over actions p(a) by replacing k with 1/p(a). The following lemma shows that this method of filling missing reward components is unbiased. Lemma 2.1 For all arms a′: E⃗r∼P |x [ra′] = E⃗r∼P |x,a∼U(1,...,k) [r′ a′(ra)]. Therefore for any hypothesis h(x), we have R(h) = E(x,⃗r)∼P,a∼U(1,...,k) h r′ h(x)(ra) i . Proof We have: E⃗r∼P |x,a∼U(1,...,k) [r′ a′(ra)] =E⃗r∼P |x k X a=1 k−1 [r′ a′(ra)] =E⃗r∼P |x k X a=1 k−1 [kraI(a′ = a)] = E⃗r∼P |x [ra′] . Lemma 2.1 implies that we can estimate reward R(h) of any hypothesis h(x) using expectation with respect to exploration samples (x, a, ra). The right hand side can then be replaced by empirical samples as P t I(h(xt) = at)ra,t for hypotheses in a hypothesis space H. The quality of this estimation can be obtained with uniform convergence learning bounds. 3 Exploration with the Epoch-Greedy algorithm The problem of treating contextual bandits as standard bandits is that the information in x is lost. That is, the optimal arm to pull should be a function of the context x, but this is not captured by the 3 standard bandits setting. An alternative approach is to regard each hypothesis h as a separate artificial “arm”, and then apply a standard bandits algorithm to these artificial arms. Using this approach, let m be the number of hypotheses, we can get a bound of O(m). However, this solution ignores the fact that many hypotheses can share the same arm so that choosing an arm yields information for many hypotheses. For this reason, with a simple algorithm, we can get a bound that depends on m logarithmically, instead of O(m) as would be the case for the standard bandits solution discussed above. As discussed earlier, the key issue in the algorithm is to determine when to explore and when to exploit, so as to achieve appropriate balance. If we are given the time horizon T in advance, and would like to optimize performance with the given T, then it is always advantageous to perform a first phase of exploration steps, followed by a second phase of exploitation steps (until time step T). The reason that there is no advantage to take any exploitation step before the last exploration step is: by switching the two steps, we can more accurately pick the optimal hypothesis in the exploitation step due to more samples from exploration. With fixed T, assume that we have taken n steps of exploration, and obtain an average regret bound of ϵn for each exploitation step at the point, then we can bound the regret of the exploration phase as n, and the exploitation phase as ϵn(T −n). The total regret is n + (T −n)ϵn. Using this bound, we shall switch from exploration to exploitation at the point n that minimizes the sum. Without knowing T in advance, but with the same generalization bound, we can run exploration/exploitation in epochs, where at the beginning of each epoch ℓ, we perform one step of exploration, followed by ⌈1/ϵn⌉steps of exploitation. We then start the next epoch. After epoch L, the total average regret is no more than PL n=1(1 + ϵn⌈1/ϵn⌉) ≤3L. Moreover, the epoch L∗containing T is no more than the optimal regret bound minn[n + (T −n)ϵn] (with known T and optimal stopping point). Therefore the performance of our method (which does not need to know T) is no worse than three time the optimal bound with known T and optimal stopping point. This motivates a modified algorithm in Figure 1. The idea described above is related to forcing in (Lai & Yakowitz, 1995). Proposition 3.1 Consider a sequence of nonnegative and monotone non-increasing numbers {ϵn}. Let L∗= min{L : PL ℓ=1(1 + ⌈1/ϵℓ⌉) ≥T}, then L∗≤min n∈[0,T ][n + (T −n)ϵn]. Proof Let n∗= arg minn∈[0,T ][n + (T −n)ϵn]. The bound is trivial if n∗≥L∗. We only need consider the case n∗≤L∗−1. By assumption, PL∗−1 ℓ=1 (1 + 1/ϵℓ) ≤T −1. Since PL∗−1 ℓ=1 1/ϵℓ≥PL∗−1 ℓ=n∗1/ϵℓ≥(L∗−n∗)1/ϵn∗, we have L∗−1 + (L∗−n∗)1/ϵn∗≤T −1. Rearranging, we have L∗≤n∗+ (T −L∗)ϵn∗. In Figure 1, s(Zn 1 ) is a sample-dependent (integer valued) exploitation step count. Proposition 3.1 suggests that choosing s(Zn 1 ) = ⌈1/ϵn(Zn 1 )⌉, where ϵn(Zn 1 ) is a sample dependent average generalization bound, yields performance comparable to the optimal bound with known time horizon T. Definition 3.1 (Epoch-Greedy Exploitation Cost) Consider a hypothesis space H consisting of hypotheses that take values in {1, 2, . . . , k}. Let Zt = (xt, at, ra,t) for i = 1, . . . , n be independent random samples, where ai is uniform randomly distributed in {1, . . . , k}, and ra,t ∈[0, 1] is the observed (random) reward. Let Zn 1 = {Z1, . . . , Zn}, and the empirical reward maximization estimator ˆh(Zn 1 ) = arg max h∈H n X t=1 ra,tI(h(xt) = at). Given any fixed n, δ ∈[0, 1], and observation Zn 1 , we denote by s(Zn 1 ) a data-dependent exploitation step count. Then the per-epoch exploitation cost is defined as: µn(H, s) = EZn 1 sup h∈H R(h) −R(ˆh(Zn 1 )) s(Zn 1 ). 4 Epoch-Greedy (s(Wℓ)) /*parameter s(Wℓ): exploitation steps*/ initialize: exploration samples W0 = {} and t1 = 1 iterate ℓ= 1, 2, . . . t = tℓ, and observe xt /*do one-step exploration*/ select an arm at ∈{1, . . . , k} uniformly at random receive reward ra,t ∈[0, 1] Wℓ= Wℓ−1 ∪{(xt, at, ra,t)} find best hypothesis ˆhℓ∈H by solving maxh∈H P (x,a,ra)∈WℓraI(h(x) = a) tℓ+1 = tℓ+ s(Wℓ) + 1 for t = tℓ+ 1, · · · , tℓ+1 −1 /*do s(Wℓ)-steps exploitation*/ select arm at = ˆhℓ(xt) receive reward ra,t ∈[0, 1] end for end iterate Figure 1: Exploration by ϵ-greedy in epochs Theorem 3.1 For all T, nℓ, L such that: T ≤L + PL ℓ=1 nℓ, the expected regret of Epoch-Greedy in Figure 1 is bounded by ∆R(Epoch-Greedy, H, T) ≤L + L X ℓ=1 µℓ(H, s) + T L X ℓ=1 P[s(Zℓ 1) < nℓ]. This theorem statement is very general, because we want to allow sample dependent bounds to be used. When sample-independent bounds are used the following simple corollary holds: Corollary 3.1 Assume we choose s(Zℓ 1) = sℓ≤⌊1/µℓ(H, 1)⌋(ℓ= 1, . . .), and let LT = arg minL{L : L + PL ℓ=1 sℓ≥T}. Then the expected regret of Epoch-Greedy in Figure 1 is bounded by ∆R(Epoch-Greedy, H, T) ≤2LT . Proof (of the main theorem) Let B be the Epoch-Greedy algorithm. One of the following events will occur: • A: s(Zℓ 1) < nℓfor some ℓ= 1, . . . , L. • B: s(Zℓ 1) ≥nℓfor all ℓ= 1, . . . , L. If event A occurs, then since each reward is in [0,1], up to time T, regret cannot be larger than T. Thus the total expected contribution of A to the regret ∆R(B, H, T) is at most TP(A) ≤T L X ℓ=1 P[s(Zℓ 1) < nℓ]. (2) If event B occurs, then tℓ+1 −tℓ≥nℓ+ 1 for ℓ= 1, . . . , L, and thus tL+1 > T. Therefore the expected contribution of B to the regret ∆R(B, H, T) is at most the sum of expected regret in the first L epochs. By definition and construction, after the first step of epoch ℓ, Wℓconsists of ℓrandom observations Zj = (xj, aj, ra,j) where aj is drawn uniformly at random from {1, . . . , k}, and j = 1, . . . , ℓ. This is independent of the number of exploitation steps before epoch ℓ. Therefore we can treat Wℓas ℓindependent samples. This means that the expected regret associated with exploitation steps in epoch ℓis µℓ(H, s). Since the exploration step in each epoch contributes at most 1 to the 5 expected regret, the total expected regret for each epoch ℓis at most 1 + µℓ(H, s). Therefore the total expected regret for epochs ℓ= 1, . . . , L is at most L + PL ℓ=1 µℓ(H, s). Combined with (2), we obtain the desired bound. In the theorem, we bound the expected regret of each exploration step by one. Clearly this assumes the worst case scenario and can often be improved. Some consequences of the theorem with specific function classes are given in Section 4. 4 Examples Theorem 3.1 is quite general. In this section, we present a few simple examples to illustrate the potential applications. 4.1 Finite hypothesis space worst case bound Consider the finite hypothesis space situation, with m = |H| < ∞. We apply Theorem 3.1 with a worst-case deviation bound. Let x1, . . . , xn ∈[0, k] be iid random variables, such that Exi ≤1, then Bernstein inequality implies that there exists a constant c0 > 0 such that ∀η ∈(0, 1), with probability 1 −η: n X i=1 xi − n X i=1 Exi ≤c0 v u u tln(1/η) n X i=1 Ex2 i + c0k ln(1/η) ≤c0 p nk ln(1/η) + c0k ln(1/η). It follows that there exists a universal constant c > 0 such that µn(H, 1) ≤c−1p k ln m/n. Therefore in Figure 1, if we choose s(Zℓ 1) = ⌊c p ℓ/(k ln m)⌋, then µℓ(H, s) ≤1: this is consistent with the choice recommended in Proposition 3.1. In order to obtain a performance bound of this scheme using Theorem 3.1, we can simply take nℓ= ⌊c p ℓ/(k ln m)⌋. This implies that P(s(Zℓ 1) < nℓ) = 0. Moreover, with this choice, for any T, we can pick an L that satisfies the condition T ≤PL ℓ=1 nℓ. It implies that there exists a universal constant c′ > 0 such that for any given T, we can take L = ⌊c′T 2/3(k ln m)1/3⌋ in Theorem 3.1. In summary, if we choose s(Zℓ 1) = ⌊c p ℓ/(k ln m)⌋in Figure 1, then ∆(Epoch-Greedy, H, T) ≤2L ≤2c′T 2/3(k ln m)1/3. Reducing the problem to standard bandits, as discussed at the beginning of Section 3, leads to a bound of O(m ln T) (Lai & Robbins, 1985; Auer et al., 2002). Therefore when m is large, the Epoch-Greedy algorithm in Figure 1 can perform significantly better. In this particular situation, Epoch-Greedy does not do as well as Exp4 in (Auer et al., 1995), which implies a regret of O( √ kT ln m). However, the advantage of Epoch-Greedy is that any learning bound can be applied. For many hypothesis classes, the ln m factor can be improved for Epoch-Greedy. In fact, a similar result can be obtained for classes with infinitely many hypotheses but finite VC dimensions. Moreover, as we will see next, under additional assumptions, it is possible to obtain much better bounds in terms of T for Epoch-Greedy, such as O(k ln m + k ln T). This extends the classical O(ln T) bound for standard bandits, and is not possible to achieve using Exp4 or simple variations of it. 6 4.2 Finite hypothesis space with unknown expected reward gap This example illustrates the importance of allowing sample-dependent s(Zℓ 1). We still assume a finite hypothesis space, with m = |H| < ∞. However, we would like to improve the performance bound by imposing additional assumptions. In particular we note that the standard bandits problem has regret of the form O(ln T) while in the worst case, our method for the contextual bandits problem has regret O(T 2/3). A natural question is then: what are the assumptions we can impose so that the Epoch-Greedy algorithm can have a regret of the form O(ln T). The main technical reason that the standard bandits problem has regret O(ln T) is that the expected reward of the best bandit and that of the second best bandit has a gap: the constant hidden in the O(ln T) bound depends on this gap, and the bound becomes trivial (infinity) when the gap approaches zero. In this example we show that a similar assumption for contextual bandits problems leads to a similar regret bound of O(ln T) for the Epoch-Greedy algorithm. Let H = {h1, . . . , hm}, and assume without loss of generality that R(h1) ≥R(h2) ≥· · · ≥ R(hm). Suppose that we know that R(h1) ≥R(h2) + ∆for some ∆> 0, but the value of ∆is not known in advance. Although ∆is not known, it can be estimated from the data Zn 1 . Let the empirical reward of h ∈H be ˆR(h|Zn 1 ) = k n n X t=1 ra,tI(h(xt) = at). Let ˆh1 be the hypothesis with highest empirical reward on Zn 1 , and ˆh2 be the hypothesis with second highest empirical reward. We define the empirical gap as ˆ∆(Zn 1 ) = ˆR(ˆh1|Zn 1 ) −ˆR(ˆh2|Zn 1 ). Let h1 be the hypothesis with the highest true expected reward, then we suffer a regret when ˆh1 ̸= h1. Again, the standard large deviation bound implies that there exists a universal constant c > 0 such that for all j ≥1: P( ˆ∆(Zn 1 ) ≥(j −1)∆, ˆh1 ̸= h1) ≤me−ck−1n(1+j2)∆2 P( ˆ∆(Zn 1 ) ≤0.5∆) ≤me−ck−1n∆2. Now we can set s(Zn 1 ) = ⌊m−1e(2k)−1cn ˆ∆(Zn 1 )2⌋. With this choice, there exists a constant c′ > 0 such that µn(H, s) ≤ ⌈∆−1⌉ X j=1 sup{s(Zn 1 ) : ˆ∆(Zn 1 ) ≤j∆}P( ˆ∆(Zn 1 ) ∈[(j −1)∆, j∆], ˆh1 ̸= h1) ≤ ⌈∆−1⌉ X j=1 m−1e(2k)−1cnj2∆2P( ˆ∆(Zn 1 ) ∈[(j −1)∆, j∆], ˆh1 ̸= h1) ≤ ⌈∆−1⌉ X j=1 e(2k)−1cnj2∆2−ck−1n(1+j2)∆2 ≤ ⌈∆−1⌉ X j=1 e−ck−1n(0.5j2+1)∆2 ≤c′p k/n∆−1e−ck−1n∆2. There exists a constant c′′ > 0 such that for any L: L X ℓ=1 µℓ(H, s) ≤L + c′ ∞ X ℓ=1 p k/ℓ∆−1e−ck−1ℓ∆2 ≤L + c′′k∆−2. 7 Now, consider any time horizon T. If we set nℓ= 0 when ℓ< L, nL = T, and L = 8k(ln m + ln(T + 1)) c∆2 , then P(s(ZL 1 ) ≤nL) ≤P( ˆ∆(ZL 1 ) ≤0.5∆) ≤me−ck−1L∆2 ≤1/T. That is, if we choose s(Zn 1 ) = ⌊m−1e(2k)−1cn ˆ∆(Zn 1 )2⌋in Figure 1, then ∆R(Epoch-Greedy, H, T) ≤2L + 1 + c′′k∆−2 ≤2 8k(ln m + ln(T + 1)) c∆2 + 1 + c′′k∆−2. The regret for this choice is O(ln T), which is better than O(T 2/3) of Section 4.1. However, the constant depends on the gap ∆which can be small. It is possible to combine the two strategies (that is, use the s(Zn 1 ) choice of Section 4.1 when ˆ∆(Zn 1 ) is small) and obtain bounds that not only work well when the gap ∆is large, but also not much worse than the bound of Section 4.1 when ∆is small. As a special case, we can apply the method in this section to solve the standard bandits problem. The O(k ln T) bound of the Epoch-Greedy method matches those more specialized algorithms for the standard bandits problem, although our algorithm has a larger constant. 5 Conclusion We consider a generalization of the multi-armed bandits problem, where observable context can be used to determine which arm to pull and investigate the sample complexity of the exploration/exploitation trade-off for the Epoch-Greedy algorithm. The Epoch-Greedy algorithm analysis leaves one important open problem behind. Epoch-Greedy is much better at dealing with large hypothesis spaces or hypothesis spaces with special structures due to its ability to employ any data-dependent sample complexity bound. However, for finite hypothesis space, in the worst case scenario, Exp4 has better dependency on T. In such situations, it’s possible that a better designed algorithm can achieve both strengths. References Auer, P., Cesa-Bianchi, N., & Fischer, P. (2002). Finite time analysis of the multi-armed bandit problem. Machine Learning, 47, 235–256. Auer, P., Cesa-Bianchi, N., Freund, Y., & Schapire, R. E. (1995). Gambling in a rigged casino: The adversarial multi-armed bandit problem. FOCS. Even-dar, E., Mannor, S., & Mansour, Y. (2006). Action elimination and stopping conditions for the multi-armed bandit and reinforcement learning problems. JMLR, 7, 1079–1105. Heckman, J. (1979). Sample selection bias as a specification error. Econometrica, 47, 153–161. Kearns, M., Mansour, Y., & Ng, A. Y. (2000). Approximate planning in large pomdps via reusable trajectories. NIPS. Lai, T., & Robbins, H. (1985). Asymptotically efficient adaptive allocation rules. Advances in Applied Mathematics, 6, 4–22. Lai, T., & Yakowitz, S. (1995). Machine learning and nonparametric bandit theory. IEEE TAC, 40, 1199–1209. Pandey, S., Agarwal, D., Chakrabarti, D., & Josifovski, V. (2007). Bandits for taxonomies: a modelbased approach. SIAM Data Mining Conference. Strehl, A. L., Mesterharm, C., Littman, M. L., & Hirsh, H. (2006). Experience-efficient learning in associative bandit problems. ICML. Wang, C.-C., Kulkarni, S. R., & Poor, H. V. (2005). Bandit problems with side observations. IEEE Transactions on Automatic Control, 50, 338–355. 8
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Using Deep Belief Nets to Learn Covariance Kernels for Gaussian Processes Ruslan Salakhutdinov and Geoffrey Hinton Department of Computer Science, University of Toronto 6 King’s College Rd, M5S 3G4, Canada rsalakhu,hinton@cs.toronto.edu Abstract We show how to use unlabeled data and a deep belief net (DBN) to learn a good covariance kernel for a Gaussian process. We first learn a deep generative model of the unlabeled data using the fast, greedy algorithm introduced by [7]. If the data is high-dimensional and highly-structured, a Gaussian kernel applied to the top layer of features in the DBN works much better than a similar kernel applied to the raw input. Performance at both regression and classification can then be further improved by using backpropagation through the DBN to discriminatively fine-tune the covariance kernel. 1 Introduction Gaussian processes (GP’s) are a widely used method for Bayesian non-linear non-parametric regression and classification [13, 16]. GP’s are based on defining a similarity or kernel function that encodes prior knowledge of the smoothness of the underlying process that is being modeled. Because of their flexibility and computational simplicity, GP’s have been successfully used in many areas of machine learning. Many real-world applications are characterized by high-dimensional, highly-structured data with a large supply of unlabeled data but a very limited supply of labeled data. Applications such as information retrieval and machine vision are examples where unlabeled data is readily available. GP’s are discriminative models by nature and within the standard regression or classification scenario, unlabeled data is of no use. Given a set of i.i.d. labeled input vectors Xl = {xn}N n=1 and their associated target labels {yn}N n=1 ∈R or {yn}N n=1 ∈{−1, 1} for regression/classification, GP’s model p(yn|xn) directly. Unless some assumptions are made about the underlying distribution of the input data X = [Xl, Xu], unlabeled data, Xu, cannot be used. Many researchers have tried to use unlabeled data by incorporating a model of p(X). For classification tasks, [11] model p(X) as a mixture P yn p(xn|yn)p(yn) and then infer p(yn|xn), [15] attempts to learn covariance kernels based on p(X), and [10] assumes that the decision boundaries should occur in regions where the data density, p(X), is low. When faced with high-dimensional, highly-structured data, however, none of the existing approaches have proved to be particularly successful. In this paper we exploit two properties of DBN’s. First, they can be learned efficiently from unlabeled data and the top-level features generally capture significant, high-order correlations in the data. Second, they can be discriminatively fine-tuned using backpropagation. We first learn a DBN model of p(X) in an entirely unsupervised way using the fast, greedy learning algorithm introduced by [7] and further investigated in [2, 14, 6]. We then use this generative model to initialize a multi-layer, non-linear mapping F(x|W), parameterized by W, with F : X →Z mapping the input vectors in X into a feature space Z. Typically the mapping F(x|W) will contain millions of parameters. The top-level features produced by this mapping allow fairly accurate reconstruction of the input, so they must contain most of the information in the input vector, but they express this information in a way that makes explicit a lot of the higher-order structure in the input data. After learning F(x|W), a natural way to define a kernel function is to set K(xi, xj) = exp (−||F(xi|W) −F(xj|W)||2). Note that the kernel is initialized in an entirely unsupervised way. The parameters W of the covariance kernel can then be fine-tuned using the labeled data by 1 maximizing the log probability of the labels with respect to W. In the final model most of the information for learning a covariance kernel will have come from modeling the input data. The very limited information in the labels will be used only to slightly adjust the layers of features already discovered by the DBN. 2 Gaussian Processes for Regression and Binary Classification For a regression task, we are given a data set D of i.i.d. labeled input vectors Xl = {xn}N n=1 and their corresponding target labels {yn}N n=1 ∈R. We are interested in the following probabilistic regression model: yn = f(xn) + ǫ, ǫ ∼N(ǫ|0, σ2) (1) A Gaussian process regression places a zero-mean GP prior over the underlying latent function f we are modeling, so that a-priori p(f|Xl) =N(f|0, K), where f = [f(x1), ..., f(xn)]T and K is the covariance matrix, whose entries are specified by the covariance function Kij = K(xi, xj). The covariance function encodes our prior notion of the smoothness of f, or the prior assumption that if two input vectors are similar according to some distance measure, their labels should be highly correlated. In this paper we will use the spherical Gaussian kernel, parameterized by θ = {α, β}: Kij = α exp −1 2β (xi −xj)T (xi −xj) (2) Integrating out the function values f, the marginal log-likelihood takes form: L = log p(y|Xl) = −N 2 log 2π −1 2 log |K + σ2I| −1 2yT (K + σ2I)−1y (3) which can then be maximized with respect to the parameters θ and σ. Given a new test point x∗, a prediction is obtained by conditioning on the observed data and θ. The distribution of the predicted value y∗at x∗takes the form: p(y∗|x∗, D, θ, σ2) = N(y∗|kT ∗(K + σ2I)−1y, k∗∗−kT ∗(K + σ2I)−1k∗+ σ2) (4) where k∗= K(x∗, Xl), and k∗∗= K(x∗, x∗). For a binary classification task, we similarly place a zero mean GP prior over the underlying latent function f, which is then passed through the logistic function g(x) = 1/(1 + exp(−x)) to define a prior p(yn = 1|xn) = g(f(xn)). Given a new test point x∗, inference is done by first obtaining the distribution over the latent function f∗= f(x∗): p(f∗|x∗, D) = Z p(f∗|x∗, Xl, f)p(f|Xl, y)df (5) which is then used to produce a probabilistic prediction: p(y∗= 1|x∗, D) = Z g(f∗)p(f∗|x∗, D)df∗ (6) The non-Gaussian likelihood makes the integral in Eq. 5 analytically intractable. In our experiments, we approximate the non-Gaussian posterior p(f|Xl, y) with a Gaussian one using expectation propagation [12]. For more thorough reviews and implementation details refer to [13, 16]. 3 Learning Deep Belief Networks (DBN’s) In this section we describe an unsupervised way of learning a DBN model of the input data X = [Xl, Xu], that contains both labeled and unlabeled data sets. A DBN can be trained efficiently by using a Restricted Boltzmann Machine (RBM) to learn one layer of hidden features at a time [7]. Welling et. al. [18] introduced a class of two-layer undirected graphical models that generalize RBM’s to exponential family distributions. This framework will allow us to model real-valued images of face patches and word-count vectors of documents. 3.1 Modeling Real-valued Data We use a conditional Gaussian distribution for modeling observed “visible” pixel values x (e.g. images of faces) and a conditional Bernoulli distribution for modeling “hidden” features h (Fig. 1): p(xi = x|h) = 1 √ 2πσi exp(− (x−bi−σi P j hjwij)2 2σ2 i ) (7) p(hj = 1|x) = g bj + P i wij xi σi (8) 2 h W Binary Hidden Features x Gaussian Visible Units W W W W W W GP Input X target y Feature Representation F(X|W) 1 RBM 1000 RBM 1000 1000 1000 2 3 1000 1000 3 RBM 1000 2 T T T 1000 1 Figure 1: Left panel: Markov random field of the generalized RBM. The top layer represents stochastic binary hidden features h and and the bottom layer is composed of linear visible units x with Gaussian noise. When using a Constrained Poisson Model, the top layer represents stochastic binary latent topic features h and the bottom layer represents the Poisson visible word-count vector x. Middle panel: Pretraining consists of learning a stack of RBM’s. Right panel: After pretraining, the RBM’s are used to initialize a covariance function of the Gaussian process, which is then fine-tuned by backpropagation. where g(x) = 1/(1+exp(−x)) is the logistic function, wij is a symmetric interaction term between input i and feature j, σ2 i is the variance of input i, and bi, bj are biases. The marginal distribution over visible vector x is: p(x) = X h exp (−E(x, h)) R u P g exp (−E(u, g))du (9) where E(x, h) is an energy term: E(x, h) = P i (xi−bi)2 2σ2 i −P j bjhj −P i,j hjwij xi σi . The parameter updates required to perform gradient ascent in the log-likelihood is obtained from Eq. 9: ∆wij = ǫ∂log p(x) ∂wij = ǫ(<zihj>data −<zihj>model) (10) where ǫ is the learning rate, zi = xi/σi, <·>data denotes an expectation with respect to the data distribution and <·>model is an expectation with respect to the distribution defined by the model. To circumvent the difficulty of computing <·>model, we use 1-step Contrastive Divergence [5]: ∆wij = ǫ(<zihj>data −<zihj>recon) (11) The expectation < zihj >data defines the expected sufficient statistics of the data distribution and is computed as zip(hj = 1|x) when the features are being driven by the observed data from the training set using Eq. 8. After stochastically activating the features, Eq. 7 is used to “reconstruct” real-valued data. Then Eq. 8 is used again to activate the features and compute <zihj>recon when the features are being driven by the reconstructed data. Throughout our experiments we set variances σ2 i = 1 for all visible units i, which facilitates learning. The learning rule for the biases is just a simplified version of Eq. 11. 3.2 Modeling Count Data with the Constrained Poisson Model We use a conditional “constrained” Poisson distribution for modeling observed “visible” word count data x and a conditional Bernoulli distribution for modeling “hidden” topic features h: p(xi = n|h) = Pois n, exp (λi + P j hjwij) P k exp λk + P j hjWkj × N , p(hj = 1|x) = g(bj + X i wijxi) (12) where Pois n, λ = e−λλn/n!, wij is a symmetric interaction term between word i and feature j, N = P i xi is the total length of the document, λi is the bias of the conditional Poisson model for word i, and bj is the bias of feature j. The Poisson rate, whose log is shifted by the weighted combination of the feature activations, is normalized and scaled up by N. We call this the “Constrained Poisson Model” since it ensures that the mean Poisson rates across all words sum up to the length of the document. This normalization is significant because it makes learning stable and it deals appropriately with documents of different lengths. 3 The marginal distribution over visible count vectors x is given in Eq. 9 with an “energy” given by E(x, h) = − X i λixi + X i log (xi!) − X j bjhj − X i,j xihjwij (13) The gradient of the log-likelihood function is: ∆wij = ǫ∂log p(v) ∂wij = ǫ(<xihj>data −<xihj>model) (14) 3.3 Greedy Recursive Learning of Deep Belief Nets A single layer of binary features is not the best way to capture the structure in the input data. We now describe an efficient way to learn additional layers of binary features. After learning the first layer of hidden features we have an undirected model that defines p(v, h) by defining a consistent pair of conditional probabilities, p(h|v) and p(v|h) which can be used to sample from the model distribution. A different way to express what has been learned is p(v|h) and p(h). Unlike a standard, directed model, this p(h) does not have its own separate parameters. It is a complicated, non-factorial prior on h that is defined implicitly by p(h|v) and p(v|h). This peculiar decomposition into p(h) and p(v|h) suggests a recursive algorithm: keep the learned p(v|h) but replace p(h) by a better prior over h, i.e. a prior that is closer to the average, over all the data vectors, of the conditional posterior over h. So after learning an undirected model, the part we keep is part of a multilayer directed model. We can sample from this average conditional posterior by simply using p(h|v) on the training data and these samples are then the “data” that is used for training the next layer of features. The only difference from learning the first layer of features is that the “visible” units of the second-level RBM are also binary [6, 3]. The learning rule provided in the previous section remains the same [5]. We could initialize the new RBM model by simply using the existing learned model but with the roles of the hidden and visible units reversed. This ensures that p(v) in our new model starts out being exactly the same as p(h) in our old one. Provided the number of features per layer does not decrease, [7] show that each extra layer increases a variational lower bound on the log probability of data. To suppress noise in the learning signal, we use the real-valued activation probabilities for the visible units of every RBM, but to prevent hidden units from transmitting more than one bit of information from the data to its reconstruction, the pretraining always uses stochastic binary values for the hidden units. The greedy, layer-by-layer training can be repeated several times to learn a deep, hierarchical model in which each layer of features captures strong high-order correlations between the activities of features in the layer below. 4 Learning the Covariance Kernel for a Gaussian Process After pretraining, the stochastic activities of the binary features in each layer are replaced by deterministic, real-valued probabilities and the DBN is used to initialize a multi-layer, non-linear mapping f(x|W) as shown in figure 1. We define a Gaussian covariance function, parameterized by θ = {α, β} and W as: Kij = α exp −1 2β ||F(xi|W) −F(xj|W)||2 (15) Note that this covariance function is initialized in an entirely unsupervised way. We can now maximize the log-likelihood of Eq. 3 with respect to the parameters of the covariance function using the labeled training data[9]. The derivative of the log-likelihood with respect to the kernel function is: ∂L ∂Ky = 1 2 K−1 y yyT K−1 y −K−1 y (16) where Ky = K +σ2I is the covariance matrix. Using the chain rule we readily obtain the necessary gradients: ∂L ∂θ = ∂L ∂Ky ∂Ky ∂θ and ∂L W = ∂L ∂Ky ∂Ky ∂F(x|W) ∂F(x|W) ∂W (17) 4 32.99 −41.15 66.38 −22.07 27.49 Unlabeled Training Data Test Data A B Figure 2: Top panel A: Randomly sampled examples of the training and test data. Bottom panel B: The same sample of the training and test images but with rectangular occlusions. Training GPstandard GP-DBNgreedy GP-DBNfine GPpca labels Sph. ARD Sph. ARD Sph. ARD Sph. ARD A 100 22.24 28.57 17.94 18.37 15.28 15.01 18.13 (10) 16.47 (10) 500 17.25 18.16 12.71 8.96 7.25 6.84 14.75 (20) 10.53 (80) 1000 16.33 16.36 11.22 8.77 6.42 6.31 14.86 (20) 10.00 (160) B 100 26.94 28.32 23.15 19.42 19.75 18.59 25.91 (10) 19.27 (20) 500 20.20 21.06 15.16 11.01 10.56 10.12 17.67 (10) 14.11 (20) 1000 19.20 17.98 14.15 10.43 9.13 9.23 16.26 (10) 11.55 (80) Table 1: Performance results on the face-orientation regression task. The root mean squared error (RMSE) on the test set is shown for each method using spherical Gaussian kernel and Gaussian kernel with ARD hyperparameters. By row: A) Non-occluded face data, B) Occluded face data. For the GPpca model, the number of principal components that performs best on the test data is shown in parenthesis. where ∂F(x|W)/∂W is computed using standard backpropagation. We also optimize the observation noise σ2. It is necessary to compute the inverse of Ky, so each gradient evaluation has O(N 3) complexity where N is the number of the labeled training cases. When learning the restricted Boltzmann machines that are composed to form the initial DBN, however, each gradient evaluation scales linearly in time and space with the number of unlabeled training cases. So the pretraining stage can make efficient use of very large sets of unlabeled data to create sensible, high-level features and when the amount of labeled data is small. Then the very limited amount of information in the labels can be used to slightly refine those features rather than to create them. 5 Experimental Results In this section we present experimental results for several regression and classification tasks that involve high-dimensional, highly-structured data. The first regression task is to extract the orientation of a face from a gray-level image of a large patch of the face. The second regression task is to map images of handwritten digits to a single real-value that is as close as possible to the integer represented by the digit in the image. The first classification task is to discriminate between images of odd digits and images of even digits. The second classification task is to discriminate between two different classes of news story based on the vector of word counts in each story. 5.1 Extracting the Orientation of a Face Patch The Olivetti face data set contains ten 64×64 images of each of forty different people. We constructed a data set of 13,000 28×28 images by randomly rotating (−90◦to +90◦), cropping, and subsampling the original 400 images. The data set was then subdivided into 12,000 training images, which contained the first 30 people, and 1,000 test images, which contained the remaining 10 people. 1,000 randomly sampled face patches from the training set were assigned an orientation label. The remaining 11,000 training images were used as unlabeled data. We also made a more difficult version of the task by occluding part of each face patch with randomly chosen rectangles. Panel A of figure 2 shows randomly sampled examples from the training and test data. For training on the Olivetti face patches we used the 784-1000-1000-1000 architecture shown in figure 1. The entire training set of 12,000 unlabeled images was used for greedy, layer-by-layer training of a DBN model. The 2.8 million parameters of the DBN model may seem excessive for 12,000 training cases, but each training case involves modeling 625 real-values rather than just a single real-valued label. Also, we only train each layer of features for a few passes through the training data and we penalize the squared weights. 5 Feature 992 Feature 312 0 0.2 0.4 0.6 0.8 1.0 1.0 0.8 0.6 0.4 0.2 1 2 3 4 5 6 0 5 10 15 20 25 30 35 40 45 log β −1 0 1 2 3 4 5 6 0 10 20 30 40 50 60 70 80 90 log β More Relevant Input Pixel Space Feature Space Figure 3: Left panel shows a scatter plot of the two most relevant features, with each point replaced by the corresponding input test image. For better visualization, overlapped images are not shown. Right panel displays the histogram plots of the learned ARD hyper-parameters log β. After the DBN has been pretrained on the unlabeled data, a GP model was fitted to the labeled data using the top-level features of the DBN model as inputs. We call this model GP-DBNgreedy. GP-DBNgreedy can be significantly improved by slightly altering the weights in the DBN. The GP model gives error derivatives for its input vectors which are the top-level features of the DBN. These derivatives can be backpropagated through the DBN to allow discriminative fine-tuning of the weights. Each time the weights in the DBN are updated, the GP model is also refitted. We call this model GP-DBNfine. For comparison, we fitted a GP model that used the pixel intensities of the labeled images as its inputs. We call this model GPstandard. We also used PCA to reduce the dimensionality of the labeled images and fitted several different GP models using the projections onto the first m principal components as the input. Since we only want a lower bound on the error of this model, we simply use the value of m that performs best on the test data. We call this model GPpca. Table 1 shows the root mean squared error (RMSE) of the predicted face orientations using all four types of GP model on varying amounts of labeled data. The results show that both GPDBNgreedy and GP-DBNfine significantly outperform a regular GP model. Indeed, GP-DBNfine with only 100 labeled training cases outperforms GPstandard with 1000. To test the robustness of our approach to noise in the input we took the same data set and created artificial rectangular occlusions (see Fig. 2, panel B). The number of rectangles per image was drawn from a Poisson with λ = 2. The top-left location, length and width of each rectangle was sampled from a uniform [0,25]. The pixel intensity of each occluding rectangle was set to the mean pixel intensity of the entire image. Table 1 shows that the performance of all models degrades, but their relative performances remain the same and GP-DBNfine on occluded data is still much better than GPstandard on non-occluded data. We have also experimented with using a Gaussian kernel with ARD hyper-parameters, which is a common practice when the input vectors are high-dimensional: Kij = α exp −1 2(xi −xj)T D(xi −xj) (18) where D is the diagonal matrix with Dii = 1/βi, so that the covariance function has a separate length-scale parameter for each dimension. ARD hyper-parameters were optimized by maximizing the marginal log-likelihood of Eq. 3. Table 1 shows that ARD hyper-parameters do not improve GPstandard, but they do slightly improve GP-DBNfine and they strongly improve GP-DBNgreedy and GPpca when there are 500 or 1000 labeled training cases. The histogram plot of log β in figure 3 reveals that there are a few extracted features that are very relevant (small β) to our prediction task. The same figure (left panel) shows a scatter plot of the two most relevant features of GP-DBNgreedy model, with each point replaced by the corresponding input test image. Clearly, these two features carry a lot of information about the orientation of the face. 6 Train GPstandard GP-DBNgreedy GP-DBNfine GPpca labels Sph. ARD Sph. ARD Sph. ARD Sph. ARD A 100 1.86 2.27 1.68 1.61 1.63 1.58 1.73 (20) 2.00 (20) 500 1.42 1.62 1.19 1.27 1.16 1.22 1.32 (40) 1.36 (20) 1000 1.25 1.36 1.07 1.14 1.03 1.10 1.19 (40) 1.22 (80) B 100 0.0884 0.1087 0.0528 0.0597 0.0501 0.0599 0.0785 (10) 0.0920 (10) 500 0.0222 0.0541 0.0100 0.0161 0.0055 0.0104 0.0160 (40) 0.0235 (20) 1000 0.0129 0.0385 0.0058 0.0059 0.0050 0.0100 0.0091 (40) 0.0127 (40) Table 2: Performance results on the digit magnitude regression task (A) and and discriminating odd vs. even digits classification task (B). The root mean squared error for regression task on the test set is shown for each method. For classification task the area under the ROC (AUROC) metric is used. For each method we show 1-AUROC on the test set. All methods were tried using both spherical Gaussian kernel, and a Gaussian kernel with ARD hyper-parameters. For the GPpca model, the number of principal components that performs best on the test data is shown in parenthesis. Number of labeled GPstandard GP-DBNgreedy GP-DBNfine cases (50% in each class) 100 0.1295 0.1180 0.0995 500 0.0875 0.0793 0.0609 1000 0.0645 0.0580 0.0458 Table 3: Performance results using the area under the ROC (AUROC) metric on the text classification task. For each method we show 1-AUROC on the test set. We suspect that the GP-DBNfine model does not benefit as much from the ARD hyper-parameters because the fine-tuning stage is already capable of turning down the activities of irrelevant top-level features. 5.2 Extracting the Magnitude Represented by a Handwritten Digit and Discriminating between Images of Odd and Even Digits The MNIST digit data set contains 60,000 training and 10,000 test 28×28 images of ten handwritten digits (0 to 9). 100 randomly sampled training images of each class were assigned a magnitude label. The remaining 59,000 training images were used as unlabeled data. As in the previous experiment, we used the 784-1000-1000-1000 architecture with the entire training set of 60,000 unlabeled digits being used for greedily pretraining the DBN model. Table 2, panel A, shows that GP-DBNfine and GP-DBNgreedy perform considerably better than GPstandard both with and without ARD hyperparameters. The same table, panel B, shows results for the classification task of discriminating between images of odd and images of even digits. We used the same labeled training set, but with each digit categorized into an even or an odd class. The same DBN model was used, so the Gaussian covariance function was initialized in exactly the same way for both regression and classification tasks. The performance of GP-DBNgreedy demonstrates that the greedily learned feature representation captures a lot of structure in the unlabeled input data which is useful for subsequent discrimination tasks, even though these tasks are unknown when the DBN is being trained. 5.3 Classifying News Stories The Reuters Corpus Volume II is an archive of 804,414 newswire stories The corpus covers four major groups: Corporate/Industrial, Economics, Government/Social, and Markets. The data was randomly split into 802,414 training and 2000 test articles. The test set contains 500 articles of each major group. The available data was already in a convenient, preprocessed format, where common stopwords were removed and all the remaining words were stemmed. We only made use of the 2000 most frequently used word stems in the training data. As a result, each document was represented as a vector containing 2000 word counts. No other preprocessing was done. For the text classification task we used a 2000-1000-1000-1000 architecture. The entire unlabeled training set of 802,414 articles was used for learning a multilayer generative model of the text documents. The bottom layer of the DBN was trained using a Constrained Poisson Model. Table 3 shows the area under the ROC curve for classifying documents belonging to the Corporate/Industrial vs. Economics groups. As expected, GP-DBNfine and GP-DBNgreedy work better than GPstandard. The results of binary discrimination between other pairs of document classes are very similar to the results presented in table 3. Our experiments using a Gaussian kernel with ARD hyper-parameters did not show any significant improvements. Examining the histograms of the length-scale parame7 ters β, we found that most of the input word-counts as well as most of the extracted features were relevant to the classification task. 6 Conclusions and Future Research In this paper we have shown how to use Deep Belief Networks to greedily pretrain and discriminatively fine-tune a covariance kernel for a Gaussian Process. The discriminative fine-tuning produces an additional improvement in performance that is comparable in magnitude to the improvement produced by using the greedily pretrained DBN. For high-dimensional, highly-structured data, this is an effective way to make use of large unlabeled data sets, especially when labeled training data is scarce. Greedily pretrained DBN’s can also be used to provide input vectors for other kernel-based methods, including SVMs [17, 8] and kernel regression [1], and our future research will concentrate on comparing our method to other kernel-based semi-supervised learning algorithms [4, 19]. Acknowledgments We thank Radford Neal for many helpful suggestions. This research was supported by NSERC, CFI and OTI. GEH is a fellow of CIAR and holds a CRC chair. References [1] J. K. Benedetti. On the nonparametric estimation of regression functions. Journal of the Royal Statistical Society series B, 39:248–253, 1977. [2] Y. Bengio and Y. Le Cun. Scaling learning algorithms towards AI. In L. Bottou, O. Chapelle, D. DeCoste, and J. Weston, editors, Large-Scale Kernel Machines. MIT Press, 2007. [3] Y. Bengio, P. Lamblin, D. Popovici, and H. Larochelle. Greedy layer-wise training of deep networks. In Advances in Neural Information Processing Systems, 2006. [4] O. Chapelle, B. Sch¨olkopf, and A. Zien. Semi-Supervised Learning. MIT Press, 2006. [5] G. E. Hinton. Training products of experts by minimizing contrastive divergence. Neural Computation, 14(8):1711–1800, 2002. [6] G. E. Hinton and R. Salakhutdinov. Reducing the dimensionality of data with neural networks. Science, 313, 2006. [7] Geoffrey E. Hinton, Simon Osindero, and Yee Whye Teh. A fast learning algorithm for deep belief nets. Neural Computation, 18(7):1527–1554, 2006. [8] F. Lauer, C. Y. Suen, and G. Bloch. A trainable feature extractor for handwritten digit recognition. Pattern Recognition, 40(6):1816–1824, 2007. [9] N. D. Lawrence and J. Qui˜nonero Candela. Local distance preservation in the GP-LVM through back constraints. In William W. Cohen and Andrew Moore, editors, ICML, volume 148, pages 513–520. ACM, 2006. [10] N. D. Lawrence and M. I. Jordan. Semi-supervised learning via gaussian processes. In NIPS, 2004. [11] N. D. Lawrence and B. Sch¨olkopf. Estimating a kernel Fisher discriminant in the presence of label noise. In Proc. 18th International Conf. on Machine Learning, pages 306–313. Morgan Kaufmann, San Francisco, CA, 2001. [12] T. P. Minka. Expectation propagation for approximate bayesian inference. In Jack Breese and Daphne Koller, editors, UAI, pages 362–369, San Francisco, CA, 2001. Morgan Kaufmann Publishers. [13] C. E. Rasmussen and C. Williams. Gaussian Processes for Machine Learning. The MIT Press, 2006. [14] R. Salakhutdinov and G. E. Hinton. Learning a nonlinear embedding by preserving class neighbourhood structure. In AI and Statistics, 2007. [15] M. Seeger. Covariance kernels from bayesian generative models. In Thomas G. Dietterich, Suzanna Becker, and Zoubin Ghahramani, editors, NIPS, pages 905–912. MIT Press, 2001. [16] M. Seeger. Gaussian processes for machine learning. Int. J. Neural Syst, 14(2):69–106, 2004. [17] V. Vapnik. Statistical Learning Theory. Wiley, 1998. [18] M. Welling, M. Rosen-Zvi, and G. Hinton. Exponential family harmoniums with an application to information retrieval. In NIPS 17, pages 1481–1488, Cambridge, MA, 2005. MIT Press. [19] Xiaojin Zhu, Jaz S. Kandola, Zoubin Ghahramani, and John D. Lafferty. Nonparametric transforms of graph kernels for semi-supervised learning. In NIPS, 2004. 8
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Kernels on Attributed Pointsets with Applications Mehul Parsana1 mehul.parsana@gmail.com Sourangshu Bhattacharya1 sourangshu@gmail.com Chiranjib Bhattacharyya1 chiru@csa.iisc.ernet.in K. R. Ramakrishnan2 krr@ee.iisc.ernet.in Abstract This paper introduces kernels on attributed pointsets, which are sets of vectors embedded in an euclidean space. The embedding gives the notion of neighborhood, which is used to define positive semidefinite kernels on pointsets. Two novel kernels on neighborhoods are proposed, one evaluating the attribute similarity and the other evaluating shape similarity. Shape similarity function is motivated from spectral graph matching techniques. The kernels are tested on three real life applications: face recognition, photo album tagging, and shot annotation in video sequences, with encouraging results. 1 Introduction In recent times, one of the major challenges in kernel methods has been design of kernels on structured data e.g. sets [9, 17, 15], graphs [8, 3], strings, automata, etc. In this paper, we propose kernels on a type of structured objects called attributed pointsets [18]. Attributed pointsets are points embedded in a euclidean space with a vector of attributes attached to each point. The embedding of points in the euclidean space yields a notion of neighborhood of each point which is exploited in designing new kernels. Also, we describe the notion of similarity between pointsets which model many real life scenarios and incorporate it in the proposed kernels. The main contribution of this paper is definition of two different kernels on neighborhoods. These neighborhood kernels are then used to define kernels on the entire pointsets. The first kernel treats the neighborhoods as sets of vectors for calculating the similarity. Second kernel calculates similarity in shape of the two neighborhoods. It is motivated using spectral graph matching techniques [16]. We demonstrate practical applications of the kernels on the well known task of face recognition [20], and two other novel tasks of tagging photo albums and annotation of shots in video sequences. For the face recognition task, we test our kernels on benchmark datasets and compare their performance with state-of-the-art algorithms. Our kernels outperform the existing methods in many cases. The kernels also perform according to expectation on the two novel applications. Section 2 defines attributed pointsets and contrasts it with related notions. Section 3 proposes two kernels and section 4 describes experimental results. 2 Definition and related work An attributed pointset [18, 1] (a.k.a. point pattern) X is sets of points in Rk with attributes or labels (real vectors in this case) attached to each point. Thus, X = {(xi, di)|i = 1 . . . n}, where xi ∈Ru and di ∈Rv, l being the dimension of the attribute vector. The number of points in a pointset, 1Dept. of Computer Science & Automation, 2Dept. of Electrical Engineering, Indian Institute of Science, Bangalore - 560012, India. 1 n, is variable. Also, for practical purposes pointsets with u = 2, 3 are of interest. The construct of pointsets are richer than sets of vectors [17] because of the structure formed by embedding of the points in a euclidean space. However, they are less general than attributed graphs because all attributed graphs cannot be embedded onto a euclidean space. Pointsets are useful in several domains including computer vision [18], computational biology [5], etc. The notion of similarity between pointsets is also different from those between sets of vectors, or graphs. The main aspect of similarity is that there should be correspondences (1-1 mappings) between the points of a pointset such that the relative positions of corresponding point are same. Also the attribute vectors of the matching points should be similar. In case of sets of vectors, the kernel function captures the similarity between aggregate properties of the two sets, such as the principle angles between spanned subspaces [17], or distance between the distributions generating the vectors [9]. Kernels on graphs try to capture similarity in the graph topology by comparing the number of similar paths [3], or comparing steady state distributions on of linear systems on graphs [8]. For example, consider recognizing faces using local descriptors calculated at some descriptor points (corner points in this case) on the face. It is necessary that subsets of descriptor points found in two images of the same face should be approximately superimposable (slight changes may be due to change of expression) and that the descriptor values for the corresponding points should be roughly same to ensure similar local features. Thus, a face can be modeled as an attributed pointset X = {(xi, di)|i = 1 . . . n}, where xi ∈R2 is the coordinate of ith descriptor point and di ∈Rv is the local descriptor vector at the ith descriptor point. Similar arguments can be provided for any object recognition task. A local descriptor based kernel was proposed for object recognition in similar setting in [12]. Suppose XA = {(xA i , dA i )|i = 1 . . . nA} and XB = {(xB i , dB i )|i = 1 . . . nB} are two pointsets. The normalized sum kernel [12] was defined as KNS(XA, XB) = 1 nAnB PnA i=1 PnB j=1(K(dA i , dB j ))p, where K(dA i , dB j ) is some kernel function on the descriptors. It was argued in [12] that raising the kernel to a high power p approximately calculates similarity between matched pairs of vectors. Using the RBF kernel KRBF (x, y) = e−∥x−y∥2 σ2 , and adjusting the parameter p in σ, we get the normalized sum kernels as: KNS(XA, XB) = 1 nAnB nA X i=1 nB X j=1 KRBF (dA i , dB j ) (1) Observe that this kernel doesn’t use the in formation in xi anywhere, and thus is actually a kernel on a set of vectors. In fact, this kernel can be derived as a special case of the set kernel proposed in [15]. The kernel K(A, B) = trace P r(AT ˆGrB) ˆFr becomes K(A, B) = P ij k(ai, bj)fij for ˆGr = I and F = P r Fr (whose entries are fij) should be positive semidefinite [15]. Thus, choosing F = 11T (all entries 1) and multiplying the kernel by 1 n2 An2 B and using KRBF as the kernel on vectors, we get back the kernel defined in (1). The normalized sum kernel is used as the basic kernel for development and validation of the new kernels proposed here. In the next section, we incorporate position xi of the points using the concept of neighborhood. 3 Kernels 3.1 Neighborhood kernels The key idea in this section is to use spatially co-occurring points of a point to improve the similarity values given by the kernel function. In other words, we hypothesize that similar points from two pointsets should also have neighboring points which are similar. Thus, for each point we define a neighborhood of the point and weight the similarity between each pair of points with the similarity between their neighborhoods. The k-neighborhood Ni of a point (xi, di) in a pointset X is defined as the set of points (including itself) that are closest to it in the embedding euclidean space. So, Ni = {(xj, dj) ∈X|∥xi −xj∥≤ ∥xi −xl∥∀(xl, dl) ̸∈Ni and |Ni| = k}. The neighborhood kernel between two points (xA i , dA i ) 2 Figure 1: Correspondences implicitly found by sum and neighborhood kernels and (xB j , dB j ) is defined as: KN((xA i , dA i ), (xB j , dB j )) = KRBF (dA i , dB j )× 1 |N A i ||N B j | X (xA s ,dA s )∈N A i X (xB t ,dB t )∈N B j KRBF (dA s , dB t ) (2) The neighborhood kernel (NK) between two pointsets XA and XB is thus defined as: KNK(XA, XB) = 1 nAnB × nA X i=1 nB X j=1 KN((xA i , dA i ), (xB j , dB j )) (3) It is easy to see that KNK is a positive semidefinite kernel function. Even though KNK is a straightforward extension, it considerably improves accuracy of KNS. Figure 1 shows values of KNS and KNK for 4 pairs of point from two pointsets modeling faces. Dark blue lines indicate best matches given by KNS while bright blue lines indicate best matches by the KNK. In both cases, KNK gives the correct match while the KNS fails. Computational complexity of KNK is O(k2n2), k being neighborhood size and n number of points. The next section proposes a kernel which uses positions of points (xi) in a neighborhood more strongly to calculate similarity in shape. 3.2 Spectral Neighborhood Kernel The kernel defined in the previous section still uses a set of vectors kernel for finding similarity between the neighborhoods. Here, we are interested in a kernel function which evaluates the similarity in relative position of the corresponding points. Since the neighborhoods being compared are of fixed size, we assume that all points in a neighborhood have a corresponding point in the other. Thus, the correspondences are given by a permutation of points in one of the neighborhoods. This problem can be formulated as the weighted graph matching problem [16], for which spectral method is one of the popular heuristics. We use the features given by spectral decomposition of adjacency matrix of the neighborhood to define a kernel function. Given a neighborhood Ni we define its adjacency matrix Ai as Ai(s, t) = e−∥xs−xt∥ α , ∀s, t|(xs, ds), (xt, dt) ∈Ni, where α is a parameter. Given two neighborhoods N A i and N B j , we are thus interested in a permutation π of the basis of adjacency matrix of one of the neighborhoods (say N B j ), such that ∥AA i −π(AB j )∥F is minimized, ∥.∥F being the frobenius norm of a matrix. It is well known that a matrix can be fully reconstructed from its spectral decomposition. Also, in the case that fewer eigenvectors are used, the equation ∥A −Pk i=1 λiζiζT i ∥2 F = Pn j=k+1 λ2 j, suggests that eigenvectors corresponding to the higher eigenvalues will give better reconstruction. We use one eigenvector corresponding to largest eigenvalue. Thus, the approximate adjacency matrix becomes ˆ A = λ1ζ1ζT 1 . Let π∗be the optimal permutation that minimizes ∥ˆ AA i −π( ˆ AB j )∥F . Note that here π applied on a matrix implies permutation of the basis. It is easy to see that same permutation is induced on basis 3 of the eigenvectors ζB j (1). Call f A i = |ζA i (1)| and f B j = |ζB j (1)|, the spectral projection vectors corresponding to neighborhoods N A i and N B j . Here ζA i (1), ζB j (1) are eigenvectors corresponding to largest eigenvalue of ˆ AA i , ˆ AB j , and |ζ(1)| is the vector of absolute values of components of ζ(1). f(s) can be thought of as projection of the sth point in the corresponding neighborhood on R1. It is equivalent to seek a permutation π∗which minimizes ∥f A i −π(f B j )∥, for comparing neighborhoods N A i and N B j . The resulting similarity score is: S(N A i , N B j ) = max π∈Π T −∥f A i −π(f B j )∥2 2 (4) where, T is a threshold for converting the distance measure to similarity, and Π is the set of all permutations. However, this similarity function is not necessarily positive semidefinite. To construct a positive semidefinite kernel giving similarity between the vectors f A i and f B j , we use the convolution kernel technique [7] on discrete structures. Let x ∈ X be a composite object formed using parts from X1, . . . , Xm. Let R be a relation over X1 × · · · × Xm × X such that R(x1, . . . , xm, x) is true if x is composed of x1, . . . , xm. Let R−1(x) = (x1, . . . , xm) ∈X1 × · · · × Xm|R(x1, . . . , xm, x) = true and K1, . . . , Km be kernels on X1, . . . , Xm, respectively. The convolution kernel K over X is defined as: K(x, y) = X (x1,...,xm)∈R−1(x),(y1,...,ym)∈R−1(y) m Y i=1 Ki(xi, yi) (5) Haussler [7] showed that if K1, . . . , Km are symmetric and positive semidefinite, so is K. For us, let X be the set of all neighborhoods and X1, . . . , Xm be the sets of spectral projections of all points from all the neighborhoods. Here, note that even if the same point appears in different neighborhoods, the appearances will be considered to be different because the projections are relative to the neighborhoods. Since, each neighborhood has size k, in our case m = k. The relation R is defined as R(f(1), . . . , f(k), N A i ) is true iff the vector (f(1), . . . , f(k)) = π(f A i ) for some permutation π. In other words, R(f(1), . . . , f(k), N A i ) is true iff f(1), . . . , f(k) are spectral projections the points of neighborhood N A i ). Also, let Ki, i = 1 . . . k all be RBF kernels with the same parameter β. Thus, from the above equation, the convolution kernel becomes K(N A i , N B j ) = k! P π∈Π e −1 β Pl l=1(f A i (l)−f B j (π(l)))2 = k! P π∈Π e −∥fA i −π(fB j )∥2 β . Dividing by the constant (k!)2, we get kernel KSN as: KSN(N A i , N B j ) = 1 k! X π∈Π e −∥fA i −π(fB j )∥2 β (6) The spectral kernel (SK) KSK between two pointsets XA and XB is thus defined as: KSK(XA, XB) = 1 nAnB nA X i=1 nB X j=1 KRBF (dA i , dB j )KSN(N A i , N B j ) (7) Following theorem relates KSN(N A i , N B j ) to S(N A i , N B j ) (eqn 4). Theorem 3.1 Let Ni and Nj be two sub-structures with spectral projection vectors f i and f j. For large enough value of T such that all points are matched. lim β→0 KSN(Ni, Nj))β = e−T k! eS(Ni,Nj) Proof: Let π∗be the permutation that gives the optimal score S(Ni, Nj). By definition, eS(Ni,Nj) = eT e−∥f i−π∗(f j)∥2. limβ→0(KSN(Ni, Nj))β = limβ→0( 1 k! P π∈Π(l) e −∥fi−π(fj )∥2 β )β = 1 k!e−∥f i−π∗(f j)∥2 limβ→0(1 + P π∈Π\{π∗} e −1 β (∥f i−π(f j)∥2−∥f i−π∗(f j)∥2))β = −1 k! e−∥f i−π∗(f j)∥2 4 Table 1: Recognition accuracy on AR face dataset (section 4.1) Smile Angry Scream Glasses Scarf Left-Light Right-Light 1-NN 96.3% 88.9% 57.0% 48.1% 3.0% 22.2% 17.8% PCA 94.1% 79.3% 44.4% 32.9% 2.2% 7.4% 7.4% LEM 78.6% 92.9% 31.3% 74.8% 47.4% 92.9% 91.1% AMM 96.0% 96.0% 56.0% 80.0% 82.0% NA NA Face-ARG 97.8% 96.3% 66.7% 80.7% 85.2% 98.5% 96.3% Sum(eq (1)) 96.19% 95.23% 83.80% 89.52% 60.00% 86.66% 80.95% NK (eq (3)) 98.09% 98.09% 85.71% 94.28% 65.71% 92.38% 86.66% SK (eq (7)) 99.04% 99.04% 86.66% 93.33% 65.71% 90.47% 84.76% □ Computational complexity of this kernel is O(k!n2), where k is neighborhood size and n is no. of descriptor points. However, since in practice only small neighborhood sizes are considered, the computation time doesn’t become prohibitive. 4 Experimental Results In order to study the effectiveness of proposed kernels for practical visual tasks, we applied them on three problems. Firstly, the kernels were applied to the well known problem of face recognition [20], and results on two benchmark datasets (AR and ORL) were compared to existing state-of-theart methods. Next we used the spectral kernel to tag images in personal photo albums using faces of people present in them. Finally, the spectral kernel was used for annotation of video sequences using faces of people present. Attribute For face recognition, faces were modeled as attributed pointsets using local gabor descriptors [10] calculated at the corner points using Harris corner point detector [6]. At each point, gabor despite for three different scales and four different orientations were calculated. Descriptors for 5 points (4 pixel neighbors and itself) were used for each of the 12 combinations, making a total of 60 descriptors per point. For image tagging and video annotation, faces were modeled as attributed pointsets using SIFT local descriptors [11], having 128 descriptors per point. The kernels were implemented in GNU C/C++. LAPACK [2] was used for calculation of eigenvectors and GNU GSL for calculation of permutations. LIBSVM [4] was used as the SVM based classifier for classifying pointsets. The face detector provided in OpenCV was used for detecting faces in album images and video frames. Dataset The AR dataset [13] is composed of color images of 135 people (75 men and 60 women). The DB includes frontal view images with different facial expressions, illumination conditions, and occlusion by sunglasses and scarf. After removing persons with corrupted images or missing any of the 8 types of required images, a total of 105 persons (56 men and 49 women) were selected. All the images were converted to greyscale and rescaled to 154×115 pixels. The ORL dataset is composed of 10 images for each of the 40 persons. The images have minor variations in pose, illumination and scale. All the 400, 112 × 92 pixel images were used for experiments. 4.1 Face Recognition in AR face DB The kernels proposed in this paper, were tested pointsets derived from images in AR face DB. Face recognition was posed as a multiclass classification problem, and SVMs were along with the proposed kernels. The AR face DB is a standard benchmark dataset, on which a recent comparison of state of the art methods for face recognition has been given in [14]. In table 1, we have restated the results provided in [14] along with the results of our kernels. All the results reported in table 1 have been obtained using one normal (no occlusion or change of expression) face image as the training set. It can be seen that for all the images showing change of expression (Smile, Angry and Scream), the pointset kernels outperform existing methods. Also, in case of occlusion of face by glasses, the 5 Table 2: Recognition accuracy on ORL dataset (section 4.2) # of training images → 1 3 5 Sum (eq (1)) 70.83% 92.50% 98.00% NK (eq (3)) 71.38% 93.57% 98.00% SK (eq (7)) 71.94% 93.92% 98.00% Figure 2: Representative cluster from tagging of album pointset kernels give better results than existing methods. However, in case of occlusion by scarf, the kernel based method do not perform as well as the Face-ARG or AMM. This failure is due to introduction of a large number of points in the scarf themselves. It was observed that about 50% of the descriptor points in the faces having scarfs were in the scarf region of the image. Summing the similarities over such a large number of extra points makes the overall kernel value noisy. The proposed approach doesn’t perform better than existing methods on images taken under extreme variation in lighting conditions. This is due to the fact that values of the local descriptors change drastically with illumination. Also, some of the corner points disappear under different lighting condition. However, performance of the kernels is comparable to the existing methods, thus demonstrating the effectiveness of modeling faces as attributed pointsets. 4.2 Recognition performance on ORL Dataset Real life problems in face recognition also show minor variations in pose, which are addressed by testing the kernels on images in the ORL dataset. The problem was posed as a multiclass classification problem and SVM was used along with the kernels for classification. Table 2 reports the recognition accuracies of all the three kernels for two different values of parameters, and for 1, 3 and 5 training images. It can be seen that even with images showing minor variations in pose, the proposed kernels perform reasonably well. Also, due to change in pose the relative position of points in the pointsets change. This is reflected in the fact that improvement due to addition of position information in kernels is minor as compared to those shown in AR dataset. For higher number of training images, the performance of all the kernels saturate at 98%. 4.3 Tagging images in personal albums based on faces The problem of tagging images in personal albums with names of people present in them, is a problem of high practical relevance [19]. The spectral kernels were used solve this problem. Images from publicly available sources like http://www.flickr.com 1 were used for experimentation. Five personal albums having 20 - 55 images each were downloaded and many images had upto 6 people. Face detector from openCV library was used to automatically detect faces in images. Detected faces are cropped and resized to 100 × 100 px resolution. 47 - 265 such faces detected from each album. To the best of our knowledge, there are no openly available techniques to benchmark our method against. Due to non-availability of training data, the problem of image tagging was posed as a clustering problem. Faces detected from the images were represented as attributed pointsets using SIFT local descriptors, and spectral kernel was evaluated on them. A threshold based clustering scheme was used on the distance metric induced by the kernel (d(x, y) = p K(x, x) + K(y, y) −2 ∗k(x, y)). Ideally, each cluster thus obtained should represent a person and images containing faces from a given cluster should be tagged with the name of that person. 1We intend to make the dataset publicly available if no copyrights are violated 6 Table 3: Face based album tagging Album no. No. of people % Identified % False +ve (Actual) (Identified) 1 2 90% 0% 2 14 6 84% 10.52% 3 8 4 66.66% 8.33% 4 4 2 83.33% 19.44% 5 3 2 80.00% 14.70% Figure 3: Keyframes of a few shots detected with annotation Table 3 reports results from tagging experiments for five albums. No. of people identified reports the number clusters having more than one faces, as singleton cluster will always be correct for that person. Thus, people appearing only once in the entire album are not reported, which reduce the no. of identified people. % identified and % false +ve are averaged over all clusters detected in the album, and are calculated for each cluster as: % identified = No. of correct faces in the cluster T otal no. of faces of the person and % false + ve = false +ves in the cluster T otal no. of faces in the cluster. It can be seen that the kernel performs reasonably well on the dataset. Figure 2 shows a representative cluster with the first 8 images as true +ves and rest as false +ves. 4.4 Video annotation based on faces The kernels were also used to perform video shot annotation based faces detected in video sequences. Experimentation was performed on videos from “News and Public affair” section of www.archive.org and music videos from www.youtube.com. Video was sampled at 1 frame per second and experimental methodology was similar section 4.3 was used on the frames. Figure 3 shows two representative shots from corresponding to two candidates from “Election 2004, presidential debate part 2”, and one from “Westlife- Seasons in the Sun” video. The faces annotating the shots are shown in the left as thumbnails. It may be noted that for videos, high pose variation did not reduce accuracy of recognition due to gradual changing of pose. The results on detecting shots were highly encouraging, thus demonstrating the varied applicability of proposed attributed pointset kernels. 5 Conclusion In this article, we propose kernels on attributed pointsets. We define the notion of neighborhood in an attributed pointset and propose two new kernels. The first kernel evaluates attribute similarities between the neighborhoods and uses the co-occurrence information to improve the performance of kernels on sets of vectors. The second kernel uses the position information more strongly and 7 matches the shapes of neighborhoods. This kernel function is motivated from spectral graph matching techniques. The proposed kernels were validated on the well known task on face recognition on two popular benchmark datasets. Results show that the current kernels perform competitively with the state-ofthe-art techniques for face recognition. The spectral kernel was also used to perform two real life tasks of tagging images in personal photo albums and annotating shots in videos. The results were encouraging in both cases. References [1] Helmut Alt and Leonidas J. Guibas. Discrete geometric shapes: Matching, interpolation, and approximation A survey. Technical Report B 96-11, 1996. [2] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen. LAPACK Users’ Guide. Society for Industrial and Applied Mathematics, Philadelphia, PA, third edition, 1999. [3] Karsten M. Borgwardt and Hans-Peter Kriegel. Shortest-path kernels on graphs. In ICDM ’05: Proceedings of the Fifth IEEE International Conference on Data Mining, pages 74–81, Washington, DC, USA, 2005. IEEE Computer Society. [4] Chih-Chung Chang and Chih-Jen Lin. LIBSVM: a library for support vector machines, 2001. Software available at http://www.csie.ntu.edu.tw/∼cjlin/libsvm. [5] Ingvar Eidhammer, Inge Jonassen, and William R. Taylor. Structure comparison and structure patterns. Journal of Computational Biology, 7(5):685–716, 2000. [6] C. Harris and M.J. Stephens. A combined corner and edge detector. In Proc. of Alvey Vision Conf., 1988. [7] David Haussler. Convolution kernels on discrete structures. Technical report, University of California, Santa Cruz, 1999. [8] Koji Tsuda Hisashi Kashima and Akihiro Inokuchi. Marginalized kernels between labeled graphs. In Twentieth International Conference on Machine Learning (ICML), 2003. [9] Risi Kondor and Tony Jebara. A kernel between sets of vectors. In Twentieth International Conference on Machine Learning (ICML), 2003. [10] Tai Sing Lee. Image representation using 2d gabor wavelets. IEEE TPAMI, 18(10):959–971, 1996. [11] D. Lowe. Distinctive image features from scale-invariant keypoints. Int. Journal of Computer Vision, 20:91–110, 2003. [12] Siwei Lyu. Mercer kernels for object recognition with local features. In IEEE CVPR, 2005. [13] A.M. Martinez and R. Benavente. The ar face database. CVC Technical Report, 24, 1998. [14] Bo Gun Park, Kyoung Mu Lee, and Sang Uk Lee. Face recognition using face-arg matching. IEEE TPAMI, 27(12):1982–1988, 2005. [15] Amnon Shashua and Tamir Hazan. Algebraic set kernels with application to inference over local image representations. In Neural Information Processing Systems (NIPS), 2004. [16] Shinji Umeyama. An eigendecomposition approach to weighted graph matching problems. IEEE transactions on pattern analysis and machine intelligence, 10(5):695–703, 1988. [17] Lior Wolf and Amnon Shashua. Learning over sets using kernel principal angles. Journal of Machine Learning Research, (4):913–931, 2003. [18] Haim J. Wolfson and Isidore Rigoutsos. Geometric hashing: An overview. IEEE Comput. Sci. Eng., 4(4):10–21, 1997. [19] L. Zhang, L. Chen, M. Li, and H. Zhang. Automated annotation of human faces in family albums, 2003. [20] W. Zhao, R. Chellappa, P. J. Phillips, and A. Rosenfeld. Face recognition: A literature survey. ACM Comput. Surv., 35(4):399–458, 2003. 8
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Testing for Homogeneity with Kernel Fisher Discriminant Analysis Za¨ıd Harchaoui LTCI, TELECOM ParisTech and CNRS 46, rue Barrault, 75634 Paris cedex 13, France zaid.harchaoui@enst.fr Francis Bach Willow Project, INRIA-ENS 45, rue d’Ulm, 75230 Paris, France francis.bach@mines.org ´Eric Moulines LTCI, TELECOM ParisTech and CNRS 46, rue Barrault, 75634 Paris cedex 13, France eric.moulines@enst.fr Abstract We propose to investigate test statistics for testing homogeneity based on kernel Fisher discriminant analysis. Asymptotic null distributions under null hypothesis are derived, and consistency against fixed alternatives is assessed. Finally, experimental evidence of the performance of the proposed approach on both artificial and real datasets is provided. 1 Introduction An important problem in statistics and machine learning consists in testing whether the distributions of two random variables are identical under the alternative that they may differ in some ways. More precisely, let {X(1) 1 , . . . , X(1) n1 } and {X(2) 1 , . . . , X(2) n2 } be independent random variables taking values in the input space (X, d), with common distributions P1 and P2, respectively. The problem consists in testing the null hypothesis H0 : P1 = P2 against the alternative HA : P1 ̸= P2. This problem arises in many applications, ranging from computational anatomy [10] to process monitoring [7]. We shall allow the input space X to be quite general, including for example finite-dimensional Euclidean spaces or more sophisticated structures such as strings or graphs (see [17]) arising in applications such as bioinformatics [4]. Traditional approaches to this problem are based on distribution functions and use a certain distance between the empirical distributions obtained from the two samples. The most popular procedures are the two-sample Kolmogorov-Smirnov tests or the Cramer-Von Mises tests, that have been the standard for addressing these issues (at least when the dimension of the input space is small, and most often when X = R). Although these tests are popular due to their simplicity, they are known to be insensitive to certain characteristics of the distribution, such as densities containing highfrequency components or local features such as bumps. The low-power of the traditional density based statistics can be improved on using test statistics based on kernel density estimators [2] and [1] and wavelet estimators [6]. Recent work [11] has shown that one could difference in means in RKHSs in order to consistently test for homogeneity. In this paper, we show that taking into account the covariance structure in the RKHS allows to obtain simple limiting distributions. The paper is organized as follows: in Section 2 and Section 3, we state the main definitions and we construct the test statistics. In Section 4, we give the asymptotic distribution of our test statistic under the null hypothesis, and investigate, the consistency and the power of the test for fixed alternatives. In 1 Section 5 we provide experimental evidence of the performance of our test statistic on both artificial and real datasets. Detailed proofs are presented in the last sections. 2 Mean and covariance in reproducing kernel Hilbert spaces We first highlight the main assumptions we make in the paper on the reproducing kernel, then introduce operator-theoretic tools for working with distributions in infinite-dimensional spaces. 2.1 Reproducing kernel Hilbert spaces Let (X, d) be a separable metric space, and denote by X the associated σ-algebra. Let X be Xvalued random variable, with probability measure P; the corresponding expectation is denoted E. Consider a Hilbert space (H, ⟨·, ·⟩H) of functions from X to R. The Hilbert space H is an RKHS if at each x ∈X, the point evaluation operator δx : H →R, which maps f ∈H to f(x) ∈R, is a bounded linear functional. To each point x ∈X, there corresponds an element Φ(x) ∈H (we call Φ the feature map) such that ⟨Φ(x), f⟩H = f(x) for all f ∈H, and ⟨Φ(x), Φ(y)⟩H = k(x, y), where k : X × X →R is a positive definite kernel. We denote by ∥f∥H = ⟨f, f⟩1/2 H the associated norm. It is assumed in the remainder that H is a separable Hilbert space. Note that this is always the case if X is a separable metric space and if the kernel is continuous (see [18]). Throughout this paper, we make the following two assumptions on the kernel: (A1) The kernel k is bounded, that is |k|∞= sup(x,y)∈X×X k(x, y) < ∞. (A2) For all probability measures P on (X, X), the RKHS associated with k(·, ·) is dense in L2(P). The asymptotic normality of our test statistics is valid without assumption (A2), while consistency results against fixed alternatives does need (A2). Assumption (A2) is true for translation-invariant kernels [8], and in particular for the Gaussian kernel on Rd [18]. Note that we do not require the compactness of X as in [18], 2.2 Mean element and covariance operator We shall need some operator-theoretic tools to define mean elements and covariance operators in RKHS. A linear operator T is said to be bounded if there is a number C such that ∥Tf∥H ≤C ∥f∥H for all f ∈H. The operator-norm of T is then defined as the infimum of such numbers C, that is ∥T∥= sup∥f∥H≤1 ∥Tf∥H (see [9]). We recall below some basic facts about first and second-order moments of RKHS-valued random variables. If R k1/2(x, x)P(dx) < ∞, the mean element µP is defined for all functions f ∈H as the unique element in H satisfying, ⟨µP, f⟩H = Pf def = Z fdP . (1) If furthermore R k(x, x)P(dx) < ∞, then the covariance operator ΣP is defined as the unique linear operator onto H satisfying for all f, g ∈H, ⟨f, ΣPg⟩H def = Z (f −Pf)(g −Pg)dP . (2) Note that when assumption (A2) is satisfied, then the map from P 7→µP is injective. The operator ΣP is a self-adjoint nonnegative trace-class operator. In the sequel, the dependence of µP and ΣP in P is omitted whenever there is no risk of confusion. Given a sample {X1, . . . , Xn}, the empirical estimates respectively of the mean element and the covariance operator are then defined using empirical moments and lead to: ˆµ = n−1 n X i=1 k(Xi, ·) , ˆΣ = n−1 n X i=1 k(Xi, ·) ⊗k(Xi, ·) −ˆµ ⊗ˆµ . (3) 2 The operator Σ is a self-adjoint nonnegative trace-class operators. Hence, it can de diagonalized in an orthonormal basis, with a spectrum composed of a strictly decreasing sequence λp > 0 tending to zero and potentially a null space N(Σ) composed of functions f in H such that R {f −Pf}2dP = 0 [5], i.e., functions which are constant in the support of P. The null space may be reduced to the null element (in particular for the Gaussian kernel), or may be infinite-dimensional. Similarly, there may be infinitely many strictly positive eigenvalues (true nonparametric case) or finitely many (underlying finite dimensional problems). 3 KFDA-based test statistic In the feature space, the two-sample homogeneity test procedure can be formulated as follows. Given {X(1) 1 , . . . , X(1) n1 } and {X(2) 1 , . . . , X(2) n2 } from distributions P1 and P2, two independent identically distributed samples respectively from P1 and P2, having mean and covariance operators respectively given by (µ1, Σ1) and (µ2, Σ2), we wish to test the null hypothesis H0, µ1 = µ2 and Σ1 = Σ2, against the alternative hypothesis HA, µ1 ̸= µ2. In this paper, we tackle the problem by using a (regularized) kernelized version of the Fisher discriminant analysis. Denote by ΣW def = (n1/n)Σ1+(n2/n)Σ2 the pooled covariance operator, where n def = n1 + n2, corresponding to the within-class covariance matrix in the finite-dimensional setting (see [14]. Let us denote ΣB def = (n1n2/n2)(µ2−µ1)⊗(µ2−µ1) the between-class covariance operator. For a = 1, 2, denote by (ˆµa, ˆΣa) respectively the empirical estimates of the mean element and the covariance operator, defined as previously stated in (3). Denote ˆΣW def = (n1/n)ˆΣ1 + (n2/n)ˆΣ2 the empirical pooled covariance estimator, and ˆΣB def = (n1n2/n2)(ˆµ2 −ˆµ1) ⊗(ˆµ2 −ˆµ1) the empirical between-class covariance operator. Let {γn}n≥0 be a sequence of strictly positive numbers. The maximum Fisher discriminant ratio serves as a basis of our test statistics: n max f∈H D f, ˆΣBf E H D f, (ˆΣW + γnI)f E H = n1n2 n
(ˆΣW + γnI)−1 2 ˆδ
2 H , (4) where I denotes the identity operator. Note that if the input space is Euclidean, e.g. X = Rd, the kernel is linear k(x, y) = x⊤y and γn = 0, this quantity matches the so-called Hotelling’s T 2statistic in the two-sample case [15]. Moreover, in practice it may be computed thanks to the kernel trick, adapted to the kernel Fisher discriminant analysis and outlined in [17, Chapter 6]. We shall make the following assumptions respectively on Σ1 and Σ2 (B1) For u = 1, 2, the eigenvalues {λp(Σu)}p≥1 satisfy P∞ p=1 λ1/2 p (Σu) < ∞. (B2) For u = 1, 2, there are infinitely many strictly positive eigenvalues {λp(Σu)}p≥1 of Σu. The statistical analysis conducted in Section 4 shall demonstrate, as γn →0 at an appropriate rate, the need to respectively recenter and rescale (a standard statistical transformation known as studentization) the maximum Fisher discriminant ratio, in order to get a theoretically well-calibrated test statistic. These roles, recentering and rescaling, will be played respectively by d1(ΣW , γ) and d2(ΣW , γ), where for a given compact operator Σ with decreasing eigenvalues λp(S), the quantity dr(Σ, γ) is defined for all q ≥1 as dr(Σ, γ) def = ( ∞ X p=1 (λp + γ)−rλr p )1/r . (5) 4 Theoretical results We consider in the sequel the following studentized test statistic: bTn(γn) = n1n2 n
(ˆΣW + γnI)−1/2ˆδ
2 H −d1(ˆΣW , γn) √ 2d2(ˆΣW , γn) . (6) 3 In this paper, we first consider the asymptotic behavior of bTn under the null hypothesis, and then against a fixed alternative. This will establish that our nonparametric test procedure is consistent in power. 4.1 Asymptotic normality under null hypothesis In this section, we derive the distribution of the test statistics under the null hypothesis H0 : P1 = P2 of homogeneity, i.e. µ1 = µ2 and Σ1 = Σ2 = Σ. As γn →0 tends to zero, Theorem 1. Assume (A1) and (B1). If P1 = P2 = P and if γn + γ−1 n n−1/2 →0, then bTn(γn) D −→N(0, 1) (7) The proof is postponed to Section 7. Under the assumptions of Theorem 1, the sequence of tests that rejects the null hypothesis when ˆTn(γn) ≥z1−α, where z1−α is the (1−α)-quantile of the standard normal distribution, is asymptotically level α. Note that the limiting distribution does not depend on the kernel nor on the regularization parameter. 4.2 Power consistency We study the power of the test based on bTn(γn) under alternative hypotheses. The minimal requirement is to to prove that this sequence of tests is consistent in power. A sequence of tests of constant level α is said to be consistent in power if the probability of accepting the null hypothesis of homogeneity goes to zero as the sample size goes to infinity under a fixed alternative. The following proposition shows that the limit is finite, strictly positive and independent of the kernel otherwise (see [8] for similar results for canonical correlation analysis). The following result gives some useful insights on
Σ−1/2 W δ
H, i.e.the population counterpart of
(ˆΣ−1/2 W + γnI)−1/2ˆδ
H on which our test statistics is based upon. Proposition 2. Assume (A1) and (A2). If γn+γ−1 n n−1/2 →0, then for any probability distributions P1 and P2,
Σ−1/2 W δ
2 H = 1 ρ1ρ2 1 − Z p1p2 ρ1p1 + ρ2p2 dν Z p1p2 ρ1p1 + ρ2p2 dρ −1 , where ν is any probability measure such that P1 and P2 are absolutely continuous w.r.t. ν and p1 and p2 are the densities of P1 and P2 with respect to ν. The norm
Σ−1/2 W δ
2 H is finite when the χ2-divergence R p−1 1 (p2 −p1)2dρ is finite. It is equal to zero if the χ2-divergence is null, that is, if and only if P1 = P2. By combining the two previous propositions, we therefore obtain the following consistency Theorem. Theorem 3. Assume (A1) and (A2). Let P1 and P2 be two distributions over (X, X), such that P2 ̸= P1. If γn + γ−1 n n−1/2 →0, then PHA( bTn(γ) > z1−α) →∞. (8) 5 Experiments In this section, we investigate the experimental performances of our test statistic KFDA, and compare it in terms of power against other nonparametric test statistics. 5.1 Artificial data We shall focus here on a particularly simple setting, in order analyze the major issues arising in applying our approach in practice. Indeed, we consider the periodic smoothing spline kernel (see 4 γ = 10−1 10−4 10−7 10−10 KFDA 0.01±0.0032 0.11±0.0062 0.98±0.0031 0.99±0.0001 MMD 0.01±0.0023 id. id. id. Table 1: Evolution of power of KFDA and MMD respectively, as γ goes to 0. [19] for a detailed derivation), for which explicit formulae are available for the eigenvalues of the corresponding covariance operator when the underlying distribution is uniform. This allows us to alleviate the issue of estimating the spectrum of the covariance operator, and weigh up the practical impact of the regularization on the power of our test statistic. Periodic smoothing spline kernel Consider X as the two-dimensional circle identified with the interval [0, 1] (with periodicity conditions). We consider the strictly positive sequence Kν = (2πν)−2m and the following norm: ∥f∥2 H = ⟨f, c0⟩2 K0 + X ν>0 ⟨f, cν⟩2 + ⟨f, sν⟩2 Kν where cν(t) = √ 2 cos 2πνt and sν(t) = √ 2 sin 2πνt for ν ≥1 and c0(t) = 1X. This is always an RKHS norm associated with the following kernel K(s, t) = (−1)m−1 (2m)! B2m((s −t) −⌊s −t⌋) where B2m is the 2m-th Bernoulli polynomial. We have B2(x) = x2 −x + 1/6. We consider the following testing problem H0 : p1 = p2 HA : p2 ̸= p2 with p1 the uniform density (i.e., the density with respect to the Lebesgue measure is equal to c0), and densities p2 = p1(c0 +.25∗c4). The covariance operator Σ(p1) has eigenvectors c0, cν, sν with eigenvalues 0 for c0 and Kν for others. Comparison with MMD We conducted experimental comparison in terms of power, for m = 2 and n = 104 and ε = 0.5. All quantities involving the eigenvalues of the covariance operator were computed from their counterparts instead of being estimated. The sampling from pn 2 was performed by inverting the cumulative distribution function. The table below displays the results, averaged over 10 Monte-Carlo runs. 5.2 Speaker verification We conducted experiments in a speaker verification task [3], on a subset of 8 female speakers using data from the NIST 2004 Speaker Recognition Evaluation. We refer the reader to [16] for instance for details on the pre-processing of data. The figure shows averaged results over all couples of speakers. For each couple of speaker, at each run we took 3000 samples of each speaker and launched our KFDA-test to decide whether samples come from the same speaker or not, and computed the type II error by comparing the prediction to ground truth. We averaged the results for 100 runs for each couple, and all couples of speaker. The level was set to α = 0.05, since the empirical level seemed to match the prescribed for this value of the level as we noticed in previous subsection. We performed the same experiments for the Maximum Mean Discrepancy and the Tajvidi-Hall test statistic (TH, [13]). We summed up the results by plotting the ROC-curve for all competing methods. Our method reaches good empirical power for a small value of the prescribed level (1 −β = 90% for α = 0.05%). Maximum Mean Discrepancy also yields good empirical performance on this task. 6 Conclusion We proposed a well-calibrated test statistic, built on kernel Fisher discriminant analysis, for which we proved that the asymptotic limit distribution under null hypothesis is standard normal distribution. Our test statistic can be readily computed from Gram matrices once a kernel is defined, and 5 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 Level Power ROC Curve KFDA MMD TH Figure 1: Comparison of ROC curves in a speaker verification task allows us to perform nonparametric hypothesis testing for homogeneity for high-dimensional data. The KFDA-test statistic yields competitive performance for speaker identification. 7 Sketch of proof of asymptotic normality under null hypothesis Outline. The proof of the asymptotic normality of the test statistics under null hypothesis follows four steps. As a first step, we derive an asymptotic approximation of the test statistics as γn + γ−1 n n−1/2 →0 , where the only remaining stochastic term is ˆδ. The test statistics is then spanned onto the eigenbasis of Σ, and decomposed into two terms Bn and Cn. The second step allows to prove the asymptotic negligibility of Bn, while the third step establishes the asymptotic normality of Cn by a martingale central limit theorem (MCLT). Step 1: bTn(γn) = ˜Tn(γn) + oP (1). First, we may prove, using perturbation results of covariance operators, that, as γn + γ−1 n n−1/2 →0 , we have bTn(γn) = (n1n2/n)
(Σ + γI)−1/2 ˆδ
2 H −d1(Σ, γ) √ 2d2(Σ, γ) + oP (1) . (9) For ease of notation, in the following, we shall often omit Σ in quantities involving it. Hence, from now on, λp, λq, d2,n stand for λp(Σ), λq(Σ), d2(Σ, γn). Define Yn,p,i def = n2 n1n 1/2 ep(X(1) i ) −E[ep(X(1) 1 )] 1 ≤i ≤n1 , − n1 n2n 1/2 ep(X(2) i−n1) −E[ep(X(2) 1 )] n1 + 1 ≤i ≤n . (10) We now give formulas for the moments of {Yn,p,i}1≤i≤n,p≥1, often used in the proof. Straightforward calculations give n X i=1 E[Yn,p,iYn,q,i] = λ1/2 p λ1/2 q δp,q , (11) while the Cauchy-Schwarz inequality and the reproducing property give Cov(Y 2 n,p,i, Y 2 n,q,i) ≤Cn−2|k|∞λ1/2 p λ1/2 q . (12) Denote Sn,p def = Pn i=1 Yn,p,i. Using Eq. (11), our test statistics now writes as ˜Tn = ( √ 2d2,n)−1An with An def = n1n2 n
(Σ + γnI)−1/2ˆδ
2 −d1,n = ∞ X p=1 (λp + γn)−1 S2 n,p −ES2 n,p = Bn + 2Cn . (13) 6 where Bn and Cn are defined as follows Bn def = ∞ X p=1 n X i=1 Y 2 n,p,i −EY 2 n,p,i , (14) Cn def = ∞ X p=1 (λp + γn)−1 n X i=1 Yn,p,i i−1 X j=1 Yn,p,j . (15) Step 2: Bn = oP (1). The proof consists in computing the variance of this term. Since the variables Yn,p,i and Yn,q,j are independent if i ̸= j, then Var(Bn) = Pn i=1 vn,i, where vn,i def = Var ∞ X p=1 (λp + γn)−1{Y 2 n,p,i −E[Y 2 n,p,i]} ! = ∞ X p,q=1 (λp + γn)−1(λq + γn)−1Cov(Y 2 n,p,i, Y 2 n,q,i) . Using Eq. (12), we get Pn i=1 vn,i ≤Cn−1γ−2 n P∞ p=1 λ1/2 p 2 where the RHS above is indeed negligible, since by assumption we have γ−1 n n−1/2 →0 and P∞ p=1 λ1/2 p < ∞. Step 3: d−1 2,nCn D −→N(0, 1/2). We use the central limit theorem (MCLT) for triangular arrays of martingale differences (see e.g. [12, Theorem 3.2]). For = 1, . . . , n, denote ξn,i def = d−1 2,n ∞ X p=1 (λp + γn)−1Yn,p,iMn,p,i−1 , where Mn,p,i def = i X j=1 Yn,p,j , (16) and let Fn,i = σ (Yn,p,j, p ∈{1, . . . , n}, j ∈{0, . . . , i}). Note that, by construction, ξn,i is a martingale increment, i.e. E [ξn,i | Fn,i−1] = 0. The first step in the proof of the CLT is to establish that s2 n = n X i=1 E ξ2 n,i Fn,i−1 P −→1/2 . (17) The second step of the proof is to establish the negligibility condition. We use [12, Theorem 3.2], which requires to establish that max1≤i≤n |ξn,i| P −→0 (smallness) and E(max1≤i≤n ξ2 n,i) is bounded in n (tightness), where ξn,i is defined in (16). We will establish the two conditions simultaneously by checking that E max 1≤i≤n ξ2 n,i = o(1) . (18) Splitting the sum s2 n, between diagonal terms Dn, and off-diagonal terms En, we have Dn = d−2 2,n ∞ X p=1 (λp + γn)−2 n X i=1 M 2 n,p,i−1E[Y 2 n,p,i] , (19) En = d−2 2,n X p̸=q (λp + γn)−1(λq + γn)−1 n X i=1 Mn,p,i−1Mn,q,i−1E[Yn,p,iYn,q,i] . (20) Consider first the diagonal terms En. We first compute its mean. Note that E[M 2 n,p,i] = Pi j=1 E[Y 2 n,p,j]. Using Eq. (11) we get ∞ X p=1 (λp + γn)−2 n X i=1 i−1 X j=1 E[Y 2 n,p,j]E[Y 2 n,p,i] = 1 2 ∞ X p=1 (λp + γn)−2 " n X i=1 E[Y 2 n,p,i] #2 − n X i=1 E2[Y 2 n,p,i] = 1 2d2 2,n 1 + O(n−1) . 7 Therefore, E[Dn] = 1/2 + o(1). Next, we may prove that Dn −E[Dn] = oP (1) is negligible, by checking that Var[Dn] = o(1). We finally consider En defined in (20), and prove that En = oP (1) using Eq. (11). This concludes the proof of Eq. (17). We finally show Eq. (18). Since |Yn,p,i| ≤n−1/2|k|1/2 ∞P-a.s we may bound max 1≤i≤n |ξn,i| ≤Cd−1 2,nn−1/2 ∞ X p=1 (λp + γn)−1 max 1≤i≤n |Mn,p,i−1| . (21) Then, the Doob inequality implies that E1/2[max1≤i≤n |Mn,p,i−1|2] ≤E1/2[M 2 n,p,n−1] ≤Cλ1/2 p . Plugging this bound in (21), the Minkowski inequality E1/2 max 1≤i≤n ξ2 n,i ≤C ( d−1 2,nγ−1 n n−1/2 ∞ X p=1 λ1/2 p ) , and the proof is concluded using the fact that γn + γ−1 n n−1/2 →0 and Assumption (B1). References [1] D. L. Allen. Hypothesis testing using an L1-distance bootstrap. The American Statistician, 51(2):145– 150, 1997. [2] N. H. Anderson, P. Hall, and D. M. 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. [3] F. Bimbot, J.-F. Bonastre, C. Fredouille, G. Gravier, I. Magrin-Chagnolleau, S. Meignier, T. Merlin, J. Ortega-Garcia, D. Petrovska-Delacretaz, and D. A. Reynolds. A tutorial on text-independent speaker verification. EURASIP, 4:430–51, 2004. [4] K. Borgwardt, A. Gretton, M. Rasch, H.-P. Kriegel, Sch¨olkopf, and A. J. Smola. Integrating structured biological data by kernel maximum mean discrepancy. 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Permutation tests for equality of distributions in high-dimensional settings. Biometrika, 89(2):359–374, 2002. [14] T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning. Springer Series in Statistics. Springer, 2001. [15] E. Lehmann and J. Romano. Testing Statistical Hypotheses (3rd ed.). Springer, 2005. [16] J. Louradour, K. Daoudi, and F. Bach. Feature space mahalanobis sequence kernels: Application to svm speaker verification. IEEE Transactions on Audio, Speech and Language Processing, 2007. To appear. [17] J. Shawe-Taylor and N. Cristianini. Kernel Methods for Pattern Analysis. Cambridge Univ. Press, 2004. [18] I. Steinwart, D. Hush, and C. Scovel. An explicit description of the reproducing kernel hilbert spaces of gaussian RBF kernels. IEEE Transactions on Information Theory, 52:4635–4643, 2006. [19] G. Wahba. Spline Models for Observational Data. SIAM, 1990. 8
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Sparse deep belief net model for visual area V2 Honglak Lee Chaitanya Ekanadham Andrew Y. Ng Computer Science Department Stanford University Stanford, CA 94305 {hllee,chaitu,ang}@cs.stanford.edu Abstract Motivated in part by the hierarchical organization of the cortex, a number of algorithms have recently been proposed that try to learn hierarchical, or “deep,” structure from unlabeled data. While several authors have formally or informally compared their algorithms to computations performed in visual area V1 (and the cochlea), little attempt has been made thus far to evaluate these algorithms in terms of their fidelity for mimicking computations at deeper levels in the cortical hierarchy. This paper presents an unsupervised learning model that faithfully mimics certain properties of visual area V2. Specifically, we develop a sparse variant of the deep belief networks of Hinton et al. (2006). We learn two layers of nodes in the network, and demonstrate that the first layer, similar to prior work on sparse coding and ICA, results in localized, oriented, edge filters, similar to the Gabor functions known to model V1 cell receptive fields. Further, the second layer in our model encodes correlations of the first layer responses in the data. Specifically, it picks up both colinear (“contour”) features as well as corners and junctions. More interestingly, in a quantitative comparison, the encoding of these more complex “corner” features matches well with the results from the Ito & Komatsu’s study of biological V2 responses. This suggests that our sparse variant of deep belief networks holds promise for modeling more higher-order features. 1 Introduction The last few years have seen significant interest in “deep” learning algorithms that learn layered, hierarchical representations of high-dimensional data. [1, 2, 3, 4]. Much of this work appears to have been motivated by the hierarchical organization of the cortex, and indeed authors frequently compare their algorithms’ output to the oriented simple cell receptive fields found in visual area V1. (E.g., [5, 6, 2]) Indeed, some of these models are often viewed as first attempts to elucidate what learning algorithm (if any) the cortex may be using to model natural image statistics. However, to our knowledge no serious attempt has been made to directly relate, such as through quantitative comparisons, the computations of these deep learning algorithms to areas deeper in the cortical hierarchy, such as to visual areas V2, V4, etc. In this paper, we develop a sparse variant of Hinton’s deep belief network algorithm, and measure the degree to which it faithfully mimics biological measurements of V2. Specifically, we take Ito & Komatsu [7]’s characterization of V2 in terms of its responses to a large class of angled bar stimuli, and quantitatively measure the degree to which the deep belief network algorithm generates similar responses. Deep architectures attempt to learn hierarchical structure, and hold the promise of being able to first learn simple concepts, and then successfully build up more complex concepts by composing together the simpler ones. For example, Hinton et al. [1] proposed an algorithm based on learning individual layers of a hierarchical probabilistic graphical model from the bottom up. Bengio et al. [3] proposed a similarly greedy algorithm, one based on autoencoders. Ranzato et al. [2] developed an energy-based hierarchical algorithm, based on a sequence of sparsified autoencoders/decoders. 1 In related work, several studies have compared models such as these, as well as nonhierarchical/non-deep learning algorithms, to the response properties of neurons in area V1. A study by van Hateren and van der Schaaf [8] showed that the filters learned by independent components analysis (ICA) [9] on natural image data match very well with the classical receptive fields of V1 simple cells. (Filters learned by sparse coding [10, 11] also similarly give responses similar to V1 simple cells.) Our work takes inspiration from the work of van Hateren and van der Schaaf, and represents a study that is done in a similar spirit, only extending the comparisons to a deeper area in the cortical hierarchy, namely visual area V2. 2 Biological comparison 2.1 Features in early visual cortex: area V1 The selectivity of neurons for oriented bar stimuli in cortical area V1 has been well documented [12, 13]. The receptive field of simple cells in V1 are localized, oriented, bandpass filters that resemble gabor filters. Several authors have proposed models that have been either formally or informally shown to replicate the gabor-like properties of V1 simple cells. Many of these algorithms, such as [10, 9, 8, 6], compute a (approximately or exactly) sparse representation of the natural stimuli data. These results are consistent with the “efficient coding hypothesis” which posits that the goal of early visual processing is to encode visual information as efficiently as possible [14]. Some hierarchical extensions of these models [15, 6, 16] are able to learn features that are more complex than simple oriented bars. For example, hierarchical sparse models of natural images have accounted for complex cell receptive fields [17], topography [18, 6], colinearity and contour coding [19]. Other models, such as [20], have also been shown to give V1 complex cell-like properties. 2.2 Features in visual cortex area V2 It remains unknown to what extent the previously described algorithms can learn higher order features that are known to be encoded further down the ventral visual pathway. In addition, the response properties of neurons in cortical areas receiving projections from area V1 (e.g., area V2) are not nearly as well documented. It is uncertain what type of stimuli cause V2 neurons to respond optimally [21]. One V2 study by [22] reported that the receptive fields in this area were similar to those in the neighboring areas V1 and V4. The authors interpreted their findings as suggestive that area V2 may serve as a place where different channels of visual information are integrated. However, quantitative accounts of responses in area V2 are few in number. In the literature, we identified two sets of quantitative data that give us a good starting point for making measurements to determine whether our algorithms may be computing similar functions as area V2. In one of these studies, Ito and Komatsu [7] investigated how V2 neurons responded to angular stimuli. They summarized each neuron’s response with a two-dimensional visualization of the stimuli set called an angle profile. By making several axial measurements within the profile, the authors were able to compute various statistics about each neuron’s selectivity for angle width, angle orientation, and for each separate line component of the angle (see Figure 1). Approximately 80% of the neurons responded to specific angle stimuli. They found neurons that were selective for only one line component of its peak angle as well as neurons selective for both line components. These neurons yielded angle profiles resembling those of Cell 2 and Cell 5 in Figure 1, respectively. In addition, several neurons exhibited a high amount of selectivity for its peak angle producing angle profiles like that of Cell 1 in Figure 1. No neurons were found that had more elongation in a diagonal axis than in the horizontal or vertical axes, indicating that neurons in V2 were not selective for angle width or orientation. Therefore, an important conclusion made from [7] was that a V2 neuron’s response to an angle stimulus is highly dependent on its responses to each individual line component of the angle. While the dependence was often observed to be simply additive, as was the case with neurons yielding profiles like those of Cells 1 and 2 in Figure 1(right), this was not always the case. 29 neurons had very small peak response areas and yielded profiles like that of Cell 1 in Figure 1(right), thus indicating a highly specific tuning to an angle stimulus. While the former responses suggest a simple linear computation of V1 neural responses, the latter responses suggest a nonlinear computation [21]. The analysis methods adopted in [7] are very useful in characterizing the response properties, and we use these methods to evaluate our own model. Another study by Hegde and Van Essen [23] studied the responses of a population of V2 neurons to complex contour and grating stimuli. They found several V2 neurons responding maximally for angles, and the distribution of peak angles for these neurons is consistent with that found by [7]. In addition, several V2 neurons responded maximally for shapes such as intersections, tri-stars, fivepoint stars, circles, and arcs of varying length. 2 Figure 1: (Images from [7]; courtesy of Ito and Komatsu) Left: Visualization of angle profiles. The upper-right and lower-left triangles contain the same stimuli. (A,B) Darkened squares correspond to stimuli that elicited a large response. The peak responses are circled. (C) The arrangement of the figure is so that one line component remains constant as one moves along any vertical or horizontal axis. (D) The angles width remains constant as one moves along a the diagonal indicated (E) The angle orientation remains constant as one moves along the diagonal indicated. After identifying the optimal stimuli for a neuron in the profile, the number of stimuli along these various axes (as in C,D,E) eliciting responses larger than 80% of the peak response measure the neuron’s tolerance to perturbations to the line components, peak angle width, and orientation, respectively. Right: Examples of 4 typical angle profiles. As before, stimuli eliciting large responses are highlighted. Cell 1 has a selective response to a stimulus, so there is no elongation along any axis. Cell 2 has one axis of elongation, indicating selectivity for one orientation. Cell 5 has two axes of elongation, and responds strongly so long as either of two edge orientations is present. Cell 4 has no clear axis of elongation. 3 Algorithm Hinton et al. [1] proposed an algorithm for learning deep belief networks, by treating each layer as a restricted Boltzmann machine (RBM) and greedily training the network one layer at a time from the bottom up [24, 1]. In general, however, RBMs tend to learn distributed, non-sparse representations. Based on results from other methods (e.g., sparse coding [10, 11], ICA [9], heavy-tailed models [6], and energy based models [2]), sparseness seems to play a key role in learning gabor-like filters. Therefore, we modify Hinton et al.’s learning algorithm to enable deep belief nets to learn sparse representations. 3.1 Sparse restricted Boltzmann machines We begin by describing the restricted Boltzmann machine (RBM), and present a modified version of it. An RBM has a set of hidden units h, a set of visible units v, and symmetric connections weights between these two layers represented by a weight matrix W. Suppose that we want to model k dimensional real-valued data using an undirected graphical model with n binary hidden units. The negative log probability of any state in the RBM is given by the following energy function:1 −log P(v, h) = E(v, h) = 1 2σ2 X i v2 i −1 σ2 X i civi + X j bjhj + X i,j viwijhj . (1) Here, σ is a parameter, hj are hidden unit variables, vi are visible unit variables. Informally, the maximum likelihood parameter estimation problem corresponds to learning wij, ci and bj so as to minimize the energy of states drawn from the data distribution, and raise the energy of states that are improbable given the data. Under this model, we can easily compute the conditional probability distributions. Holding either h or v fixed, we can sample from the other as follows: P(vi|h) = N ci + P j wijhj, σ2 , (2) P(hj|v) = logistic 1 σ2 (bj + P i wijvi) . (3) 1Due to space constraints, we present an energy function only for the case of real-valued visible units. It is also straightforward to formulate a sparse RBM with binary-valued visible units; for example, we can write the energy function as E(v, h) = −1/σ2(P i civi + P j bjhj + P i,j viwijhj) (see also [24]). 3 Here, N(·) is the gaussian density, and logistic(·) is the logistic function. For training the parameters of the model, the objective is to maximize the log-likelihood of the data. We also want hidden unit activations to be sparse; thus, we add a regularization term that penalizes a deviation of the expected activation of the hidden units from a (low) fixed level p.2 Thus, given a training set {v(1), . . . , v(m)} comprising m examples, we pose the following optimization problem: minimize{wij,ci,bj} −Pm l=1 log P h P(v(l), h(l)) + λ Pn j=1 | p −1 m Pm l=1 E[h(l) j |v(l)] |2, (4) where E[·] is the conditional expectation given the data, λ is a regularization constant, and p is a constant controlling the sparseness of the hidden units hj. Thus, our objective is the sum of a log-likelihood term and a regularization term. In principle, we can apply gradient descent to this problem; however, computing the gradient of the log-likelihood term is expensive. Fortunately, the contrastive divergence learning algorithm gives an efficient approximation to the gradient of the loglikelihood [25]. Building upon this, on each iteration we can apply the contrastive divergence update rule, followed by one step of gradient descent using the gradient of the regularization term.3 The details of our procedure are summarized in Algorithm 1. Algorithm 1 Sparse RBM learning algorithm 1. Update the parameters using contrastive divergence learning rule. More specifically, wij := wij + α(⟨vihj⟩data −⟨vihj⟩recon) ci := ci + α(⟨vi⟩data −⟨vi⟩recon) bj := bj + α(⟨bj⟩data −⟨bj⟩recon), where α is a learning rate, and ⟨·⟩recon is an expectation over the reconstruction data, estimated using one iteration of Gibbs sampling (as in Equations 2,3). 2. Update the parameters using the gradient of the regularization term. 3. Repeat Steps 1 and 2 until convergence. 3.2 Learning deep networks using sparse RBM Once a layer of the network is trained, the parameters wij, bj, ci’s are frozen and the hidden unit values given the data are inferred. These inferred values serve as the “data” used to train the next higher layer in the network. Hinton et al. [1] showed that by repeatedly applying such a procedure, one can learn a multilayered deep belief network. In some cases, this iterative “greedy” algorithm can further be shown to be optimizing a variational bound on the data likelihood, if each layer has at least as many units as the layer below (although in practice this is not necessary to arrive at a desirable solution; see [1] for a detailed discussion). In our experiments using natural images, we learn a network with two hidden layers, with each layer learned using the sparse RBM algorithm described in Section 3.1. 4 Visualization 4.1 Learning “strokes” from handwritten digits Figure 2: Bases learned from MNIST data We applied the sparse RBM algorithm to the MNIST handwritten digit dataset.4 We learned a sparse RBM with 69 visible units and 200 hidden units. The learned bases are shown in Figure 2. (Each basis corresponds to one column of the weight matrix W left-multiplied by the unwhitening matrix.) Many bases found by the algorithm roughly represent different “strokes” of which handwritten digits are comprised. This is consistent 2Less formally, this regularization ensures that the “firing rate” of the model neurons (corresponding to the latent random variables hj) are kept at a certain (fairly low) level, so that the activations of the model neurons are sparse. Similar intuition was also used in other models (e.g., see Olshausen and Field [10]). 3To increase computational efficiency, we made one additional change. Note that the regularization term is defined using a sum over the entire training set; if we use stochastic gradient descent or mini-batches (small subsets of the training data) to estimate this term, it results in biased estimates of the gradient. To ameliorate this, we used mini-batches, but in the gradient step that tries to minimize the regularization term, we update only the bias terms bj’s (which directly control the degree to which the hidden units are activated, and thus their sparsity), instead of updating all the parameters bj and wij’s. 4Downloaded from http://yann.lecun.com/exdb/mnist/. Each pixel was normalized to the unit interval, and we used PCA whitening to reduce the dimension to 69 principal components for computational efficiency. (Similar results were obtained without whitening.) 4 Figure 3: 400 first layer bases learned from the van Hateren natural image dataset, using our algorithm. Figure 4: Visualization of 200 second layer bases (model V2 receptive fields), learned from natural images. Each small group of 3-5 (arranged in a row) images shows one model V2 unit; the leftmost patch in the group is a visualization of the model V2 basis, and is obtained by taking a weighted linear combination of the first layer “V1” bases to which it is connected. The next few patches in the group show the first layer bases that have the strongest weight connection to the model V2 basis. with results obtained by applying different algorithms to learn sparse representations of this data set (e.g., [2, 5]). 4.2 Learning from natural images We also applied the algorithm to a training set a set of 14-by-14 natural image patches, taken from a dataset compiled by van Hateren.5 We learned a sparse RBM model with 196 visible units and 400 hidden units. The learned bases are shown in Figure 3; they are oriented, gabor-like bases and resemble the receptive fields of V1 simple cells.6 4.3 Learning a two-layer model of natural images using sparse RBMs We further learned a two-layer network by stacking one sparse RBM on top of another (see Section 3.2 for details.)7 After learning, the second layer weights were quite sparse—most of the weights were very small, and only a few were either highly positive or highly negative. Positive 5The images were obtained from http://hlab.phys.rug.nl/imlib/index.html. We used 100,000 14-by-14 image patches randomly sampled from an ensemble of 2000 images; each subset of 200 patches was used as a mini-batch. 6Most other authors’ experiments to date using regular (non-sparse) RBMs, when trained on such data, seem to have learned relatively diffuse, unlocalized bases (ones that do not represent oriented edge filters). While sensitive to the parameter settings and requiring a long training time, we found that it is possible in some cases to get a regular RBM to learn oriented edge filter bases as well. But in our experiments, even in these cases we found that repeating this process to build a two layer deep belief net (see Section 4.3) did not encode a significant number of corners/angles, unlike one trained using the sparse RBM; therefore, it showed significantly worse match to the Ito & Komatsu statistics. For example, the fraction of model V2 neurons that respond strongly to a pair of edges near right angles (formally, have peak angle in the range 60-120 degrees) was 2% for the regular RBM, whereas it was 17% for the sparse RBM (and Ito & Komatsu reported 22%). See Section 5.1 for more details. 7For the results reported in this paper, we trained the second layer sparse RBM with real-valued visible units; however, the results were very similar when we trained the second layer sparse RBM with binary-valued visible units (except that the second layer weights became less sparse). 5 Figure 5: Top: Visualization of four learned model V2 neurons. (Visualization in each row of four or five patches follows format in Figure 4.) Bottom: Angle stimulus response profile for model V2 neurons in the top row. The 36*36 grid of stimuli follows [7], in which the orientation of two lines are varied to form different angles. As in Figure 1, darkened patches represent stimuli to which the model V2 neuron responds strongly; also, a small black square indicates the overall peak response. weights represent excitatory connections between model V1 and model V2 units, whereas negative elements represent inhibitory connections. By visualizing the second layer bases as shown in Figure 4, we observed bases that encoded co-linear first layer bases as well as edge junctions. This shows that by extending the sparse RBM to two layers and using greedy learning, the model is able to learn bases that encode contours, angles, and junctions of edges. 5 Evaluation experiments We now more quantitatively compare the algorithm’s learned responses to biological measurements.8 5.1 Method: Ito-Komatsu paper protocol We now describe the procedure we used to compare our model with the experimental data in [7]. We generated a stimulus set consisting of the same set of angles (pairs of edges) as [7]. To identify the “center” of each model neuron’s receptive field, we translate all stimuli densely over the 14x14 input image patch, and identify the position at which the maximum response is elicited. All measures are then taken with all angle stimuli centered at this position.9 Using these stimuli, we compute the hidden unit probabilities from our model V1 and V2 neurons. In other words, for each stimulus we compute the first hidden layer activation probabilities, then feed this probability as data to the second hidden layer and compute the activation probabilities again in the same manner. Following a protocol similar to [7], we also eliminate from consideration the model neurons that do not respond strongly to corners and edges.10 Some representative results are shown in Figure 5. (The four angle profiles shown are fairly typical of those obtained in our experiments.) We see that all the V2 bases in Figure 5 have maximal response when its strongest V1-basis components are aligned with the stimulus. Thus, some of these bases do indeed seem to encode edge junctions or crossings. We also compute similar summary statistics as [7] (described in Figure 1(C,D,E)), that more quantitatively measure the distribution of V2 or model V2 responses to the different angle stimuli. Figure 6 plots the responses of our model, together with V2 data taken from [7]. Along many dimensions, the results from our model match that from the Macaque V2 fairly well. 8The results we report below were very insensitive to the choices of σ and λ. We set σ to 0.4 and 0.05 for the first and second layers (chosen to be on the same scale as the standard deviation of the data and the first-layer activations), and λ = 1/p in each layer. We used p = 0.02 and 0.05 for the first and second layers. 9Other details: The stimulus set is created by generating a binary-mask image, that is then scaled to normalize contrast. To determine this scaling constant, we used single bar images by translating and rotating to all possible positions, and fixed the constant such that the top 0.5% (over all translations and rotations) of the stimuli activate the model V1 cells above 0.5. This normalization step corrects for the RBM having been trained on a data distribution (natural images) that had very different contrast ranges than our test stimulus set. 10In detail, we generated a set of random low-frequency stimulus, by generating small random KxK (K=2,3,4) images with each pixel drawn from a standard normal distribution, and rescaled the image using bicubic interpolation to 14x14 patches. These stimulus are scaled such that about 5% of the V2 bases fires maximally to these random stimuli. We then exclude the V2 bases that are maximally activated to these random stimuli from the subsequent analysis. 6 15 45 75 105 135 165 0 0.1 0.2 0.3 0.4 0.5 peak angles sparse DBN Ito & Komatsu 1 2 3 4 5 6 7 8 9 1011 0 0.05 0.1 0.15 0.2 primary line axis sparse DBN Ito & Komatsu 1 2 3 4 5 6 7 8 9 1011 0 0.1 0.2 0.3 0.4 0.5 secondary line axis sparse DBN Ito & Komatsu 1 2 3 4 5 6 7 8 9 1011 0 0.2 0.4 0.6 0.8 angle width axis sparse DBN Ito & Komatsu 1 2 3 4 5 6 7 8 9 1011 0 0.2 0.4 0.6 0.8 1 angle orientation axis sparse DBN Ito & Komatsu Figure 6: Images show distributions over stimulus response statistics (averaged over 10 trials) from our algorithm (blue) and in data taken from [7] (green). The five figures show respectively (i) the distribution over peak angle response (ranging from 0 to 180 degrees; each bin represents a range of 30 degrees), (ii) distribution over tolerance to primary line component (Figure 1C, in dominant vertical or horizontal direction), (iii) distribution over tolerance to secondary line component (Figure 1C, in non-dominant direction), (iv) tolerance to angle width (Figure 1D), (v) tolerance to angle orientation (Figure 1E). See Figure 1 caption, and [7], for details. Figure 7: Visualization of a number of model V2 neurons that maximally respond to various complex stimuli. Each row of seven images represents one V2 basis. In each row, the leftmost image shows a linear combination of the top three weighted V1 components that comprise the V2 basis; the next three images show the top three optimal stiimuli; and the last three images show the top three weighted V1 bases. The V2 bases shown in the figures maximally respond to acute angles (left), obtuse angles (middle), and tri-stars and junctions (right). 5.2 Complex shaped model V2 neurons Our second experiment represents a comparison to a subset of the results described in Hegde and van Essen [23]. We generated a stimulus set comprising some [23]’s complex shaped stimuli: angles, single bars, tri-stars (three line segments that meet at a point), and arcs/circles, and measured the response of the second layer of our sparse RBM model to these stimuli.11 We observe that many V2 bases are activated mainly by one of these different stimulus classes. For example, some model V2 neurons activate maximally to single bars; some maximally activate to (acute or obtuse) angles; and others to tri-stars (see Figure 7). Further, the number of V2 bases that are maximally activated by acute angles is significantly larger than the number of obtuse angles, and the number of V2 bases that respond maximally to the tri-stars was much smaller than both preceding cases. This is also consistent with the results described in [23]. 6 Conclusions We presented a sparse variant of the deep belief network model. When trained on natural images, this model learns local, oriented, edge filters in the first layer. More interestingly, the second layer captures a variety of both colinear (“contour”) features as well as corners and junctions, that in a quantitative comparison to measurements of V2 taken by Ito & Komatsu, appeared to give responses that were similar along several dimensions. This by no means indicates that the cortex is a sparse RBM, but perhaps is more suggestive of contours, corners and junctions being fundamental to the statistics of natural images.12 Nonetheless, we believe that these results also suggest that sparse 11All the stimuli were 14-by-14 pixel image patches. We applied the protocol described in Section 5.1 to the stimulus data, to compute the model V1 and V2 responses. 12In preliminary experiments, we also found that when these ideas are applied to self-taught learning [26] (in which one may use unlabeled data to identify features that are then useful for some supervised learning task), using a two-layer sparse RBM usually results in significantly better features for object recognition than using only a one-layer network. 7 deep learning algorithms, such as our sparse variant of deep belief nets, hold promise for modeling higher-order features such as might be computed in the ventral visual pathway in the cortex. Acknowledgments We give warm thanks to Minami Ito, Geoffrey Hinton, Chris Williams, Rajat Raina, Narut Sereewattanawoot, and Austin Shoemaker for helpful discussions. Support from the Office of Naval Research under MURI N000140710747 is gratefully acknowledged. References [1] G. E. Hinton, S. Osindero, and Y.-W. Teh. A fast learning algorithm for deep belief nets. Neural Computation, 18(7):1527–1554, 2006. [2] M. Ranzato, C. Poultney, S. Chopra, and Y. LeCun. Efficient learning of sparse representations with an energy-based model. In NIPS, 2006. [3] Y. Bengio, P. Lamblin, D. Popovici, and H. Larochelle. Greedy layer-wise training of deep networks. In NIPS, 2006. [4] 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 ICML, 2007. [5] G. E. Hinton, S. Osindero, and K. Bao. Learning causally linked MRFs. In AISTATS, 2005. [6] S. Osindero, M. Welling, and G. E. Hinton. Topographic product models applied to natural scene statistics. Neural Computation, 18:381–344, 2006. [7] M. Ito and H. Komatsu. Representation of angles embedded within contour stimuli in area v2 of macaque monkeys. The Journal of Neuroscience, 24(13):3313–3324, 2004. [8] J. H. van Hateren and A. van der Schaaf. Independent component filters of natural images compared with simple cells in primary visual cortex. Proc.R.Soc.Lond. B, 265:359–366, 1998. [9] A. J. Bell and T. J. Sejnowski. The ‘independent components’ of natural scenes are edge filters. Vision Research, 37(23):3327–3338, 1997. [10] B. A. Olshausen and D. J. Field. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature, 381:607–609, 1996. [11] H. Lee, , A. Battle, R. Raina, and A. Y. Ng. Efficient sparse coding algorithms. In NIPS, 2007. [12] D. Hubel and T. Wiesel. Receptive fields and functional architecture of monkey striate cortex. 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Gutmann, and P. O. Hoyer. Statistical model of natural stimuli predicts edge-like pooling of spatial frequency channels in v2. BMC Neuroscience, 6:12, 2005. [20] L. Wiskott and T. Sejnowski. Slow feature analysis: Unsupervised learning of invariances. Neural Computation, 14(4):715–770, 2002. [21] G. Boynton and J. Hegde. Visual cortex: The continuing puzzle of area v2. Current Biology, 14(13):R523–R524, 2004. [22] J. B. Levitt, D. C. Kiper, and J. A. Movshon. Receptive fields and functional architecture of macaque v2. Journal of Neurophysiology, 71(6):2517–2542, 1994. [23] J. Hegde and D.C. Van Essen. Selectivity for complex shapes in primate visual area v2. Journal of Neuroscience, 20:RC61–66, 2000. [24] G. E. Hinton and R. R. Salakhutdinov. Reducing the dimensionality of data with neural networks. Science, 313(5786):504–507, 2006. [25] G. E. Hinton. Training products of experts by minimizing contrastive divergence. Neural Computation, 14:1771–1800, 2002. [26] R. Raina, A. Battle, H. 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Loop Series and Bethe Variational Bounds in Attractive Graphical Models Erik B. Sudderth and Martin J. Wainwright Electrical Engineering & Computer Science, University of California, Berkeley sudderth@eecs.berkeley.edu, wainwrig@eecs.berkeley.edu Alan S. Willsky Electrical Engineering & Computer Science, Massachusetts Institute of Technology willsky@mit.edu Abstract Variational methods are frequently used to approximate or bound the partition or likelihood function of a Markov random field. Methods based on mean field theory are guaranteed to provide lower bounds, whereas certain types of convex relaxations provide upper bounds. In general, loopy belief propagation (BP) provides often accurate approximations, but not bounds. We prove that for a class of attractive binary models, the so–called Bethe approximation associated with any fixed point of loopy BP always lower bounds the true likelihood. Empirically, this bound is much tighter than the naive mean field bound, and requires no further work than running BP. We establish these lower bounds using a loop series expansion due to Chertkov and Chernyak, which we show can be derived as a consequence of the tree reparameterization characterization of BP fixed points. 1 Introduction Graphical models are widely used in many areas, including statistical machine learning, computer vision, bioinformatics, and communications. Such applications typically require computationally efficient methods for (approximately) solving various problems, including computing marginal distributions and likelihood functions. The variational framework provides a suite of candidate methods, including mean field approximations [3, 9], the sum–product or belief propagation (BP) algorithm [11, 14], Kikuchi and cluster variational methods [23], and related convex relaxations [21]. The likelihood or partition function of an undirected graphical model is of fundamental interest in many contexts, including parameter estimation, error bounds in hypothesis testing, and combinatorial enumeration. In rough terms, particular variational methods can be understood as solving optimization problems whose optima approximate the log partition function. For mean field methods, this optimal value is desirably guaranteed to lower bound the true likelihood [9]. For other methods, including the Bethe variational problem underlying loopy BP [23], optima may either over–estimate or under–estimate the truth. Although “convexified” relaxations of the Bethe problem yield upper bounds [21], to date the best known lower bounds on the partition function are based on mean field theory. Recent work has studied loop series expansions [2, 4] of the partition function, which generate better approximations but not, in general, bounds. Several existing theoretical results show that loopy BP, and the corresponding Bethe approximation, have desirable properties for graphical models with long cycles [15] or sufficiently weak dependencies [6, 7, 12, 19]. However, these results do not explain the excellent empirical performance of BP in many graphs with short cycles, like the nearest–neighbor grids arising in spatial statistics and low–level vision [3, 18, 22]. Such models often encode “smoothness” priors, and thus have attractive interactions which encourage connected variables to share common values. The first main contribution of this paper is to demonstrate a family of attractive models for which the Bethe variational method always yields lower bounds on the true likelihood. Although we focus on models with binary variables (but arbitrary order of interactions), we suspect that some ideas are more generally applicable. For such models, these lower bounds are easily computed from any fixed point of loopy BP, and empirically improve substantially on naive mean field bounds. 1 Our second main contribution lies in the route used to establish the Bethe lower bounds. In particular, Sec. 3 uses the reparameterization characterization of BP fixed points [20] to provide a simple derivation for the loop series expansion of Chertkov and Chernyak [2]. The Bethe approximation is the first term in this representation of the true partition function. Sec. 4 then identifies attractive models for which all terms in this expansion are positive, thus establishing the Bethe lower bound. We conclude with empirical results demonstrating the accuracy of this bound, and discuss implications for future analysis and applications of loopy BP. 2 Undirected Graphical Models Given an undirected graph G = (V, E), with edges (s, t) ∈E connecting n vertices s ∈V , a graphical model associates each node with a random variable Xs taking values xs ∈X. For pairwise Markov random fields (MRFs) as in Fig. 1, the joint distribution of x := {xs | s ∈V } is specified via a normalized product of local compatibility functions: p(x) = 1 Z(ψ) Y s∈V ψs(xs) Y (s,t)∈E ψst(xs, xt) (1) The partition function Z(ψ) := P x∈X n Q s ψs(xs) Q (s,t) ψst(xs, xt), whose value depends on the compatibilities ψ, is defined so that p(x) is properly normalized. We also consider distributions defined by hypergraphs G = (V, C), where each hyperedge c ∈C connects some subset of the vertices (c ⊂V ). Letting xc := {xs | s ∈c}, the corresponding joint distribution equals p(x) = 1 Z(ψ) Y s∈V ψs(xs) Y c∈C ψc(xc) (2) where as before Z(ψ) = P x∈X n Q s ψs(xs) Q c ψc(xc). Such higher–order random fields are conveniently described by the bipartite factor graphs [11] of Fig. 2. In statistical physics, the partition function arises in the study of how physical systems respond to changes in external stimuli or temperature [23]. Alternatively, when compatibility functions are parameterized by exponential families [20], log Z(ψ) is the family’s cumulant generating function, and thus intrinsically related to the model’s marginal statistics. For directed Bayesian networks (which can be factored as in eq. (2)), Z(ψ) is the marginal likelihood of observed data, and plays a central role in learning and model selection [9]. However, for general graphs coupling discrete random variables, the cost of exactly evaluating Z(ψ) grows exponentially with n [8]. Computationally tractable families of bounds on the true partition function are thus of great practical interest. 2.1 Attractive Discrete Random Fields In this paper, we focus on binary random vectors x ∈{0, 1}n. We say that a pairwise MRF, with compatibility functions ψst :{0, 1}2 →R+, has attractive interactions if ψst(0, 0) ψst(1, 1) ≥ψst(0, 1) ψst(1, 0) (3) for each edge (s, t) ∈E. Intuitively, this condition requires all potentials to place greater weight on configurations where neighboring variables take the same value. Our later analysis is based on pairwise marginal distributions τst(xs, xt), which we parameterize as follows: τst(xs, xt) = 1 −τs −τt + τst τt −τst τs −τst τst τs := Eτst[Xs] τst := Eτst[XsXt] (4) We let Eτst[·] denote expectation with respect to τst(xs, xt), so that τst is the probability that Xs = Xt = 1. This normalized matrix is attractive, satisfying eq. (3), if and only if τst ≥τsτt. For binary variables, the pairwise MRF of eq. (1) provides one representation of a general, inhomogeneous Ising model. In the statistical physics literature, Ising models are typically expressed by coupling random spins zs ∈{−1, +1} with symmetric potentials log ψst(zs, zt) = θstzszt. The attractiveness condition of eq. (3) then becomes θst ≥0, and the resulting model has ferromagnetic interactions. Furthermore, pairwise MRFs satisfy the regularity condition of [10], and thus allow tractable MAP estimation via graph cuts [5], if and only if they are attractive. Even for attractive models, however, calculation of the partition function in non–planar graphs is #P–complete [8]. To define families of higher–order attractive potentials, we first consider a probability distribution τc(xc) on k = |c| binary variables. Generalizing eq. (4), we parameterize such distributions by the 2 following collection of 2k −1 mean parameters: τa := Eτc Y s∈a Xs ∅̸= a ⊆c (5) For example, τstu(xs, xt, xu) would be parameterized by {τs, τt, τu, τst, τsu, τtu, τstu}. For any subset a ⊆c, we then define the following central moment statistic: κa := Eτc Y s∈a (Xs −τs) ∅̸= a ⊆c (6) Note that κs = 0, while κst = Covτ(Xs, Xt) = τst −τsτt. The third–order central moment then equals the cumulant κstu = τstu −τstτu −τsuτt −τtuτs + 2τsτtτu. Given these definitions, we say that a probability distribution τc(xc) is attractive if the central moments associated with all subsets a ⊆c of binary variables are non–negative (κa ≥0). Similarly, a compatibility function ψc(xc) is attractive if the probability distribution attained by normalizing its values has non–negative central moments. For example, the following potential is easily shown to satisfy this condition for all degrees k = |c|, and any scalar θc > 0: log ψc(x1, . . . , xk) = θc x1 = x2 = · · · = xk −θc otherwise (7) 2.2 Belief Propagation and the Bethe Variational Principle Many applications of graphical models require estimates of the posterior marginal distributions of individual variables τs(xs) or factors τc(xc). Loopy belief propagation (BP) approximates these marginals via a series of messages passed among nodes of the graphical model [14, 23]. Let Γ(s) denote the set of factors which depend on Xs, or equivalently the neighbors of node s in the corresponding factor graph. The BP algorithm then iterates the following message updates: ¯msc(xs) ←ψs(xs) Y d∈Γ(s)\c mds(xs) mcs(xs) ← X xc\s ψc(xc) Y t∈c\s ¯mtc(xt) (8) The left–hand expression updates the message ¯msc(xs) passed from variable node s to factor c. New outgoing messages mcs(xs) from factor c to each s ∈c are then determined by marginalizing the incoming messages from other nodes. At any iteration, appropriately normalized products of these messages define estimates of the desired marginals: τs(xs) ∝ψs(xs) Y c∈Γ(s) mcs(xs) τc(xc) ∝ψc(xc) Y t∈c ¯mtc(xt) (9) In tree–structured graphs, BP defines a dynamic programming recursion which converges to the exact marginals after finitely many iterations [11, 14]. In graphs with cycles, however, convergence is not guaranteed, and pseudo–marginals computed via eq. (9) are (often good) approximations. A wide range of inference algorithms can be derived via variational approximations [9] to the true partition function. Loopy BP is implicitly associated with the following Bethe approximation: log Zβ(ψ; τ) = X s∈V X xs τs(xs) log ψs(xs) + X c∈C X xc τc(xc) log ψc(xc) − X s∈V X xs τs(xs) log τs(xs) − X c∈C X xc τc(xc) log τc(xc) Q t∈c τt(xt) (10) Fixed points of loopy BP correspond to stationary points of this Bethe approximation [23], subject to the local marginalization constraints P xc\s τc(xc) = τs(xs). 3 Reparameterization and Loop Series Expansions As discussed in Sec. 2.2, any BP fixed point is in one–to–one correspondence with a set {τs, τc} of pseudo–marginals associated with each of the graph’s nodes s ∈V and factors c ∈C. These pseudo–marginals then lead to an alternative parameterization [20] of the factor graph of eq. (2): p(x) = 1 Z(τ) Y s∈V τs(xs) Y c∈C τc(xc) Q t∈c τt(xt) (11) For pairwise MRFs, the reparameterized compatibility functions equal τst(xs, xt)/τs(xs)τt(xt). The BP algorithm effectively searches for reparameterizations which are tree–consistent, so that 3 τc(xc) is the exact marginal distribution of Xc for any tree (or forest) embedded in the original graph [20]. In later sections, we take expectations with respect to τc(xc) of functions f(xc) defined over individual factors. Although these pseudo–marginals will in general not equal the true marginals pc(xc), BP fixed points ensure local consistency so that Eτc[f(Xc)] is well–defined. Using eq. (10), it is easily shown that the Bethe approximation Zβ(τ; τ) = 1 for any joint distribution defined by reparameterized potentials as in eq. (11). For simplicity, the remainder of this paper focuses on reparameterized models of this form, and analyzes properties of the corresponding exact partition function Z(τ). The resulting expansions and bounds are then related to the original MRF’s partition function via the positive constant Z(ψ)/Z(τ) = Zβ(ψ; τ) of eq. (10). Recently, Chertkov and Chernyak proposed a finite loop series expansion [2] of the partition function, whose first term coincides with the Bethe approximation. They provide two derivations: one applies a trigonometric identity to Fourier representations of binary variables, while the second employs a saddle point approximation obtained via an auxiliary field of complex variables. The gauge transformations underlying these derivations are a type of reparameterization, but their form is complicated by auxiliary variables and extraneous degrees of freedom. In this section, we show that the fixed point characterization of eq. (11) leads to a more direct, and arguably simpler, derivation. 3.1 Pairwise Loop Series Expansions We begin by developing a loop series expansion for pairwise MRFs. Given an undirected graph G = (V, E), and some subset F ⊆E of the graph’s edges, let ds(F) denote the degree (number of neighbors) of node s in the subgraph induced by F. As illustrated in Fig. 1, any subset F for which all nodes s ∈V have degree ds(F) ̸= 1 defines a generalized loop [2]. The partition function for any binary, pairwise MRF can then be expanded via an associated set of loop corrections. Proposition 1. Consider a pairwise MRF defined on an undirected G = (V, E), with reparameterized potentials as in eq. (11). The associated partition function then equals Z(τ) = 1 + X ∅̸=F ⊆E βF Y s∈V Eτs h (Xs −τs)ds(F )i βF := Y (s,t)∈F βst (12) βst := τst −τsτt τs(1 −τs)τt(1 −τt) = Covτst(Xs, Xt) Varτs(Xs) Varτt(Xt) (13) where only generalized loops F lead to non–zero terms in the sum of eq. (12), and Eτs (Xs −τs)d = τs(1 −τs) (1 −τs)d−1 + (−1)d (τs)d−1 (14) are central moments of the binary variables at individual nodes. Proof. To establish the expansion of eq. (12), we exploit the following polynomial representation of reparameterized pairwise compatibility functions: τst(xs, xt) τs(xs)τt(xt) = 1 + βst(xs −τs)(xt −τt) (15) As verified in [17], this expression is satisfied for any (xs, xt) ∈{0, 1}2 if βst is defined as in eq. (13). For attractive models satisfying eq. (3), βst ≥0 for all edges. Using E˜τ[·] to denote expectation with respect to the fully factorized distribution ˜τ(x) = Q s τs(xs), we then have Z(τ) = X x∈{0,1}n Y s∈V τs(xs) Y (s,t)∈E τst(xs, xt) τs(xs)τt(xt) = E˜τ Y (s,t)∈E τst(Xs, Xt) τs(Xs)τt(Xt) = E˜τ Y (s,t)∈E 1 + βst(Xs −τs)(Xt −τt) (16) Expanding this polynomial via the expectation operator’s linearity, we recover one term for each non–empty subset F ⊆E of the graph’s edges: Z(τ) = 1 + X ∅̸=F ⊆E E˜τ Y (s,t)∈F βst(Xs −τs)(Xt −τt) (17) The expression in eq. (12) then follows from the independence structure of ˜τ(x), and standard formulas for the moments of Bernoulli random variables. To evaluate these terms, note that if ds(F) = 1, it follows that Eτs[Xs −τs] = 0. There is thus one loop correction for each generalized loop F, in which all connected nodes have degree at least two. 4 Figure 1: A pairwise MRF coupling ten binary variables (left), and the nine generalized loops in its loop series expansion (right). For attractive potentials, two of the generalized loops may have negative signs (second & third from right), while the core graph of Thm. 1 contains eight variables (far right). Figure 1 illustrates the set of generalized loops associated with a particular pairwise MRF. These loops effectively define corrections to the Bethe estimate Z(τ) ≈1 of the partition function for reparameterized models. Tree–structured graphs do not contain any non–trivial generalized loops, and the Bethe variational approximation is thus exact. The loop expansion formulas of [2] can be precisely recovered by transforming binary variables to a spin representation, and refactoring terms from the denominator of edge weights βst to adjacent vertices. Explicit computation of these loop corrections is in general intractable; for example, fully connected graphs with n ≥5 nodes have more than 2n generalized loops. In some cases, accounting for a small set of significant loop corrections may lead to improved approximations to Z(ψ) [4], or more accurate belief estimates for LDPC codes [1]. We instead use the series expansion of Prop. 1 to establish analytic properties of BP fixed points. 3.2 Factor Graph Loop Series Expansions We now extend the loop series expansion to higher–order MRFs defined on hypergraphs G = (V, C). Let E = {(s, c) | c ∈C, s ∈c} denote the set of edges in the factor graph representation of this MRF. As illustrated in Fig. 2, we define a generalized loop to be a subset F ⊆E of edges such that all connected factor and variable nodes have degree at least two. Proposition 2. Consider any factor graph G = (V, C) with reparameterized potentials as in eq. (11), and associated edges E. The partition function then equals Z(τ) = 1 + X ∅̸=F ⊆E βF Y s∈V Eτs h (Xs −τs)ds(F )i βF := Y c∈C βac(F ) (18) βa := κa Q t∈a τt(1 −τt) = Eτc Q s∈a(Xs −τs) Q t∈a Varτt(Xt) (19) where ac(F) := {s ∈c | (s, c) ∈F} denotes the subset of variables linked to factor node c by the edges in F. Only generalized loops F lead to non–zero terms in the sum of eq. (18). Proof. As before, we employ a polynomial representation of the reparameterized factors in eq. (11): τc(xc) Q t∈c τt(xt) = 1 + X a⊆c,|a|≥2 βa Y s∈a (xs −τs) (20) For factor graphs with attractive reparameterized potentials, the constant βa ≥0 for all a ⊆c. Note that this representation, which is derived in [17], reduces to that of eq. (15) when c = {s, t}. Single–variable subsets are excluded in eq. (20) because κs = Eτs[Xs −τs] = 0. Applying eq. (20) as in our earlier derivation for pairwise MRFs (see eq. (16)), we may express the partition function of the reparameterized factor graph as follows: Z(τ) = E˜τ Y c∈C τc(Xc) Q t∈c τt(Xt) = E˜τ Y c∈C 1 + X ∅̸=a⊆c βa Y s∈a (Xs −τs) (21) Note that βa = 0 for any subset where |a| = 1. There is then a one–to–one correspondence between variable node subsets a ⊆c, and subsets {(s, c) | s ∈a} of the factor graph’s edges E. Expanding this expression by F ⊆E, it follows that each factor c ∈C contributes a term corresponding to the chosen subset ac(F) of its edges: Z(τ) = 1 + X ∅̸=F ⊆E E˜τ Y c∈C βac(F ) Y s∈ac(F ) (Xs −τs) (22) Note that β∅= 1. Equation (18) then follows from the independence properties of ˜τ(x). For a term in this loop series to be non–zero, there must be no degree one variables, since Eτs[Xs −τs] = 0. In addition, the definition of βa implies that there can be no degree one factor nodes. 5 Figure 2: A factor graph (left) with three binary variables (circles) and four factor nodes (squares), and the thirteen generalized loops in its loop series expansion (right, along with the full graph). 4 Lower Bounds in Attractive Binary Models The Bethe approximation underlying loopy BP differs from mean field methods [9], which lower bound the true log partition function Z(ψ), in two key ways. First, while the Bethe entropy (second line of eq. (10)) is exact for tree–structured graphs, it approximates (rather than bounds) the true entropy in graphs with cycles. Second, the marginalization condition imposed by loopy BP relaxes (rather than strengthens) the global constraints characterizing valid distributions [21]. Nevertheless, we now show that for a large family of attractive graphical models, the Bethe approximation Zβ(ψ; τ) of eq. (10) lower bounds Z(ψ). In contrast with mean field methods, these bounds hold only at appropriate BP fixed points, not for arbitrarily chosen pseudo–marginals τc(xc). 4.1 Partition Function Bounds for Pairwise Graphical Models Consider a pairwise MRF defined on G = (V, E), as in eq. (1). Let VH ⊆V denote the set of nodes which either belong to some cycle in G, or lie on a path (sequence of edges) connecting two cycles. We then define the core graph H = (VH, EH) as the node–induced subgraph obtained by discarding edges from nodes outside VH, so that EH = {(s, t) ∈E | s, t ∈VH}. The unique core graph H underlying any graph G can be efficiently constructed by iteratively pruning degree one nodes, or leaves, until all remaining nodes have two or more neighbors. The following theorem identifies conditions under which all terms in the loop series expansion must be non–negative. Theorem 1. Let H = (VH, EH) be the core graph for a pairwise binary MRF, with attractive potentials satisfying eq. (3). Consider any BP fixed point for which all nodes s ∈VH with three or more neighbors in H have marginals τs ≤1 2 (or equivalently, τs ≥1 2). The corresponding Bethe variational approximation Zβ(ψ; τ) then lower bounds the true partition function Z(ψ). Proof. It is sufficient to show that Z(τ) ≥1 for any reparameterized pairwise MRF, as in eq. (11). From eq. (9), note that loopy BP estimates the pseudo–marginal τst(xs, xt) via the product of ψst(xs, xt) with message functions of single variables. For this reason, attractive pairwise compatibilities always lead to BP fixed points with attractive pseudo–marginals satisfying τst ≥τsτt. Consider the pairwise loop series expansion of eq. (12). As shown by eq. (13), attractive models lead to edge weights βst ≥0. It is thus sufficient to show that Q s Eτs (Xs −τs)ds(F ) ≥0 for each generalized loop F ⊆E. Suppose first that the graph has a single cycle, and thus exactly one non–zero generalized loop F. Because all connected nodes in this cycle have degree two, the bound follows because Eτs (Xs −τs)2 ≥0. More generally, we clearly have Z(τ) ≥1 in graphs where every generalized loop F associates an even number of neighbors ds(F) with each node. Focusing on generalized loops containing nodes with odd degree d ≥3, eq. (14) implies that Eτs (Xs −τs)d ≥0 for marginals satisfying 1 −τs ≥τs. For BP fixed points in which τs ≤1 2 for all nodes, we thus have Z(τ) ≥1. In particular, the symmetric fixed point τs = 1 2 leads to uniformly positive generalized loop corrections. More generally, the marginals of nodes s for which ds(F) ≤2 for every generalized loop F do not influence the expansion’s positivity. Theorem 1 discards these nodes by examining the topology of the core graph H (see Fig. 1 for an example). For fixed points where τs ≥1 2 for all nodes, we rewrite the polynomial in the loop expansion of eq. (15) as (1 + βst(τs −xs)(τt −xt)), and employ an analogous line of reasoning. In addition to establishing Thm. 1, our arguments show that the true partition function monotonically increases as additional edges, with attractive reparameterized potentials as in eq. (11), are added to a graph with fixed pseudo–marginals τs ≤ 1 2. For such models, the accumulation of particular loop corrections, as explored by [4], produces a sequence of increasingly tight bounds on Z(ψ). In addition, we note that the conditions required by Thm. 1 are similar to those underlying classical 6 correlation inequalities [16] from the statistical physics literature. Indeed, the Griffiths–Kelly– Sherman (GKS) inequality leads to an alternative proof in cases where τs = 1 2 for all nodes. For attractive Ising models in which some nodes have marginals τs > 1 2 and others τt < 1 2, the loop series expansion may contain negative terms. For small graphs like that in Fig. 1, it is possible to use upper bounds on the edge weights βst, which follow from τst ≤min(τs, τt), to cancel negative loop corrections with larger positive terms. As confirmed by the empirical results in Sec. 4.3, the lower bound Z(ψ) ≥Zβ(ψ; τ) thus continues to hold for many (perhaps all) attractive Ising models with less homogeneous marginal biases. 4.2 Partition Function Bounds for Factor Graphs Given a factor graph G = (V, C) relating binary variables, define a core graph H = (VH, CH) by excluding variable and factor nodes which are not members of any generalized loops. As in Sec. 2.2, let Γ(s) denote the set of factor nodes neighboring variable node s in the core graph H. Theorem 2. Let H = (VH, CH) be the core graph for a binary factor graph, and consider an attractive BP fixed point for which one of the following conditions holds: (i) τs ≤1 2 for all nodes s ∈VH with |Γ(s)| ≥3, and κa ≥0 for all a ⊆c, c ∈CH. (ii) τs ≥1 2 for all nodes s ∈VH with |Γ(s)| ≥3, and (−1)|a|κa ≥0 for all a ⊆c, c ∈CH. The Bethe approximation Zβ(ψ; τ) then lower bounds the true partition function Z(ψ). For the case where τs ≤ 1 2, the proof of this theorem is a straightforward generalization of the arguments in Sec. 4.1. When τs ≥1 2, we replace all (xs −τs) terms by (τs −xs) in the expansion of eq. (20), and again recover uniformly positive loop corrections. For any given BP fixed point, the conditions of Thm. 2 are easy to verify. For factor graphs, it is more challenging to determine which compatibility functions ψc(xc) necessarily lead to attractive fixed points. For symmetric potentials as in eq. (7), however, one can show that the conditions on κa, a ⊆c are necessarily satisfied whenever all variable nodes s ∈VH have the same bias. 4.3 Empirical Comparison of Mean Field and Bethe Lower Bounds In this section, we compare the accuracy of the Bethe variational bounds established by Thm. 1 to those produced by a naive, fully factored mean field approximation [3, 9]. Using the spin representation zs ∈{−1, +1}, we examine Ising models with attractive pairwise potentials log ψst(zs, zt) = θstzszt of varying strengths θst ≥0. We first examine a 2D torus, with potentials of uniform strength θst = ¯θ and no local observations. For such MRFs, the exact partition function may be computed via Onsager’s classical eigenvector method [13]. As shown in Fig. 3(a), for moderate ¯θ the Bethe bound Zβ(ψ; τ) is substantially tighter than mean field. For large ¯θ, only two states (all spins “up” or “down”) have significant probability, so that Z(ψ) ≈2 exp(¯θ|E|). In this regime, loopy BP exhibits “symmetry breaking” [6], and converges to one of these states at random with corresponding bound Zβ(ψ; τ) ≈exp(¯θ|E|). As verified in Fig. 3(a), as ¯θ →∞the difference log Z(ψ) −log Zβ(ψ; τ) ≈log 2 ≈0.69 thus remains bounded. We also consider a set of random 10 × 10 nearest–neighbor grids, with inhomogeneous pairwise potentials sampled according to |θst| ∼N 0, ¯θ 2 , and observation potentials log ψs(zs) = θszs, |θs| ∼N 0, 0.12 . For each candidate ¯θ, we sample 100 random MRFs, and plot the average difference log Zβ(ψ; τ) −log Z(ψ) between the true partition function and the BP (or mean field) fixed point reached from a random initialization. Fig. 3(b) first considers MRFs where θs > 0 for all nodes, so that the conditions of Thm. 1 are satisfied for all BP fixed points. For these models, the Bethe bound is extremely accurate. In Fig. 3(c), we also consider MRFs where the observation potentials θs are of mixed signs. Although this sometimes leads to BP fixed points with negative associated loop corrections, the Bethe variational approximation nevertheless always lower bounds the true partition function in these examples. We hypothesize that this bound in fact holds for all attractive, binary pairwise MRFs, regardless of the observation potentials. 5 Discussion We have provided an alternative, direct derivation of the partition function’s loop series expansion, based on the reparameterization characterization of BP fixed points. We use this expansion to prove that the Bethe approximation lower bounds the true partition function in a family of binary attractive 7 0 0.2 0.4 0.6 0.8 1 −70 −60 −50 −40 −30 −20 −10 0 10 Edge Strength Difference from True Log Partition Belief Propagation Mean Field 0 0.2 0.4 0.6 0.8 1 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 Edge Strength Difference from True Log Partition Belief Propagation Mean Field 0 0.2 0.4 0.6 0.8 1 −8 −6 −4 −2 0 2 Edge Strength Difference from True Log Partition Belief Propagation Mean Field (a) (b) (c) Figure 3: Bethe (dark blue, top) and naive mean field (light green, bottom) lower bounds on log Z(ψ) for three families of attractive, pairwise Ising models. (a) 30 × 30 torus with no local observations and homogeneous potentials. (b) 10 × 10 grid with random, inhomogeneous potentials and all pseudo–marginals τs > 1 2, satisfying the conditions of Thm. 1. (c) 10 × 10 grid with random, inhomogeneous potentials and pseudo–marginals of mixed biases. Empirically, the Bethe lower bound also holds for these models. models. These results have potential implications for the suitability of loopy BP in approximate parameter estimation [3], as well as its convergence dynamics. We are currently exploring generalizations of our results to other families of attractive, or “nearly” attractive, graphical models. Acknowledgments The authors thank Yair Weiss for suggesting connections to loop series expansions, and helpful conversations. Funding provided by Army Research Office Grant W911NF-05-1-0207, National Science Foundation Grant DMS-0528488, and NSF Career Grant CCF-0545862. References [1] M. Chertkov and V. Y. Chernyak. Loop calculus helps to improve belief propagation and linear programming decodings of low density parity check codes. In Allerton Conf., 2006. [2] M. Chertkov and V. Y. Chernyak. Loop series for discrete statistical models on graphs. J. Stat. Mech., 2006:P06009, June 2006. [3] B. J. Frey and N. Jojic. A comparison of algorithms for inference and learning in probabilistic graphical models. IEEE Trans. PAMI, 27(9):1392–1416, Sept. 2005. [4] V. G´omez, J. M. Mooij, and H. J. Kappen. Truncating the loop series expansion for BP. JMLR, 8:1987– 2016, 2007. [5] D. M. Greig, B. T. Porteous, and A. H. Seheult. Exact maximum a posteriori estimation for binary images. J. R. Stat. Soc. B, 51(2):271–279, 1989. [6] T. Heskes. On the uniqueness of loopy belief propagation fixed points. Neural Comp., 16:2379–2413, 2004. [7] A. T. Ihler, J. W. Fisher, and A. S. Willsky. Loopy belief propagation: Convergence and effects of message errors. JMLR, 6:905–936, 2005. [8] M. Jerrum and A. Sinclair. Polynomial-time approximation algorithms for the Ising model. SIAM J. Comput., 22(5):1087–1116, Oct. 1993. [9] M. I. Jordan, Z. Ghahramani, T. S. Jaakkola, and L. K. Saul. An introduction to variational methods for graphical models. Machine Learning, 37:183–233, 1999. [10] V. Kolmogorov and R. Zabih. What energy functions can be minimized via graph cuts? IEEE Trans. PAMI, 26(2):147–159, Feb. 2004. [11] F. R. Kschischang, B. J. Frey, and H.-A. Loeliger. Factor graphs and the sum–product algorithm. IEEE Trans. IT, 47(2):498–519, Feb. 2001. [12] J. M. Mooij and H. J. Kappen. Sufficient conditions for convergence of loopy belief propagation. In UAI 21, pages 396–403. AUAI Press, 2005. [13] L. Onsager. Crystal statistics I: A two–dimensional model with an order–disorder transition. Physical Review, 65:117–149, 1944. [14] J. Pearl. Probabilistic Reasoning in Intelligent Systems. Morgan Kaufman, San Mateo, 1988. [15] T. J. Richardson and R. L. Urbanke. The capacity of low-density parity-check codes under messagepassing decoding. IEEE Trans. IT, 47(2):599–618, Feb. 2001. [16] S. B. Shlosman. Correlation inequalities and their applications. J. Math. Sci., 15(2):79–101, Jan. 1981. [17] E. B. Sudderth, M. J. Wainwright, and A. S. Willsky. Loop series and Bethe variational bounds in attractive graphical models. UC Berkeley, EECS department technical report, in preparation, 2008. [18] M. F. Tappen and W. T. Freeman. Comparison of graph cuts with belief propagation for stereo, using identical MRF parameters. In ICCV, volume 2, pages 900–907, 2003. [19] S. C. Tatikonda and M. I. Jordan. Loopy belief propagation and Gibbs measures. In UAI 18, pages 493–500. Morgan Kaufmann, 2002. [20] M. J. Wainwright, T. S. Jaakkola, and A. S. Willsky. Tree–based reparameterization framework for analysis of sum–product and related algorithms. IEEE Trans. IT, 49(5):1120–1146, May 2003. [21] M. J. Wainwright, T. S. Jaakkola, and A. S. Willsky. A new class of upper bounds on the log partition function. IEEE Trans. IT, 51(7):2313–2335, July 2005. [22] Y. Weiss. Comparing the mean field method and belief propagation for approximate inference in MRFs. In D. Saad and M. Opper, editors, Advanced Mean Field Methods. MIT Press, 2001. [23] J. S. Yedidia, W. T. Freeman, and Y. Weiss. Constructing free energy approximations and generalized belief propagation algorithms. IEEE Trans. IT, 51(7):2282–2312, July 2005. 8
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Second Order Bilinear Discriminant Analysis for single-trial EEG analysis Christoforos Christoforou Department of Computer Science The Graduate Center of the City University of New York 365 Fifth Avenue New York, NY 10016-4309 cchristoforou@gc.cuny.edu Paul Sajda Department of Biomedical Engineering Columbia University 351 Engineering Terrace Building, MC 8904 1210 Amsterdam Avenue New York, NY 10027 ps629@columbia.edu Lucas C. Parra Department of Biomedical Engineering The City College of The City University of New York Convent Avenue 138th Street New York,NY 10031, USA parra@ccny.cuny.edu Abstract Traditional analysis methods for single-trial classification of electroencephalography (EEG) focus on two types of paradigms: phase locked methods, in which the amplitude of the signal is used as the feature for classification, e.g. event related potentials; and second order methods, in which the feature of interest is the power of the signal, e.g. event related (de)synchronization. The procedure for deciding which paradigm to use is ad hoc and is typically driven by knowledge of the underlying neurophysiology. Here we propose a principled method, based on a bilinear model, in which the algorithm simultaneously learns the best first and second order spatial and temporal features for classification of EEG. The method is demonstrated on simulated data as well as on EEG taken from a benchmark data used to test classification algorithms for brain computer interfaces. 1 Introduction 1.1 Utility of discriminant analysis in EEG Brain computer interface (BCI) algorithms [1][2][3][4] aim to decode brain activity, on a singletrial basis, in order to provide a direct control pathway between a user’s intentions and a computer. Such an interface could provide “locked in patients” a more direct and natural control over a neuroprosthesis or other computer applications [2]. Further, by providing an additional communication 1 channel for healthy individuals, BCI systems can be used to increase productivity and efficiency in high-throughput tasks [5, 6]. Single-trial discriminant analysis has also been used as a research tool to study the neural correlates of behavior. By extracting activity that differs maximally between two experimental conditions, the typically low signal-noise ratio of EEG can be overcome. The resulting discriminant components can be used to identify the spatial origin and time course of stimulus/response specific activity, while the improved SNR can be leveraged to correlate variability of neural activity across trials to behavioral variability and behavioral performance [7, 5]. In essence, discriminant analysis adds to the existing set of multi-variate statistical tools commonly used in neuroscience research (ANOVA, Hoteling T 2, Wilks’ Λ test). 1.2 Linear and quadratic approaches In EEG the signal-to-noise ratio of individual channels is low, often at -20dB or less. To overcome this limitation, all analysis methods perform some form of averaging, either across repeated trials, across time, or across electrodes. Traditional EEG analysis averages signals across many repeated trials for individual electrodes. A conventional method is to average the measured potentials following stimulus presentation, thereby canceling uncorrelated noise that is not reproducible from one trial to the next. This averaged activity, called an event related potential (ERP), captures activity that is time-locked to the stimulus presentation but cancels evoked oscillatory activity that is not locked in phase to the timing of the stimulus. Alternatively, many studies compute the oscillatory activity in specific frequency bands by filtering and squaring the signal prior to averaging. Thus, changes in oscillatory activity are termed event related synchronization or desynchronization (ERS/ERD). Surprisingly, discriminant analysis methods developed thus far by the machine learning community have followed this dichotomy: First order methods in which the amplitude of the EEG signal is considered to be the feature of interest in classification – corresponding to ERP – and second order methods in which the power of the feature is considered to be of importance for classification – corresponding to ERS/ERD. First order methods include temporal filtering + thresholding [2], hierarchical linear classifiers [5] and bilinear discriminant analysis [8, 9]. Second order methods include the logistic regression with a quadratic term [11] and the well known common spatial patterns method (CSP) [10] and its variants: common spatio-spectral patterns (CSSP)[12], and common sparse spectral spatial patterns (CSSSP)[13] . Choosing what kind of features to use traditionally has been an ad hoc process motivated by knowledge of the underlying neurophysiology and task. From a machine-learning point of view, it seems limiting to commit a priori to only one type of feature. Instead it would be desirable for the analysis method to extract the relevant neurophysiological activity de novo with minimal prior expectations. In this paper we present a new framework that combines both the first order features and the second order features in the analysis of EEG. We use a bilinear formulation which can simultaneously extract spatial linear components as well as temporal (filtered) features. 2 Second order bilinear discriminant analysis 2.1 Problem setting Given a set of sample points D = {Xn, yn}N n=1, X ∈RD × T , y ∈{−1, 1} , where Xn corresponds to the EEG signal of D channels and T sample points and yn indicate the class that corresponds to one of two conditions (e.g. right or left hand imaginary movement, stimulus versus control conditions, etc.), the task is then to predict the class label y for an unobserved trial X. 2.2 Second order bilinear model Define a function, f(X; θ) = C Trace(UTXV) + (1 −C) Trace(ΛAT(XB)(XB)TA) (1) where θ = {U ∈RD × R, V ∈RT × R, A ∈RD × K B ∈RT × T ′} are the parameters of the model, Λ ∈diag({−1, 1}) a given diagonal matrix with elements {−1, 1} and C ∈[0, 1]. We consider the 2 following discriminative model; we model the log-odds ratio of the posterior class probability to be the sum of a bilinear function with respect to the EEG signal amplitude and linear with respect to the second order statistics of the EEG signal: log P(y = +1|X) P(y = −1|X) = f(X|θ) (2) 2.2.1 Interpretation of the model The first term of the equation (1) can be interpreted as a spatio-temporal projection of the signal, under the bilinear model, and captures the first order statistics of the signal. Specifically, the columns ur of U represent R linear projections in space (rows of X). Similarly, each of the R columns of vk in matrix V represent linear projections in time (columns of X). By re-writing the term as: Trace(UTXV) = Trace(VUTX) = Trace(WTX) (3) where we defined W = UVT, it is easy to see that the bilinear projection is a linear combination of elements of X with the rank −R constrained on W. This expression is linear in X and thus captures directly the amplitude of the signal directly. In particular, the polarity of the signal (positive evoked response versus negative evoked response) will contribute significantly to discrimination if it is consistent across trials. This term, therefore, captures phase locked event related potentials in the EEG signal. The second term of equation (1), is a projection of the power of the filtered signal, which captures the second order statistics of the signal. As before, each column of matrix A and B, represent components that project the data in space and time respectively. Depending on the structure one enforces in matrix B different interpretations of the model can be archived. In the general case where no structure on B is assumed, the model captures a linear combination of the elements of a rank −T ′ second order matrix approximation of the signal Σ = XB(XB)T. In the case where Toeplitz structure is enforced on B, then B defines a temporal filter on the signal and the model captures the linear combination of the power of the second order matrix of the filtered signal. For example if B is fixed to a Toeplitz matrix with coefficients corresponding to a 8Hz-12Hz band pass filter, then the second term is able to extract differences in the alpha-band which is known to be modulated during motor related tasks. Further, by learning B from the data, we may be able to identify new frequency bands that have so far not been identified in novel experimental paradigms. The spatial weights A together with the Trace operation ensure that the power is measured, not in individual electrodes, but in some component space that may reflect activity distributed across several electrodes. Finally, the scaling factor λ (which may seem superfluous given the available degrees of freedom) is necessary once regularization terms are added to the log-likelihood function. 2.3 Logistic regression We use a logistic Rregression (LR) formalism as it is particularly convenient when imposing additional statistical properties on the matrices U, V, A, B such as smoothness or sparseness. In addition, in our experience, LR performs well in strongly overlapping high-dimensional datasets and is insensitive to outliers, the later being of particular concern when including quadratic features. Under the logistic regression model (2) the class posterior probability P(y|X; θ) is modeled as P(y|X; θ) = 1 1 + e−y(f(X;θ)+wo) (4) and the resulting log likelihood is given by L(θ) = − N X n=1 log(1 + e−y(f(Xn;θ)+wo)) (5) We minimize the negative log likelihood and add a log-prior on each of the columns of U, V and A and parameters of B that act as a regularization term, which is written as: argmin U,V,A,B,wo −L(θ) − R X r=1 (log p(ur) + log p(vr)) − K X k=1 log p(ak) − T ′ X t=1 log(p(bt)) (6) 3 where the log-priors are given for each of the parameters as log p(uk) = uT k K(u)uk , log p(vk) = uT k K(v)uk, log p(ak) = aT k K(a)ak and log p(bk) = bT k K(b)bk. K(u) ∈RD×D, K(v) ∈RT ×T , K(a) ∈RD×D, K(b) ∈RT ×T are kernel matrices that control the smoothness of the parameter space. Details on the regularization procedure can be found in [8]. Analytic gradients of the log likelihood (5) with respect to the various parameters are given by: ∂L(θ) ∂ur = N X n=1 ynπ(Xn)Xnvr (7) ∂L(θ) ∂vr = N X n=1 ynπ(Xn)urXn (8) ∂L(θ) ∂ar = 2 N X n=1 ynπ(Xn)Λr,r(XnB)(XnB)Tar (9) ∂L(θ) ∂bt = 2 N X n=1 ynπ(Xn)XTAΛATXbt (10) where we define π(Xn) = 1 −P(y|X) = e−y(f(Xn;θ)+wo) 1 + e−y(f(Xn;θ)+wo) (11) where ui, vi, ai, and bi correspond to the ith columns of U, V, A, B respectively. 2.4 Fourier Basis for B If matrix B is constrained to have a circular toepliz structure then it can be represented as B = F−1DF, where F−1 denotes the inverse Fourier matrix, and D is a diagonal complex-valued matrix of Fourier coefficients. In such a case, we can re-write equations (9) and (10) as ∂L(θ) ∂ar = 2 N X n=1 ynπ(Xn)Λr,r(XnF−1 ˆDF−TXT n)ar (12) ∂L(θ) ∂di = 2 N X n=1 ynπ(Xn)(F−TXT nAΛATXnF−1)i,idi (13) (14) where ˆD = DDT, and the parameters are now optimized with respect to Fourier coefficients di = Di,i. An iterative minimization procedure can be used to solve the above minimization. 3 Results 3.1 Simulated data In order to validate our method and its ability to capture both linear and second order features, we generated simulated data that contained both types of features; namely ERP type of features and ERS/ERD type of features. The simulated signals were generated with a signal to noise ratio of −20dB which is a typical noise level for EEG. A total of 28 channels, 500 ms long signals and at a sampling frequency of 100Hz where generated, resulting in a matrix of X of 28 by 50 elements, for each trial. Data corresponding to a total of 1000 trials were generated; 500 trials contained only zero mean Gaussian noise (representing baseline conditions), with the other 500 trials having the signal of interest added to the noise (representing the stimulus condition): For channels 1-9 the signal was composed of a 10Hz sinusoid with random phase in each of the nine channels, and across trials. The 4 0 10 20 30 −0.5 0 0.5 1 1.5 channels U component 0 50 100 150 200 250 300 350 400 450 500 −0.1 0 0.1 0.2 0.3 0.4 time(m/s) amplitude V Component 0 10 20 30 −0.5 0 0.5 1 1.5 channels A component 0 50 100 150 200 250 300 350 400 450 500 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 time (m/s) amplitute B component Figure 1: Spatial and temporal component extracted on simulated data for the linear term (top) and quadratic term (bottom). sinusoids were scaled to match the −20dB SNR. This simulates an ERS type feature. For channels 10-18, a peak represented by a half cycle sinusoid was added at approximately 400 ms, which T simulates an ERP type feature. The extracted components are shown in Figure 1. The linear component U (in this case only a column vector) has non-zero coefficients for channels 10 to 18 only, showing that the method correctly identified the ERP activity. Furthermore, the associated temporal component V has a temporal profile that matches the time course of the simulated evoked response. Similarly, the second order components A have non-zero weights for only channels 1-9 showing that the method also identified the spatial distribution of the non-phase locked activity. The temporal filter B was trained in the frequency domain and the resulting filter is shown here in the time domain. It exhibits a dominant 10Hz component, which is indeed the frequency of the non-phase locked activity. 3.2 BCI competition dataset To evaluate the performance of the proposed method on real data we applied the algorithm to an EEG data set that was made available through The BCI Competition 2003 ([14], Data Set IV). EEG was recorded on 28 channels for a single subject performing self-paced key typing, that is, pressing the corresponding keys with the index and little fingers in a self-chosen order and timing (i.e. self-paced). Key-presses occurred at an average speed of 1 key per second. Trial matrices were extracted by epoching the data starting 630ms before each key-press. A total of 416 epochs were recorded, each of length 500ms. For the competition, the first 316 epochs were to be used for classifier training, while the remaining 100 epochs were to be used as a test set. Data were recorded at 1000 Hz with a pass-band between 0.05 and 200 Hz, then downsampled to 100Hz sampling rate. For this experiment, the matrix B was fixed to a Toeplitz structure that encodes a 10Hz33Hz bandpass filter and only the parameters U, V, A and w0 were trained. The number of columns of U and V were set to 1, where two columns were used for A. The temporal filter was selected based on prior knowledge of the relevant frequency band. This demonstrates the flexibility of our approach to either incorporate prior knowledge when available or extract it from 5 U component 0 100 200 300 400 500 −0.1 −0.05 0 0.05 0.1 time (m/s) V component First Column of A Second Column of A Figure 2: Spatial and temporal component (top), and two spatial components for second order features (bottom) learned on the benchmark dataset data otherwise. Regularization parameters where chosen via a five fold cross validation procedure (details can be found in [8]). The resulting components for this dataset are shown in Figure 2. Benchmark performance was measured on the test set which had not been used during either training or cross validation. The number of misclassified trials in the test set was 13 which places our method on a new first place given the results of the competition which can be found online http://ida.first.fraunhofer.de/projects/bci/competition ii/results/index.html ([14]). Hence, our method works as a classifier producing a state-of-the art result on a realistic data set. The receiveroperator characteristic (ROC) curve for cross validation and for the independent testset are shown in Figure 3. Figure 3.2 also shows the contribution of the linear and quadratic terms for every trial for the two types of key-presses. The figure shows that the two terms provide independent information and that in this case the optimal relative weighting factor is C ≈0.5. 4 Conclusion In this paper we have presented a framework for uncovering spatial as well as temporal features in EEG that combine the two predominant paradigms used in EEG analysis: event related potentials and oscillatory power. These represent phase locked activity (where polarity of the activity matters), and non-phase locked activity (where only the power of the signal is relevant). We used the probabilistic formalism of logistic regression that readily incorporates prior probabilities to regularize the increased number of parameters. We have evaluated the proposed method on both simulated data, and a real BCI benchmark dataset, achieving state-of-the-art classification performance. The proposed method provides a basis for various future directions. For example, different sets of basis functions (other than a Fourier basis) can be enforced on the temporal decomposition of the data through the matrix B (e.g. wavelet basis). Further, the method can be easily generalized to 6 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 False positive rate True positive rate AUC : 0.96 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 False positive rate True positive rate AUC : 0.935 #errors:13 Figure 3: ROC curve with area under the curve 0.96 for the cross validation on the benchmark dataset (left). ROC curve with area under the curve 0.93, on the independent test set, for the benchmark dataset. There were a total of 13 errors on unseen data, which is less than any of the results previously reported, placing this method in first place in the benchmark ranking. −20 −10 0 10 −15 −10 −5 0 5 10 Training Set first order term second order term −15 −10 −5 0 5 10 −10 −5 0 5 first order term second order term Testing set Figure 4: Scatter plot of the first order term vs second order term of the model, on the training and testing set for the benchmark dataset (’+’ left key, and ’o’ right key). It is clear that the two types of features contain independent information that can help improve the classification performance. 7 multi-class problems by using a multinomial distribution on y. Finally, different regularizations (i.e L1 norm, L2 norm) can be applied to the different types of parameters of the model. References [1] J. R. Wolpaw, N. Birbaumer, D. J. McFarland, G. Pfurtscheller, and T. 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Convex Relaxations of Latent Variable Training Yuhong Guo and Dale Schuurmans Department of Computing Science University of Alberta {yuhong, dale}@cs.ualberta.ca Abstract We investigate a new, convex relaxation of an expectation-maximization (EM) variant that approximates a standard objective while eliminating local minima. First, a cautionary result is presented, showing that any convex relaxation of EM over hidden variables must give trivial results if any dependence on the missing values is retained. Although this appears to be a strong negative outcome, we then demonstrate how the problem can be bypassed by using equivalence relations instead of value assignments over hidden variables. In particular, we develop new algorithms for estimating exponential conditional models that only require equivalence relation information over the variable values. This reformulation leads to an exact expression for EM variants in a wide range of problems. We then develop a semidefinite relaxation that yields global training by eliminating local minima. 1 Introduction Few algorithms are better known in machine learning and statistics than expectation-maximization (EM) [5]. One reason is that EM solves a common problem—learning from incomplete data—that occurs in almost every area of applied statistics. Equally well known to the algorithm itself, however, is the fact that EM suffers from shortcomings. Here it is important to distinguish between the EM algorithm (essentially a coordinate descent procedure [10]) and the objective it optimizes (marginal observed or conditional hidden likelihood). Only one problem is due to the algorithm itself: since it is a simple coordinate descent, EM suffers from slow (linear) convergence and therefore can require a large number of iterations to reach a solution. Standard optimization algorithms such as quasi-Newton methods can, in principle, require exponentially fewer iterations to achieve the same accuracy (once close enough to a well behaved solution) [2, 11]. Nevertheless, EM converges quickly in many circumstances [12, 13]. The main problems attributed to EM are not problems with the algorithm per se, but instead are properties of the objective it optimizes. In particular, the standard objectives tackled by EM are not convex in any standard probability model (e.g. the exponential family). Non-convexity immediately creates the risk of local minima, which unfortunately is not just a theoretical concern: EM often does not produce very good results in practice, and can sometimes fail to improve significantly upon initial parameter settings [9]. For example, the field of unsupervised grammar induction [8] has been thwarted in its attempts to use EM for decades and is still unable to infer useful syntactic models of natural language from raw unlabeled text. We present a convex relaxation of EM for a standard training criterion and a general class of models in an attempt to understand whether local minima are really a necessary aspect of unsupervised learning. Convex relaxations have been a popular topic in machine learning recently [4, 16]. In this paper, we propose a convex relaxation of EM that can be applied to a general class of directed graphical models, including mixture models and Bayesian networks, in the presence of hidden variables. There are some technical barriers to overcome in achieving an effective convex relaxation however. First, as we will show, any convex relaxation of EM must produce trivial results if it maintains any dependence on the values of hidden variables. Although this result suggests that any convex relaxation of EM cannot succeed, we subsequently show that the problem can be overcome by working with equivalence relations over the values of the hidden variables, rather than the missing values themselves. Although equivalence relations provide an easy way to solve the symmetry collapsing problem, they do not immediately yield a convex EM formulation, because the underlying estimation principles for directed graphical models have not been formulated in these terms. Our main technical contribution therefore is a reformulation of standard estimation principles for exponential conditional models in terms of equivalence relations on variable values, rather than the variable values themselves. Given an adequate reformulation of the core estimation principle, developing a useful convex relaxation of EM becomes possible. 1.1 EM Variants Before proceeding, it is important to first clarify the precise EM variant we address. In fact, there are many EM variants that optimize different criteria. Let z = (x, y) denote a complete observation, where x refers to the observed part of the data and y refers to the unobserved part; and let w refer to the parameters of the underlying probability model, P(x, y|w). (Here we consider discrete probability distributions just for simplicity of the discussion.) Joint and conditional EM algorithms are naive “self-supervised” training procedures that alternate between inferring the values of the missing variables and optimizing the parameters of the model (joint EM update) y(k+1) = arg max y P(x, y|w(k)) w(k+1) = arg max w P(x, y(k+1)|w) (conditional EM update) y(k+1) = arg max y P(y|x, w(k)) w(k+1) = arg max w P(y(k+1)|x, w) These are clearly coordinate descent procedures that make monotonic progress in their objectives, P(x, y|w) and P(y|x, w). Moreover, the criteria being optimized are in fact well motivated objectives for unsupervised training: joint EM is frequently used in statistical natural language processing (where it is referred to as “Viterbi EM” [3, 7]); the conditional form has been used in [16]. The primary problem with these iterations is not that they optimize approximate or unjustified criteria, but rather that they rapidly get stuck in poor local maxima due to the extreme updates made on y. By far, the more common form of EM—contributing the very name expectation-maximization—is given by (marginal EM update) q(k+1) y = P(y|x, w(k)) w(k+1) = arg max w P y q(k+1) y log P(x, y|w) where qy is a distribution over possible missing values. Although it is not immediately obvious what this iteration optimizes, it has long been known that it monotonically improves the marginal likelihood P(x|w) [5]. [10] later showed that the E-step could be generalized to maxqy P y qy log P(x, y|w(k))/qy . Due to the softer qy update, the standard EM update does not as converge as rapidly to a local maximum as the joint and conditional variants; however, as a result, it tends to find better local maxima. Marginal EM has subsequently become the dominant form of EM algorithm in the literature (although joint EM is still frequently used in statistical NLP [3, 7]). Nevertheless, none of the training criteria are jointly convex in the optimization variables, thus these iterations are only guaranteed to find local maxima. Independent of the updates, the three training criteria are not equivalent nor equally well motivated. In fact, for most applications we are more interested in acquiring an accurate conditional P(y|x, w), rather than optimizing the marginal P(x|w) [16]. Of the three training criteria therefore (joint, conditional and marginal), marginal likelihood appears to be the least relevant to learning predictive models. Nevertheless, the convex relaxation techniques we propose can be applied to all three objectives. For simplicity we will focus on maximizing joint likelihood in this paper, since it incorporates aspects of both marginal and conditional training. Conveniently, joint and marginal EM pose nearly identical optimization problems: (joint EM objective) arg max w max y P(x, y|w) = arg max w max qy P y qy log P(x, y|w) (marg. EM objective) arg max w P y P(x, y|w) = arg max w max qy P y qy log P(x, y|w) +H(qy) where qy is a distribution over possible missing values [10]. Therefore, much of the analysis we provide for joint EM also applies to marginal EM, leaving only a separate convex relaxation of the entropy term that can be conducted independently. We will also primarily consider the hidden variable case and assume a fixed set of random variables Y1, ..., Yℓis always unobserved, and a fixed set of variables Xℓ+1, ..., Xn is always observed. The technique remains extendable to the general missing value case however. 2 A Cautionary Result for Convexity Our focus in this paper will be to develop a jointly convex relaxation to the minimization problem posed by joint EM min y min w −P i log P(xi, yi|w) (1) One obvious issue we must face is to relax the discrete constraints on the assignments y. However, the challenge is deeper than this. In the hidden variable case—when the same variables are missing in each observation—there is a complete symmetry between the missing values. In particular, for any optimal solution (y, w) there must be other, equivalent solutions (y′, w′) corresponding to a permutation of the hidden variable values. Unfortunately, this form of solution symmetry has devastating consequences for any convex relaxation: Assume one attempts to use any jointly convex relaxation f(qy, w) of the standard loglikelihood objective (1), where the the missing variable assignment y has been relaxed into a continuous probabilistic assignment qy (like standard EM). Lemma 1 If f is strictly convex and invariant to permutations of unobserved variable values, then the global minimum of f, (q∗ y, w∗), must satisfy q∗ y = uniform. Proof: Assume (qy, w) is a global minimum of f but qy ̸= uniform. Then there must be some permutation of the missing values, Π, such that the alternative (q′ y, w′) = (Π(qy), Π(w)) satisfies q′ y ̸= qy. But by the permutation invariance of f, this implies f(qy, w) = f(q′ y, w′). By the strict convexity of f, we then have f α(qy, w) + (1 −α)(q′ y, w′) < αf(qy, w)+(1−α)f(q′ y, w′) = f(qy, w), for 0 < α < 1, contradicting the global optimality of f(qy, w). Therefore, any convex relaxation of (1) that uses a distribution qy over missing values and does not make arbitrary distinctions can never do anything but produce a uniform distribution over the hidden variable values. (The same is true for marginal and conditional versions of EM.) Moreover, any non-strictly convex relaxation must admit the uniform distribution as a possible solution. This trivialization is perhaps the main reason why standard EM objectives have not been previously convexified. (Note that standard coordinate descent algorithms simply break the symmetry arbitrarily and descend into some local solution.) This negative result seems to imply that no useful convex relaxation of EM is possible in the hidden variable case. However, our key observation is that a convex relaxation expressed in terms of an equivalence relation over the missing values avoids this symmetry breaking problem. In particular, equivalence relations exactly collapse the unresolvable symmetries in this context, while still representing useful structure over the hidden assignments. Representations based on equivalence relations are a useful tool for unsupervised learning that has largely been overlooked (with some exceptions [4, 15]). Our goal in this paper, therefore, will be to reformulate standard training objectives to use only equivalence relations on hidden variable values. 3 Directed Graphical Models We will derive a convex relaxation framework for a general class of probability models—namely, directed models—that includes mixture models and discrete Bayesian networks as special cases. A directed model defines a joint probability distribution over a set of random variables Z1, ..., Zn by exploiting the chain rule of probability to decompose the joint into a product of locally normalized conditional distributions P(z|w) = Qn j=1 P(zj|zπ(j), wj). Here, π(j) ⊆{1, ..., j −1}, and wj is the set of parameters defining conditional distribution j. Furthermore, we will assume an exponential family representation for the conditional distributions P(zj|zπ(j), wj) = exp w⊤ j φj(zj, zπ(j)) −A(wj, zπ(j)) , where A(wj, zπ(j)) = log P a exp w⊤ j φj(a, zπ(j)) and φj(zj, zπ(j)) denotes a vector of features evaluated on the value of the child and its parents. For simplicity, we will initially restrict our discussion to discrete Bayesian networks, but then reintroduce continuous random variables later. A discrete Bayesian network is just a directed model where the conditional distributions are represented by a sparse feature vector indicating the identity of the child-parent configuration φj(zj, zπ(j)) = (...1(zj=a,zπ(j)=b)...)⊤. That is, there is a single indicator feature for each local configuration (a, b). A particularly convenient property of directed models is that the complete data likelihood decomposes into an independent sum of local loglikelihoods P i log P(zi|w) = P j P i w⊤ j φj(zi j, zi π(j)) −A(wj, zi π(j)) (2) Thus the problem of solving for a maximum likelihood set of parameters, given complete training data, amounts to solving a set of independent log-linear regression problems, one for each variable Zj. To simplify notation, consider one of the log-linear regression problems in (2) and drop the subscript j. Then, using a matrix notation we can rewrite the jth local optimization problem as min W P i A(W, Φi:) −tr(ΦWY ⊤) where W ∈IRc×v, Φ ∈{0, 1}t×c, and Y ∈{0, 1}t×v, such that t is the number of training examples, v is the number of possible values for the child variable, c is the number of possible configurations for the parent variables, and tr is the matrix trace. To explain this notation, note that Y and Φ are indicator matrices that have a single 1 in each row, where Y indicates the value of the child variable, and Φ indicates the specific configuration of the parent values, respectively; i.e. Y 1 = 1 and Φ1 = 1, where 1 denotes the vector of all 1s. (This matrix notation greatly streamlines the presentation below.) We also use the notation Φi: to denote the ith row vector in Φ. Here, the log normalization factor is given by A(W, Φi:) = log P a exp (Φi:W1a), where 1a denotes a sparse vector with a single 1 in position a. Below, we will consider a regularized form of the objective, and thereby work with the maximum a posteriori (MAP) form of the problem min W P i A(W, Φi:) −tr(ΦWY ⊤) + α 2 tr(W ⊤W) (3) This provides the core estimation principle at the heart of Bayesian network parameter learning. However, for our purposes it suffers from a major drawback: (3) is not expressed in terms of equivalence relations between the variable values. Rather it is expressed in terms of direct indicators of specific variable values in specific examples—which will lead to a trivial outcome if we attempt any convex relaxation. Instead, we require a fundamental reformulation of (3) to remove the value dependence and replace it with a dependence only on equivalence relationships. 4 Log-linear Regression on Equivalence Relations The first step in reformulating (3) in terms of equivalence relations is to derive its dual. Lemma 2 An equivalent optimization problem to (3) is max Θ −tr(Θ log Θ⊤) −1 2αtr (Y −Θ)⊤ΦΦ⊤(Y −Θ) subject to Θ ≥0, Θ1 = 1 (4) Proof: The proof follows a standard derivation, which we sketch; see e.g. [14]. First, by considering the Fenchel conjugate of A it can be shown that A(W, Φi:) = max Θi: tr(Θ⊤ i: Φi:W) −Θi: log Θ⊤ i: subject to Θi: ≥0, Θi:1 = 1 Substituting this in (3) and then invoking the strong minimax property [1] allows one to show that (3) is equivalent to max Θ min W −tr(Θ log Θ⊤) −tr((Y −Θ)⊤ΦW) + α 2 tr(W ⊤W) subject to Θ ≥0, Θ1 = 1 Finally, the inner minimization can be solved by setting W = 1 αΦ⊤(Y −Θ), yielding (4). Interestingly, deriving the dual has already achieved part of the desired result: the parent configurations now only enter the problem through the kernel matrix K = ΦΦ⊤. For Bayesian networks this kernel matrix is in fact an equivalence relation between parent configurations: Φ is a 0-1 indicator matrix with a single 1 in each row, implying that Kij = 1 iff Φi: = Φj:, and Kij = 0 otherwise. But more importantly, K can be re-expressed as a function of the individual equivalence relations on each of the parent variables. Let Y p ∈{0, 1}t×vp indicate the value of a parent variable Zp for each training example. That is, Y p i: is a 1 × vp sparse row vector with a single 1 indicating the value of variable Zp in example i. Then M p = Y pY p⊤defines an equivalence relation over the assignments to variable Zp, since M p ij = 1 if Y p i: = Y p j: and M p ij = 0 otherwise. It is not hard to see that the equivalence relation over complete parent configurations, K = ΦΦ⊤, is equal to the componentwise (Hadamard) product of the individual equivalence relations for each parent variable. That is, K = ΦΦ⊤= M 1 ◦M 2 ◦· · · ◦M p, since Kij = 1 iff M 1 ij = 1 and M 2 ij = 1 and ... M p ij = 1. Unfortunately, the dual problem (4) is still expressed in terms of the indicator matrix Y over child variable values, which is still not acceptable. We still need to reformulate (4) in terms of the equivalence relation matrix M = Y Y ⊤. Consider an alternative dual parameterization Ω∈IRt×t such that Ω≥0, Ω1 = 1, and ΩY = Θ. (Note that Θ ∈IRt×v, for v < t, and therefore Ωis larger than Θ. Also note that as long as every child value occurs at least once in the training set, Y has full rank v. If not, then the child variable effectively has fewer values, and we could simply reduce Y until it becomes full rank again without affecting the objective (3).) Therefore, since Y is full rank, for any Θ, some Ωmust exist that achieves ΩY = Θ. Then we can relate the primal parameters to this larger set of dual parameters by the relation W = 1 αΦ⊤(I −Ω)Y . (Even though Ωis larger than Θ, they can only express the same realizable set of parameters W.) To simplify notation, let B = I −Ω and note the relation W = 1 αΦ⊤BY . If we reparameterize the original problem using this relation, then it is possible to show that an equivalent optimization problem to (3) is given by min B P i A(B, Φi: −tr(KBM) + 1 2αtr(B⊤KBM) subject to B ≤I, B1 = 0 (5) where K = ΦΦ⊤and M = Y Y ⊤are equivalence relations on the parent configurations and child values respectively. The formulation (5) is now almost completely expressed in terms of equivalence relations over the data, except for one subtle problem: the log normalization factor A(B, Φi:) = log P a exp 1 αΦi:Φ⊤BY 1a still directly depends on the label indicator matrix Y . Our key technical lemma is that this log normalization factor can be re-expressed to depend on the equivalence relation matrix M alone. Lemma 3 A(B, Φi:) = log P j exp 1 αKi:BM:j −log 1⊤M:j Proof: The main observation is that an equivalence relation over value indicators, M = Y Y ⊤, consists of columns copied from Y . That is, for all j, M:j = Y:a for some a corresponding to the child value in example j. Let y(j) denote the child value in example j and let βi: = 1 αKi:B. Then P a exp 1 αΦi:Φ⊤BY 1a = P a exp(βi:Y:a) = P a P j:y(j)=a 1 |{ℓ:y(ℓ)=a}| exp(βi:M:j) = P j 1 |{ℓ:y(ℓ)=y(j)}| exp(βi:M:j) = P j 1 1⊤M:j exp(βi:M:j) = P j exp(βi:M:j −log 1⊤M:j) Using Lemma 3 one can show that the dual problem to (5) is given by the following. Theorem 1 An equivalent optimization problem to (3) is max Λ≥0,Λ1=1 −tr(Λ log Λ⊤) −1⊤Λ log(M1) −1 2αtr((I −Λ)⊤K(I −Λ)M) (6) where K = M 1 ◦· · · ◦M p for parent variables Z1, ..., Zp. Proof: This follows the same derivation as Lemma 2, modified by taking into account the extra term introduced by Lemma 3. First, considering the Fenchel conjugate of A, it can be shown that A(B, Φi:) = max Λi:≥0,Λi:1=1 1 αKi:BMΛ⊤ i: −Λi: log Λ⊤ i: −Λi: log(M1) Substituting this in (5) and then invoking the strong minimax property [1] allows one to show that (5) is equivalent to max Λ≥0,Λ1=1 min B≤I,B1=0 −tr(Λ log Λ⊤) −1⊤Λ log(M1) −1 αtr((I −Λ)⊤KBM) + 1 2αtr(B⊤KBM) Finally, the inner minimization on B can be solved by setting B = I −Λ, yielding (6). This gives our key result: the log-linear regression (3) is equivalent to (6), which is now expressed strictly in terms of equivalence relations over the parent configurations and child values. That is, the value indicators, Φ and Y , have been successfully eliminated from the formulation. Given a solution Λ∗to (6), the optimal model parameters W ∗for (3) can be recovered via W ∗= ! αΦ⊤(I −Λ∗)Y . 5 Convex Relaxation of Joint EM The equivalence relation form of log-linear regression can be used to derive useful relaxations of EM variants for directed models. In particular, by exploiting Theorem 1, we can now re-express the regularized form of the joint EM objective (1) strictly in terms of equivalence relations over the hidden variable values min {Y h} X j min wj −log P(zi j|zi π(j), wj) + α 2 w⊤ j wj (7) = min {M h} X j max Λj≥0,Λj1=1−tr(Λj log Λ⊤ j ) −1⊤Λj log(M j1) −1 2αtr (I −Λj)⊤Kj(I −Λj)M j (8) subject to M h = Y hY h⊤, Y h ∈{0, 1}t×vh, Y h1 = 1 (9) where h ranges over the hidden variables, and Kj = M j1 ◦· · · ◦M jp for the parent variables Zj1, ..., Zjp of Zj. Note that (8) is an exact reformulation of the joint EM objective (7); no relaxation has yet been introduced. Another nice property of the objective in (8) is that is it concave in each Λj and convex in each M h individually (a maximum of convex functions is convex [2]). Therefore, (8) appears as though it might admit an efficient algorithmic solution. However, one difficulty in solving the resulting optimization problem is the constraints. Although the constraints imposed in (9) are not convex, there is a natural convex relaxation suggested by the following. Lemma 4 (9) is equivalent to: M ∈{0, 1}t×t, diag(M) = 1, M = M ⊤, M ⪰0, rank(M) = v. A natural convex relaxation of (9) can therefore be obtained by relaxing the discreteness constraint and dropping the nonconvex rank constraint, yielding M h ∈[0, 1]t×t, diag(M h) = 1, M h = M h⊤, M h ⪰0 (10) Optimizing the exact objective in (8) subject to the relaxed convex constraints (10) provides the foundation for our approach to convexifying EM. Note that since (8) and (10) are expressed solely in terms of equivalence relations, and do not depend on the specific values of hidden variables in any way, this formulation is not subject to the triviality result of Lemma 1. However, there are still some details left to consider. First, if there is only a single hidden variable then (8) is convex with respect to the single matrix variable M h. This result immediately provides a convex EM training algorithm for various applications, such as for mixture models for example (see the note regarding continuous random variables below). Second, if there are multiple hidden variables that are separated from each other (none are neighbors, nor share a common child) then the formulation (8) remains convex and can be directly applied. On the other hand, if hidden variables are connected in any way, either by sharing a parent-child relationship or having a common child, then (8) is no longer jointly convex because the trace term is no longer linear in the matrix variables {M h}. In this case, we can restore convexity by further relaxing the problem: To illustrate, if there are multiple hidden parents Zp1, ..., Zpℓfor a given child, then the combined equivalence relation M p1 ◦· · · ◦M pℓis a Hadamard product of the individual matrices. A convex formulation can be recovered by introducing an auxiliary matrix variable ˜ M to replace M p1 ◦· · ·◦M pℓin (8) and adding the set of linear constraints ˜ Mij ≤M p ij for p ∈{p1, ..., pℓ}, ˜ Mij ≥M p1 ij + · · · + M pℓ ij −ℓ+ 1 to approximate the componentwise ’and’. A similar relaxation can also be applied when a child is hidden concurrently with hidden parent variables. Continuous Variables The formulation in (8) can be applied to directed models with continuous random variables, provided that all hidden variables remain discrete. If every continuous random variable is observed, then the subproblems on these variables can be kept in their natural formulations, and hence still solved. This extension is sufficient to allow the formulation to handle Gaussian mixture models, for example. Unfortunately, the techniques developed in this paper do not apply to the situation where there are continuous hidden variables. Recovering the Model Parameters Once the relaxed equivalence relation matrices {M h} have been obtained, the parameters of they underlying probability model need to be recovered. At an Bayesian Fully Supervised Viterbi EM Convex EM networks Train Test Train Test Train Test Synth1 7.23 ±.06 7.90 ±.04 11.29 ±.44 11.73 ±.38 8.96 ±.24 9.16 ±.21 Synth2 4.24 ±.04 4.50 ±.03 6.02 ±.20 6.41 ±.23 5.27 ±.18 5.55 ±.19 Synth3 4.93 ±.02 5.32 ±.05 7.81 ±.35 8.18 ±.33 6.23 ±.18 6.41 ±.14 Diabetes 5.23 ±.04 5.53 ±.04 6.70 ±.27 7.07 ±.23 6.51 ±.35 6.50 ±.28 Pima 5.07 ±.03 5.32 ±.03 6.74 ±.34 6.93 ±.21 5.81 ±.07 6.03 ±.09 Cancer 2.18 ±.05 2.31 ±.02 3.90 ±.31 3.94 ±.29 2.98 ±.19 3.06 ±.16 Alarm 10.23 ±.16 12.30 ±.06 11.94 ±.32 13.75 ±.17 11.74 ±.25 13.62 ±.20 Asian 2.17 ±.05 2.33 ±.02 2.21 ±.05 2.36 ±.03 2.70 ±.14 2.78 ±.12 Table 1: Results on synthetic and real-world Bayesian networks: average loss ± standard deviation optimal solution to (8), one not only obtains {M h}, but also the associated set of dual parameters {Λj}. Therefore, we can recover the primal parameters Wj from the dual parameters Λj by using the relationship Wj = 1 αΦ⊤ j (I−Λj)Y j established above, which only requires availability of a label assignment matrix Y j. For observed variables, Y j is known, and therefore the model parameters can be immediately recovered. For hidden variables, we first need to compute a rank vh factorization of M h. Let V = UΣ1/2 where U and Σ are the top vh eigenvector and eigenvalue matrices of the centered matrix HM hH, such that H = I −1 t 11⊤. One simple idea to recover ˆYh from V is to run k-means on the rows of V and construct the indicator matrix. A more elegant approach would be to use a randomized rounding scheme [6], which also produces a deterministic ˆYh, but provides some guarantees about how well ˆYh ˆY ⊤ h approximates M h. Note however that V is an approximation of Y h where the row vectors have been re-centered on the origin in a rotated coordinate system. Therefore, a simpler approach is just to map the rows of V back onto the simplex by translating the mean back to the simplex center and rotation the coordinates back into the positive orthant. 6 Experimental Results An important question to ask is whether the relaxed, convex objective (8) is in fact over-relaxed, and whether important structure in the original marginal likelihood objective has been lost as a result. To investigate this question, we conducted a set of experiments to evaluate our convex approach compared to the standard Viterbi (i.e. joint) EM algorithm, and to supervised training on fully observed data. Our experiments are conducted using both synthetic Bayesian networks and real networks, while measuring the trained models by their logloss produced on the fully observed training data and testing data. All the results reported in this paper are averages over 10 times repeats. The test size for the experiments is 1000, the training size is 100 without specification. For a fair comparison, we used 10 random restarts for Viterbi EM to help avoid poor local optima. For the synthetic experiments, we constructed three Bayesian networks: (1) Bayesian network 1 (Synth1) is a three layer network with 9 variables, where the two nodes in the middle layer are picked as hidden variables; (2) Bayesian network 2 (Synth2) is a network with 6 variables and 6 edges, where a node with 2 parents and 2 children is picked as hidden variable; (3) Bayesian network 3 (Synth3) is a Naive Bayes model with 7 variables, where the parent node is selected as the hidden variable. The parameters are generated in a discriminative way to produce models with apparent causal relations between the connected nodes. We performed experiments on these three synthetic networks using varying training sizes: 50, 100 and 150. Due to space limits, we only report the results for training size 100 in Table 1. Besides these three synthetic Bayesian networks, we also ran experiments using real UCI data, where we used Naive Bayes as the model structure, and set the class variables to be hidden. The middle two rows of the Table 1 show the results on two UCI data sets. Here we can see that the convex relaxation was successful at preserving structure in the EM objective, and in fact, generally performed much better than the Viterbi EM algorithm, particularly in the case (Synth1) where there was two hidden variables. Not surprisingly, supervised training on the complete data performed better than the EM methods, but generally demonstrated a larger gap between training and test losses than the EM methods. Similar results were obtained for both larger and smaller training sample sizes. For the UCI experiments, the results are very similar to the synthetic networks, showing good results again for the convex EM relaxation. Finally, we conducted additional experiments on three real world Bayesian networks: Alarm, Cancer and Asian (downloaded from http://www.norsys.com/networklibrary.html). We picked one well connected node from each model to serve as the hidden variable, and generated data by sampling from the models. Table 1 shows the experimental results for these three Bayesian networks. Here we can see that the convex EM relaxation performed well on the Cancer and Alarm networks. Since we only picked one hidden variable from the 37 variables in Alarm, it is understandable that any potential advantage for the convex approach might not be large. Nevertheless, a slight advantage is still detected here. Much weaker results are obtained on the Asian network however. We are still investigating what aspects of the problem are responsible for the poorer approximation in this case. 7 Conclusion We have presented a new convex relaxation of EM that obtains generally effective results in simple experimental comparisons to a standard joint EM algorithm (Viterbi EM), on both synthetic and real problems. This new approach was facilitated by a novel reformulation of log-linear regression that refers only to equivalence relation information on the data, and thereby allows us to avoid the symmetry breaking problem that blocks naive convexification strategies from working. One shortcoming of the proposed technique however is that it cannot handle continuous hidden variables; this remains a direction for future research. In one experiment, weaker approximation quality was obtained, and this too is the subject of further investigation. References [1] J. Borwein and A. Lewis. Convex Analysis and Nonlinear Optimization. Springer, 2000. [2] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge U. Press, 2004. [3] S. Chen. Models for grapheme-to-phoneme conversion. In Eurospeech, 2003. [4] T. De Bie and N. Cristianini. Fast SDP relaxations of graph cut clustering, transduction, and other combinatorial problems. Journal of Machine Learning Research, 7, 2006. [5] A. Dempster, N. Laird, and D. Rubin. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society. Series B, 39(1):1–38, 1977. [6] M. Goemans and D. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. JACM, 42(6):1115–1145, 1995. [7] S. Goldwater and M. Johnson. Bias in learning syllable structure. In Proc. CONLL, 2005. [8] D. Klein and C. Manning. Corpus-based induction of syntactic structure: Models of dependency and constituency. In Proceedings ACL, 2004. [9] B. Merialdo. Tagging text with a probabilistic model. Comput. Ling., 20(2):155–171, 1994. [10] R. Neal and G. Hinton. A view of the em algorithm that justifies incremental, sparse, and other variants. In M. Jordan, editor, Learning in Graphical Models. Kluwer, 1998. [11] J. Nocedal and S. Wright. Numerical Optimization. Springer, 1999. [12] R. Salakhutdinov, S. Roweis, and Z. Ghahramani. Optimization with EM and expectationconjugate-gradient. In Proceedings ICML, 2003. [13] N. Srebro, G. Shakhnarovich, and S. Roweis. An investigation of computational and informational limits in gaussian mixture clustering. In Proceedings ICML, 2006. [14] M. Wainwright and M. Jordan. Graphical models, exponential families, and variational inference. Technical Report TR-649, UC Berkeley, Dept. Statistics, 2003. [15] L. Xu, J. Neufeld, B. Larson, and D. Schuurmans. Max margin clustering. In NIPS 17, 2004. [16] L. Xu, D. Wilkinson, F. Southey, and D. Schuurmans. Discriminative unsupervised learning of structured predictors. In Proceedings ICML, 2006.
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